Abstract
In this paper, we introduce the notion of -contractive multivalued mappings to generalize and extend the notion of α-ψ-contractive mappings to closed valued multifunctions. We investigate the existence of fixed points for such mappings. We also construct an example to show that our result is more general than the results of α-ψ-contractive closed valued multifunctions.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
Recently, Samet et al. [1] introduced the notions of α-ψ-contractive and α-admissible self-mappings and proved some fixed-point results for such mappings in complete metric spaces. Karapınar and Samet [2] generalized these notions and obtained some fixed-point results. Asl et al. [3] extended these notions to multifunctions by introducing the notions of -ψ-contractive and -admissible mappings and proved some fixed-point results. Some results in this direction are also given in [4–6]. Ali and Kamran [7] further generalized the notion of -ψ-contractive mappings and obtained some fixed-point theorems for multivalued mappings. Salimi et al. [8] modified the notions of α-ψ-contractive and α-admissible self-mappings by introducing another function η and established some fixed-point theorems for such mappings in complete metric spaces. N. Hussain et al. [9] extended these modified notions to multivalued mappings. Recently, Mohammadi and Rezapour [10] showed that the results obtained by Salimi et al. [8] follow from corresponding results for α-ψ-contractive mappings. More recently, Berzig and Karapinar [11] proved that the first main result of Salimi et al. [8] follows from a result of Karapınar and Samet [2]. The purpose of this paper is to introduce the notion of -contractive multivalued mappings to generalize and extend the notion of α-ψ-contractive mappings to closed valued multifunctions and to provide fixed-point theorems for -contractive multivalued mappings in complete metric spaces.
We recollect the following definitions, for the sake of completeness. Let be a metric space. We denote by the class of all nonempty closed and bounded subsets of X and by the class of all nonempty closed subsets of X. For every , let
Such a map H is called the generalized Hausdorff metric induced by the metric d. Let Ψ be a set 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 . More details as regards such a function can be found in e.g. [1, 2].
Definition 1.1 [3]
Let be a metric space and be a mapping. A mapping is -admissible if
where .
2 Main results
We begin this section by considering a family Ξ of functions satisfying the following conditions:
-
(i)
ξ is continuous;
-
(ii)
ξ is nondecreasing on ;
-
(iii)
and for all ;
-
(iv)
ξ is subadditive.
Example 2.1 Suppose that is a Lebesgue integrable mapping which is summable on each compact subset of , for each , , and for each , we have
Define by for each . Then .
Lemma 2.2 Let is a metric space and let . Then is a metric space.
Lemma 2.3 Let be a metric space, let and let . Assume that there exists such that . Then there exists such that
where .
Proof By hypothesis we have . We choose
By the definition of an infimum, since is a metric space, it follows that there exists such that
□
Definition 2.4 Let be a metric space. A mapping is called -contractive if there exist three functions , and such that
where .
In case when is strictly increasing, the -contractive mapping is called a strictly -contractive mapping.
Theorem 2.5 Let be a complete metric space and let be a strictly -contractive mapping satisfying the following assumptions:
-
(i)
G is an -admissible mapping;
-
(ii)
there exist and such that ;
-
(iii)
G is continuous.
Then G has a fixed point.
Proof By hypothesis, there exist and such that . If , then we have nothing to prove. Let . If , then is a fixed point. Let . Then from equation (2.1), we have
since . Assume that . Then from equation (2.2), we have
which is a contradiction. Hence, . Then from equation (2.2), we have
For by Lemma 2.3, there exists such that
From equations (2.4) and (2.5), we have
Applying ψ in equation (2.6), we have
Put . Then . Since G is an -admissible mapping, then . Thus we have . If , then is a fixed point. Let . Then from equation (2.1), we have
since . Assume that . Then from equation (2.8), we have
which is a contradiction. Hence, . Then from equation (2.8), we have
For by Lemma 2.3, there exists such that
From equations (2.10) and (2.11), we have
Applying ψ in equation (2.12), we have
Put . Then . Since G is an -admissible mapping, then . Thus we have . If , then is a fixed point. Let . Then from equation (2.1), we have
since . Assume that . Then from equation (2.14), we have
which is a contradiction to our assumption. Hence, . Then from equation (2.14), we have
For by Lemma 2.3, there exists such that
From equations (2.16) and (2.17), we have
Continuing the same process, we get a sequence in X such that , , , and
Let , we have
Since , we have
This implies that
Hence is a Cauchy sequence in . By completeness of , there exists such that as . Since G is continuous, we have
Thus . □
Theorem 2.6 Let be a complete metric space and let be a strictly -contractive mapping satisfying the following assumptions:
-
(i)
G is an -admissible mapping;
-
(ii)
there exist and such that ;
-
(iii)
if is a sequence in X with as and for each , then we have for each .
Then G has a fixed point.
Proof Following the proof of Theorem 2.5, we know that is a Cauchy sequence in X with as and for each . By hypothesis (iii), we have for each . Then from equation (2.1), we have
Suppose that .
We let . Taking we can find such that
Moreover, as is a Cauchy sequence, there exists such that
Furthermore,
As . Taking we can find such that
It follows from equations (2.23), (2.24), (2.25), and (2.26) that
for . Moreover, for , by the triangle inequality, we have
Letting in the above inequality, we have
This is not possible if . Therefore, we have , which implies that , i.e., . □
Example 2.7 Let be endowed with the usual metric d. Define by
and by
Take and for each . Then G is an -contractive mapping. For and we have . Also, for each with , we have . Moreover, for any sequence in X with as and for each , we have for each . Therefore, all conditions of Theorem 2.6 are satisfied and G has infinitely many fixed points. Note that Nadler’s fixed-point theorem is not applicable here; see, for example, and .
3 Consequences
It can be seen, by restricting and taking for each in Theorems 2.5 and 2.6, that:
-
Theorem 2.1 and Theorem 2.2 of Samet et al.[1] are special cases of Theorem 2.5 and Theorem 2.6, respectively;
-
Theorem 2.3 of Asl et al.[3] is a special case of Theorem 2.6;
-
Theorem 2.1 of Amiri et al.[5] is a special case of Theorem 2.5;
-
Theorem 2.1 of Salimi et al.[8] is a special case of Theorems 2.5 and 2.6.
Further, it can be seen, by considering and for each , that
-
Theorem 3.1 and Theorem 3.4 of Mohammadi et al.[4] are special cases of our results;
-
Theorem 2.2 of Amiri et al.[5] is a special case of Theorem 2.6, when is sublinear.
Remark 3.1 Observe that, in case , ψ may be a nondecreasing function in Theorem 2.5 and Theorem 2.6.
Remark 3.2 Note that in Example 2.7, . Therefore, one may not apply the aforementioned results and, as a consequence, conclude that G has a fixed point.
References
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Asl JH, Rezapour S, Shahzad N: On fixed points of α - ψ -contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212 10.1186/1687-1812-2012-212
Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α - ψ -Ciric generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24 10.1186/1687-1812-2013-24
Amiri P, Rezapour S, Shahzad N: Fixed points of generalized α - ψ -contractions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2013. 10.1007/s13398-013-0123-9
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Salimi P, Latif A, Hussain N: Modified α - ψ -contractive mappings with applications. Fixed Point Theory Appl. 2013., 2013: Article ID 151 10.1186/1687-1812-2013-151
Hussain N, Salimi P, Latif A: Fixed point results for single and set-valued α - η - ψ -contractive mappings. Fixed Point Theory Appl. 2013., 2013: Article ID 212 10.1186/1687-1812-2013-212
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Ali, M.U., Kamran, T. & Karapınar, E. -contractive multivalued mappings. Fixed Point Theory Appl 2014, 7 (2014). https://doi.org/10.1186/1687-1812-2014-7
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DOI: https://doi.org/10.1186/1687-1812-2014-7