© M. Abbas and D. Ðorić. 2010
Received: 21 August 2010
Accepted: 18 October 2010
Published: 20 October 2010
We introduce the class of generalized -weak contractive set-valued mappings on a metric space. We establish that such mappings have a unique common end point under certain weak conditions. The theorem obtained generalizes several recent results on single-valued as well as certain set-valued mappings.
1. Introduction and Preliminaries
Alber and Guerre-Delabriere  defined weakly contractive maps on a Hilbert space and established a fixed point theorem for such a map. Afterwards, Rhoades , using the notion of weakly contractive maps, obtained a fixed point theorem in a complete metric space. Dutta and Choudhury  generalized the weak contractive condition and proved a fixed point theorem for a selfmap, which in turn generalizes theorem 1 in  and the corresponding result in . The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity. Beg and Abbas  obtained a common fixed point theorem extending weak contractive condition for two maps. In this direction, Zhang and Song  introduced the concept of a generalized -weak contraction condition and obtained a common fixed point for two maps, and Ðorić  proved a common fixed point theorem for generalized -weak contractions. On the other hand, there are many theorems in the existing literature which deal with fixed point of multivalued mappings. In some cases, multivalued mapping defined on a nonempty set assumes a compact value for each in . There are the situations when, for each in , is assumed to be closed and bounded subset of . To prove existence of fixed point of such mappings, it is essential for mappings to satisfy certain contractive conditions which involve Hausdorff metric.
The aim of this paper is to obtain the common end point, a special case of fixed point, of two multivlaued mappings without appeal to continuity of any map involved therein. It is also noted that our results do not require any commutativity condition to prove an existence of common end point of two mappings. These results extend, unify, and improve the earlier comparable results of a number of authors.
2. Main Results
In this section, we established an end point theorem which is a generalization of fixed point theorem for generalized -weak contractions. The idea is in line with Theorem 2.1 in  and theorem 1 in .
The proof is completed.
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