- Research Article
- Open Access
Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a Q-Function
© N. Hussain et al. 2011
Received: 20 August 2010
Accepted: 16 September 2010
Published: 28 September 2010
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.
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 , Nieto and Rodríguez-López [28, 29], Ran and Reurings , and Agarwal et al.  presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham  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 .
Recently, Al-Homidan et al.  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  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  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  introduced the following notions of a mixed monotone mapping and a coupled fixed point.
Definition 1.1 (Bhaskar and Lakshmikantham ).
Definition 1.2 (Bhaskar and Lakshmikantham ).
The main theoretical result of Lakshmikantham and Ćirić in  is the following coupled fixed point theorem.
Theorem 1.3 (Lakshmikantham and Ćirić [24, Theorem ]).
Remark 1.6 (see ).
then a -function is called a -function, introduced by Lin and Du . It has been shown in that every -distance or -function, introduced and studied by Kada et al. , is a -function. In fact, if we consider as a metric space and replace by the following condition:
then a -function is called a -distance on . Several examples of -distance are given in . 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.
The following lemma lists some properties of a -function on which are similar to that of a -function (see ).
Lemma 1.8 (see ).
2. Main Results
Analogous with Definition 1.1, Lakshmikantham and Ćirić  introduced the following concept of a mixed -monotone mapping.
Definition 2.1 (Lakshmikantham and Ćirić ).
Definition 2.2 (see ).
Definition 2.3 (see ).
Following theorem is the main result of this paper.
The following example illustrates Theorem 2.4.
Also suppose that either
Also, suppose that either
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.
The first and third author are grateful to DSR, King Abdulaziz University for supporting research project no. (3-74/430).
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