- Mujahid Abbas
^{1}and - Dragan Ðorić
^{2}Email author

**2010**:509658

https://doi.org/10.1155/2010/509658

© M. Abbas and D. Ðorić. 2010

**Received: **21 August 2010

**Accepted: **18 October 2010

**Published: **20 October 2010

## Abstract

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.

## Keywords

## 1. Introduction and Preliminaries

Alber and Guerre-Delabriere [1] defined weakly contractive maps on a Hilbert space and established a fixed point theorem for such a map. Afterwards, Rhoades [2], using the notion of weakly contractive maps, obtained a fixed point theorem in a complete metric space. Dutta and Choudhury [3] generalized the weak contractive condition and proved a fixed point theorem for a selfmap, which in turn generalizes theorem 1 in [2] and the corresponding result in [1]. The study of common fixed points of mappings satisfying certain contractive conditions has been at the center of vigorous research activity. Beg and Abbas [4] obtained a common fixed point theorem extending weak contractive condition for two maps. In this direction, Zhang and Song [5] introduced the concept of a generalized -weak contraction condition and obtained a common fixed point for two maps, and Ðorić [6] 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.

for . A point is called a fixed point of if . If there exists a point such that , then is termed as an end point of the mapping .

## 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 [6] and theorem 1 in [5].

Definition 2.1.

*generalized*

*-weak contraction*if the inequality

holds for all and for given functions .

Theorem 2.2.

Let be a complete metric space, and let be two set-valued mappings that satisfy the property of generalized -weak contraction, where

(a) is a continuous monotone nondecreasing function with if and only if ,

(b) is a lower semicontinuous function with if and only if

then there exists the unique point such that .

Proof.

*The sequences*

*and*

*are convergent*. Suppose that is an odd number. Substituting and in (2.1) and using properties of functions and , we obtain

which is a contradiction with .

From (2.4) and (2.29), we conclude that is a Cauchy sequence.

In complete metric space , there exists such that as .

and using an argument similar to the above, we conclude that or .

The proof is completed.

The Theorem 2.2 established that set-valued mappings and under weak condition (2.1) have the unique common end point . Now, we give an example to support our result.

Example 2.3.

From Tables 1 and 2, it is easy to verify that mappings and satisfy condition (2.1).

Remark 2.4.

The Theorem 2.2 generalizes recent results on single-valued weak contractions given in [3, 5, 6]. The example above shows that function in (2.1) gives an improvement over condition (2.1) in [5].

## Authors’ Affiliations

## References

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**Some theorems on weakly contractive maps.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(4):2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleMATHGoogle Scholar - Dutta PN, Choudhury BS:
**A generalisation of contraction principle in metric spaces.***Fixed Point Theory and Applications*2008,**2008:**- 8.Google Scholar - Beg I, Abbas M:
**Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition.***Fixed Point Theory and Applications*2006,**2006:**-7.Google Scholar - Zhang Q, Song Y:
**Fixed point theory for generalized****-weak contractions.***Applied Mathematics Letters*2009,**22**(1):75–78. 10.1016/j.aml.2008.02.007MathSciNetView ArticleMATHGoogle Scholar - Ðorić D:
**Common fixed point for generalized****-weak contractions.***Applied Mathematics Letters*2009,**22**(12):1896–1900. 10.1016/j.aml.2009.08.001MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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