Fixed Points of Generalized Contractive Maps
© A. Latif and A. A. N. Abdou. 2009
Received: 13 October 2008
Accepted: 27 January 2009
Published: 3 February 2009
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.
A multivalued map is called
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.  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 ).
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  have obtained the following general form of the Nadler's theorem.
Theorem 1.2 (see ).
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 . Without using the concept of the Hausdorff metric, they proved the following result.
Theorem 1.3 (see ).
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  generalized Theorem 1.3 as follows:
Theorem 1.4 (see ).
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.  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 , 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 .
The metric is a -distance on .
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 ).
Let and be sequences in and let and be sequences in converging to Then, for the w-distance on the following hold for every :
if and for any then converges to ;
if for any with then is a Cauchy sequence;
if for any then is a Cauchy sequence.
Lemma 1.6 (see ).
Let be a closed subset of and let be a w-distance on Suppose that there exists such that . Then (where )
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 . In particular, if we take a constant map then the map is weakly contractive (in short, -contractive) , and further if we take then we obtain and is contractive .
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 , Latif and Albar , and Klim and Wardowski .
First, we prove key lemma in the setting of metric spaces.
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.
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.
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 .
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 ).
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.
for every with Then
which is impossible and hence .
Corollary 2.5 (see ).
for every with Then
The authors thank the referees for their valuable comments and suggestions.
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