Abstract

The aim of this work is to introduce double controlled dislocated quasi-metric type spaces and to obtain fixed point results for a pair of multivalued mappings satisfying Kannan-type double controlled contraction in such spaces. An example has been built and a remark has been given which shows that how our result can be used when a corresponding new result in dislocated quasi -metric type spaces cannot be used. Our results generalize and extend many existing results in the literature.

1. Introduction and Preliminaries

The field of fixed point theory covers both pure and applied mathematics. Fixed point theory is a special branch of functional analysis and its results are used to show the solution existence of different mathematical models. A multivalued mapping from to the subsets of has a fixed point , if . If we take elements of instead of subsets of , then represents a single valued mapping from to . A single valued mapping has a fixed point if . Fixed point results for multivalued mappings generalize the results for single-valued mappings. Fixed point results of multivalued mappings have applications in engineering, economics, Nash equilibria, and game theory [1–4]. Due to its importance, many interesting results have been proved in the setting of multivalued mappings, for example, see [5–14].

By excluding one and a half condition, out of three conditions of a metric space, we obtain dislocated quasi-metric space [11]. Complete dislocated quasi-metric space is a generalization of 0-complete and complete quasi-partial metric space [15–17]. Dislocated quasi-metric also generalizes dislocated metric, partial metric, and quasi-metric. Fixed point results in dislocated quasi-metric space can be seen in [18–22].

Ran and Reurings [23] gave a fixed point result with an order and obtained solution to matrix equations as an application. Nieto et al. [24] gave an extension to the result in [23] for ordered mappings and used it to give a unique solution for ODE with periodic boundary conditions. Altun et al. [25] introduced a new approach to common fixed point of mappings, satisfying a generalized contraction with a new restriction in a complete ordered metric space.

On the contrary, Kamran et al. [26] introduced a new concept of generalized b metric space, named as extended b-metric spaces, see also [27]. They replaced the parameter in triangle inequality by the control function . Recently, Mlaiki et al. [27] generalized the triangle inequality in b-metric space by using control function in a different style and introduced controlled metric type spaces. Very recently, Abdeljawad et al. [28] generalized the concept of controlled metric type space by introducing two control functions and and establishing double controlled metric type space. In this paper, we extend the result of Altun et al. [25] in four different ways by using(i)Multivalued mappings instead of single-valued mappings(ii)Kannan-type contraction instead of Banach-type contraction(iii)Left -sequentially complete double controlled dislocated quasi-metric type space instead of complete metric space(iv)A function which generalized the partial order relation

Our results unify, extend, and generalize several comparable results in the existing literature. We give the following definitions and results which will be useful to understand the paper.

Definition 1. (see [28]). Given noncomparable functions . Suppose that satisfies the following: (q1) if and only if  (q2)  (q3) , for all Then, is called double controlled metric type with the functions and , and the pair is called double controlled metric type space with the functions .

Theorem 1 (see [28]). Let be a complete double controlled metric type space with the functions and let be a given mapping. Suppose that the following conditions are satisfied.
There exists such that

For , choose . Assume that

In addition, for every , we have

Then, has a unique fixed point .

Definition 2. Given noncomparable functions . If satisfiesfor all , then is called a double controlled dislocated quasi-metric type with the functions and and is called a double controlled dislocated quasi-metric type space. If then is called a controlled quasi-metric type space.

Remark 1. Any dislocated quasi-metric space or double controlled metric-type space is double controlled dislocated quasi-metric type space, but the converse is not true in general. Also, a controlled dislocated quasi-metric type space is also double controlled quasi-metric type space. The converse is not true in general (see Examples 1 and 2).

Example 1. Let . Define by , , , , , , , , and .
Define as , , , , , , , ,
, , , and . It is obvious that is double controlled dislocated quasi-metric type for all . It is clear that is not double controlled metric-type space. Also, it is not controlled dislocated quasi-metric type. Indeed,

Definition 3. Let be a double controlled dislocated quasi-metric type space:(i)A sequence in is called left -Cauchy if , such that and .(ii)A sequence is double controlled dislocated quasi and converges to if or for any , there exists , such that, for all and . In this case, is called a -limit of .(iii) is called left -sequentially complete if every left -Cauchy sequence in converges to a point such that .

Definition 4. Let be a double controlled dislocated quasi-metric type space. Let be a nonempty subset of and let . An element is called a best approximation in ifIf each has at least one best approximation in , then is called a proximinal set. We denote the set of all proximinal subsets of by .

Definition 5. (see [11]). The function , defined byis called double controlled dislocated quasi-Hausdorff metric type on . Also, is known as double controlled dislocated quasi-Hausdorff metric type space.
Following similar arguments of Lemma 1.7 given by Shoaib [11], we obtain the following lemma.

Lemma 1. Let be a double controlled dislocated quasi-metric type space. Let be a double controlled dislocated quasi-Hausdorff metric type space on . Then, for all and for each , there exists , such that and .

2. Main Result

Let be a double controlled complete dislocated quasi-metric type space and and be multifunctions on . Let be an element such that and . Let be such that and . Let be such that and so on. Thus, we construct a sequence of points in such that and , with , , , and , where . We denote this iterative sequence by . We say that is a sequence in generated by . If , then we say that is a sequence in generated by . Let , define and .

Definition 6. Let be a nonempty set, be a mapping, and be the multivalued mappings; then, the pair is said to be a pair of -Alt multivalued mapping, if

Definition 7. Let be a complete double controlled dislocated quasi-metric type space and be a pair of -Alt multivalued mapping. Then, is called Kannan-type double controlled contraction, if for every two consecutive points belonging to the range of an iterative sequence with and , and we havewhere . Also, the terms of the sequence satisfy the following:

Theorem 2. Let be a left -sequentially complete double controlled dislocated quasi-metric type space. be the pair of Kannan-type double controlled contraction. Assume that(i)The set contains and is closed.(ii)For every , we have

Then, . Also, if (9) holds for each , then and have a common fixed point in and .

Proof. As be an arbitrary element of , from condition (i) . Let be the iterative sequence in generated by a point . Since , and . As is -Alt multivalued mapping, so . Now, , , and imply that . By induction, we deduce that and , for all . Now, by Lemma 1, we haveAs , , and , then by using condition (9) in inequality (13), we haveAs , , and , then by using condition (9) in inequality (12), we obtainAs , , and , then by using condition (9), we obtainUsing (16) in (15), we haveAs , , and , then by using condition (9), we obtainFrom (17) and (18), we haveUsing (19) in (15), we haveContinuing in this way, we obtainSimilarly, by using (14)–(16) and continuing in this way, we obtainCombining inequalities (21) and (22), we haveNow, to prove that is a left Cauchy sequence, for all natural numbers , we haveHence, we havewhereBy using ratio test and inequality (10), it can be proved easily that the infinite series is convergent. Letting in (25), yieldsSo, the sequence is a left Cauchy sequence. Since is a left -sequentially double controlled complete dislocated quasi-metric type space, there exists such that andSince is a left -sequentially complete and is closed subset of , so is also left -sequentially complete. As for all . So, is a subsequence of contained in . By completeness of and uniqueness of limit, , that is,Now, we suppose that . By Lemma 1, we haveAs and , by using (9), we haveTaking on both sides of inequality (31), we obtainNow,Taking of inequality (33) and using inequality (11) and (28), we obtainUsing inequalities (11) and (32) in inequality (34), we obtainIt is a contradiction; therefore,Now, suppose that . By Lemma 1, we haveNow, and , so by (9), we haveTaking on both sides of inequality (38), we obtainNow,Taking of inequality (40) and using inequalities (11) and (28), we obtainBy using inequality (39) in inequality (41), we obtainIt is a contradiction. Hence, , so . Now,This implies as . As and . So, Definition 6 impliesNow, by Lemma 1, we haveAs, and , so by (9), we haveTaking on both sides of inequality (46), we obtainLetting and using inequality (27), we haveSince,by taking and using inequality (11) and (46), we obtainBy using inequalities (11) and (48), we obtainIt is a contradiction. Thus, . Similar to the arguments above, we obtainHence, . Hence, is a common fixed point for and .