Research ArticleCoupled Fixed Point Theorems with Rational Type ContractiveCondition in a Partially Ordered 119866-Metric Space

K Chakrabarti

Department of Mathematics Scottish Church College 1 amp 3 Urquhart Square Kolkata 700 006 India

Correspondence should be addressed to K Chakrabarti kcmathscottishchurchacin

Received 29 May 2014 Accepted 15 September 2014 Published 29 September 2014

Academic Editor S T Ali

Copyright copy 2014 K Chakrabarti This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Coupled fixed point theorems for a map satisfying mixed monotone property and a nonlinear rational type contractive conditionare established in a partially ordered 119866-metric space The conditions for uniqueness of the coupled fixed point are discussed Wealso present results for the existence of coupled coincidence points of two maps

1 Introduction

The idea of weakening the contractive condition in a metricspace by introducing partial order in the space and consid-ering monotone functions satisfying contractive conditionswas first developed by Ran and Reurings [1] Later this wasextended by Bhaskar and Lakshmikantham [2] to prove acoupled fixed point theorem for functions satisfying mixedmonotone property Since then there has been considerableinterest in the development of coupled fixed point theoremsin partially orderedmetric spaceswith a variety of contractiveconditions [3ndash18]

Nonlinear contractive conditions were considered in [46 19] In particular a rational type contractive condition wasconsidered by Jaggi [19] in a complete metric space and thiswas extended to a partially ordered complete metric space byHarjani et al [6] to prove some fixed point theorems

Some coupled fixed point theorems in partially orderedcomplete119866-metric spaces were developed byChoudhury andMaity [8] and Saadati et al [9] The contractive conditionsused in [8] were extensions of that used by Bhaskar andLakshmikantham [2] into a 119866-metric space A new conceptof an Ω distance was introduced in [9]

In this paper we develop a coupled fixed point theoremusing a rational type nonlinear contractive condition in apartially ordered complete 119866-metric space The condition issimilar to the rational type contractive condition of Ciric et al[3] andmay be considered as a generalization of the condition

given in [3]We also find conditions for the uniqueness of thecoupled fixed point Finally we consider the conditions forexistence of coupled coincidence points

We begin by introducing the basic definitions and notionsused in the paper

2 General Preliminaries

Throughout this work⪯will denote a partial order relation onsome given set For any two elements 119909 119910 in some partiallyordered set endowed with the partial order relation ⪯ 119909 ⪯ 119910

and 119910 ⪰ 119909 are equivalent Also by 119909 ≺ 119910 we mean 119909 ⪯ 119910 and119909 = 119910

Definition 1 (see [20]) Let 119883 be a nonempty set and let 119866 119883 times 119883 times 119883 rarr R+ be a function satisfying the followingproperties

(1) 119866(119909 119910 119911) = 0 if 119909 = 119910 = 119911(2) 119866(119909 119909 119910) gt 0 for all 119909 119910 isin 119883 with 119909 = 119910(3) 119866(119909 119909 119910) le 119866(119909 119910 119911) for all 119909 119910 119911 isin 119883 with 119911 = 119910(4) 119866(119909 119910 119911) = 119866(119909 119911 119910) = 119866(119910 119911 119909) = sdot sdot sdot (symmetry

in all three variables)(5) 119866(119909 119910 119911) le 119866(119909 119886 119886)+119866(119886 119910 119911) for all 119909 119910 119911 119886 isin 119883

(rectangle inequality)Then 119866 is called a generalized metric or more specifically a119866-metric on119883 and the pair (119883 119866) is called a 119866-metric space

Hindawi Publishing CorporationJournal of MathematicsVolume 2014 Article ID 785357 7 pageshttpdxdoiorg1011552014785357

2 Journal of Mathematics

Theorem 2 (see [20]) Let (119883 119866) be a119866-metric space then forany 119909 119910 119911 119886 isin 119883 it follows that

(1) if 119866(119909 119910 119911) = 0 then 119909 = 119910 = 119911(2) 119866(119909 119910 119911) le 119866(119909 119909 119910) + 119866(119909 119909 119911)(3) 119866(119909 119910 119910) le 2119866(119910 119909 119909)(4) 119866(119909 119910 119911) le 119866(119909 119886 119911) + 119866(119886 119910 119911)

Definition 3 (see [20]) Let (119883 119866) be a 119866-metric space Thesequence 119909

119899 sube 119883 is 119866-convergent to 119909 if for any arbitrary

120598 gt 0 there is a positive integer 119873 such that 119866(119909 119909119899 119909119899) lt 120598

for 119899 ge 119873 that is if lim119899rarrinfin

119866(119909 119909119899 119909119899) = 0

Theorem 4 (see [20]) Let (119883 119866) be a 119866-metric space thenfor a sequence 119909

119899 sube 119883 and a point 119909 isin 119883 the following are

equivalent

(1) 119909119899is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

(4) 119866(119909119898 119909119899 119909) rarr 0 as119898 119899 rarr infin

Theorem 5 (see [20]) Let (119883 119866) be a 119866-metric space Thenthe function 119866(119909 119910 119911) is jointly continuous in all three of itsvariables

Remark 6 This means if 119909119899 119910119898 and 119911

119897 are sequences

in 119883 such that lim119899rarrinfin

119909119899

= 119909 lim119898rarrinfin

119910119898

= 119910 andlim119897rarrinfin

119911119897= 119911 then 119866(119909

119899 119910119898 119911119897) rarr 119866(119909 119910 119911) as 119897 119898 119899 rarr

infin

Definition 7 (see [20]) Let (119883 119866) be a 119866-metric space Thensequence 119909

119899 sube 119883 is said to be 119866-Cauchy if for every 120598 gt 0

there exists a positive integer 119873 such that 119866(119909119899 119909119898 119909119897) lt 120598

for all 119899119898 119897 ge 119873

Theorem8 (see [20]) In a119866-metric space (119883 119866) the followingare equivalent

(1) the sequence 119909119899is 119866-Cauchy

(2) for every 120598 gt 0 there exists a positive integer 119873 suchthat 119866(119909

119899 119909119898 119909119898) lt 120598 for all 119899119898 gt 119873

Definition 9 (see [20]) A 119866-metric space (119883 119866) is said tobe 119866-complete if and only if every 119866-Cauchy sequence is119866-convergent in (119883 119866)

Definition 10 (see [2]) Let (119883 ⪯) be a partially ordered setand 119879 119883 times 119883 rarr 119883 Then 119879 is said to have mixed mono-tone property if 119879(119909 119910) is monotone nondecreasing in 119909 andmonotone nonincreasing in 119910 That is for all 119909 119910 isin 119883

1199091 1199092isin 119883 119909

1⪯ 1199092997904rArr 119879 (119909

1 119910) ⪯ 119879 (119909

2 119910)

1199101 1199102isin 119883 119910

1⪯ 1199102997904rArr 119879 (119909 119910

1) ⪰ 119879 (119909 119910

2)

(1)

Definition 11 (see [4]) Let (119883 ⪯) be a partially ordered set and119879 119883 times119883 rarr 119883 and 119892 119883 rarr 119883 We say 119879 has the mixed 119892-monotone property if 119879 is monotone 119892-nondecreasing in its

first argument and ismonotone119892-nonincreasing in its secondargument That is for all 119909 119910 isin 119883

1199091 1199092isin 119883 119892119909

1⪯ 1198921199092997904rArr 119879 (119909

1 119910) ⪯ 119879 (119909

2 119910)

1199101 1199102isin 119883 119892119910

1⪯ 1198921199102997904rArr 119879 (119909 119910

1) ⪰ 119879 (119909 119910

2)

(2)

Definition 12 (see [2]) An element (119909 119910) isin 119883 times 119883 is called acoupled fixed point of a map 119879 119883 times 119883 rarr 119883 if 119879(119909 119910) = 119909

and 119879(119910 119909) = 119910

Definition 13 (see [4]) An element (119909 119910) isin 119883 times 119883 is called acoupled coincidence point of the maps 119879 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if 119879(119909 119910) = 119892119909 and 119879(119910 119909) = 119892119910

Definition 14 (see [4]) The maps 119879 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 are said to be commutative if 119892(119879(119909 119910)) =

119879(119892119909 119892119910)

Definition 15 For a map 119879 119883 times 119883 rarr 119883 by 1198792(119909 119910) wemean 119879(119879(119909 119910) 119879(119910 119909)) Similarly we define 119879119899(119909 119910) 119899 ge 3

3 Main Results

Our main results are presented in this section We firstdevelop a rational type contractive condition on a partiallyordered complete 119866-metric space and give a coupled fixedpoint theorem for a map satisfying this condition

We start with a partially ordered set (119883 ⪯) and supposethat there is a 119866-metric on 119883 so that (119883 119866) is a complete119866-metric space We induce partial ordering on 119883 times 119883 bydemanding that for any (119909 119910) (119906 V) isin 119883 times 119883 (119909 119910) ⪯

(119906 V) hArr 119909 ⪯ 119906 119910 ⪰ V

Theorem16 Let (119883 ⪯) be a partially ordered set and let119866 be ageneralizedmetric on119883 such that (119883 119866) is a complete119866-metricspace Suppose 119879 119883times119883 rarr 119883 is a continuous mapping on119883having the mixed monotone property Suppose also that for all(119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911)

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120572 (119866 (119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119908 119879 (119908 119911) 119879 (119908 119911)) times ([119866 (119909 119906 119908)]2

)minus1

)

+ 120573119866 (119909 119906 119908)

(3)

where 8120572 + 120573 lt 1 If there exists 1199090 1199100

isin 119883 such that1199090⪯ 119879(119909

0 1199100) and 119910

0⪰ 119879(119910

0 1199090) then 119879 has a coupled

fixed point (119909lowast 119910lowast) isin 119883 times 119883 That is (119909lowast 119910lowast) satisfies 119909lowast =119879(119909lowast 119910lowast) 119910lowast= 119879(119910

lowast 119909lowast)

Proof Suppose that there exists 1199090 1199100isin 119883 such that 119909

0⪯

119879(1199090 1199100) and 119910

0⪰ 119879(119910

0 1199090) We write 119909

1= 119879(119909

0 1199100)

1199101= 119879(119910

0 1199090) and define 119909

119899+1= 119879(119909

119899 119910119899) 119910119899+1

= 119879(119910119899 119909119899)

Journal of Mathematics 3

119899 ge 1 From the conditions of the theorem and the mixedmonotone property it easily follows that

1199091= 119879 (119909

0 1199100) ⪰ 1199090

1199092= 119879 (119909

1 1199101) ⪰ 119879 (119909

0 1199100) = 1199091

(4)

This gives

1199090⪯ 1199091⪯ 1199092⪯ sdot sdot sdot ⪯ 119909

119899⪯ 119909119899+1

⪯ sdot sdot sdot (5)

Similarly proceeding with

1199101= 119879 (119910

0 1199090) ⪯ 1199100

1199102= 119879 (119910

1 1199091) ⪯ 119879 (119910

0 1199090) = 1199101

(6)

we find

1199100⪰ 1199101⪰ 1199102⪰ sdot sdot sdot ⪰ 119910

119899⪰ 119910119899+1

⪰ sdot sdot sdot (7)

Considering the sequence 119909119899 and using (3) we have

119866 (119909119899+1

119909119899 119909119899)

= 119866 (119879 (119909119899 119910119899) 119879 (119909

119899minus1 119910119899minus1

) 119879 (119909119899minus1

119910119899minus1

))

le 120572119866 (119909119899 119909119899+1

119909119899+1

) 119866 (119909119899minus1

119909119899 119909119899) 119866 (119909

119899minus1 119909119899 119909119899)

[119866 (119909119899 119909119899minus1

119909119899minus1

)]2

+ 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

(8)

Now using inequality (3) from Theorem 2 119866(119909 119910 119910) le

2119866(119910 119909 119909) Setting 119909 = 119909119899minus1

and 119910 = 119909119899in this we find

119866 (119909119899minus1

119909119899 119909119899) le 2119866 (119909

119899 119909119899minus1

119909119899minus1

) (9)

Using this (8) becomes

119866 (119909119899+1

119909119899 119909119899)

le 4120572119866 (119909119899 119909119899+1

119909119899+1

) 119866 (119909119899minus1

119909119899 119909119899) 119866 (119909

119899minus1 119909119899 119909119899)

[119866 (119909119899minus1

119909119899 119909119899)]2

+ 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

= 4120572119866 (119909119899 119909119899+1

119909119899+1

) + 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

le 8120572119866 (119909119899+1

119909119899 119909119899) + 120573119866 (119909

119899 119909119899minus1

119909119899minus1

)

(10)

where in the last step we have used inequality (3) fromTheorem 2 Rearranging and simplifying this we get

119866 (119909119899+1

119909119899 119909119899) le (

120573

1 minus 8120572)119866 (119909

119899 119909119899minus1

119909119899minus1

) (11)

Evidently the condition that (11) is contractive is 120573(1minus8120572) lt1 that is

8120572 + 120573 lt 1 (12)

Similarly considering 119866(119910119899+1

119910119899 119910119899) = 119866(119879(119910

119899 119909119899) 119879(119910

119899minus1

119909119899minus1

) 119879(119910119899minus1

119909119899minus1

)) and arguing as above we find

119866 (119910119899+1

119910119899 119910119899) le (

120573

1 minus 8120572)119866 (119910

119899 119910119899minus1

119910119899minus1

) (13)

With condition (12) we find (13) is contractiveLet 119889119899= 119866(119909

119899+1 119909119899 119909119899) and 119896 = (120573(1 minus 8120572)) Using (11)

we get

119889119899le 119896119889119899minus1

le 1198962

119889119899minus2

le sdot sdot sdot le 119896119899

1198890 (14)

If 1198890= 0 then 119866(119879(119909

0 1199100) 1199090 1199090) = 119866(119909

1 1199090 1199090) = 119889

0=

0 But this means 119879(1199090 1199100) = 119909

0 Similarly writing 120575

119899=

119866(119910119899+1

119910119899 119910119899) we find

120575119899le 119896120575119899minus1

le 1198962

120575119899minus2

le sdot sdot sdot le 119896119899

1205750 (15)

If 1205750= 0 in addition to 119889

0= 0 we deduce similarly that

119879(1199100 1199090) = 1199100 So if 119889

0= 1205750= 0 (119909

0 1199100) is a coupled fixed

pointHowever if 119889

0gt 0 for 119898 gt 119899 using inequality (5) of

Definition 1 we get

119866 (119909119898 119909119899 119909119899) le 119866 (119909

119899+1 119909119899 119909119899) + 119866 (119909

119898 119909119899+1

119909119899+1

)

le 119866 (119909119899+1

119909119899 119909119899) + 119866 (119909

119899+2 119909119899+1

119909119899+1

)

+ 119866 (119909119898 119909119899+2

119909119899+2

)

le 119866 (119909119899+1

119909119899 119909119899) + 119866 (119909

119899+2 119909119899+1

119909119899+1

)

+ 119866 (119909119899+3

119909119899+2

119909119899+2

) + 119866 (119909119898 119909119898minus1

119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ 119896119899+2

+ sdot sdot sdot + 119896119898minus1

] 1198890

= 119896119899

[1 + 119896 + 1198962

+ sdot sdot sdot + 119896119898minus119899minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin since 119896 lt 1

(16)

So 119909119899 is a 119866-Cauchy sequence in 119883 Next considering

119866(119910119898 119910119899 119910119899) and arguing as above we can show that 119910

119899 is

also a 119866-Cauchy sequence in 119883 119866-Completeness of (119883 119866)now implies that there are points 119909lowast 119910lowast isin 119883 such that 119909

119899rarr

119909lowast and 119910

119899rarr 119910lowast as 119899 rarr infin

We next show that (119909lowast 119910lowast) is a coupled fixed point of 119879Using the fact that 119879 is continuous on 119883 and 119866 as a metric iscontinuous in each of its variables we have

119866 (119879 (119909lowast

119910lowast

) 119909lowast

119909lowast

) = 119866 ( lim119899rarrinfin

119879 (119909119899 119910119899) 119909lowast

119909lowast

)

= 119866( lim119899rarrinfin

119909119899+1

119909lowast

119909lowast

)

= lim119899rarrinfin

119866 (119909119899+1

119909lowast

119909lowast

)

= 0 since 119909119899997888rarr 119909lowast as 119899 997888rarr infin

(17)

4 Journal of Mathematics

But this means 119879(119909lowast 119910lowast) = 119909lowast Similarly by considering

119866(119879(119910lowast 119909lowast) 119910lowast 119910lowast) and repeating the arguments used to

derive (17) we can show that 119879(119910lowast 119909lowast) = 119910lowast This proves

(119909lowast 119910lowast) is a coupled fixed point of 119879

The W map was introduced by Chen [21] and a subclassofW functions was defined by Chakrabarti in [22]

Definition 17 (see [22]) We call 120593 R+ rarr R+ a function ofclassW

120573if there is a 120573 such that 0 lt 120573 lt 1 and the following

conditions are satisfied

(1) 120593(119905) le 120573119905 for all 119905 gt 0 and 120593(0) = 0

(2) lim119905119899rarr119905

inf 120593(119905119899) le 120573119905 for all 119905 gt 0

Using Definition 17 we obtain the following as a general-ization of Theorem 16

Theorem 18 Let (119883 ⪯) be a partially ordered set and let 119866be a metric on119883 such that (119883 119866) is a complete119866-metric spaceSuppose119879 119883times119883 rarr 119883 is a continuous mapping on119883 havingthe mixed monotone property For some given 0 lt 120572 120573 lt 1 let120601 isin W

120572and 120595 isin W

120573where 8120572 + 120573 lt 1 Suppose also that for

all (119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911)

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120601 (119866 (119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119909 119906 119908)]2

)minus1

)

+ 120595 (119866 (119909 119906 119908))

(18)

If there exists 1199090 1199100isin 119883 such that 119909

0⪯ 119879(119909

0 1199100) and 119910

0⪰

119879(1199100 1199090) then 119879 has a coupled fixed point (119909lowast 119910lowast) isin 119883 times 119883

That is (119909lowast 119910lowast) satisfies 119909lowast = 119879(119909lowast 119910lowast) 119910lowast= 119879(119910

lowast 119909lowast)

Proof Since 120601 isin W120572and 120595 isin W

120573 it follows from

Definition 17 that 120601(119905) le 120572119905 and 120595(119905) le 120573119905 for all 119905 gt 0Inequality (18) now becomes equivalent to inequality (3) ofTheorem 16 and the proof is immediate

Example 19 Let 119883 = [0infin) and consider the function 119866

119883 times 119883 times 119883 rarr R+ defined by

119866 (119909 119910 119911) = 0 if 119909 = 119910 = 119911

max 119909 119910 119911 otherwise(19)

Then (119883 119866) is a complete 119866-metric space [23] We define apartial order ⪯ on119883 by the following for any 119909 119910 isin 119883 119909 ⪯ 119910

if 119909 ge 119910 Also let 119865 119883 times 119883 rarr 119883 be defined by

119865 (119909 119910) = 1 if 119909 ⪯ 119910

0 otherwise(20)

Suppose 119909 119910 119906 V 119908 119911 isin 119883 satisfy 119911 ⪯ V ⪯ 119910 ⪯ 119909 ⪯ 119906 ⪯ 119908

with nonzero 119909 119906 119908 but are otherwise arbitrary Then we

have 119908 le 119906 le 119909 le 119910 le V le 119911 So the left side of (3) is119866(0 0 0) = 0 The right side of (3)

= 120572119866 (119909 0 0) 119866 (119906 0 0) 119866 (119908 0 0)

[119866 (119909 119906 119908)]2

+ 120573119866 (119909 119906 119908)

= 120572119909119906119908

1199092+ 120573119908

=120572119906119908 + 120573119909119908

119909gt 0 with 120572 =

1

16 120573 =

1

3

(21)

If 1199090= 0 and 119910

0= 1 then 119909

0⪯ 119865(119909

0 1199100) and 119910

0⪰ 119865(119910

0 1199090)

So all conditions of Theorem 16 are satisfied Easily we findthat (0 1) is a coupled fixed point of 119865 Similarly (1 0) is acoupled fixed point

In the next theorem we provide conditions underwhich the coupled fixed point of the map 119879 established inTheorem 16 is unique

Theorem 20 Suppose that the conditions of Theorem 16 arevalid In addition suppose that for each (119909 119910) (119906 V) isin 119883 times 119883

there is a (119908 119911) isin 119883 times 119883 which is comparable to (119909 119910) and(119906 V) Then 119879 has a unique coupled fixed point

Proof Suppose that (119909lowast 119910lowast) (1199091015840 1199101015840) isin 119883 times 119883 are coupledfixed points

Case 1 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are comparable

119866(119879 (119909lowast

119910lowast

) 119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))

le 120572 (119866 (119909lowast

119879 (119909lowast

119910lowast

) 119879 (119909lowast

119910lowast

))

times [119866 (1199091015840

119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))]2

times ([119866 (119909lowast

1199091015840

1199091015840

)]2

)

minus1

)

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120572

119866 (119909lowast 119909lowast 119909lowast) [119866 (119909

1015840 1199091015840 1199091015840)]2

[119866 (119909lowast 1199091015840 1199091015840)]2

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120573119866 (119909lowast

1199091015840

1199091015840

)

(22)

This is equivalent to 119866(119909lowast 1199091015840 1199091015840) le 120573119866(119909lowast 1199091015840 1199091015840) However

this is a contradiction since 120573 lt 1 So we must have 119909lowast = 1199091015840

Similarly considering 119866(119879(119910lowast 119909lowast) 119879(119910

1015840 1199091015840) 119879(119910

1015840 1199091015840)) we

easily show that 119910lowast = 1199101015840 This shows that (119909lowast 119910lowast) = (119909

1015840 1199101015840)

so the coupled fixed point is unique

Case 2 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are not comparable by the

condition of the theorem there is a (119906 V) isin 119883times119883 comparable

Journal of Mathematics 5

to (119909lowast 119910lowast) and (119909

1015840 1199101015840) If there is a positive integer 119899

0such

that 1198791198990(119906 V) = (119909lowast 119910lowast) then

1198791198990 (119906 V) = (119909

lowast

119910lowast

)

1198791198990+1

(119906 V) = 119879 (119909lowast

119910lowast

) = 119909lowast

1198791198990+2

(119906 V) = 1198792

(119909lowast

119910lowast

) = 119879 (119879 (119909lowast

119910lowast

) 119879 (119910lowast

119909lowast

))

= 119879 (119909lowast

119910lowast

) = 119909lowast

(23)

So 119879119899(119906 V) = 119909

lowast for 119899 ge 1198990and hence 119879119899(119906 V) rarr 119909

lowast as119899 rarr infin

On the other hand if no such 1198990exists we have that for

any 119899 ge 1

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

= 119866 (119879119899

(119906 V) 119879119899 (119909lowast 119910lowast) 119879119899 (119909lowast 119910lowast))

le 120572 (119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V))

times [119866 (119879119899minus1

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

))]2

times ( [119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast)

119879119899minus1

(119909lowast

119910lowast

))]2

)

minus1

)

+ 120573119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast) 119879119899minus1 (119909lowast 119910lowast)) (24)

where we have used the fact that 119879119899(119909 119910) = 119879(119879119899minus1

(119909

119910) 119879119899minus1

(119910 119909)) for any (119909 119910) isin 119883 times 119883 Since (119909lowast 119910lowast) is acoupled fixed point of 119879 119879119899(119909lowast 119910lowast) = 119909

lowast for all 119899 ge 1 andfrom (24) we now deduce that

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

le 120572

119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V)) [119866 (119909lowast 119909lowast 119909lowast)]2

[119866 (119879119899minus1 (119906 V) 119909lowast 119909lowast)]2

+ 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119899minus1

119866 (119879 (119906 V) 119909lowast 119909lowast) 997888rarr 0 as 119899 997888rarr infin

(25)

since 120573 lt 1This proves119879119899(119906 V) rarr 119909lowast as 119899 rarr infin Similarly

we can show that 119879119899(V 119906) rarr 119910lowast as 119899 rarr infin Replacing 119909lowast

with 1199091015840 and 119910lowast with 1199101015840 and repeating the above arguments wecan deduce 119879119899(119906 V) rarr 119909

1015840 and 119879119899(V 119906) rarr 119910

1015840 as 119899 rarr infinBut thismeans 119909lowast = 119909

1015840 and119910lowast = 1199101015840 So the (119909lowast 119910lowast) = (119909

1015840 1199101015840)

and the coupled fixed point is unique

We next establish the conditions under which two maps119879 119883times119883 rarr 119883 and 119892 119883 rarr 119883 have a coupled coincidencepoint

Theorem 21 Let (119883 ⪯) be a partially ordered set and let 119866 bea metric on 119883 such that (119883 119866) is a complete 119866-metric spaceLet 119879 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 be continuous map-ping on 119883 such that 119879 has the mixed 119892-monotone propertySuppose that 119879(119883 times 119883) sube 119892(119883) 119892 commutes with 119879 and for(119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911) and119892119909 ⪯ 119892119906 ⪯ 119892119908 119892119910 ⪰ 119892V ⪰ 119892119911

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120572 (119866 (119892119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119892119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119892119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119892119909 119892119906 119892119908)]2

)minus1

)

+ 120573119866 (119892119909 119892119906 119892119908)

(26)

where 8120572 + 120573 lt 1 If there exists 1199090 1199100

isin 119883 such that1198921199090⪯ 119879(119909

0 1199100) and 119892119910

0⪰ 119879(119910

0 1199090) then 119879 has a coupled

coincidence point (119909lowast 119910lowast) isin 119883 times 119883 That is (119909lowast 119910lowast) satisfies119892119909lowast= 119879(119909

lowast 119910lowast) 119892119910lowast= 119879(119910

lowast 119909lowast)

Proof Since119879(119883times119883) sube 119892(119883)we can choose 1199091 1199101isin 119883 such

that 1198921199091= 119879(119909

0 1199100) 1198921199101= 119879(119910

0 1199090) For similar reasons

1199092 1199102isin 119883 can be found such that 119892119909

2= 119879(119909

1 1199101) 1198921199102=

119879(1199101 1199091) Due to the mixed 119892-monotone property of 119879 we

have 1198921199090⪯ 1198921199091⪯ 1198921199092and 119892119910

2⪯ 1198921199101⪯ 1198921199100 In general it

can be shown that [4] for 119899 ge 0119892119909119899= 119879 (119909

119899minus1 119910119899minus1

) ⪯ 119892119909119899+1

= 119879 (119909119899 119910119899)

119892119910119899+1

= 119879 (119910119899 119909119899) ⪯ 119892119910

119899= 119879 (119909

119899minus1 119910119899minus1

)

(27)

Now by the same arguments used to deduce (10) we have

119866 (119892119909119899+1

119892119909119899 119892119909119899)

le 4120572119866 (119892119909

119899 119892119909119899+1

119892119909119899+1

) [119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

[119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

= 4120572119866 (119892119909119899 119892119909119899+1

119892119909119899+1

) + 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

le 8120572119866 (119892119909119899+1

119892119909119899 119892119909119899)

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

(28)

This gives

119866 (119892119909119899+1

119892119909119899 119892119909119899) le (

120573

1 minus 8120572)119866 (119892119909

119899 119892119909119899minus1

119909119899minus1

)

(29)

Since 8120572 + 120573 lt 1 119896 = 120573(1 minus 8120572) lt 1 Writing 119889119899=

119866(119892119909119899+1

119892119909119899 119892119909119899) we find from (28) that

119889119899le 119896119889119899minus1

le 1198962

119889119899minus1

le sdot sdot sdot le 119896119899

1198890 (30)

6 Journal of Mathematics

If 1198890= 0 then119866(119892119909

1 1198921199090 1198921199090) = 119866(119879(119909

0 1199100) 1198921199090 1198921199090) = 0

giving 1198921199090= 119879(119909

0 1199100) so 119909

0is a coincidence point However

if 1198890gt 0 we have for119898 gt 119899

119866 (119892119909119898 119892119909119899 119892119909119899)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119898 119892119909119899+1

119892119909119899+1

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ 119866 (119892119909119898 119892119909119899+2

119892119909119899+2

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ sdot sdot sdot + 119866 (119892119909119898 119892119909119898minus1

119892119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ sdot sdot sdot + 119896119898minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin

(31)

since 119896 lt 1 This shows that 119892119909119899 is a Cauchy sequence in

(119883 119866) and 119866-completeness of (119883 119866) ensures a point 119909lowast isin 119883

such that 119892119909119899rarr 119909lowast as 119899 rarr infin

Replacing 119909119899by 119910119899for all 119899 ge 0 we get the analogue of

(29)

119866 (119892119910119899+1

119892119910119899 119892119910119899) le (

120573

1 minus 8120572)119866 (119892119910

119899 119892119910119899minus1

119910119899minus1

) (32)

Next writing 120575119899= 119866(119892119910

119899+1 119892119910119899 119892119910119899) we find as in (30) that

120575119899le 119896120575119899minus1

le 1198962

120575119899minus1

le sdot sdot sdot le 119896119899

1205750 (33)

As before if 1205750= 0 119866(119892119910

1 1198921199100 1198921199100) = 119866(119879(119910

0 1199090) 1198921199100

1198921199100) = 0 giving 119879(119910

0 1199090) = 119892119910

0so that 119910

0is a coinci-dence

point If in addition 1198890= 0 we have 119892119909

0= 119879(119909

0 1199100) and

1198921199100= 119879(119910

0 1199090) So (119909

0 1199100) is a coupled coincidence point

Proceeding as in (31) we can show further that 119892119910119899 is

a Cauchy sequence in (119883 119866) and due to 119866-completeness of(119883 119866) there is a point 119910lowast isin 119883 such that 119892119910

119899rarr 119910lowast as 119899 rarr

infinFinally we prove that (119909lowast 119910lowast) is a coupled coincident

point Since 119879 and 119892 commute we have

119892 (119892119909119899+1

) = 119892 (119879 (119909119899 119910119899)) = 119879 (119892119909

119899 119892119910119899)

119892 (119892119910119899+1

) = 119892 (119879 (119910119899 119909119899)) = 119879 (119892119910

119899 119892119909119899)

(34)

Taking limits as 119899 rarr infin in (34) and noting that 119879 and 119892 arerespectively continuous on119883 times 119883 and119883 we get

119892119909lowast

= lim119892 (119892119909119899+1

) = lim119892 (119879 (119909119899 119910119899)) = lim119879 (119892119909

119899 119892119910119899)

119892119910lowast

= lim119892 (119892119910119899+1

) = lim119892 (119879 (119910119899 119909119899)) = lim119879 (119892119910

119899 119892119909119899)

(35)

Next we observe that 119866 as metric is continuous in all itsvariables This finally leads to

119866 (119879 (119909lowast

119910lowast

) 119892119909lowast

119892119909lowast

) = 119866 (lim119879 (119892119909119899 119892119910119899) 119892119909lowast

119892119909lowast

)

= 119866 (119892119909lowast

119892119909lowast

119892119909lowast

)

= 0

(36)

So 119892119909lowast = 119879(119909lowast 119910lowast) Similarly we show that 119892119910lowast = 119879(119910

lowast 119909lowast)

This proves (119909lowast 119910lowast) is a coupled coincidence point

4 Conclusion

To summarize we have introduced a rational type contractivecondition in a 119866-metric space and proved some coupledfixed point theorems for maps satisfying mixed monotoneproperty We established the conditions for uniqueness of thecoupled fixed point Conditions for the existence of coupledcoincidence points of two maps are also deduced

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[2] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006

[3] L Ciric M O Olatinwo D Gopal and G Akinbo ldquoCoupledfixed point theorems for mappings satisfying a contractivecondition of rational type on a partially ordered metric spacerdquoAdvances in Fixed Point Theory vol 2 no 1 pp 1ndash8 2012

[4] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[5] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010

[6] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[7] V Berinde ldquoGeneralized coupled fixed point theorems formixed monotone mappings in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no18 pp 7347ndash7355 2011

[8] B S Choudhury and P Maity ldquoCoupled fixed point resultsin generalized metric spacesrdquo Mathematical and ComputerModelling vol 54 no 1-2 pp 73ndash79 2011

Journal of Mathematics 7

[9] R Saadati S M Vaezpour P Vetro and B E RhoadesldquoFixed point theorems in generalized partially ordered 119866-met-ric spacesrdquo Mathematical and Computer Modelling vol 52 no5-6 pp 797ndash801 2010

[10] Z Mustafa J R Roshan and V Parvaneh ldquoCoupled coinci-dence point results for (120595 120593)-weakly contractive mappings inpartially ordered 119866

119887-metric spacesrdquo Fixed Point Theory and

Applications vol 2013 article 206 2013[11] ZMustafa J R Roshan andV Parvaneh ldquoExistence of a tripled

coincidence point in ordered G119887-metric spaces and applications

to a system of integral equationsrdquo Journal of Inequalities andApplications vol 2013 article no 453 2013

[12] V ParvanehA Razani and J R Roshan ldquoCommonfixed pointsof six mappings in partially ordered 119866-metric spacesrdquo Math-ematical Sciences vol 7 article 18 2013

[13] A Razani and V Parvaneh ldquoOn generalized weakly G-contractive mappings in partially ordered G-metric spacesrdquoAbstract and Applied Analysis vol 2012 Article ID 701910 18pages 2012

[14] Z Mustafa V Parvaneh M Abbas and J Rezaei RoshanldquoSome coincidence point results for generalized (120595 120593)-weaklycontractive mappings in ordered G-metric spaces Fixed PointTheory and Applicationsrdquo Fixed Point Theory and Applicationsvol 2013 article 326 2013

[15] M A Kutbi N Hussain J R Roshan and V ParvanehldquoCoupled and tripled coincidence point results with applicationto Fredholm integral equationsrdquo Abstract and Applied Analysisvol 2014 Article ID 568718 18 pages 2014

[16] N Hussain V Parvaneh and J R Roshan ldquoFixed point resultsfor G-120572-contractive maps with application to boundary valueproblemsrdquo ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[17] A Latif N Hussain J R Roshan and V Parvaneh ldquoA uni-fication of Gminusmetric partial metric and b-metric spacesrdquoAbstract andApplied Analysis vol 2014 Article ID 180698 2014

[18] L Ciric S M Alsulami V Parvaneh and J R Roshan ldquoSomefixed point results in ordered 119866

119901-metric spacesrdquo Fixed Point

Theory and Applications vol 2013 article 317 2013[19] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journal

of Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977[20] Z Mustafa and B Sims ldquoA new approach to generalized metric

spacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash296 2006

[21] C-M Chen ldquoCommon fixed-point theorems in completegeneralized metric spacesrdquo Journal of Applied Mathematics vol2012 Article ID 945915 14 pages 2012

[22] K Chakrabarti ldquoFixed point theorems in G-metric spaces withW mapsrdquoMathematical Sciences Letters vol 2 pp 29ndash35 2013

[23] E Karapınar and R P Agarwal ldquoFurther fixed point results on119866-metric spacesrdquo Fixed PointTheory andApplications vol 2013article 154 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Mathematics

Theorem 2 (see [20]) Let (119883 119866) be a119866-metric space then forany 119909 119910 119911 119886 isin 119883 it follows that

(1) if 119866(119909 119910 119911) = 0 then 119909 = 119910 = 119911(2) 119866(119909 119910 119911) le 119866(119909 119909 119910) + 119866(119909 119909 119911)(3) 119866(119909 119910 119910) le 2119866(119910 119909 119909)(4) 119866(119909 119910 119911) le 119866(119909 119886 119911) + 119866(119886 119910 119911)

Definition 3 (see [20]) Let (119883 119866) be a 119866-metric space Thesequence 119909

119899 sube 119883 is 119866-convergent to 119909 if for any arbitrary

120598 gt 0 there is a positive integer 119873 such that 119866(119909 119909119899 119909119899) lt 120598

for 119899 ge 119873 that is if lim119899rarrinfin

119866(119909 119909119899 119909119899) = 0

Theorem 4 (see [20]) Let (119883 119866) be a 119866-metric space thenfor a sequence 119909

119899 sube 119883 and a point 119909 isin 119883 the following are

equivalent

(1) 119909119899is 119866-convergent to 119909

(2) 119866(119909119899 119909119899 119909) rarr 0 as 119899 rarr infin

(3) 119866(119909119899 119909 119909) rarr 0 as 119899 rarr infin

(4) 119866(119909119898 119909119899 119909) rarr 0 as119898 119899 rarr infin

Theorem 5 (see [20]) Let (119883 119866) be a 119866-metric space Thenthe function 119866(119909 119910 119911) is jointly continuous in all three of itsvariables

Remark 6 This means if 119909119899 119910119898 and 119911

119897 are sequences

in 119883 such that lim119899rarrinfin

119909119899

= 119909 lim119898rarrinfin

119910119898

= 119910 andlim119897rarrinfin

119911119897= 119911 then 119866(119909

119899 119910119898 119911119897) rarr 119866(119909 119910 119911) as 119897 119898 119899 rarr

infin

Definition 7 (see [20]) Let (119883 119866) be a 119866-metric space Thensequence 119909

119899 sube 119883 is said to be 119866-Cauchy if for every 120598 gt 0

there exists a positive integer 119873 such that 119866(119909119899 119909119898 119909119897) lt 120598

for all 119899119898 119897 ge 119873

Theorem8 (see [20]) In a119866-metric space (119883 119866) the followingare equivalent

(1) the sequence 119909119899is 119866-Cauchy

(2) for every 120598 gt 0 there exists a positive integer 119873 suchthat 119866(119909

119899 119909119898 119909119898) lt 120598 for all 119899119898 gt 119873

Definition 9 (see [20]) A 119866-metric space (119883 119866) is said tobe 119866-complete if and only if every 119866-Cauchy sequence is119866-convergent in (119883 119866)

Definition 10 (see [2]) Let (119883 ⪯) be a partially ordered setand 119879 119883 times 119883 rarr 119883 Then 119879 is said to have mixed mono-tone property if 119879(119909 119910) is monotone nondecreasing in 119909 andmonotone nonincreasing in 119910 That is for all 119909 119910 isin 119883

1199091 1199092isin 119883 119909

1⪯ 1199092997904rArr 119879 (119909

1 119910) ⪯ 119879 (119909

2 119910)

1199101 1199102isin 119883 119910

1⪯ 1199102997904rArr 119879 (119909 119910

1) ⪰ 119879 (119909 119910

2)

(1)

Definition 11 (see [4]) Let (119883 ⪯) be a partially ordered set and119879 119883 times119883 rarr 119883 and 119892 119883 rarr 119883 We say 119879 has the mixed 119892-monotone property if 119879 is monotone 119892-nondecreasing in its

first argument and ismonotone119892-nonincreasing in its secondargument That is for all 119909 119910 isin 119883

1199091 1199092isin 119883 119892119909

1⪯ 1198921199092997904rArr 119879 (119909

1 119910) ⪯ 119879 (119909

2 119910)

1199101 1199102isin 119883 119892119910

1⪯ 1198921199102997904rArr 119879 (119909 119910

1) ⪰ 119879 (119909 119910

2)

(2)

Definition 12 (see [2]) An element (119909 119910) isin 119883 times 119883 is called acoupled fixed point of a map 119879 119883 times 119883 rarr 119883 if 119879(119909 119910) = 119909

and 119879(119910 119909) = 119910

Definition 13 (see [4]) An element (119909 119910) isin 119883 times 119883 is called acoupled coincidence point of the maps 119879 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if 119879(119909 119910) = 119892119909 and 119879(119910 119909) = 119892119910

Definition 14 (see [4]) The maps 119879 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 are said to be commutative if 119892(119879(119909 119910)) =

119879(119892119909 119892119910)

Definition 15 For a map 119879 119883 times 119883 rarr 119883 by 1198792(119909 119910) wemean 119879(119879(119909 119910) 119879(119910 119909)) Similarly we define 119879119899(119909 119910) 119899 ge 3

3 Main Results

Our main results are presented in this section We firstdevelop a rational type contractive condition on a partiallyordered complete 119866-metric space and give a coupled fixedpoint theorem for a map satisfying this condition

We start with a partially ordered set (119883 ⪯) and supposethat there is a 119866-metric on 119883 so that (119883 119866) is a complete119866-metric space We induce partial ordering on 119883 times 119883 bydemanding that for any (119909 119910) (119906 V) isin 119883 times 119883 (119909 119910) ⪯

(119906 V) hArr 119909 ⪯ 119906 119910 ⪰ V

Theorem16 Let (119883 ⪯) be a partially ordered set and let119866 be ageneralizedmetric on119883 such that (119883 119866) is a complete119866-metricspace Suppose 119879 119883times119883 rarr 119883 is a continuous mapping on119883having the mixed monotone property Suppose also that for all(119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911)

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120572 (119866 (119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119908 119879 (119908 119911) 119879 (119908 119911)) times ([119866 (119909 119906 119908)]2

)minus1

)

+ 120573119866 (119909 119906 119908)

(3)

where 8120572 + 120573 lt 1 If there exists 1199090 1199100

isin 119883 such that1199090⪯ 119879(119909

0 1199100) and 119910

0⪰ 119879(119910

0 1199090) then 119879 has a coupled

fixed point (119909lowast 119910lowast) isin 119883 times 119883 That is (119909lowast 119910lowast) satisfies 119909lowast =119879(119909lowast 119910lowast) 119910lowast= 119879(119910

lowast 119909lowast)

Proof Suppose that there exists 1199090 1199100isin 119883 such that 119909

0⪯

119879(1199090 1199100) and 119910

0⪰ 119879(119910

0 1199090) We write 119909

1= 119879(119909

0 1199100)

1199101= 119879(119910

0 1199090) and define 119909

119899+1= 119879(119909

119899 119910119899) 119910119899+1

= 119879(119910119899 119909119899)

Journal of Mathematics 3

119899 ge 1 From the conditions of the theorem and the mixedmonotone property it easily follows that

1199091= 119879 (119909

0 1199100) ⪰ 1199090

1199092= 119879 (119909

1 1199101) ⪰ 119879 (119909

0 1199100) = 1199091

(4)

This gives

1199090⪯ 1199091⪯ 1199092⪯ sdot sdot sdot ⪯ 119909

119899⪯ 119909119899+1

⪯ sdot sdot sdot (5)

Similarly proceeding with

1199101= 119879 (119910

0 1199090) ⪯ 1199100

1199102= 119879 (119910

1 1199091) ⪯ 119879 (119910

0 1199090) = 1199101

(6)

we find

1199100⪰ 1199101⪰ 1199102⪰ sdot sdot sdot ⪰ 119910

119899⪰ 119910119899+1

⪰ sdot sdot sdot (7)

Considering the sequence 119909119899 and using (3) we have

119866 (119909119899+1

119909119899 119909119899)

= 119866 (119879 (119909119899 119910119899) 119879 (119909

119899minus1 119910119899minus1

) 119879 (119909119899minus1

119910119899minus1

))

le 120572119866 (119909119899 119909119899+1

119909119899+1

) 119866 (119909119899minus1

119909119899 119909119899) 119866 (119909

119899minus1 119909119899 119909119899)

[119866 (119909119899 119909119899minus1

119909119899minus1

)]2

+ 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

(8)

Now using inequality (3) from Theorem 2 119866(119909 119910 119910) le

2119866(119910 119909 119909) Setting 119909 = 119909119899minus1

and 119910 = 119909119899in this we find

119866 (119909119899minus1

119909119899 119909119899) le 2119866 (119909

119899 119909119899minus1

119909119899minus1

) (9)

Using this (8) becomes

119866 (119909119899+1

119909119899 119909119899)

le 4120572119866 (119909119899 119909119899+1

119909119899+1

) 119866 (119909119899minus1

119909119899 119909119899) 119866 (119909

119899minus1 119909119899 119909119899)

[119866 (119909119899minus1

119909119899 119909119899)]2

+ 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

= 4120572119866 (119909119899 119909119899+1

119909119899+1

) + 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

le 8120572119866 (119909119899+1

119909119899 119909119899) + 120573119866 (119909

119899 119909119899minus1

119909119899minus1

)

(10)

where in the last step we have used inequality (3) fromTheorem 2 Rearranging and simplifying this we get

119866 (119909119899+1

119909119899 119909119899) le (

120573

1 minus 8120572)119866 (119909

119899 119909119899minus1

119909119899minus1

) (11)

Evidently the condition that (11) is contractive is 120573(1minus8120572) lt1 that is

8120572 + 120573 lt 1 (12)

Similarly considering 119866(119910119899+1

119910119899 119910119899) = 119866(119879(119910

119899 119909119899) 119879(119910

119899minus1

119909119899minus1

) 119879(119910119899minus1

119909119899minus1

)) and arguing as above we find

119866 (119910119899+1

119910119899 119910119899) le (

120573

1 minus 8120572)119866 (119910

119899 119910119899minus1

119910119899minus1

) (13)

With condition (12) we find (13) is contractiveLet 119889119899= 119866(119909

119899+1 119909119899 119909119899) and 119896 = (120573(1 minus 8120572)) Using (11)

we get

119889119899le 119896119889119899minus1

le 1198962

119889119899minus2

le sdot sdot sdot le 119896119899

1198890 (14)

If 1198890= 0 then 119866(119879(119909

0 1199100) 1199090 1199090) = 119866(119909

1 1199090 1199090) = 119889

0=

0 But this means 119879(1199090 1199100) = 119909

0 Similarly writing 120575

119899=

119866(119910119899+1

119910119899 119910119899) we find

120575119899le 119896120575119899minus1

le 1198962

120575119899minus2

le sdot sdot sdot le 119896119899

1205750 (15)

If 1205750= 0 in addition to 119889

0= 0 we deduce similarly that

119879(1199100 1199090) = 1199100 So if 119889

0= 1205750= 0 (119909

0 1199100) is a coupled fixed

pointHowever if 119889

0gt 0 for 119898 gt 119899 using inequality (5) of

Definition 1 we get

119866 (119909119898 119909119899 119909119899) le 119866 (119909

119899+1 119909119899 119909119899) + 119866 (119909

119898 119909119899+1

119909119899+1

)

le 119866 (119909119899+1

119909119899 119909119899) + 119866 (119909

119899+2 119909119899+1

119909119899+1

)

+ 119866 (119909119898 119909119899+2

119909119899+2

)

le 119866 (119909119899+1

119909119899 119909119899) + 119866 (119909

119899+2 119909119899+1

119909119899+1

)

+ 119866 (119909119899+3

119909119899+2

119909119899+2

) + 119866 (119909119898 119909119898minus1

119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ 119896119899+2

+ sdot sdot sdot + 119896119898minus1

] 1198890

= 119896119899

[1 + 119896 + 1198962

+ sdot sdot sdot + 119896119898minus119899minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin since 119896 lt 1

(16)

So 119909119899 is a 119866-Cauchy sequence in 119883 Next considering

119866(119910119898 119910119899 119910119899) and arguing as above we can show that 119910

119899 is

also a 119866-Cauchy sequence in 119883 119866-Completeness of (119883 119866)now implies that there are points 119909lowast 119910lowast isin 119883 such that 119909

119899rarr

119909lowast and 119910

119899rarr 119910lowast as 119899 rarr infin

We next show that (119909lowast 119910lowast) is a coupled fixed point of 119879Using the fact that 119879 is continuous on 119883 and 119866 as a metric iscontinuous in each of its variables we have

119866 (119879 (119909lowast

119910lowast

) 119909lowast

119909lowast

) = 119866 ( lim119899rarrinfin

119879 (119909119899 119910119899) 119909lowast

119909lowast

)

= 119866( lim119899rarrinfin

119909119899+1

119909lowast

119909lowast

)

= lim119899rarrinfin

119866 (119909119899+1

119909lowast

119909lowast

)

= 0 since 119909119899997888rarr 119909lowast as 119899 997888rarr infin

(17)

4 Journal of Mathematics

But this means 119879(119909lowast 119910lowast) = 119909lowast Similarly by considering

119866(119879(119910lowast 119909lowast) 119910lowast 119910lowast) and repeating the arguments used to

derive (17) we can show that 119879(119910lowast 119909lowast) = 119910lowast This proves

(119909lowast 119910lowast) is a coupled fixed point of 119879

The W map was introduced by Chen [21] and a subclassofW functions was defined by Chakrabarti in [22]

Definition 17 (see [22]) We call 120593 R+ rarr R+ a function ofclassW

120573if there is a 120573 such that 0 lt 120573 lt 1 and the following

conditions are satisfied

(1) 120593(119905) le 120573119905 for all 119905 gt 0 and 120593(0) = 0

(2) lim119905119899rarr119905

inf 120593(119905119899) le 120573119905 for all 119905 gt 0

Using Definition 17 we obtain the following as a general-ization of Theorem 16

Theorem 18 Let (119883 ⪯) be a partially ordered set and let 119866be a metric on119883 such that (119883 119866) is a complete119866-metric spaceSuppose119879 119883times119883 rarr 119883 is a continuous mapping on119883 havingthe mixed monotone property For some given 0 lt 120572 120573 lt 1 let120601 isin W

120572and 120595 isin W

120573where 8120572 + 120573 lt 1 Suppose also that for

all (119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911)

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120601 (119866 (119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119909 119906 119908)]2

)minus1

)

+ 120595 (119866 (119909 119906 119908))

(18)

If there exists 1199090 1199100isin 119883 such that 119909

0⪯ 119879(119909

0 1199100) and 119910

0⪰

119879(1199100 1199090) then 119879 has a coupled fixed point (119909lowast 119910lowast) isin 119883 times 119883

That is (119909lowast 119910lowast) satisfies 119909lowast = 119879(119909lowast 119910lowast) 119910lowast= 119879(119910

lowast 119909lowast)

Proof Since 120601 isin W120572and 120595 isin W

120573 it follows from

Definition 17 that 120601(119905) le 120572119905 and 120595(119905) le 120573119905 for all 119905 gt 0Inequality (18) now becomes equivalent to inequality (3) ofTheorem 16 and the proof is immediate

Example 19 Let 119883 = [0infin) and consider the function 119866

119883 times 119883 times 119883 rarr R+ defined by

119866 (119909 119910 119911) = 0 if 119909 = 119910 = 119911

max 119909 119910 119911 otherwise(19)

Then (119883 119866) is a complete 119866-metric space [23] We define apartial order ⪯ on119883 by the following for any 119909 119910 isin 119883 119909 ⪯ 119910

if 119909 ge 119910 Also let 119865 119883 times 119883 rarr 119883 be defined by

119865 (119909 119910) = 1 if 119909 ⪯ 119910

0 otherwise(20)

Suppose 119909 119910 119906 V 119908 119911 isin 119883 satisfy 119911 ⪯ V ⪯ 119910 ⪯ 119909 ⪯ 119906 ⪯ 119908

with nonzero 119909 119906 119908 but are otherwise arbitrary Then we

have 119908 le 119906 le 119909 le 119910 le V le 119911 So the left side of (3) is119866(0 0 0) = 0 The right side of (3)

= 120572119866 (119909 0 0) 119866 (119906 0 0) 119866 (119908 0 0)

[119866 (119909 119906 119908)]2

+ 120573119866 (119909 119906 119908)

= 120572119909119906119908

1199092+ 120573119908

=120572119906119908 + 120573119909119908

119909gt 0 with 120572 =

1

16 120573 =

1

3

(21)

If 1199090= 0 and 119910

0= 1 then 119909

0⪯ 119865(119909

0 1199100) and 119910

0⪰ 119865(119910

0 1199090)

So all conditions of Theorem 16 are satisfied Easily we findthat (0 1) is a coupled fixed point of 119865 Similarly (1 0) is acoupled fixed point

In the next theorem we provide conditions underwhich the coupled fixed point of the map 119879 established inTheorem 16 is unique

Theorem 20 Suppose that the conditions of Theorem 16 arevalid In addition suppose that for each (119909 119910) (119906 V) isin 119883 times 119883

there is a (119908 119911) isin 119883 times 119883 which is comparable to (119909 119910) and(119906 V) Then 119879 has a unique coupled fixed point

Proof Suppose that (119909lowast 119910lowast) (1199091015840 1199101015840) isin 119883 times 119883 are coupledfixed points

Case 1 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are comparable

119866(119879 (119909lowast

119910lowast

) 119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))

le 120572 (119866 (119909lowast

119879 (119909lowast

119910lowast

) 119879 (119909lowast

119910lowast

))

times [119866 (1199091015840

119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))]2

times ([119866 (119909lowast

1199091015840

1199091015840

)]2

)

minus1

)

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120572

119866 (119909lowast 119909lowast 119909lowast) [119866 (119909

1015840 1199091015840 1199091015840)]2

[119866 (119909lowast 1199091015840 1199091015840)]2

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120573119866 (119909lowast

1199091015840

1199091015840

)

(22)

This is equivalent to 119866(119909lowast 1199091015840 1199091015840) le 120573119866(119909lowast 1199091015840 1199091015840) However

this is a contradiction since 120573 lt 1 So we must have 119909lowast = 1199091015840

Similarly considering 119866(119879(119910lowast 119909lowast) 119879(119910

1015840 1199091015840) 119879(119910

1015840 1199091015840)) we

easily show that 119910lowast = 1199101015840 This shows that (119909lowast 119910lowast) = (119909

1015840 1199101015840)

so the coupled fixed point is unique

Case 2 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are not comparable by the

condition of the theorem there is a (119906 V) isin 119883times119883 comparable

Journal of Mathematics 5

to (119909lowast 119910lowast) and (119909

1015840 1199101015840) If there is a positive integer 119899

0such

that 1198791198990(119906 V) = (119909lowast 119910lowast) then

1198791198990 (119906 V) = (119909

lowast

119910lowast

)

1198791198990+1

(119906 V) = 119879 (119909lowast

119910lowast

) = 119909lowast

1198791198990+2

(119906 V) = 1198792

(119909lowast

119910lowast

) = 119879 (119879 (119909lowast

119910lowast

) 119879 (119910lowast

119909lowast

))

= 119879 (119909lowast

119910lowast

) = 119909lowast

(23)

So 119879119899(119906 V) = 119909

lowast for 119899 ge 1198990and hence 119879119899(119906 V) rarr 119909

lowast as119899 rarr infin

On the other hand if no such 1198990exists we have that for

any 119899 ge 1

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

= 119866 (119879119899

(119906 V) 119879119899 (119909lowast 119910lowast) 119879119899 (119909lowast 119910lowast))

le 120572 (119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V))

times [119866 (119879119899minus1

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

))]2

times ( [119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast)

119879119899minus1

(119909lowast

119910lowast

))]2

)

minus1

)

+ 120573119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast) 119879119899minus1 (119909lowast 119910lowast)) (24)

where we have used the fact that 119879119899(119909 119910) = 119879(119879119899minus1

(119909

119910) 119879119899minus1

(119910 119909)) for any (119909 119910) isin 119883 times 119883 Since (119909lowast 119910lowast) is acoupled fixed point of 119879 119879119899(119909lowast 119910lowast) = 119909

lowast for all 119899 ge 1 andfrom (24) we now deduce that

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

le 120572

119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V)) [119866 (119909lowast 119909lowast 119909lowast)]2

[119866 (119879119899minus1 (119906 V) 119909lowast 119909lowast)]2

+ 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119899minus1

119866 (119879 (119906 V) 119909lowast 119909lowast) 997888rarr 0 as 119899 997888rarr infin

(25)

since 120573 lt 1This proves119879119899(119906 V) rarr 119909lowast as 119899 rarr infin Similarly

we can show that 119879119899(V 119906) rarr 119910lowast as 119899 rarr infin Replacing 119909lowast

with 1199091015840 and 119910lowast with 1199101015840 and repeating the above arguments wecan deduce 119879119899(119906 V) rarr 119909

1015840 and 119879119899(V 119906) rarr 119910

1015840 as 119899 rarr infinBut thismeans 119909lowast = 119909

1015840 and119910lowast = 1199101015840 So the (119909lowast 119910lowast) = (119909

1015840 1199101015840)

and the coupled fixed point is unique

We next establish the conditions under which two maps119879 119883times119883 rarr 119883 and 119892 119883 rarr 119883 have a coupled coincidencepoint

Theorem 21 Let (119883 ⪯) be a partially ordered set and let 119866 bea metric on 119883 such that (119883 119866) is a complete 119866-metric spaceLet 119879 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 be continuous map-ping on 119883 such that 119879 has the mixed 119892-monotone propertySuppose that 119879(119883 times 119883) sube 119892(119883) 119892 commutes with 119879 and for(119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911) and119892119909 ⪯ 119892119906 ⪯ 119892119908 119892119910 ⪰ 119892V ⪰ 119892119911

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120572 (119866 (119892119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119892119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119892119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119892119909 119892119906 119892119908)]2

)minus1

)

+ 120573119866 (119892119909 119892119906 119892119908)

(26)

where 8120572 + 120573 lt 1 If there exists 1199090 1199100

isin 119883 such that1198921199090⪯ 119879(119909

0 1199100) and 119892119910

0⪰ 119879(119910

0 1199090) then 119879 has a coupled

coincidence point (119909lowast 119910lowast) isin 119883 times 119883 That is (119909lowast 119910lowast) satisfies119892119909lowast= 119879(119909

lowast 119910lowast) 119892119910lowast= 119879(119910

lowast 119909lowast)

Proof Since119879(119883times119883) sube 119892(119883)we can choose 1199091 1199101isin 119883 such

that 1198921199091= 119879(119909

0 1199100) 1198921199101= 119879(119910

0 1199090) For similar reasons

1199092 1199102isin 119883 can be found such that 119892119909

2= 119879(119909

1 1199101) 1198921199102=

119879(1199101 1199091) Due to the mixed 119892-monotone property of 119879 we

have 1198921199090⪯ 1198921199091⪯ 1198921199092and 119892119910

2⪯ 1198921199101⪯ 1198921199100 In general it

can be shown that [4] for 119899 ge 0119892119909119899= 119879 (119909

119899minus1 119910119899minus1

) ⪯ 119892119909119899+1

= 119879 (119909119899 119910119899)

119892119910119899+1

= 119879 (119910119899 119909119899) ⪯ 119892119910

119899= 119879 (119909

119899minus1 119910119899minus1

)

(27)

Now by the same arguments used to deduce (10) we have

119866 (119892119909119899+1

119892119909119899 119892119909119899)

le 4120572119866 (119892119909

119899 119892119909119899+1

119892119909119899+1

) [119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

[119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

= 4120572119866 (119892119909119899 119892119909119899+1

119892119909119899+1

) + 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

le 8120572119866 (119892119909119899+1

119892119909119899 119892119909119899)

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

(28)

This gives

119866 (119892119909119899+1

119892119909119899 119892119909119899) le (

120573

1 minus 8120572)119866 (119892119909

119899 119892119909119899minus1

119909119899minus1

)

(29)

Since 8120572 + 120573 lt 1 119896 = 120573(1 minus 8120572) lt 1 Writing 119889119899=

119866(119892119909119899+1

119892119909119899 119892119909119899) we find from (28) that

119889119899le 119896119889119899minus1

le 1198962

119889119899minus1

le sdot sdot sdot le 119896119899

1198890 (30)

6 Journal of Mathematics

If 1198890= 0 then119866(119892119909

1 1198921199090 1198921199090) = 119866(119879(119909

0 1199100) 1198921199090 1198921199090) = 0

giving 1198921199090= 119879(119909

0 1199100) so 119909

0is a coincidence point However

if 1198890gt 0 we have for119898 gt 119899

119866 (119892119909119898 119892119909119899 119892119909119899)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119898 119892119909119899+1

119892119909119899+1

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ 119866 (119892119909119898 119892119909119899+2

119892119909119899+2

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ sdot sdot sdot + 119866 (119892119909119898 119892119909119898minus1

119892119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ sdot sdot sdot + 119896119898minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin

(31)

since 119896 lt 1 This shows that 119892119909119899 is a Cauchy sequence in

(119883 119866) and 119866-completeness of (119883 119866) ensures a point 119909lowast isin 119883

such that 119892119909119899rarr 119909lowast as 119899 rarr infin

Replacing 119909119899by 119910119899for all 119899 ge 0 we get the analogue of

(29)

119866 (119892119910119899+1

119892119910119899 119892119910119899) le (

120573

1 minus 8120572)119866 (119892119910

119899 119892119910119899minus1

119910119899minus1

) (32)

Next writing 120575119899= 119866(119892119910

119899+1 119892119910119899 119892119910119899) we find as in (30) that

120575119899le 119896120575119899minus1

le 1198962

120575119899minus1

le sdot sdot sdot le 119896119899

1205750 (33)

As before if 1205750= 0 119866(119892119910

1 1198921199100 1198921199100) = 119866(119879(119910

0 1199090) 1198921199100

1198921199100) = 0 giving 119879(119910

0 1199090) = 119892119910

0so that 119910

0is a coinci-dence

point If in addition 1198890= 0 we have 119892119909

0= 119879(119909

0 1199100) and

1198921199100= 119879(119910

0 1199090) So (119909

0 1199100) is a coupled coincidence point

Proceeding as in (31) we can show further that 119892119910119899 is

a Cauchy sequence in (119883 119866) and due to 119866-completeness of(119883 119866) there is a point 119910lowast isin 119883 such that 119892119910

119899rarr 119910lowast as 119899 rarr

infinFinally we prove that (119909lowast 119910lowast) is a coupled coincident

point Since 119879 and 119892 commute we have

119892 (119892119909119899+1

) = 119892 (119879 (119909119899 119910119899)) = 119879 (119892119909

119899 119892119910119899)

119892 (119892119910119899+1

) = 119892 (119879 (119910119899 119909119899)) = 119879 (119892119910

119899 119892119909119899)

(34)

Taking limits as 119899 rarr infin in (34) and noting that 119879 and 119892 arerespectively continuous on119883 times 119883 and119883 we get

119892119909lowast

= lim119892 (119892119909119899+1

) = lim119892 (119879 (119909119899 119910119899)) = lim119879 (119892119909

119899 119892119910119899)

119892119910lowast

= lim119892 (119892119910119899+1

) = lim119892 (119879 (119910119899 119909119899)) = lim119879 (119892119910

119899 119892119909119899)

(35)

Next we observe that 119866 as metric is continuous in all itsvariables This finally leads to

119866 (119879 (119909lowast

119910lowast

) 119892119909lowast

119892119909lowast

) = 119866 (lim119879 (119892119909119899 119892119910119899) 119892119909lowast

119892119909lowast

)

= 119866 (119892119909lowast

119892119909lowast

119892119909lowast

)

= 0

(36)

So 119892119909lowast = 119879(119909lowast 119910lowast) Similarly we show that 119892119910lowast = 119879(119910

lowast 119909lowast)

This proves (119909lowast 119910lowast) is a coupled coincidence point

4 Conclusion

To summarize we have introduced a rational type contractivecondition in a 119866-metric space and proved some coupledfixed point theorems for maps satisfying mixed monotoneproperty We established the conditions for uniqueness of thecoupled fixed point Conditions for the existence of coupledcoincidence points of two maps are also deduced

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[2] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006

[3] L Ciric M O Olatinwo D Gopal and G Akinbo ldquoCoupledfixed point theorems for mappings satisfying a contractivecondition of rational type on a partially ordered metric spacerdquoAdvances in Fixed Point Theory vol 2 no 1 pp 1ndash8 2012

[4] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[5] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010

[6] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[7] V Berinde ldquoGeneralized coupled fixed point theorems formixed monotone mappings in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no18 pp 7347ndash7355 2011

[8] B S Choudhury and P Maity ldquoCoupled fixed point resultsin generalized metric spacesrdquo Mathematical and ComputerModelling vol 54 no 1-2 pp 73ndash79 2011

Journal of Mathematics 7

[9] R Saadati S M Vaezpour P Vetro and B E RhoadesldquoFixed point theorems in generalized partially ordered 119866-met-ric spacesrdquo Mathematical and Computer Modelling vol 52 no5-6 pp 797ndash801 2010

[10] Z Mustafa J R Roshan and V Parvaneh ldquoCoupled coinci-dence point results for (120595 120593)-weakly contractive mappings inpartially ordered 119866

119887-metric spacesrdquo Fixed Point Theory and

Applications vol 2013 article 206 2013[11] ZMustafa J R Roshan andV Parvaneh ldquoExistence of a tripled

coincidence point in ordered G119887-metric spaces and applications

to a system of integral equationsrdquo Journal of Inequalities andApplications vol 2013 article no 453 2013

[12] V ParvanehA Razani and J R Roshan ldquoCommonfixed pointsof six mappings in partially ordered 119866-metric spacesrdquo Math-ematical Sciences vol 7 article 18 2013

[13] A Razani and V Parvaneh ldquoOn generalized weakly G-contractive mappings in partially ordered G-metric spacesrdquoAbstract and Applied Analysis vol 2012 Article ID 701910 18pages 2012

[14] Z Mustafa V Parvaneh M Abbas and J Rezaei RoshanldquoSome coincidence point results for generalized (120595 120593)-weaklycontractive mappings in ordered G-metric spaces Fixed PointTheory and Applicationsrdquo Fixed Point Theory and Applicationsvol 2013 article 326 2013

[15] M A Kutbi N Hussain J R Roshan and V ParvanehldquoCoupled and tripled coincidence point results with applicationto Fredholm integral equationsrdquo Abstract and Applied Analysisvol 2014 Article ID 568718 18 pages 2014

[16] N Hussain V Parvaneh and J R Roshan ldquoFixed point resultsfor G-120572-contractive maps with application to boundary valueproblemsrdquo ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[17] A Latif N Hussain J R Roshan and V Parvaneh ldquoA uni-fication of Gminusmetric partial metric and b-metric spacesrdquoAbstract andApplied Analysis vol 2014 Article ID 180698 2014

[18] L Ciric S M Alsulami V Parvaneh and J R Roshan ldquoSomefixed point results in ordered 119866

119901-metric spacesrdquo Fixed Point

Theory and Applications vol 2013 article 317 2013[19] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journal

of Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977[20] Z Mustafa and B Sims ldquoA new approach to generalized metric

spacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash296 2006

[21] C-M Chen ldquoCommon fixed-point theorems in completegeneralized metric spacesrdquo Journal of Applied Mathematics vol2012 Article ID 945915 14 pages 2012

[22] K Chakrabarti ldquoFixed point theorems in G-metric spaces withW mapsrdquoMathematical Sciences Letters vol 2 pp 29ndash35 2013

[23] E Karapınar and R P Agarwal ldquoFurther fixed point results on119866-metric spacesrdquo Fixed PointTheory andApplications vol 2013article 154 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 3

119899 ge 1 From the conditions of the theorem and the mixedmonotone property it easily follows that

1199091= 119879 (119909

0 1199100) ⪰ 1199090

1199092= 119879 (119909

1 1199101) ⪰ 119879 (119909

0 1199100) = 1199091

(4)

This gives

1199090⪯ 1199091⪯ 1199092⪯ sdot sdot sdot ⪯ 119909

119899⪯ 119909119899+1

⪯ sdot sdot sdot (5)

Similarly proceeding with

1199101= 119879 (119910

0 1199090) ⪯ 1199100

1199102= 119879 (119910

1 1199091) ⪯ 119879 (119910

0 1199090) = 1199101

(6)

we find

1199100⪰ 1199101⪰ 1199102⪰ sdot sdot sdot ⪰ 119910

119899⪰ 119910119899+1

⪰ sdot sdot sdot (7)

Considering the sequence 119909119899 and using (3) we have

119866 (119909119899+1

119909119899 119909119899)

= 119866 (119879 (119909119899 119910119899) 119879 (119909

119899minus1 119910119899minus1

) 119879 (119909119899minus1

119910119899minus1

))

le 120572119866 (119909119899 119909119899+1

119909119899+1

) 119866 (119909119899minus1

119909119899 119909119899) 119866 (119909

119899minus1 119909119899 119909119899)

[119866 (119909119899 119909119899minus1

119909119899minus1

)]2

+ 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

(8)

Now using inequality (3) from Theorem 2 119866(119909 119910 119910) le

2119866(119910 119909 119909) Setting 119909 = 119909119899minus1

and 119910 = 119909119899in this we find

119866 (119909119899minus1

119909119899 119909119899) le 2119866 (119909

119899 119909119899minus1

119909119899minus1

) (9)

Using this (8) becomes

119866 (119909119899+1

119909119899 119909119899)

le 4120572119866 (119909119899 119909119899+1

119909119899+1

) 119866 (119909119899minus1

119909119899 119909119899) 119866 (119909

119899minus1 119909119899 119909119899)

[119866 (119909119899minus1

119909119899 119909119899)]2

+ 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

= 4120572119866 (119909119899 119909119899+1

119909119899+1

) + 120573119866 (119909119899 119909119899minus1

119909119899minus1

)

le 8120572119866 (119909119899+1

119909119899 119909119899) + 120573119866 (119909

119899 119909119899minus1

119909119899minus1

)

(10)

where in the last step we have used inequality (3) fromTheorem 2 Rearranging and simplifying this we get

119866 (119909119899+1

119909119899 119909119899) le (

120573

1 minus 8120572)119866 (119909

119899 119909119899minus1

119909119899minus1

) (11)

Evidently the condition that (11) is contractive is 120573(1minus8120572) lt1 that is

8120572 + 120573 lt 1 (12)

Similarly considering 119866(119910119899+1

119910119899 119910119899) = 119866(119879(119910

119899 119909119899) 119879(119910

119899minus1

119909119899minus1

) 119879(119910119899minus1

119909119899minus1

)) and arguing as above we find

119866 (119910119899+1

119910119899 119910119899) le (

120573

1 minus 8120572)119866 (119910

119899 119910119899minus1

119910119899minus1

) (13)

With condition (12) we find (13) is contractiveLet 119889119899= 119866(119909

119899+1 119909119899 119909119899) and 119896 = (120573(1 minus 8120572)) Using (11)

we get

119889119899le 119896119889119899minus1

le 1198962

119889119899minus2

le sdot sdot sdot le 119896119899

1198890 (14)

If 1198890= 0 then 119866(119879(119909

0 1199100) 1199090 1199090) = 119866(119909

1 1199090 1199090) = 119889

0=

0 But this means 119879(1199090 1199100) = 119909

0 Similarly writing 120575

119899=

119866(119910119899+1

119910119899 119910119899) we find

120575119899le 119896120575119899minus1

le 1198962

120575119899minus2

le sdot sdot sdot le 119896119899

1205750 (15)

If 1205750= 0 in addition to 119889

0= 0 we deduce similarly that

119879(1199100 1199090) = 1199100 So if 119889

0= 1205750= 0 (119909

0 1199100) is a coupled fixed

pointHowever if 119889

0gt 0 for 119898 gt 119899 using inequality (5) of

Definition 1 we get

119866 (119909119898 119909119899 119909119899) le 119866 (119909

119899+1 119909119899 119909119899) + 119866 (119909

119898 119909119899+1

119909119899+1

)

le 119866 (119909119899+1

119909119899 119909119899) + 119866 (119909

119899+2 119909119899+1

119909119899+1

)

+ 119866 (119909119898 119909119899+2

119909119899+2

)

le 119866 (119909119899+1

119909119899 119909119899) + 119866 (119909

119899+2 119909119899+1

119909119899+1

)

+ 119866 (119909119899+3

119909119899+2

119909119899+2

) + 119866 (119909119898 119909119898minus1

119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ 119896119899+2

+ sdot sdot sdot + 119896119898minus1

] 1198890

= 119896119899

[1 + 119896 + 1198962

+ sdot sdot sdot + 119896119898minus119899minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin since 119896 lt 1

(16)

So 119909119899 is a 119866-Cauchy sequence in 119883 Next considering

119866(119910119898 119910119899 119910119899) and arguing as above we can show that 119910

119899 is

also a 119866-Cauchy sequence in 119883 119866-Completeness of (119883 119866)now implies that there are points 119909lowast 119910lowast isin 119883 such that 119909

119899rarr

119909lowast and 119910

119899rarr 119910lowast as 119899 rarr infin

We next show that (119909lowast 119910lowast) is a coupled fixed point of 119879Using the fact that 119879 is continuous on 119883 and 119866 as a metric iscontinuous in each of its variables we have

119866 (119879 (119909lowast

119910lowast

) 119909lowast

119909lowast

) = 119866 ( lim119899rarrinfin

119879 (119909119899 119910119899) 119909lowast

119909lowast

)

= 119866( lim119899rarrinfin

119909119899+1

119909lowast

119909lowast

)

= lim119899rarrinfin

119866 (119909119899+1

119909lowast

119909lowast

)

= 0 since 119909119899997888rarr 119909lowast as 119899 997888rarr infin

(17)

4 Journal of Mathematics

But this means 119879(119909lowast 119910lowast) = 119909lowast Similarly by considering

119866(119879(119910lowast 119909lowast) 119910lowast 119910lowast) and repeating the arguments used to

derive (17) we can show that 119879(119910lowast 119909lowast) = 119910lowast This proves

(119909lowast 119910lowast) is a coupled fixed point of 119879

The W map was introduced by Chen [21] and a subclassofW functions was defined by Chakrabarti in [22]

Definition 17 (see [22]) We call 120593 R+ rarr R+ a function ofclassW

120573if there is a 120573 such that 0 lt 120573 lt 1 and the following

conditions are satisfied

(1) 120593(119905) le 120573119905 for all 119905 gt 0 and 120593(0) = 0

(2) lim119905119899rarr119905

inf 120593(119905119899) le 120573119905 for all 119905 gt 0

Using Definition 17 we obtain the following as a general-ization of Theorem 16

Theorem 18 Let (119883 ⪯) be a partially ordered set and let 119866be a metric on119883 such that (119883 119866) is a complete119866-metric spaceSuppose119879 119883times119883 rarr 119883 is a continuous mapping on119883 havingthe mixed monotone property For some given 0 lt 120572 120573 lt 1 let120601 isin W

120572and 120595 isin W

120573where 8120572 + 120573 lt 1 Suppose also that for

all (119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911)

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120601 (119866 (119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119909 119906 119908)]2

)minus1

)

+ 120595 (119866 (119909 119906 119908))

(18)

If there exists 1199090 1199100isin 119883 such that 119909

0⪯ 119879(119909

0 1199100) and 119910

0⪰

119879(1199100 1199090) then 119879 has a coupled fixed point (119909lowast 119910lowast) isin 119883 times 119883

That is (119909lowast 119910lowast) satisfies 119909lowast = 119879(119909lowast 119910lowast) 119910lowast= 119879(119910

lowast 119909lowast)

Proof Since 120601 isin W120572and 120595 isin W

120573 it follows from

Definition 17 that 120601(119905) le 120572119905 and 120595(119905) le 120573119905 for all 119905 gt 0Inequality (18) now becomes equivalent to inequality (3) ofTheorem 16 and the proof is immediate

Example 19 Let 119883 = [0infin) and consider the function 119866

119883 times 119883 times 119883 rarr R+ defined by

119866 (119909 119910 119911) = 0 if 119909 = 119910 = 119911

max 119909 119910 119911 otherwise(19)

Then (119883 119866) is a complete 119866-metric space [23] We define apartial order ⪯ on119883 by the following for any 119909 119910 isin 119883 119909 ⪯ 119910

if 119909 ge 119910 Also let 119865 119883 times 119883 rarr 119883 be defined by

119865 (119909 119910) = 1 if 119909 ⪯ 119910

0 otherwise(20)

Suppose 119909 119910 119906 V 119908 119911 isin 119883 satisfy 119911 ⪯ V ⪯ 119910 ⪯ 119909 ⪯ 119906 ⪯ 119908

with nonzero 119909 119906 119908 but are otherwise arbitrary Then we

have 119908 le 119906 le 119909 le 119910 le V le 119911 So the left side of (3) is119866(0 0 0) = 0 The right side of (3)

= 120572119866 (119909 0 0) 119866 (119906 0 0) 119866 (119908 0 0)

[119866 (119909 119906 119908)]2

+ 120573119866 (119909 119906 119908)

= 120572119909119906119908

1199092+ 120573119908

=120572119906119908 + 120573119909119908

119909gt 0 with 120572 =

1

16 120573 =

1

3

(21)

If 1199090= 0 and 119910

0= 1 then 119909

0⪯ 119865(119909

0 1199100) and 119910

0⪰ 119865(119910

0 1199090)

So all conditions of Theorem 16 are satisfied Easily we findthat (0 1) is a coupled fixed point of 119865 Similarly (1 0) is acoupled fixed point

In the next theorem we provide conditions underwhich the coupled fixed point of the map 119879 established inTheorem 16 is unique

Theorem 20 Suppose that the conditions of Theorem 16 arevalid In addition suppose that for each (119909 119910) (119906 V) isin 119883 times 119883

there is a (119908 119911) isin 119883 times 119883 which is comparable to (119909 119910) and(119906 V) Then 119879 has a unique coupled fixed point

Proof Suppose that (119909lowast 119910lowast) (1199091015840 1199101015840) isin 119883 times 119883 are coupledfixed points

Case 1 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are comparable

119866(119879 (119909lowast

119910lowast

) 119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))

le 120572 (119866 (119909lowast

119879 (119909lowast

119910lowast

) 119879 (119909lowast

119910lowast

))

times [119866 (1199091015840

119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))]2

times ([119866 (119909lowast

1199091015840

1199091015840

)]2

)

minus1

)

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120572

119866 (119909lowast 119909lowast 119909lowast) [119866 (119909

1015840 1199091015840 1199091015840)]2

[119866 (119909lowast 1199091015840 1199091015840)]2

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120573119866 (119909lowast

1199091015840

1199091015840

)

(22)

This is equivalent to 119866(119909lowast 1199091015840 1199091015840) le 120573119866(119909lowast 1199091015840 1199091015840) However

this is a contradiction since 120573 lt 1 So we must have 119909lowast = 1199091015840

Similarly considering 119866(119879(119910lowast 119909lowast) 119879(119910

1015840 1199091015840) 119879(119910

1015840 1199091015840)) we

easily show that 119910lowast = 1199101015840 This shows that (119909lowast 119910lowast) = (119909

1015840 1199101015840)

so the coupled fixed point is unique

Case 2 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are not comparable by the

condition of the theorem there is a (119906 V) isin 119883times119883 comparable

Journal of Mathematics 5

to (119909lowast 119910lowast) and (119909

1015840 1199101015840) If there is a positive integer 119899

0such

that 1198791198990(119906 V) = (119909lowast 119910lowast) then

1198791198990 (119906 V) = (119909

lowast

119910lowast

)

1198791198990+1

(119906 V) = 119879 (119909lowast

119910lowast

) = 119909lowast

1198791198990+2

(119906 V) = 1198792

(119909lowast

119910lowast

) = 119879 (119879 (119909lowast

119910lowast

) 119879 (119910lowast

119909lowast

))

= 119879 (119909lowast

119910lowast

) = 119909lowast

(23)

So 119879119899(119906 V) = 119909

lowast for 119899 ge 1198990and hence 119879119899(119906 V) rarr 119909

lowast as119899 rarr infin

On the other hand if no such 1198990exists we have that for

any 119899 ge 1

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

= 119866 (119879119899

(119906 V) 119879119899 (119909lowast 119910lowast) 119879119899 (119909lowast 119910lowast))

le 120572 (119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V))

times [119866 (119879119899minus1

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

))]2

times ( [119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast)

119879119899minus1

(119909lowast

119910lowast

))]2

)

minus1

)

+ 120573119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast) 119879119899minus1 (119909lowast 119910lowast)) (24)

where we have used the fact that 119879119899(119909 119910) = 119879(119879119899minus1

(119909

119910) 119879119899minus1

(119910 119909)) for any (119909 119910) isin 119883 times 119883 Since (119909lowast 119910lowast) is acoupled fixed point of 119879 119879119899(119909lowast 119910lowast) = 119909

lowast for all 119899 ge 1 andfrom (24) we now deduce that

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

le 120572

119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V)) [119866 (119909lowast 119909lowast 119909lowast)]2

[119866 (119879119899minus1 (119906 V) 119909lowast 119909lowast)]2

+ 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119899minus1

119866 (119879 (119906 V) 119909lowast 119909lowast) 997888rarr 0 as 119899 997888rarr infin

(25)

since 120573 lt 1This proves119879119899(119906 V) rarr 119909lowast as 119899 rarr infin Similarly

we can show that 119879119899(V 119906) rarr 119910lowast as 119899 rarr infin Replacing 119909lowast

with 1199091015840 and 119910lowast with 1199101015840 and repeating the above arguments wecan deduce 119879119899(119906 V) rarr 119909

1015840 and 119879119899(V 119906) rarr 119910

1015840 as 119899 rarr infinBut thismeans 119909lowast = 119909

1015840 and119910lowast = 1199101015840 So the (119909lowast 119910lowast) = (119909

1015840 1199101015840)

and the coupled fixed point is unique

We next establish the conditions under which two maps119879 119883times119883 rarr 119883 and 119892 119883 rarr 119883 have a coupled coincidencepoint

Theorem 21 Let (119883 ⪯) be a partially ordered set and let 119866 bea metric on 119883 such that (119883 119866) is a complete 119866-metric spaceLet 119879 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 be continuous map-ping on 119883 such that 119879 has the mixed 119892-monotone propertySuppose that 119879(119883 times 119883) sube 119892(119883) 119892 commutes with 119879 and for(119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911) and119892119909 ⪯ 119892119906 ⪯ 119892119908 119892119910 ⪰ 119892V ⪰ 119892119911

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120572 (119866 (119892119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119892119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119892119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119892119909 119892119906 119892119908)]2

)minus1

)

+ 120573119866 (119892119909 119892119906 119892119908)

(26)

where 8120572 + 120573 lt 1 If there exists 1199090 1199100

isin 119883 such that1198921199090⪯ 119879(119909

0 1199100) and 119892119910

0⪰ 119879(119910

0 1199090) then 119879 has a coupled

coincidence point (119909lowast 119910lowast) isin 119883 times 119883 That is (119909lowast 119910lowast) satisfies119892119909lowast= 119879(119909

lowast 119910lowast) 119892119910lowast= 119879(119910

lowast 119909lowast)

Proof Since119879(119883times119883) sube 119892(119883)we can choose 1199091 1199101isin 119883 such

that 1198921199091= 119879(119909

0 1199100) 1198921199101= 119879(119910

0 1199090) For similar reasons

1199092 1199102isin 119883 can be found such that 119892119909

2= 119879(119909

1 1199101) 1198921199102=

119879(1199101 1199091) Due to the mixed 119892-monotone property of 119879 we

have 1198921199090⪯ 1198921199091⪯ 1198921199092and 119892119910

2⪯ 1198921199101⪯ 1198921199100 In general it

can be shown that [4] for 119899 ge 0119892119909119899= 119879 (119909

119899minus1 119910119899minus1

) ⪯ 119892119909119899+1

= 119879 (119909119899 119910119899)

119892119910119899+1

= 119879 (119910119899 119909119899) ⪯ 119892119910

119899= 119879 (119909

119899minus1 119910119899minus1

)

(27)

Now by the same arguments used to deduce (10) we have

119866 (119892119909119899+1

119892119909119899 119892119909119899)

le 4120572119866 (119892119909

119899 119892119909119899+1

119892119909119899+1

) [119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

[119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

= 4120572119866 (119892119909119899 119892119909119899+1

119892119909119899+1

) + 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

le 8120572119866 (119892119909119899+1

119892119909119899 119892119909119899)

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

(28)

This gives

119866 (119892119909119899+1

119892119909119899 119892119909119899) le (

120573

1 minus 8120572)119866 (119892119909

119899 119892119909119899minus1

119909119899minus1

)

(29)

Since 8120572 + 120573 lt 1 119896 = 120573(1 minus 8120572) lt 1 Writing 119889119899=

119866(119892119909119899+1

119892119909119899 119892119909119899) we find from (28) that

119889119899le 119896119889119899minus1

le 1198962

119889119899minus1

le sdot sdot sdot le 119896119899

1198890 (30)

6 Journal of Mathematics

If 1198890= 0 then119866(119892119909

1 1198921199090 1198921199090) = 119866(119879(119909

0 1199100) 1198921199090 1198921199090) = 0

giving 1198921199090= 119879(119909

0 1199100) so 119909

0is a coincidence point However

if 1198890gt 0 we have for119898 gt 119899

119866 (119892119909119898 119892119909119899 119892119909119899)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119898 119892119909119899+1

119892119909119899+1

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ 119866 (119892119909119898 119892119909119899+2

119892119909119899+2

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ sdot sdot sdot + 119866 (119892119909119898 119892119909119898minus1

119892119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ sdot sdot sdot + 119896119898minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin

(31)

since 119896 lt 1 This shows that 119892119909119899 is a Cauchy sequence in

(119883 119866) and 119866-completeness of (119883 119866) ensures a point 119909lowast isin 119883

such that 119892119909119899rarr 119909lowast as 119899 rarr infin

Replacing 119909119899by 119910119899for all 119899 ge 0 we get the analogue of

(29)

119866 (119892119910119899+1

119892119910119899 119892119910119899) le (

120573

1 minus 8120572)119866 (119892119910

119899 119892119910119899minus1

119910119899minus1

) (32)

Next writing 120575119899= 119866(119892119910

119899+1 119892119910119899 119892119910119899) we find as in (30) that

120575119899le 119896120575119899minus1

le 1198962

120575119899minus1

le sdot sdot sdot le 119896119899

1205750 (33)

As before if 1205750= 0 119866(119892119910

1 1198921199100 1198921199100) = 119866(119879(119910

0 1199090) 1198921199100

1198921199100) = 0 giving 119879(119910

0 1199090) = 119892119910

0so that 119910

0is a coinci-dence

point If in addition 1198890= 0 we have 119892119909

0= 119879(119909

0 1199100) and

1198921199100= 119879(119910

0 1199090) So (119909

0 1199100) is a coupled coincidence point

Proceeding as in (31) we can show further that 119892119910119899 is

a Cauchy sequence in (119883 119866) and due to 119866-completeness of(119883 119866) there is a point 119910lowast isin 119883 such that 119892119910

119899rarr 119910lowast as 119899 rarr

infinFinally we prove that (119909lowast 119910lowast) is a coupled coincident

point Since 119879 and 119892 commute we have

119892 (119892119909119899+1

) = 119892 (119879 (119909119899 119910119899)) = 119879 (119892119909

119899 119892119910119899)

119892 (119892119910119899+1

) = 119892 (119879 (119910119899 119909119899)) = 119879 (119892119910

119899 119892119909119899)

(34)

Taking limits as 119899 rarr infin in (34) and noting that 119879 and 119892 arerespectively continuous on119883 times 119883 and119883 we get

119892119909lowast

= lim119892 (119892119909119899+1

) = lim119892 (119879 (119909119899 119910119899)) = lim119879 (119892119909

119899 119892119910119899)

119892119910lowast

= lim119892 (119892119910119899+1

) = lim119892 (119879 (119910119899 119909119899)) = lim119879 (119892119910

119899 119892119909119899)

(35)

Next we observe that 119866 as metric is continuous in all itsvariables This finally leads to

119866 (119879 (119909lowast

119910lowast

) 119892119909lowast

119892119909lowast

) = 119866 (lim119879 (119892119909119899 119892119910119899) 119892119909lowast

119892119909lowast

)

= 119866 (119892119909lowast

119892119909lowast

119892119909lowast

)

= 0

(36)

So 119892119909lowast = 119879(119909lowast 119910lowast) Similarly we show that 119892119910lowast = 119879(119910

lowast 119909lowast)

This proves (119909lowast 119910lowast) is a coupled coincidence point

4 Conclusion

To summarize we have introduced a rational type contractivecondition in a 119866-metric space and proved some coupledfixed point theorems for maps satisfying mixed monotoneproperty We established the conditions for uniqueness of thecoupled fixed point Conditions for the existence of coupledcoincidence points of two maps are also deduced

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[2] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006

[3] L Ciric M O Olatinwo D Gopal and G Akinbo ldquoCoupledfixed point theorems for mappings satisfying a contractivecondition of rational type on a partially ordered metric spacerdquoAdvances in Fixed Point Theory vol 2 no 1 pp 1ndash8 2012

[4] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[5] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010

[6] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[7] V Berinde ldquoGeneralized coupled fixed point theorems formixed monotone mappings in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no18 pp 7347ndash7355 2011

[8] B S Choudhury and P Maity ldquoCoupled fixed point resultsin generalized metric spacesrdquo Mathematical and ComputerModelling vol 54 no 1-2 pp 73ndash79 2011

Journal of Mathematics 7

[9] R Saadati S M Vaezpour P Vetro and B E RhoadesldquoFixed point theorems in generalized partially ordered 119866-met-ric spacesrdquo Mathematical and Computer Modelling vol 52 no5-6 pp 797ndash801 2010

[10] Z Mustafa J R Roshan and V Parvaneh ldquoCoupled coinci-dence point results for (120595 120593)-weakly contractive mappings inpartially ordered 119866

119887-metric spacesrdquo Fixed Point Theory and

Applications vol 2013 article 206 2013[11] ZMustafa J R Roshan andV Parvaneh ldquoExistence of a tripled

coincidence point in ordered G119887-metric spaces and applications

to a system of integral equationsrdquo Journal of Inequalities andApplications vol 2013 article no 453 2013

[12] V ParvanehA Razani and J R Roshan ldquoCommonfixed pointsof six mappings in partially ordered 119866-metric spacesrdquo Math-ematical Sciences vol 7 article 18 2013

[13] A Razani and V Parvaneh ldquoOn generalized weakly G-contractive mappings in partially ordered G-metric spacesrdquoAbstract and Applied Analysis vol 2012 Article ID 701910 18pages 2012

[14] Z Mustafa V Parvaneh M Abbas and J Rezaei RoshanldquoSome coincidence point results for generalized (120595 120593)-weaklycontractive mappings in ordered G-metric spaces Fixed PointTheory and Applicationsrdquo Fixed Point Theory and Applicationsvol 2013 article 326 2013

[15] M A Kutbi N Hussain J R Roshan and V ParvanehldquoCoupled and tripled coincidence point results with applicationto Fredholm integral equationsrdquo Abstract and Applied Analysisvol 2014 Article ID 568718 18 pages 2014

[16] N Hussain V Parvaneh and J R Roshan ldquoFixed point resultsfor G-120572-contractive maps with application to boundary valueproblemsrdquo ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[17] A Latif N Hussain J R Roshan and V Parvaneh ldquoA uni-fication of Gminusmetric partial metric and b-metric spacesrdquoAbstract andApplied Analysis vol 2014 Article ID 180698 2014

[18] L Ciric S M Alsulami V Parvaneh and J R Roshan ldquoSomefixed point results in ordered 119866

119901-metric spacesrdquo Fixed Point

Theory and Applications vol 2013 article 317 2013[19] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journal

of Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977[20] Z Mustafa and B Sims ldquoA new approach to generalized metric

spacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash296 2006

[21] C-M Chen ldquoCommon fixed-point theorems in completegeneralized metric spacesrdquo Journal of Applied Mathematics vol2012 Article ID 945915 14 pages 2012

[22] K Chakrabarti ldquoFixed point theorems in G-metric spaces withW mapsrdquoMathematical Sciences Letters vol 2 pp 29ndash35 2013

[23] E Karapınar and R P Agarwal ldquoFurther fixed point results on119866-metric spacesrdquo Fixed PointTheory andApplications vol 2013article 154 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Mathematics

But this means 119879(119909lowast 119910lowast) = 119909lowast Similarly by considering

119866(119879(119910lowast 119909lowast) 119910lowast 119910lowast) and repeating the arguments used to

derive (17) we can show that 119879(119910lowast 119909lowast) = 119910lowast This proves

(119909lowast 119910lowast) is a coupled fixed point of 119879

The W map was introduced by Chen [21] and a subclassofW functions was defined by Chakrabarti in [22]

Definition 17 (see [22]) We call 120593 R+ rarr R+ a function ofclassW

120573if there is a 120573 such that 0 lt 120573 lt 1 and the following

conditions are satisfied

(1) 120593(119905) le 120573119905 for all 119905 gt 0 and 120593(0) = 0

(2) lim119905119899rarr119905

inf 120593(119905119899) le 120573119905 for all 119905 gt 0

Using Definition 17 we obtain the following as a general-ization of Theorem 16

Theorem 18 Let (119883 ⪯) be a partially ordered set and let 119866be a metric on119883 such that (119883 119866) is a complete119866-metric spaceSuppose119879 119883times119883 rarr 119883 is a continuous mapping on119883 havingthe mixed monotone property For some given 0 lt 120572 120573 lt 1 let120601 isin W

120572and 120595 isin W

120573where 8120572 + 120573 lt 1 Suppose also that for

all (119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911)

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120601 (119866 (119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119909 119906 119908)]2

)minus1

)

+ 120595 (119866 (119909 119906 119908))

(18)

If there exists 1199090 1199100isin 119883 such that 119909

0⪯ 119879(119909

0 1199100) and 119910

0⪰

119879(1199100 1199090) then 119879 has a coupled fixed point (119909lowast 119910lowast) isin 119883 times 119883

That is (119909lowast 119910lowast) satisfies 119909lowast = 119879(119909lowast 119910lowast) 119910lowast= 119879(119910

lowast 119909lowast)

Proof Since 120601 isin W120572and 120595 isin W

120573 it follows from

Definition 17 that 120601(119905) le 120572119905 and 120595(119905) le 120573119905 for all 119905 gt 0Inequality (18) now becomes equivalent to inequality (3) ofTheorem 16 and the proof is immediate

Example 19 Let 119883 = [0infin) and consider the function 119866

119883 times 119883 times 119883 rarr R+ defined by

119866 (119909 119910 119911) = 0 if 119909 = 119910 = 119911

max 119909 119910 119911 otherwise(19)

Then (119883 119866) is a complete 119866-metric space [23] We define apartial order ⪯ on119883 by the following for any 119909 119910 isin 119883 119909 ⪯ 119910

if 119909 ge 119910 Also let 119865 119883 times 119883 rarr 119883 be defined by

119865 (119909 119910) = 1 if 119909 ⪯ 119910

0 otherwise(20)

Suppose 119909 119910 119906 V 119908 119911 isin 119883 satisfy 119911 ⪯ V ⪯ 119910 ⪯ 119909 ⪯ 119906 ⪯ 119908

with nonzero 119909 119906 119908 but are otherwise arbitrary Then we

have 119908 le 119906 le 119909 le 119910 le V le 119911 So the left side of (3) is119866(0 0 0) = 0 The right side of (3)

= 120572119866 (119909 0 0) 119866 (119906 0 0) 119866 (119908 0 0)

[119866 (119909 119906 119908)]2

+ 120573119866 (119909 119906 119908)

= 120572119909119906119908

1199092+ 120573119908

=120572119906119908 + 120573119909119908

119909gt 0 with 120572 =

1

16 120573 =

1

3

(21)

If 1199090= 0 and 119910

0= 1 then 119909

0⪯ 119865(119909

0 1199100) and 119910

0⪰ 119865(119910

0 1199090)

So all conditions of Theorem 16 are satisfied Easily we findthat (0 1) is a coupled fixed point of 119865 Similarly (1 0) is acoupled fixed point

In the next theorem we provide conditions underwhich the coupled fixed point of the map 119879 established inTheorem 16 is unique

Theorem 20 Suppose that the conditions of Theorem 16 arevalid In addition suppose that for each (119909 119910) (119906 V) isin 119883 times 119883

there is a (119908 119911) isin 119883 times 119883 which is comparable to (119909 119910) and(119906 V) Then 119879 has a unique coupled fixed point

Proof Suppose that (119909lowast 119910lowast) (1199091015840 1199101015840) isin 119883 times 119883 are coupledfixed points

Case 1 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are comparable

119866(119879 (119909lowast

119910lowast

) 119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))

le 120572 (119866 (119909lowast

119879 (119909lowast

119910lowast

) 119879 (119909lowast

119910lowast

))

times [119866 (1199091015840

119879 (1199091015840

1199101015840

) 119879 (1199091015840

1199101015840

))]2

times ([119866 (119909lowast

1199091015840

1199091015840

)]2

)

minus1

)

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120572

119866 (119909lowast 119909lowast 119909lowast) [119866 (119909

1015840 1199091015840 1199091015840)]2

[119866 (119909lowast 1199091015840 1199091015840)]2

+ 120573119866 (119909lowast

1199091015840

1199091015840

)

= 120573119866 (119909lowast

1199091015840

1199091015840

)

(22)

This is equivalent to 119866(119909lowast 1199091015840 1199091015840) le 120573119866(119909lowast 1199091015840 1199091015840) However

this is a contradiction since 120573 lt 1 So we must have 119909lowast = 1199091015840

Similarly considering 119866(119879(119910lowast 119909lowast) 119879(119910

1015840 1199091015840) 119879(119910

1015840 1199091015840)) we

easily show that 119910lowast = 1199101015840 This shows that (119909lowast 119910lowast) = (119909

1015840 1199101015840)

so the coupled fixed point is unique

Case 2 If (119909lowast 119910lowast) and (1199091015840 1199101015840) are not comparable by the

condition of the theorem there is a (119906 V) isin 119883times119883 comparable

Journal of Mathematics 5

to (119909lowast 119910lowast) and (119909

1015840 1199101015840) If there is a positive integer 119899

0such

that 1198791198990(119906 V) = (119909lowast 119910lowast) then

1198791198990 (119906 V) = (119909

lowast

119910lowast

)

1198791198990+1

(119906 V) = 119879 (119909lowast

119910lowast

) = 119909lowast

1198791198990+2

(119906 V) = 1198792

(119909lowast

119910lowast

) = 119879 (119879 (119909lowast

119910lowast

) 119879 (119910lowast

119909lowast

))

= 119879 (119909lowast

119910lowast

) = 119909lowast

(23)

So 119879119899(119906 V) = 119909

lowast for 119899 ge 1198990and hence 119879119899(119906 V) rarr 119909

lowast as119899 rarr infin

On the other hand if no such 1198990exists we have that for

any 119899 ge 1

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

= 119866 (119879119899

(119906 V) 119879119899 (119909lowast 119910lowast) 119879119899 (119909lowast 119910lowast))

le 120572 (119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V))

times [119866 (119879119899minus1

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

))]2

times ( [119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast)

119879119899minus1

(119909lowast

119910lowast

))]2

)

minus1

)

+ 120573119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast) 119879119899minus1 (119909lowast 119910lowast)) (24)

where we have used the fact that 119879119899(119909 119910) = 119879(119879119899minus1

(119909

119910) 119879119899minus1

(119910 119909)) for any (119909 119910) isin 119883 times 119883 Since (119909lowast 119910lowast) is acoupled fixed point of 119879 119879119899(119909lowast 119910lowast) = 119909

lowast for all 119899 ge 1 andfrom (24) we now deduce that

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

le 120572

119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V)) [119866 (119909lowast 119909lowast 119909lowast)]2

[119866 (119879119899minus1 (119906 V) 119909lowast 119909lowast)]2

+ 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119899minus1

119866 (119879 (119906 V) 119909lowast 119909lowast) 997888rarr 0 as 119899 997888rarr infin

(25)

since 120573 lt 1This proves119879119899(119906 V) rarr 119909lowast as 119899 rarr infin Similarly

we can show that 119879119899(V 119906) rarr 119910lowast as 119899 rarr infin Replacing 119909lowast

with 1199091015840 and 119910lowast with 1199101015840 and repeating the above arguments wecan deduce 119879119899(119906 V) rarr 119909

1015840 and 119879119899(V 119906) rarr 119910

1015840 as 119899 rarr infinBut thismeans 119909lowast = 119909

1015840 and119910lowast = 1199101015840 So the (119909lowast 119910lowast) = (119909

1015840 1199101015840)

and the coupled fixed point is unique

We next establish the conditions under which two maps119879 119883times119883 rarr 119883 and 119892 119883 rarr 119883 have a coupled coincidencepoint

Theorem 21 Let (119883 ⪯) be a partially ordered set and let 119866 bea metric on 119883 such that (119883 119866) is a complete 119866-metric spaceLet 119879 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 be continuous map-ping on 119883 such that 119879 has the mixed 119892-monotone propertySuppose that 119879(119883 times 119883) sube 119892(119883) 119892 commutes with 119879 and for(119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911) and119892119909 ⪯ 119892119906 ⪯ 119892119908 119892119910 ⪰ 119892V ⪰ 119892119911

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120572 (119866 (119892119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119892119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119892119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119892119909 119892119906 119892119908)]2

)minus1

)

+ 120573119866 (119892119909 119892119906 119892119908)

(26)

where 8120572 + 120573 lt 1 If there exists 1199090 1199100

isin 119883 such that1198921199090⪯ 119879(119909

0 1199100) and 119892119910

0⪰ 119879(119910

0 1199090) then 119879 has a coupled

coincidence point (119909lowast 119910lowast) isin 119883 times 119883 That is (119909lowast 119910lowast) satisfies119892119909lowast= 119879(119909

lowast 119910lowast) 119892119910lowast= 119879(119910

lowast 119909lowast)

Proof Since119879(119883times119883) sube 119892(119883)we can choose 1199091 1199101isin 119883 such

that 1198921199091= 119879(119909

0 1199100) 1198921199101= 119879(119910

0 1199090) For similar reasons

1199092 1199102isin 119883 can be found such that 119892119909

2= 119879(119909

1 1199101) 1198921199102=

119879(1199101 1199091) Due to the mixed 119892-monotone property of 119879 we

have 1198921199090⪯ 1198921199091⪯ 1198921199092and 119892119910

2⪯ 1198921199101⪯ 1198921199100 In general it

can be shown that [4] for 119899 ge 0119892119909119899= 119879 (119909

119899minus1 119910119899minus1

) ⪯ 119892119909119899+1

= 119879 (119909119899 119910119899)

119892119910119899+1

= 119879 (119910119899 119909119899) ⪯ 119892119910

119899= 119879 (119909

119899minus1 119910119899minus1

)

(27)

Now by the same arguments used to deduce (10) we have

119866 (119892119909119899+1

119892119909119899 119892119909119899)

le 4120572119866 (119892119909

119899 119892119909119899+1

119892119909119899+1

) [119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

[119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

= 4120572119866 (119892119909119899 119892119909119899+1

119892119909119899+1

) + 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

le 8120572119866 (119892119909119899+1

119892119909119899 119892119909119899)

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

(28)

This gives

119866 (119892119909119899+1

119892119909119899 119892119909119899) le (

120573

1 minus 8120572)119866 (119892119909

119899 119892119909119899minus1

119909119899minus1

)

(29)

Since 8120572 + 120573 lt 1 119896 = 120573(1 minus 8120572) lt 1 Writing 119889119899=

119866(119892119909119899+1

119892119909119899 119892119909119899) we find from (28) that

119889119899le 119896119889119899minus1

le 1198962

119889119899minus1

le sdot sdot sdot le 119896119899

1198890 (30)

6 Journal of Mathematics

If 1198890= 0 then119866(119892119909

1 1198921199090 1198921199090) = 119866(119879(119909

0 1199100) 1198921199090 1198921199090) = 0

giving 1198921199090= 119879(119909

0 1199100) so 119909

0is a coincidence point However

if 1198890gt 0 we have for119898 gt 119899

119866 (119892119909119898 119892119909119899 119892119909119899)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119898 119892119909119899+1

119892119909119899+1

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ 119866 (119892119909119898 119892119909119899+2

119892119909119899+2

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ sdot sdot sdot + 119866 (119892119909119898 119892119909119898minus1

119892119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ sdot sdot sdot + 119896119898minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin

(31)

since 119896 lt 1 This shows that 119892119909119899 is a Cauchy sequence in

(119883 119866) and 119866-completeness of (119883 119866) ensures a point 119909lowast isin 119883

such that 119892119909119899rarr 119909lowast as 119899 rarr infin

Replacing 119909119899by 119910119899for all 119899 ge 0 we get the analogue of

(29)

119866 (119892119910119899+1

119892119910119899 119892119910119899) le (

120573

1 minus 8120572)119866 (119892119910

119899 119892119910119899minus1

119910119899minus1

) (32)

Next writing 120575119899= 119866(119892119910

119899+1 119892119910119899 119892119910119899) we find as in (30) that

120575119899le 119896120575119899minus1

le 1198962

120575119899minus1

le sdot sdot sdot le 119896119899

1205750 (33)

As before if 1205750= 0 119866(119892119910

1 1198921199100 1198921199100) = 119866(119879(119910

0 1199090) 1198921199100

1198921199100) = 0 giving 119879(119910

0 1199090) = 119892119910

0so that 119910

0is a coinci-dence

point If in addition 1198890= 0 we have 119892119909

0= 119879(119909

0 1199100) and

1198921199100= 119879(119910

0 1199090) So (119909

0 1199100) is a coupled coincidence point

Proceeding as in (31) we can show further that 119892119910119899 is

a Cauchy sequence in (119883 119866) and due to 119866-completeness of(119883 119866) there is a point 119910lowast isin 119883 such that 119892119910

119899rarr 119910lowast as 119899 rarr

infinFinally we prove that (119909lowast 119910lowast) is a coupled coincident

point Since 119879 and 119892 commute we have

119892 (119892119909119899+1

) = 119892 (119879 (119909119899 119910119899)) = 119879 (119892119909

119899 119892119910119899)

119892 (119892119910119899+1

) = 119892 (119879 (119910119899 119909119899)) = 119879 (119892119910

119899 119892119909119899)

(34)

Taking limits as 119899 rarr infin in (34) and noting that 119879 and 119892 arerespectively continuous on119883 times 119883 and119883 we get

119892119909lowast

= lim119892 (119892119909119899+1

) = lim119892 (119879 (119909119899 119910119899)) = lim119879 (119892119909

119899 119892119910119899)

119892119910lowast

= lim119892 (119892119910119899+1

) = lim119892 (119879 (119910119899 119909119899)) = lim119879 (119892119910

119899 119892119909119899)

(35)

Next we observe that 119866 as metric is continuous in all itsvariables This finally leads to

119866 (119879 (119909lowast

119910lowast

) 119892119909lowast

119892119909lowast

) = 119866 (lim119879 (119892119909119899 119892119910119899) 119892119909lowast

119892119909lowast

)

= 119866 (119892119909lowast

119892119909lowast

119892119909lowast

)

= 0

(36)

So 119892119909lowast = 119879(119909lowast 119910lowast) Similarly we show that 119892119910lowast = 119879(119910

lowast 119909lowast)

This proves (119909lowast 119910lowast) is a coupled coincidence point

4 Conclusion

To summarize we have introduced a rational type contractivecondition in a 119866-metric space and proved some coupledfixed point theorems for maps satisfying mixed monotoneproperty We established the conditions for uniqueness of thecoupled fixed point Conditions for the existence of coupledcoincidence points of two maps are also deduced

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[2] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006

[3] L Ciric M O Olatinwo D Gopal and G Akinbo ldquoCoupledfixed point theorems for mappings satisfying a contractivecondition of rational type on a partially ordered metric spacerdquoAdvances in Fixed Point Theory vol 2 no 1 pp 1ndash8 2012

[4] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[5] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010

[6] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[7] V Berinde ldquoGeneralized coupled fixed point theorems formixed monotone mappings in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no18 pp 7347ndash7355 2011

[8] B S Choudhury and P Maity ldquoCoupled fixed point resultsin generalized metric spacesrdquo Mathematical and ComputerModelling vol 54 no 1-2 pp 73ndash79 2011

Journal of Mathematics 7

[9] R Saadati S M Vaezpour P Vetro and B E RhoadesldquoFixed point theorems in generalized partially ordered 119866-met-ric spacesrdquo Mathematical and Computer Modelling vol 52 no5-6 pp 797ndash801 2010

[10] Z Mustafa J R Roshan and V Parvaneh ldquoCoupled coinci-dence point results for (120595 120593)-weakly contractive mappings inpartially ordered 119866

119887-metric spacesrdquo Fixed Point Theory and

Applications vol 2013 article 206 2013[11] ZMustafa J R Roshan andV Parvaneh ldquoExistence of a tripled

coincidence point in ordered G119887-metric spaces and applications

to a system of integral equationsrdquo Journal of Inequalities andApplications vol 2013 article no 453 2013

[12] V ParvanehA Razani and J R Roshan ldquoCommonfixed pointsof six mappings in partially ordered 119866-metric spacesrdquo Math-ematical Sciences vol 7 article 18 2013

[13] A Razani and V Parvaneh ldquoOn generalized weakly G-contractive mappings in partially ordered G-metric spacesrdquoAbstract and Applied Analysis vol 2012 Article ID 701910 18pages 2012

[14] Z Mustafa V Parvaneh M Abbas and J Rezaei RoshanldquoSome coincidence point results for generalized (120595 120593)-weaklycontractive mappings in ordered G-metric spaces Fixed PointTheory and Applicationsrdquo Fixed Point Theory and Applicationsvol 2013 article 326 2013

[15] M A Kutbi N Hussain J R Roshan and V ParvanehldquoCoupled and tripled coincidence point results with applicationto Fredholm integral equationsrdquo Abstract and Applied Analysisvol 2014 Article ID 568718 18 pages 2014

[16] N Hussain V Parvaneh and J R Roshan ldquoFixed point resultsfor G-120572-contractive maps with application to boundary valueproblemsrdquo ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[17] A Latif N Hussain J R Roshan and V Parvaneh ldquoA uni-fication of Gminusmetric partial metric and b-metric spacesrdquoAbstract andApplied Analysis vol 2014 Article ID 180698 2014

[18] L Ciric S M Alsulami V Parvaneh and J R Roshan ldquoSomefixed point results in ordered 119866

119901-metric spacesrdquo Fixed Point

Theory and Applications vol 2013 article 317 2013[19] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journal

of Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977[20] Z Mustafa and B Sims ldquoA new approach to generalized metric

spacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash296 2006

[21] C-M Chen ldquoCommon fixed-point theorems in completegeneralized metric spacesrdquo Journal of Applied Mathematics vol2012 Article ID 945915 14 pages 2012

[22] K Chakrabarti ldquoFixed point theorems in G-metric spaces withW mapsrdquoMathematical Sciences Letters vol 2 pp 29ndash35 2013

[23] E Karapınar and R P Agarwal ldquoFurther fixed point results on119866-metric spacesrdquo Fixed PointTheory andApplications vol 2013article 154 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 5

to (119909lowast 119910lowast) and (119909

1015840 1199101015840) If there is a positive integer 119899

0such

that 1198791198990(119906 V) = (119909lowast 119910lowast) then

1198791198990 (119906 V) = (119909

lowast

119910lowast

)

1198791198990+1

(119906 V) = 119879 (119909lowast

119910lowast

) = 119909lowast

1198791198990+2

(119906 V) = 1198792

(119909lowast

119910lowast

) = 119879 (119879 (119909lowast

119910lowast

) 119879 (119910lowast

119909lowast

))

= 119879 (119909lowast

119910lowast

) = 119909lowast

(23)

So 119879119899(119906 V) = 119909

lowast for 119899 ge 1198990and hence 119879119899(119906 V) rarr 119909

lowast as119899 rarr infin

On the other hand if no such 1198990exists we have that for

any 119899 ge 1

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

= 119866 (119879119899

(119906 V) 119879119899 (119909lowast 119910lowast) 119879119899 (119909lowast 119910lowast))

le 120572 (119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V))

times [119866 (119879119899minus1

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

) 119879119899

(119909lowast

119910lowast

))]2

times ( [119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast)

119879119899minus1

(119909lowast

119910lowast

))]2

)

minus1

)

+ 120573119866 (119879119899minus1

(119906 V) 119879119899minus1 (119909lowast 119910lowast) 119879119899minus1 (119909lowast 119910lowast)) (24)

where we have used the fact that 119879119899(119909 119910) = 119879(119879119899minus1

(119909

119910) 119879119899minus1

(119910 119909)) for any (119909 119910) isin 119883 times 119883 Since (119909lowast 119910lowast) is acoupled fixed point of 119879 119879119899(119909lowast 119910lowast) = 119909

lowast for all 119899 ge 1 andfrom (24) we now deduce that

119866 (119879119899

(119906 V) 119909lowast 119909lowast)

le 120572

119866 (119879119899minus1

(119906 V) 119879119899 (119906 V) 119879119899 (119906 V)) [119866 (119909lowast 119909lowast 119909lowast)]2

[119866 (119879119899minus1 (119906 V) 119909lowast 119909lowast)]2

+ 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119866 (119879119899minus1

(119906 V) 119909lowast 119909lowast)

= 120573119899minus1

119866 (119879 (119906 V) 119909lowast 119909lowast) 997888rarr 0 as 119899 997888rarr infin

(25)

since 120573 lt 1This proves119879119899(119906 V) rarr 119909lowast as 119899 rarr infin Similarly

we can show that 119879119899(V 119906) rarr 119910lowast as 119899 rarr infin Replacing 119909lowast

with 1199091015840 and 119910lowast with 1199101015840 and repeating the above arguments wecan deduce 119879119899(119906 V) rarr 119909

1015840 and 119879119899(V 119906) rarr 119910

1015840 as 119899 rarr infinBut thismeans 119909lowast = 119909

1015840 and119910lowast = 1199101015840 So the (119909lowast 119910lowast) = (119909

1015840 1199101015840)

and the coupled fixed point is unique

We next establish the conditions under which two maps119879 119883times119883 rarr 119883 and 119892 119883 rarr 119883 have a coupled coincidencepoint

Theorem 21 Let (119883 ⪯) be a partially ordered set and let 119866 bea metric on 119883 such that (119883 119866) is a complete 119866-metric spaceLet 119879 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 be continuous map-ping on 119883 such that 119879 has the mixed 119892-monotone propertySuppose that 119879(119883 times 119883) sube 119892(119883) 119892 commutes with 119879 and for(119909 119910) (119906 V) (119908 119911) isin 119883 times 119883 with (119909 119910) ⪯ (119906 V) ⪯ (119908 119911) and119892119909 ⪯ 119892119906 ⪯ 119892119908 119892119910 ⪰ 119892V ⪰ 119892119911

119866 (119879 (119909 119910) 119879 (119906 V) 119879 (119908 119911))

le 120572 (119866 (119892119909 119879 (119909 119910) 119879 (119909 119910)) 119866 (119892119906 119879 (119906 V) 119879 (119906 V))

times 119866 (119892119908 119879 (119908 119911) 119879 (119908 119911)) ([119866 (119892119909 119892119906 119892119908)]2

)minus1

)

+ 120573119866 (119892119909 119892119906 119892119908)

(26)

where 8120572 + 120573 lt 1 If there exists 1199090 1199100

isin 119883 such that1198921199090⪯ 119879(119909

0 1199100) and 119892119910

0⪰ 119879(119910

0 1199090) then 119879 has a coupled

coincidence point (119909lowast 119910lowast) isin 119883 times 119883 That is (119909lowast 119910lowast) satisfies119892119909lowast= 119879(119909

lowast 119910lowast) 119892119910lowast= 119879(119910

lowast 119909lowast)

Proof Since119879(119883times119883) sube 119892(119883)we can choose 1199091 1199101isin 119883 such

that 1198921199091= 119879(119909

0 1199100) 1198921199101= 119879(119910

0 1199090) For similar reasons

1199092 1199102isin 119883 can be found such that 119892119909

2= 119879(119909

1 1199101) 1198921199102=

119879(1199101 1199091) Due to the mixed 119892-monotone property of 119879 we

have 1198921199090⪯ 1198921199091⪯ 1198921199092and 119892119910

2⪯ 1198921199101⪯ 1198921199100 In general it

can be shown that [4] for 119899 ge 0119892119909119899= 119879 (119909

119899minus1 119910119899minus1

) ⪯ 119892119909119899+1

= 119879 (119909119899 119910119899)

119892119910119899+1

= 119879 (119910119899 119909119899) ⪯ 119892119910

119899= 119879 (119909

119899minus1 119910119899minus1

)

(27)

Now by the same arguments used to deduce (10) we have

119866 (119892119909119899+1

119892119909119899 119892119909119899)

le 4120572119866 (119892119909

119899 119892119909119899+1

119892119909119899+1

) [119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

[119866 (119892119909119899minus1

119892119909119899 119892119909119899)]2

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

= 4120572119866 (119892119909119899 119892119909119899+1

119892119909119899+1

) + 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

le 8120572119866 (119892119909119899+1

119892119909119899 119892119909119899)

+ 120573119866 (119892119909119899 119892119909119899minus1

119892119909119899minus1

)

(28)

This gives

119866 (119892119909119899+1

119892119909119899 119892119909119899) le (

120573

1 minus 8120572)119866 (119892119909

119899 119892119909119899minus1

119909119899minus1

)

(29)

Since 8120572 + 120573 lt 1 119896 = 120573(1 minus 8120572) lt 1 Writing 119889119899=

119866(119892119909119899+1

119892119909119899 119892119909119899) we find from (28) that

119889119899le 119896119889119899minus1

le 1198962

119889119899minus1

le sdot sdot sdot le 119896119899

1198890 (30)

6 Journal of Mathematics

If 1198890= 0 then119866(119892119909

1 1198921199090 1198921199090) = 119866(119879(119909

0 1199100) 1198921199090 1198921199090) = 0

giving 1198921199090= 119879(119909

0 1199100) so 119909

0is a coincidence point However

if 1198890gt 0 we have for119898 gt 119899

119866 (119892119909119898 119892119909119899 119892119909119899)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119898 119892119909119899+1

119892119909119899+1

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ 119866 (119892119909119898 119892119909119899+2

119892119909119899+2

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ sdot sdot sdot + 119866 (119892119909119898 119892119909119898minus1

119892119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ sdot sdot sdot + 119896119898minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin

(31)

since 119896 lt 1 This shows that 119892119909119899 is a Cauchy sequence in

(119883 119866) and 119866-completeness of (119883 119866) ensures a point 119909lowast isin 119883

such that 119892119909119899rarr 119909lowast as 119899 rarr infin

Replacing 119909119899by 119910119899for all 119899 ge 0 we get the analogue of

(29)

119866 (119892119910119899+1

119892119910119899 119892119910119899) le (

120573

1 minus 8120572)119866 (119892119910

119899 119892119910119899minus1

119910119899minus1

) (32)

Next writing 120575119899= 119866(119892119910

119899+1 119892119910119899 119892119910119899) we find as in (30) that

120575119899le 119896120575119899minus1

le 1198962

120575119899minus1

le sdot sdot sdot le 119896119899

1205750 (33)

As before if 1205750= 0 119866(119892119910

1 1198921199100 1198921199100) = 119866(119879(119910

0 1199090) 1198921199100

1198921199100) = 0 giving 119879(119910

0 1199090) = 119892119910

0so that 119910

0is a coinci-dence

point If in addition 1198890= 0 we have 119892119909

0= 119879(119909

0 1199100) and

1198921199100= 119879(119910

0 1199090) So (119909

0 1199100) is a coupled coincidence point

Proceeding as in (31) we can show further that 119892119910119899 is

a Cauchy sequence in (119883 119866) and due to 119866-completeness of(119883 119866) there is a point 119910lowast isin 119883 such that 119892119910

119899rarr 119910lowast as 119899 rarr

infinFinally we prove that (119909lowast 119910lowast) is a coupled coincident

point Since 119879 and 119892 commute we have

119892 (119892119909119899+1

) = 119892 (119879 (119909119899 119910119899)) = 119879 (119892119909

119899 119892119910119899)

119892 (119892119910119899+1

) = 119892 (119879 (119910119899 119909119899)) = 119879 (119892119910

119899 119892119909119899)

(34)

Taking limits as 119899 rarr infin in (34) and noting that 119879 and 119892 arerespectively continuous on119883 times 119883 and119883 we get

119892119909lowast

= lim119892 (119892119909119899+1

) = lim119892 (119879 (119909119899 119910119899)) = lim119879 (119892119909

119899 119892119910119899)

119892119910lowast

= lim119892 (119892119910119899+1

) = lim119892 (119879 (119910119899 119909119899)) = lim119879 (119892119910

119899 119892119909119899)

(35)

Next we observe that 119866 as metric is continuous in all itsvariables This finally leads to

119866 (119879 (119909lowast

119910lowast

) 119892119909lowast

119892119909lowast

) = 119866 (lim119879 (119892119909119899 119892119910119899) 119892119909lowast

119892119909lowast

)

= 119866 (119892119909lowast

119892119909lowast

119892119909lowast

)

= 0

(36)

So 119892119909lowast = 119879(119909lowast 119910lowast) Similarly we show that 119892119910lowast = 119879(119910

lowast 119909lowast)

This proves (119909lowast 119910lowast) is a coupled coincidence point

4 Conclusion

To summarize we have introduced a rational type contractivecondition in a 119866-metric space and proved some coupledfixed point theorems for maps satisfying mixed monotoneproperty We established the conditions for uniqueness of thecoupled fixed point Conditions for the existence of coupledcoincidence points of two maps are also deduced

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[2] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006

[3] L Ciric M O Olatinwo D Gopal and G Akinbo ldquoCoupledfixed point theorems for mappings satisfying a contractivecondition of rational type on a partially ordered metric spacerdquoAdvances in Fixed Point Theory vol 2 no 1 pp 1ndash8 2012

[4] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[5] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010

[6] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[7] V Berinde ldquoGeneralized coupled fixed point theorems formixed monotone mappings in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no18 pp 7347ndash7355 2011

[8] B S Choudhury and P Maity ldquoCoupled fixed point resultsin generalized metric spacesrdquo Mathematical and ComputerModelling vol 54 no 1-2 pp 73ndash79 2011

Journal of Mathematics 7

[9] R Saadati S M Vaezpour P Vetro and B E RhoadesldquoFixed point theorems in generalized partially ordered 119866-met-ric spacesrdquo Mathematical and Computer Modelling vol 52 no5-6 pp 797ndash801 2010

[10] Z Mustafa J R Roshan and V Parvaneh ldquoCoupled coinci-dence point results for (120595 120593)-weakly contractive mappings inpartially ordered 119866

119887-metric spacesrdquo Fixed Point Theory and

Applications vol 2013 article 206 2013[11] ZMustafa J R Roshan andV Parvaneh ldquoExistence of a tripled

coincidence point in ordered G119887-metric spaces and applications

to a system of integral equationsrdquo Journal of Inequalities andApplications vol 2013 article no 453 2013

[12] V ParvanehA Razani and J R Roshan ldquoCommonfixed pointsof six mappings in partially ordered 119866-metric spacesrdquo Math-ematical Sciences vol 7 article 18 2013

[13] A Razani and V Parvaneh ldquoOn generalized weakly G-contractive mappings in partially ordered G-metric spacesrdquoAbstract and Applied Analysis vol 2012 Article ID 701910 18pages 2012

[14] Z Mustafa V Parvaneh M Abbas and J Rezaei RoshanldquoSome coincidence point results for generalized (120595 120593)-weaklycontractive mappings in ordered G-metric spaces Fixed PointTheory and Applicationsrdquo Fixed Point Theory and Applicationsvol 2013 article 326 2013

[15] M A Kutbi N Hussain J R Roshan and V ParvanehldquoCoupled and tripled coincidence point results with applicationto Fredholm integral equationsrdquo Abstract and Applied Analysisvol 2014 Article ID 568718 18 pages 2014

[16] N Hussain V Parvaneh and J R Roshan ldquoFixed point resultsfor G-120572-contractive maps with application to boundary valueproblemsrdquo ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[17] A Latif N Hussain J R Roshan and V Parvaneh ldquoA uni-fication of Gminusmetric partial metric and b-metric spacesrdquoAbstract andApplied Analysis vol 2014 Article ID 180698 2014

[18] L Ciric S M Alsulami V Parvaneh and J R Roshan ldquoSomefixed point results in ordered 119866

119901-metric spacesrdquo Fixed Point

Theory and Applications vol 2013 article 317 2013[19] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journal

of Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977[20] Z Mustafa and B Sims ldquoA new approach to generalized metric

spacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash296 2006

[21] C-M Chen ldquoCommon fixed-point theorems in completegeneralized metric spacesrdquo Journal of Applied Mathematics vol2012 Article ID 945915 14 pages 2012

[22] K Chakrabarti ldquoFixed point theorems in G-metric spaces withW mapsrdquoMathematical Sciences Letters vol 2 pp 29ndash35 2013

[23] E Karapınar and R P Agarwal ldquoFurther fixed point results on119866-metric spacesrdquo Fixed PointTheory andApplications vol 2013article 154 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Mathematics

If 1198890= 0 then119866(119892119909

1 1198921199090 1198921199090) = 119866(119879(119909

0 1199100) 1198921199090 1198921199090) = 0

giving 1198921199090= 119879(119909

0 1199100) so 119909

0is a coincidence point However

if 1198890gt 0 we have for119898 gt 119899

119866 (119892119909119898 119892119909119899 119892119909119899)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119898 119892119909119899+1

119892119909119899+1

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ 119866 (119892119909119898 119892119909119899+2

119892119909119899+2

)

le 119866 (119892119909119899+1

119892119909119899 119892119909119899) + 119866 (119892119909

119899+2 119892119909119899+1

119892119909119899+1

)

+ sdot sdot sdot + 119866 (119892119909119898 119892119909119898minus1

119892119909119898minus1

)

= 119889119899+ 119889119899+1

+ 119889119899+2

+ sdot sdot sdot + 119889119898minus1

le [119896119899

+ 119896119899+1

+ sdot sdot sdot + 119896119898minus1

] 1198890

lt119896119899

1 minus 1198961198890997888rarr 0 as 119899 997888rarr infin

(31)

since 119896 lt 1 This shows that 119892119909119899 is a Cauchy sequence in

(119883 119866) and 119866-completeness of (119883 119866) ensures a point 119909lowast isin 119883

such that 119892119909119899rarr 119909lowast as 119899 rarr infin

Replacing 119909119899by 119910119899for all 119899 ge 0 we get the analogue of

(29)

119866 (119892119910119899+1

119892119910119899 119892119910119899) le (

120573

1 minus 8120572)119866 (119892119910

119899 119892119910119899minus1

119910119899minus1

) (32)

Next writing 120575119899= 119866(119892119910

119899+1 119892119910119899 119892119910119899) we find as in (30) that

120575119899le 119896120575119899minus1

le 1198962

120575119899minus1

le sdot sdot sdot le 119896119899

1205750 (33)

As before if 1205750= 0 119866(119892119910

1 1198921199100 1198921199100) = 119866(119879(119910

0 1199090) 1198921199100

1198921199100) = 0 giving 119879(119910

0 1199090) = 119892119910

0so that 119910

0is a coinci-dence

point If in addition 1198890= 0 we have 119892119909

0= 119879(119909

0 1199100) and

1198921199100= 119879(119910

0 1199090) So (119909

0 1199100) is a coupled coincidence point

Proceeding as in (31) we can show further that 119892119910119899 is

a Cauchy sequence in (119883 119866) and due to 119866-completeness of(119883 119866) there is a point 119910lowast isin 119883 such that 119892119910

119899rarr 119910lowast as 119899 rarr

infinFinally we prove that (119909lowast 119910lowast) is a coupled coincident

point Since 119879 and 119892 commute we have

119892 (119892119909119899+1

) = 119892 (119879 (119909119899 119910119899)) = 119879 (119892119909

119899 119892119910119899)

119892 (119892119910119899+1

) = 119892 (119879 (119910119899 119909119899)) = 119879 (119892119910

119899 119892119909119899)

(34)

Taking limits as 119899 rarr infin in (34) and noting that 119879 and 119892 arerespectively continuous on119883 times 119883 and119883 we get

119892119909lowast

= lim119892 (119892119909119899+1

) = lim119892 (119879 (119909119899 119910119899)) = lim119879 (119892119909

119899 119892119910119899)

119892119910lowast

= lim119892 (119892119910119899+1

) = lim119892 (119879 (119910119899 119909119899)) = lim119879 (119892119910

119899 119892119909119899)

(35)

Next we observe that 119866 as metric is continuous in all itsvariables This finally leads to

119866 (119879 (119909lowast

119910lowast

) 119892119909lowast

119892119909lowast

) = 119866 (lim119879 (119892119909119899 119892119910119899) 119892119909lowast

119892119909lowast

)

= 119866 (119892119909lowast

119892119909lowast

119892119909lowast

)

= 0

(36)

So 119892119909lowast = 119879(119909lowast 119910lowast) Similarly we show that 119892119910lowast = 119879(119910

lowast 119909lowast)

This proves (119909lowast 119910lowast) is a coupled coincidence point

4 Conclusion

To summarize we have introduced a rational type contractivecondition in a 119866-metric space and proved some coupledfixed point theorems for maps satisfying mixed monotoneproperty We established the conditions for uniqueness of thecoupled fixed point Conditions for the existence of coupledcoincidence points of two maps are also deduced

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] A C M Ran and M C B Reurings ldquoA fixed point theoremin partially ordered sets and some applications to matrixequationsrdquo Proceedings of the American Mathematical Societyvol 132 no 5 pp 1435ndash1443 2004

[2] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysisTheoryMethodsampApplications vol 65 no 7 pp 1379ndash1393 2006

[3] L Ciric M O Olatinwo D Gopal and G Akinbo ldquoCoupledfixed point theorems for mappings satisfying a contractivecondition of rational type on a partially ordered metric spacerdquoAdvances in Fixed Point Theory vol 2 no 1 pp 1ndash8 2012

[4] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquoNonlinearAnalysisTheoryMethodsampApplications vol70 no 12 pp 4341ndash4349 2009

[5] J Harjani and K Sadarangani ldquoGeneralized contractions inpartially ordered metric spaces and applications to ordinarydifferential equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 72 no 3-4 pp 1188ndash1197 2010

[6] J Harjani B Lopez and K Sadarangani ldquoA fixed point theoremfor mappings satisfying a contractive condition of rationaltype on a partially ordered metric spacerdquo Abstract and AppliedAnalysis vol 2010 Article ID 190701 8 pages 2010

[7] V Berinde ldquoGeneralized coupled fixed point theorems formixed monotone mappings in partially ordered metric spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 74 no18 pp 7347ndash7355 2011

[8] B S Choudhury and P Maity ldquoCoupled fixed point resultsin generalized metric spacesrdquo Mathematical and ComputerModelling vol 54 no 1-2 pp 73ndash79 2011

Journal of Mathematics 7

[9] R Saadati S M Vaezpour P Vetro and B E RhoadesldquoFixed point theorems in generalized partially ordered 119866-met-ric spacesrdquo Mathematical and Computer Modelling vol 52 no5-6 pp 797ndash801 2010

[10] Z Mustafa J R Roshan and V Parvaneh ldquoCoupled coinci-dence point results for (120595 120593)-weakly contractive mappings inpartially ordered 119866

119887-metric spacesrdquo Fixed Point Theory and

Applications vol 2013 article 206 2013[11] ZMustafa J R Roshan andV Parvaneh ldquoExistence of a tripled

coincidence point in ordered G119887-metric spaces and applications

to a system of integral equationsrdquo Journal of Inequalities andApplications vol 2013 article no 453 2013

[12] V ParvanehA Razani and J R Roshan ldquoCommonfixed pointsof six mappings in partially ordered 119866-metric spacesrdquo Math-ematical Sciences vol 7 article 18 2013

[13] A Razani and V Parvaneh ldquoOn generalized weakly G-contractive mappings in partially ordered G-metric spacesrdquoAbstract and Applied Analysis vol 2012 Article ID 701910 18pages 2012

[14] Z Mustafa V Parvaneh M Abbas and J Rezaei RoshanldquoSome coincidence point results for generalized (120595 120593)-weaklycontractive mappings in ordered G-metric spaces Fixed PointTheory and Applicationsrdquo Fixed Point Theory and Applicationsvol 2013 article 326 2013

[15] M A Kutbi N Hussain J R Roshan and V ParvanehldquoCoupled and tripled coincidence point results with applicationto Fredholm integral equationsrdquo Abstract and Applied Analysisvol 2014 Article ID 568718 18 pages 2014

[16] N Hussain V Parvaneh and J R Roshan ldquoFixed point resultsfor G-120572-contractive maps with application to boundary valueproblemsrdquo ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[17] A Latif N Hussain J R Roshan and V Parvaneh ldquoA uni-fication of Gminusmetric partial metric and b-metric spacesrdquoAbstract andApplied Analysis vol 2014 Article ID 180698 2014

[18] L Ciric S M Alsulami V Parvaneh and J R Roshan ldquoSomefixed point results in ordered 119866

119901-metric spacesrdquo Fixed Point

Theory and Applications vol 2013 article 317 2013[19] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journal

of Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977[20] Z Mustafa and B Sims ldquoA new approach to generalized metric

spacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash296 2006

[21] C-M Chen ldquoCommon fixed-point theorems in completegeneralized metric spacesrdquo Journal of Applied Mathematics vol2012 Article ID 945915 14 pages 2012

[22] K Chakrabarti ldquoFixed point theorems in G-metric spaces withW mapsrdquoMathematical Sciences Letters vol 2 pp 29ndash35 2013

[23] E Karapınar and R P Agarwal ldquoFurther fixed point results on119866-metric spacesrdquo Fixed PointTheory andApplications vol 2013article 154 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Mathematics 7

[9] R Saadati S M Vaezpour P Vetro and B E RhoadesldquoFixed point theorems in generalized partially ordered 119866-met-ric spacesrdquo Mathematical and Computer Modelling vol 52 no5-6 pp 797ndash801 2010

[10] Z Mustafa J R Roshan and V Parvaneh ldquoCoupled coinci-dence point results for (120595 120593)-weakly contractive mappings inpartially ordered 119866

119887-metric spacesrdquo Fixed Point Theory and

Applications vol 2013 article 206 2013[11] ZMustafa J R Roshan andV Parvaneh ldquoExistence of a tripled

coincidence point in ordered G119887-metric spaces and applications

to a system of integral equationsrdquo Journal of Inequalities andApplications vol 2013 article no 453 2013

[12] V ParvanehA Razani and J R Roshan ldquoCommonfixed pointsof six mappings in partially ordered 119866-metric spacesrdquo Math-ematical Sciences vol 7 article 18 2013

[13] A Razani and V Parvaneh ldquoOn generalized weakly G-contractive mappings in partially ordered G-metric spacesrdquoAbstract and Applied Analysis vol 2012 Article ID 701910 18pages 2012

[14] Z Mustafa V Parvaneh M Abbas and J Rezaei RoshanldquoSome coincidence point results for generalized (120595 120593)-weaklycontractive mappings in ordered G-metric spaces Fixed PointTheory and Applicationsrdquo Fixed Point Theory and Applicationsvol 2013 article 326 2013

[15] M A Kutbi N Hussain J R Roshan and V ParvanehldquoCoupled and tripled coincidence point results with applicationto Fredholm integral equationsrdquo Abstract and Applied Analysisvol 2014 Article ID 568718 18 pages 2014

[16] N Hussain V Parvaneh and J R Roshan ldquoFixed point resultsfor G-120572-contractive maps with application to boundary valueproblemsrdquo ScientificWorld Journal vol 2014 Article ID 58596414 pages 2014

[17] A Latif N Hussain J R Roshan and V Parvaneh ldquoA uni-fication of Gminusmetric partial metric and b-metric spacesrdquoAbstract andApplied Analysis vol 2014 Article ID 180698 2014

[18] L Ciric S M Alsulami V Parvaneh and J R Roshan ldquoSomefixed point results in ordered 119866

119901-metric spacesrdquo Fixed Point

Theory and Applications vol 2013 article 317 2013[19] D S Jaggi ldquoSome unique fixed point theoremsrdquo Indian Journal

of Pure and AppliedMathematics vol 8 no 2 pp 223ndash230 1977[20] Z Mustafa and B Sims ldquoA new approach to generalized metric

spacesrdquo Journal of Nonlinear and Convex Analysis vol 7 no 2pp 289ndash296 2006

[21] C-M Chen ldquoCommon fixed-point theorems in completegeneralized metric spacesrdquo Journal of Applied Mathematics vol2012 Article ID 945915 14 pages 2012

[22] K Chakrabarti ldquoFixed point theorems in G-metric spaces withW mapsrdquoMathematical Sciences Letters vol 2 pp 29ndash35 2013

[23] E Karapınar and R P Agarwal ldquoFurther fixed point results on119866-metric spacesrdquo Fixed PointTheory andApplications vol 2013article 154 2013

Submit your manuscripts athttpwwwhindawicom

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of