# Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces

- Abdul Latif
^{1}Email author and - Afrah A. N. Abdou
^{2}

**2009**:432130

**DOI: **10.1155/2009/432130

© A. Latif and A. A. N. Abdou. 2009

**Received: **17 May 2009

**Accepted: **10 August 2009

**Published: **19 August 2009

## Abstract

The concept of generalized contractive multimaps in the setting of metric spaces is introduced, and the existence of fixed points for such maps is guaranteed under certain conditions. Consequently, our results either generalize or improve a number of fixed point results including the corresponding recent fixed point results of Ciric (2008), Latif-Albar (2008), Klim-Wardowski (2007), and Feng-Liu (2006). Examples are also given.

## 1. Introduction and Preliminaries

Let be a metric space, a collection of nonempty subsets of , a collection of nonempty closed bounded subsets of , a collection of nonempty closed subsets of a collection of nonempty compact subsets of and the Hausdorff metric induced by Then for any

where

An element
is called a *fixed point* of a multivalued map
if
. We denote
A sequence
in
is called an
of
at
if
for all
.

A map
is called *lower semicontinuous* if for any sequence
with
it implies that
.

Using the concept of Hausdorff metric, Nadler [1] established the following fixed point result for multivalued contraction maps, known as Nadler's contraction principle which in turn is a generalization of the well-known Banach contraction principle.

Theorem 1.1 (see [1]).

Let be a complete metric space and let be a contraction map. Then

Using the concept of the Hausdorff metric, many authors have generalized Nadler's contraction principle in many directions. 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 [2] extended Nadler's fixed point theorem without using the concept of Hausdorff metric. They proved the following result.

Theorem 1.2.

Then provided a real-valued function on , is lower semicontinuous.

Recently, Klim and Wardowski [3] generalized Theorem 1.2 and proved the following two results.

Theorem 1.3.

Let be a complete metric space and let . Assume that the following conditions hold:

Then provided a real-valued function on , is lower semicontinuous.

Theorem 1.4.

Let be a complete metric space and let . Assume that the following conditions hold:

Then provided a real-valued function on , is lower semicontinuous.

Note that Theorem 1.3 generalizes Nadler's contraction principle and Theorem 1.2. Most recently, Ciric [4] obtained some interesting fixed point results which extend and generalize the cited results. Namely, [4, Theorem 5] generalizes [5, Theorem 5], [4, Theorem 6] generalizes [4, Theorems 1.2, 1.3], and [3, theorem 7] generalizes Theorem 1.4.

In [6], Kada et al. introduced the concept of -distance on a metric space as follows:

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

()

()a map is lower semicontinuous; that is, if a sequence in with , then ;

()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 [6]. Many other examples and properties of the -distance can be found in [6, 7].

The following lemma is crucial for the proofs of our results.

Lemma 1.5 (see [8]).

Let be a closed subset of and be a w-distance on Suppose that there exists such that . Then where

Most recently, the authors of this paper generalized Latif and Albar [9, Theorem 1.3] as follows.

Theorem 1.6 (see [10]).

where and is a function from to with for every Suppose that a real-valued function on defined by is lower semicontinuous. Then there exists such that Further, if then .

The aim of this paper is to present some more general results on the existence of fixed points for multivalued maps satisfying certain conditions. Our results unify and generalize the corresponding results of Mizoguchi and Takahashi [5], Klim and Wardowski [3], Latif and Abdou [10], Ciric [4], Feng and Liu [2], Latif and Albar [9] and several others.

## 2. The Results

First we prove a theorem which is a generalization of Ciric [4, Theorem 5] and due to Klim and Wardowski [3, Theorem 1.4].

Theorem 2.1.

Let be a complete metric space with a -distance Let be a multivalued map. Assume that the following conditions hold:

(iii)the map defined by is lower semicontinuous.

Then there exists such that Further if then

Proof.

and thus, Since and is closed, it follows from Lemma 1.5 that

We also have the following interesting result by replacing the hypothesis (iii) of Theorem 2.1 with another natural condition.

Theorem 2.2.

for every with Then

Proof.

which is impossible and hence

Now, we present an improved version of Ciric [4, Theorem 6] and which also generalizes due to Latif and Abdou [10, Theorem 1.6] and due to Klim and Wardowski [3, Theorem 1.3].

Theorem 2.3.

Let be a complete metric space with a -distance Let , be a multivalued map. Assume that the following condition hold:

(iii)the map defined by is lower semicontinuous.

Then there exists such that Further if then

Proof.

Thus we conclude that is a Cauchy sequence. Now, proceeding the proof of Theorem 2.1, we get some such that and

Following the proof of Theorem 2.2, we can obtain the following result.

Theorem 2.4.

for every with Then

Now, we present a result which is a generalization of Theorem 1.4 due to Klim and Wardowski [3] and Ciric [4, Theorem 7].

Theorem 2.5.

Let be a complete metric space with a -distance Let be a multivalued map. Assume that the following conditions hold:

(i)there exists a function such that for each

(iii)the map defined by is lower semicontinuous.

Then there exists such that Further if then

Proof.

thus, Further by closedness of and since it follows from Lemma 1.5 that

## 3. Examples

The following example shows that Theorem 2.1 is a genuine generalization of Ciric [4, Theorem 5].

Example 3.1.

Thus, also satisfies all the conditions of Theorem 2.1 for Hence it follows from Theorem 2.1 that Note that Clearly, does not satisfy the hypotheses of Ciric [4, Theorem 5] because is not the metric .

Finally, we present an example which shows that Theorem 2.5 is a genuine generalization of Theorem 1.4 due to Klim-Wardowski [3].

Example 3.2.

Therefore, all assumptions of Theorem 2.5 are satisfied and Note that is not compact for all and the -distance is not a metric so do not satisfy the hypotheses of Theorem 1.4.

## Declarations

### Acknowledgment

The authors thank the referees for their valuable comments.

## Authors’ Affiliations

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