$ \alpha $-Admissible mapping in $ C^{*} $-algebra-valued b-metric spaces and fixed point theorems

On the α−ψ−Contractive Mappings in C∗-Algebra Valued b-Rectangular Metric Spaces and Fixed Point Theorems

“…In this paper, inspired by the work done in [9], we introduce the notion of α − ψ−contraction and establish some new fixed point theorems for mappings in the setting of complete C * -algebra valued rectangular b-metric spaces.…”

“…Definition 2.7. [9] Let Ψ be the set of positive functions ψ : A + → A + satisfying the following conditions :…”

“…Definition 2.10. [9] Let X be a nonempty set and α : X × X → A + be a function, wesay that the (i) T is α− admissible (ii) There exists x 0 ∈ X such that α(x 0 , T x 0 ) I (iii) for all x, y ∈ X ,there exists z ∈ X such that α(x, z) I and α(y , z) I…”

“…, T y ) I,where I the unit of A. Definition 2.11 [9]. Let (X, A, d) be a C * -algebra valued b− metric space and T : X → X is mapping, we say that T is an α − ψ− contractive mapping if there exist two functions α : X × X → A + and ψ ∈ Ψ such that α(x, y )d(T x, T y ) ψ(d(x, y )), for all x, y ∈ X3.…”

Common fixed point theorem for pair of quasi triangular α-orbital admissible mappings in complete metric space with application

“…Samet et al [25] found the conception of α − ψ contraction type mapping and take advantage of their new concept to established and found several fixed point theorems. Several inventors used the concept of α-admissible mapping to established new results in many spaces [10,21,22,26,30]. In 2014, Popescu [20] found the two new concept α-orbital admissible and triangular α-orbital admissible and gave the result each α-admissible mapping is an α-orbital admissible mapping and each triangular α-admissible mapping is an triangular α-orbital admissible mapping.…”