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Fixed Points of Generalized Contractive Maps
Fixed Point Theory and Applications volumeÂ 2009, ArticleÂ number:Â 487161 (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.
1. Introduction
Throughout this paper, unless otherwise specified, is a metric space with metric . Let , and denote the collection of nonempty subsets of , nonempty closed subsets of , and nonempty closed bounded subsets of , respectively. Let be the Hausdorff metric on , that is,
A multivalued map is called
(i)contraction [1] if for a fixed constant and for each
(ii)generalized contraction [2] if for any
where is a function from to with for every ;
(iii)contractive [3] if there exist constants such that for any there is satisfying
where ;
(iv)generalized contractive [4] if there exist such that for any there is satisfying
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 wellknown 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 realvalued 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 realvalued 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 [7â€“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 wdistance on the following hold for every :
(a)if and for any then in particular, if and then ;

(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 wdistance on Suppose that there exists such that . Then (where )
We say a multivalued map is 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.
Since for each , is closed, the set is nonempty for any Let be an arbitrary but fixed element of . Since is generalized contractive, there is such that
Using (2.1) and (2.2), we have
Similarly, there is such that
Using (2.4) and (2.5), we have
From (2.5) and (2.1), it follows that
Continuing this process, we get an orbit of in such that
Using (2.8), we get
and thus for all
Note that the sequences and are decreasing, and thus convergent. Now, by the definition of the function there exists such that
Thus, for any there exists such that
and thus for all we have
Also, it follows from (2.9) that for all ,
where Note that for all we have
and thus
Now, since we have and hence the decreasing sequence converges to . Now, we show that is a Cauchy sequence. Note that for all ,
where Now, for any
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 realvalued function on defined by is lower semicontinous. Then there exists such that Further, if then .
Proof.
Since is a generalized contractive map, it follows from Lemma 2.1 that there exists a Cauchy sequence in such that the decreasing sequence converges to 0. Due to the completeness of , there exists some such that Since is lower semicontinuous, we have
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 realvalued 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.
Let be a complete space and let be a generalized contractive map. Assume that
for every with Then
Proof.
By Lemma 2.1, there exists an orbit of , which is a Cauchy sequence in . Due to the completeness of , there exists such that Since is lower semicontinuous and it follows from the proof of Lemma 2.1 that for all
where Also, we get
Assume that Then, we have
which is impossible and hence .
Corollary 2.5 (see [8]).
Let be a complete space and let be contractive map. Assume that
for every with Then
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The authors thank the referees for their valuable comments and suggestions.
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Latif, A., Abdou, A.A.N. Fixed Points of Generalized Contractive Maps. Fixed Point Theory Appl 2009, 487161 (2009). https://doi.org/10.1155/2009/487161
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DOI: https://doi.org/10.1155/2009/487161