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On -contractive multi-valued mappings
Fixed Point Theory and Applications volume 2013, Article number: 137 (2013)
Abstract
In this paper, we generalize the contractive condition for multi-valued mappings given by Asl, Rezapour and Shahzad in 2012. We establish some fixed point theorems for multi-valued mappings from a complete metric space to the space of closed or bounded subsets of the metric space satisfying generalized -contractive condition.
MSC:47H10, 54H25.
1 Introduction
Samet et al. [1] introduced the notion of α-ψ-contractive self-mappings of a metric space. Recently, Asl et al. [2] introduced the notion of -ψ-contractive mappings to extend the notion α-ψ-contractive mappings. In this paper, we generalize the notion of -ψ-contractive mappings and prove some fixed point theorems for such mappings.
Let Ψ be a family of nondecreasing functions, such that for each , where is the n th iterate of ψ. It is known that for each , we have for all and for [1]. Let be a metric space. A mapping is called α-ψ-contractive if there exist two functions and such that for each . A mapping is called α-admissible [1] if . We denote by the space of all nonempty subsets of X, by the space of all nonempty bounded subsets of X and by the space of all nonempty closed subsets of X. For and , . For every , . When , we denote by . For every , let
Such a map H is called generalized Hausdorff metric induced by d. Let be an ordered metric space and . We say that if for each and , we have . We give a few definitions and the result due to Asl et al. [2] for convenience.
Definition 1.1 [2]
Let be a metric space and let be a mapping. A mapping is -admissible if , where .
Definition 1.2 [2]
Let be a metric space. A mapping is called -ψ-contractive if there exist two functions and such that
for all .
Theorem 1.3 [2]
Let be a complete metric space, let be a function, let be a strictly increasing map and T be a closed-valued, -admissible and -ψ-contractive multi-function on X. Suppose that there exist and such that . Assume that if is a sequence in X such that for all n and , then for all n. Then G has a fixed point.
2 Main results
We begin this section by introducing the following definition.
Definition 2.1 Let be a metric space and let be a mapping. We say that G is generalized -contractive if there exists such that
for each and , where .
Note that an -ψ-contractive mapping is generalized -contractive. In case when is strictly increasing, generalized -contractive is called strictly generalized -contractive. The following lemma is inspired by [[3], Lemma 2.2].
Lemma 2.2 Let be a metric space and . Then, for each with and , there exists an element such that
Proof It is given that . Choose
Then, by using the definition of , it follows that there exists such that
□
Lemma 2.3 Let be a metric space and . Assume that there exists a sequence in X such that and . Then x is a fixed point of G if and only if the function is lower semi-continuous at x.
Proof Suppose is lower semi-continuous at x, then
By the closedness of G it follows that . Conversely, suppose that x is a fixed point of G, then . □
Theorem 2.4 Let be a complete metric space and let be an -admissible strictly generalized -contractive mapping. Assume that there exist and such that . Then x is a fixed point of G if and only if is lower semi-continuous at x.
Proof By the hypothesis, there exist and such that . If , then we have nothing to prove. Let . If , then is a fixed point. Let . Since G is -admissible, so , we have
For given by Lemma 2.2, there exists such that
It follows from (2.3), (2.4) and (2.1) that
It is clear that and . Thus . Since ψ is strictly increasing, by (2.5), we have
Put , then . If , then is a fixed point. Let , then by Lemma 2.2, there exists such that
It is clear that , and . Now put . Then . If , then is a fixed point. Let . Then by Lemma 2.2 there exists such that
By continuing the same process, we get a sequence in X such that . Also, , and or
For each , we have
Since , it follows that is a Cauchy sequence in X. Thus there is such that . Letting in (2.6), we have
The rest of the proof follows from Lemma 2.3. □
Example 2.5 Let be endowed with the usual metric d. Define and by
and
Let for all . For each and , we have
Hence G is a strictly generalized -contractive mapping. Clearly, G is -admissible. Also, we have and such that . Therefore, all conditions of Theorem 2.4 are satisfied and G has infinitely many fixed points. Note that Theorem 1.3 in Section 1 is not applicable here. For example, take and .
Corollary 2.6 Let be a complete ordered metric space, be a strictly increasing map and be a mapping such that for each and with , we have
Also, assume that
-
(i)
there exist and such that ,
-
(ii)
if , then .
Then x is a fixed point of G if and only if is lower semi-continuous at x.
Proof Define by
By using condition (i) and the definition of α, we have . Also, from condition (ii), we have implies ; by using the definitions of α and , we have implies . Moreover, it is easy to check that G is a strictly generalized -contractive mapping. Therefore, by Theorem 2.4, x is a fixed point of G if and only if is lower semi-continuous at x. □
Definition 2.7 Let be a metric space and be a mapping. We say that G is a generalized -contractive mapping if there exists such that
for each and , where .
Lemma 2.8 Let be a metric space and . Assume that there exists a sequence in X such that and . Then if and only if the function is lower semi-continuous at x.
Proof Suppose that is lower semi-continuous at x, then
Hence, because implies . Conversely, suppose that . Then . □
Theorem 2.9 Let be a complete metric space and let be an -admissible generalized -contractive mapping. Assume that there exist and such that . Then there exists such that if and only if is lower semi-continuous at x.
Proof By the hypothesis of the theorem, there exist and such that . Assume that , for otherwise, is a fixed point. Let . As G is -admissible, we have . Then
Since , there is . Then
From (2.12) and (2.13), we have
Since ψ is nondecreasing, we have
As , we have . Since , there is . Assume that , for otherwise, is a fixed point of G. Then
Since ψ is nondecreasing, we have
By continuing in this way, we get a sequence in X such that and for . Further we have
For each , we have
Since , it follows that is a Cauchy sequence in X. As X is complete, there exists such that . Letting in (2.18), we have
The rest of the proof follows from Lemma 2.8. □
Example 2.10 Let be endowed with the usual metric d. Define and by
and
Let for all . For each and , we have
Hence G is a generalized -contractive mapping. Clearly, G is -admissible. Also, we have and such that . Therefore, all conditions of Theorem 2.9 are satisfied and G has infinitely many fixed points.
Corollary 2.11 Let be a complete ordered metric space, and be a mapping such that for each and with , we have
Also, assume that
-
(i)
there exists such that , i.e., there exists such that ,
-
(ii)
if , then .
Then there exists such that if and only if is lower semi-continuous at x.
Proof Define by
By using condition (i) and the definition of α, we have . Also, from condition (ii), we have implies , by using the definitions of α and , we have implies . Moreover, it is easy to check that G is a generalized -contractive mapping. Therefore, by Theorem 2.9, there exists such that if and only if is lower semi-continuous at x. □
References
Samet B, Vetro C, Vetro P: Fixed point theorems for α - ψ -contractive type mappings. Nonlinear Anal. 2012, 75: 2154–2165. 10.1016/j.na.2011.10.014
Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212. doi:10.1186/1687–1812–2012–212
Kamran T: Mizoguchi-Takahashi’s type fixed point theorem. Comput. Math. Appl. 2009, 57: 507–511.
Acknowledgements
Authors are grateful to referees for their suggestions and careful reading.
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Ali, M.U., Kamran, T. On -contractive multi-valued mappings. Fixed Point Theory Appl 2013, 137 (2013). https://doi.org/10.1186/1687-1812-2013-137
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DOI: https://doi.org/10.1186/1687-1812-2013-137
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