- Research Article
- Open Access

# Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a *Q*-Function

- N Hussain
^{1}Email author, - MH Shah
^{2}and - MA Kutbi
^{1}

**2011**:703938

https://doi.org/10.1155/2011/703938

© N. Hussain et al. 2011

**Received:**20 August 2010**Accepted:**16 September 2010**Published:**28 September 2010

## Abstract

Using the concept of a mixed *g*-monotone mapping, we prove some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a *Q*-function *q*. The presented theorems are generalizations of the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham (2006), Lakshmikantham and Ćirić (2009) and many others.

## Keywords

- Equilibrium Problem
- Fixed Point Theorem
- Monotone Property
- Couple Fixed Point
- Couple Coincidence Point

## 1. Introduction

The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directions (cf. [1–31]). Recently, Bhaskar and Lakshmikantham [8], Nieto and Rodríguez-López [28, 29], Ran and Reurings [30], and Agarwal et al. [1] presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham [8] noted that their theorem can be used to investigate a large class of problems and discussed the existence and uniqueness of solution for a periodic boundary value problem. For more on metric fixed point theory, the reader may consult the book [22].

Recently, Al-Homidan et al. [2] introduced the concept of a -function defined on a quasi-metric space which generalizes the notions of a -function and a -distance and establishes the existence of the solution of equilibrium problem (see also [3–7]). The aim of this paper is to extend the results of Lakshmikantham and Ćirić [24] for a mixed monotone nonlinear contractive mapping in the setting of partially ordered quasi-metric spaces with a -function . We prove some coupled coincidence and coupled common fixed point theorems for a pair of mappings. Our results extend the recent coupled fixed point theorems due to Lakshmikantham and Ćirić [24] and many others.

Recall that if is a partially ordered set and such that for implies , then a mapping is said to be nondecreasing. Similarly, a nonincreasing mapping is defined. Bhaskar and Lakshmikantham [8] introduced the following notions of a mixed monotone mapping and a coupled fixed point.

Definition 1.1 (Bhaskar and Lakshmikantham [8]).

Definition 1.2 (Bhaskar and Lakshmikantham [8]).

The main theoretical result of Lakshmikantham and Ćirić in [24] is the following coupled fixed point theorem.

Theorem 1.3 (Lakshmikantham and Ćirić [24, Theorem ]).

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(a) is continuous or

(b) has the following property:

(i) if a nondecreasing sequence ,then for all

(ii) if a nonincreasing sequence ,then for all

that is, and have a coupled coincidence.

Definition 1.4.

Let be a nonempty set. A real-valued function is said to be quasi-metric on if

for all

if and only if

for all .

The pair is called a quasi-metric space.

Definition 1.5.

Let be a quasi-metric space. A mapping is called a -function on if the following conditions are satisfied:

for all

if and is a sequence in such that it converges to a point (with respect to the quasi-metric) and for some then ;

for any , there exists such that , and implies that

Remark 1.6 (see [2]).

If is a metric space, and in addition to the following condition is also satisfied:

for any sequence in with and if there exists a sequence in such that then

then a -function is called a -function, introduced by Lin and Du [27]. It has been shown in [27]that every -distance or -function, introduced and studied by Kada et al. [21], is a -function. In fact, if we consider as a metric space and replace by the following condition:

for any , the function is lower semicontinuous,

then a -function is called a -distance on . Several examples of -distance are given in [21]. It is easy to see that if is lower semicontinuous, then holds. Hence, it is obvious that every -function is a -function and every -function is a -function, but the converse assertions do not hold.

- (a)

- (b)

Then is a -function on However, is neither a -function nor a -function, because is not a metric space.

The following lemma lists some properties of a
-function on
which are similar to that of a
-function (see [21])**.**

Lemma 1.8 (see [2]).

Let be a -function on Let and be sequences in , and let and be such that they converge to and Then, the following hold:

(1) if and for all , then . In particular, if and , then ;

(2) if and for all , then converges to ;

(3) if for all with , then is a Cauchy sequence;

(4) if for all , then is a Cauchy sequence;

(5) if are -functions on , then is also a -function on .

## 2. Main Results

Analogous with Definition 1.1, Lakshmikantham and Ćirić [24] introduced the following concept of a mixed -monotone mapping.

Definition 2.1 (Lakshmikantham and Ćirić [24]).

Note that if is the identity mapping, then Definition 2.1 reduces to Definition 1.1.

Definition 2.2 (see [24]).

Definition 2.3 (see [24]).

for all

Following theorem is the main result of this paper.

Theorem 2.4.

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(a) is continuous or

(b) has the following property:

(i) if a nondecreasing sequence , then for all

(ii) if a nonincreasing sequence , then for all

that is, and have a coupled coincidence.

Proof.

We now show that and

Case 1.

*Suppose that the assumption (a) holds*. Taking the limit as in (2.26), and using the continuity of , we get

Case 2.

*Suppose that the assumption (b) holds*. Let . Now, since is continuous, is nondecreasing with for all , and is nonincreasing with for all , so is nondecreasing, that is,

with , for all .

Therefore, by the triangle inequality and ( ), we have for

Case 3.

Case 4.

Hence, by Lemma 1.8, and Thus, and have a coupled coincidence point.

The following example illustrates Theorem 2.4.

Example 2.5.

and , are both continuous on their domains and . Let be such that and There are four possibilities for (2.5) to hold. We first compute expression on the left of (2.5) for these cases:

Thus, satisfies the contraction condition (2.5) of Theorem 2.4. Now, suppose that be, respectively, nondecreasing and nonincreasing sequences such that and , then by Theorem 2.4, and for all

Therefore, by Theorem 2.4, there exists such that and

Corollary 2.6.

for all for which and Suppose that and is continuous and commutes with , and also suppose that either

(a) is continuous or

(b) has the following properties:

(i) if a nondecreasing sequence , then for all

(ii) if a nonincreasing sequence , then for all .

that is, and have a coupled coincidence.

Proof.

Taking in Theorem 2.4, we obtain Corollary 2.6.

From Theorem 2.4, it follows that the set of coupled coincidences is nonempty.

Theorem 2.7.

Proof.

Therefore, is a coupled common fixed point of and To prove the uniqueness, assume that is another coupled common fixed point. Then, by (2.55), we have and

Corollary 2.8.

Also suppose that either

(a) is continuous or

(b) has the following properties:

(i) if a nondecreasing sequence , then for all

(ii) if a non-increasing sequence , then for all

Furthermore, if are comparable, then that is,

Proof.

where . Hence, by Lemma 1.8, that is,

Corollary 2.9.

Also, suppose that either

(a) is continuous or

(b) has the following properties:

(i) if a nondecreasing sequence , then for all

(ii) if a nonincreasing sequence , then for all

Furthermore, if are comparable, then that is,

Proof.

Taking in Corollary 2.8, we obtain Corollary 2.9.

Remark 2.10.

As an application of fixed point results, the existence of a solution to the equilibrium problem was considered in [2–7]. It would be interesting to solve Ekeland-type variational principle, Ky Fan type best approximation problem and equilibrium problem utilizing recent results on coupled fixed points and coupled coincidence points.

## Declarations

### Acknowledgment

The first and third author are grateful to DSR, King Abdulaziz University for supporting research project no. (3-74/430).

## Authors’ Affiliations

## References

- Agarwal RP, El-Gebeily MA, O'Regan D:
**Generalized contractions in partially ordered metric spaces.***Applicable Analysis*2008,**87**(1):109–116. 10.1080/00036810701556151MATHMathSciNetView ArticleGoogle Scholar - Al-Homidan S, Ansari QH, Yao J-C:
**Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(1):126–139. 10.1016/j.na.2007.05.004MATHMathSciNetView ArticleGoogle Scholar - Ansari QH:
**Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory.***Journal of Mathematical Analysis and Applications*2007,**334**(1):561–575. 10.1016/j.jmaa.2006.12.076MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Konnov IV, Yao JC:
**On generalized vector equilibrium problems.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(1):543–554. 10.1016/S0362-546X(01)00199-7MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Siddiqi AH, Wu SY:
**Existence and duality of generalized vector equilibrium problems.***Journal of Mathematical Analysis and Applications*2001,**259**(1):115–126. 10.1006/jmaa.2000.7397MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Yao J-C:
**An existence result for the generalized vector equilibrium problem.***Applied Mathematics Letters*1999,**12**(8):53–56. 10.1016/S0893-9659(99)00121-4MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Yao J-C:
**A fixed point theorem and its applications to a system of variational inequalities.***Bulletin of the Australian Mathematical Society*1999,**59**(3):433–442. 10.1017/S0004972700033116MATHMathSciNetView ArticleGoogle Scholar - Bhaskar TG, Lakshmikantham V:
**Fixed point theorems in partially ordered metric spaces and applications.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(7):1379–1393. 10.1016/j.na.2005.10.017MATHMathSciNetView ArticleGoogle Scholar - Bhaskar TG, Lakshmikantham V, Vasundhara Devi J:
**Monotone iterative technique for functional differential equations with retardation and anticipation.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(10):2237–2242. 10.1016/j.na.2006.03.013MATHMathSciNetView ArticleGoogle Scholar - Boyd DW, Wong JSW:
**On nonlinear contractions.***Proceedings of the American Mathematical Society*1969,**20:**458–464. 10.1090/S0002-9939-1969-0239559-9MATHMathSciNetView ArticleGoogle Scholar - Ćirić LjB:
**A generalization of Banach's contraction principle.***Proceedings of the American Mathematical Society*1974,**45**(2):267–273.MATHMathSciNetGoogle Scholar - Ćirić L:
**Fixed point theorems for multi-valued contractions in complete metric spaces.***Journal of Mathematical Analysis and Applications*2008,**348**(1):499–507. 10.1016/j.jmaa.2008.07.062MATHMathSciNetView ArticleGoogle Scholar - Ćirić L, Hussain N, Cakić N:
**Common fixed points for Ćirić type**f**-weak contraction with applications.***Publicationes Mathematicae Debrecen*2010,**76**(1–2):31–49.MATHMathSciNetGoogle Scholar - Ćirić LjB, Ume JS:
**Multi-valued non-self-mappings on convex metric spaces.***Nonlinear Analysis: Theory, Methods & Applications*2005,**60**(6):1053–1063. 10.1016/j.na.2004.09.057MATHMathSciNetView ArticleGoogle Scholar - Gajić L, Rakočević V:
**Quasicontraction nonself-mappings on convex metric spaces and common fixed point theorems.***Fixed Point Theory and Applications*2005, (3):365–375.Google Scholar - Guo DJ, Lakshmikantham V:
*Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering*.*Volume 5*. Academic Press, Boston, Mass, USA; 1988:viii+275.MATHGoogle Scholar - Heikkilä S, Lakshmikantham V:
*Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 181*. Marcel Dekker, New York, NY, USA; 1994:xii+514.MATHGoogle Scholar - Hussain N:
**Common fixed points in best approximation for Banach operator pairs with Ćirić type**I**-contractions.***Journal of Mathematical Analysis and Applications*2008,**338**(2):1351–1363. 10.1016/j.jmaa.2007.06.008MATHMathSciNetView ArticleGoogle Scholar - Hussain N, Berinde V, Shafqat N:
**Common fixed point and approximation results for generalized**ϕ**-contractions.***Fixed Point Theory*2009,**10**(1):111–124.MATHMathSciNetGoogle Scholar - Hussain N, Khamsi MA:
**On asymptotic pointwise contractions in metric spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(10):4423–4429. 10.1016/j.na.2009.02.126MATHMathSciNetView ArticleGoogle Scholar - Kada O, Suzuki T, Takahashi W:
**Nonconvex minimization theorems and fixed point theorems in complete metric spaces.***Mathematica Japonica*1996,**44**(2):381–391.MATHMathSciNetGoogle Scholar - Khamsi MA, Kirk WA:
*An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics*. Wiley-Interscience, New York, NY, USA; 2001:x+302.View ArticleMATHGoogle Scholar - Lakshmikantham V, Bhaskar TG, Vasundhara Devi J:
*Theory of Set Differential Equations in Metric Spaces*. Cambridge Scientific Publishers, Cambridge, UK; 2006:x+204.MATHGoogle Scholar - Lakshmikantham V, Ćirić L:
**Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**70**(12):4341–4349. 10.1016/j.na.2008.09.020MATHMathSciNetView ArticleGoogle Scholar - Lakshmikantham V, Köksal S:
*Monotone Flows and Rapid Convergence for Nonlinear Partial Differential Equations, Series in Mathematical Analysis and Applications*.*Volume 7*. Taylor & Francis, London, UK; 2003:x+318.MATHGoogle Scholar - Lakshmikantham V, Vatsala AS:
**General uniqueness and monotone iterative technique for fractional differential equations.***Applied Mathematics Letters*2008,**21**(8):828–834. 10.1016/j.aml.2007.09.006MATHMathSciNetView ArticleGoogle Scholar - Lin L-J, Du W-S:
**Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces.***Journal of Mathematical Analysis and Applications*2006,**323**(1):360–370. 10.1016/j.jmaa.2005.10.005MATHMathSciNetView ArticleGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.***Order*2005,**22**(3):223–239. 10.1007/s11083-005-9018-5MATHMathSciNetView ArticleGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations.***Acta Mathematica Sinica (English Series)*2007,**23**(12):2205–2212. 10.1007/s10114-005-0769-0MATHMathSciNetView ArticleGoogle Scholar - Ran ACM, Reurings MCB:
**A fixed point theorem in partially ordered sets and some applications to matrix equations.***Proceedings of the American Mathematical Society*2004,**132**(5):1435–1443. 10.1090/S0002-9939-03-07220-4MATHMathSciNetView ArticleGoogle Scholar - Ray BK:
**On Ciric's fixed point theorem.***Fundamenta Mathematicae*1977,**94**(3):221–229.MATHMathSciNetGoogle Scholar

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