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Some results on fixed points of α-ψ-Ciric generalized multifunctions
Fixed Point Theory and Applications volume 2013, Article number: 24 (2013)
Abstract
In 2012, Samet, Vetro and Vetro introduced α-ψ-contractive mappings and gave some results on a fixed point of the mappings (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012). In fact, their technique generalized some ordered fixed point results (see (Alikhani et al. in Filomat, 2012, to appear) and (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012)). By using the main idea of (Samet et al. in Nonlinear Anal. 75:2154-2165, 2012), we give some new results for α-ψ-Ciric generalized multifunctions and some related self-maps. Also, we give an affirmative answer to a recent open problem which was raised by Haghi, Rezapour and Shahzad in 2012.
1 Introduction
In 2012, Samet, Vetro and Vetro introduced α-ψ-contractive mappings and gave fixed point results for such mappings [1]. Their results generalized some ordered fixed point results (see [1]). Immediately, using their idea, some authors presented fixed point results in the field (see, for example, [2]). Denote by Ψ the family of nondecreasing functions such that for each . It is well known that for all . Let be a metric space, be a mapping and . A multivalued operator is said to be β-ψ-contractive whenever for all , where H is the Hausdorff distance (see [2]). Alikhani, Rezapour and Shahzad proved fixed point results for β-ψ-contractive multifunctions [2]. Let be a metric space, be a mapping and . We say that is an α-ψ-Ciric generalized multifunction if
for all . One can find an idea of this notion in [3]. Also, we say that the self-map F on X is α-admissible whenever implies [1]. In this paper, we give fixed point results for α-ψ-Ciric generalized multifunctions.
In 2012, Haghi, Rezapour and Shahzad proved that some fixed point generalizations are not real ones [4]. Here, by presenting a result and an example, we are going to show that obtained results in this new field are real generalizations in respect to old similar results in the literature (compare the next result and the example with main results of [5] and [6]). The following result has been proved in [7].
Lemma 1.1 Let be a complete metric space, be a function, and T be a self-map on X such that
for all . Suppose that T is α-admissible and there exists such that . Assume that if is a sequence in X such that for all n and , then for all n. Then T has a fixed point.
Here, we give the following example which shows that Lemma 1.1 holds while we cannot use some similar old results, for example, Theorem 1 of [5].
Example 1.1 Let , , and . Now, put and define the self-map by whenever , whenever and whenever . Put for all . If and , then . Now, we consider two cases. If , then we have
Hence, . If , then
Hence, . If , then
If , then . Define whenever and otherwise. Also, put for all . Then it is easy to see that for all . Also, . If , then and so . Thus, T is α-admissible. It is easy to show that if is a sequence in M such that for all n and , then for all n. Now, by using Theorem 1.1, T has a fixed point. Now, we show that Theorem 1 in [5] does not apply here. Put and . Then , , , , , . Thus, and so , where .
2 An affirmative answer for an open problem
After providing some results on fixed points of quasi-contractions on normal cone metric spaces by Ilic and Rakocevic in 2009 [8], Kadelburg, Radenovic and Rakocevic generalized the results by considering an additional assumption and deleting the assumption on normality [9]. In 2011, Haghi, Rezapour and Shahzad proved same results without the additional assumption and for [10]. Then Amini-Harandi proved a result on the existence of fixed points of set-valued quasi-contraction maps in metric spaces by using the technique of [10] (see [11]). But similar to [9], he could prove it only for [11]. In 2012, Haghi, Rezapour and Shahzad proved the main result of [11] by using a simple method [12]. Also, they introduced quasi-contraction type multifunctions and showed that the main result of [11] holds for quasi-contraction type multifunctions. They raised an open problem about the difference between quasi-contraction and quasi-contraction type multifunctions [12]. In this section, we give a positive answer to the question. Let be a metric space. Recall that the multifunction is called quasi-contraction whenever there exists such that
for all [11]. Also, a multifunction is called quasi-contraction type whenever there exists such that
for all [12]. It is clear that each quasi-contraction type multifunction is a quasi-contractive multifunction. In 2012, Haghi, Rezapour and Shahzad raised this question that (see [12]): Is there any quasi-contractive multifunction which is not quasi-contraction type? By providing the following example, we give a positive answer to the problem.
Example 2.1 Let and . Define by whenever , whenever and whenever . We show that T is a quasi-contraction while it is not a quasi-contractive type multifunction. Put for all . First, we show that T is a quasi-contraction. It is easy to check that whenever or or . If and , then and . Hence, . If and , then and . Hence,
If and , then and . Hence, . Therefore, T is a quasi-contraction with . Now, put and . Then we have , , and . Hence, . Also, we have . Thus, for all . This shows that T is not a quasi-contractive type multifunction.
3 Main results
Now, we are ready to state and prove our main results. Let be a metric space, be a mapping and be a multifunction. We say that X satisfies the condition () whenever for each sequence in X with for all n and , there exists a subsequence of such that for all k (see [13] for the idea of this notion). Recall that T is continuous whenever for all sequence in X with . Also, we say that T is α-admissible whenever for each and with , we have for all . Note that this notion is different from the notion of -admissible multifunctions which has been provided in [7]. But, by providing a similar proof to that of Theorem 2.1 in [7], we can prove the following result.
Theorem 3.1 Let be a complete metric space, be a function, be a strictly increasing map and be an α-admissible multifunction such that for all and there exist and with . If T is continuous or X satisfies the condition (), then T has a fixed point.
Corollary 3.2 Let be a complete metric space, be a strictly increasing map, and be a multifunction such that for all with . Suppose that there exist and such that . Assume that for each and with , we have for all . If T is continuous or X satisfies the condition (), then T has a fixed point.
Proof Define by whenever and otherwise. Then, by using Theorem 3.1, T has a fixed point. □
Let be an ordered set and . We say that whenever for each there exists such that .
Corollary 3.3 Let be a complete ordered metric space, be a strictly increasing map and be a multifunction such that for all with or . Suppose that there exist and such that or . Assume that for each and with or , we have or for all . If T is continuous or X satisfies the condition (), then T has a fixed point.
Now, we give the following result.
Theorem 3.4 Let be a complete metric space, be a function, be a strictly increasing map and be an α-admissible α-ψ-Ciric generalized multifunction, and let there exist and with . If T is continuous, then T has a fixed point.
Proof If , then we have nothing to prove. Let . Then we have
If , then and so we get . Thus, , which is a contradiction. Hence, we obtain and so . If , then is a fixed point of T. Let and . Then
Put . Then and . Hence, there exists such that and so . It is clear that . Put . Then and we have
Similarly, we should have and so we get
If , then is a fixed point of T. Let . Then . Hence, there exists such that . It is clear that and . Put . Then . Also, we have
By continuing this process, we obtain a sequence in X such that , and for all n. Let . Then
and so is a Cauchy sequence in X. Hence, there exists such that . If T is continuous, then
and so . □
Corollary 3.5 Let be a complete metric space, be a strictly increasing map, and be a multifunction such that
for all with . Suppose that there exist and such that . Assume that for each and with , we have for all . If T is continuous, then T has a fixed point.
Corollary 3.6 Let be a complete ordered metric space, be a strictly increasing map and be a multifunction such that
for all with or . Suppose that there exist and such that or . Assume that for each and with or , we have or for all . If T is continuous, then T has a fixed point.
Now, we give the following result about a fixed point of self-maps on complete metric spaces.
Theorem 3.7 Let be a complete metric space, be a mapping, be a continuous and nondecreasing map such that for all and T be a self-map on X such that
for all . Assume that there exists such that for all with . Suppose that T is continuous or for all whenever . Then T has a fixed point.
Proof It is easy to check that for all . Let for all . If for some n, then is a fixed point of T. Suppose that for all n. Then we have
Since does not hold, we get . Since ϕ is nondecreasing, we have
for all n. Hence, . If is not a Cauchy sequence, then there exists and subsequences and of with such that for all i. For each , put . Then we have
and so . But we have
and so . Thus, we get
and so . On the other hand, we have
and so . This contradiction shows that is a Cauchy sequence. Since X is complete, there exists such that . If T is continuous, then we get
Now, suppose that for all whenever . Then
Thus, . □
Here, we give the following example to show that there are discontinuous mappings satisfying the conditions of Theorem 3.7.
Example 3.1 Let and . Define by whenever , whenever and whenever . Also, define the mappings and by , whenever and otherwise. An easy calculation shows us that
for all . Put . Since for all , for all with and for all whenever . Thus, the map T satisfies the conditions of Theorem 3.7. Note that is a fixed point of T.
Corollary 3.8 Let be a complete ordered metric space, be a continuous and nondecreasing map such that for all and T be a self-map on X such that
for all comparable elements . Assume that there exists such that and are comparable for all with . Suppose that T is continuous or and x are comparable for all whenever . Then T has a fixed point.
Proof Define the mapping by whenever x and y are comparable and otherwise. Then, by using Theorem 3.7, T has a fixed point. □
Corollary 3.9 Let be a complete ordered metric space, , be a continuous and nondecreasing map such that for all and T be a self-map on X such that
for all which are comparable with z. Assume that there exists such that and are comparable with z for all with . Suppose that T is continuous or and x are comparable with z for all whenever . Then T has a fixed point.
Proof Define the mapping by whenever x and y are comparable with z and otherwise. Then, by using Theorem 3.7, T has a fixed point. □
Finally, by using [14], we can find also some equivalent conditions for some presented results. We give two following results in this way. This shows us the importance of the main results of [14]. Also, Example 3.1 leads us to the fact that there are discontinuous mappings satisfying the conditions of the following results.
Proposition 3.10 Let be a complete metric space, be a mapping, be a lower semi-continuous function, be a map such that and for all and T be a self-map on X such that
for all . Assume that there exists such that for all with . Suppose that T is continuous or for all whenever . Then T has a fixed point.
Proposition 3.11 Let be a complete metric space, be a mapping, be a nondecreasing map which is continuous from right at each point, be a map such that for all and T be a self-map on X such that for all . Assume that there exists such that for all with . Suppose that T is continuous or for all whenever . Then T has a fixed point.
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The authors express their gratitude to the referees for their helpful suggestions concerning the final version of this paper.
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Mohammadi, B., Rezapour, S. & Shahzad, N. Some results on fixed points of α-ψ-Ciric generalized multifunctions. Fixed Point Theory Appl 2013, 24 (2013). https://doi.org/10.1186/1687-1812-2013-24
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DOI: https://doi.org/10.1186/1687-1812-2013-24