New Types of Contraction for Multivalued Mappings and Related Fixed Point Results in Abstract Spaces
In this article, we establish fixed point results for a pair of multivalued mappings satisfying generalized contraction on a sequence in dislocated -quasi metric spaces and Khan type contraction on a sequence in -quasi metric spaces. An example and an application have been discussed. Our results modify and generalize many existing results in literature.
1. Introduction and Preliminaries
A point is said to be a fixed point of a multivalued/self-mapping , if Fixed point theory has a large number of applications, for example, [1–4]. Czewick  initiated the study of fixed point in b-metric spaces. Many authors used the concept of b-metric spaces to prove the existence and the uniqueness of a fixed point for several contraction mappings [6–9]. Furthermore, dislocated quasi-metric spaces [10–13] generalized abstract spaces such as dislocated metric spaces  and quasi-metric spaces [15–17]. Recently, Klin-eam and Suanoom  introduced the concept of dislocated -quasi metric spaces. Fixed point results in complete dislocated -quasi metric spaces can be seen in [19, 20].
Wardowski  generalized many fixed point results in a beautiful way by introducing contraction (see also [6, 22–30]). Nadler  extended Banach’s contraction mapping principle to a fundamental fixed point theorem for multivalued mappings. Since then, an interesting and rich fixed point theory for such mappings was developed in many directions; see [32–36]. The results of single valued mappings can be generalized by using multivalued mappings. Results for multivalued mappings have applications in engineering, Nash equilibria, and game theory [37–40]. Rasham et al.  obtained fixed point results for a pair of multivalued contractive mappings, which are extensions of some multivalued fixed point results.
This paper introduces new types of contractions on a sequence and generalizes many recent results. An example has been given to show how our results are valid when the others fail. An application has been given to obtain a solution of a system of integral equations.
Definition 1 (see ). Let be a nonempty set and a real number. A mapping is called a dislocated quasi -metric (or simply -metric), if the following conditions hold for any (a)If , then ;(b) The pair is called a dislocated quasi -metric space (in short dislocated -quasi-metric space).
The following remarks can be observed
(a) If , then a dislocated -quasi-metric space becomes a dislocated quasi-metric space ;
(b) if and implies , then becomes a quasi-metric space ;
(c) if and implies , then becomes a -metric space .
Example 2 (see , let and ). Define by for Then is a -metric space with . But it is not a quasi -metric space. Also it is not a dislocated -metric space. It is obvious that is neither -metric space nor dislocated quasi-metric space.
Definition 3 (see ). Let be a dislocated -quasi-metric space. Let be a sequence in , and then
(a) is called Cauchy if , such that (respectively ,
(b) dislocated quasi -converges (for short -converges) to , if or for any , there exists , such that for all , and . In this case is called a -limit of
(c) is called complete if every Cauchy sequence in converges to a point .
Definition 4 (see ). Let be a dislocated -quasi metric space. Let be a nonempty subset of 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.
It is clear that if , then But if , then or may not equal zero. We denote by the set of all proximinal subsets of
Definition 5 (see ). The function , defined by is called dislocated quasi Hausdorff metric on Also is known as dislocated quasi Hausdorff -metric space, where is the proximinal subset of
(F1) is strictly increasing, that is, for all such that implies ;
(F2) for each sequence of positive numbers, if and only if ;
(F3) there exists such that .
(F4) For each sequence of positive real numbers and such that for each , and some , we have , for each .
Lemma 6. Let be a dislocated -quasi-metric space. Let be the dislocated quasi Hausdorff b-metric space on Then, for all and for each , there exists , such that and , where and
Lemma 7 (see ). Let be a b-metric space and let be any sequence in for which there exist and such that , Then is a Cauchy sequence in .
Lemma 8. Let be a dislocated -quasi metric space, and let be any sequence in for which there exist and such thatfor each Then is a Cauchy sequence in
2. Main Result
Let be a dislocated -quasi metric space, and be multifunctions on . Let be an element such that , . Let be such that , 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 us introduce the following definition:
Definition 9. Let be a dislocated -quasi-metric space and be two multivalued mappings. The pair is called a contraction, if there exists and such that for every two consecutive points belonging to the range of an iterative sequence with , we havewhereAnd we now prove the following main result.
Theorem 10. Let be a complete dislocated -quasi-metric with and be a contraction. Then Also, if (5) holds for each , then and have a common fixed point in and
Proof. Let be the iterative sequence in generated by a point . If for some , then Clearly, if , then Also implies So, and Now, implies and implies So, and is a common fixed point of and . So the proof is completed in this case. Now, let for all By Lemma 6, we haveandFrom (11), (F1) and using condition (5), we get From (6), we have If , then which is a contradiction due to (F1) and Therefore,From (11), (F1) and using condition (5), we get By using (15) and (F1), we get Combining (16) and (19), we getBy using (10) and (5), we have From (6), we have If , then we obtain which is a contradiction due to (F1). Therefore,By using (10) and (5), we have From (24), , soCombining (24) and (26), we getCombining (20) and (27), we getBy Lemma 8, is a Cauchy sequence in Since is a complete dislocated -quasi-metric space, so there exists such that ; that is,Now, suppose , and then , so By using Lemma 6 and (5), we have Since is strictly increasing, we have Taking on both sides, we getFrom (6) Taking limit as , and by using (29), we getUsing inequality (35) in (33), we getNow, Taking limit as ,Using inequalities (29) and (36) in (38), we get This is a contradiction, so Now, suppose , and then there exists such that for all By Lemma 6 , so for all Following similar arguments as above, we getNow, Taking limit as , and using inequalities (29) and (41), we get which is a contradiction, so Hence Similarly by using (29), Lemma 6, and the inequality we can show that Similarly, Hence, the pair has a common fixed point in Now, This implies that Hence the proof is completed.
Now, let us introduce the following example.
Example 11. Let and if , and , if Then is a dislocated -quasi-metric space with . Define the mappings as follows:
Case 1. If holds. Define the function by for all and As , and by taking , we define the sequence in generated by Also, Now, if , we have Also Case (i). If , and then we have This implies that Case (ii). Similarly, if , and then we have Hence,
Case 2. If holds. where Case (i). If , and then we have so Case (ii). Similarly, if , and then we have