Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 495864 11 pageshttpdxdoiorg1011552013495864

Research ArticleStationary PointsmdashI One-Dimensional p-FuzzyDynamical Systems

Joao de Deus M Silva1 Jefferson Leite2 Rodney C Bassanezi3 and Moiseis S Cecconello4

1 CCET UFMA 65085-558 Sao Luis MA Brazil2 DEMAT CCN UFPI 64063040 Teresina PI Brazil3 CMCC UFABC 09210-170 Santo Andre SP Brazil4DMAT ICET UFMT 78075-202 Cuiaba MT Brazil

Correspondence should be addressed to Jefferson Leite jleiteufpiedubr

Received 13 April 2013 Accepted 1 August 2013

Academic Editor Toly Chen

Copyright copy 2013 Joao de Deus M Silva et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

p-Fuzzy dynamical systems are variational systems whose dynamic is obtained by means of a Mamdani type fuzzy rule-basedsystem In this paper we will show the 1-dimensional p-fuzzy dynamical systems andwill present theorems that establish conditionsof existence and uniqueness of stationary points Besides the obtained analytical results we will present examples that illustrate andconfirm the obtained mathematical results

1 Introduction

Variational equations or deterministic dynamical systems(difference and differential equations) constitute a power-ful tool for modeling when the state variables depend onvariations throughout time The efficiency of a deterministicmodel depends on knowledge of the relationships betweenvariables and their variations In addition in many situa-tions such relations are only partially known therefore themodeling with deterministic variational systems or evenwith stochastic ones may not be adequate In additionfuzzy systems derived from deterministic models whichhave subjectivity regarding some parameters are not appro-priate when we have only incomplete information of thephenomenon being analyzed Thus the use of a rule-basedsystem can be adopted as an alternative formodeling partiallyknown phenomena or those carried with imprecision

Fuzzy rule-based systems have been used with success insome areas as control decision taking recognition systemsand so forth This success is due to its simplicity andinterrelation with humans way of reasoning Fuzzy rule-based systems are conceptually simple [1] Such systemsare basically threefold an input (fuzzifier) an inferencemechanism composed of a base of fuzzy rules together with

an inference method and finally an output (defuzzifier) stage(see Figure 1)

There are twomain types of fuzzy rule-based systems theMamdani fuzzy systems and theTakagi-Sugeno fuzzy systems[2] The main characteristic of the Mamdani type systemsis that both the antecedent and consequent are expressedby linguistic terms while in the Takagi-Sugeno type systemsonly the antecedent is expressed by linguistic terms and theconsequent is expressed by functions

The Takagi-Sugeno fuzzy systems are more restrictivethan the Mamdani fuzzy systems because they require an apriori function in the output However due to the existenceof theoretical methods for Takagi-Sugeno fuzzy systemsstability analysis [3ndash8] the latter has become more used Onthe other hand Mamdani type systems are used as a ldquoblackboxrdquo and are still criticized because they lack a study on itsstability [9 10]

Fuzzy variational equations have been used in differentmethods Some attempts to contemplate subjectivity originalfrom aleatory processes such as Hukuhararsquos derivative differ-ential inclusions andZadehrsquos extension [11] have been alreadyproposed In thesemethods the adopted process for studyingthe variational systems is always derived from deterministicclassical systems

2 Journal of Applied Mathematics

Defuzzifier

rulesbase

Fuzzy

methodInference

xo

y

y

xo

Fuzzifier

Figure 1 Architecture of a fuzzy rule-based system

In this paper we will present the p-fuzzy systems whosedynamics is not based in formal concepts of variations orig-inated from derivative or explicit differences or differentialinclusions In the p-fuzzy dynamical systems the dynamic(iterative process) is obtained by means of a Mamdanirsquos fuzzyrule-based system The main advantage of this method withrespect to the other ones is the simplicity of the involvedmathematics just because the interactive method is suppliedby the Mamdani controller

Formally a p-fuzzy system in R119899 is a discrete dynamicsystem

119909119896+1

= 119865 (119909119896)

119909119900given and 119909

119896isin R119899

(1)

where the 119865 function is given by 119865(119909119896) = 119909

119896+ Δ(119909

119896) and

Δ(119909119896) isin R119899 is obtained by means of a fuzzy rule-based

system that is Δ(119909119894) is the defuzzification value of the rule-

based system The architecture of a p-fuzzy system can bevisualized in Figure 2

The name p-fuzzy dynamical systems or purely fuzzy waschosen to differentiate it from other fuzzy systems given byvariational equations

In this paper we will focus on the one-dimensional p-fuzzy systems which are always associated with a Mamdanifuzzy system where the defuzzification method is the cen-troid We have chosen this method because it is widely usedand more general to deal with weight mean of linguisticvariables [12 pages 242 243]

Analogous to the inhibited variational models in whichone has stationary solutions our objective is to present resultsthat establish the necessary and sufficient conditions for theexistence of a stationary point

Fuzzy rule-based systemxk Δ(xk)

Mathematical modelxk+1 = xk + Δ(xk)

Figure 2 Architecture of a p-fuzzy system

1

x

AkAkminus1Ai+1Ai

middot middot middotmiddot middot middot

A1 A2

Figure 3 Family of successive fuzzy subsets

2 Preliminaries

In this section we introduce the main concepts for thedevelopment of the work presented in this paper

21 Definitions

Definition 1 (support) Let 119860 be a fuzzy subset of 119883 thesupport of119860 denoted supp(119860) is the crisp subset of119883whoseelements all have nonzero membership grades in 119860 that is

supp (119860) = 119909 isin 119883119860 (119909) gt 0 (2)

Definition 2 (120572-cut) An 120572-level set of a fuzzy subset119860 of119883 isa crisp set denoted by [119860]120572 and defined by

(i) [119860]0 = supp(119860) (120572 = 0)(ii) [119860]120572 = 119909 isin 119883119860(119909) ge 120572 if 120572 isin (0 1]

Definition 3 (fuzzy number) A fuzzy number119860 is a fuzzy set119860 sub R satisfying the following conditions

(i) [119860]120572 = forall120572 isin [0 1](ii) [119860]120572 is a closed interval forall120572 isin [0 1](iii) the supp(119860) is bounded

Definition 4 Consider a one-dimensional p-fuzzy systemgiven by (1)

Consider that 119909lowast is a stationary point of (1) if

119865 (119909lowast

) = 119909lowast

+ Δ (119909lowast

) = 119909lowast

lArrrArr Δ(119909lowast

) = 0 (3)

Definition 5 Let 1198601198941le119894le119896

be any finite family of normal fuzzysubsets associated with the fuzzy variable 119909 Assume that1198601198941le119894le119896

is a family of successive fuzzy subsets (Figure 3) if

(i) supp(119860119894) cap supp(119860

119894+1) = for each 1 le 119894 lt 119896

Journal of Applied Mathematics 3

n

m

c1 x xz1 zo z2c2

Alowast

r s

Ai Ai+1

b 0 a

B

f

fminus1(m) gminus1(n)

g

R

CCR1 if x is Ai then Δ isR2 if x is Ai+1 then Δ is B

Δx

Δ

Figure 4 Mamdanirsquos inference process for 119860lowast of the type (119860119894 119860 i+1) rarr (119862 119861)

(ii) ⋂119895=119894119894+2

supp(119860119895) has at maximum only one element

for each 1 le 119894 lt 119896 minus 1 that is supp(119860119894) cap

supp(119860119894+2) = 120601 if and only if max119909 isin supp(119860

119894) =

min119909 isin supp(119860119894+2)

(iii) ⋃119894=1119896

supp(119860119894) = 119880 where 119880 is the domain of the

fuzzy variable 119909

(iv) given 1199111isin supp(119860

119894) and 119911

2isin supp(119860

119894+1) if 119860

119894(1199111) =

1 and 119860119894+1(1199112) = 1 then necessarily 119911

1lt 1199112for each

1 le 119894 lt 119896

Definition 6 Consider a family of successive fuzzy subsets1198601198941le119894le119896

that describe the antecedent of a fuzzy systemassociated with the p-fuzzy system (1) We say that 119860lowast is anequilibrium viable set of (1) if 119860lowast contains stationary pointsof (1)

If for some 1 le 119894 lt 119896 there are 1199111 1199112isin [119860119894cup 119860119894+1]0 such

that Δ(1199111) sdot Δ(119911

2) lt 0 then 119860lowast is given by 119860lowast = [119860

119894cap119860119894+1]0

If for all 1199111 1199112isin [119860119894cup 119860119894+1]0 1 le 119894 lt 119896 Δ(119911

1) sdot Δ(119911

2) gt 0

then 119860lowast = [119860119896]0

A p-fuzzy system depends on the fuzzy system associatedwith it that is it depends on the rule-base on the infer-ence method and on the defuzzification method used InDefinition 6 a sufficient condition forΔ(119911

1) sdotΔ(1199112) lt 0 is that

the p-fuzzy system be associated with a fuzzy system whoserule-base in 119860lowast = [119860

119894cup 119860119894+1]0 is of the type

1198771 if 119909 is 119860

119894then Δ is 119861

1198772 if 119909 is 119860

119894+1then Δ is 119862

where supp(119861) sub Rminus and supp(119862) sub R+ or vice versaWhen supp(119861) sub Rminus and supp(119862) sub R+ we have that theequilibrium viable set 119860lowast is of the type (119860

119894 119860119894+1) rarr (119862 119861)

If on the other hand we have that supp(119861) sub R+ andsupp(119862) sub Rminus we say that 119860lowast is of the type (119860

119894 119860119894+1) rarr

(119861 119862)To understand the dynamics of the p-fuzzy system we

need to understand how the rule-based system works morespecifically given 119909 isin 119860

lowast how is Δ(119909) obtained In thefollowing section we will describe this process

22 Output Defuzzification of the Fuzzy System Let 119860lowast =[119860119894cap 119860119894+1]0

= [1198881 1198882] be an equilibrium viable set of the

p-fuzzy system To facilitate the notation we will indicate

by 119903 the membership function of 119860119894 by 119904 the membership

function of 119860119894+1

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(4)

and by 119891 and 119892 the membership functions of 119862 and 119863

(Figure 4) respectively Assume that the p-fuzzy system inthe equilibrium viable set 119860lowast is of the type (119860

119894 119860119894+1) rarr

(119862 119861)For each 119909 isin 119860

lowast Δ(119909) is the R region centroid abscisewith R limited by the membership function of the fuzzyoutput Δ119909 (see Figure 4) Thus

Δ (119909) = (int

119891minus1(119898)

119887

119905119891 (119905) 119889119905 + int

0

119891minus1(119898)

119898119905119889119905 + int

119892minus1(119899)

0

119899119905 119889119905

+int

119886

119892minus1(119899)

119905119892 (119905) 119889119905)

times (int

119899

0

119892minus1

(119905) 119889119905 minus int

119898

0

119891minus1

(119905) 119889119905)

minus1

(5)

where (119899119898) = (119903(119909) 119904(119909)) Equation (5) can be rewritten as

Δ (119909) =

ℎ1(119899) + ℎ

2(119898)

119860 (119898 119899)

(6)

where

ℎ1(119899) = int

119892minus1(119899)

0

119899119905 119889119905 + int

119886

119892minus1(119899)

119905119892 (119905) 119889119905 (7)

ℎ2(119898) = int

119891minus1(119898)

119887

119905119891 (119905) 119889119905 + int

0

119891minus1(119898)

119898119905119889119905 (8)

119860 (119898 119899) = int

119899

0

119892minus1

(119905) 119889119905 minus int

119898

0

119891minus1

(119905) 119889119905 (9)

23 Preliminary Results In this section wewill present someimportant technical results for the main demonstrations ofthe theorems that establish sufficient conditions for unique-ness and existence of the stationary point of the p-fuzzysystems Here the presented results are technical enoughand are important only for demonstrating the theorems in

4 Journal of Applied Mathematics

the next section Therefore the reader will be able withoutloss of continuity to skip them if desired

The following results are referred to the equilibriumviable set of the type (119860

119894 119860119894+1) rarr (119862 119861) (Figure 4) From

now on it will be assumed that the functions 119903 119904 119891 and 119892(which represent themembership functions120583

119860119894120583119860119894+1

120583119861and

120583119862 resp) are continuous

Lemma 7 The function ℎ1(7) is increasing and its range is

given by 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Proof Since 119892 is continuous in [0 119886] (119892minus1 is limited in [0 119886])then the function ℎ

1is differentiable and

ℎ1015840

1(119899) = (int

119892minus1(119899)

0

119899119905 119889119905)

1015840

+ (int

119886

119892minus1(119899)

119905119892 (119905) 119889119905)

1015840

(10)

Using the derivative properties

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899(int

119892minus1(119899)

0

119905119889119905)

1015840

minus (int

119892minus1(119899)

119886

119905119892 (119905) 119889119905)

1015840

(11)

and using the chain rule and the fundamental theorem ofcalculus we obtain

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

minus 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

(12)

Hence

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 =

(119892minus1

(119899))

2

2

gt 0(13)

Therefore ℎ1is increasing

Since

ℎ1(0) = int

119892minus1(0)

0

0119905119889119905 + int

119886

119892minus1(0)

119905119892 (119905) 119889119905 = int

119886

119886

119905119892 (119905) 119889119905 = 0

ℎ1(1) = int

119892minus1(1)

0

119905119889119905 + int

119886

119892minus1(1)

119905119892 (119905) 119889119905 = int

119886

0

119905119892 (119905) 119889119905

(14)

then 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Lemma 8 The function ℎ2is decreasing and its range is given

by 119868119898(ℎ2) = [int

0

119887

119905119891(119905)dt 0]

Proof Analogous to the demonstration of Lemma 7

Lemma 9 Let 120601 119868 = [1198891 1198892] rarr R be a function of the class

1198622 If 12060110158401015840(119911) gt 0 forall119911 isin (119889

1 1198892) and 120601(119889

1) lt 0 then 120601 has at

maximum one root in 119868

1

k

b fminus1(k) 0 minusfminus1(k)

gminus1(k)

minusb a

f g

Figure 5 System p-fuzzy output with 119892(119905) gt 119891(minus119905)

Proof of Lemma 9 Assume there are 1199111 1199112isin 119868 (119911

1lt 1199112) such

that 120601(1199111) = 120601(119911

2) = 0 Since 12060110158401015840(119911) gt 0 we have that 120601 is not

constant Hence by Rollersquos Theorem exist119888 isin (1199111 1199112) such that

1206011015840

(119888) = 0 Hence 119888 is a minimum point because 12060110158401015840(119888) gt 0But 120601(119889

1) lt 0 rArr 120601(119888) gt 0 Since 120601 is continuous exist119911

119900isin

(1199111 1199112) such that 120601(119888) gt 120601(119911

119900) gt 0 it is nonsense Therefore

120601 has at maximum one root

Lemma 10 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then 119892minus1(119896) gt

minus119891minus1

(119896)forall119896 isin [0 1]

Proof The proof is simple (see Figure 5)

Lemma 11 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then for119898 119899 isin [0 1]with119898 le 119899 one has that

Δ (119909) =

ℎ1(119899) + ℎ

2(119898)

119860 (119898 119899)

gt 0 (15)

Proof The proof is simple

Lemma 12 If 119892(119905) lt 119891(minus119905) forall119905 isin [0 119886] and119898 119899 isin [0 1] 119898 ge

119899 then Δ(119909) lt 0

Proof It is analogous to the proof of Lemma 11

3 Stationary Point

In this section we will enunciate and prove a theorem thatguarantees the existence of at least one stationary point foreach equilibrium viable set of the p-fuzzy system For thiswe will use again Figure 4 tomotivate the presented results inthis section

Theorem 13 (existence) Let 119878 be a p-fuzzy system and 119860lowast anequilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861)

Then there is at least one stationary point of 119878 in 119860lowast That isexist119909lowast

isin 119860lowast such that Δ(119909lowast) = 0

Journal of Applied Mathematics 5

Proof Given 119909 isin 119860lowast fromDefinition 4 then 119909 is a stationarypoint if and only if

Δ (119909) = 0 lArrrArr ℎ1(119899) + ℎ

2(119898) = 0 (16)

If 120583119860119894(1198881) = 0 then

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(0) + ℎ

2(0) = 0

(17)

and therefore 1198881is a stationary point If 120583

119860119894+1(1198882) = 0 one has

that Δ(1198882) = 0 hence 119888

2is a stationary point Now assume

that 120583119860119894(1198881) gt 0 and 120583

119860119894+1(1198882) gt 0 Since 120583

119860119894+1(1198881) = 0

then from Lemmas 7 and 8 one has that ℎ1(120583119860119894(1198881)) gt 0 and

ℎ2(120583119860119894+1

(1198881)) = 0 Therefore

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(120583119860119894(1198881)) gt 0

(18)

Thus from Lemma 7

Δ (1198882) = ℎ1(120583119860119894(1198882)) + ℎ

2(120583119860119894+1

(1198882)) = ℎ

2(120583119860119894+1

(1198882)) lt 0

(19)

Since Δ is continuous by Bolzanorsquos Intermediate ValueTheorem exist119909lowast isin [119888

1 1198882] such that Δ(119909lowast) = 0 therefore 119909lowast

is a stationary point

Remark 14 If in Theorem 13 we consider 119860lowast as an equilib-rium viable set of the type (119860

119894 119860119894+1) rarr (119861 119862) the result is

analogous that is there exists a stationary point 119909lowast isin 119860lowast

31 Local Stationary Points-Symmetrical Output If 119860lowast is anequilibrium viable set where the membership functions ofthe consequents 119861 and 119862 are symmetrical functions thenthe stationary point in119860lowast is unique Except when 120583

119860119894(1198881) = 0

or 120583119860119894+1

(1198882) = 0 possibly occurs Then we have the following

proposition

Proposition 15 Let 119878 be a p-fuzzy system and 119860lowast an equi-librium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the

membership functions of 119861 and 119862 respectively 120583119861and 120583

119862 are

monotonous and symmetric that is 120583119862(119905) = 120583

119861(minus119905) then there

exists an equilibrium point in 119860lowast

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (20)

Proof Since 120583119861(119905) = 120583

119862(minus119905) then 120583

119861(minus119886) = 120583

119862(119886) = 0 =

120583119861(119887) rArr 119887 = minus119886 because 120583

119861U is monotonous Yet we have

that 120583119861(119905) = 120583

119862(minus119905) rArr 120583

minus1

119861(120583119862(119905)) = minus119905 = minus120583

minus1

119861(120583119861(119905)) rArr

120583minus1

119861(119910) = minus120583

minus1

119862(119910)

Then Δ(119911119900) = 0 if and only if ℎ

1(119899) = minusℎ

2(119898) Since

119887 = minus119886 from (8) then we obtain

ℎ2(119898) = int

120583minus1

119861(119898)

minus119886

119905120583119861(119905) 119889119905 + int

0

120583minus1

119861(119898)

119898119905119889119905 (21)

If we perform a change in the variable 119906 = minus119905 we have

ℎ2(119898) = int

minus120583minus1

119861(119898)

a119906120583119861(minus119906) 119889119906 + int

0

minus120583minus1

119861(119898)

119898119906119889119906

997904rArr ℎ2(119898) = int

120583minus1

119862(119898)

119886

119906120583119862(119906) 119889119906 + int

0

120583minus1

119862(119898)

119898119906119889119906 = minusℎ1(119898)

(22)

That is ℎ2= minusℎ

1 Hence ℎ

1(119899) = minusℎ

2(119898) hArr ℎ

1(119899) =

ℎ1(119898) hArr 119898 = 119899 (because ℎ

1is increasing Lemma 7) that

proves the proposition

Remark 16 If 120583119860119894(1198881) = 0 then 119909lowast = max

119909isin119860lowast[min(120583

119860119894(119909)

120583119860119894+1

(119909))] is the only stationary point in 119860lowast Besides that ifthe system 119878 is of the type (119860

119894 119860119894+1) rarr (119861 119862) then the result

of Proposition 15 is the same

4 Uniqueness of the Stationary Point

In this section we will enunciate and prove theorems thatestablish condition for uniqueness of the stationary point ofa one-dimensional p-fuzzy system Initially we will considera simpler case when 119860lowast sub [119911

1 1199112] (Figure 4) where

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(23)

Theorem 17 Let 119878 be a p-fuzzy system and119860lowast an equilibriumviable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the functions

120583119860119894and120583119860119894+1

are piecewisemonotonous and119860lowast sub [1199111 1199112] then

there exists only one stationary point in 119860lowast

Proof Given 119909 isin 119860lowast one has

Δ (119909) = ℎ1(119899) + ℎ

2(119898) = ℎ

1(120583119860119894(119909)) + ℎ

2(120583119860119894+1

(119909))

(24)

Using Lemmas 7 and 8 and the chain rule we find that thederivative of Δ is

Δ1015840

(119909) =

[120583minus1

119862(120583119860119894(119909))]

2

2

1205831015840

119860119894

(119909)

minus

[120583minus1

119861(120583119860119894+1

(119909))]

2

2

1205831015840

119860119894+1

(119909)

(25)

Since in 119860lowast = [1198881 1198882]120583119860119894

is not increasing and 120583119860119894+1

is notdecreasing then 1205831015840

119860119894

(119909) le 0 and 1205831015840119860119894+1

(119909) ge 0 and besidesthat if 1205831015840

119860119894

(119909) = 0 we have that 1205831015840119860119894+1

(119909) = 0 and if 1205831015840119860119894+1

(119909) =

0 we obtain 1205831015840

119860119894

(119909) = 0 Then from (25) Δ1015840(119909) lt 0 Thisshows that Δ is decreasing From Theorem 13 there exists astationary point in 119860lowast then this point is unique

Now consider the more general case when119860lowast sub [1199111 1199112]

and divide it into two theorems Initially consider the casethat the membership functions 120583

119862and 120583

119861are such that

120583119862(119905) gt 120583

119861(minus119905) next consider the case where 120583

119862(119905) lt

120583119861(minus119905)

6 Journal of Applied Mathematics

h1(1)

minush2(1)

h2(1)

h1

h2

hminus11 (minush2(1)) 1

Figure 6 Functions ℎ1and ℎ

2

m

n

1

1

1205751

1205752

s(z1)

s(c2)

120585

r(z0)r(c1)r(z2)hminus11 (minush2(1))

Figure 7 Functions 120585 1205751and 120575

2

41 Case 1 120583119862(119905) gt 120583

119861(minus119905)

Theorem 18 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) gt 120583

119861(minus119905) forall119905 isin (0 minus119887)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) lt (119901

3

1199023

) forall119901 isin supp(119861) 119902 isin

supp(119862) and 120583119861(119901) gt 120583

119862(119902)

(iii) [1205831015840119860119894+1

(119909)1205831015840

119860119894

(119909)] le 0 forall119909 isin (119911119900 1198882) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

(119911119900 1198882]

Proof For the sake of simplicity notation we make 119903 = 120583119860119894

119904 = 120583119860119894+1

119891 = 120583119861and 119892 = 120583

119862

Initially we see that given 119909 isin (119911119900 1198882] (Figure 4) 119909

determines only one (119899119898) isin [0 1]2 such that 119899 = 119903(119909)

and 119898 = 119904(119909) By monotonicity of 119903 we have that for each119899 isin [0 119903(119911

119900)) there exists only one 119898 isin [0 1] such that

119899 = 119903(119909) and 119898 = 119904(119909) That is each (119899119898) in this situationdetermines only one 119909 isin (119911

119900 1198882]

By Theorem 13 there exists a stationary point 119909lowast isin

[1198881 1198882] = [1198881 119911119900] cup (119911119900 1198882] Given 119909 isin [119888

1 119911119900] rArr 119898 = 119904(119909) le

119899 = 119903(119909) Then by Lemma 11 119909lowast notin [1198881 119911119900] rArr 119909

lowast

isin (119911119900 1198882]

That is equivalent to the existence of only one (119899lowast 119898lowast) with119899lowast

isin [0 119903(119911119900)) such that119867(119899lowast 119898lowast) = 0

Since for each 119899 isin [0 119903(119911119900)) there exists only one 119898 isin

[0 1] such that 119899 = 119903(119909) and 119898 = 119904(119909) then we may definea function 120575

2 [0 119903(119911

119900)) rarr [0 1] such that 119898 = 120575

2(119899)

(Figure 7) We observe that 1205752is continuous because 119903 and

119904 are continuous Using the chain rule we get the derivative of1205752to be

1205751015840

2(119899) =

1199041015840

(119909)

1199031015840(119909)

(iii)rArr 12057510158401015840

2(119899) le 0 forall119899 isin 119863

1205752 (26)

By Lemmas 7 and 8 ℎ1and minusℎ

2are increasing and by

condition (i) it follows that (Figure 6)

int

119886

0

119905119892 (119905) 119889119905 gt minusint

0

119887

119905119891 (119905) 119889119905 lArrrArr ℎ1(1) gt minusℎ

2(1) (27)

Then given 119899 isin [0 ℎminus1

1(minusℎ2(1))] there exists only one 119898 isin

[0 1] such that

ℎ1(119899) = minusℎ

2(119898) lArrrArr ℎ

1(119899) + ℎ

2(119898) = 0 (28)

Therefore we may define an injective function 120585 119898 = 120585(119899)

(Figure 7) so that the inverse range of 0 by119867 is given by

119867minus1

(0) = (119899119898) 119898 = 120585 (119899) (29)

where 119867 [0 ℎminus1

1(minusℎ2(1))] times [0 1] rarr R is given by

119867(119899119898) = ℎ1(119899) + ℎ

2(119898)

Since 120597119867120597119899 = ℎ10158401(119899) = (119892

minus1

(119899))

2

2 gt 0 (Lemma 7) and120597119867120597119898 = ℎ

1015840

2(119898) = minus(119891

minus1

(119898))

2

2 lt 0 (Lemma 8) then bythe Implicity Function Theorem 120585 is 119896 times differentiableand besides that

1205851015840

(119899) = minus

119889ℎ1119899

119889ℎ2119898

= [

119892minus1

(119899)

119891minus1(119898)

]

2

gt 0

forall119899 isin (0 ℎminus1

1(minusℎ2(1))) 119898 isin (0 1) 119898 = 120585 (119899)

(30)

Thus 120585 is a strictly increasing function and since119867(0 0) = 0and 119867(ℎminus1

1(minusℎ2(1)) 1) = 0 then 119863

120585= [0 ℎ

minus1

1(ℎ2(1))] and

119868119898120585= [0 1]Given 119898 119899 isin (0 1) there is only one 119901 isin (119887 0) such that

119901 = 119891minus1

(119898) and there exists only one 119902 isin (0 119886) such that 119902 =119892minus1

(119899) since by assumption119891 and 119892 are strictly monotonousFrom Lemma 11 we have that 119898 le 119899 rArr 119867(119898 119899) gt 0

Hence119867(119898 119899) = 0 rArr 119898 gt 119899 Therefore we are interested inthe pairs (119898 119899) such that119898 gt 119899 Thus we have

119898 gt 119899 lArrrArr 119891(119901) gt 119892 (119902) (31)

Journal of Applied Mathematics 7

1 1

0 5 40 60 90 100x

A1 A2 B

Δminus3 minus2 minus120576 0 2 3

m1 + 120576

m1

C

R1 if x is A1 then Δ is CR2 if x is A2 then Δ is B

Figure 8 p-Fuzzy system existence of more than one stationary point

Since 119891 and 119892 are monotonous by Lagrangersquos Medium ValueTheorem we obtain

119901 = 119891minus1

(119898) lArrrArr (119891minus1

)

1015840

(119898) =

1

1198911015840(119901)

(32)

119902 = 119892minus1

(119899) lArrrArr (119892minus1

)

1015840

(119899) =

1

1198921015840(119902)

(33)

Then

119898 gt 119899

(31)

lArr997904997904rArr 119891(119901) gt 119892 (119902)

(ii)rArr

1198921015840

(119902)

1198911015840(119901)

lt

1199013

1199023

(32)119890(33)

lArr997904997904997904997904997904rArr

(119891minus1

)

1015840

(119898)

(119892minus1)1015840

(119899)

lt

[119891minus1

(119898)]

3

[119892minus1(119899)]3

(34)

Therefore

119898 gt 119899 997904rArr (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

gt 0

(35)

By differentiation of (30) we obtain

12058510158401015840

(119899) =

minus2119892minus1

(119899)

[119891minus1(119898)]5

times (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

(36)

and since minus2119892minus1(119899)[119891minus1(119898)]5 gt 0 forall119898 119899 isin (0 1) from (35)we have

12058510158401015840

(119899) gt 0 forall119899 isin

119900

119863120585 (37)

Now we take 119868 = 119863120585cap 1198631205752= 119863120585cap [0 119903(119911

119900)) and we define

the function 120601 119868 rarr [0 1] such that

120601 (119899) = 120585 (119899) minus 1205752(119899) (38)

Then from (26) and (37) we have 12060110158401015840(119899) gt 0 forall119899 isin

119900

119868 Since120585(0) = 0 and by condition (iii) of Theorem 18 it followsthat 120575

2(0) gt 0 then we have 120601(0) lt 0 Consequently from

Lemma 9 we have that there is only one 119899lowast isin 119868 such that

120601 (119899lowast

) = 0

(38)

lArr997904997904rArr 120585 (119899lowast

) = 1205752(119899lowast

) (39)

Since 120585 = 119867minus1(0) then we obtain

0 = 119867 (119899lowast

120585 (119899lowast

))

(39)

= 119867 (119899lowast

1205752(119899lowast

)) (40)

So there exists only one 119909lowast isin (119911119900 1198882] 119899lowast = 119903(119909

lowast

) and119898lowast

= 1205752(119899lowast

) = 119904(119909lowast

) such that

Δ (119909lowast

) =

119867 (119899lowast

119898lowast

)

119860 (119899lowast 119898lowast)

= 0 (41)

This finally proves the theorem

42 Case 2 120583119862(119905)lt120583119861(minus119905)

Theorem 19 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861 and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) lt 120583

119861(minus119905) forall119905 isin (0 119886)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 forall119901 isin supp(119861) 119902 isin supp(119862)

and 120583119861(119901) lt 120583

119862(119902)

(iii) [1205831015840119860119894

(119909)1205831015840

119860119894+1

(119909)] le 0 forall119909 isin (1198881 119911119900) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

[1198881 119911119900)

Proof It is analogous to the proof of Theorem 18

43 Some Comments about Uniqueness Theorems When wedo not have 120583

119862(119905) gt 120583

119861(minus119905) or 120583

119862(119905) lt 120583

119861(minus119905) it is impossible

to establish general conditions for uniqueness of stationarypoints For example consider the p-fuzzy system in Figure 8This system has an equilibrium viable set 119860lowast = [119860

1cap 1198602]0

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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2 Journal of Applied Mathematics

Defuzzifier

rulesbase

Fuzzy

methodInference

xo

y

y

xo

Fuzzifier

Figure 1 Architecture of a fuzzy rule-based system

In this paper we will present the p-fuzzy systems whosedynamics is not based in formal concepts of variations orig-inated from derivative or explicit differences or differentialinclusions In the p-fuzzy dynamical systems the dynamic(iterative process) is obtained by means of a Mamdanirsquos fuzzyrule-based system The main advantage of this method withrespect to the other ones is the simplicity of the involvedmathematics just because the interactive method is suppliedby the Mamdani controller

Formally a p-fuzzy system in R119899 is a discrete dynamicsystem

119909119896+1

= 119865 (119909119896)

119909119900given and 119909

119896isin R119899

(1)

where the 119865 function is given by 119865(119909119896) = 119909

119896+ Δ(119909

119896) and

Δ(119909119896) isin R119899 is obtained by means of a fuzzy rule-based

system that is Δ(119909119894) is the defuzzification value of the rule-

based system The architecture of a p-fuzzy system can bevisualized in Figure 2

The name p-fuzzy dynamical systems or purely fuzzy waschosen to differentiate it from other fuzzy systems given byvariational equations

In this paper we will focus on the one-dimensional p-fuzzy systems which are always associated with a Mamdanifuzzy system where the defuzzification method is the cen-troid We have chosen this method because it is widely usedand more general to deal with weight mean of linguisticvariables [12 pages 242 243]

Analogous to the inhibited variational models in whichone has stationary solutions our objective is to present resultsthat establish the necessary and sufficient conditions for theexistence of a stationary point

Fuzzy rule-based systemxk Δ(xk)

Mathematical modelxk+1 = xk + Δ(xk)

Figure 2 Architecture of a p-fuzzy system

1

x

AkAkminus1Ai+1Ai

middot middot middotmiddot middot middot

A1 A2

Figure 3 Family of successive fuzzy subsets

2 Preliminaries

In this section we introduce the main concepts for thedevelopment of the work presented in this paper

21 Definitions

Definition 1 (support) Let 119860 be a fuzzy subset of 119883 thesupport of119860 denoted supp(119860) is the crisp subset of119883whoseelements all have nonzero membership grades in 119860 that is

supp (119860) = 119909 isin 119883119860 (119909) gt 0 (2)

Definition 2 (120572-cut) An 120572-level set of a fuzzy subset119860 of119883 isa crisp set denoted by [119860]120572 and defined by

(i) [119860]0 = supp(119860) (120572 = 0)(ii) [119860]120572 = 119909 isin 119883119860(119909) ge 120572 if 120572 isin (0 1]

Definition 3 (fuzzy number) A fuzzy number119860 is a fuzzy set119860 sub R satisfying the following conditions

(i) [119860]120572 = forall120572 isin [0 1](ii) [119860]120572 is a closed interval forall120572 isin [0 1](iii) the supp(119860) is bounded

Definition 4 Consider a one-dimensional p-fuzzy systemgiven by (1)

Consider that 119909lowast is a stationary point of (1) if

119865 (119909lowast

) = 119909lowast

+ Δ (119909lowast

) = 119909lowast

lArrrArr Δ(119909lowast

) = 0 (3)

Definition 5 Let 1198601198941le119894le119896

be any finite family of normal fuzzysubsets associated with the fuzzy variable 119909 Assume that1198601198941le119894le119896

is a family of successive fuzzy subsets (Figure 3) if

(i) supp(119860119894) cap supp(119860

119894+1) = for each 1 le 119894 lt 119896

Journal of Applied Mathematics 3

n

m

c1 x xz1 zo z2c2

Alowast

r s

Ai Ai+1

b 0 a

B

f

fminus1(m) gminus1(n)

g

R

CCR1 if x is Ai then Δ isR2 if x is Ai+1 then Δ is B

Δx

Δ

Figure 4 Mamdanirsquos inference process for 119860lowast of the type (119860119894 119860 i+1) rarr (119862 119861)

(ii) ⋂119895=119894119894+2

supp(119860119895) has at maximum only one element

for each 1 le 119894 lt 119896 minus 1 that is supp(119860119894) cap

supp(119860119894+2) = 120601 if and only if max119909 isin supp(119860

119894) =

min119909 isin supp(119860119894+2)

(iii) ⋃119894=1119896

supp(119860119894) = 119880 where 119880 is the domain of the

fuzzy variable 119909

(iv) given 1199111isin supp(119860

119894) and 119911

2isin supp(119860

119894+1) if 119860

119894(1199111) =

1 and 119860119894+1(1199112) = 1 then necessarily 119911

1lt 1199112for each

1 le 119894 lt 119896

Definition 6 Consider a family of successive fuzzy subsets1198601198941le119894le119896

that describe the antecedent of a fuzzy systemassociated with the p-fuzzy system (1) We say that 119860lowast is anequilibrium viable set of (1) if 119860lowast contains stationary pointsof (1)

If for some 1 le 119894 lt 119896 there are 1199111 1199112isin [119860119894cup 119860119894+1]0 such

that Δ(1199111) sdot Δ(119911

2) lt 0 then 119860lowast is given by 119860lowast = [119860

119894cap119860119894+1]0

If for all 1199111 1199112isin [119860119894cup 119860119894+1]0 1 le 119894 lt 119896 Δ(119911

1) sdot Δ(119911

2) gt 0

then 119860lowast = [119860119896]0

A p-fuzzy system depends on the fuzzy system associatedwith it that is it depends on the rule-base on the infer-ence method and on the defuzzification method used InDefinition 6 a sufficient condition forΔ(119911

1) sdotΔ(1199112) lt 0 is that

the p-fuzzy system be associated with a fuzzy system whoserule-base in 119860lowast = [119860

119894cup 119860119894+1]0 is of the type

1198771 if 119909 is 119860

119894then Δ is 119861

1198772 if 119909 is 119860

119894+1then Δ is 119862

where supp(119861) sub Rminus and supp(119862) sub R+ or vice versaWhen supp(119861) sub Rminus and supp(119862) sub R+ we have that theequilibrium viable set 119860lowast is of the type (119860

119894 119860119894+1) rarr (119862 119861)

If on the other hand we have that supp(119861) sub R+ andsupp(119862) sub Rminus we say that 119860lowast is of the type (119860

119894 119860119894+1) rarr

(119861 119862)To understand the dynamics of the p-fuzzy system we

need to understand how the rule-based system works morespecifically given 119909 isin 119860

lowast how is Δ(119909) obtained In thefollowing section we will describe this process

22 Output Defuzzification of the Fuzzy System Let 119860lowast =[119860119894cap 119860119894+1]0

= [1198881 1198882] be an equilibrium viable set of the

p-fuzzy system To facilitate the notation we will indicate

by 119903 the membership function of 119860119894 by 119904 the membership

function of 119860119894+1

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(4)

and by 119891 and 119892 the membership functions of 119862 and 119863

(Figure 4) respectively Assume that the p-fuzzy system inthe equilibrium viable set 119860lowast is of the type (119860

119894 119860119894+1) rarr

(119862 119861)For each 119909 isin 119860

lowast Δ(119909) is the R region centroid abscisewith R limited by the membership function of the fuzzyoutput Δ119909 (see Figure 4) Thus

Δ (119909) = (int

119891minus1(119898)

119887

119905119891 (119905) 119889119905 + int

0

119891minus1(119898)

119898119905119889119905 + int

119892minus1(119899)

0

119899119905 119889119905

+int

119886

119892minus1(119899)

119905119892 (119905) 119889119905)

times (int

119899

0

119892minus1

(119905) 119889119905 minus int

119898

0

119891minus1

(119905) 119889119905)

minus1

(5)

where (119899119898) = (119903(119909) 119904(119909)) Equation (5) can be rewritten as

Δ (119909) =

ℎ1(119899) + ℎ

2(119898)

119860 (119898 119899)

(6)

where

ℎ1(119899) = int

119892minus1(119899)

0

119899119905 119889119905 + int

119886

119892minus1(119899)

119905119892 (119905) 119889119905 (7)

ℎ2(119898) = int

119891minus1(119898)

119887

119905119891 (119905) 119889119905 + int

0

119891minus1(119898)

119898119905119889119905 (8)

119860 (119898 119899) = int

119899

0

119892minus1

(119905) 119889119905 minus int

119898

0

119891minus1

(119905) 119889119905 (9)

23 Preliminary Results In this section wewill present someimportant technical results for the main demonstrations ofthe theorems that establish sufficient conditions for unique-ness and existence of the stationary point of the p-fuzzysystems Here the presented results are technical enoughand are important only for demonstrating the theorems in

4 Journal of Applied Mathematics

the next section Therefore the reader will be able withoutloss of continuity to skip them if desired

The following results are referred to the equilibriumviable set of the type (119860

119894 119860119894+1) rarr (119862 119861) (Figure 4) From

now on it will be assumed that the functions 119903 119904 119891 and 119892(which represent themembership functions120583

119860119894120583119860119894+1

120583119861and

120583119862 resp) are continuous

Lemma 7 The function ℎ1(7) is increasing and its range is

given by 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Proof Since 119892 is continuous in [0 119886] (119892minus1 is limited in [0 119886])then the function ℎ

1is differentiable and

ℎ1015840

1(119899) = (int

119892minus1(119899)

0

119899119905 119889119905)

1015840

+ (int

119886

119892minus1(119899)

119905119892 (119905) 119889119905)

1015840

(10)

Using the derivative properties

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899(int

119892minus1(119899)

0

119905119889119905)

1015840

minus (int

119892minus1(119899)

119886

119905119892 (119905) 119889119905)

1015840

(11)

and using the chain rule and the fundamental theorem ofcalculus we obtain

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

minus 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

(12)

Hence

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 =

(119892minus1

(119899))

2

2

gt 0(13)

Therefore ℎ1is increasing

Since

ℎ1(0) = int

119892minus1(0)

0

0119905119889119905 + int

119886

119892minus1(0)

119905119892 (119905) 119889119905 = int

119886

119886

119905119892 (119905) 119889119905 = 0

ℎ1(1) = int

119892minus1(1)

0

119905119889119905 + int

119886

119892minus1(1)

119905119892 (119905) 119889119905 = int

119886

0

119905119892 (119905) 119889119905

(14)

then 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Lemma 8 The function ℎ2is decreasing and its range is given

by 119868119898(ℎ2) = [int

0

119887

119905119891(119905)dt 0]

Proof Analogous to the demonstration of Lemma 7

Lemma 9 Let 120601 119868 = [1198891 1198892] rarr R be a function of the class

1198622 If 12060110158401015840(119911) gt 0 forall119911 isin (119889

1 1198892) and 120601(119889

1) lt 0 then 120601 has at

maximum one root in 119868

1

k

b fminus1(k) 0 minusfminus1(k)

gminus1(k)

minusb a

f g

Figure 5 System p-fuzzy output with 119892(119905) gt 119891(minus119905)

Proof of Lemma 9 Assume there are 1199111 1199112isin 119868 (119911

1lt 1199112) such

that 120601(1199111) = 120601(119911

2) = 0 Since 12060110158401015840(119911) gt 0 we have that 120601 is not

constant Hence by Rollersquos Theorem exist119888 isin (1199111 1199112) such that

1206011015840

(119888) = 0 Hence 119888 is a minimum point because 12060110158401015840(119888) gt 0But 120601(119889

1) lt 0 rArr 120601(119888) gt 0 Since 120601 is continuous exist119911

119900isin

(1199111 1199112) such that 120601(119888) gt 120601(119911

119900) gt 0 it is nonsense Therefore

120601 has at maximum one root

Lemma 10 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then 119892minus1(119896) gt

minus119891minus1

(119896)forall119896 isin [0 1]

Proof The proof is simple (see Figure 5)

Lemma 11 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then for119898 119899 isin [0 1]with119898 le 119899 one has that

Δ (119909) =

ℎ1(119899) + ℎ

2(119898)

119860 (119898 119899)

gt 0 (15)

Proof The proof is simple

Lemma 12 If 119892(119905) lt 119891(minus119905) forall119905 isin [0 119886] and119898 119899 isin [0 1] 119898 ge

119899 then Δ(119909) lt 0

Proof It is analogous to the proof of Lemma 11

3 Stationary Point

In this section we will enunciate and prove a theorem thatguarantees the existence of at least one stationary point foreach equilibrium viable set of the p-fuzzy system For thiswe will use again Figure 4 tomotivate the presented results inthis section

Theorem 13 (existence) Let 119878 be a p-fuzzy system and 119860lowast anequilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861)

Then there is at least one stationary point of 119878 in 119860lowast That isexist119909lowast

isin 119860lowast such that Δ(119909lowast) = 0

Journal of Applied Mathematics 5

Proof Given 119909 isin 119860lowast fromDefinition 4 then 119909 is a stationarypoint if and only if

Δ (119909) = 0 lArrrArr ℎ1(119899) + ℎ

2(119898) = 0 (16)

If 120583119860119894(1198881) = 0 then

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(0) + ℎ

2(0) = 0

(17)

and therefore 1198881is a stationary point If 120583

119860119894+1(1198882) = 0 one has

that Δ(1198882) = 0 hence 119888

2is a stationary point Now assume

that 120583119860119894(1198881) gt 0 and 120583

119860119894+1(1198882) gt 0 Since 120583

119860119894+1(1198881) = 0

then from Lemmas 7 and 8 one has that ℎ1(120583119860119894(1198881)) gt 0 and

ℎ2(120583119860119894+1

(1198881)) = 0 Therefore

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(120583119860119894(1198881)) gt 0

(18)

Thus from Lemma 7

Δ (1198882) = ℎ1(120583119860119894(1198882)) + ℎ

2(120583119860119894+1

(1198882)) = ℎ

2(120583119860119894+1

(1198882)) lt 0

(19)

Since Δ is continuous by Bolzanorsquos Intermediate ValueTheorem exist119909lowast isin [119888

1 1198882] such that Δ(119909lowast) = 0 therefore 119909lowast

is a stationary point

Remark 14 If in Theorem 13 we consider 119860lowast as an equilib-rium viable set of the type (119860

119894 119860119894+1) rarr (119861 119862) the result is

analogous that is there exists a stationary point 119909lowast isin 119860lowast

31 Local Stationary Points-Symmetrical Output If 119860lowast is anequilibrium viable set where the membership functions ofthe consequents 119861 and 119862 are symmetrical functions thenthe stationary point in119860lowast is unique Except when 120583

119860119894(1198881) = 0

or 120583119860119894+1

(1198882) = 0 possibly occurs Then we have the following

proposition

Proposition 15 Let 119878 be a p-fuzzy system and 119860lowast an equi-librium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the

membership functions of 119861 and 119862 respectively 120583119861and 120583

119862 are

monotonous and symmetric that is 120583119862(119905) = 120583

119861(minus119905) then there

exists an equilibrium point in 119860lowast

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (20)

Proof Since 120583119861(119905) = 120583

119862(minus119905) then 120583

119861(minus119886) = 120583

119862(119886) = 0 =

120583119861(119887) rArr 119887 = minus119886 because 120583

119861U is monotonous Yet we have

that 120583119861(119905) = 120583

119862(minus119905) rArr 120583

minus1

119861(120583119862(119905)) = minus119905 = minus120583

minus1

119861(120583119861(119905)) rArr

120583minus1

119861(119910) = minus120583

minus1

119862(119910)

Then Δ(119911119900) = 0 if and only if ℎ

1(119899) = minusℎ

2(119898) Since

119887 = minus119886 from (8) then we obtain

ℎ2(119898) = int

120583minus1

119861(119898)

minus119886

119905120583119861(119905) 119889119905 + int

0

120583minus1

119861(119898)

119898119905119889119905 (21)

If we perform a change in the variable 119906 = minus119905 we have

ℎ2(119898) = int

minus120583minus1

119861(119898)

a119906120583119861(minus119906) 119889119906 + int

0

minus120583minus1

119861(119898)

119898119906119889119906

997904rArr ℎ2(119898) = int

120583minus1

119862(119898)

119886

119906120583119862(119906) 119889119906 + int

0

120583minus1

119862(119898)

119898119906119889119906 = minusℎ1(119898)

(22)

That is ℎ2= minusℎ

1 Hence ℎ

1(119899) = minusℎ

2(119898) hArr ℎ

1(119899) =

ℎ1(119898) hArr 119898 = 119899 (because ℎ

1is increasing Lemma 7) that

proves the proposition

Remark 16 If 120583119860119894(1198881) = 0 then 119909lowast = max

119909isin119860lowast[min(120583

119860119894(119909)

120583119860119894+1

(119909))] is the only stationary point in 119860lowast Besides that ifthe system 119878 is of the type (119860

119894 119860119894+1) rarr (119861 119862) then the result

of Proposition 15 is the same

4 Uniqueness of the Stationary Point

In this section we will enunciate and prove theorems thatestablish condition for uniqueness of the stationary point ofa one-dimensional p-fuzzy system Initially we will considera simpler case when 119860lowast sub [119911

1 1199112] (Figure 4) where

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(23)

Theorem 17 Let 119878 be a p-fuzzy system and119860lowast an equilibriumviable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the functions

120583119860119894and120583119860119894+1

are piecewisemonotonous and119860lowast sub [1199111 1199112] then

there exists only one stationary point in 119860lowast

Proof Given 119909 isin 119860lowast one has

Δ (119909) = ℎ1(119899) + ℎ

2(119898) = ℎ

1(120583119860119894(119909)) + ℎ

2(120583119860119894+1

(119909))

(24)

Using Lemmas 7 and 8 and the chain rule we find that thederivative of Δ is

Δ1015840

(119909) =

[120583minus1

119862(120583119860119894(119909))]

2

2

1205831015840

119860119894

(119909)

minus

[120583minus1

119861(120583119860119894+1

(119909))]

2

2

1205831015840

119860119894+1

(119909)

(25)

Since in 119860lowast = [1198881 1198882]120583119860119894

is not increasing and 120583119860119894+1

is notdecreasing then 1205831015840

119860119894

(119909) le 0 and 1205831015840119860119894+1

(119909) ge 0 and besidesthat if 1205831015840

119860119894

(119909) = 0 we have that 1205831015840119860119894+1

(119909) = 0 and if 1205831015840119860119894+1

(119909) =

0 we obtain 1205831015840

119860119894

(119909) = 0 Then from (25) Δ1015840(119909) lt 0 Thisshows that Δ is decreasing From Theorem 13 there exists astationary point in 119860lowast then this point is unique

Now consider the more general case when119860lowast sub [1199111 1199112]

and divide it into two theorems Initially consider the casethat the membership functions 120583

119862and 120583

119861are such that

120583119862(119905) gt 120583

119861(minus119905) next consider the case where 120583

119862(119905) lt

120583119861(minus119905)

6 Journal of Applied Mathematics

h1(1)

minush2(1)

h2(1)

h1

h2

hminus11 (minush2(1)) 1

Figure 6 Functions ℎ1and ℎ

2

m

n

1

1

1205751

1205752

s(z1)

s(c2)

120585

r(z0)r(c1)r(z2)hminus11 (minush2(1))

Figure 7 Functions 120585 1205751and 120575

2

41 Case 1 120583119862(119905) gt 120583

119861(minus119905)

Theorem 18 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) gt 120583

119861(minus119905) forall119905 isin (0 minus119887)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) lt (119901

3

1199023

) forall119901 isin supp(119861) 119902 isin

supp(119862) and 120583119861(119901) gt 120583

119862(119902)

(iii) [1205831015840119860119894+1

(119909)1205831015840

119860119894

(119909)] le 0 forall119909 isin (119911119900 1198882) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

(119911119900 1198882]

Proof For the sake of simplicity notation we make 119903 = 120583119860119894

119904 = 120583119860119894+1

119891 = 120583119861and 119892 = 120583

119862

Initially we see that given 119909 isin (119911119900 1198882] (Figure 4) 119909

determines only one (119899119898) isin [0 1]2 such that 119899 = 119903(119909)

and 119898 = 119904(119909) By monotonicity of 119903 we have that for each119899 isin [0 119903(119911

119900)) there exists only one 119898 isin [0 1] such that

119899 = 119903(119909) and 119898 = 119904(119909) That is each (119899119898) in this situationdetermines only one 119909 isin (119911

119900 1198882]

By Theorem 13 there exists a stationary point 119909lowast isin

[1198881 1198882] = [1198881 119911119900] cup (119911119900 1198882] Given 119909 isin [119888

1 119911119900] rArr 119898 = 119904(119909) le

119899 = 119903(119909) Then by Lemma 11 119909lowast notin [1198881 119911119900] rArr 119909

lowast

isin (119911119900 1198882]

That is equivalent to the existence of only one (119899lowast 119898lowast) with119899lowast

isin [0 119903(119911119900)) such that119867(119899lowast 119898lowast) = 0

Since for each 119899 isin [0 119903(119911119900)) there exists only one 119898 isin

[0 1] such that 119899 = 119903(119909) and 119898 = 119904(119909) then we may definea function 120575

2 [0 119903(119911

119900)) rarr [0 1] such that 119898 = 120575

2(119899)

(Figure 7) We observe that 1205752is continuous because 119903 and

119904 are continuous Using the chain rule we get the derivative of1205752to be

1205751015840

2(119899) =

1199041015840

(119909)

1199031015840(119909)

(iii)rArr 12057510158401015840

2(119899) le 0 forall119899 isin 119863

1205752 (26)

By Lemmas 7 and 8 ℎ1and minusℎ

2are increasing and by

condition (i) it follows that (Figure 6)

int

119886

0

119905119892 (119905) 119889119905 gt minusint

0

119887

119905119891 (119905) 119889119905 lArrrArr ℎ1(1) gt minusℎ

2(1) (27)

Then given 119899 isin [0 ℎminus1

1(minusℎ2(1))] there exists only one 119898 isin

[0 1] such that

ℎ1(119899) = minusℎ

2(119898) lArrrArr ℎ

1(119899) + ℎ

2(119898) = 0 (28)

Therefore we may define an injective function 120585 119898 = 120585(119899)

(Figure 7) so that the inverse range of 0 by119867 is given by

119867minus1

(0) = (119899119898) 119898 = 120585 (119899) (29)

where 119867 [0 ℎminus1

1(minusℎ2(1))] times [0 1] rarr R is given by

119867(119899119898) = ℎ1(119899) + ℎ

2(119898)

Since 120597119867120597119899 = ℎ10158401(119899) = (119892

minus1

(119899))

2

2 gt 0 (Lemma 7) and120597119867120597119898 = ℎ

1015840

2(119898) = minus(119891

minus1

(119898))

2

2 lt 0 (Lemma 8) then bythe Implicity Function Theorem 120585 is 119896 times differentiableand besides that

1205851015840

(119899) = minus

119889ℎ1119899

119889ℎ2119898

= [

119892minus1

(119899)

119891minus1(119898)

]

2

gt 0

forall119899 isin (0 ℎminus1

1(minusℎ2(1))) 119898 isin (0 1) 119898 = 120585 (119899)

(30)

Thus 120585 is a strictly increasing function and since119867(0 0) = 0and 119867(ℎminus1

1(minusℎ2(1)) 1) = 0 then 119863

120585= [0 ℎ

minus1

1(ℎ2(1))] and

119868119898120585= [0 1]Given 119898 119899 isin (0 1) there is only one 119901 isin (119887 0) such that

119901 = 119891minus1

(119898) and there exists only one 119902 isin (0 119886) such that 119902 =119892minus1

(119899) since by assumption119891 and 119892 are strictly monotonousFrom Lemma 11 we have that 119898 le 119899 rArr 119867(119898 119899) gt 0

Hence119867(119898 119899) = 0 rArr 119898 gt 119899 Therefore we are interested inthe pairs (119898 119899) such that119898 gt 119899 Thus we have

119898 gt 119899 lArrrArr 119891(119901) gt 119892 (119902) (31)

Journal of Applied Mathematics 7

1 1

0 5 40 60 90 100x

A1 A2 B

Δminus3 minus2 minus120576 0 2 3

m1 + 120576

m1

C

R1 if x is A1 then Δ is CR2 if x is A2 then Δ is B

Figure 8 p-Fuzzy system existence of more than one stationary point

Since 119891 and 119892 are monotonous by Lagrangersquos Medium ValueTheorem we obtain

119901 = 119891minus1

(119898) lArrrArr (119891minus1

)

1015840

(119898) =

1

1198911015840(119901)

(32)

119902 = 119892minus1

(119899) lArrrArr (119892minus1

)

1015840

(119899) =

1

1198921015840(119902)

(33)

Then

119898 gt 119899

(31)

lArr997904997904rArr 119891(119901) gt 119892 (119902)

(ii)rArr

1198921015840

(119902)

1198911015840(119901)

lt

1199013

1199023

(32)119890(33)

lArr997904997904997904997904997904rArr

(119891minus1

)

1015840

(119898)

(119892minus1)1015840

(119899)

lt

[119891minus1

(119898)]

3

[119892minus1(119899)]3

(34)

Therefore

119898 gt 119899 997904rArr (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

gt 0

(35)

By differentiation of (30) we obtain

12058510158401015840

(119899) =

minus2119892minus1

(119899)

[119891minus1(119898)]5

times (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

(36)

and since minus2119892minus1(119899)[119891minus1(119898)]5 gt 0 forall119898 119899 isin (0 1) from (35)we have

12058510158401015840

(119899) gt 0 forall119899 isin

119900

119863120585 (37)

Now we take 119868 = 119863120585cap 1198631205752= 119863120585cap [0 119903(119911

119900)) and we define

the function 120601 119868 rarr [0 1] such that

120601 (119899) = 120585 (119899) minus 1205752(119899) (38)

Then from (26) and (37) we have 12060110158401015840(119899) gt 0 forall119899 isin

119900

119868 Since120585(0) = 0 and by condition (iii) of Theorem 18 it followsthat 120575

2(0) gt 0 then we have 120601(0) lt 0 Consequently from

Lemma 9 we have that there is only one 119899lowast isin 119868 such that

120601 (119899lowast

) = 0

(38)

lArr997904997904rArr 120585 (119899lowast

) = 1205752(119899lowast

) (39)

Since 120585 = 119867minus1(0) then we obtain

0 = 119867 (119899lowast

120585 (119899lowast

))

(39)

= 119867 (119899lowast

1205752(119899lowast

)) (40)

So there exists only one 119909lowast isin (119911119900 1198882] 119899lowast = 119903(119909

lowast

) and119898lowast

= 1205752(119899lowast

) = 119904(119909lowast

) such that

Δ (119909lowast

) =

119867 (119899lowast

119898lowast

)

119860 (119899lowast 119898lowast)

= 0 (41)

This finally proves the theorem

42 Case 2 120583119862(119905)lt120583119861(minus119905)

Theorem 19 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861 and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) lt 120583

119861(minus119905) forall119905 isin (0 119886)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 forall119901 isin supp(119861) 119902 isin supp(119862)

and 120583119861(119901) lt 120583

119862(119902)

(iii) [1205831015840119860119894

(119909)1205831015840

119860119894+1

(119909)] le 0 forall119909 isin (1198881 119911119900) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

[1198881 119911119900)

Proof It is analogous to the proof of Theorem 18

43 Some Comments about Uniqueness Theorems When wedo not have 120583

119862(119905) gt 120583

119861(minus119905) or 120583

119862(119905) lt 120583

119861(minus119905) it is impossible

to establish general conditions for uniqueness of stationarypoints For example consider the p-fuzzy system in Figure 8This system has an equilibrium viable set 119860lowast = [119860

1cap 1198602]0

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

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Differential EquationsInternational Journal of

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

n

m

c1 x xz1 zo z2c2

Alowast

r s

Ai Ai+1

b 0 a

B

f

fminus1(m) gminus1(n)

g

R

CCR1 if x is Ai then Δ isR2 if x is Ai+1 then Δ is B

Δx

Δ

Figure 4 Mamdanirsquos inference process for 119860lowast of the type (119860119894 119860 i+1) rarr (119862 119861)

(ii) ⋂119895=119894119894+2

supp(119860119895) has at maximum only one element

for each 1 le 119894 lt 119896 minus 1 that is supp(119860119894) cap

supp(119860119894+2) = 120601 if and only if max119909 isin supp(119860

119894) =

min119909 isin supp(119860119894+2)

(iii) ⋃119894=1119896

supp(119860119894) = 119880 where 119880 is the domain of the

fuzzy variable 119909

(iv) given 1199111isin supp(119860

119894) and 119911

2isin supp(119860

119894+1) if 119860

119894(1199111) =

1 and 119860119894+1(1199112) = 1 then necessarily 119911

1lt 1199112for each

1 le 119894 lt 119896

Definition 6 Consider a family of successive fuzzy subsets1198601198941le119894le119896

that describe the antecedent of a fuzzy systemassociated with the p-fuzzy system (1) We say that 119860lowast is anequilibrium viable set of (1) if 119860lowast contains stationary pointsof (1)

If for some 1 le 119894 lt 119896 there are 1199111 1199112isin [119860119894cup 119860119894+1]0 such

that Δ(1199111) sdot Δ(119911

2) lt 0 then 119860lowast is given by 119860lowast = [119860

119894cap119860119894+1]0

If for all 1199111 1199112isin [119860119894cup 119860119894+1]0 1 le 119894 lt 119896 Δ(119911

1) sdot Δ(119911

2) gt 0

then 119860lowast = [119860119896]0

A p-fuzzy system depends on the fuzzy system associatedwith it that is it depends on the rule-base on the infer-ence method and on the defuzzification method used InDefinition 6 a sufficient condition forΔ(119911

1) sdotΔ(1199112) lt 0 is that

the p-fuzzy system be associated with a fuzzy system whoserule-base in 119860lowast = [119860

119894cup 119860119894+1]0 is of the type

1198771 if 119909 is 119860

119894then Δ is 119861

1198772 if 119909 is 119860

119894+1then Δ is 119862

where supp(119861) sub Rminus and supp(119862) sub R+ or vice versaWhen supp(119861) sub Rminus and supp(119862) sub R+ we have that theequilibrium viable set 119860lowast is of the type (119860

119894 119860119894+1) rarr (119862 119861)

If on the other hand we have that supp(119861) sub R+ andsupp(119862) sub Rminus we say that 119860lowast is of the type (119860

119894 119860119894+1) rarr

(119861 119862)To understand the dynamics of the p-fuzzy system we

need to understand how the rule-based system works morespecifically given 119909 isin 119860

lowast how is Δ(119909) obtained In thefollowing section we will describe this process

22 Output Defuzzification of the Fuzzy System Let 119860lowast =[119860119894cap 119860119894+1]0

= [1198881 1198882] be an equilibrium viable set of the

p-fuzzy system To facilitate the notation we will indicate

by 119903 the membership function of 119860119894 by 119904 the membership

function of 119860119894+1

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(4)

and by 119891 and 119892 the membership functions of 119862 and 119863

(Figure 4) respectively Assume that the p-fuzzy system inthe equilibrium viable set 119860lowast is of the type (119860

119894 119860119894+1) rarr

(119862 119861)For each 119909 isin 119860

lowast Δ(119909) is the R region centroid abscisewith R limited by the membership function of the fuzzyoutput Δ119909 (see Figure 4) Thus

Δ (119909) = (int

119891minus1(119898)

119887

119905119891 (119905) 119889119905 + int

0

119891minus1(119898)

119898119905119889119905 + int

119892minus1(119899)

0

119899119905 119889119905

+int

119886

119892minus1(119899)

119905119892 (119905) 119889119905)

times (int

119899

0

119892minus1

(119905) 119889119905 minus int

119898

0

119891minus1

(119905) 119889119905)

minus1

(5)

where (119899119898) = (119903(119909) 119904(119909)) Equation (5) can be rewritten as

Δ (119909) =

ℎ1(119899) + ℎ

2(119898)

119860 (119898 119899)

(6)

where

ℎ1(119899) = int

119892minus1(119899)

0

119899119905 119889119905 + int

119886

119892minus1(119899)

119905119892 (119905) 119889119905 (7)

ℎ2(119898) = int

119891minus1(119898)

119887

119905119891 (119905) 119889119905 + int

0

119891minus1(119898)

119898119905119889119905 (8)

119860 (119898 119899) = int

119899

0

119892minus1

(119905) 119889119905 minus int

119898

0

119891minus1

(119905) 119889119905 (9)

23 Preliminary Results In this section wewill present someimportant technical results for the main demonstrations ofthe theorems that establish sufficient conditions for unique-ness and existence of the stationary point of the p-fuzzysystems Here the presented results are technical enoughand are important only for demonstrating the theorems in

4 Journal of Applied Mathematics

the next section Therefore the reader will be able withoutloss of continuity to skip them if desired

The following results are referred to the equilibriumviable set of the type (119860

119894 119860119894+1) rarr (119862 119861) (Figure 4) From

now on it will be assumed that the functions 119903 119904 119891 and 119892(which represent themembership functions120583

119860119894120583119860119894+1

120583119861and

120583119862 resp) are continuous

Lemma 7 The function ℎ1(7) is increasing and its range is

given by 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Proof Since 119892 is continuous in [0 119886] (119892minus1 is limited in [0 119886])then the function ℎ

1is differentiable and

ℎ1015840

1(119899) = (int

119892minus1(119899)

0

119899119905 119889119905)

1015840

+ (int

119886

119892minus1(119899)

119905119892 (119905) 119889119905)

1015840

(10)

Using the derivative properties

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899(int

119892minus1(119899)

0

119905119889119905)

1015840

minus (int

119892minus1(119899)

119886

119905119892 (119905) 119889119905)

1015840

(11)

and using the chain rule and the fundamental theorem ofcalculus we obtain

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

minus 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

(12)

Hence

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 =

(119892minus1

(119899))

2

2

gt 0(13)

Therefore ℎ1is increasing

Since

ℎ1(0) = int

119892minus1(0)

0

0119905119889119905 + int

119886

119892minus1(0)

119905119892 (119905) 119889119905 = int

119886

119886

119905119892 (119905) 119889119905 = 0

ℎ1(1) = int

119892minus1(1)

0

119905119889119905 + int

119886

119892minus1(1)

119905119892 (119905) 119889119905 = int

119886

0

119905119892 (119905) 119889119905

(14)

then 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Lemma 8 The function ℎ2is decreasing and its range is given

by 119868119898(ℎ2) = [int

0

119887

119905119891(119905)dt 0]

Proof Analogous to the demonstration of Lemma 7

Lemma 9 Let 120601 119868 = [1198891 1198892] rarr R be a function of the class

1198622 If 12060110158401015840(119911) gt 0 forall119911 isin (119889

1 1198892) and 120601(119889

1) lt 0 then 120601 has at

maximum one root in 119868

1

k

b fminus1(k) 0 minusfminus1(k)

gminus1(k)

minusb a

f g

Figure 5 System p-fuzzy output with 119892(119905) gt 119891(minus119905)

Proof of Lemma 9 Assume there are 1199111 1199112isin 119868 (119911

1lt 1199112) such

that 120601(1199111) = 120601(119911

2) = 0 Since 12060110158401015840(119911) gt 0 we have that 120601 is not

constant Hence by Rollersquos Theorem exist119888 isin (1199111 1199112) such that

1206011015840

(119888) = 0 Hence 119888 is a minimum point because 12060110158401015840(119888) gt 0But 120601(119889

1) lt 0 rArr 120601(119888) gt 0 Since 120601 is continuous exist119911

119900isin

(1199111 1199112) such that 120601(119888) gt 120601(119911

119900) gt 0 it is nonsense Therefore

120601 has at maximum one root

Lemma 10 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then 119892minus1(119896) gt

minus119891minus1

(119896)forall119896 isin [0 1]

Proof The proof is simple (see Figure 5)

Lemma 11 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then for119898 119899 isin [0 1]with119898 le 119899 one has that

Δ (119909) =

ℎ1(119899) + ℎ

2(119898)

119860 (119898 119899)

gt 0 (15)

Proof The proof is simple

Lemma 12 If 119892(119905) lt 119891(minus119905) forall119905 isin [0 119886] and119898 119899 isin [0 1] 119898 ge

119899 then Δ(119909) lt 0

Proof It is analogous to the proof of Lemma 11

3 Stationary Point

In this section we will enunciate and prove a theorem thatguarantees the existence of at least one stationary point foreach equilibrium viable set of the p-fuzzy system For thiswe will use again Figure 4 tomotivate the presented results inthis section

Theorem 13 (existence) Let 119878 be a p-fuzzy system and 119860lowast anequilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861)

Then there is at least one stationary point of 119878 in 119860lowast That isexist119909lowast

isin 119860lowast such that Δ(119909lowast) = 0

Journal of Applied Mathematics 5

Proof Given 119909 isin 119860lowast fromDefinition 4 then 119909 is a stationarypoint if and only if

Δ (119909) = 0 lArrrArr ℎ1(119899) + ℎ

2(119898) = 0 (16)

If 120583119860119894(1198881) = 0 then

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(0) + ℎ

2(0) = 0

(17)

and therefore 1198881is a stationary point If 120583

119860119894+1(1198882) = 0 one has

that Δ(1198882) = 0 hence 119888

2is a stationary point Now assume

that 120583119860119894(1198881) gt 0 and 120583

119860119894+1(1198882) gt 0 Since 120583

119860119894+1(1198881) = 0

then from Lemmas 7 and 8 one has that ℎ1(120583119860119894(1198881)) gt 0 and

ℎ2(120583119860119894+1

(1198881)) = 0 Therefore

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(120583119860119894(1198881)) gt 0

(18)

Thus from Lemma 7

Δ (1198882) = ℎ1(120583119860119894(1198882)) + ℎ

2(120583119860119894+1

(1198882)) = ℎ

2(120583119860119894+1

(1198882)) lt 0

(19)

Since Δ is continuous by Bolzanorsquos Intermediate ValueTheorem exist119909lowast isin [119888

1 1198882] such that Δ(119909lowast) = 0 therefore 119909lowast

is a stationary point

Remark 14 If in Theorem 13 we consider 119860lowast as an equilib-rium viable set of the type (119860

119894 119860119894+1) rarr (119861 119862) the result is

analogous that is there exists a stationary point 119909lowast isin 119860lowast

31 Local Stationary Points-Symmetrical Output If 119860lowast is anequilibrium viable set where the membership functions ofthe consequents 119861 and 119862 are symmetrical functions thenthe stationary point in119860lowast is unique Except when 120583

119860119894(1198881) = 0

or 120583119860119894+1

(1198882) = 0 possibly occurs Then we have the following

proposition

Proposition 15 Let 119878 be a p-fuzzy system and 119860lowast an equi-librium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the

membership functions of 119861 and 119862 respectively 120583119861and 120583

119862 are

monotonous and symmetric that is 120583119862(119905) = 120583

119861(minus119905) then there

exists an equilibrium point in 119860lowast

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (20)

Proof Since 120583119861(119905) = 120583

119862(minus119905) then 120583

119861(minus119886) = 120583

119862(119886) = 0 =

120583119861(119887) rArr 119887 = minus119886 because 120583

119861U is monotonous Yet we have

that 120583119861(119905) = 120583

119862(minus119905) rArr 120583

minus1

119861(120583119862(119905)) = minus119905 = minus120583

minus1

119861(120583119861(119905)) rArr

120583minus1

119861(119910) = minus120583

minus1

119862(119910)

Then Δ(119911119900) = 0 if and only if ℎ

1(119899) = minusℎ

2(119898) Since

119887 = minus119886 from (8) then we obtain

ℎ2(119898) = int

120583minus1

119861(119898)

minus119886

119905120583119861(119905) 119889119905 + int

0

120583minus1

119861(119898)

119898119905119889119905 (21)

If we perform a change in the variable 119906 = minus119905 we have

ℎ2(119898) = int

minus120583minus1

119861(119898)

a119906120583119861(minus119906) 119889119906 + int

0

minus120583minus1

119861(119898)

119898119906119889119906

997904rArr ℎ2(119898) = int

120583minus1

119862(119898)

119886

119906120583119862(119906) 119889119906 + int

0

120583minus1

119862(119898)

119898119906119889119906 = minusℎ1(119898)

(22)

That is ℎ2= minusℎ

1 Hence ℎ

1(119899) = minusℎ

2(119898) hArr ℎ

1(119899) =

ℎ1(119898) hArr 119898 = 119899 (because ℎ

1is increasing Lemma 7) that

proves the proposition

Remark 16 If 120583119860119894(1198881) = 0 then 119909lowast = max

119909isin119860lowast[min(120583

119860119894(119909)

120583119860119894+1

(119909))] is the only stationary point in 119860lowast Besides that ifthe system 119878 is of the type (119860

119894 119860119894+1) rarr (119861 119862) then the result

of Proposition 15 is the same

4 Uniqueness of the Stationary Point

In this section we will enunciate and prove theorems thatestablish condition for uniqueness of the stationary point ofa one-dimensional p-fuzzy system Initially we will considera simpler case when 119860lowast sub [119911

1 1199112] (Figure 4) where

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(23)

Theorem 17 Let 119878 be a p-fuzzy system and119860lowast an equilibriumviable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the functions

120583119860119894and120583119860119894+1

are piecewisemonotonous and119860lowast sub [1199111 1199112] then

there exists only one stationary point in 119860lowast

Proof Given 119909 isin 119860lowast one has

Δ (119909) = ℎ1(119899) + ℎ

2(119898) = ℎ

1(120583119860119894(119909)) + ℎ

2(120583119860119894+1

(119909))

(24)

Using Lemmas 7 and 8 and the chain rule we find that thederivative of Δ is

Δ1015840

(119909) =

[120583minus1

119862(120583119860119894(119909))]

2

2

1205831015840

119860119894

(119909)

minus

[120583minus1

119861(120583119860119894+1

(119909))]

2

2

1205831015840

119860119894+1

(119909)

(25)

Since in 119860lowast = [1198881 1198882]120583119860119894

is not increasing and 120583119860119894+1

is notdecreasing then 1205831015840

119860119894

(119909) le 0 and 1205831015840119860119894+1

(119909) ge 0 and besidesthat if 1205831015840

119860119894

(119909) = 0 we have that 1205831015840119860119894+1

(119909) = 0 and if 1205831015840119860119894+1

(119909) =

0 we obtain 1205831015840

119860119894

(119909) = 0 Then from (25) Δ1015840(119909) lt 0 Thisshows that Δ is decreasing From Theorem 13 there exists astationary point in 119860lowast then this point is unique

Now consider the more general case when119860lowast sub [1199111 1199112]

and divide it into two theorems Initially consider the casethat the membership functions 120583

119862and 120583

119861are such that

120583119862(119905) gt 120583

119861(minus119905) next consider the case where 120583

119862(119905) lt

120583119861(minus119905)

6 Journal of Applied Mathematics

h1(1)

minush2(1)

h2(1)

h1

h2

hminus11 (minush2(1)) 1

Figure 6 Functions ℎ1and ℎ

2

m

n

1

1

1205751

1205752

s(z1)

s(c2)

120585

r(z0)r(c1)r(z2)hminus11 (minush2(1))

Figure 7 Functions 120585 1205751and 120575

2

41 Case 1 120583119862(119905) gt 120583

119861(minus119905)

Theorem 18 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) gt 120583

119861(minus119905) forall119905 isin (0 minus119887)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) lt (119901

3

1199023

) forall119901 isin supp(119861) 119902 isin

supp(119862) and 120583119861(119901) gt 120583

119862(119902)

(iii) [1205831015840119860119894+1

(119909)1205831015840

119860119894

(119909)] le 0 forall119909 isin (119911119900 1198882) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

(119911119900 1198882]

Proof For the sake of simplicity notation we make 119903 = 120583119860119894

119904 = 120583119860119894+1

119891 = 120583119861and 119892 = 120583

119862

Initially we see that given 119909 isin (119911119900 1198882] (Figure 4) 119909

determines only one (119899119898) isin [0 1]2 such that 119899 = 119903(119909)

and 119898 = 119904(119909) By monotonicity of 119903 we have that for each119899 isin [0 119903(119911

119900)) there exists only one 119898 isin [0 1] such that

119899 = 119903(119909) and 119898 = 119904(119909) That is each (119899119898) in this situationdetermines only one 119909 isin (119911

119900 1198882]

By Theorem 13 there exists a stationary point 119909lowast isin

[1198881 1198882] = [1198881 119911119900] cup (119911119900 1198882] Given 119909 isin [119888

1 119911119900] rArr 119898 = 119904(119909) le

119899 = 119903(119909) Then by Lemma 11 119909lowast notin [1198881 119911119900] rArr 119909

lowast

isin (119911119900 1198882]

That is equivalent to the existence of only one (119899lowast 119898lowast) with119899lowast

isin [0 119903(119911119900)) such that119867(119899lowast 119898lowast) = 0

Since for each 119899 isin [0 119903(119911119900)) there exists only one 119898 isin

[0 1] such that 119899 = 119903(119909) and 119898 = 119904(119909) then we may definea function 120575

2 [0 119903(119911

119900)) rarr [0 1] such that 119898 = 120575

2(119899)

(Figure 7) We observe that 1205752is continuous because 119903 and

119904 are continuous Using the chain rule we get the derivative of1205752to be

1205751015840

2(119899) =

1199041015840

(119909)

1199031015840(119909)

(iii)rArr 12057510158401015840

2(119899) le 0 forall119899 isin 119863

1205752 (26)

By Lemmas 7 and 8 ℎ1and minusℎ

2are increasing and by

condition (i) it follows that (Figure 6)

int

119886

0

119905119892 (119905) 119889119905 gt minusint

0

119887

119905119891 (119905) 119889119905 lArrrArr ℎ1(1) gt minusℎ

2(1) (27)

Then given 119899 isin [0 ℎminus1

1(minusℎ2(1))] there exists only one 119898 isin

[0 1] such that

ℎ1(119899) = minusℎ

2(119898) lArrrArr ℎ

1(119899) + ℎ

2(119898) = 0 (28)

Therefore we may define an injective function 120585 119898 = 120585(119899)

(Figure 7) so that the inverse range of 0 by119867 is given by

119867minus1

(0) = (119899119898) 119898 = 120585 (119899) (29)

where 119867 [0 ℎminus1

1(minusℎ2(1))] times [0 1] rarr R is given by

119867(119899119898) = ℎ1(119899) + ℎ

2(119898)

Since 120597119867120597119899 = ℎ10158401(119899) = (119892

minus1

(119899))

2

2 gt 0 (Lemma 7) and120597119867120597119898 = ℎ

1015840

2(119898) = minus(119891

minus1

(119898))

2

2 lt 0 (Lemma 8) then bythe Implicity Function Theorem 120585 is 119896 times differentiableand besides that

1205851015840

(119899) = minus

119889ℎ1119899

119889ℎ2119898

= [

119892minus1

(119899)

119891minus1(119898)

]

2

gt 0

forall119899 isin (0 ℎminus1

1(minusℎ2(1))) 119898 isin (0 1) 119898 = 120585 (119899)

(30)

Thus 120585 is a strictly increasing function and since119867(0 0) = 0and 119867(ℎminus1

1(minusℎ2(1)) 1) = 0 then 119863

120585= [0 ℎ

minus1

1(ℎ2(1))] and

119868119898120585= [0 1]Given 119898 119899 isin (0 1) there is only one 119901 isin (119887 0) such that

119901 = 119891minus1

(119898) and there exists only one 119902 isin (0 119886) such that 119902 =119892minus1

(119899) since by assumption119891 and 119892 are strictly monotonousFrom Lemma 11 we have that 119898 le 119899 rArr 119867(119898 119899) gt 0

Hence119867(119898 119899) = 0 rArr 119898 gt 119899 Therefore we are interested inthe pairs (119898 119899) such that119898 gt 119899 Thus we have

119898 gt 119899 lArrrArr 119891(119901) gt 119892 (119902) (31)

Journal of Applied Mathematics 7

1 1

0 5 40 60 90 100x

A1 A2 B

Δminus3 minus2 minus120576 0 2 3

m1 + 120576

m1

C

R1 if x is A1 then Δ is CR2 if x is A2 then Δ is B

Figure 8 p-Fuzzy system existence of more than one stationary point

Since 119891 and 119892 are monotonous by Lagrangersquos Medium ValueTheorem we obtain

119901 = 119891minus1

(119898) lArrrArr (119891minus1

)

1015840

(119898) =

1

1198911015840(119901)

(32)

119902 = 119892minus1

(119899) lArrrArr (119892minus1

)

1015840

(119899) =

1

1198921015840(119902)

(33)

Then

119898 gt 119899

(31)

lArr997904997904rArr 119891(119901) gt 119892 (119902)

(ii)rArr

1198921015840

(119902)

1198911015840(119901)

lt

1199013

1199023

(32)119890(33)

lArr997904997904997904997904997904rArr

(119891minus1

)

1015840

(119898)

(119892minus1)1015840

(119899)

lt

[119891minus1

(119898)]

3

[119892minus1(119899)]3

(34)

Therefore

119898 gt 119899 997904rArr (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

gt 0

(35)

By differentiation of (30) we obtain

12058510158401015840

(119899) =

minus2119892minus1

(119899)

[119891minus1(119898)]5

times (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

(36)

and since minus2119892minus1(119899)[119891minus1(119898)]5 gt 0 forall119898 119899 isin (0 1) from (35)we have

12058510158401015840

(119899) gt 0 forall119899 isin

119900

119863120585 (37)

Now we take 119868 = 119863120585cap 1198631205752= 119863120585cap [0 119903(119911

119900)) and we define

the function 120601 119868 rarr [0 1] such that

120601 (119899) = 120585 (119899) minus 1205752(119899) (38)

Then from (26) and (37) we have 12060110158401015840(119899) gt 0 forall119899 isin

119900

119868 Since120585(0) = 0 and by condition (iii) of Theorem 18 it followsthat 120575

2(0) gt 0 then we have 120601(0) lt 0 Consequently from

Lemma 9 we have that there is only one 119899lowast isin 119868 such that

120601 (119899lowast

) = 0

(38)

lArr997904997904rArr 120585 (119899lowast

) = 1205752(119899lowast

) (39)

Since 120585 = 119867minus1(0) then we obtain

0 = 119867 (119899lowast

120585 (119899lowast

))

(39)

= 119867 (119899lowast

1205752(119899lowast

)) (40)

So there exists only one 119909lowast isin (119911119900 1198882] 119899lowast = 119903(119909

lowast

) and119898lowast

= 1205752(119899lowast

) = 119904(119909lowast

) such that

Δ (119909lowast

) =

119867 (119899lowast

119898lowast

)

119860 (119899lowast 119898lowast)

= 0 (41)

This finally proves the theorem

42 Case 2 120583119862(119905)lt120583119861(minus119905)

Theorem 19 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861 and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) lt 120583

119861(minus119905) forall119905 isin (0 119886)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 forall119901 isin supp(119861) 119902 isin supp(119862)

and 120583119861(119901) lt 120583

119862(119902)

(iii) [1205831015840119860119894

(119909)1205831015840

119860119894+1

(119909)] le 0 forall119909 isin (1198881 119911119900) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

[1198881 119911119900)

Proof It is analogous to the proof of Theorem 18

43 Some Comments about Uniqueness Theorems When wedo not have 120583

119862(119905) gt 120583

119861(minus119905) or 120583

119862(119905) lt 120583

119861(minus119905) it is impossible

to establish general conditions for uniqueness of stationarypoints For example consider the p-fuzzy system in Figure 8This system has an equilibrium viable set 119860lowast = [119860

1cap 1198602]0

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

the next section Therefore the reader will be able withoutloss of continuity to skip them if desired

The following results are referred to the equilibriumviable set of the type (119860

119894 119860119894+1) rarr (119862 119861) (Figure 4) From

now on it will be assumed that the functions 119903 119904 119891 and 119892(which represent themembership functions120583

119860119894120583119860119894+1

120583119861and

120583119862 resp) are continuous

Lemma 7 The function ℎ1(7) is increasing and its range is

given by 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Proof Since 119892 is continuous in [0 119886] (119892minus1 is limited in [0 119886])then the function ℎ

1is differentiable and

ℎ1015840

1(119899) = (int

119892minus1(119899)

0

119899119905 119889119905)

1015840

+ (int

119886

119892minus1(119899)

119905119892 (119905) 119889119905)

1015840

(10)

Using the derivative properties

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899(int

119892minus1(119899)

0

119905119889119905)

1015840

minus (int

119892minus1(119899)

119886

119905119892 (119905) 119889119905)

1015840

(11)

and using the chain rule and the fundamental theorem ofcalculus we obtain

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 + 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

minus 119899119892minus1

(119899) (119892minus1

)

1015840

(119899)

(12)

Hence

ℎ1015840

1(119899) = int

119892minus1(119899)

0

119905119889119905 =

(119892minus1

(119899))

2

2

gt 0(13)

Therefore ℎ1is increasing

Since

ℎ1(0) = int

119892minus1(0)

0

0119905119889119905 + int

119886

119892minus1(0)

119905119892 (119905) 119889119905 = int

119886

119886

119905119892 (119905) 119889119905 = 0

ℎ1(1) = int

119892minus1(1)

0

119905119889119905 + int

119886

119892minus1(1)

119905119892 (119905) 119889119905 = int

119886

0

119905119892 (119905) 119889119905

(14)

then 119868119898(ℎ1) = [0 int

119886

0

119905119892(119905)119889119905]

Lemma 8 The function ℎ2is decreasing and its range is given

by 119868119898(ℎ2) = [int

0

119887

119905119891(119905)dt 0]

Proof Analogous to the demonstration of Lemma 7

Lemma 9 Let 120601 119868 = [1198891 1198892] rarr R be a function of the class

1198622 If 12060110158401015840(119911) gt 0 forall119911 isin (119889

1 1198892) and 120601(119889

1) lt 0 then 120601 has at

maximum one root in 119868

1

k

b fminus1(k) 0 minusfminus1(k)

gminus1(k)

minusb a

f g

Figure 5 System p-fuzzy output with 119892(119905) gt 119891(minus119905)

Proof of Lemma 9 Assume there are 1199111 1199112isin 119868 (119911

1lt 1199112) such

that 120601(1199111) = 120601(119911

2) = 0 Since 12060110158401015840(119911) gt 0 we have that 120601 is not

constant Hence by Rollersquos Theorem exist119888 isin (1199111 1199112) such that

1206011015840

(119888) = 0 Hence 119888 is a minimum point because 12060110158401015840(119888) gt 0But 120601(119889

1) lt 0 rArr 120601(119888) gt 0 Since 120601 is continuous exist119911

119900isin

(1199111 1199112) such that 120601(119888) gt 120601(119911

119900) gt 0 it is nonsense Therefore

120601 has at maximum one root

Lemma 10 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then 119892minus1(119896) gt

minus119891minus1

(119896)forall119896 isin [0 1]

Proof The proof is simple (see Figure 5)

Lemma 11 If 119892(119905) gt 119891(minus119905) forall119905 isin [0 minus119887] then for119898 119899 isin [0 1]with119898 le 119899 one has that

Δ (119909) =

ℎ1(119899) + ℎ

2(119898)

119860 (119898 119899)

gt 0 (15)

Proof The proof is simple

Lemma 12 If 119892(119905) lt 119891(minus119905) forall119905 isin [0 119886] and119898 119899 isin [0 1] 119898 ge

119899 then Δ(119909) lt 0

Proof It is analogous to the proof of Lemma 11

3 Stationary Point

In this section we will enunciate and prove a theorem thatguarantees the existence of at least one stationary point foreach equilibrium viable set of the p-fuzzy system For thiswe will use again Figure 4 tomotivate the presented results inthis section

Theorem 13 (existence) Let 119878 be a p-fuzzy system and 119860lowast anequilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861)

Then there is at least one stationary point of 119878 in 119860lowast That isexist119909lowast

isin 119860lowast such that Δ(119909lowast) = 0

Journal of Applied Mathematics 5

Proof Given 119909 isin 119860lowast fromDefinition 4 then 119909 is a stationarypoint if and only if

Δ (119909) = 0 lArrrArr ℎ1(119899) + ℎ

2(119898) = 0 (16)

If 120583119860119894(1198881) = 0 then

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(0) + ℎ

2(0) = 0

(17)

and therefore 1198881is a stationary point If 120583

119860119894+1(1198882) = 0 one has

that Δ(1198882) = 0 hence 119888

2is a stationary point Now assume

that 120583119860119894(1198881) gt 0 and 120583

119860119894+1(1198882) gt 0 Since 120583

119860119894+1(1198881) = 0

then from Lemmas 7 and 8 one has that ℎ1(120583119860119894(1198881)) gt 0 and

ℎ2(120583119860119894+1

(1198881)) = 0 Therefore

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(120583119860119894(1198881)) gt 0

(18)

Thus from Lemma 7

Δ (1198882) = ℎ1(120583119860119894(1198882)) + ℎ

2(120583119860119894+1

(1198882)) = ℎ

2(120583119860119894+1

(1198882)) lt 0

(19)

Since Δ is continuous by Bolzanorsquos Intermediate ValueTheorem exist119909lowast isin [119888

1 1198882] such that Δ(119909lowast) = 0 therefore 119909lowast

is a stationary point

Remark 14 If in Theorem 13 we consider 119860lowast as an equilib-rium viable set of the type (119860

119894 119860119894+1) rarr (119861 119862) the result is

analogous that is there exists a stationary point 119909lowast isin 119860lowast

31 Local Stationary Points-Symmetrical Output If 119860lowast is anequilibrium viable set where the membership functions ofthe consequents 119861 and 119862 are symmetrical functions thenthe stationary point in119860lowast is unique Except when 120583

119860119894(1198881) = 0

or 120583119860119894+1

(1198882) = 0 possibly occurs Then we have the following

proposition

Proposition 15 Let 119878 be a p-fuzzy system and 119860lowast an equi-librium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the

membership functions of 119861 and 119862 respectively 120583119861and 120583

119862 are

monotonous and symmetric that is 120583119862(119905) = 120583

119861(minus119905) then there

exists an equilibrium point in 119860lowast

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (20)

Proof Since 120583119861(119905) = 120583

119862(minus119905) then 120583

119861(minus119886) = 120583

119862(119886) = 0 =

120583119861(119887) rArr 119887 = minus119886 because 120583

119861U is monotonous Yet we have

that 120583119861(119905) = 120583

119862(minus119905) rArr 120583

minus1

119861(120583119862(119905)) = minus119905 = minus120583

minus1

119861(120583119861(119905)) rArr

120583minus1

119861(119910) = minus120583

minus1

119862(119910)

Then Δ(119911119900) = 0 if and only if ℎ

1(119899) = minusℎ

2(119898) Since

119887 = minus119886 from (8) then we obtain

ℎ2(119898) = int

120583minus1

119861(119898)

minus119886

119905120583119861(119905) 119889119905 + int

0

120583minus1

119861(119898)

119898119905119889119905 (21)

If we perform a change in the variable 119906 = minus119905 we have

ℎ2(119898) = int

minus120583minus1

119861(119898)

a119906120583119861(minus119906) 119889119906 + int

0

minus120583minus1

119861(119898)

119898119906119889119906

997904rArr ℎ2(119898) = int

120583minus1

119862(119898)

119886

119906120583119862(119906) 119889119906 + int

0

120583minus1

119862(119898)

119898119906119889119906 = minusℎ1(119898)

(22)

That is ℎ2= minusℎ

1 Hence ℎ

1(119899) = minusℎ

2(119898) hArr ℎ

1(119899) =

ℎ1(119898) hArr 119898 = 119899 (because ℎ

1is increasing Lemma 7) that

proves the proposition

Remark 16 If 120583119860119894(1198881) = 0 then 119909lowast = max

119909isin119860lowast[min(120583

119860119894(119909)

120583119860119894+1

(119909))] is the only stationary point in 119860lowast Besides that ifthe system 119878 is of the type (119860

119894 119860119894+1) rarr (119861 119862) then the result

of Proposition 15 is the same

4 Uniqueness of the Stationary Point

In this section we will enunciate and prove theorems thatestablish condition for uniqueness of the stationary point ofa one-dimensional p-fuzzy system Initially we will considera simpler case when 119860lowast sub [119911

1 1199112] (Figure 4) where

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(23)

Theorem 17 Let 119878 be a p-fuzzy system and119860lowast an equilibriumviable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the functions

120583119860119894and120583119860119894+1

are piecewisemonotonous and119860lowast sub [1199111 1199112] then

there exists only one stationary point in 119860lowast

Proof Given 119909 isin 119860lowast one has

Δ (119909) = ℎ1(119899) + ℎ

2(119898) = ℎ

1(120583119860119894(119909)) + ℎ

2(120583119860119894+1

(119909))

(24)

Using Lemmas 7 and 8 and the chain rule we find that thederivative of Δ is

Δ1015840

(119909) =

[120583minus1

119862(120583119860119894(119909))]

2

2

1205831015840

119860119894

(119909)

minus

[120583minus1

119861(120583119860119894+1

(119909))]

2

2

1205831015840

119860119894+1

(119909)

(25)

Since in 119860lowast = [1198881 1198882]120583119860119894

is not increasing and 120583119860119894+1

is notdecreasing then 1205831015840

119860119894

(119909) le 0 and 1205831015840119860119894+1

(119909) ge 0 and besidesthat if 1205831015840

119860119894

(119909) = 0 we have that 1205831015840119860119894+1

(119909) = 0 and if 1205831015840119860119894+1

(119909) =

0 we obtain 1205831015840

119860119894

(119909) = 0 Then from (25) Δ1015840(119909) lt 0 Thisshows that Δ is decreasing From Theorem 13 there exists astationary point in 119860lowast then this point is unique

Now consider the more general case when119860lowast sub [1199111 1199112]

and divide it into two theorems Initially consider the casethat the membership functions 120583

119862and 120583

119861are such that

120583119862(119905) gt 120583

119861(minus119905) next consider the case where 120583

119862(119905) lt

120583119861(minus119905)

6 Journal of Applied Mathematics

h1(1)

minush2(1)

h2(1)

h1

h2

hminus11 (minush2(1)) 1

Figure 6 Functions ℎ1and ℎ

2

m

n

1

1

1205751

1205752

s(z1)

s(c2)

120585

r(z0)r(c1)r(z2)hminus11 (minush2(1))

Figure 7 Functions 120585 1205751and 120575

2

41 Case 1 120583119862(119905) gt 120583

119861(minus119905)

Theorem 18 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) gt 120583

119861(minus119905) forall119905 isin (0 minus119887)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) lt (119901

3

1199023

) forall119901 isin supp(119861) 119902 isin

supp(119862) and 120583119861(119901) gt 120583

119862(119902)

(iii) [1205831015840119860119894+1

(119909)1205831015840

119860119894

(119909)] le 0 forall119909 isin (119911119900 1198882) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

(119911119900 1198882]

Proof For the sake of simplicity notation we make 119903 = 120583119860119894

119904 = 120583119860119894+1

119891 = 120583119861and 119892 = 120583

119862

Initially we see that given 119909 isin (119911119900 1198882] (Figure 4) 119909

determines only one (119899119898) isin [0 1]2 such that 119899 = 119903(119909)

and 119898 = 119904(119909) By monotonicity of 119903 we have that for each119899 isin [0 119903(119911

119900)) there exists only one 119898 isin [0 1] such that

119899 = 119903(119909) and 119898 = 119904(119909) That is each (119899119898) in this situationdetermines only one 119909 isin (119911

119900 1198882]

By Theorem 13 there exists a stationary point 119909lowast isin

[1198881 1198882] = [1198881 119911119900] cup (119911119900 1198882] Given 119909 isin [119888

1 119911119900] rArr 119898 = 119904(119909) le

119899 = 119903(119909) Then by Lemma 11 119909lowast notin [1198881 119911119900] rArr 119909

lowast

isin (119911119900 1198882]

That is equivalent to the existence of only one (119899lowast 119898lowast) with119899lowast

isin [0 119903(119911119900)) such that119867(119899lowast 119898lowast) = 0

Since for each 119899 isin [0 119903(119911119900)) there exists only one 119898 isin

[0 1] such that 119899 = 119903(119909) and 119898 = 119904(119909) then we may definea function 120575

2 [0 119903(119911

119900)) rarr [0 1] such that 119898 = 120575

2(119899)

(Figure 7) We observe that 1205752is continuous because 119903 and

119904 are continuous Using the chain rule we get the derivative of1205752to be

1205751015840

2(119899) =

1199041015840

(119909)

1199031015840(119909)

(iii)rArr 12057510158401015840

2(119899) le 0 forall119899 isin 119863

1205752 (26)

By Lemmas 7 and 8 ℎ1and minusℎ

2are increasing and by

condition (i) it follows that (Figure 6)

int

119886

0

119905119892 (119905) 119889119905 gt minusint

0

119887

119905119891 (119905) 119889119905 lArrrArr ℎ1(1) gt minusℎ

2(1) (27)

Then given 119899 isin [0 ℎminus1

1(minusℎ2(1))] there exists only one 119898 isin

[0 1] such that

ℎ1(119899) = minusℎ

2(119898) lArrrArr ℎ

1(119899) + ℎ

2(119898) = 0 (28)

Therefore we may define an injective function 120585 119898 = 120585(119899)

(Figure 7) so that the inverse range of 0 by119867 is given by

119867minus1

(0) = (119899119898) 119898 = 120585 (119899) (29)

where 119867 [0 ℎminus1

1(minusℎ2(1))] times [0 1] rarr R is given by

119867(119899119898) = ℎ1(119899) + ℎ

2(119898)

Since 120597119867120597119899 = ℎ10158401(119899) = (119892

minus1

(119899))

2

2 gt 0 (Lemma 7) and120597119867120597119898 = ℎ

1015840

2(119898) = minus(119891

minus1

(119898))

2

2 lt 0 (Lemma 8) then bythe Implicity Function Theorem 120585 is 119896 times differentiableand besides that

1205851015840

(119899) = minus

119889ℎ1119899

119889ℎ2119898

= [

119892minus1

(119899)

119891minus1(119898)

]

2

gt 0

forall119899 isin (0 ℎminus1

1(minusℎ2(1))) 119898 isin (0 1) 119898 = 120585 (119899)

(30)

Thus 120585 is a strictly increasing function and since119867(0 0) = 0and 119867(ℎminus1

1(minusℎ2(1)) 1) = 0 then 119863

120585= [0 ℎ

minus1

1(ℎ2(1))] and

119868119898120585= [0 1]Given 119898 119899 isin (0 1) there is only one 119901 isin (119887 0) such that

119901 = 119891minus1

(119898) and there exists only one 119902 isin (0 119886) such that 119902 =119892minus1

(119899) since by assumption119891 and 119892 are strictly monotonousFrom Lemma 11 we have that 119898 le 119899 rArr 119867(119898 119899) gt 0

Hence119867(119898 119899) = 0 rArr 119898 gt 119899 Therefore we are interested inthe pairs (119898 119899) such that119898 gt 119899 Thus we have

119898 gt 119899 lArrrArr 119891(119901) gt 119892 (119902) (31)

Journal of Applied Mathematics 7

1 1

0 5 40 60 90 100x

A1 A2 B

Δminus3 minus2 minus120576 0 2 3

m1 + 120576

m1

C

R1 if x is A1 then Δ is CR2 if x is A2 then Δ is B

Figure 8 p-Fuzzy system existence of more than one stationary point

Since 119891 and 119892 are monotonous by Lagrangersquos Medium ValueTheorem we obtain

119901 = 119891minus1

(119898) lArrrArr (119891minus1

)

1015840

(119898) =

1

1198911015840(119901)

(32)

119902 = 119892minus1

(119899) lArrrArr (119892minus1

)

1015840

(119899) =

1

1198921015840(119902)

(33)

Then

119898 gt 119899

(31)

lArr997904997904rArr 119891(119901) gt 119892 (119902)

(ii)rArr

1198921015840

(119902)

1198911015840(119901)

lt

1199013

1199023

(32)119890(33)

lArr997904997904997904997904997904rArr

(119891minus1

)

1015840

(119898)

(119892minus1)1015840

(119899)

lt

[119891minus1

(119898)]

3

[119892minus1(119899)]3

(34)

Therefore

119898 gt 119899 997904rArr (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

gt 0

(35)

By differentiation of (30) we obtain

12058510158401015840

(119899) =

minus2119892minus1

(119899)

[119891minus1(119898)]5

times (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

(36)

and since minus2119892minus1(119899)[119891minus1(119898)]5 gt 0 forall119898 119899 isin (0 1) from (35)we have

12058510158401015840

(119899) gt 0 forall119899 isin

119900

119863120585 (37)

Now we take 119868 = 119863120585cap 1198631205752= 119863120585cap [0 119903(119911

119900)) and we define

the function 120601 119868 rarr [0 1] such that

120601 (119899) = 120585 (119899) minus 1205752(119899) (38)

Then from (26) and (37) we have 12060110158401015840(119899) gt 0 forall119899 isin

119900

119868 Since120585(0) = 0 and by condition (iii) of Theorem 18 it followsthat 120575

2(0) gt 0 then we have 120601(0) lt 0 Consequently from

Lemma 9 we have that there is only one 119899lowast isin 119868 such that

120601 (119899lowast

) = 0

(38)

lArr997904997904rArr 120585 (119899lowast

) = 1205752(119899lowast

) (39)

Since 120585 = 119867minus1(0) then we obtain

0 = 119867 (119899lowast

120585 (119899lowast

))

(39)

= 119867 (119899lowast

1205752(119899lowast

)) (40)

So there exists only one 119909lowast isin (119911119900 1198882] 119899lowast = 119903(119909

lowast

) and119898lowast

= 1205752(119899lowast

) = 119904(119909lowast

) such that

Δ (119909lowast

) =

119867 (119899lowast

119898lowast

)

119860 (119899lowast 119898lowast)

= 0 (41)

This finally proves the theorem

42 Case 2 120583119862(119905)lt120583119861(minus119905)

Theorem 19 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861 and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) lt 120583

119861(minus119905) forall119905 isin (0 119886)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 forall119901 isin supp(119861) 119902 isin supp(119862)

and 120583119861(119901) lt 120583

119862(119902)

(iii) [1205831015840119860119894

(119909)1205831015840

119860119894+1

(119909)] le 0 forall119909 isin (1198881 119911119900) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

[1198881 119911119900)

Proof It is analogous to the proof of Theorem 18

43 Some Comments about Uniqueness Theorems When wedo not have 120583

119862(119905) gt 120583

119861(minus119905) or 120583

119862(119905) lt 120583

119861(minus119905) it is impossible

to establish general conditions for uniqueness of stationarypoints For example consider the p-fuzzy system in Figure 8This system has an equilibrium viable set 119860lowast = [119860

1cap 1198602]0

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

Proof Given 119909 isin 119860lowast fromDefinition 4 then 119909 is a stationarypoint if and only if

Δ (119909) = 0 lArrrArr ℎ1(119899) + ℎ

2(119898) = 0 (16)

If 120583119860119894(1198881) = 0 then

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(0) + ℎ

2(0) = 0

(17)

and therefore 1198881is a stationary point If 120583

119860119894+1(1198882) = 0 one has

that Δ(1198882) = 0 hence 119888

2is a stationary point Now assume

that 120583119860119894(1198881) gt 0 and 120583

119860119894+1(1198882) gt 0 Since 120583

119860119894+1(1198881) = 0

then from Lemmas 7 and 8 one has that ℎ1(120583119860119894(1198881)) gt 0 and

ℎ2(120583119860119894+1

(1198881)) = 0 Therefore

Δ (1198881) = ℎ1(120583119860119894(1198881)) + ℎ

2(120583119860119894+1

(1198881)) = ℎ

1(120583119860119894(1198881)) gt 0

(18)

Thus from Lemma 7

Δ (1198882) = ℎ1(120583119860119894(1198882)) + ℎ

2(120583119860119894+1

(1198882)) = ℎ

2(120583119860119894+1

(1198882)) lt 0

(19)

Since Δ is continuous by Bolzanorsquos Intermediate ValueTheorem exist119909lowast isin [119888

1 1198882] such that Δ(119909lowast) = 0 therefore 119909lowast

is a stationary point

Remark 14 If in Theorem 13 we consider 119860lowast as an equilib-rium viable set of the type (119860

119894 119860119894+1) rarr (119861 119862) the result is

analogous that is there exists a stationary point 119909lowast isin 119860lowast

31 Local Stationary Points-Symmetrical Output If 119860lowast is anequilibrium viable set where the membership functions ofthe consequents 119861 and 119862 are symmetrical functions thenthe stationary point in119860lowast is unique Except when 120583

119860119894(1198881) = 0

or 120583119860119894+1

(1198882) = 0 possibly occurs Then we have the following

proposition

Proposition 15 Let 119878 be a p-fuzzy system and 119860lowast an equi-librium viable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the

membership functions of 119861 and 119862 respectively 120583119861and 120583

119862 are

monotonous and symmetric that is 120583119862(119905) = 120583

119861(minus119905) then there

exists an equilibrium point in 119860lowast

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (20)

Proof Since 120583119861(119905) = 120583

119862(minus119905) then 120583

119861(minus119886) = 120583

119862(119886) = 0 =

120583119861(119887) rArr 119887 = minus119886 because 120583

119861U is monotonous Yet we have

that 120583119861(119905) = 120583

119862(minus119905) rArr 120583

minus1

119861(120583119862(119905)) = minus119905 = minus120583

minus1

119861(120583119861(119905)) rArr

120583minus1

119861(119910) = minus120583

minus1

119862(119910)

Then Δ(119911119900) = 0 if and only if ℎ

1(119899) = minusℎ

2(119898) Since

119887 = minus119886 from (8) then we obtain

ℎ2(119898) = int

120583minus1

119861(119898)

minus119886

119905120583119861(119905) 119889119905 + int

0

120583minus1

119861(119898)

119898119905119889119905 (21)

If we perform a change in the variable 119906 = minus119905 we have

ℎ2(119898) = int

minus120583minus1

119861(119898)

a119906120583119861(minus119906) 119889119906 + int

0

minus120583minus1

119861(119898)

119898119906119889119906

997904rArr ℎ2(119898) = int

120583minus1

119862(119898)

119886

119906120583119862(119906) 119889119906 + int

0

120583minus1

119862(119898)

119898119906119889119906 = minusℎ1(119898)

(22)

That is ℎ2= minusℎ

1 Hence ℎ

1(119899) = minusℎ

2(119898) hArr ℎ

1(119899) =

ℎ1(119898) hArr 119898 = 119899 (because ℎ

1is increasing Lemma 7) that

proves the proposition

Remark 16 If 120583119860119894(1198881) = 0 then 119909lowast = max

119909isin119860lowast[min(120583

119860119894(119909)

120583119860119894+1

(119909))] is the only stationary point in 119860lowast Besides that ifthe system 119878 is of the type (119860

119894 119860119894+1) rarr (119861 119862) then the result

of Proposition 15 is the same

4 Uniqueness of the Stationary Point

In this section we will enunciate and prove theorems thatestablish condition for uniqueness of the stationary point ofa one-dimensional p-fuzzy system Initially we will considera simpler case when 119860lowast sub [119911

1 1199112] (Figure 4) where

1199111= min119909isinsupp(119860119894)

119903 (119909) = 1 1199112= max119909isinsupp(119860119894+1)

119904 (119909) = 1

(23)

Theorem 17 Let 119878 be a p-fuzzy system and119860lowast an equilibriumviable set of 119878 of the type (119860

119894 119860119894+1) rarr (119862 119861) If the functions

120583119860119894and120583119860119894+1

are piecewisemonotonous and119860lowast sub [1199111 1199112] then

there exists only one stationary point in 119860lowast

Proof Given 119909 isin 119860lowast one has

Δ (119909) = ℎ1(119899) + ℎ

2(119898) = ℎ

1(120583119860119894(119909)) + ℎ

2(120583119860119894+1

(119909))

(24)

Using Lemmas 7 and 8 and the chain rule we find that thederivative of Δ is

Δ1015840

(119909) =

[120583minus1

119862(120583119860119894(119909))]

2

2

1205831015840

119860119894

(119909)

minus

[120583minus1

119861(120583119860119894+1

(119909))]

2

2

1205831015840

119860119894+1

(119909)

(25)

Since in 119860lowast = [1198881 1198882]120583119860119894

is not increasing and 120583119860119894+1

is notdecreasing then 1205831015840

119860119894

(119909) le 0 and 1205831015840119860119894+1

(119909) ge 0 and besidesthat if 1205831015840

119860119894

(119909) = 0 we have that 1205831015840119860119894+1

(119909) = 0 and if 1205831015840119860119894+1

(119909) =

0 we obtain 1205831015840

119860119894

(119909) = 0 Then from (25) Δ1015840(119909) lt 0 Thisshows that Δ is decreasing From Theorem 13 there exists astationary point in 119860lowast then this point is unique

Now consider the more general case when119860lowast sub [1199111 1199112]

and divide it into two theorems Initially consider the casethat the membership functions 120583

119862and 120583

119861are such that

120583119862(119905) gt 120583

119861(minus119905) next consider the case where 120583

119862(119905) lt

120583119861(minus119905)

6 Journal of Applied Mathematics

h1(1)

minush2(1)

h2(1)

h1

h2

hminus11 (minush2(1)) 1

Figure 6 Functions ℎ1and ℎ

2

m

n

1

1

1205751

1205752

s(z1)

s(c2)

120585

r(z0)r(c1)r(z2)hminus11 (minush2(1))

Figure 7 Functions 120585 1205751and 120575

2

41 Case 1 120583119862(119905) gt 120583

119861(minus119905)

Theorem 18 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) gt 120583

119861(minus119905) forall119905 isin (0 minus119887)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) lt (119901

3

1199023

) forall119901 isin supp(119861) 119902 isin

supp(119862) and 120583119861(119901) gt 120583

119862(119902)

(iii) [1205831015840119860119894+1

(119909)1205831015840

119860119894

(119909)] le 0 forall119909 isin (119911119900 1198882) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

(119911119900 1198882]

Proof For the sake of simplicity notation we make 119903 = 120583119860119894

119904 = 120583119860119894+1

119891 = 120583119861and 119892 = 120583

119862

Initially we see that given 119909 isin (119911119900 1198882] (Figure 4) 119909

determines only one (119899119898) isin [0 1]2 such that 119899 = 119903(119909)

and 119898 = 119904(119909) By monotonicity of 119903 we have that for each119899 isin [0 119903(119911

119900)) there exists only one 119898 isin [0 1] such that

119899 = 119903(119909) and 119898 = 119904(119909) That is each (119899119898) in this situationdetermines only one 119909 isin (119911

119900 1198882]

By Theorem 13 there exists a stationary point 119909lowast isin

[1198881 1198882] = [1198881 119911119900] cup (119911119900 1198882] Given 119909 isin [119888

1 119911119900] rArr 119898 = 119904(119909) le

119899 = 119903(119909) Then by Lemma 11 119909lowast notin [1198881 119911119900] rArr 119909

lowast

isin (119911119900 1198882]

That is equivalent to the existence of only one (119899lowast 119898lowast) with119899lowast

isin [0 119903(119911119900)) such that119867(119899lowast 119898lowast) = 0

Since for each 119899 isin [0 119903(119911119900)) there exists only one 119898 isin

[0 1] such that 119899 = 119903(119909) and 119898 = 119904(119909) then we may definea function 120575

2 [0 119903(119911

119900)) rarr [0 1] such that 119898 = 120575

2(119899)

(Figure 7) We observe that 1205752is continuous because 119903 and

119904 are continuous Using the chain rule we get the derivative of1205752to be

1205751015840

2(119899) =

1199041015840

(119909)

1199031015840(119909)

(iii)rArr 12057510158401015840

2(119899) le 0 forall119899 isin 119863

1205752 (26)

By Lemmas 7 and 8 ℎ1and minusℎ

2are increasing and by

condition (i) it follows that (Figure 6)

int

119886

0

119905119892 (119905) 119889119905 gt minusint

0

119887

119905119891 (119905) 119889119905 lArrrArr ℎ1(1) gt minusℎ

2(1) (27)

Then given 119899 isin [0 ℎminus1

1(minusℎ2(1))] there exists only one 119898 isin

[0 1] such that

ℎ1(119899) = minusℎ

2(119898) lArrrArr ℎ

1(119899) + ℎ

2(119898) = 0 (28)

Therefore we may define an injective function 120585 119898 = 120585(119899)

(Figure 7) so that the inverse range of 0 by119867 is given by

119867minus1

(0) = (119899119898) 119898 = 120585 (119899) (29)

where 119867 [0 ℎminus1

1(minusℎ2(1))] times [0 1] rarr R is given by

119867(119899119898) = ℎ1(119899) + ℎ

2(119898)

Since 120597119867120597119899 = ℎ10158401(119899) = (119892

minus1

(119899))

2

2 gt 0 (Lemma 7) and120597119867120597119898 = ℎ

1015840

2(119898) = minus(119891

minus1

(119898))

2

2 lt 0 (Lemma 8) then bythe Implicity Function Theorem 120585 is 119896 times differentiableand besides that

1205851015840

(119899) = minus

119889ℎ1119899

119889ℎ2119898

= [

119892minus1

(119899)

119891minus1(119898)

]

2

gt 0

forall119899 isin (0 ℎminus1

1(minusℎ2(1))) 119898 isin (0 1) 119898 = 120585 (119899)

(30)

Thus 120585 is a strictly increasing function and since119867(0 0) = 0and 119867(ℎminus1

1(minusℎ2(1)) 1) = 0 then 119863

120585= [0 ℎ

minus1

1(ℎ2(1))] and

119868119898120585= [0 1]Given 119898 119899 isin (0 1) there is only one 119901 isin (119887 0) such that

119901 = 119891minus1

(119898) and there exists only one 119902 isin (0 119886) such that 119902 =119892minus1

(119899) since by assumption119891 and 119892 are strictly monotonousFrom Lemma 11 we have that 119898 le 119899 rArr 119867(119898 119899) gt 0

Hence119867(119898 119899) = 0 rArr 119898 gt 119899 Therefore we are interested inthe pairs (119898 119899) such that119898 gt 119899 Thus we have

119898 gt 119899 lArrrArr 119891(119901) gt 119892 (119902) (31)

Journal of Applied Mathematics 7

1 1

0 5 40 60 90 100x

A1 A2 B

Δminus3 minus2 minus120576 0 2 3

m1 + 120576

m1

C

R1 if x is A1 then Δ is CR2 if x is A2 then Δ is B

Figure 8 p-Fuzzy system existence of more than one stationary point

Since 119891 and 119892 are monotonous by Lagrangersquos Medium ValueTheorem we obtain

119901 = 119891minus1

(119898) lArrrArr (119891minus1

)

1015840

(119898) =

1

1198911015840(119901)

(32)

119902 = 119892minus1

(119899) lArrrArr (119892minus1

)

1015840

(119899) =

1

1198921015840(119902)

(33)

Then

119898 gt 119899

(31)

lArr997904997904rArr 119891(119901) gt 119892 (119902)

(ii)rArr

1198921015840

(119902)

1198911015840(119901)

lt

1199013

1199023

(32)119890(33)

lArr997904997904997904997904997904rArr

(119891minus1

)

1015840

(119898)

(119892minus1)1015840

(119899)

lt

[119891minus1

(119898)]

3

[119892minus1(119899)]3

(34)

Therefore

119898 gt 119899 997904rArr (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

gt 0

(35)

By differentiation of (30) we obtain

12058510158401015840

(119899) =

minus2119892minus1

(119899)

[119891minus1(119898)]5

times (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

(36)

and since minus2119892minus1(119899)[119891minus1(119898)]5 gt 0 forall119898 119899 isin (0 1) from (35)we have

12058510158401015840

(119899) gt 0 forall119899 isin

119900

119863120585 (37)

Now we take 119868 = 119863120585cap 1198631205752= 119863120585cap [0 119903(119911

119900)) and we define

the function 120601 119868 rarr [0 1] such that

120601 (119899) = 120585 (119899) minus 1205752(119899) (38)

Then from (26) and (37) we have 12060110158401015840(119899) gt 0 forall119899 isin

119900

119868 Since120585(0) = 0 and by condition (iii) of Theorem 18 it followsthat 120575

2(0) gt 0 then we have 120601(0) lt 0 Consequently from

Lemma 9 we have that there is only one 119899lowast isin 119868 such that

120601 (119899lowast

) = 0

(38)

lArr997904997904rArr 120585 (119899lowast

) = 1205752(119899lowast

) (39)

Since 120585 = 119867minus1(0) then we obtain

0 = 119867 (119899lowast

120585 (119899lowast

))

(39)

= 119867 (119899lowast

1205752(119899lowast

)) (40)

So there exists only one 119909lowast isin (119911119900 1198882] 119899lowast = 119903(119909

lowast

) and119898lowast

= 1205752(119899lowast

) = 119904(119909lowast

) such that

Δ (119909lowast

) =

119867 (119899lowast

119898lowast

)

119860 (119899lowast 119898lowast)

= 0 (41)

This finally proves the theorem

42 Case 2 120583119862(119905)lt120583119861(minus119905)

Theorem 19 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861 and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) lt 120583

119861(minus119905) forall119905 isin (0 119886)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 forall119901 isin supp(119861) 119902 isin supp(119862)

and 120583119861(119901) lt 120583

119862(119902)

(iii) [1205831015840119860119894

(119909)1205831015840

119860119894+1

(119909)] le 0 forall119909 isin (1198881 119911119900) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

[1198881 119911119900)

Proof It is analogous to the proof of Theorem 18

43 Some Comments about Uniqueness Theorems When wedo not have 120583

119862(119905) gt 120583

119861(minus119905) or 120583

119862(119905) lt 120583

119861(minus119905) it is impossible

to establish general conditions for uniqueness of stationarypoints For example consider the p-fuzzy system in Figure 8This system has an equilibrium viable set 119860lowast = [119860

1cap 1198602]0

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

h1(1)

minush2(1)

h2(1)

h1

h2

hminus11 (minush2(1)) 1

Figure 6 Functions ℎ1and ℎ

2

m

n

1

1

1205751

1205752

s(z1)

s(c2)

120585

r(z0)r(c1)r(z2)hminus11 (minush2(1))

Figure 7 Functions 120585 1205751and 120575

2

41 Case 1 120583119862(119905) gt 120583

119861(minus119905)

Theorem 18 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) gt 120583

119861(minus119905) forall119905 isin (0 minus119887)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) lt (119901

3

1199023

) forall119901 isin supp(119861) 119902 isin

supp(119862) and 120583119861(119901) gt 120583

119862(119902)

(iii) [1205831015840119860119894+1

(119909)1205831015840

119860119894

(119909)] le 0 forall119909 isin (119911119900 1198882) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

(119911119900 1198882]

Proof For the sake of simplicity notation we make 119903 = 120583119860119894

119904 = 120583119860119894+1

119891 = 120583119861and 119892 = 120583

119862

Initially we see that given 119909 isin (119911119900 1198882] (Figure 4) 119909

determines only one (119899119898) isin [0 1]2 such that 119899 = 119903(119909)

and 119898 = 119904(119909) By monotonicity of 119903 we have that for each119899 isin [0 119903(119911

119900)) there exists only one 119898 isin [0 1] such that

119899 = 119903(119909) and 119898 = 119904(119909) That is each (119899119898) in this situationdetermines only one 119909 isin (119911

119900 1198882]

By Theorem 13 there exists a stationary point 119909lowast isin

[1198881 1198882] = [1198881 119911119900] cup (119911119900 1198882] Given 119909 isin [119888

1 119911119900] rArr 119898 = 119904(119909) le

119899 = 119903(119909) Then by Lemma 11 119909lowast notin [1198881 119911119900] rArr 119909

lowast

isin (119911119900 1198882]

That is equivalent to the existence of only one (119899lowast 119898lowast) with119899lowast

isin [0 119903(119911119900)) such that119867(119899lowast 119898lowast) = 0

Since for each 119899 isin [0 119903(119911119900)) there exists only one 119898 isin

[0 1] such that 119899 = 119903(119909) and 119898 = 119904(119909) then we may definea function 120575

2 [0 119903(119911

119900)) rarr [0 1] such that 119898 = 120575

2(119899)

(Figure 7) We observe that 1205752is continuous because 119903 and

119904 are continuous Using the chain rule we get the derivative of1205752to be

1205751015840

2(119899) =

1199041015840

(119909)

1199031015840(119909)

(iii)rArr 12057510158401015840

2(119899) le 0 forall119899 isin 119863

1205752 (26)

By Lemmas 7 and 8 ℎ1and minusℎ

2are increasing and by

condition (i) it follows that (Figure 6)

int

119886

0

119905119892 (119905) 119889119905 gt minusint

0

119887

119905119891 (119905) 119889119905 lArrrArr ℎ1(1) gt minusℎ

2(1) (27)

Then given 119899 isin [0 ℎminus1

1(minusℎ2(1))] there exists only one 119898 isin

[0 1] such that

ℎ1(119899) = minusℎ

2(119898) lArrrArr ℎ

1(119899) + ℎ

2(119898) = 0 (28)

Therefore we may define an injective function 120585 119898 = 120585(119899)

(Figure 7) so that the inverse range of 0 by119867 is given by

119867minus1

(0) = (119899119898) 119898 = 120585 (119899) (29)

where 119867 [0 ℎminus1

1(minusℎ2(1))] times [0 1] rarr R is given by

119867(119899119898) = ℎ1(119899) + ℎ

2(119898)

Since 120597119867120597119899 = ℎ10158401(119899) = (119892

minus1

(119899))

2

2 gt 0 (Lemma 7) and120597119867120597119898 = ℎ

1015840

2(119898) = minus(119891

minus1

(119898))

2

2 lt 0 (Lemma 8) then bythe Implicity Function Theorem 120585 is 119896 times differentiableand besides that

1205851015840

(119899) = minus

119889ℎ1119899

119889ℎ2119898

= [

119892minus1

(119899)

119891minus1(119898)

]

2

gt 0

forall119899 isin (0 ℎminus1

1(minusℎ2(1))) 119898 isin (0 1) 119898 = 120585 (119899)

(30)

Thus 120585 is a strictly increasing function and since119867(0 0) = 0and 119867(ℎminus1

1(minusℎ2(1)) 1) = 0 then 119863

120585= [0 ℎ

minus1

1(ℎ2(1))] and

119868119898120585= [0 1]Given 119898 119899 isin (0 1) there is only one 119901 isin (119887 0) such that

119901 = 119891minus1

(119898) and there exists only one 119902 isin (0 119886) such that 119902 =119892minus1

(119899) since by assumption119891 and 119892 are strictly monotonousFrom Lemma 11 we have that 119898 le 119899 rArr 119867(119898 119899) gt 0

Hence119867(119898 119899) = 0 rArr 119898 gt 119899 Therefore we are interested inthe pairs (119898 119899) such that119898 gt 119899 Thus we have

119898 gt 119899 lArrrArr 119891(119901) gt 119892 (119902) (31)

Journal of Applied Mathematics 7

1 1

0 5 40 60 90 100x

A1 A2 B

Δminus3 minus2 minus120576 0 2 3

m1 + 120576

m1

C

R1 if x is A1 then Δ is CR2 if x is A2 then Δ is B

Figure 8 p-Fuzzy system existence of more than one stationary point

Since 119891 and 119892 are monotonous by Lagrangersquos Medium ValueTheorem we obtain

119901 = 119891minus1

(119898) lArrrArr (119891minus1

)

1015840

(119898) =

1

1198911015840(119901)

(32)

119902 = 119892minus1

(119899) lArrrArr (119892minus1

)

1015840

(119899) =

1

1198921015840(119902)

(33)

Then

119898 gt 119899

(31)

lArr997904997904rArr 119891(119901) gt 119892 (119902)

(ii)rArr

1198921015840

(119902)

1198911015840(119901)

lt

1199013

1199023

(32)119890(33)

lArr997904997904997904997904997904rArr

(119891minus1

)

1015840

(119898)

(119892minus1)1015840

(119899)

lt

[119891minus1

(119898)]

3

[119892minus1(119899)]3

(34)

Therefore

119898 gt 119899 997904rArr (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

gt 0

(35)

By differentiation of (30) we obtain

12058510158401015840

(119899) =

minus2119892minus1

(119899)

[119891minus1(119898)]5

times (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

(36)

and since minus2119892minus1(119899)[119891minus1(119898)]5 gt 0 forall119898 119899 isin (0 1) from (35)we have

12058510158401015840

(119899) gt 0 forall119899 isin

119900

119863120585 (37)

Now we take 119868 = 119863120585cap 1198631205752= 119863120585cap [0 119903(119911

119900)) and we define

the function 120601 119868 rarr [0 1] such that

120601 (119899) = 120585 (119899) minus 1205752(119899) (38)

Then from (26) and (37) we have 12060110158401015840(119899) gt 0 forall119899 isin

119900

119868 Since120585(0) = 0 and by condition (iii) of Theorem 18 it followsthat 120575

2(0) gt 0 then we have 120601(0) lt 0 Consequently from

Lemma 9 we have that there is only one 119899lowast isin 119868 such that

120601 (119899lowast

) = 0

(38)

lArr997904997904rArr 120585 (119899lowast

) = 1205752(119899lowast

) (39)

Since 120585 = 119867minus1(0) then we obtain

0 = 119867 (119899lowast

120585 (119899lowast

))

(39)

= 119867 (119899lowast

1205752(119899lowast

)) (40)

So there exists only one 119909lowast isin (119911119900 1198882] 119899lowast = 119903(119909

lowast

) and119898lowast

= 1205752(119899lowast

) = 119904(119909lowast

) such that

Δ (119909lowast

) =

119867 (119899lowast

119898lowast

)

119860 (119899lowast 119898lowast)

= 0 (41)

This finally proves the theorem

42 Case 2 120583119862(119905)lt120583119861(minus119905)

Theorem 19 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861 and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) lt 120583

119861(minus119905) forall119905 isin (0 119886)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 forall119901 isin supp(119861) 119902 isin supp(119862)

and 120583119861(119901) lt 120583

119862(119902)

(iii) [1205831015840119860119894

(119909)1205831015840

119860119894+1

(119909)] le 0 forall119909 isin (1198881 119911119900) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

[1198881 119911119900)

Proof It is analogous to the proof of Theorem 18

43 Some Comments about Uniqueness Theorems When wedo not have 120583

119862(119905) gt 120583

119861(minus119905) or 120583

119862(119905) lt 120583

119861(minus119905) it is impossible

to establish general conditions for uniqueness of stationarypoints For example consider the p-fuzzy system in Figure 8This system has an equilibrium viable set 119860lowast = [119860

1cap 1198602]0

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

1 1

0 5 40 60 90 100x

A1 A2 B

Δminus3 minus2 minus120576 0 2 3

m1 + 120576

m1

C

R1 if x is A1 then Δ is CR2 if x is A2 then Δ is B

Figure 8 p-Fuzzy system existence of more than one stationary point

Since 119891 and 119892 are monotonous by Lagrangersquos Medium ValueTheorem we obtain

119901 = 119891minus1

(119898) lArrrArr (119891minus1

)

1015840

(119898) =

1

1198911015840(119901)

(32)

119902 = 119892minus1

(119899) lArrrArr (119892minus1

)

1015840

(119899) =

1

1198921015840(119902)

(33)

Then

119898 gt 119899

(31)

lArr997904997904rArr 119891(119901) gt 119892 (119902)

(ii)rArr

1198921015840

(119902)

1198911015840(119901)

lt

1199013

1199023

(32)119890(33)

lArr997904997904997904997904997904rArr

(119891minus1

)

1015840

(119898)

(119892minus1)1015840

(119899)

lt

[119891minus1

(119898)]

3

[119892minus1(119899)]3

(34)

Therefore

119898 gt 119899 997904rArr (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

gt 0

(35)

By differentiation of (30) we obtain

12058510158401015840

(119899) =

minus2119892minus1

(119899)

[119891minus1(119898)]5

times (119891minus1

)

1015840

(119898) [119892minus1

(119899)]

3

minus (119892minus1

)

1015840

(119899) [119891minus1

(119898)]

3

(36)

and since minus2119892minus1(119899)[119891minus1(119898)]5 gt 0 forall119898 119899 isin (0 1) from (35)we have

12058510158401015840

(119899) gt 0 forall119899 isin

119900

119863120585 (37)

Now we take 119868 = 119863120585cap 1198631205752= 119863120585cap [0 119903(119911

119900)) and we define

the function 120601 119868 rarr [0 1] such that

120601 (119899) = 120585 (119899) minus 1205752(119899) (38)

Then from (26) and (37) we have 12060110158401015840(119899) gt 0 forall119899 isin

119900

119868 Since120585(0) = 0 and by condition (iii) of Theorem 18 it followsthat 120575

2(0) gt 0 then we have 120601(0) lt 0 Consequently from

Lemma 9 we have that there is only one 119899lowast isin 119868 such that

120601 (119899lowast

) = 0

(38)

lArr997904997904rArr 120585 (119899lowast

) = 1205752(119899lowast

) (39)

Since 120585 = 119867minus1(0) then we obtain

0 = 119867 (119899lowast

120585 (119899lowast

))

(39)

= 119867 (119899lowast

1205752(119899lowast

)) (40)

So there exists only one 119909lowast isin (119911119900 1198882] 119899lowast = 119903(119909

lowast

) and119898lowast

= 1205752(119899lowast

) = 119904(119909lowast

) such that

Δ (119909lowast

) =

119867 (119899lowast

119898lowast

)

119860 (119899lowast 119898lowast)

= 0 (41)

This finally proves the theorem

42 Case 2 120583119862(119905)lt120583119861(minus119905)

Theorem 19 (uniqueness) Let 119878 be a p-fuzzy system and119860lowast an equilibrium viable set of 119878 of the type (119860

119894 119860119894+1) rarr

(119862 119861) If the functions 120583119860119894 120583119860119894+1

120583119861 and 120583

119862are continuously

differentiable 120583119860119894

and 120583119860119894+1

are piecewise monotonous 120583119861and

120583119862are strictly monotonous such that

(i) 120583119862(119905) lt 120583

119861(minus119905) forall119905 isin (0 119886)

(ii) 1205831015840119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 forall119901 isin supp(119861) 119902 isin supp(119862)

and 120583119861(119901) lt 120583

119862(119902)

(iii) [1205831015840119860119894

(119909)1205831015840

119860119894+1

(119909)] le 0 forall119909 isin (1198881 119911119900) 120583119860119894(119909) = 120583

119860119894+1(119909)

Then 119878 has only one stationary point 119909lowast in 119860lowast and 119909lowast isin

[1198881 119911119900)

Proof It is analogous to the proof of Theorem 18

43 Some Comments about Uniqueness Theorems When wedo not have 120583

119862(119905) gt 120583

119861(minus119905) or 120583

119862(119905) lt 120583

119861(minus119905) it is impossible

to establish general conditions for uniqueness of stationarypoints For example consider the p-fuzzy system in Figure 8This system has an equilibrium viable set 119860lowast = [119860

1cap 1198602]0

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Applied Mathematics

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 9 Function Δ with1198981= 120576 = 01

The sets that describe the input variable havemembershipfunctions 120583

1198601and 120583

1198602

1205831198601(119909) =

1

40

119909 if 0 lt 119909 le 40

minus1

50

119909 +

9

5

if 40 lt 119909 le 900 otherwise

1205831198602(119909) =

1

55

119909 minus

1

11

if 5 lt 119909 le 60

minus1

40

119909 +

5

2

if 60 lt 119909 le 1000 otherwise

(42)

and the fuzzy sets that describe the output variable havemembership functions 120583

119862(119905) = (minus12)119905 + 1 and

120583119861(119905)

=

1

3

119905 + 1 if minus 3 lt 119905 le 3 (1198981minus 1)

120576

3 (1198981minus 1)

119905 + 1198981+ 120576 if 3 (119898

1minus 1) lt 119905

le

3120576 (120576 + 1198981minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

1

120576

119905 + 1 if3120576 (120576 + 119898

1minus 1) (119898

1minus 1)

3 (1198981minus 1) + 120576

2

lt 119905 le 0

0 otherwise(43)

For example if we take in (43) 1198981= 01 and 120576 = 01

the p-fuzzy systemobtained has three stationary points in119860lowastwhich can be visualized in Figure 9 which depicts the graphicof function Δ

Remark 20 Note that the function 120583119861is not differentiable in

all points into supp(119861) which was a requirement made inthe previous cases However it can be clearly constructed afunction 120583

119861derivable in all points of supp(119861) For example

0 10 20 30 40 50 60 70 80 90

0

05

1

minus15

minus1

minus05

Figure 10 Function Δ with 120576 = 041198981= 01

if we substitute the second sentence of 120583119861by an adequate

fourth degree polynomial obviously 120583119861will be derivable into

supp(119861)

Remark 21 If we take for example 120576 = 03 and 1198981=

03 we have that the obtained p-fuzzy system has only onestationary point (Figure 10)This shows thatTheorems 18 and19 establish only sufficient conditions for uniqueness of thestationary point

44 Uniqueness for Triangular and Trapezoidal MembershipFunctions In this section we will list some important con-sequences of Theorems 18 and 19 We will also show thatfor triangular and trapezoidal membership functions thestationary point is only one in 119860lowast However before doing solet us take a look at the following lemmas

Lemma 22 If 120583119861(119901) gt 120583

119862(119902) then 119902 gt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof In fact we have that 120583119861(119901) gt 120583

119862(119902) rArr 119898 gt 119899 hence

using Lemma 10 and the fact that minus120583minus1119861

isincreasing once 120583119861

is increasing then we have that

119902 = 120583minus1

119862(119899) gt minus120583

minus1

119861(119899) gt minus120583

minus1

119861(119898) = minus119901 (44)

Lemma 23 If 120583119861(119901) lt 120583

119862(119902) then 119902 lt minus119901 where 119901 = 120583minus1

119861(119898)

and 119902 = 120583minus1119862(119899)

Proof Analogous to previous proof

Corollary 24 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894 120583119860119894+1

120583119861and 120583119862are triangular fuzzy

numbers then 119878 has only one stationary point in 119860lowast

Proof We will prove the case where 119878 is (120583119860119894 120583119860119894+1

) rarr

(120583119862 120583119861) If 119878 is (120583

119860119894 120583119860119894+1

) rarr (120583119861 120583119862) then the proof is

analogous

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 9

0 50 100 150 200 250 300

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

x

A1 A2 A3 A4 A5 A6

Figure 11 Population (119909)

If 119886 = 119887 then 120583119860119894

and 120583119860119894+1

are symmetrical and byProposition 15 the stationary point is only one

119909lowast

= max119909isin119860lowast

[min (120583119860119894(119909) 120583

119860119894+1(119909))] (45)

Assume that 119886 gt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862satisfy

Theorem 18 In fact (i) and (iii) are trivial Since 120583119861(119905) =

minus(1119887)119905 + 1 and 120583119862(119905) = minus(1119886)119905 + 1 then 120583

119861(119901) gt 120583

119862(119902) rArr

119887119886 lt 119901119902 rArr 1205831015840

119862(119902)1205831015840

119861(119901) lt 119901119902 From Lemma 22 we

have that 119902 gt minus119901 rArr 119901119902 lt 1199013

1199023 and therefore we obtain

1205831015840

119862(119902)1205831015840

119861(119901) lt 119901

3

1199023 which satisfies (ii)

Now assume that 119886 lt 119887 then 120583119860119894 120583119860119894+1

120583119861 and 120583

119862

satisfy the Theorem 19 In fact (i) and (iii) are trivial and120583119861(119901) lt 120583

119862(119902) rArr 119887119886 gt 119901119902 rArr 120583

1015840

119862(119902)1205831015840

119861(119901) gt 119901119902 From

Lemma 23 we get 119902 lt minus119901 rArr 119901119902 gt 1199013

1199023 and therefore

1205831015840

119862(119902)1205831015840

119861(119901) gt 119901

3

1199023 which satisfies (ii) This concludes the

proof

Corollary 25 Let 119878 be a p-fuzzy system and 119860lowast an equilib-rium viable set of 119878 If 120583

119860119894and 120583

119860119894+1are trapezoidal fuzzy

numbers and 120583119861and 120583

119862are triangular fuzzy numbers then

119878 has only one stationary point in 119860lowast

Proof It is analogous to the proof of Corollary 24

5 Examples

In this section we will present some computational exper-iments that confirm the mathematical theory presented inthe previous sections The experiments had been carried outwith Matlab software For the experiments we will considerinhibited one-dimensional p-fuzzy systems These systemscan be used to model situations where the state variable isincreasing (resp decreasing) with a carrying capacity (resplower bound) These situations in population dynamics aredescribed by inhibited models such as Gompertzrsquos modelVerhulstrsquos model von Bertallanffyrsquos models and AsymptoticExponential model

The inhibited one-dimensional p-fuzzy systems are com-posed of the variables ldquoPopulationrdquo (Figure 11) and ldquoVaria-tionrdquo (Figure 12) The rule-base of these systems is

(1) if Population is low (1198601) thenVariation is low positive

(119862)

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 12 Variation (Δ) 120583119862(119905) = 120583

119861(minus119905)

(2) if Population is medium low (1198602) then Variation is

medium positive (119863)(3) if Population is medium (119860

3) then Variation is high

positive (119864)(4) if Population is medium high (119860

4) then Variation is

medium positive (119863)(5) if Population is high (119860

5) then Variation is low

positive (119862)(6) if Population is the highest (119860

6) then Variation is low

negative (119861)

51 Example 1 In this system the membership functions of 119861and 119862 are 120583

119861(119905) = 119905 + 1 and 120583

119862(119905) = 1 minus 119905 These functions are

symmetric (Figure 12) Observing the rules we can identify anequilibrium viable set119860lowast = [200 280] where119860lowast = 119860

5cap1198606

in which membership functions are given by size

1205831198605(119909) =

1

50

(119909 minus 150) if 150 lt 119909 le 200

1 if 200 lt 119909 le 220minus1

60

(119909 minus 280) if 220 lt 119909 le 270

0 otherwise

1205831198606(119909) =

1

70

(119909 minus 200) if 200 lt 119909 le 270

1 if 270 lt 119909 le 3000 otherwise

(46)

A simple calculation shows that 1205831198605

cap 1205831198606

= 24307which is the stationary point of the system as shown inProposition 15 This is the same result as that obtained fromnumerical experiments in Figure 13 where it is possibleto observe the solution of the p-fuzzy system with initialcondition 119909

119900= 50 converging to the stationary point 119909lowast =

24307 (dotted line curve)

52 Example 2 (Application) Losses caused by fungi attacksreach up to the amazing and worrisome figure of 20 ofthe total harvested fruits Among the recognized pathogens

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Applied Mathematics

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 13 Equilibrium of the p-fuzzy systems

0 1 2 3 4 5

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

Δ

B C D E

minus2 minus1

Figure 14 Population (119860)

of these degenerative processes are the Colletotrichum sp(bitter putrefaction) Alternaria sp and Fusarium sp (carpelarputrefaction) Alternaria alternata (black putrefaction) andPenicillium sp (blue mould) In general putrefaction occursalways after harvesting when the apples are kept in boxesawaiting the refrigeration process

Let us consider a situation where an apple box filled withapproximately 3000 fruits exists and there is a rotten apple inthe center of the box that will contaminate the other applesThe dispersion of the disease occurs through the contact ofthe rotten apple with a healthy one If we want to modelthe dispersion of the disease we can only use the intuitionbecause the available information we have is that after 119899 daysall the apples will be contaminated We do not have in handa ldquoforce of infection of the diseaserdquo parameter and we knowlittle about the possible contacts between the rotten apple anda healthy one Thus any mathematical model or simulationof the phenomenon only will produce an approximate resultOn the other hand if we simply use the intuition we canformulate some relative rules to the dispersion process suchas the following

ldquoIf the rotten apple population is ldquolowrdquo then thevariation (incidence) of rotten apples is ldquosmallrdquordquo

Let us consider then the p-fuzzy systemwith the followinglinguistic variables

300

250

200

150

100

0

x

0 200 400 600 800 1000 1200

Iteration

xlowast = 27449xlowast = 24307xlowast = 20642

Figure 15 Variation (Δ)

0 500 1000 1500 2000 2500 3000

0

02

04

06

08

1

Deg

ree o

f mem

bers

hip

L Ml M Mh H

A

Figure 16 Solutions p-fuzzy system and adjust model

(i) 119860 ldquorotten apple populationrdquo (Figure 14) L low Mlmedium low M medium medium high H high

(ii) Δ ldquoVariationrdquo (incidence of the disease Figure 15) Ssmall M medium G great

where the rule-base is

(1) if Population (119860) is low (L) then Variation is small(S)

(2) if Population (119860) is medium low (L) then Variation ismedium (M)

(3) if Population (119860) is medium (M) then Variation isgreat (G)

(4) if Population (119860) is medium high (M) then Variationis medium (M)

(5) if Population (119860) is medium (G) then Variation issmall (S)

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 11

This p-fuzzy system can be used to adjust the parametersof a deterministic differential equation model of the type

119889119860

119889119905

= 119903119860(1 minus

119860

119870

) (47)

for example by the method of least squares In Figure 16 it ispossible to observe the solution of the p-fuzzy system (dottedline curve) and the adjust model (continuous line curve)

Acknowledgment

The authors would like to thank CAPES for the financialsupport

References

[1] R Fuller Neural Fuzzy Systems Abo Akademi University 1995[2] T Takagi and M Sugeno ldquoFuzzy identification of systems and

its applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[3] F Cuesta F Gordillo J Aracil and A Ollero ldquoStabilityanalysis of nonlinearmultivariable Takagi-Sugeno fuzzy controlsystemsrdquo IEEE Transactions on Fuzzy Systems vol 7 no 5 pp508ndash520 1999

[4] G Feng ldquoStability analysis of discrete-time fuzzy dynamicsystems based on piecewise lyapunov functionsrdquo IEEE Trans-actions on Fuzzy Systems vol 12 no 1 pp 22ndash28 2004

[5] M M Gupta ldquoEnergetic stability of dynamic systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 15 no 61985

[6] M Johansson A Rantzer and K-E Arzen ldquoPiecewisequadratic stability of fuzzy systemsrdquo IEEETransactions on FuzzySystems vol 7 no 6 pp 713ndash722 1999

[7] K TanakaM F Griffin andH OWang ldquoAn approach to fuzzycontrol of nonlinear systems stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[8] K Tanaka and M Sano ldquoConcept of stability margin for fuzzysystems and design of robust fuzzy controllersrdquo in Proceedingsof the 2nd IEEE International Conference on Fuzzy Systems pp29ndash34 San Francisco Calif USA April 1993

[9] H Ying ldquoStructure and stability analysis of general mamdanifuzzy dynamic modelsrdquo International Journal of IntelligentSystems vol 20 no 1 pp 103ndash125 2005

[10] M Sugeno and T Taniguchi ldquoOn Improvement of stabilityconditions for continuous mamdani-like fuzzy systemsrdquo IEEETransactions on Systems Man and Cybernetics B vol 34 no 1pp 120ndash131 2004

[11] M T Misukoshi Estabilidade de sistemas dinamicos fuzzy [Tesede Doutorado] IMECC-Unicamp Campinas-SP 2004

[12] H T Nguyen and E A Walker A First Course in Fuzzy LogicChapman amp HallCRC Boca Raton Fla USA 3rd edition2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of