© Mujahid Abbas et al. 2010
Received: 7 April 2010
Accepted: 18 October 2010
Published: 24 October 2010
We introduce the concept of a -compatible mapping to obtain a coupled coincidence point and a coupled point of coincidence for nonlinear contractive mappings in partially ordered metric spaces equipped with -distances. Related coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend, and unify several well-known comparable results in the literature.
1. Introduction and Preliminaries
In 1996, Kada et al.  introduced the notion of -distance. They elaborated, with the help of examples, that the concept of -distance is general than that of metric on a nonempty set. They also proved a generalization of Caristi fixed point theorem employing the definition of -distance on a complete metric space. Recently, Ilić and Rakočević  obtained fixed point and common fixed point theorems in terms of -distance on complete metric spaces (see also [3–9]).
The metric is a -distance on . For more examples of -distances, we refer to .
The next Lemma is crucial in the proof of our results.
Lemma 1.3 (see ).
Bhaskar and Lakshmikantham in  introduced the concept of coupled fixed point of a mapping and investigated some coupled fixed point theorems in partially ordered sets. They also discussed an application of their result by investigating the existence and uniqueness of solution for a periodic boundary value problem. Sabetghadam et al. in  introduced this concept in cone metric spaces. They investigated some coupled fixed point theorems in cone metric spaces. Recently, Lakshmikantham and Ćirić  proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces which extend the coupled fixed point theorem given in . The following are some other definitions needed in the sequel.
Definition 1.4 . (see ).
Kada et al.  gave an example to show that is not symmetric in general. We denote by and , respectively, the class of all -distances on and the class of all -distances on which are symmetric for comparable elements in . Also in the sequel, we will consider that and are comparable with respect to ordering in if and .
2. Coupled Coincidence Point
3. Coupled Common Fixed Point
Let all the hypotheses of Theorem 2.1 (resp., Theorem 2.2) hold with . If for every there exists that is comparable to and with respect to ordering in , then there exists a unique coupled point of coincidence of and . Moreover if and are -compatible, then and have a unique coupled common fixed point.
The present version of the paper owes much to the precise and kind remarks of the learned referees.
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