Fixed point theorems for modified generalized F-contraction in G-metric spaces

Abstract

In this paper, we introduce new notions of generalized F-contractions of type(S) and type(M) in G-metric spaces. Some common fixed point theorems are proved using these notions. A suitable example is also provided to support our results.

Advancements in Fixed Point Results of Generalized Metric Spaces: A Survey

“…Kumar and Arora [69] introduced new notions of generalized F-contractions of type (S) and type (M) in G-metric spaces and established related fixed point theorems.…”

“…Consequently [69], denote by S G the family of all functions F : (0, ∞) −→ R which satisfy the following conditions: x m F(x) = 0. Definition 112.…”

“…Definition 113. [69] Let (X, G) be a G-metric space. A mapping T : X −→ X is said to be a modified generalized F-contraction of type (M) if F ∈ M G and there exists τ > 0 such that for all x, y, z ∈ X, G(T x, Ty, T z) > 0 ⇒ τ + F(G(T x, Ty, T z)) ≤ F(S(x, y, z)), where S(x, y, z) = max G(x, Ty, Ty), G(y, T x, T x), G(y, T z, T z), G(z, Ty, Ty), G(z, T x, T x), G(x, T z, T z) .…”

“…Theorem 121. [69] Let (X, G) be a complete G-metric space and T : X −→ X be a modified generalized F-contraction of type (S). Then T has a unique fixed point u ∈ X and the sequence {T n x 0 } n∈N converges to u. Theorem 122.…”

“…Then T has a unique fixed point u ∈ X and the sequence {T n x 0 } n∈N converges to u. Theorem 122. [69] Let (X, G) be a complete G-metric space and T : X −→ X be a modified generalized F-contraction of type (M). Then T has a unique fixed point u ∈ X and the sequence {T n x 0 } n∈N converges to u.…”