- Research Article
- Open Access

# Fixed Points of Generalized Contractive Maps

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

**2009**:487161

https://doi.org/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

We prove some results on the existence of fixed points for multivalued generalized -contractive maps not involving the extended Hausdorff metric. Consequently, several known fixed point results are either generalized or improved.

## Keywords

- Fixed Point Theorem
- Nonempty Subset
- Lower Semicontinuous
- Cauchy Sequence
- Contraction Principle

## 1. Introduction

A multivalued map is called

where is a function from to with for every ;

where ;

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)
The metric is a -distance on .

- (b)
Let be normed space with norm Then the functions defined by and for every , are -distance.

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*
*:*

- (b)
if and for any then converges to ;

- (c)
if for any with then is a Cauchy sequence;

- (d)
if for any then is a Cauchy sequence.

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.

for every with Then

Proof.

which is impossible and hence .

Corollary 2.5 (see [8]).

for every with Then

## 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

## Copyright

This article is published under license to BioMed Central Ltd. 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.