# Fixed Points of Generalized Contractive Maps

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

**2009**:487161

**DOI: **10.1155/2009/487161

© A. Latif and A. A. N. Abdou. 2009

**Received: **13 October 2008

**Accepted: **27 January 2009

**Published: **3 February 2009

## Abstract

## 1. Introduction

where is a function from to with for every ;

where is a function from to with for every

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
imply that
.

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

Theorem 1.1 (see [1]).

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

This result has been generalized in many directions. For instance, Mizoguchi and Takahashi [2] have obtained the following general form of the Nadler's theorem.

Theorem 1.2 (see [2]).

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

Another extension of Nadler's result obtained recently by Feng and Liu [3]. Without using the concept of the Hausdorff metric, they proved the following result.

Theorem 1.3 (see [3]).

Let X be a complete space and let be a multivalued contractive map. Suppose that a real-valued function on , , is lower semicontinuous. Then

Most recently, Klim and Wardowski [4] generalized Theorem 1.3 as follows:

Theorem 1.4 (see [4]).

Let X be a complete metric space and let be a multivalued generalized contractive map such that a real-valued function on , is lower semicontinuous. Then

Recently, Kada et al. [5] introduced the concept of -distance on a metric space as follows.

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

() a map is lower semicontinuous;

() for any there exists such that and imply

Using the concept of -distance, they improved Caristi's fixed point theorem, Ekland's variational principle, and Takahashi's existence theorem. In [6], Susuki and Takahashi proved a fixed point theorem for contractive type multivalued maps with respect to -distance. See also [7–12].

Let us give some examples of -distance [5].

- (a)
- (b)

The following lemmas concerning -distance are crucial for the proofs of our results.

Lemma 1.5 (see [5]).

*Let*
*and*
*be sequences in*
*and let*
*and*
*be sequences in*
*converging to*
*Then, for the w-distance*
*on*
*the following hold for every*
*:*

Lemma 1.6 (see [9]).

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

*generalized*

*-contractive*if there exist a -distance on and a constant such that for any there is satisfying

where and is a function from to with for every

Note that if we take then the definition of generalized -contractive map reduces to the definition of generalized contractive map due to Klim and Wardowski [4]. In particular, if we take a constant map then the map is weakly contractive (in short, -contractive) [8], and further if we take then we obtain and is contractive [3].

In this paper, using the concept of -distance, we first establish key lemma and then obtain fixed point results for multivalued generalized -contractive maps not involving the extended Hausdorff metric. Our results either generalize or improve a number of fixed point results including the corresponding results of Feng and Liu [3], Latif and Albar [8], and Klim and Wardowski [4].

## 2. Results

First, we prove key lemma in the setting of metric spaces.

Lemma 2.1.

*Let*
*be a generalized*
*-contractive map. Then, there exists an orbit*
*of*
*in*
*such that the sequence of nonnegative real numbers*
*is decreasing to zero and the sequence*
*is Cauchy.*

Proof.

and thus by Lemma 1.5, is a Cauchy sequence.

Using Lemma 2.1, we obtain the following fixed point result which is an improved version of Theorem 1.4 and contains Theorem 1.3 as a special case.

Theorem 2.2.

*Let*
*be a complete space and let*
*be a generalized*
*-contractive map. Suppose that a real-valued function*
*on*
*defined by*
*is lower semicontinous. Then there exists*
*such that*
*Further, if*
*then*
.

Proof.

and thus, Since and is closed, it follows from Lemma 1.6 that

As a consequence, we also obtain the following fixed point result.

Corollary 2.3 (see [8]).

Let be a complete space and let be a -contractive map. If the real-valued function on defined by is lower semicontinous, then there exists such that Further, if then

Applying Lemma 2.1, we also obtain a fixed point result for multivalued generalized -contractive map satisfying another suitable condition.

Theorem 2.4.

Proof.

which is impossible and hence .

Corollary 2.5 (see [8]).

## Declarations

### Acknowledgment

The authors thank the referees for their valuable comments and suggestions.

## Authors’ Affiliations

## References

- Nadler SB Jr.:
**Multi-valued contraction mappings.***Pacific Journal of Mathematics*1969,**30**(2):475–488.MathSciNetView ArticleMATHGoogle Scholar - Mizoguchi N, Takahashi W:
**Fixed point theorems for multivalued mappings on complete metric spaces.***Journal of Mathematical Analysis and Applications*1989,**141**(1):177–188. 10.1016/0022-247X(89)90214-XMathSciNetView ArticleMATHGoogle Scholar - Feng Y, Liu S:
**Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings.***Journal of Mathematical Analysis and Applications*2006,**317**(1):103–112. 10.1016/j.jmaa.2005.12.004MathSciNetView ArticleMATHGoogle Scholar - Klim D, Wardowski D:
**Fixed point theorems for set-valued contractions in complete metric spaces.***Journal of Mathematical Analysis and Applications*2007,**334**(1):132–139. 10.1016/j.jmaa.2006.12.012MathSciNetView ArticleMATHGoogle Scholar - Kada O, Suzuki T, Takahashi W:
**Nonconvex minimization theorems and fixed point theorems in complete metric spaces.***Mathematica Japonica*1996,**44**(2):381–391.MathSciNetMATHGoogle Scholar - Suzuki T, Takahashi W:
**Fixed point theorems and characterizations of metric completeness.***Topological Methods in Nonlinear Analysis*1996,**8**(2):371–382.MathSciNetMATHGoogle Scholar - Ansari QH:
**Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory.***Journal of Mathematical Analysis and Applications*2007,**334**(1):561–575. 10.1016/j.jmaa.2006.12.076MathSciNetView ArticleMATHGoogle Scholar - Latif A, Albar WA:
**Fixed point results in complete metric spaces.***Demonstratio Mathematica*2008,**41**(1):145–150.MathSciNetMATHGoogle Scholar - Lin L-J, Du W-S:
**Some equivalent formulations of the generalized Ekeland's variational principle and their applications.***Nonlinear Analysis: Theory, Methods & Applications*2007,**67**(1):187–199. 10.1016/j.na.2006.05.006MathSciNetView ArticleMATHGoogle Scholar - Suzuki T:
**Several fixed point theorems in complete metric spaces.***Yokohama Mathematical Journal*1997,**44**(1):61–72.MathSciNetMATHGoogle Scholar - Takahashi W:
*Nonlinear Functional Analysis: Fixed Point Theory and Its Application*. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar - Ume JS, Lee BS, Cho SJ:
**Some results on fixed point theorems for multivalued mappings in complete metric spaces.***International Journal of Mathematics and Mathematical Sciences*2002,**30**(6):319–325. 10.1155/S0161171202110350MathSciNetView ArticleMATHGoogle Scholar

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