Abdul Latif and Marwan A. Kutbi

Copyright © 2009 Abdul Latif and Marwan A. Kutbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using the generalized Caristi's fixed point theorems we prove the existence of fixed points for self and nonself multivalued weakly w-contractive maps. Consequently, Our results either improve or generalize the corresponding fixed point results due to Latif (2007), Bae (2003), Suzuki, and Takahashi (1996) and others.

1. Introduction

It is well known that Caristi's fixed point theorem [1] is equivalent to Ekland variational principle [2], which is nowadays is an important tool in nonlinear analysis. Most recently, many authors studied and generalized Caristi's fixed point theorem to various directions. For example, see [3–6] and references therein.

Using the concept of Hausdorff metric, Nadler Jr. [7] has proved multivalued version of the Banach contraction principle which states that each closed bounded valued contraction map on a complete metric space, has a fixed point. Recently, Bae [4] introduced a notion of multivalued weakly contractive maps and applying generalized Caristi's fixed point theorems he proved several fixed point results for such maps in the setting of metric and Banach spaces. Many authors have been using the Hausdorff metric to obtain fixed point results for multivalued maps on metric spaces, but, in fact for most cases the existence part of the results can be proved without using the concept of Hausdorff metric.

Recently, using the concept of -distance [8], Suzuki and Takahashi [9] introduced a notion of multivalued weakly contractive(in short, -contractive) maps and improved the Nadler's fixed point result without using the concept of Hausdorff metric. Most recently, Latif [10] generalized the fixed point result of Suzuki and Takahashi [9, Theorem 1]. Some interesting examples and fixed point results concerning -distance can be found in [6, 11–15] and references therein.

In this paper, introducing a notion of multivalued weakly -contractive maps, we prove some fixed point results for self and nonself multivalued maps. Our results either improve or generalize the corresponding results due to Latif [10], Bae [4], Mizoguchi and Takahashi [16], Suzuki and Takahashi [9], Husain and Latif [17], Kaneko [18] and many others.

2. Preliminaries

Let be a metric space with metric . We use to denote the collection of all nonempty subsets of and for the collection of all nonempty closed subsets of . Recall that a real-valued function defined on is said to be lower (upper) semicontinuous if for any sequence with imply that ().

Introducing the following notion of –distance, Kada et al. [8] improved the Caristi's fixed point theorem, Ekland variational principle, and Takahashi existence theorem.

A function is called a - on if it satisfies the following for any :

() () a map is lower semicontinuous; () for any there exists such that and imply

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 [8].

Let be a nonempty subset of . A multivalued map is called -contractive [9] if there exist a -distance on and a constant such that for any and there is satisfying(2.1)In particular, if we take , then -contractive map is a contractive type map [17].

We say is weakly -contractive if there exists a -distance on such that for any and there is with(2.2)where is a function from to such that is positive on and

In particular, if we take for a constant with then a weakly -contractive map is -contractive. If we define for and , then is a function from to with for every . Also we get(2.3)that is, the weakly -contractive map is generalized -contraction [10].

We say a multivalued map is -inward if for each w-, where w- is the -inward set of at , which consists all the elements such that either or there exists with and

In particular, if we take , then -inward set is known as metrically inward set [4].

A point is called a fixed point of if and the set of all fixed points of is denoted by Fix.

In the sequel, otherwise specified, we will assume that is lower semicontinuous function, is positive function on and and is a -distance on .

Using the concept of -distance, Kada et al. [8] have generalized Caristi's fixed point theorem as follows.Theorem 2.1. Let be a complete metric space. Let be a map such that for each (2.4)Then, there exists such that and

Now, we state generalized Caristi's fixed point theorems which are variant to the results of Bae [4, Theorem 2.1 and Corollary 2.5].Theorem 2.2. Let be a complete metric space. Let be a map such that for each , (2.5)where is an upper semicontinuous function from the right. Then, has a fixed point such that Theorem 2.3. Let be a complete metric space. Let be lower semicontinuous function such that for and (2.6)Let be a map such that for each and (2.7)Then, has a fixed point such that

Suzuki and Takahashi [9] have proved the following fixed point result which is an improved version of the multivalued contraction principle due to Nadler Jr. [7].Theorem 2.4. Let be a complete metric space. Then each multivalued -contractive map has a fixed point.

3. Main Results

Without using the Hausdorff metric, we prove the following fixed point result for multivalued self map. Theorem 3.1. Let be a complete metric space and let be a weakly -contractive map for which is lower semicontinuous from the right and . Then has a fixed point.

Proof. Let be the graph of . Clearly, is a closed subset of . Define a metric on by(3.1)Then is a complete metric space and is -distance on . Now, define by for all and by(3.2)Then is lower semicontinuous and is upper semicontinuous from the right because is lower semicontinuous from the right. Define by(3.3)Then is a -distance on (see [14, page 47]. Now, suppose Fix=. Then for each , we have . Since there is such that(3.4)Since , we have(3.5)also, note that(3.6)Define a function by , then we get(3.7)Thus, by Theorem 2.2, has a fixed point, which is impossible. Hence, must has a fixed point. This completes the proof.

As a consequence, we obtain the following recent fixed point result of Latif [10, Theorem 2.2].

Corollary 3.2. Let be a complete metric space. Let be a map such that for any and there is with (3.8) where is function from to with for every Then has a fixed point.

Proof. Define by(3.9)Then for all is lower semicontinuous from the right (see [19]). Also note that(3.10)and for each , we have(3.11)It follows from (3.8) and (3.11) that(3.12)Thus is weakly -contractive map for which is lower semicontinuous from the right and . Therefore, by Theorem 3.1, has a fixed point.

Remark 3.3. (a) Theorem 3.1 generalizes Theorem 2.4 of Suzuki and Takahashi [9]. Indeed, consider for a constant with Theorem 3.1 also generalizes and improves the fixed point result of Bae [4, Theorem 3.1] .

(b) Corollary 3.2 generalizes fixed point result of Husain and Latif [17, Theorem 2.3] and improves [16, Theorem 5]. Moreover, it improves and generalizes [18, Theorem 1].

Without using the Hausdorff metric, we prove the following fixed point result for nonself multivalued maps with respect to -distance.

Theorem 3.4. Let be a closed subset of a complete metric space and let be a weakly -contractive map for which is lower semicontinuous and . Then has a fixed point provided is -inward on .

Proof. Let , and be the same as in the proof of Theorem 3.1. Suppose Fix. Then, for each we have . Since w- there exists with and(3.13) Since the map is weakly -contractive, there exists such that(3.14)where is lower semicontinuous and . From (3.13) and (3.14), we get(3.15)Thus,(3.16)Since , we have(3.17)and hence, we get(3.18)Now, define a function by . Then from (3.18) we get(3.19)and using (3.16), we obtain(3.20)Thus by Theorem 2.3, has a fixed point, which is impossible. Hence, it follows that must has a fixed point.

Using the same method as in the proof of Corollary 3.2, we can obtain the following fixed point result for nonself generalized -contractions.

Corollary 3.5. Let be a closed subset of a complete metric space and let be a map satisfying inequality (3.8) for which is upper semicontinuous. Then has a fixed point provided is -inward on .

Remark 3.6. (a) Our Theorem 3.4 and Corollary 3.5 improve the results of Bae [4, Theorem 3.3 and Corollary 3.4], respectively.

(b) The analogue of all the results of this section can be established with respect to -distance [20].

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