Fixed Point Results in Quasimetric Spaces
© A. Latif and S. A. Al-Mezel. 2011
Received: 21 August 2010
Accepted: 5 October 2010
Published: 14 October 2010
1. Introduction and Preliminaries
Using the Hausdorff metric, Nadler Jr.  has established a multivalued version of the well-known Banach contraction principle in the setting of metric spaces as follows.
Without using the Hausdorff metric, Feng and Liu  generalized Nadler's contraction principle as follows.
In , Kada et al. introduced the concept of -distance in the setting of metric spaces as follows.
Note that in general for , and not either of the implications necessarily holds. Clearly, the metric is a -distance on . Many other examples and properties of -distances are given in .
In , Suzuki and Takahashi improved Nadler contraction principle (Theorem 1.1) as follows.
Recently, Latif and Albar  generalized Theorem 1.2 with respect to -distance (see, Theorem in ), and Latif  proved a fixed point result with respect to -distance ( see, Theorem in ) which contains Theorem 1.3 as a special case.
A nonempty set together with a quasimetric (i.e., not necessarily symmetric) is called a quasimetric space. In the setting of a quasimetric spaces, Al-Homidan et al.  introduced the concept of a -function on quasimetric spaces which generalizes the notion of a -distance.
Using the concept -function, Al-Homidan et al.  recently studied an equilibrium version of the Ekeland-type variational principle. They also generalized Nadler's fixed point theorem (Theorem 1.1) in the setting of quasimetric spaces as follows.
Clearly, t he class of weakly - contractive maps contains the class of weakly contractive maps, and the class of generalized -contractive maps contains the classes of generalized -contraction maps , -contractive maps , and -contractive maps .
In this paper, we prove some new fixed point results in the setting of quasimetric spaces for weakly -contractive and generalized -contractive multivalued maps. Consequently, our results either improve or generalize many known results including the above stated fixed point results.
2. The Results
Now, we prove the following useful lemma.
Let be a complete quasimetric space and let be a generalized -contractive map, then there exists an orbit of at , such that the sequence of nonnegative numbers is decreasing to zero and is a Cauchy sequence.
Finally, we conclude with the following remarks concerning our results related to the known fixed point results.
(3)Theorem 2.3 also generalizes Theorem in  in several ways.
(4)Corollary 2.4 improves and generalizes Theorem in .
The authors thank the referees for their kind comments. The authors also thank King Abdulaziz University and the Deanship of Scientific Research for the research Grant no. 3-35/429.
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