Fixed point results for Geraghty quasi-contraction type mappings in dislocated quasi-metric spaces

Abstract

In this paper, fixed point results for a newly introduced Geraghty quasi-contraction type mappings are proved in more general metric spaces called T-orbitally complete dislocated quasi-metric spaces. Geraghty quasi-contraction type mappings generalize, among others, Ciric’s quasi-contraction mappings and other Geraghty quasi-contractive type mappings in the literature. Fixed point results are obtained without imposing a continuity condition on the mapping, thereby further generalizing some other related work in the literature. An example is given to show the validity of results obtained.

Best Proximity Points for Generalized Geraghty Quasi-Contraction Type Mappings in Metric Spaces

“…Numerous results on best proximity point theory were studied by several authors ( [1], [3], [4], [5]) imposing sufficient conditions that would assure the existence and uniqueness of such points. These results are generalizations of the contraction principle and other contractive mappings ( [2], [6], [8], [16], [21], [22], [24]) in the case of self-mappings, which reduces to a fixed point if the mapping under consideration is a self-mapping. The notion of best proximity point was introduced in [14], the class of proximal quasi contraction mappings was introduced in [11] and thereafter, several known results were derived ( [10], [12], [13]).…”

“…Best proximity pair theorems analyse the conditions under which the optimization problem, namely min x∈A d(x, T x) has a solution and is known to have applications in game theory. For additional information on best proximity point, see [7], [9], [10], [11], [12], [13], [14], [15], [17], [18], [20], [23]. Definition 1.1 [4].…”

“…Recently, using these class of functions, Umudu et al [22] introduced a new class of quasicontraction type mappings called generalized α-φ-Geraghty quasi-contraction type mappings and proved the existence of its unique fixed point as follows. Definition 1.3 [22]. Let (X, d) be a metric space and α :…”

Geraghty-Ciric type mappings with an application

“…Firstly, we restate the class of mappings introduced in Umudu et al [23] and fixed point results as follows: Definition 2.1. Let (X, d) be a metric space and γ : X ×X → R + .…”

“…Since then, many authors have generalized and extended this result in diverse ways see ( [5], [13], [12], [17] and [23]). Meanwhile, Ciric [7] defined the following concepts and proved the following fixed point result.…”