Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces
© A. Latif and A. A. N. Abdou. 2009
Received: 17 May 2009
Accepted: 10 August 2009
Published: 19 August 2009
The concept of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, our results either generalize or improve a number of fixed point results including the corresponding recent fixed point results of Ciric (2008), Latif-Albar (2008), Klim-Wardowski (2007), and Feng-Liu (2006). Examples are also given.
1. Introduction and Preliminaries
Let be a metric space, a collection of nonempty subsets of , a collection of nonempty closed bounded subsets of , a collection of nonempty closed subsets of a collection of nonempty compact subsets of and the Hausdorff metric induced by Then for any
Using the concept of Hausdorff metric, Nadler  established the following fixed point result for multivalued contraction maps, known as Nadler's contraction principle which in turn is a generalization of the well-known Banach contraction principle.
Theorem 1.1 (see ).
Using the concept of the Hausdorff metric, many authors have generalized Nadler's contraction principle in many directions. But, in fact for most cases the existence part of the results can be proved without using the concept of Hausdorff metric. Recently, Feng and Liu  extended Nadler's fixed point theorem without using the concept of Hausdorff metric. They proved the following result.
Recently, Klim and Wardowski  generalized Theorem 1.2 and proved the following two results.
Note that Theorem 1.3 generalizes Nadler's contraction principle and Theorem 1.2. Most recently, Ciric  obtained some interesting fixed point results which extend and generalize the cited results. Namely, [4, Theorem 5] generalizes [5, Theorem 5], [4, Theorem 6] generalizes [4, Theorems 1.2, 1.3], and [3, theorem 7] generalizes Theorem 1.4.
In , Kada et al. introduced the concept of -distance on a metric space as follows:
Note that, in general for , and not either of the implications necessarily hold. Clearly, the metric is a -distance on . Let be a normed space. Then the functions defined by and for all are -distances . Many other examples and properties of the -distance can be found in [6, 7].
The following lemma is crucial for the proofs of our results.
Lemma 1.5 (see ).
Most recently, the authors of this paper generalized Latif and Albar [9, Theorem 1.3] as follows.
Theorem 1.6 (see ).
The aim of this paper is to present some more general results on the existence of fixed points for multivalued maps satisfying certain conditions. Our results unify and generalize the corresponding results of Mizoguchi and Takahashi , Klim and Wardowski , Latif and Abdou , Ciric , Feng and Liu , Latif and Albar  and several others.
2. The Results
We also have the following interesting result by replacing the hypothesis (iii) of Theorem 2.1 with another natural condition.
Following the proof of Theorem 2.2, we can obtain the following result.
The following example shows that Theorem 2.1 is a genuine generalization of Ciric [4, Theorem 5].
Thus, also satisfies all the conditions of Theorem 2.1 for Hence it follows from Theorem 2.1 that Note that Clearly, does not satisfy the hypotheses of Ciric [4, Theorem 5] because is not the metric .
Finally, we present an example which shows that Theorem 2.5 is a genuine generalization of Theorem 1.4 due to Klim-Wardowski .
The authors thank the referees for their valuable comments.
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