Some New Weakly Contractive Type Multimaps and Fixed Point Results in Metric Spaces
© A. Latif and A. A. N. Abdou. 2009
Received: 20 September 2009
Accepted: 10 December 2009
Published: 3 February 2010
Some new weakly contractive type multimaps in the setting of metric spaces are introduced, and we prove some results on the existence of fixed points for such maps under certain conditions. Our results extend and improve several known results including the corresponding recent fixed point results of Pathak and Shahzad (2009), Latif and Abdou (2009), Latif and Albar (2008), Cirić (2008), Feng and Liu (2006), and Klim and Wardowski (2007).
Let be a metric space. Let denote a collection of nonempty subsets of , a collection of nonempty closed subsets of and a collection of nonempty closed bounded subsets of Let be the Hausdorff metric with respect to , that is,
Using the concept of Hausdorff metric, Nadler  established the following multivalued version of the Banach contraction principle.
This result has been generalized in many directions. For instance, Mizoguchi and Takahashi  have obtained the following general form of the Nadler's theorem.
Many authors have been using the Hausdorff metric to obtain fixed point results for multivalued maps. 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 the Hausdorff metric. They proved the following result.
Recently, Klim and Wardowski  generalized Theorem 1.3 as follows.
In , Kada et al. introduced the concept of -distance on a metric space as follows.
Note that, in general for , and neither 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 [10, 11].
Lemma 1.5 (see ).
Lemma 1.6 (see ).
Using the concept of -distance, the authors of this paper most recently extended and generalized Theorem 1.4 and [8, Theorem ] as follows.
Theorem 1.7 (see ).
If and is continuous at , then due to the following two facts must be continuous at each point of . First, every sub-additive and continuous function at 0 such that is right upper and left lower semicontinuous . Second, each nondecreasing function is left upper and right lower semicontinuous.
Assuming that the function is continuous and satisfies the conditions (i) and (ii) above, Zhang  proved some fixed point results for single-valued maps which satisfy some contractive type condition involving such function . Recently, using Pathak and Shahzad  generalized Theorem 1.4.
In this paper, we prove some results on the existence of fixed points for contractive type multimaps involving the function where and the function is a -distance on a metric space Our results either generalize or improve several known fixed point results in the setting of metric spaces, (see Remarks 2.3 and 2.6).
2. The Results
Theorem 2.1 extends and generalizes Theorem 1.7. Indeed, if we consider for each in Theorem 2.1, then we can get Theorem 1.7 due to Latif and Abdou [13, Theorem ].
Theorem 2.1 contains Theorem of Pathak and Shahzad  as a special case.
Corollary 2.2 extends and generalizes Theorem of Latif and Albar .
We have also the following fixed point result which generalizes [13, Theorem ].
Following the same method as in the proof of Theorem 2.4, we can obtain the following fixed point result.
Thus, all the hypotheses of Theorem 2.1 are satisfied and clearly we have Now, if we consider , then all the hypotheses of Theorem 2.5 are also satisfied. Note that in the above example the -distance is not a metric .
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