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.
Highlights
Introduction and preliminariesGeraghty [1] generalized the Banach [2] contraction mapping in metric spaces by using an auxiliary function instead of a constant.Let F be the family of all functions β : [0, ∞) → [0, 1) which satisfy the condition lim n→∞ β = implies lim n→∞ tnUsing such a function, Geraghty [1] proved the following theorem.Theorem 1.1 ([1]) Let (X, d) be a complete metric space and let T be a self-mapping on X
As an improvement of α-admissible maps introduced by Samet et al [24] and Karapínar et al [26], Popescu [5] introduced the following concepts, which were used to prove the existence and uniqueness of fixed point results in a complete metric space
The purpose of this paper is to prove some fixed point results in dislocated quasi-metric space using new concepts of Geraghty quasi-contraction type self-mappings that the authors just introduced and proved fixed point results in the context of metric spaces [27]
Summary
Asadi [8] proved some fixed point results satisfying certain contraction principles on a convex metric space. Some other papers have been published containing fixed point results for selfmappings with different contraction conditions in metric spaces and their generalizations including dislocated metric spaces and dislocated quasi-metric spaces (see [17,18,19,20,21,22,23,24,25,26]). As an improvement of α-admissible maps introduced by Samet et al [24] and Karapínar et al [26], Popescu [5] introduced the following concepts, which were used to prove the existence and uniqueness of fixed point results in a complete metric space.
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