Fixed Points of Generalized Contractive Maps
© A. Latif and A. A. N. Abdou. 2009
Received: 13 October 2008
Accepted: 27 January 2009
Published: 3 February 2009
Using the concept of Hausdorff metric, Nadler Jr.  established the following fixed point result for multivalued contraction maps which in turn is a generalization of the well-known Banach contraction principle.
Theorem 1.1 (see ).
This result has been generalized in many directions. For instance, Mizoguchi and Takahashi  have obtained the following general form of the Nadler's theorem.
Theorem 1.2 (see ).
Another extension of Nadler's result obtained recently by Feng and Liu . Without using the concept of the Hausdorff metric, they proved the following result.
Theorem 1.3 (see ).
Most recently, Klim and Wardowski  generalized Theorem 1.3 as follows:
Theorem 1.4 (see ).
Recently, Kada et al.  introduced the concept of -distance on a metric space as follows.
Using the concept of -distance, they improved Caristi's fixed point theorem, Ekland's variational principle, and Takahashi's existence theorem. In , Susuki and Takahashi proved a fixed point theorem for contractive type multivalued maps with respect to -distance. See also [7–12].
Let us give some examples of -distance .
Lemma 1.5 (see ).
Lemma 1.6 (see ).
Note that if we take then the definition of generalized -contractive map reduces to the definition of generalized contractive map due to Klim and Wardowski . In particular, if we take a constant map then the map is weakly contractive (in short, -contractive) , and further if we take then we obtain and is contractive .
In this paper, using the concept of -distance, we first establish key lemma and then obtain fixed point results for multivalued generalized -contractive maps not involving the extended Hausdorff metric. Our results either generalize or improve a number of fixed point results including the corresponding results of Feng and Liu , Latif and Albar , and Klim and Wardowski .
First, we prove key lemma in the setting of metric spaces.
Using Lemma 2.1, we obtain the following fixed point result which is an improved version of Theorem 1.4 and contains Theorem 1.3 as a special case.
As a consequence, we also obtain the following fixed point result.
Corollary 2.3 (see ).
Corollary 2.5 (see ).
The authors thank the referees for their valuable comments and suggestions.
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