Abstract
In this paper, we introduce the notions of α-admissible and α-ψ-contractive type condition for nonself multivalued mappings. We establish fixed point theorems using these new notions along with a new condition. Moreover, we have constructed examples to show that our new condition is different from the corresponding existing conditions in the literature.
MSC: 47H10, 54H25.
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1 Introduction and preliminaries
In the last decades, metric fixed point theory has been appreciated by a number of authors who have extended the celebrated Banach fixed point theorem for various contractive mapping in the context of different abstract spaces; see, for example, [1–32]. Among them, we mention the interesting fixed point theorems of Samet et al. [20]. In this paper [20], the authors introduced the notions of α-ψ-contractive mappings and investigated the existence and uniqueness of a fixed point for such mappings. Further, they showed that several well-known fixed point theorems can be derived from the fixed point theorem of α-ψ-contractive mappings. Following this paper, Karapınar and Samet [21] generalized the notion α-ψ-contractive mappings and obtained a fixed point for this generalized version. On the other hand, Asl et al. [22] characterized the notions of α-ψ-contractive mapping and α-admissible mappings with the notions of -ψ-contractive and -admissible mappings to investigate the existence of a fixed point for a multivalued function. Afterward, Ali and Kamran [23] generalized the notion of -ψ-contractive mappings and obtained further fixed point results for multivalued mappings. Some results in this direction in the context of various abstract spaces were also given by the authors [24–28, 33–36]. The purpose of this paper is to prove fixed point theorems for nonself multivalued -contractive type mappings using a new condition.
Let Ψ be the family of functions , known in the literature as Bianchini-Grandolfi gauge functions (see, e.g., [30–32]), satisfying the following conditions:
() ψ is nondecreasing;
() for all , where is the n th iterate of ψ.
Notice that such functions are also known as -comparison functions in some sources (see, e.g., [29]).
It is easily proved that if , then for any and for (see, e.g., [20, 29]). Let be a metric space. A mapping is called α-ψ-contractive type if there exist two functions and such that
for each . A mapping is called α-admissible [20] if
We denote by the space of all nonempty subsets of X and by the space of all nonempty closed subsets of X. For and , . For every , let
Such a map H is called a generalized Hausdorff metric induced by d. We use the following lemma in our results.
Lemma 1.1 [23]
Let be a metric space and . Then, for each with and , there exists an element such that
Let be an ordered metric space and . We say that if for each and , we have .
2 Main results
We begin this section with the following definition which is a modification of the notion of α-admissible.
Definition 2.1 Let be a metric space and let D be a nonempty subset of X. A mapping is called α-admissible if there exists a mapping such that
for each and .
Definition 2.2 Let be a metric space and let D be a nonempty subset of X. We say that is an -contractive type mapping on D if there exist and satisfying the following conditions:
-
(i)
for all ,
-
(ii)
for each , we have
(2.1)
where .
Note that if in the above definition is a strictly increasing function, then is said to be a strictly -contractive type mapping on D.
Theorem 2.3 Let be a metric space, let D be a nonempty subset of X which is complete with respect to the metric induced by d, and let G be a strictly -contractive type mapping on D. Assume that the following conditions hold:
-
(i)
G is an α-admissible map;
-
(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 . From (2.1) we have
since and . Assume that . Then from (2.2) we have
a contradiction to our assumption. Thus . Then from (2.2) we have
For by Lemma 1.1, there exists such that
Applying ψ in (2.5), we have
Put . Then . Since G is an α-admissible mapping, . If , then is a fixed point. Let . From (2.1) we have
since and . Assume that . Then from (2.7) we have
a contradiction to our assumption. Thus . Then from (2.7) we have
For by Lemma 1.1, there exists such that
Applying ψ in (2.10), we have
Put . Then . Since G is an α-admissible mapping, . If , then is a fixed point. Let . From (2.1) we have
since and . Assume that . Then from (2.12) we have
a contradiction to our assumption. Thus . Then from (2.12) we have
For by Lemma 1.1, there exists such that
Applying ψ in (2.15), we have
Continuing the same process, we get a sequence in D such that , , , and
For , we have
Since , it follows that is a Cauchy sequence in D. Since D is complete, there exists such that as . By the continuity of G, we have
□
Theorem 2.4 Let be a metric space, D be a nonempty subset of X which is complete with respect to the metric induced by d, and let G be a strictly -contractive type mapping on D. Assume that the following conditions hold:
-
(i)
G is an α-admissible map;
-
(ii)
there exist and such that ;
-
(iii)
either
-
(a)
for any sequence in D such that as and for each , ,
or
-
(b)
for any sequence in D such that as and for each , for each .
-
(a)
Then G has a fixed point.
Proof Following the proof of Theorem 2.3, there exists a Cauchy sequence in D with as and for each . Suppose that . From (2.1) we have
Letting in (2.18), we have
Since , by condition (iii)(a), we have
Further, it is clear that . Then from (2.20) we have
which is impossible. Thus . If we use (iii)(b), then from (2.1) we have
Letting in (2.21), we have
which is impossible. Thus . □
Example 2.5 Let be endowed with the usual metric d, and let . Define by
and by
Clearly, for each . Let for each . To see that G is a strictly -contractive type mapping on D, we consider the following cases.
Case (i) When , we have
Case (ii) When and , we have
Case (iii) Otherwise, we have
where .
Thus, G is a strictly -contractive type mapping on D. For , we have , then , thus for each and . Further, for any sequence in D such that as and for each , . Therefore, all the conditions of Theorem 2.4 hold and G has a fixed point.
Corollary 2.6 Let be an ordered metric space, let be a nonempty subset of X which is complete with respect to the metric induced by d. Let be a mapping such that for each and for each with , we have
where and ψ is an increasing function in Ψ. Also, assume that the following conditions hold:
-
(i)
there exist and such that ;
-
(ii)
if then ;
-
(iii)
either
-
(a)
G is continuous,
or
-
(b)
for any sequence in D such that as and for each , as ,
or
-
(c)
for any sequence in D such that as and for each , for each .
-
(a)
Then G has a fixed point.
Proof Define by
By using condition (i) and the definition of α, we have . Also, from condition (ii), we have that implies ; by using the definitions of α and , we have that implies for each and . Moreover, it is easy to check that G is a strictly -contractive type mapping on D. Therefore, all the conditions of Theorem 2.3 (or Theorem 2.4 for (iii)(b), (iii)(c)) hold, hence G has a fixed point. □
Remark 2.7 Condition (a), in the statement of Theorem 2.4, was introduced by Samet et al. [20]. In Theorem 2.4 we introduce a new condition (b). The following examples show that (a) and (b) are independent conditions.
Example 2.8 Let . Consider for each , then as . Define by
Now, we have for each and for each . Thus condition (a) holds but . Thus condition (b) does not hold.
Example 2.9 Let . Consider for each , then as . Define by
Now, we have for each and . Then . Thus condition (b) holds but for , we have ; for , we have ; for , we have , which implies that for each . Thus condition (a) does not hold.
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Ali, M.U., Kamran, T. & Karapınar, E. A new approach to -contractive nonself multivalued mappings. J Inequal Appl 2014, 71 (2014). https://doi.org/10.1186/1029-242X-2014-71
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DOI: https://doi.org/10.1186/1029-242X-2014-71