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};
