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.