α-ψ-Geraghty contractions on generalized metric spaces

Abstract

In this work, we introduce the class of α-ψ-Geraghty contraction as well as generalized α-ψ-Geraghty contraction mappings in the context of generalized metric spaces where ψ is an auxiliary function which does not require the subadditive property and set up some fixed point results for both classes individually. Our results will extend, improve and generalize several existing results in the literature. MSC: 46T99; 47H10; 54H25

Fixed Point Theorems for Set-ValuedL-Contractions in Branciari Distance Spaces

“…In many papers, for example, [2][3][4][5][6][7][8], it is called generalized metric space, Branciari metric space, or rectangular metric space. However, these names do not reflect and indicate the meaning well of the notion of Branciari distance spaces because Branciari distance can not reduce to the standard metric.…”

“…After that, a lot of authors, for example, [2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references therein, obtained fixed point results in such spaces. Jain et al [16] obtained fixed point results in extended Branciari b-distance spaces [17] by defining the notion of certain contractive conditions, and they gave an application to nonlinear fractional differential equations.…”

“…Sarma et al [15] and Samet [14] (see also [3,4,8,18]) show that Branciari distances have the following topological disadvantages.…”

Various contractions in generalized metric space

“…In 2014, Asadi et al [39] exercise on the concept of Geraghty [35] and Samet et al [10] and they developed a new class of contractive mapping namely α − ψ Geraghty contraction. Further they investigated the existence and uniqueness of fixed point theorem in gms.…”

“…T is triangular α-admissible;Various Contractions in Generalized Metric Space 123 3. there existsx 0 ∈ X such that α(x 0 , T x 0 ) ≥ 1 and α(x 0 , T 2 x 0 ) ≥ 1; 4. T is continuous.Then T has a fixed point x * in X and {T n x 0 } converges to x * .Asadi et al[39] also introduced the notion of generalized α − ψ Geraghty contraction, which is stronger version of α − ψ Geraghty contraction: Definition 1.45. Let (X, d) be a generalized metric space, and let α : X × X → R be a function.A map T : X → X is called generalized α − ψ-Geraghty contraction mapping if there exists Geraghty function β such that for all x, y ∈ X,α(x, y)ψ(d(T x, T y)) ≤ β(ψ(M (x, y)))ψ(M (x, y)), where M (x, y) = max{d(x, y), d(x, T x), d(y, T y)} and ψ ∈ Ψ.Theorem 1.46.…”