Fixed point theory for simulation functions in G-metric spaces: A novel approach

Abstract

In this paper, with the aid of simulation mapping 𝜂: [0,∞) × [0,∞)->ℝ, we prove some Lemmas and fixed point result for generalized 𝒵 − contraction of the mapping 𝑔: 𝑋->𝑋 satisfying the following conditions:
𝜂(𝒢(𝑔𝑥, 𝑔𝑦, 𝑔𝑧),ℳ(𝑥, 𝑦, 𝑧)) ≥ 0,
for all 𝑥, 𝑦, 𝑧 ∈ 𝑋, where
ℳ(𝑥, 𝑦, 𝑧) = max {𝒢(𝑥, 𝑔𝑦, 𝑔𝑦), 𝒢(𝑦, 𝑔𝑥, 𝑔𝑥), 𝒢(𝑦, 𝑔𝑧, 𝑔𝑧), 𝒢(𝑧, 𝑔𝑦, 𝑔𝑦), 𝒢(𝑧, 𝑔𝑥, 𝑔𝑥), 𝒢(𝑥, 𝑔𝑧, 𝑔𝑧)}. and (𝑋, 𝒢) is a 𝒢 − metric space. An example is also given to support our results.