© C.-M. Chen and T.-H. Chang. 2009
Received: 25 March 2009
Accepted: 19 June 2009
Published: 14 July 2009
1. Introduction and Preliminarie
In 1929, Knaster et al.  had proved the well-known theorem on -simplex. Besides, in 1961, Fan  had generalized the theorem to an infinite dimensional topological vector space. Later, Amini et al.  had introduced the class of -type mappings on metric spaces and established some fixed point theorems for this class. In this paper, we define a weaker Meir-Keeler type function and establish the fixed point theorems for a weaker Meir-Keeler type -set contraction in metric spaces.
Throughout this paper, by we denote the set of all real nonnegative numbers, while is the set of all natural numbers. We digress briefly to list some notations and review some definitions. Let and be two Hausdorff topological spaces, and let be a set-valued mapping. Then is said to be closed if its graph is closed. is said to be compact if the image of under is contained in a compact subset of . If is a nonempty subset of , then denotes the class of all nonempty finite subsets of . And, the following notations are used:
(ii) is said to be subadmissible , if for each , .
In 1996, Chang and Yen  introduced the family on the topological vector spaces and got results about fixed point theorems, coincidence theorems, and its applications on this family. Later, Amini et al.  introduced the following concept of the property on a subadmissible subset of a metric space .
Let be an nonempty subadmissible subset of a metric space , and let a topological space. If are two set-valued mappings such that for any , , then is called a generalized mapping with respect to . If the set-valued mapping satisfies the requirement that for any generalized mapping with respest to , the family has finite intersection property, then is said to have the property. The class is denoted to be the set has the property .
Recall the notion of the Meir-Keeler type function. A function is said to be a Meir-Keeler type function (see ), if for each , there exists such that for with , we have .
We now define a new weaker Meir-Keeler type function as follows.
A function is said to be upper semicontinuous, if for each , . Recall also that is said to be a comparison function (see ) if it is increasing and . As a consequence, we also have that for each , , and , is continuous at . We generalize the comparison function to be the other form, as follows.
We now are going to give the axiomatic definition for the measure of noncompactness in a complete metric space.
2. Main Results
Using the conception of the weaker Meir-Keeler type function, we establish the following important theorem.
Let be a nonempty bounded subadmissible subset of a metric space . If is a weaker Meir-Keeler type -set contraction with for each , is nonicreasing, then contains a precompact subadmissible subset with .
Since is nonincreasing, it must converge to some with ; that is, . We now claim that . On the contrary, assume that .Then by the definition of the weaker Meir-Keeler type function, there exists such that for each with , there exists such that .Since , there exists such that , for all . Thus, we conclude that . So we get a contradiction. So , and so .
Applying Proposition 1.3, 1.4, and Remark 1.5, we are easy to conclude the following corollary.
The proof is similar to the proof of Theorem 2.1; we omit it.
Following the concepts of the family (see ), we immediately have the following Lemma 2.6.
By the same process of Theorem 2.1, we get a weaker Meir-Keeler type precompact-inducing subadmissible subset of . Since and for each , we have for each . Since as , by the above Lemma 2.6, we have that is a nonempty compact subset of .
Choose and such that . From the definition of , it follows that , for each . Since , , we have , which implies that . Therefore, . This contradicts to . Hence, is a generalized mapping with respect to .
Therefore, we have proved that for each , there exists an such that . Let . Since and is compact, we may assume that converges to some , then also converges to . Since is closed, we have . This completes the proof.
The -spaces, in an abstract convex space setting, were introduced by Amini et al.. An abstract convex space consists of a nonempty topological space and a family of subsets of such that and belong to and is closed under arbitrary intersection. Let be an abstract convex space, and let a topological space. A map is called a -mapping if there exists a multifunction such that
The mapping is called a companion mapping of . Furthermore, if the abstract convex space which has a uniformity and has an open symmetric base family , then is called a -space if for each entourage , there exists a -mapping such that . Following the conceptions of the abstract convex space and the -space, we are easy to know that a bounded metric space is an important example of the abstract convex space, and if and , then is also a -space.
Applying Theorem of Amini et al. , we can deduce the following theorem in metric spaces.
Now we let be an identity mapping, all of the the conditions of Theorem of Amini et al.  are fulfilled, and we can obtain the results.
Applying Theorems 2.1 and 2.10, we can conclude the following fixed point theorems.
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