Abstract

The purpose of this paper is to find out fixed point results for semi--dominated multivalued mappings fulfilling a new generalized locally dominated multivalued contractive condition on a closed ball in complete dislocated metric space. Example and application both are given to show the novelty of our results. Our results merge, extend, and infer many results.

1. Introduction and Preliminaries

Fixed point theory has a foundational role in functional analysis. Banach [1] established the fundamental fixed point theorem, which has played a significant role in different fields of applied mathematics. Due to its significance, a large number of authors have proven many interesting multiplications of his result (see [1–33]). Many authors introduced fixed point theorems in complete dislocated metric space. The idea of dislocated topology has been applied in the field of logic programming semantics (see [14]).

Wardowski [33] introduced new type of contraction called contraction and showed a new generalized fixed point theorem. He generalized many previous fixed point results in different directions. Many other useful results on contractions can be seen in [3–5, 11, 12, 15, 18, 22, 24–27]. In this paper, we recalled the concept of contraction to obtain some common fixed point results for semi--dominated multivalued mappings on proximinal sets satisfying a new type rational -contraction in the context of complete dislocated metric spaces. We have also given an example in which the mapping is not -admissible but it fulfills the condition of -dominated.

Definition 1 (see [14]). Let be a nonempty set. A mapping is called a dislocated metric (or simply -metric) if the following conditions hold, for any :(i)If , then (ii)(iii)

Then is called a dislocated metric on , and the pair is called dislocated metric space or metric space. It is clear that if , then, from (i), . But if , may not be We use DMS instead of dislocated metric space.

Definition 2 (see [14]). Let be a DMS:(i)A sequence in is called a Cauchy sequence if given ; there corresponds such that for all one has or (ii)A sequence is said to be dislocated—converges (for short -converges) to if In this case is called a -limit of (iii) is called complete if every Cauchy sequence in converges to a point such that .

Definition 3 (see [29]). Let be a nonempty subset of DMS and let An element is called a best approximation in if

If each has at least one best approximation in , then is called a proximinal set. We denote as the set of all closed proximinal subsets of

Definition 4 (see [29]). The function , defined by is called dislocated Hausdorff metric on

Definition 5 (see [29]). Let be a multivalued mapping and . Let ; we say that is semi--admissible on , whenever implies that , for all , , where If , then we say that is -admissible on .

Definition 6. Let be a DMS. Let be multivalued mapping and . Let , and we say that is semi--dominated on , whenever for all , where If , then we say that is -dominated on If is a self-mapping, then is semi--dominated on , whenever for all

Definition 7 (see [33]). Let be a metric space. A mapping is said to be an -contraction if there exists such thatwhere is a mapping satisfying the following conditions:

(F1) is strictly increasing; that is, for all , such that , .

(F2) For each sequence of positive numbers, if and only if .

(F3) There exists such that .

Lemma 8. Let be a DMS. Let be a dislocated Hausdorff metric space on Then, for all and for each , there exists satisfying ; then .

Example 9 (see [14]). If , then defines a dislocated metric on .

Example 10 (see [33]). The family of is not empty:(1)(2)(3)

Example 11. Let Define the mapping by Define the multivalued mappings by Suppose that and As , then . Now, , and this means that the pair is not -admissible. Also, and This implies that and are not -admissible individually. Now, , for all Hence is -dominated mapping. Similarly, Hence it is clear that and are -dominated but not -admissible.

2. Main Result

Let be a DMS, let , and let be the multifunctions on . Let be an element such that Let be such that Let be such that Continuing this method, we get a sequence of points in such that and , where . Also , and We denote this iterative sequence by We say that is a sequence in generated by

Theorem 12. Let be a complete DMS. Suppose that there exists a function Let and let be the semi-dominated mappings on Assume that, for some ,for all with either or , where and andwhere and Then is a sequence in , for all , and Also if inequality (6) holds for and either or for all , then and have common fixed point in .

Proof. Consider a sequence From (7), we get It follows that Let for some . If , where , then since are semi-dominated mappings on , and As , this implies that Also , so Now, by using Lemma 8, we have this implies that for all As is strictly increasing, we have which implies thatHere HenceNow, Thus, Hence , for all Continuing this process, we get this implies that for all As is strictly increasing, we have which implies that Here HenceConsequently, which implies that This implies thatAnd so By , we find thatWe shall prove that is Cauchy in So it suffices to show that We argue by contradiction. Suppose that there exist and sequences and of natural numbers such thatBy triangular inequality, we haveFrom (24), there exist such that, for all , Combining (26) with (27) yieldsAs are semi-dominated mappings on , and , for all Now, by using Lemma 8 and condition (6), we getwhich implies that As , we get we deduce that which is a contradiction. Thus, is a Cauchy sequence in Since is a complete metric space, so there exist such that as ; thenSince and , by using Lemma 8 and inequality (6), we have By using (33), we get which implies that which is a contradiction; hence or Similarly, by using Lemma 8, inequality (6), and the inequality we can show that Hence, and have a common fixed point in Now, This implies that