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Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces
Fixed Point Theory and Applications volume 2009, Article number: 432130 (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.
Let be a complete metric space and let be a map such that for any fixed constants and for each there is satisfying the following conditions:
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:
(i)there exist a number and a function such that for each
(ii)for any there is satisfying
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:
(i)there exists a function such that for each
(ii)for any there is satisfying
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]).
Let be a complete metric space with a -distance . Let be a multivalued map satisfying that for any constant and for each there is such that
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:
(i)there exists a function such that for each
(ii)for any there exists satisfying
(iii)the map defined by is lower semicontinuous.
Then there exists such that Further if then
Proof.
let be any initial point. Then there exists such that
From (2.3) we get
Define a function by
Using the facts that for each and we have
From (2.4) and (2.5), we have
Similarly, for , there exists such that
Thus
Continuing this process we can get an orbit of in satisfying the following:
for each integer . Since for each and from (2.12), we have for all
Thus the sequence of nonnegative real numbers is decreasing and bounded below, thus convergent. Therefore, there is some such that
From (2.11), as for all we get
Thus, we conclude that the sequence of nonnegative reals is bounded. Therefore, there is some such that
Note that for each so we have Now we will show that Suppose that Then we get
Now consider Suppose to the contrary, that Then and so from (2.14) and (2.16) there is a positive integer such that
Then from (2.19), (2.11) and (2.18), we get
Thus for all
that is,
and we get
Thus for all
Thus, from (2.12) and (2.24), we get
where Clearly as From (2.18) and (2.25), we have for any
Since and there is a positive integer such that Now, since for each by (2.26) we have
a contradiction. Hence, our assumption is wrong. Thus Now we will show that Since then from (2.16) we can read as
so, there exists a subsequence of such that
Now from (2.7) we have
and from (2.12), we have
Taking the limit as and using (2.14), we get
If we suppose that then from last inequality, we have
which contradicts with (2.30). Thus Then from (2.14) and (2.15), we have
and thus
Now, let
Then by (2.7), Let be such that Then there is some such that
Thus it follows from (2.12),
By induction we get
Now, using (2.15) and (2.39), we have
Now, we show that is a Cauchy sequence, for all we get
Hence we conclude, as that is Cauchy sequence. Due to the completeness of , there exists some such that . Since is lower semicontinuous and from (2.34), we have
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.
Suppose that all the hypotheses of Theorem 2.1 except (iii) hold. Assume that
for every with Then
Proof.
Following the proof of Theorem 2.1, there exists a Cauchy sequence with Due to the completeness of , there exists such that Since is lower semicontinuous and it follows for all
Assume that Then, we have
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:
(i)there exist functions and with nondecreasing such that
(ii)for any there exists satisfying the following conditions:
(iii)the map defined by is lower semicontinuous.
Then there exists such that Further if then
Proof.
Let be an arbitrary, then there exists such that
From (2.48) we have
Define a function by
Since we have
Thus from (2.49)
Similarly, there exists such that
Then by definition of we get
Continuing this process, we get an orbit of at such that
Thus
Since for all we get
Thus the sequence of nonnegative real numbers is decreasing and bounded below, thus convergent. Now, we want to show that the sequence is also decreasing. Suppose to the contrary, that then as is nondecreasing, we have
Now using (2.56), (2.57) and (2.60) with , we get
a contradiction. Thus the sequences is decreasing. Now let
Thus by (2.52), Then for any there exists such that
So, from (2.58), for all we get
Thus by induction we get for all
Since from (2.56) and (2.65), we have
for all Note that Now, we show that is a Cauchy sequence. For all we have
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.
Suppose that all the hypotheses of Theorem 2.3 except (iii) hold. Assume that
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
(ii)for any there exists satisfying
(iii)the map defined by is lower semicontinuous.
Then there exists such that Further if then
Proof.
Let be any initial point. Then from (ii) we can choose such that
Using the analogous method like in the proof of Lemma  2.1 [10], we obtain the existence of Cauchy sequence such that and satisfying
Consequently, there exists such that . Since is lower semicontinuous, we have
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.
Let with the usual metric . Define a function , by
Clearly, is a -distance on and . Let be such that
Define now as follows
Note that
and is lower semicontinuous. Moreover for each we have Take then we have
Further, note that
Hence, for all , satisfies all the conditions of Theorem 2.1. Now, if then we have and
Note that for there is such that
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.
Let with the usual metric . Define a function , by
Then is a -distance on Note that . Now, for any real number define by
and define a constant function by
Note that for all . And for each we have
Thus, is continuous. Now for each there exists satisfying
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.
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The authors thank the referees for their valuable comments.
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Latif, A., Abdou, A.A.N. Fixed Point Results for Generalized Contractive Multimaps in Metric Spaces. Fixed Point Theory Appl 2009, 432130 (2009). https://doi.org/10.1155/2009/432130
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DOI: https://doi.org/10.1155/2009/432130