Coincidence and fixed points of multivalued F -contractions in generalized metric space with application
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Journal of Fixed Point Theory and Applications
Coincidence and fixed points of multivalued F -contractions in generalized metric space with application Naeem Saleem , Iram Iqbal, Bilal Iqbal and Stojan Radenov´ıc Abstract. The aim of the paper was to prove some new fixed point theorems, coincidence point theorems and common fixed point theorems for multivalued F -contractions involving a binary relation that is not necessarily a partial order, in the context of generalized metric spaces (in the sense of Jleli and Samet). We also prove existence of common solutions to integral inclusions. Mathematics Subject Classification. 46N40, 47H10, 54H25, 46T99. Keywords. Fixed point, common fixed point, coincidence point, periodic point, F -contraction, JS-generalized metric space.
1. Introduction The Banach’s fixed point theorem (in short BFPT) [5] provides a strong foundation on which metric fixed point theory has been developed. In the past few decades, many authors have extended and generalized the BFPT in several ways. Recently, Wardowski in [39] opened a new window for researchers by introducing a concept of F -contraction and proved a fixed point theorem. Further, Altun et al. [3] broadened this idea for multivalued F -contraction. To more in this direction, consult [1,4,12,15–18,20,23,31,38,40]. Nadler proved the multivalued version of BFPT in [28]. Nadler’s fixed point theorem has attained the attention of several researchers. For a study to this direction and more references, see [6–9,14,22,24,33–35]. The fixed point theorem for multivalued mapping without using generalized Hausdorff distance is proved by Feng and Liu in [11]. On the other hand, notion of standard metric space is generalized in several ways (see [10,13,25–27,29,30,37]). A new generalization of metric spaces was given by Jleli and Samet in [19]; it recapitulates a huge class of topological spaces including b-metric spaces, standard metric spaces, dislocated 0123456789().: V,-vol
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metric spaces and modular spaces. They extended fixed point results including BFPT, Ciri´c’s fixed point theorem and a fixed point result due to Ran and Reurings. Further, in [21], Karapinar et al. obtained fixed point theorems under very general contractive conditions in generalized metric spaces (in the sense of Jleli and Samet). Recently, in [2], Altun et al. obtained a Feng-Liu’s type fixed point theorem in the setting of generalized metric spaces (in the sense of Jleli and Samet). In this paper, we prove fixed point theorems for multivalued F -contractions in the context of generalized metric space in the manner of Jleli and Samet.
2. Preliminaries In the sequel, N = {0, 1, 2, 3, ...} indicates the set of all non-negative integers, R, indicates the set of all real numbers. Let ψ be a self mapping on nonempty set U . A binary relation on U is a nonempty subset R of the Cartesian product U × U . For simplicity, we denote uRv if (u, v) ∈ R. The notions of reflexivity, transitivity, antisymmetry, preorder and partial order can be found in [36]. The t
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