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Research ArticleSome Stochastic Functional Differential Equations withInfinite Delay A Result on Existence and Uniqueness ofSolutions in a Concrete Fading Memory Space
Hassane Bouzahir Brahim Benaid and Chafai Imzegouan
LISTI ENSA Ibn Zohr University PO Box 1136 Agadir Morocco
Correspondence should be addressed to Hassane Bouzahir hbouzahiryahoofr
Received 4 February 2017 Accepted 2 April 2017 Published 16 April 2017
Academic Editor Chuanzhi Bai
Copyright copy 2017 Hassane Bouzahir 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
This paper is devoted to existence and uniqueness of solutions for some stochastic functional differential equations with infinitedelay in a fading memory phase space
1 Introduction
Let | sdot | denote the Euclidian norm in R119899 If 119860 is a vector ora matrix its transpose is denoted by 1198601015840 and its trace norm isrepresented by |119860| = (Trace(1198601015840119860))12 Let 119886 and 119887 (119886 or 119887) be theminimum (maximum) for 119886 119887 isin R
Let (ΩF 119875) be a complete probability space with afiltration F119905119905ge0 satisfying the usual conditions that is it isright continuous andF1199050 contains all 119875-null sets
M2((minusinfin119879]R119899) denotes the family of all F119905-meas-urable R119899 valued processes 119909(119905) 119905 isin (minusinfin119879] such that119864(int119879minusinfin
|119909(119905)|2119889119905) lt infinAssume that119882(119905) is an119898-dimensional Brownianmotion
which is defined on (ΩF 119875) that is 119882(119905) = (1198821(119905)1198822(119905) 119882119898(119905))1015840Let 119862120583 = 120593 isin 119862(minusinfin 0]R119899) lim120579rarrminusinfin119890120583120579120593(120579) exists in
R119899 denote the family of continuous functions 120593 defined on(minusinfin 0] with norm |120593|120583 = sup120579le0119890120583120579|120593(120579)|Consider the 119899-dimensional stochastic functional differ-
ential equation119889119909 (119905) = 119891 (119909119905 119905) 119889119905 + 119892 (119909119905 119905) 119889119882 (119905) 1199050 le 119905 le 119879 (1)
where 119909119905 (minusinfin 0] rarr R119899 120579 997891rarr 119909119905(120579) = 119909(119905 + 120579) minusinfin lt120579 le 0 can be regarded as a 119862120583-value stochastic process and119891 119862120583 times [1199050 119879] rarr R119899 and 119892 119862120583 times [1199050 119879] rarr R119899times119898 are Borelmeasurable
The initial data of the stochastic process is defined on(minusinfin 1199050] That is the initial value 1199091199050 = 120585 = 120585(120579) minusinfin lt120579 le 0 is a F1199050-measurable and 119862120583-value random variablesuch that 120585 isin M2(119862120583)
Our aim in this paper is to study existence and unique-ness of solutions to stochastic functional differential equa-tions with infinite delay of type (1) in a fading memory phasespace
2 Preliminary
The theory of partial functional differential equations withdelay has attracted widespread attention However when thedelay is infinite one of the fundamental tasks is the choiceof a suitable phase space B A large variety of phase spacescould be utilized to build an appropriate theory for any classof functional differential equations with infinite delay One ofthe reasons for a best choice is to guarantee that the historyfunction 119905 rarr 119909119905 is continuous if 119909 (minusinfin 119886] rarr R119899 is con-tinuous (where 119886 gt 0) In general the selection of the phasespace plays an important role in the study of both qualitativeand quantitative analysis of solutions Sometimes it becomesdesirable to approach the problem purely axiomatically Thefirst axiomatic approach was introduced by Coleman andMizel in [1] After this paper many contributions have beenpublished by various authors until 1978 when Hale and Kato
HindawiChinese Journal of MathematicsVolume 2017 Article ID 8219175 9 pageshttpsdoiorg10115520178219175
2 Chinese Journal of Mathematics
organized the study of functional differential equations withinfinite delay in [2] They assumed thatB is a normed linearspace of functions mapping (minusinfin 0] into a Banach space(119883 |sdot|) endowedwith a norm |sdot|B and satisfying the followingaxioms(1198601) There exist a positive constant 119867 and functions119870(sdot)119872(sdot) [0 +infin) rarr [0 +infin) with 119870 being continuousand 119872 being locally bounded such that for any 120590 isin R and119886 gt 0 if 119909 (minusinfin 120590+119886] rarr 119883 119909120590 isin B and 119909(sdot) is continuouson [120590 120590+119886] then for all 119905 in [120590 120590+119886] the following conditionshold
(i) 119909119905 isin B(ii) |119909(119905)| le 119867|119909119905|B(iii) |119909119905|B le 119870(119905 minus 120590)sup120590le119904le119905|119909(119904)| + 119872(119905 minus 120590)|119909120590|B
(1198602) For the function 119909(sdot) in (1198601) 119905 rarr 119909119905 is aB-valuedcontinuous function for 119905 in [120590 120590 + 119886](1198603)The spaceB is complete
Later on the concept of fading and uniform fadingmemory spaces has been adopted as the best choice
For 120593 isin B 119905 ge 0 and 120579 le 0 we define the linear operatorO(119905) by
[O (119905) 120593] (120579) = 120593 (0) if 119905 + 120579 ge 0120593 (119905 + 120579) if 119905 + 120579 lt 0 (2)
(O(119905))119905ge0 is exactly the solution semigroup associatedwith thefollowing trivial equation
119889119889119905119906 (119905) = 01199060 = 120593
(3)
We define
O0 (119905) = O (119905)B0
where B0 fl 120593 isin B 120593 (0) = 0 (4)
Let C00 be the set of continuous functions 120593 (minusinfin 0] rarr 119883with compact support We recall the following axiom(1198604) If a uniformly bounded sequence (120593119899)119899ge0 in C00converges to a function 120593 compactly on (minusinfin 0] then 120593 isin Band |120593119899 minus 120593|B rarr 0Definition 1
(1) B is called a fading memory space if it satisfies theaxioms (1198601) (1198602) (1198603) (1198604) and |O0(119905)120593| rarr 0 as119905 rarr +infin for all 120593 isin B0
(2) B is called a uniform fading memory space if it satis-fies the axioms (1198601) (1198602) (1198603) (1198604) and |O0(119905)| rarr0 as 119905 rarr +infin
Examples We recall the definitions of some standard exam-ples of phase spacesB
We start first with the phase space of 119883-valued boundedcontinuous functions 120593 defined on (minusinfin 0] that isBC((minusinfin 0] 119883) with norm |120593|BC = supminusinfinlt120579le0|120593(120579)|
(1) Let
BU = 120593 isin BC ((minusinfin 0] 119883) 120593 is bounded uniformly continuous (5)
whereBC is the space of all bounded continuous functionsmapping (minusinfin 0] into 119883 provided with the uniform normtopology
(2) Let 120583 isin R and
C120583
= 120593 isin C ((minusinfin 0] 119883) lim120579rarrminusinfin
119890120583120579120593 (120579) exists in119883 (6)
provided with the norm
10038161003816100381610038161205931003816100381610038161003816120583 = supminusinfinlt120579le0
119890120583120579 1003816100381610038161003816120593 (120579)1003816100381610038161003816 (7)
(3) For any continuous function 119892 (minusinfin 0] rarr [0 +infin)we define
C119892 = 120593 isin C ((minusinfin 0] 119883) 1003816100381610038161003816120593 (120579)1003816100381610038161003816119892 (120579) is bounded on (minusinfin 0]
C0119892 = 120593 isin C119892 ((minusinfin 0] 119883) lim
120579rarrminusinfin
1003816100381610038161003816120593 (120579)1003816100381610038161003816119892 (120579) = 0 (8)
endowed with the norm
10038161003816100381610038161205931003816100381610038161003816119892 = supminusinfinlt120579le0
1003816100381610038161003816120593 (120579)1003816100381610038161003816119892 (120579) (9)
Consider the following conditions on 119892(1198921) supminusinfinlt120579leminus119905(119892(119905 + 120579)119892(120579)) is locally bounded for119905 ge 0(1198922) lim120579rarrminusinfin119892(120579) = +infin
(1198923) lim119905rarr+infinsupminusinfinlt120579leminus119905(119892(119905 + 120579)119892(120579)) = 0Properties of each phase space are summarized in Table 1For other examples properties and details about phase
spaces we refer to the book by Hino et al [3]Fengying and Ke [4] discussed existence and uniqueness
of solutions to stochastic functional differential equationwithinfinite delay in the phase space of bounded continuousfunctions 120593 defined on (minusinfin 0] with values in R119899 that isBC((minusinfin 0]R119899) with norm |120593|BC = supminusinfinlt120579le0|120593(120579)|Lemma2 (page 22 in [3]) If the phase spaceB satisfies axiom(1198604) thenBC((minusinfin 0]R119899) is included inB
Chinese Journal of Mathematics 3
Table 1
(1198601) (1198602) (1198603) (1198604) Uniform fading memory spaceBC Yes No Yes No NoBU Yes Yes Yes No NoC119892 Under (1198921) Under (1198921) Yes Under (1198922) Under (1198923)C0119892 Under (1198921) Under (1198921) Yes Under (1198922) Under (1198923)C120583 Yes Yes Yes Under 120583 gt 0 Under 120583 gt 0
3 Existence and Uniqueness
Lemma 3 (see [4]) If 119901 ge 2 119892 isin 1198712([1199050 119879]R119899times119898) such that119864int1198791199050|119892(119904)|119901119889119904 lt infin then
119864 100381610038161003816100381610038161003816100381610038161003816int119879
1199050
119892 (119904) 119889119882 (119904)100381610038161003816100381610038161003816100381610038161003816119901
le (119901 (119901 minus 1)2 )1199012 119879(119901minus2)2119864int119879
1199050
1003816100381610038161003816119892 (119904)1003816100381610038161003816119901 119889119904(10)
Lemma 4 (Borel-Cantelli page 487 in [5]) If 119864119899 is asequence of events and
infinsum119899=1
119875 (119864119899) lt infin (11)
then
119875 (119864119899 119894119900) = 0 (12)
where 119894119900 is an abbreviation for ldquoinfinitively oftenrdquo
Definitions 1 R119899-value stochastic process 119909(119905) defined onminusinfin lt 119905 le 119879 is called the solution of (1) with initial data1199091199050 if 119909(119905) has the following properties(i) 119909(119905) is continuous and 119909(119905)1199050le119905le119879 isF119905-adapted(ii) 119891(119909119905 119905) isin L1([1199050 119879]R119899) and 119892(119909119905 119905) isin L2([1199050119879]R119899times119898)(iii) 1199091199050 = 120585 for each 1199050 le 119905 le 119879119909 (119905) = 120585 (0) + int119905
1199050
119891 (119909119904 119904) 119889119904+ int1199051199050
119892 (119909119904 119904) 119889119882 (119904) almost surely (as) (13)
119909(119905) is called unique solution if any other solution 119909(119905) isdistinguishable with 119909(119905) that is
119875 119909 (119905) = 119909 (119905) for any 1199050 le 119905 le 119879 = 1 (14)
Now we establish existence and uniqueness of solutionsfor (1) with initial data 1199091199050 We suppose a uniform Lipschitzcondition and a weak linear growth condition
Theorem 5 Assume that there exist two positive number 119870and 119870 such that
(i) for any 120593 120595 isin 119862120583 and 119905 isin [1199050 119879] it follows that1003816100381610038161003816119891 (120593 119905) minus 119891 (120595 119905)10038161003816100381610038162 or 1003816100381610038161003816119892 (120593 119905) minus 119892 (120595 119905)10038161003816100381610038162le 119870 1003816100381610038161003816120593 minus 12059510038161003816100381610038162120583
(15)
(ii) for any 119905 isin [1199050 119879] it follows that 119891(0 119905) 119892(0 119905) isin1198712(119862120583) such that 1003816100381610038161003816119891 (0 119905)10038161003816100381610038162 or 1003816100381610038161003816119892 (0 119905)10038161003816100381610038162 le 119870 (16)
Then problem (1) with initial data 1199091199050 = 120585 isin M2((minusinfin 0]R119899) has a unique solution 119909(119905) Moreover 119909(119905) isin M2((minusinfin119879]R119899)Lemma 6 Let (15) and (16) hold If 119909(119905) is the solution of (1)with initial data 1199091199050 = 120585 then
119864( sup1199050le119905le119879
|119909 (119905)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (17)
where119862 = 3119864|120585|2120583 +6119870(119879minus1199050 +1)(119879minus1199050)+6119870(119879minus1199050 +1)(119879minus1199050)119864|120585|2120583
Moreover if 120585 isin M2((minusinfin 0]R119899) then 119909(119905) isinM2((minusinfin119879]R119899)Proof For each number 119902 ge 1 define the stopping time
120591119902 = 119879 and inf 119905 isin [1199050 119879] 10038161003816100381610038161199091199051003816100381610038161003816120583 ge 119902 (18)
Obviously as 119902 rarr infin 120591119902 119879 as Let 119909119902(119905) = 119909(119905 and 120591119902) 119905 isin[1199050 119879] and then 119909119902(119905) satisfy the following equation119909119902 (119905) = 120585 (0) + int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119904+ int1199051199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904) (19)
Using the elementary inequality (119886 + 119887 + 119888)2 le 3(1198862 + 1198872 + 1198882)we get
1003816100381610038161003816119909119902 (119905)10038161003816100381610038162 le 3 100381610038161003816100381612058510038161003816100381610038162120583 + 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(20)
4 Chinese Journal of Mathematics
Taking the expectation on both sides and using the Holderinequality Lemma3 and (15) and (16) we get for all 119905 in [1199050 119879]
119864 1003816100381610038161003816119909119902 (119905)10038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583
+ 3 (119905 minus 1199050) 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904) + 119891 (0 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904) + 119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
(1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162
+ 1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 + 1003816100381610038161003816119891 (0 119904)10038161003816100381610038162+ 1003816100381610038161003816119892 (0 119904)10038161003816100381610038162) 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
(119870 1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 + 119870)119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050+ 1) (119905 minus 1199050) + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 119889119904(21)
We have also for each 119905 in [1199050 119879]
sup1199050le119904le119905
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 = sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 1003816100381610038161003816119909119902 (119904 + 120579)10038161003816100381610038162
= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le1199050
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 1003816100381610038161003816119909119902 (119903 minus 1199050 + 1199050)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 100381610038161003816100381610038161199091199021199050 (119903 minus 1199050)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119904 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 1198902120583(1199050minus119904) 100381610038161003816100381612058510038161003816100381610038162120583+ sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
(22)
Chinese Journal of Mathematics 5
Letting 119905 = 119879 we get119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162)le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119879 minus 1199050 + 1) (119879 minus 1199050)
+ 6119870 (119879 minus 1199050 + 1) 119864int1198791199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
le 119862 + 6119870 (119879 minus 1199050 + 1)int1198791199050
119864( sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
(23)
where119862 = 3119864|120585|2120583+6119870(119879minus1199050+1)(119879minus1199050)+6119870(119879minus1199050+1)(119879minus1199050)119864|120585|2120583
By the Gronwall inequality we infer
119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (24)
That is
119864( sup1199050le119904le119879
10038161003816100381610038161003816119909 (119904 and 120591119902)100381610038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (25)
Consequently
119864( sup1199050le119904le120591119902
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (26)
Letting 119902 rarr infin that implies the following inequality
119864( sup1199050le119904le119879
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (27)
Now to prove the second part or the lemma suppose that120585 isin M2((minusinfin 0]R119899) Then
119864( supminusinfinle119905le119879
|119909 (119905)|2) = 119864( supminusinfinle119905le1199050
|119909 (119905)|2)
+ 119864( sup1199050le119905le119879
|119909 (119905)|2)
le 119864( supminusinfinle119905le1199050
|119909 (119905)|2)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119905minus1199050le0
1003816100381610038161003816119909 (119905 minus 1199050 + 1199050)10038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119904le0
100381610038161003816100381610038161199091199050 (119904)100381610038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864 100381610038161003816100381612058510038161003816100381610038162 + 1198621198906119870(119879minus1199050+1)(119879minus1199050)lt infin
(28)
The demonstration of the lemma is complete
Proof of Theorem 5 We begin by checking uniqueness ofsolution Let 119909(119905) and 119909(119905) be two solutions of (1) byLemma 6 119909(119905) and 119909(119905) isin M2((minusinfin119879]R119899) Note that
119909 (119905) minus 119909 (119905) = int1199051199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 119889119904+ int1199051199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904) (29)
By the elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) one thengets
|119909 (119905) minus 119909 (119905)|2le 2 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(30)
By Holder inequality Lemma 3 and (15) and (16) we have
119864 |119909 (119905) minus 119909 (119905)|2le 2 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119904 119904) minus 119891 (119909119904 119904)10038161003816100381610038162 119889119904+ 2119864int119905
1199050
1003816100381610038161003816119892 (119909119904 119904) minus 119892 (119909119904 119904)10038161003816100381610038162 119889119904le 2119870 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904+ 2119870119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904le 2119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904
(31)
From the fact 1199091199050(119904) = 1199091199050(119904) = 120585(119904) 119904 isin (minusinfin 0] andsup1199050le119904le119905
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583= sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 |119909 (119904 + 120579) minus 119909 (119904 + 120579)|2= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) |119909 (119903) minus 119909 (119903)|2
6 Chinese Journal of Mathematics
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt120579le0
1198902120583(120579+1199050) 100381610038161003816100381610038161199091199050 (120579) minus 1199091199050 (120579)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 11989021199050120583 1003816100381610038161003816120585 minus 12058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 sup1199050le119903le119904
1198902120583119904 |119909 (119903) minus 119909 (119903)|2le sup1199050le119903le119905
|119909 (119903) minus 119909 (119903)|2 (32)
We have
119864( sup1199050le119904le119905
|119909 (119904) minus 119909 (119904)|2)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
|119909 (119903) minus 119909 (119903)|2)119889119904(33)
Applying the Gronwall inequality yields
119864 (|119909 (119905) minus 119909 (119905)|2) = 0 1199050 le 119905 le 119879 (34)
The above expression means that 119909(119905) = 119909(119905) as for 1199050 le 119905 le119879 Therefore for all minusinfin lt 119905 le 119879 119909(119905) = 119909(119905) as the proofof uniqueness is complete
Next to check the existence define 11990901199050 = 120585 and 1199090(119905) =120585(0) for 1199050 le 119905 le 119879 Let 1199091198961199050 = 120585 119896 = 1 2 and define Picardsequence
119909119896 (119905) = 120585 (0) + int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) 1199050 le 119905 le 119879(35)
Obviously 1199090(119905) isin M2((minusinfin119879]R119899) By induction we cansee that 119909119896(119905) isin M2((minusinfin119879]R119899)
In fact by elementary inequality (119886+119887+119888)2 le 3(1198862+1198872+1198882)10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3 1003816100381610038161003816120585 (0)10038161003816100381610038162 + 3 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119891 (119909119896minus1119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119896minus1119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(36)
From the Holder inequality and Lemma 3 we have
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896minus1119904 119904) minus 119891 (0 119904) + 119891 (0 119904)100381610038161003816100381610038162 119889119904+ 3119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896minus1119904 119904) minus 119892 (0 119904) + 119892 (0 119904)100381610038161003816100381610038162 119889119904(37)
Again the elementary inequality (119886 + 119887)2 le 21198862 + 21198872 (22)(15) and (16) imply that
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583+ 3 (119905 minus 1199050 + 1) 119864int119905
1199050
(2119870 10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
119889119904+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904le 1198621 + 1198622+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904
(38)
where 1198621 = 3119864|120585|2120583 + 6119870(119905 minus 1199050 + 1)(119905 minus 1199050) and 1198622 = 6119870(119905 minus1199050 + 1)(119905 minus 1199050)119864|120585|2120583
Hence for any ℓ ge 1 one can derive that
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162)le 1198621 + 1198622
+ 6119870 (119905 minus 1199050 + 1)int1199051199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904(39)
Note that
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896minus1 (119904)100381610038161003816100381610038162 = max 119864 1003816100381610038161003816120585 (0)10038161003816100381610038162 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 le max 119864 100381610038161003816100381612058510038161003816100381610038162120583 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓ (119904)100381610038161003816100381610038162 = max119864 100381610038161003816100381612058510038161003816100381610038162120583 max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162 le 119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162
(40)
Chinese Journal of Mathematics 7
and then
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162) le 1198621 + 1198622 + 6119870 (119905 minus 1199050 + 1)
sdot int1199051199050
(119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162))119889119904 le 1198623+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162)119889119904
(41)
where 1198623 = 1198621 + 21198622From the Gronwall inequality we have
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) (42)
Since 119896 is arbitrary
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) 1199050 le 119905 le 119879 119896 ge 1 (43)
From the Holder inequality Lemma 3 and (15) and (16) as ina similar earlier inequality one then has
119864 100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (1199090119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162 + 2119864 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119892 (1199090119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2 (119905 minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904
+ 2119864int1199051199050
10038161003816100381610038161003816119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904
le 2 (119905 minus 1199050 + 1) 119864int1199051199050
(2119870 100381610038161003816100381610038161199090119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050)
+ 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050) 119864 100381610038161003816100381612058510038161003816100381610038162120583
(44)
That is
119864( sup1199050le119904le119905
100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162)le 4 (119905 minus 1199050 + 1) (119905 minus 1199050) (119870 + 119870119864 100381610038161003816100381612058510038161003816100381610038162120583) fl 119877
(45)
By similar arguments as above we also have
119864 100381610038161003816100381610038161199092 (119905) minus 1199091 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (1199091119904 119904) minus 119891 (1199090119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (1199091119904 119904) minus 119892 (1199090119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904 le 2 (119905 minus 1199050+ 1) 119864int119905
1199050
[10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162
+ 10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162] 119889119904 le 2119870 (119905 minus 1199050+ 1) 119864int119905
1199050
100381610038161003816100381610038161199091119904 minus 1199090119904 100381610038161003816100381610038162120583 119889119904
(46)
Then
119864( sup1199050le119904le119905
100381610038161003816100381610038161199092 (119904) minus 1199091 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199091 (119903) minus 1199090 (119903)100381610038161003816100381610038162)119889119904le 2119877119870 (119905 minus 1199050 + 1) (119905 minus 1199050) = 119877119872(119905 minus 1199050)
(47)
where119872 = 2119870(119905 minus 1199050 + 1) Similarly
119864( sup1199050le119904le119905
100381610038161003816100381610038161199093 (119904) minus 1199092 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199092 (119903) minus 1199091 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877119872(119904 minus 1199050) 119889119904 le 119877 [119872 (119905 minus 1199050)]22
(48)
Continue this process to find that
119864( sup1199050le119904le119905
100381610038161003816100381610038161199094 (119904) minus 1199093 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199093 (119903) minus 1199092 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]22 119889119904 le 119877 [119872(119905 minus 1199050)]36
(49)
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
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2 Chinese Journal of Mathematics
organized the study of functional differential equations withinfinite delay in [2] They assumed thatB is a normed linearspace of functions mapping (minusinfin 0] into a Banach space(119883 |sdot|) endowedwith a norm |sdot|B and satisfying the followingaxioms(1198601) There exist a positive constant 119867 and functions119870(sdot)119872(sdot) [0 +infin) rarr [0 +infin) with 119870 being continuousand 119872 being locally bounded such that for any 120590 isin R and119886 gt 0 if 119909 (minusinfin 120590+119886] rarr 119883 119909120590 isin B and 119909(sdot) is continuouson [120590 120590+119886] then for all 119905 in [120590 120590+119886] the following conditionshold
(i) 119909119905 isin B(ii) |119909(119905)| le 119867|119909119905|B(iii) |119909119905|B le 119870(119905 minus 120590)sup120590le119904le119905|119909(119904)| + 119872(119905 minus 120590)|119909120590|B
(1198602) For the function 119909(sdot) in (1198601) 119905 rarr 119909119905 is aB-valuedcontinuous function for 119905 in [120590 120590 + 119886](1198603)The spaceB is complete
Later on the concept of fading and uniform fadingmemory spaces has been adopted as the best choice
For 120593 isin B 119905 ge 0 and 120579 le 0 we define the linear operatorO(119905) by
[O (119905) 120593] (120579) = 120593 (0) if 119905 + 120579 ge 0120593 (119905 + 120579) if 119905 + 120579 lt 0 (2)
(O(119905))119905ge0 is exactly the solution semigroup associatedwith thefollowing trivial equation
119889119889119905119906 (119905) = 01199060 = 120593
(3)
We define
O0 (119905) = O (119905)B0
where B0 fl 120593 isin B 120593 (0) = 0 (4)
Let C00 be the set of continuous functions 120593 (minusinfin 0] rarr 119883with compact support We recall the following axiom(1198604) If a uniformly bounded sequence (120593119899)119899ge0 in C00converges to a function 120593 compactly on (minusinfin 0] then 120593 isin Band |120593119899 minus 120593|B rarr 0Definition 1
(1) B is called a fading memory space if it satisfies theaxioms (1198601) (1198602) (1198603) (1198604) and |O0(119905)120593| rarr 0 as119905 rarr +infin for all 120593 isin B0
(2) B is called a uniform fading memory space if it satis-fies the axioms (1198601) (1198602) (1198603) (1198604) and |O0(119905)| rarr0 as 119905 rarr +infin
Examples We recall the definitions of some standard exam-ples of phase spacesB
We start first with the phase space of 119883-valued boundedcontinuous functions 120593 defined on (minusinfin 0] that isBC((minusinfin 0] 119883) with norm |120593|BC = supminusinfinlt120579le0|120593(120579)|
(1) Let
BU = 120593 isin BC ((minusinfin 0] 119883) 120593 is bounded uniformly continuous (5)
whereBC is the space of all bounded continuous functionsmapping (minusinfin 0] into 119883 provided with the uniform normtopology
(2) Let 120583 isin R and
C120583
= 120593 isin C ((minusinfin 0] 119883) lim120579rarrminusinfin
119890120583120579120593 (120579) exists in119883 (6)
provided with the norm
10038161003816100381610038161205931003816100381610038161003816120583 = supminusinfinlt120579le0
119890120583120579 1003816100381610038161003816120593 (120579)1003816100381610038161003816 (7)
(3) For any continuous function 119892 (minusinfin 0] rarr [0 +infin)we define
C119892 = 120593 isin C ((minusinfin 0] 119883) 1003816100381610038161003816120593 (120579)1003816100381610038161003816119892 (120579) is bounded on (minusinfin 0]
C0119892 = 120593 isin C119892 ((minusinfin 0] 119883) lim
120579rarrminusinfin
1003816100381610038161003816120593 (120579)1003816100381610038161003816119892 (120579) = 0 (8)
endowed with the norm
10038161003816100381610038161205931003816100381610038161003816119892 = supminusinfinlt120579le0
1003816100381610038161003816120593 (120579)1003816100381610038161003816119892 (120579) (9)
Consider the following conditions on 119892(1198921) supminusinfinlt120579leminus119905(119892(119905 + 120579)119892(120579)) is locally bounded for119905 ge 0(1198922) lim120579rarrminusinfin119892(120579) = +infin
(1198923) lim119905rarr+infinsupminusinfinlt120579leminus119905(119892(119905 + 120579)119892(120579)) = 0Properties of each phase space are summarized in Table 1For other examples properties and details about phase
spaces we refer to the book by Hino et al [3]Fengying and Ke [4] discussed existence and uniqueness
of solutions to stochastic functional differential equationwithinfinite delay in the phase space of bounded continuousfunctions 120593 defined on (minusinfin 0] with values in R119899 that isBC((minusinfin 0]R119899) with norm |120593|BC = supminusinfinlt120579le0|120593(120579)|Lemma2 (page 22 in [3]) If the phase spaceB satisfies axiom(1198604) thenBC((minusinfin 0]R119899) is included inB
Chinese Journal of Mathematics 3
Table 1
(1198601) (1198602) (1198603) (1198604) Uniform fading memory spaceBC Yes No Yes No NoBU Yes Yes Yes No NoC119892 Under (1198921) Under (1198921) Yes Under (1198922) Under (1198923)C0119892 Under (1198921) Under (1198921) Yes Under (1198922) Under (1198923)C120583 Yes Yes Yes Under 120583 gt 0 Under 120583 gt 0
3 Existence and Uniqueness
Lemma 3 (see [4]) If 119901 ge 2 119892 isin 1198712([1199050 119879]R119899times119898) such that119864int1198791199050|119892(119904)|119901119889119904 lt infin then
119864 100381610038161003816100381610038161003816100381610038161003816int119879
1199050
119892 (119904) 119889119882 (119904)100381610038161003816100381610038161003816100381610038161003816119901
le (119901 (119901 minus 1)2 )1199012 119879(119901minus2)2119864int119879
1199050
1003816100381610038161003816119892 (119904)1003816100381610038161003816119901 119889119904(10)
Lemma 4 (Borel-Cantelli page 487 in [5]) If 119864119899 is asequence of events and
infinsum119899=1
119875 (119864119899) lt infin (11)
then
119875 (119864119899 119894119900) = 0 (12)
where 119894119900 is an abbreviation for ldquoinfinitively oftenrdquo
Definitions 1 R119899-value stochastic process 119909(119905) defined onminusinfin lt 119905 le 119879 is called the solution of (1) with initial data1199091199050 if 119909(119905) has the following properties(i) 119909(119905) is continuous and 119909(119905)1199050le119905le119879 isF119905-adapted(ii) 119891(119909119905 119905) isin L1([1199050 119879]R119899) and 119892(119909119905 119905) isin L2([1199050119879]R119899times119898)(iii) 1199091199050 = 120585 for each 1199050 le 119905 le 119879119909 (119905) = 120585 (0) + int119905
1199050
119891 (119909119904 119904) 119889119904+ int1199051199050
119892 (119909119904 119904) 119889119882 (119904) almost surely (as) (13)
119909(119905) is called unique solution if any other solution 119909(119905) isdistinguishable with 119909(119905) that is
119875 119909 (119905) = 119909 (119905) for any 1199050 le 119905 le 119879 = 1 (14)
Now we establish existence and uniqueness of solutionsfor (1) with initial data 1199091199050 We suppose a uniform Lipschitzcondition and a weak linear growth condition
Theorem 5 Assume that there exist two positive number 119870and 119870 such that
(i) for any 120593 120595 isin 119862120583 and 119905 isin [1199050 119879] it follows that1003816100381610038161003816119891 (120593 119905) minus 119891 (120595 119905)10038161003816100381610038162 or 1003816100381610038161003816119892 (120593 119905) minus 119892 (120595 119905)10038161003816100381610038162le 119870 1003816100381610038161003816120593 minus 12059510038161003816100381610038162120583
(15)
(ii) for any 119905 isin [1199050 119879] it follows that 119891(0 119905) 119892(0 119905) isin1198712(119862120583) such that 1003816100381610038161003816119891 (0 119905)10038161003816100381610038162 or 1003816100381610038161003816119892 (0 119905)10038161003816100381610038162 le 119870 (16)
Then problem (1) with initial data 1199091199050 = 120585 isin M2((minusinfin 0]R119899) has a unique solution 119909(119905) Moreover 119909(119905) isin M2((minusinfin119879]R119899)Lemma 6 Let (15) and (16) hold If 119909(119905) is the solution of (1)with initial data 1199091199050 = 120585 then
119864( sup1199050le119905le119879
|119909 (119905)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (17)
where119862 = 3119864|120585|2120583 +6119870(119879minus1199050 +1)(119879minus1199050)+6119870(119879minus1199050 +1)(119879minus1199050)119864|120585|2120583
Moreover if 120585 isin M2((minusinfin 0]R119899) then 119909(119905) isinM2((minusinfin119879]R119899)Proof For each number 119902 ge 1 define the stopping time
120591119902 = 119879 and inf 119905 isin [1199050 119879] 10038161003816100381610038161199091199051003816100381610038161003816120583 ge 119902 (18)
Obviously as 119902 rarr infin 120591119902 119879 as Let 119909119902(119905) = 119909(119905 and 120591119902) 119905 isin[1199050 119879] and then 119909119902(119905) satisfy the following equation119909119902 (119905) = 120585 (0) + int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119904+ int1199051199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904) (19)
Using the elementary inequality (119886 + 119887 + 119888)2 le 3(1198862 + 1198872 + 1198882)we get
1003816100381610038161003816119909119902 (119905)10038161003816100381610038162 le 3 100381610038161003816100381612058510038161003816100381610038162120583 + 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(20)
4 Chinese Journal of Mathematics
Taking the expectation on both sides and using the Holderinequality Lemma3 and (15) and (16) we get for all 119905 in [1199050 119879]
119864 1003816100381610038161003816119909119902 (119905)10038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583
+ 3 (119905 minus 1199050) 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904) + 119891 (0 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904) + 119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
(1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162
+ 1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 + 1003816100381610038161003816119891 (0 119904)10038161003816100381610038162+ 1003816100381610038161003816119892 (0 119904)10038161003816100381610038162) 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
(119870 1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 + 119870)119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050+ 1) (119905 minus 1199050) + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 119889119904(21)
We have also for each 119905 in [1199050 119879]
sup1199050le119904le119905
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 = sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 1003816100381610038161003816119909119902 (119904 + 120579)10038161003816100381610038162
= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le1199050
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 1003816100381610038161003816119909119902 (119903 minus 1199050 + 1199050)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 100381610038161003816100381610038161199091199021199050 (119903 minus 1199050)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119904 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 1198902120583(1199050minus119904) 100381610038161003816100381612058510038161003816100381610038162120583+ sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
(22)
Chinese Journal of Mathematics 5
Letting 119905 = 119879 we get119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162)le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119879 minus 1199050 + 1) (119879 minus 1199050)
+ 6119870 (119879 minus 1199050 + 1) 119864int1198791199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
le 119862 + 6119870 (119879 minus 1199050 + 1)int1198791199050
119864( sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
(23)
where119862 = 3119864|120585|2120583+6119870(119879minus1199050+1)(119879minus1199050)+6119870(119879minus1199050+1)(119879minus1199050)119864|120585|2120583
By the Gronwall inequality we infer
119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (24)
That is
119864( sup1199050le119904le119879
10038161003816100381610038161003816119909 (119904 and 120591119902)100381610038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (25)
Consequently
119864( sup1199050le119904le120591119902
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (26)
Letting 119902 rarr infin that implies the following inequality
119864( sup1199050le119904le119879
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (27)
Now to prove the second part or the lemma suppose that120585 isin M2((minusinfin 0]R119899) Then
119864( supminusinfinle119905le119879
|119909 (119905)|2) = 119864( supminusinfinle119905le1199050
|119909 (119905)|2)
+ 119864( sup1199050le119905le119879
|119909 (119905)|2)
le 119864( supminusinfinle119905le1199050
|119909 (119905)|2)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119905minus1199050le0
1003816100381610038161003816119909 (119905 minus 1199050 + 1199050)10038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119904le0
100381610038161003816100381610038161199091199050 (119904)100381610038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864 100381610038161003816100381612058510038161003816100381610038162 + 1198621198906119870(119879minus1199050+1)(119879minus1199050)lt infin
(28)
The demonstration of the lemma is complete
Proof of Theorem 5 We begin by checking uniqueness ofsolution Let 119909(119905) and 119909(119905) be two solutions of (1) byLemma 6 119909(119905) and 119909(119905) isin M2((minusinfin119879]R119899) Note that
119909 (119905) minus 119909 (119905) = int1199051199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 119889119904+ int1199051199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904) (29)
By the elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) one thengets
|119909 (119905) minus 119909 (119905)|2le 2 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(30)
By Holder inequality Lemma 3 and (15) and (16) we have
119864 |119909 (119905) minus 119909 (119905)|2le 2 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119904 119904) minus 119891 (119909119904 119904)10038161003816100381610038162 119889119904+ 2119864int119905
1199050
1003816100381610038161003816119892 (119909119904 119904) minus 119892 (119909119904 119904)10038161003816100381610038162 119889119904le 2119870 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904+ 2119870119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904le 2119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904
(31)
From the fact 1199091199050(119904) = 1199091199050(119904) = 120585(119904) 119904 isin (minusinfin 0] andsup1199050le119904le119905
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583= sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 |119909 (119904 + 120579) minus 119909 (119904 + 120579)|2= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) |119909 (119903) minus 119909 (119903)|2
6 Chinese Journal of Mathematics
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt120579le0
1198902120583(120579+1199050) 100381610038161003816100381610038161199091199050 (120579) minus 1199091199050 (120579)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 11989021199050120583 1003816100381610038161003816120585 minus 12058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 sup1199050le119903le119904
1198902120583119904 |119909 (119903) minus 119909 (119903)|2le sup1199050le119903le119905
|119909 (119903) minus 119909 (119903)|2 (32)
We have
119864( sup1199050le119904le119905
|119909 (119904) minus 119909 (119904)|2)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
|119909 (119903) minus 119909 (119903)|2)119889119904(33)
Applying the Gronwall inequality yields
119864 (|119909 (119905) minus 119909 (119905)|2) = 0 1199050 le 119905 le 119879 (34)
The above expression means that 119909(119905) = 119909(119905) as for 1199050 le 119905 le119879 Therefore for all minusinfin lt 119905 le 119879 119909(119905) = 119909(119905) as the proofof uniqueness is complete
Next to check the existence define 11990901199050 = 120585 and 1199090(119905) =120585(0) for 1199050 le 119905 le 119879 Let 1199091198961199050 = 120585 119896 = 1 2 and define Picardsequence
119909119896 (119905) = 120585 (0) + int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) 1199050 le 119905 le 119879(35)
Obviously 1199090(119905) isin M2((minusinfin119879]R119899) By induction we cansee that 119909119896(119905) isin M2((minusinfin119879]R119899)
In fact by elementary inequality (119886+119887+119888)2 le 3(1198862+1198872+1198882)10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3 1003816100381610038161003816120585 (0)10038161003816100381610038162 + 3 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119891 (119909119896minus1119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119896minus1119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(36)
From the Holder inequality and Lemma 3 we have
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896minus1119904 119904) minus 119891 (0 119904) + 119891 (0 119904)100381610038161003816100381610038162 119889119904+ 3119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896minus1119904 119904) minus 119892 (0 119904) + 119892 (0 119904)100381610038161003816100381610038162 119889119904(37)
Again the elementary inequality (119886 + 119887)2 le 21198862 + 21198872 (22)(15) and (16) imply that
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583+ 3 (119905 minus 1199050 + 1) 119864int119905
1199050
(2119870 10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
119889119904+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904le 1198621 + 1198622+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904
(38)
where 1198621 = 3119864|120585|2120583 + 6119870(119905 minus 1199050 + 1)(119905 minus 1199050) and 1198622 = 6119870(119905 minus1199050 + 1)(119905 minus 1199050)119864|120585|2120583
Hence for any ℓ ge 1 one can derive that
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162)le 1198621 + 1198622
+ 6119870 (119905 minus 1199050 + 1)int1199051199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904(39)
Note that
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896minus1 (119904)100381610038161003816100381610038162 = max 119864 1003816100381610038161003816120585 (0)10038161003816100381610038162 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 le max 119864 100381610038161003816100381612058510038161003816100381610038162120583 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓ (119904)100381610038161003816100381610038162 = max119864 100381610038161003816100381612058510038161003816100381610038162120583 max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162 le 119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162
(40)
Chinese Journal of Mathematics 7
and then
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162) le 1198621 + 1198622 + 6119870 (119905 minus 1199050 + 1)
sdot int1199051199050
(119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162))119889119904 le 1198623+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162)119889119904
(41)
where 1198623 = 1198621 + 21198622From the Gronwall inequality we have
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) (42)
Since 119896 is arbitrary
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) 1199050 le 119905 le 119879 119896 ge 1 (43)
From the Holder inequality Lemma 3 and (15) and (16) as ina similar earlier inequality one then has
119864 100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (1199090119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162 + 2119864 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119892 (1199090119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2 (119905 minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904
+ 2119864int1199051199050
10038161003816100381610038161003816119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904
le 2 (119905 minus 1199050 + 1) 119864int1199051199050
(2119870 100381610038161003816100381610038161199090119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050)
+ 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050) 119864 100381610038161003816100381612058510038161003816100381610038162120583
(44)
That is
119864( sup1199050le119904le119905
100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162)le 4 (119905 minus 1199050 + 1) (119905 minus 1199050) (119870 + 119870119864 100381610038161003816100381612058510038161003816100381610038162120583) fl 119877
(45)
By similar arguments as above we also have
119864 100381610038161003816100381610038161199092 (119905) minus 1199091 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (1199091119904 119904) minus 119891 (1199090119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (1199091119904 119904) minus 119892 (1199090119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904 le 2 (119905 minus 1199050+ 1) 119864int119905
1199050
[10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162
+ 10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162] 119889119904 le 2119870 (119905 minus 1199050+ 1) 119864int119905
1199050
100381610038161003816100381610038161199091119904 minus 1199090119904 100381610038161003816100381610038162120583 119889119904
(46)
Then
119864( sup1199050le119904le119905
100381610038161003816100381610038161199092 (119904) minus 1199091 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199091 (119903) minus 1199090 (119903)100381610038161003816100381610038162)119889119904le 2119877119870 (119905 minus 1199050 + 1) (119905 minus 1199050) = 119877119872(119905 minus 1199050)
(47)
where119872 = 2119870(119905 minus 1199050 + 1) Similarly
119864( sup1199050le119904le119905
100381610038161003816100381610038161199093 (119904) minus 1199092 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199092 (119903) minus 1199091 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877119872(119904 minus 1199050) 119889119904 le 119877 [119872 (119905 minus 1199050)]22
(48)
Continue this process to find that
119864( sup1199050le119904le119905
100381610038161003816100381610038161199094 (119904) minus 1199093 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199093 (119903) minus 1199092 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]22 119889119904 le 119877 [119872(119905 minus 1199050)]36
(49)
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
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Chinese Journal of Mathematics 3
Table 1
(1198601) (1198602) (1198603) (1198604) Uniform fading memory spaceBC Yes No Yes No NoBU Yes Yes Yes No NoC119892 Under (1198921) Under (1198921) Yes Under (1198922) Under (1198923)C0119892 Under (1198921) Under (1198921) Yes Under (1198922) Under (1198923)C120583 Yes Yes Yes Under 120583 gt 0 Under 120583 gt 0
3 Existence and Uniqueness
Lemma 3 (see [4]) If 119901 ge 2 119892 isin 1198712([1199050 119879]R119899times119898) such that119864int1198791199050|119892(119904)|119901119889119904 lt infin then
119864 100381610038161003816100381610038161003816100381610038161003816int119879
1199050
119892 (119904) 119889119882 (119904)100381610038161003816100381610038161003816100381610038161003816119901
le (119901 (119901 minus 1)2 )1199012 119879(119901minus2)2119864int119879
1199050
1003816100381610038161003816119892 (119904)1003816100381610038161003816119901 119889119904(10)
Lemma 4 (Borel-Cantelli page 487 in [5]) If 119864119899 is asequence of events and
infinsum119899=1
119875 (119864119899) lt infin (11)
then
119875 (119864119899 119894119900) = 0 (12)
where 119894119900 is an abbreviation for ldquoinfinitively oftenrdquo
Definitions 1 R119899-value stochastic process 119909(119905) defined onminusinfin lt 119905 le 119879 is called the solution of (1) with initial data1199091199050 if 119909(119905) has the following properties(i) 119909(119905) is continuous and 119909(119905)1199050le119905le119879 isF119905-adapted(ii) 119891(119909119905 119905) isin L1([1199050 119879]R119899) and 119892(119909119905 119905) isin L2([1199050119879]R119899times119898)(iii) 1199091199050 = 120585 for each 1199050 le 119905 le 119879119909 (119905) = 120585 (0) + int119905
1199050
119891 (119909119904 119904) 119889119904+ int1199051199050
119892 (119909119904 119904) 119889119882 (119904) almost surely (as) (13)
119909(119905) is called unique solution if any other solution 119909(119905) isdistinguishable with 119909(119905) that is
119875 119909 (119905) = 119909 (119905) for any 1199050 le 119905 le 119879 = 1 (14)
Now we establish existence and uniqueness of solutionsfor (1) with initial data 1199091199050 We suppose a uniform Lipschitzcondition and a weak linear growth condition
Theorem 5 Assume that there exist two positive number 119870and 119870 such that
(i) for any 120593 120595 isin 119862120583 and 119905 isin [1199050 119879] it follows that1003816100381610038161003816119891 (120593 119905) minus 119891 (120595 119905)10038161003816100381610038162 or 1003816100381610038161003816119892 (120593 119905) minus 119892 (120595 119905)10038161003816100381610038162le 119870 1003816100381610038161003816120593 minus 12059510038161003816100381610038162120583
(15)
(ii) for any 119905 isin [1199050 119879] it follows that 119891(0 119905) 119892(0 119905) isin1198712(119862120583) such that 1003816100381610038161003816119891 (0 119905)10038161003816100381610038162 or 1003816100381610038161003816119892 (0 119905)10038161003816100381610038162 le 119870 (16)
Then problem (1) with initial data 1199091199050 = 120585 isin M2((minusinfin 0]R119899) has a unique solution 119909(119905) Moreover 119909(119905) isin M2((minusinfin119879]R119899)Lemma 6 Let (15) and (16) hold If 119909(119905) is the solution of (1)with initial data 1199091199050 = 120585 then
119864( sup1199050le119905le119879
|119909 (119905)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (17)
where119862 = 3119864|120585|2120583 +6119870(119879minus1199050 +1)(119879minus1199050)+6119870(119879minus1199050 +1)(119879minus1199050)119864|120585|2120583
Moreover if 120585 isin M2((minusinfin 0]R119899) then 119909(119905) isinM2((minusinfin119879]R119899)Proof For each number 119902 ge 1 define the stopping time
120591119902 = 119879 and inf 119905 isin [1199050 119879] 10038161003816100381610038161199091199051003816100381610038161003816120583 ge 119902 (18)
Obviously as 119902 rarr infin 120591119902 119879 as Let 119909119902(119905) = 119909(119905 and 120591119902) 119905 isin[1199050 119879] and then 119909119902(119905) satisfy the following equation119909119902 (119905) = 120585 (0) + int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119904+ int1199051199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904) (19)
Using the elementary inequality (119886 + 119887 + 119888)2 le 3(1198862 + 1198872 + 1198882)we get
1003816100381610038161003816119909119902 (119905)10038161003816100381610038162 le 3 100381610038161003816100381612058510038161003816100381610038162120583 + 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(20)
4 Chinese Journal of Mathematics
Taking the expectation on both sides and using the Holderinequality Lemma3 and (15) and (16) we get for all 119905 in [1199050 119879]
119864 1003816100381610038161003816119909119902 (119905)10038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583
+ 3 (119905 minus 1199050) 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904) + 119891 (0 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904) + 119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
(1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162
+ 1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 + 1003816100381610038161003816119891 (0 119904)10038161003816100381610038162+ 1003816100381610038161003816119892 (0 119904)10038161003816100381610038162) 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
(119870 1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 + 119870)119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050+ 1) (119905 minus 1199050) + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 119889119904(21)
We have also for each 119905 in [1199050 119879]
sup1199050le119904le119905
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 = sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 1003816100381610038161003816119909119902 (119904 + 120579)10038161003816100381610038162
= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le1199050
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 1003816100381610038161003816119909119902 (119903 minus 1199050 + 1199050)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 100381610038161003816100381610038161199091199021199050 (119903 minus 1199050)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119904 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 1198902120583(1199050minus119904) 100381610038161003816100381612058510038161003816100381610038162120583+ sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
(22)
Chinese Journal of Mathematics 5
Letting 119905 = 119879 we get119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162)le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119879 minus 1199050 + 1) (119879 minus 1199050)
+ 6119870 (119879 minus 1199050 + 1) 119864int1198791199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
le 119862 + 6119870 (119879 minus 1199050 + 1)int1198791199050
119864( sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
(23)
where119862 = 3119864|120585|2120583+6119870(119879minus1199050+1)(119879minus1199050)+6119870(119879minus1199050+1)(119879minus1199050)119864|120585|2120583
By the Gronwall inequality we infer
119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (24)
That is
119864( sup1199050le119904le119879
10038161003816100381610038161003816119909 (119904 and 120591119902)100381610038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (25)
Consequently
119864( sup1199050le119904le120591119902
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (26)
Letting 119902 rarr infin that implies the following inequality
119864( sup1199050le119904le119879
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (27)
Now to prove the second part or the lemma suppose that120585 isin M2((minusinfin 0]R119899) Then
119864( supminusinfinle119905le119879
|119909 (119905)|2) = 119864( supminusinfinle119905le1199050
|119909 (119905)|2)
+ 119864( sup1199050le119905le119879
|119909 (119905)|2)
le 119864( supminusinfinle119905le1199050
|119909 (119905)|2)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119905minus1199050le0
1003816100381610038161003816119909 (119905 minus 1199050 + 1199050)10038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119904le0
100381610038161003816100381610038161199091199050 (119904)100381610038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864 100381610038161003816100381612058510038161003816100381610038162 + 1198621198906119870(119879minus1199050+1)(119879minus1199050)lt infin
(28)
The demonstration of the lemma is complete
Proof of Theorem 5 We begin by checking uniqueness ofsolution Let 119909(119905) and 119909(119905) be two solutions of (1) byLemma 6 119909(119905) and 119909(119905) isin M2((minusinfin119879]R119899) Note that
119909 (119905) minus 119909 (119905) = int1199051199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 119889119904+ int1199051199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904) (29)
By the elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) one thengets
|119909 (119905) minus 119909 (119905)|2le 2 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(30)
By Holder inequality Lemma 3 and (15) and (16) we have
119864 |119909 (119905) minus 119909 (119905)|2le 2 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119904 119904) minus 119891 (119909119904 119904)10038161003816100381610038162 119889119904+ 2119864int119905
1199050
1003816100381610038161003816119892 (119909119904 119904) minus 119892 (119909119904 119904)10038161003816100381610038162 119889119904le 2119870 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904+ 2119870119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904le 2119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904
(31)
From the fact 1199091199050(119904) = 1199091199050(119904) = 120585(119904) 119904 isin (minusinfin 0] andsup1199050le119904le119905
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583= sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 |119909 (119904 + 120579) minus 119909 (119904 + 120579)|2= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) |119909 (119903) minus 119909 (119903)|2
6 Chinese Journal of Mathematics
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt120579le0
1198902120583(120579+1199050) 100381610038161003816100381610038161199091199050 (120579) minus 1199091199050 (120579)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 11989021199050120583 1003816100381610038161003816120585 minus 12058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 sup1199050le119903le119904
1198902120583119904 |119909 (119903) minus 119909 (119903)|2le sup1199050le119903le119905
|119909 (119903) minus 119909 (119903)|2 (32)
We have
119864( sup1199050le119904le119905
|119909 (119904) minus 119909 (119904)|2)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
|119909 (119903) minus 119909 (119903)|2)119889119904(33)
Applying the Gronwall inequality yields
119864 (|119909 (119905) minus 119909 (119905)|2) = 0 1199050 le 119905 le 119879 (34)
The above expression means that 119909(119905) = 119909(119905) as for 1199050 le 119905 le119879 Therefore for all minusinfin lt 119905 le 119879 119909(119905) = 119909(119905) as the proofof uniqueness is complete
Next to check the existence define 11990901199050 = 120585 and 1199090(119905) =120585(0) for 1199050 le 119905 le 119879 Let 1199091198961199050 = 120585 119896 = 1 2 and define Picardsequence
119909119896 (119905) = 120585 (0) + int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) 1199050 le 119905 le 119879(35)
Obviously 1199090(119905) isin M2((minusinfin119879]R119899) By induction we cansee that 119909119896(119905) isin M2((minusinfin119879]R119899)
In fact by elementary inequality (119886+119887+119888)2 le 3(1198862+1198872+1198882)10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3 1003816100381610038161003816120585 (0)10038161003816100381610038162 + 3 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119891 (119909119896minus1119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119896minus1119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(36)
From the Holder inequality and Lemma 3 we have
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896minus1119904 119904) minus 119891 (0 119904) + 119891 (0 119904)100381610038161003816100381610038162 119889119904+ 3119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896minus1119904 119904) minus 119892 (0 119904) + 119892 (0 119904)100381610038161003816100381610038162 119889119904(37)
Again the elementary inequality (119886 + 119887)2 le 21198862 + 21198872 (22)(15) and (16) imply that
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583+ 3 (119905 minus 1199050 + 1) 119864int119905
1199050
(2119870 10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
119889119904+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904le 1198621 + 1198622+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904
(38)
where 1198621 = 3119864|120585|2120583 + 6119870(119905 minus 1199050 + 1)(119905 minus 1199050) and 1198622 = 6119870(119905 minus1199050 + 1)(119905 minus 1199050)119864|120585|2120583
Hence for any ℓ ge 1 one can derive that
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162)le 1198621 + 1198622
+ 6119870 (119905 minus 1199050 + 1)int1199051199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904(39)
Note that
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896minus1 (119904)100381610038161003816100381610038162 = max 119864 1003816100381610038161003816120585 (0)10038161003816100381610038162 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 le max 119864 100381610038161003816100381612058510038161003816100381610038162120583 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓ (119904)100381610038161003816100381610038162 = max119864 100381610038161003816100381612058510038161003816100381610038162120583 max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162 le 119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162
(40)
Chinese Journal of Mathematics 7
and then
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162) le 1198621 + 1198622 + 6119870 (119905 minus 1199050 + 1)
sdot int1199051199050
(119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162))119889119904 le 1198623+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162)119889119904
(41)
where 1198623 = 1198621 + 21198622From the Gronwall inequality we have
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) (42)
Since 119896 is arbitrary
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) 1199050 le 119905 le 119879 119896 ge 1 (43)
From the Holder inequality Lemma 3 and (15) and (16) as ina similar earlier inequality one then has
119864 100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (1199090119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162 + 2119864 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119892 (1199090119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2 (119905 minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904
+ 2119864int1199051199050
10038161003816100381610038161003816119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904
le 2 (119905 minus 1199050 + 1) 119864int1199051199050
(2119870 100381610038161003816100381610038161199090119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050)
+ 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050) 119864 100381610038161003816100381612058510038161003816100381610038162120583
(44)
That is
119864( sup1199050le119904le119905
100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162)le 4 (119905 minus 1199050 + 1) (119905 minus 1199050) (119870 + 119870119864 100381610038161003816100381612058510038161003816100381610038162120583) fl 119877
(45)
By similar arguments as above we also have
119864 100381610038161003816100381610038161199092 (119905) minus 1199091 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (1199091119904 119904) minus 119891 (1199090119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (1199091119904 119904) minus 119892 (1199090119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904 le 2 (119905 minus 1199050+ 1) 119864int119905
1199050
[10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162
+ 10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162] 119889119904 le 2119870 (119905 minus 1199050+ 1) 119864int119905
1199050
100381610038161003816100381610038161199091119904 minus 1199090119904 100381610038161003816100381610038162120583 119889119904
(46)
Then
119864( sup1199050le119904le119905
100381610038161003816100381610038161199092 (119904) minus 1199091 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199091 (119903) minus 1199090 (119903)100381610038161003816100381610038162)119889119904le 2119877119870 (119905 minus 1199050 + 1) (119905 minus 1199050) = 119877119872(119905 minus 1199050)
(47)
where119872 = 2119870(119905 minus 1199050 + 1) Similarly
119864( sup1199050le119904le119905
100381610038161003816100381610038161199093 (119904) minus 1199092 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199092 (119903) minus 1199091 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877119872(119904 minus 1199050) 119889119904 le 119877 [119872 (119905 minus 1199050)]22
(48)
Continue this process to find that
119864( sup1199050le119904le119905
100381610038161003816100381610038161199094 (119904) minus 1199093 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199093 (119903) minus 1199092 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]22 119889119904 le 119877 [119872(119905 minus 1199050)]36
(49)
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
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4 Chinese Journal of Mathematics
Taking the expectation on both sides and using the Holderinequality Lemma3 and (15) and (16) we get for all 119905 in [1199050 119879]
119864 1003816100381610038161003816119909119902 (119905)10038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119902119904 119904) 119868[1199050 120591119902] (119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583
+ 3 (119905 minus 1199050) 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904) + 119891 (0 119904)10038161003816100381610038162 119889119904+ 3119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904) + 119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904+ 6119864int119905
1199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050)sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
1003816100381610038161003816119891 (0 119904)10038161003816100381610038162 119889119904 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
1003816100381610038161003816119892 (0 119904)10038161003816100381610038162 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)
sdot 119864int1199051199050
(1003816100381610038161003816119891 (119909119902119904 119904) minus 119891 (0 119904)10038161003816100381610038162
+ 1003816100381610038161003816119892 (119909119902119904 119904) minus 119892 (0 119904)10038161003816100381610038162 + 1003816100381610038161003816119891 (0 119904)10038161003816100381610038162+ 1003816100381610038161003816119892 (0 119904)10038161003816100381610038162) 119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6 (119905 minus 1199050 + 1)sdot 119864int1199051199050
(119870 1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 + 119870)119889119904 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050+ 1) (119905 minus 1199050) + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 119889119904(21)
We have also for each 119905 in [1199050 119879]
sup1199050le119904le119905
1003816100381610038161003816119909119902119904 10038161003816100381610038162120583 = sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 1003816100381610038161003816119909119902 (119904 + 120579)10038161003816100381610038162
= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le1199050
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 1003816100381610038161003816119909119902 (119903 minus 1199050 + 1199050)10038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903minus1199050le0
1198902120583119903 100381610038161003816100381610038161199091199021199050 (119903 minus 1199050)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le sup1199050le119904le119905
119890minus2120583119904 11989021205831199050 100381610038161003816100381612058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119904 1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 1198902120583(1199050minus119904) 100381610038161003816100381612058510038161003816100381610038162120583+ sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162 le 100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119905
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162
(22)
Chinese Journal of Mathematics 5
Letting 119905 = 119879 we get119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162)le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119879 minus 1199050 + 1) (119879 minus 1199050)
+ 6119870 (119879 minus 1199050 + 1) 119864int1198791199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
le 119862 + 6119870 (119879 minus 1199050 + 1)int1198791199050
119864( sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
(23)
where119862 = 3119864|120585|2120583+6119870(119879minus1199050+1)(119879minus1199050)+6119870(119879minus1199050+1)(119879minus1199050)119864|120585|2120583
By the Gronwall inequality we infer
119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (24)
That is
119864( sup1199050le119904le119879
10038161003816100381610038161003816119909 (119904 and 120591119902)100381610038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (25)
Consequently
119864( sup1199050le119904le120591119902
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (26)
Letting 119902 rarr infin that implies the following inequality
119864( sup1199050le119904le119879
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (27)
Now to prove the second part or the lemma suppose that120585 isin M2((minusinfin 0]R119899) Then
119864( supminusinfinle119905le119879
|119909 (119905)|2) = 119864( supminusinfinle119905le1199050
|119909 (119905)|2)
+ 119864( sup1199050le119905le119879
|119909 (119905)|2)
le 119864( supminusinfinle119905le1199050
|119909 (119905)|2)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119905minus1199050le0
1003816100381610038161003816119909 (119905 minus 1199050 + 1199050)10038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119904le0
100381610038161003816100381610038161199091199050 (119904)100381610038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864 100381610038161003816100381612058510038161003816100381610038162 + 1198621198906119870(119879minus1199050+1)(119879minus1199050)lt infin
(28)
The demonstration of the lemma is complete
Proof of Theorem 5 We begin by checking uniqueness ofsolution Let 119909(119905) and 119909(119905) be two solutions of (1) byLemma 6 119909(119905) and 119909(119905) isin M2((minusinfin119879]R119899) Note that
119909 (119905) minus 119909 (119905) = int1199051199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 119889119904+ int1199051199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904) (29)
By the elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) one thengets
|119909 (119905) minus 119909 (119905)|2le 2 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(30)
By Holder inequality Lemma 3 and (15) and (16) we have
119864 |119909 (119905) minus 119909 (119905)|2le 2 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119904 119904) minus 119891 (119909119904 119904)10038161003816100381610038162 119889119904+ 2119864int119905
1199050
1003816100381610038161003816119892 (119909119904 119904) minus 119892 (119909119904 119904)10038161003816100381610038162 119889119904le 2119870 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904+ 2119870119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904le 2119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904
(31)
From the fact 1199091199050(119904) = 1199091199050(119904) = 120585(119904) 119904 isin (minusinfin 0] andsup1199050le119904le119905
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583= sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 |119909 (119904 + 120579) minus 119909 (119904 + 120579)|2= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) |119909 (119903) minus 119909 (119903)|2
6 Chinese Journal of Mathematics
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt120579le0
1198902120583(120579+1199050) 100381610038161003816100381610038161199091199050 (120579) minus 1199091199050 (120579)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 11989021199050120583 1003816100381610038161003816120585 minus 12058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 sup1199050le119903le119904
1198902120583119904 |119909 (119903) minus 119909 (119903)|2le sup1199050le119903le119905
|119909 (119903) minus 119909 (119903)|2 (32)
We have
119864( sup1199050le119904le119905
|119909 (119904) minus 119909 (119904)|2)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
|119909 (119903) minus 119909 (119903)|2)119889119904(33)
Applying the Gronwall inequality yields
119864 (|119909 (119905) minus 119909 (119905)|2) = 0 1199050 le 119905 le 119879 (34)
The above expression means that 119909(119905) = 119909(119905) as for 1199050 le 119905 le119879 Therefore for all minusinfin lt 119905 le 119879 119909(119905) = 119909(119905) as the proofof uniqueness is complete
Next to check the existence define 11990901199050 = 120585 and 1199090(119905) =120585(0) for 1199050 le 119905 le 119879 Let 1199091198961199050 = 120585 119896 = 1 2 and define Picardsequence
119909119896 (119905) = 120585 (0) + int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) 1199050 le 119905 le 119879(35)
Obviously 1199090(119905) isin M2((minusinfin119879]R119899) By induction we cansee that 119909119896(119905) isin M2((minusinfin119879]R119899)
In fact by elementary inequality (119886+119887+119888)2 le 3(1198862+1198872+1198882)10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3 1003816100381610038161003816120585 (0)10038161003816100381610038162 + 3 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119891 (119909119896minus1119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119896minus1119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(36)
From the Holder inequality and Lemma 3 we have
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896minus1119904 119904) minus 119891 (0 119904) + 119891 (0 119904)100381610038161003816100381610038162 119889119904+ 3119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896minus1119904 119904) minus 119892 (0 119904) + 119892 (0 119904)100381610038161003816100381610038162 119889119904(37)
Again the elementary inequality (119886 + 119887)2 le 21198862 + 21198872 (22)(15) and (16) imply that
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583+ 3 (119905 minus 1199050 + 1) 119864int119905
1199050
(2119870 10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
119889119904+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904le 1198621 + 1198622+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904
(38)
where 1198621 = 3119864|120585|2120583 + 6119870(119905 minus 1199050 + 1)(119905 minus 1199050) and 1198622 = 6119870(119905 minus1199050 + 1)(119905 minus 1199050)119864|120585|2120583
Hence for any ℓ ge 1 one can derive that
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162)le 1198621 + 1198622
+ 6119870 (119905 minus 1199050 + 1)int1199051199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904(39)
Note that
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896minus1 (119904)100381610038161003816100381610038162 = max 119864 1003816100381610038161003816120585 (0)10038161003816100381610038162 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 le max 119864 100381610038161003816100381612058510038161003816100381610038162120583 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓ (119904)100381610038161003816100381610038162 = max119864 100381610038161003816100381612058510038161003816100381610038162120583 max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162 le 119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162
(40)
Chinese Journal of Mathematics 7
and then
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162) le 1198621 + 1198622 + 6119870 (119905 minus 1199050 + 1)
sdot int1199051199050
(119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162))119889119904 le 1198623+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162)119889119904
(41)
where 1198623 = 1198621 + 21198622From the Gronwall inequality we have
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) (42)
Since 119896 is arbitrary
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) 1199050 le 119905 le 119879 119896 ge 1 (43)
From the Holder inequality Lemma 3 and (15) and (16) as ina similar earlier inequality one then has
119864 100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (1199090119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162 + 2119864 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119892 (1199090119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2 (119905 minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904
+ 2119864int1199051199050
10038161003816100381610038161003816119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904
le 2 (119905 minus 1199050 + 1) 119864int1199051199050
(2119870 100381610038161003816100381610038161199090119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050)
+ 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050) 119864 100381610038161003816100381612058510038161003816100381610038162120583
(44)
That is
119864( sup1199050le119904le119905
100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162)le 4 (119905 minus 1199050 + 1) (119905 minus 1199050) (119870 + 119870119864 100381610038161003816100381612058510038161003816100381610038162120583) fl 119877
(45)
By similar arguments as above we also have
119864 100381610038161003816100381610038161199092 (119905) minus 1199091 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (1199091119904 119904) minus 119891 (1199090119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (1199091119904 119904) minus 119892 (1199090119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904 le 2 (119905 minus 1199050+ 1) 119864int119905
1199050
[10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162
+ 10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162] 119889119904 le 2119870 (119905 minus 1199050+ 1) 119864int119905
1199050
100381610038161003816100381610038161199091119904 minus 1199090119904 100381610038161003816100381610038162120583 119889119904
(46)
Then
119864( sup1199050le119904le119905
100381610038161003816100381610038161199092 (119904) minus 1199091 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199091 (119903) minus 1199090 (119903)100381610038161003816100381610038162)119889119904le 2119877119870 (119905 minus 1199050 + 1) (119905 minus 1199050) = 119877119872(119905 minus 1199050)
(47)
where119872 = 2119870(119905 minus 1199050 + 1) Similarly
119864( sup1199050le119904le119905
100381610038161003816100381610038161199093 (119904) minus 1199092 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199092 (119903) minus 1199091 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877119872(119904 minus 1199050) 119889119904 le 119877 [119872 (119905 minus 1199050)]22
(48)
Continue this process to find that
119864( sup1199050le119904le119905
100381610038161003816100381610038161199094 (119904) minus 1199093 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199093 (119903) minus 1199092 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]22 119889119904 le 119877 [119872(119905 minus 1199050)]36
(49)
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
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Chinese Journal of Mathematics 5
Letting 119905 = 119879 we get119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162)le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119879 minus 1199050 + 1) (119879 minus 1199050)
+ 6119870 (119879 minus 1199050 + 1) 119864int1198791199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
le 119862 + 6119870 (119879 minus 1199050 + 1)int1198791199050
119864( sup1199050le119903le119879
1003816100381610038161003816119909119902 (119903)10038161003816100381610038162)119889119903
(23)
where119862 = 3119864|120585|2120583+6119870(119879minus1199050+1)(119879minus1199050)+6119870(119879minus1199050+1)(119879minus1199050)119864|120585|2120583
By the Gronwall inequality we infer
119864( sup1199050le119904le119879
1003816100381610038161003816119909119902 (119904)10038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (24)
That is
119864( sup1199050le119904le119879
10038161003816100381610038161003816119909 (119904 and 120591119902)100381610038161003816100381610038162) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (25)
Consequently
119864( sup1199050le119904le120591119902
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (26)
Letting 119902 rarr infin that implies the following inequality
119864( sup1199050le119904le119879
|119909 (119904)|2) le 1198621198906119870(119879minus1199050+1)(119879minus1199050) (27)
Now to prove the second part or the lemma suppose that120585 isin M2((minusinfin 0]R119899) Then
119864( supminusinfinle119905le119879
|119909 (119905)|2) = 119864( supminusinfinle119905le1199050
|119909 (119905)|2)
+ 119864( sup1199050le119905le119879
|119909 (119905)|2)
le 119864( supminusinfinle119905le1199050
|119909 (119905)|2)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119905minus1199050le0
1003816100381610038161003816119909 (119905 minus 1199050 + 1199050)10038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864( supminusinfinle119904le0
100381610038161003816100381610038161199091199050 (119904)100381610038161003816100381610038162)+ 1198621198906119870(119879minus1199050+1)(119879minus1199050)
le 119864 100381610038161003816100381612058510038161003816100381610038162 + 1198621198906119870(119879minus1199050+1)(119879minus1199050)lt infin
(28)
The demonstration of the lemma is complete
Proof of Theorem 5 We begin by checking uniqueness ofsolution Let 119909(119905) and 119909(119905) be two solutions of (1) byLemma 6 119909(119905) and 119909(119905) isin M2((minusinfin119879]R119899) Note that
119909 (119905) minus 119909 (119905) = int1199051199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 119889119904+ int1199051199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904) (29)
By the elementary inequality (119886 + 119887)2 le 2(1198862 + 1198872) one thengets
|119909 (119905) minus 119909 (119905)|2le 2 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
[119891 (119909119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(30)
By Holder inequality Lemma 3 and (15) and (16) we have
119864 |119909 (119905) minus 119909 (119905)|2le 2 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119891 (119909119904 119904) minus 119891 (119909119904 119904)10038161003816100381610038162 119889119904+ 2119864int119905
1199050
1003816100381610038161003816119892 (119909119904 119904) minus 119892 (119909119904 119904)10038161003816100381610038162 119889119904le 2119870 (119905 minus 1199050) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904+ 2119870119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904le 2119870 (119905 minus 1199050 + 1) 119864int119905
1199050
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583 119889119904
(31)
From the fact 1199091199050(119904) = 1199091199050(119904) = 120585(119904) 119904 isin (minusinfin 0] andsup1199050le119904le119905
1003816100381610038161003816119909119904 minus 11990911990410038161003816100381610038162120583= sup1199050le119904le119905
supminusinfinlt120579le0
1198902120583120579 |119909 (119904 + 120579) minus 119909 (119904 + 120579)|2= sup1199050le119904le119905
supminusinfinlt119903le119904
1198902120583(119903minus119904) |119909 (119903) minus 119909 (119903)|2
6 Chinese Journal of Mathematics
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt120579le0
1198902120583(120579+1199050) 100381610038161003816100381610038161199091199050 (120579) minus 1199091199050 (120579)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 11989021199050120583 1003816100381610038161003816120585 minus 12058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 sup1199050le119903le119904
1198902120583119904 |119909 (119903) minus 119909 (119903)|2le sup1199050le119903le119905
|119909 (119903) minus 119909 (119903)|2 (32)
We have
119864( sup1199050le119904le119905
|119909 (119904) minus 119909 (119904)|2)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
|119909 (119903) minus 119909 (119903)|2)119889119904(33)
Applying the Gronwall inequality yields
119864 (|119909 (119905) minus 119909 (119905)|2) = 0 1199050 le 119905 le 119879 (34)
The above expression means that 119909(119905) = 119909(119905) as for 1199050 le 119905 le119879 Therefore for all minusinfin lt 119905 le 119879 119909(119905) = 119909(119905) as the proofof uniqueness is complete
Next to check the existence define 11990901199050 = 120585 and 1199090(119905) =120585(0) for 1199050 le 119905 le 119879 Let 1199091198961199050 = 120585 119896 = 1 2 and define Picardsequence
119909119896 (119905) = 120585 (0) + int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) 1199050 le 119905 le 119879(35)
Obviously 1199090(119905) isin M2((minusinfin119879]R119899) By induction we cansee that 119909119896(119905) isin M2((minusinfin119879]R119899)
In fact by elementary inequality (119886+119887+119888)2 le 3(1198862+1198872+1198882)10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3 1003816100381610038161003816120585 (0)10038161003816100381610038162 + 3 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119891 (119909119896minus1119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119896minus1119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(36)
From the Holder inequality and Lemma 3 we have
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896minus1119904 119904) minus 119891 (0 119904) + 119891 (0 119904)100381610038161003816100381610038162 119889119904+ 3119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896minus1119904 119904) minus 119892 (0 119904) + 119892 (0 119904)100381610038161003816100381610038162 119889119904(37)
Again the elementary inequality (119886 + 119887)2 le 21198862 + 21198872 (22)(15) and (16) imply that
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583+ 3 (119905 minus 1199050 + 1) 119864int119905
1199050
(2119870 10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
119889119904+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904le 1198621 + 1198622+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904
(38)
where 1198621 = 3119864|120585|2120583 + 6119870(119905 minus 1199050 + 1)(119905 minus 1199050) and 1198622 = 6119870(119905 minus1199050 + 1)(119905 minus 1199050)119864|120585|2120583
Hence for any ℓ ge 1 one can derive that
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162)le 1198621 + 1198622
+ 6119870 (119905 minus 1199050 + 1)int1199051199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904(39)
Note that
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896minus1 (119904)100381610038161003816100381610038162 = max 119864 1003816100381610038161003816120585 (0)10038161003816100381610038162 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 le max 119864 100381610038161003816100381612058510038161003816100381610038162120583 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓ (119904)100381610038161003816100381610038162 = max119864 100381610038161003816100381612058510038161003816100381610038162120583 max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162 le 119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162
(40)
Chinese Journal of Mathematics 7
and then
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162) le 1198621 + 1198622 + 6119870 (119905 minus 1199050 + 1)
sdot int1199051199050
(119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162))119889119904 le 1198623+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162)119889119904
(41)
where 1198623 = 1198621 + 21198622From the Gronwall inequality we have
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) (42)
Since 119896 is arbitrary
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) 1199050 le 119905 le 119879 119896 ge 1 (43)
From the Holder inequality Lemma 3 and (15) and (16) as ina similar earlier inequality one then has
119864 100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (1199090119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162 + 2119864 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119892 (1199090119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2 (119905 minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904
+ 2119864int1199051199050
10038161003816100381610038161003816119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904
le 2 (119905 minus 1199050 + 1) 119864int1199051199050
(2119870 100381610038161003816100381610038161199090119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050)
+ 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050) 119864 100381610038161003816100381612058510038161003816100381610038162120583
(44)
That is
119864( sup1199050le119904le119905
100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162)le 4 (119905 minus 1199050 + 1) (119905 minus 1199050) (119870 + 119870119864 100381610038161003816100381612058510038161003816100381610038162120583) fl 119877
(45)
By similar arguments as above we also have
119864 100381610038161003816100381610038161199092 (119905) minus 1199091 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (1199091119904 119904) minus 119891 (1199090119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (1199091119904 119904) minus 119892 (1199090119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904 le 2 (119905 minus 1199050+ 1) 119864int119905
1199050
[10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162
+ 10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162] 119889119904 le 2119870 (119905 minus 1199050+ 1) 119864int119905
1199050
100381610038161003816100381610038161199091119904 minus 1199090119904 100381610038161003816100381610038162120583 119889119904
(46)
Then
119864( sup1199050le119904le119905
100381610038161003816100381610038161199092 (119904) minus 1199091 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199091 (119903) minus 1199090 (119903)100381610038161003816100381610038162)119889119904le 2119877119870 (119905 minus 1199050 + 1) (119905 minus 1199050) = 119877119872(119905 minus 1199050)
(47)
where119872 = 2119870(119905 minus 1199050 + 1) Similarly
119864( sup1199050le119904le119905
100381610038161003816100381610038161199093 (119904) minus 1199092 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199092 (119903) minus 1199091 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877119872(119904 minus 1199050) 119889119904 le 119877 [119872 (119905 minus 1199050)]22
(48)
Continue this process to find that
119864( sup1199050le119904le119905
100381610038161003816100381610038161199094 (119904) minus 1199093 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199093 (119903) minus 1199092 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]22 119889119904 le 119877 [119872(119905 minus 1199050)]36
(49)
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
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6 Chinese Journal of Mathematics
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 supminusinfinlt120579le0
1198902120583(120579+1199050) 100381610038161003816100381610038161199091199050 (120579) minus 1199091199050 (120579)100381610038161003816100381610038162
or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 11989021199050120583 1003816100381610038161003816120585 minus 12058510038161003816100381610038162120583or sup1199050le119903le119904
1198902120583119903 |119909 (119903) minus 119909 (119903)|2
le sup1199050le119904le119905
119890minus2120583119904 sup1199050le119903le119904
1198902120583119904 |119909 (119903) minus 119909 (119903)|2le sup1199050le119903le119905
|119909 (119903) minus 119909 (119903)|2 (32)
We have
119864( sup1199050le119904le119905
|119909 (119904) minus 119909 (119904)|2)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
|119909 (119903) minus 119909 (119903)|2)119889119904(33)
Applying the Gronwall inequality yields
119864 (|119909 (119905) minus 119909 (119905)|2) = 0 1199050 le 119905 le 119879 (34)
The above expression means that 119909(119905) = 119909(119905) as for 1199050 le 119905 le119879 Therefore for all minusinfin lt 119905 le 119879 119909(119905) = 119909(119905) as the proofof uniqueness is complete
Next to check the existence define 11990901199050 = 120585 and 1199090(119905) =120585(0) for 1199050 le 119905 le 119879 Let 1199091198961199050 = 120585 119896 = 1 2 and define Picardsequence
119909119896 (119905) = 120585 (0) + int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) 1199050 le 119905 le 119879(35)
Obviously 1199090(119905) isin M2((minusinfin119879]R119899) By induction we cansee that 119909119896(119905) isin M2((minusinfin119879]R119899)
In fact by elementary inequality (119886+119887+119888)2 le 3(1198862+1198872+1198882)10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3 1003816100381610038161003816120585 (0)10038161003816100381610038162 + 3 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119891 (119909119896minus1119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 3 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119892 (119909119896minus1119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
(36)
From the Holder inequality and Lemma 3 we have
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 3 (119905 minus 1199050)sdot 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896minus1119904 119904) minus 119891 (0 119904) + 119891 (0 119904)100381610038161003816100381610038162 119889119904+ 3119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896minus1119904 119904) minus 119892 (0 119904) + 119892 (0 119904)100381610038161003816100381610038162 119889119904(37)
Again the elementary inequality (119886 + 119887)2 le 21198862 + 21198872 (22)(15) and (16) imply that
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 3119864 100381610038161003816100381612058510038161003816100381610038162120583+ 3 (119905 minus 1199050 + 1) 119864int119905
1199050
(2119870 10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 3119864 100381610038161003816100381612058510038161003816100381610038162120583 + 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
119889119904+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
10038161003816100381610038161003816119909119896minus1119904 100381610038161003816100381610038162120583 119889119904 le 1198621+ 6119870 (119905 minus 1199050 + 1) 119864int119905
1199050
(100381610038161003816100381612058510038161003816100381610038162120583 + sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904le 1198621 + 1198622+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904
(38)
where 1198621 = 3119864|120585|2120583 + 6119870(119905 minus 1199050 + 1)(119905 minus 1199050) and 1198622 = 6119870(119905 minus1199050 + 1)(119905 minus 1199050)119864|120585|2120583
Hence for any ℓ ge 1 one can derive that
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162)le 1198621 + 1198622
+ 6119870 (119905 minus 1199050 + 1)int1199051199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896minus1 (119903)100381610038161003816100381610038162)119889119904(39)
Note that
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896minus1 (119904)100381610038161003816100381610038162 = max 119864 1003816100381610038161003816120585 (0)10038161003816100381610038162 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 le max 119864 100381610038161003816100381612058510038161003816100381610038162120583 119864 100381610038161003816100381610038161199091 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓminus1 (119904)100381610038161003816100381610038162 119864 10038161003816100381610038161003816119909ℓ (119904)100381610038161003816100381610038162 = max119864 100381610038161003816100381612058510038161003816100381610038162120583 max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162 le 119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162
(40)
Chinese Journal of Mathematics 7
and then
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162) le 1198621 + 1198622 + 6119870 (119905 minus 1199050 + 1)
sdot int1199051199050
(119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162))119889119904 le 1198623+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162)119889119904
(41)
where 1198623 = 1198621 + 21198622From the Gronwall inequality we have
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) (42)
Since 119896 is arbitrary
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) 1199050 le 119905 le 119879 119896 ge 1 (43)
From the Holder inequality Lemma 3 and (15) and (16) as ina similar earlier inequality one then has
119864 100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (1199090119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162 + 2119864 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119892 (1199090119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2 (119905 minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904
+ 2119864int1199051199050
10038161003816100381610038161003816119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904
le 2 (119905 minus 1199050 + 1) 119864int1199051199050
(2119870 100381610038161003816100381610038161199090119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050)
+ 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050) 119864 100381610038161003816100381612058510038161003816100381610038162120583
(44)
That is
119864( sup1199050le119904le119905
100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162)le 4 (119905 minus 1199050 + 1) (119905 minus 1199050) (119870 + 119870119864 100381610038161003816100381612058510038161003816100381610038162120583) fl 119877
(45)
By similar arguments as above we also have
119864 100381610038161003816100381610038161199092 (119905) minus 1199091 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (1199091119904 119904) minus 119891 (1199090119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (1199091119904 119904) minus 119892 (1199090119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904 le 2 (119905 minus 1199050+ 1) 119864int119905
1199050
[10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162
+ 10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162] 119889119904 le 2119870 (119905 minus 1199050+ 1) 119864int119905
1199050
100381610038161003816100381610038161199091119904 minus 1199090119904 100381610038161003816100381610038162120583 119889119904
(46)
Then
119864( sup1199050le119904le119905
100381610038161003816100381610038161199092 (119904) minus 1199091 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199091 (119903) minus 1199090 (119903)100381610038161003816100381610038162)119889119904le 2119877119870 (119905 minus 1199050 + 1) (119905 minus 1199050) = 119877119872(119905 minus 1199050)
(47)
where119872 = 2119870(119905 minus 1199050 + 1) Similarly
119864( sup1199050le119904le119905
100381610038161003816100381610038161199093 (119904) minus 1199092 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199092 (119903) minus 1199091 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877119872(119904 minus 1199050) 119889119904 le 119877 [119872 (119905 minus 1199050)]22
(48)
Continue this process to find that
119864( sup1199050le119904le119905
100381610038161003816100381610038161199094 (119904) minus 1199093 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199093 (119903) minus 1199092 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]22 119889119904 le 119877 [119872(119905 minus 1199050)]36
(49)
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Chinese Journal of Mathematics 7
and then
max1le119896leℓ
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896 (119904)100381610038161003816100381610038162) le 1198621 + 1198622 + 6119870 (119905 minus 1199050 + 1)
sdot int1199051199050
(119864 100381610038161003816100381612058510038161003816100381610038162120583 + max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162))119889119904 le 1198623+ 6119870 (119905 minus 1199050 + 1)int119905
1199050
max1le119896leℓ
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903)100381610038161003816100381610038162)119889119904
(41)
where 1198623 = 1198621 + 21198622From the Gronwall inequality we have
max1le119896leℓ
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) (42)
Since 119896 is arbitrary
119864 10038161003816100381610038161003816119909119896 (119905)100381610038161003816100381610038162 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) 1199050 le 119905 le 119879 119896 ge 1 (43)
From the Holder inequality Lemma 3 and (15) and (16) as ina similar earlier inequality one then has
119864 100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
119891 (1199090119904 119904) 1198891199041003816100381610038161003816100381610038161003816100381610038162 + 2119864 100381610038161003816100381610038161003816100381610038161003816int
119905
1199050
119892 (1199090119904 119904) 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2 (119905 minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904
+ 2119864int1199051199050
10038161003816100381610038161003816119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904
le 2 (119905 minus 1199050 + 1) 119864int1199051199050
(2119870 100381610038161003816100381610038161199090119904 100381610038161003816100381610038162120583 + 2119870)119889119904le 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050)
+ 4119870 (119905 minus 1199050 + 1) (119905 minus 1199050) 119864 100381610038161003816100381612058510038161003816100381610038162120583
(44)
That is
119864( sup1199050le119904le119905
100381610038161003816100381610038161199091 (119905) minus 1199090 (119905)100381610038161003816100381610038162)le 4 (119905 minus 1199050 + 1) (119905 minus 1199050) (119870 + 119870119864 100381610038161003816100381612058510038161003816100381610038162120583) fl 119877
(45)
By similar arguments as above we also have
119864 100381610038161003816100381610038161199092 (119905) minus 1199091 (119905)100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (1199091119904 119904) minus 119891 (1199090119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (1199091119904 119904) minus 119892 (1199090119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162 119889119904 le 2 (119905 minus 1199050+ 1) 119864int119905
1199050
[10038161003816100381610038161003816119891 (1199091119904 119904) minus 119891 (1199090119904 119904)100381610038161003816100381610038162
+ 10038161003816100381610038161003816119892 (1199091119904 119904) minus 119892 (1199090119904 119904)100381610038161003816100381610038162] 119889119904 le 2119870 (119905 minus 1199050+ 1) 119864int119905
1199050
100381610038161003816100381610038161199091119904 minus 1199090119904 100381610038161003816100381610038162120583 119889119904
(46)
Then
119864( sup1199050le119904le119905
100381610038161003816100381610038161199092 (119904) minus 1199091 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199091 (119903) minus 1199090 (119903)100381610038161003816100381610038162)119889119904le 2119877119870 (119905 minus 1199050 + 1) (119905 minus 1199050) = 119877119872(119905 minus 1199050)
(47)
where119872 = 2119870(119905 minus 1199050 + 1) Similarly
119864( sup1199050le119904le119905
100381610038161003816100381610038161199093 (119904) minus 1199092 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199092 (119903) minus 1199091 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877119872(119904 minus 1199050) 119889119904 le 119877 [119872 (119905 minus 1199050)]22
(48)
Continue this process to find that
119864( sup1199050le119904le119905
100381610038161003816100381610038161199094 (119904) minus 1199093 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119864( sup1199050le119903le119904
100381610038161003816100381610038161199093 (119903) minus 1199092 (119903)100381610038161003816100381610038162)119889119904
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]22 119889119904 le 119877 [119872(119905 minus 1199050)]36
(49)
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Chinese Journal of Mathematics
Now we claim that for any 119896 ge 0119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1 (119904) minus 119909119896 (119904)100381610038161003816100381610038162) le 119877 [119872(119905 minus 1199050)]119896119896 1199050 le 119905 le 119879
(50)
So for 119896 = 0 1 2 3 inequality (50) holds We suppose that(50) holds for some 119896 and check (50) for 119896 + 1 In fact
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 2119870 (119905 minus 1199050 + 1)int1199051199050
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+1119904 minus 119909119896119904 100381610038161003816100381610038162120583)119889119904
le 119872int1199051199050
( sup1199050le119903le119904
10038161003816100381610038161003816119909119896+1 (119903) minus 119909119896 (119903)100381610038161003816100381610038162)119889119904(51)
From (50)
119864( sup1199050le119904le119905
10038161003816100381610038161003816119909119896+2 (119904) minus 119909119896+1 (119904)100381610038161003816100381610038162)
le 119872int1199051199050
119877 [119872(119904 minus 1199050)]119896119896 119889119904 le 119877 [119872(119905 minus 1199050)]119896+1(119896 + 1) (52)
whichmeans that (50) holds for 119896+1Therefore by induction(50) holds for any 119896 ge 0
Next to verify 119909119896(119905) converge to 119909(119905) in 1198712 with 119909(119905) inM2((minusinfin119879]R119899) and 119909(119905) is the solution of (1) with initialdata 1199091199050 For (50) taking 119905 = 119879 then
119864( sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)100381610038161003816100381610038162) le 119877 [119872(119879 minus 1199050)]119896119896 (53)
By the Chebyshev inequality
119875 sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 gt 12119896
le 119877 [4119872(119879 minus 1199050)]119896119896 (54)
By using Alembertrsquos rule we show that sum+infin119896=0(119877[4119872(119879 minus1199050)]119896119896) convergeThat is sum+infin119896=0(119877[4119872(119879 minus 1199050)]119896119896) lt infin and by Borel-
Cantellirsquos lemma for almost all 120596 isin Ω there exists a positiveinteger 1198960 = 1198960(120596) such that
sup1199050le119905le119879
10038161003816100381610038161003816119909119896+1 (119905) minus 119909119896 (119905)10038161003816100381610038161003816 le 12119896 as 119896 ge 1198960 (55)
and then 119909119896(119905) is also a Cauchy sequence in 1198712 Hence119909119896(119905) converges uniformly and let 119909(119905) be its limit for any119905 isin (minusinfin119879] since 119909119896(119905) is continuous on 119905 isin (minusinfin119879] andF119905 adapted 119909(119905) is also continuous andF119905 adapted
So as 119896 rarr +infin 119909119896(119905) rarr 119909(119905) in 1198712 That is 119864|119909119896(119905) minus119909(119905)|2 rarr 0 as 119896 rarr infinThen from (43)
119864 |119909 (119905)|2 le 11986231198906119870(119879minus1199050+1)(119879minus1199050) forall1199050 le 119905 le 119879 (56)
and therefore
119864int119879minusinfin
|119909 (119904)|2 119889119904 = 119864int1199050minusinfin
|119909 (119904)|2 119889119904 + 119864int1198791199050
|119909 (119904)|2 119889119904le 119864int0minusinfin
1003816100381610038161003816120585 (119904)10038161003816100381610038162 119889119904+ int1198791199050
11986231198906119870(119879minus1199050+1)(119879minus1199050)119889119904 lt infin(57)
That is 119909(119905) isin M2((minusinfin119879]R119899)Now to show that 119909(119905) satisfies (1)119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 119889119904
+ int1199051199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162
le 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119891 (119909119896119904 119904) minus 119891 (119909119904 119904)] 1198891199041003816100381610038161003816100381610038161003816100381610038162
+ 2119864 100381610038161003816100381610038161003816100381610038161003816int119905
1199050
[119892 (119909119896119904 119904) minus 119892 (119909119904 119904)] 119889119882 (119904)1003816100381610038161003816100381610038161003816100381610038162 le 2 (119905
minus 1199050) 119864int1199051199050
10038161003816100381610038161003816119891 (119909119896119904 119904) minus 119891 (119909119904 119904)100381610038161003816100381610038162 119889119904+ 2119864int119905
1199050
10038161003816100381610038161003816119892 (119909119896119904 119904) minus 119892 (119909119904 119904)100381610038161003816100381610038162 119889119904le 119872119864int119905
1199050
10038161003816100381610038161003816119909119896119904 minus 119909119904100381610038161003816100381610038162120583 119889119904le 119872int119905
1199050
119864( sup1199050le119903le119904
10038161003816100381610038161003816119909119896 (119903) minus 119909 (119903)100381610038161003816100381610038162)119889119904le 119872int119879
1199050
119864 10038161003816100381610038161003816119909119896 (119904) minus 119909 (119904)100381610038161003816100381610038162 119889119904
(58)
Noting that the sequence 119909119896 rarr 119909(119905) means that for anygiven 120576 gt 0 there exists 1198960 such that 119896 ge 1198960 for any 119905 isin(minusinfin119879] one then deduces that
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 lt 120576int1198791199050
119864 10038161003816100381610038161003816119909119896 (119905) minus 119909 (119905)100381610038161003816100381610038162 119889119904 lt (119879 minus 1199050) 120576(59)
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
Submit your manuscripts athttpswwwhindawicom
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 201
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
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Chinese Journal of Mathematics 9
which means that for any 119905 isin [1199050 119879] one hasint1199051199050
119891 (119909119896119904 119904) 119889119904 997888rarr int1199051199050
119891 (119909119904 119904) 119889119904int1199051199050
119892 (119909119896119904 119904) 119889119882 (119904) 997888rarr int1199051199050
119892 (119909119904 119904) 119889119882 (119904) in 1198712(60)
For 1199050 le 119905 le 119879 taking limits on both sides of (35) we deducethat
lim119896rarrinfin
119909119896 (119905) = 120585 (0) + lim119896rarrinfin
int1199051199050
119891 (119909119896minus1119904 119904) 119889119904+ lim119896rarrinfin
int1199051199050
119892 (119909119896minus1119904 119904) 119889119882 (119904) (61)
and consequently
119909 (119905) = 120585 (0) + int1199051199050
119891 (119909119904 119904) 119889119904 + int1199051199050
119892 (119909119904 119904) 119889119882 (119904)1199050 le 119905 le 119879
(62)
Finally 119909(119905) is the solution of (1) and the demonstration ofexistence is complete
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] B D Coleman and V J Mizel ldquoOn the general theory of fadingmemoryrdquo Archive for Rational Mechanics and Analysis vol 29pp 18ndash31 1968
[2] J K Hale and J Kato ldquoPhase space for retarded equations withinfinite delayrdquo Funkcialaj Ekvacioj Serio Internacia vol 21 no1 pp 11ndash41 1978
[3] Y Hino T Naito N Van and J S Shin Almost PeriodicSolutions of Differential Equations in Banach Spaces vol 15 ofStability and Control Theory Methods and Applications Tayloramp Francis London UK 2000
[4] W Fengying and W Ke ldquoThe existence and uniqueness of thesolution for stochastic functional differential equations withinfinite delayrdquo Journal of Mathematical Analysis and Applica-tions vol 331 no 1 pp 516ndash531 2007
[5] S Resnick Adventures in Stochastic Processes Springer Sci-ence+Business Media New York NY USA 3rd edition 2002
Submit your manuscripts athttpswwwhindawicom
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 201
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
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpswwwhindawicom
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 201
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
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of