On common fixed point theorems for non-self hybrid mappings in convex metric spaces
Introduction
Let be a metric space and be the set of all non-empty, closed and bounded subsets of X. Denote by H the Hausdorff metric induced by the metric d and for any and , set .
Extending the Banach contraction principle, Nadler [15] and Markin [14] first initiated the study of fixed point theorems for multi-valued contraction self-mappings. The Banach principle has been developed by many authors [4], [5], [6], [7], [11]. Fixed point theorems for self-mappings have found applications in diverse disciplines of mathematics, engineering and economics.
Assad and Kirk [3] have observed that in convex spaces there occur cases where the involved function is not necessarily a self-mapping of a closed subset, and they are the first to study non-self multi-valued contraction mappings in a metric space , convex in the sense of Menger (that is, for each x,y in X with there exists z in X, , such that ). In recent years, this technique has been further developed, and fixed and common fixed points of non-self mappings have been studied by many authors [6], [8], [9], [10], [12], [13], [16], [17]. Some of these results have found applications (cf. [3], [17], [18]). In numerical mathematics especially the restricted condition is preferred instead of where K is a closed subset of X, and is the boundary of K.
Recently Imdad and Khan [13] generalized the result of Assad [2]. They proved the following coincidence point theorem for multi-valued non-self mappings. Theorem 1 [13] Let be a complete metrically convex metric space, K a non-empty closed subset of X. Let and satisfying ; ; ; ; and and are compatible pairs; , S and T are continuous on K.
where , for all with , , and
Then as well as has a point of coincidence.
In this paper, we shall introduce the concept of a contractive type non-self mappings which satisfy a new contractive condition, and prove common fixed point theorems in convex metric spaces. Our theorems generalize and improve the theorems of Ahmad and Imdad [1], Imdad and Khan [12], [13], Assad [2] and several others theorems. To accomplish that, we shall use slightly improved version of methods of proofs than used by Imdad and Khan and others in the fixed point theory for non-self mappings.
Section snippets
Results
Recall that if X and Y are non-empty sets and and are given mappings, then a point is said to be a coincidence of T and F, if .
We shall prove the following coincidence point theorem for two pairs of single-valued and multi-valued non-self mappings. Theorem 2 Let be a metrically convex metric space and K be a non-empty closed subset of X. Let mappings and satisfy the following condition:
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