Fixed Point Theorems for Multivalued Mappings Involving 𝛼-Function

Abstract

We obtain some fixed point theorems with error estimates for multivalued mappings satisfying a new --contractive type condition. Our theorems generalize many existing fixed point theorems, including some fixed point theorems proved for --contractive type conditions.

1. Introduction

Samet et al. [1] introduced and studied the notions of --contractive and -admissible self-mappings and obtained some well-known fixed point and coupled fixed point theorems in complete metric spaces as consequences. Karapınar and Samet [2] generalized these notions and obtained some results as an extension of the results of Samet et al. [1] and those contained therein. Asl et al. [3] extended these notions to multifunctions by introducing the notions of --contractive and -admissible mappings and obtained some fixed point theorems. Ali and Kamran [4] further generalized the notion of --contractive mappings and obtained some fixed point theorems for multivalued mappings. Related results in this direction are also given in [5–9]. In addition to that, Ali et al. [10] introduced the notion of -contractive multivalued mappings to generalize and extend the notion of --contractive mappings to closed valued multifunctions and proved some fixed point theorems for such mappings in complete metric spaces. For details on fixed point theory for multivalued mappings, we refer to [11–16]. The purpose of this paper is to establish some fixed point theorems for a new type of --contractive condition for multivalued mappings that also provides convergence rate and error estimates.

We recall the following definitions and results, for the sake of completeness. Let be a metric space. For each and , . We denote by the class of all nonempty closed subsets of . For every , let Such a map is called a generalized Hausdorff metric induced by . A point is said to be a fixed point of if . If, for , there exists a sequence in such that , then is said to be an orbit of . A mapping is said to be -orbitally lower semicontinuous if is a sequence in and implies . Throughout this paper denotes an interval on containing , that is, an interval of the form , , or and denotes the polynomial . We use the abbreviation for the th iterate of a function .

Definition 1 (see [17]). Let . A function is said to be a gauge function of order on if it satisfies the following conditions:(i) for all and ;(ii) for all .

It is easy to see that the first condition of Definition 1 is equivalent to the following: and is nondecreasing on .

Definition 2 (see [17]). A nondecreasing function is said to be a Bianchini-Grandolfi gauge function [18] on if

Remark 3. A function satisfying (2) can be used as a rate of convergence [19] on . Also note that satisfies the following functional equation:

Remark 4 (see [17]). Every gauge function of order on is a Bianchini-Grandolfi gauge function on .

Lemma 5. Let be a metric space. Let and . Then, for each , there exists such that .

Lemma 6 (see [17]). Let be a gauge function of order on . If is a nonnegative and nondecreasing function on satisfying then it has the following properties:(i) for all ;(ii) for all and .Moreover, for each , we have(iii) for all ,(iv) for all .

Definition 7 (see [3]). Let be a metric space and let be a mapping. A mapping is said to be an -admissible if where .

2. Main Results

Theorem 8. Let be a complete metric space and let be an -admissible mapping such that for all and , with , where is a Bianchini-Grandolfi gauge function on . Moreover, the strict inequality holds when . Suppose that there exists such that and , for some . Then,(i)there exists an orbit of in and such that ;(ii) is a fixed point of if and only if the function is -orbitally lower semicontinuous at .

Proof. Consider . We assume that , for otherwise is a fixed point of . Define , where is defined by (2). Since, from (3), , we have Notice that . It follows from (6) that . By hypothesis, we have . We can choose an such that Thus, we have It follows from Lemma 5 that there exists such that We assume that , for otherwise is a fixed point of . From inequalities (9) and (10), we have Note that . Also, we have , since Since is an -admissible, then we have . Now choose such that Thus, we have It again follows from Lemma 5 that there exists such that We assume that , for otherwise is a fixed point of . From (11), (14), and (15), we have Note that . Also, we have , since Repeating the above argument, inductively we obtain a sequence such that We claim that is a Cauchy sequence. For , from (20) we have By using (2), it follows from (22) that is a Cauchy sequence. Thus, there exists with as . Since , from (6), (19), and (20), we have Letting , from (23), we get Suppose is -orbitally lower semicontinuous at ; then, Hence, , since is closed. Conversely, if is fixed point of , then .

Example 9. Let be endowed with the usual metric and let . Define by and define by Take for each . Let ; then, we have such that and . As we know, for . Then, we have whenever . Thus, is an -admissible mapping. For and , from (6), we have for and , we have Hence, (6) holds for each and with . Therefore, all the conditions of Theorem 8 hold and hence has a fixed point.

Example 10. Let be endowed with the usual metric and let . Define by and define by Take for each . Let ; then, we have such that and . As we know, for . Then, we have whenever . Thus, is an -admissible mapping. For and , from (6), we have for otherwise we have Hence, (6) holds for each and with . Therefore, all the conditions of Theorem 8 hold and hence has a fixed point.

Theorem 11. Let be a complete metric space and let be an -admissible mapping such that for all and , with , where is a gauge function of order on and is a nondecreasing function defined by (4). Moreover, the strict inequality holds when . Suppose that there exists such that and , for some . Then,(i)there exists an orbit of in that converges with rate of convergence at least to a point , where and is defined by (2);(ii)for all , we have the following a priori estimate: where ;(iii)for all , we have the following a posteriori estimate: (iv)for all , we have where ;(v) is a fixed point of if and only if the function is -orbitally lower semicontinuous at .

Proof. (i) Following the proof of Theorem 8, we have an orbit of at in such that and .
(ii) For , by using (20) and Lemma 6(iii), we have Taking fixed and letting , we get Note that
Since , therefore since . Thus, we have Substituting this in (39), we get
(iii) For , from (39), we have Putting , , and , we have
Putting and , we have since . Now, by Lemma 6(iv), we have which means that For , from (46), we have
(iv) For , by using (20) and Lemma 6(iii), we have
(v) The proof follows from the same arguments as in the proof of Theorem 8.

Corollary 12. Let be complete metric space and let be an -admissible mapping such that for all , with , where is a gauge function of order on an interval . Suppose that there exists in such that and for some . Suppose that for any sequence in such that as with , for each , implies , for each . Then, the following statements hold true:(i)there exists an orbit of in that converges to a fixed point , where and is defined by (2);(ii)the estimates (35)–(37) are valid.

Theorem 13. Let be a complete metric space and let be a continuous and -admissible mapping such that with strict inequality holds if , where is a gauge function of the first order on and Suppose that there exists such that and for some . Then, the following statements hold true:(i)there exists an orbit of in that converges to a fixed point of ;(ii)for , we have the following a priori estimate: where and is a nondecreasing function defined by (4);(iii)for all , we have the following a posteriori estimate:

Proof. Consider . Define , where is defined by (2). Since, from (3), , we have Assume that , for otherwise and is a fixed point of . From (53), we have . By hypothesis, we have . We can choose such that Thus, we have It follows from Lemma 5 that there exists such that From the last two inequalities, we have since . Assume that . From (61), we have which is not possible. Thus, . From (61), we have Proceeding inductively in a similar way as in Theorem 8, we obtain the sequence in such that as . Since is continuous, by taking limit as , we have . Estimates (35) and (36) become (55) and (56) for .

Remark 14. Note that our results generalize [3, Theorem 2.1]; [5, Theorem 3.4]; [17, Theorems 4.1 and 4.2; and Corollary 4.5]; [20, Theorems 2.1 and 2.8; and Corollary 2.12]; [21, Theorem 2.1]; [22, Theorems 2.11 and 2.15]; and [23, Theorems 2.1 and 2.2].

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The third author therefore acknowledges with thanks the DSR for financial support. The authors are grateful to the referees for useful comments.