Unique common fixed point of a family of self-maps with same type contractive condition in 2-metric space

Abstract

In this paper, we prove that a family of self-maps {T i, j } i, j∈N in 2-metric space has a unique common fixed point if (i) {T i, j } i, j∈N satisfies the same type contractive condition for each j ∈ N; (ii) T m,μ · T n,ν = T n,ν · T m,μ for all m, n, μ, ν ∈ N with μ = ν. Our main result generalizes and improves many known unique common fixed point theorems in 2-metric spaces.There have appeared many unique common fixed point theorems for self-maps { f i } i∈N with some contractive condition on 2-metric spaces. But most of them held under subsidiary conditions [1−4] , for example: commutativity of { f i } i∈N or uniform boudedness of { f i } i∈N at some point, and so on. but in [5], the author obtained similar result under removing the above subsidiary conditions. The result generalized and improved many same type unique common fixed point theorems.In this paper, we will prove that a family of self-maps {T i, j } i, j∈N satisfying same type contractive condition on 2-metric spaces have a unique common fixed point if {T i, j } i, j∈N satisfyies the condition (ii). Now, we give some concepts and lemmas [4−5] . Definition 1. A 2-metric space (X , d) consists of a nonempty set X and a function d :X × X × X → [0, +∞) such that (i) for distinct elements x, y ∈ X , there exists u ∈ X such that d(x, y, u) = 0;(ii) d(x, y, z) = 0 if and only if at least two elements in {x, y, z} are equal;(iii) d(x, y, z) = d (u, v, w), where {u, v, w} is any permutation of {x, y, z};

New Unique Common Fixed Point Results for Four Mappings with Ф-Contractive Type in 2-Metric Spaces

“…But most of them held under subsidiary conditions [1][2][3], for examples: commutativity of mappings or uniform boundness of mappings at some point, and so on. In [4][5][6][7][8], the author obtained similar results for infinite mappings with generalized contractive or quasi-contractive conditions under removing the above subsidiary conditions. These results generalized and improved many same type unique common fixed point theorems.…”

Uniqueness of Common Fixed Points for a Family of Mappings with <i>Φ</i>-Contractive Condition in 2-Metric Spaces

“…The result generalized and improved many same type unique common fixed point theorems. Recently, the author discussed unique common fixed point theorems for a family of contractive or quasi-contractive type mappings on 2-metric spaces, see [6][7][8], these results improve the above known common fixed point theorems.…”

Suzuki-Type Fixed Point Results in b2-Metric Spaces

“…Many generalized spaces of Metric space have been established. Among them, b-metric [7] and 2-metric [8] have been extensively researched. Both of these metrics of those spaces are not continuous functions of its variables.…”