Abstract
Recently, Aydi et al. [On fixed point results for α-implicit contractions in quasi-metric spaces and consequences, Nonlinear Anal. Model. Control, 21(1):40–56, 2016] proved some fixed point results involving α-implicit contractive conditions in quasi-b-metric spaces. In this paper we extend and improve these results and derive some new fixed point theorems for implicit contractions in ordered quasi-b-metric spaces. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.
Highlights
Introduction and preliminariesIt is always recognized that the contraction mapping principle proved in the Ph.D. dissertation of Banach in 1920, see [6], is one of the most significant theorems in functional analysis and its applications in other branches of mathematics
Aydi et al [On fixed point results for α-implicit contractions in quasi-metric spaces and consequences, Nonlinear Anal
In this paper we extend and improve these results and derive some new fixed point theorems for implicit contractions in ordered quasi-b-metric spaces
Summary
It is always recognized that the contraction mapping principle proved in the Ph.D. dissertation of Banach in 1920, see [6], is one of the most significant theorems in functional analysis and its applications in other branches of mathematics. Various authors established fixed and common fixed point results for different classes of mappings defined in some generalized metric spaces [34, 38]. We recall that Samet et al [35] introduced the notion of α-ψ-contractive mapping for establishing some fixed point results in the setting of complete metric spaces; this paper is at the basis of an intensive research in fixed point theory in the last years, see, for example, [8, 21, 24, 32]. In the setting of quasib-metric space, we give some fixed point results for a class of self-mappings that satisfy an α-implicit contractive condition. (r) If is a partial order on X, X is regular if for each sequence {xn} ⊂ X such that xn → x ∈ X, and xn−1 and xn are comparable for all n ∈ N, there exists a subsequence {xn(k)} of {xn} such that xn(k) and x are comparable for all k ∈ N
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