Fixed points of mappings satisfying contractive condition of integral type in modular spaces endowed with a graph

Abstract

Jachymski (Proc. Am. Math. Soc. 136:1359Soc. 136: -1373Soc. 136: , 2008) gave a modified version of a Banach fixed point theorem on a metric space endowed with a graph. The aim of this paper is to present fixed point results of mappings satisfying integral type contractive conditions in the framework of modular spaces endowed with a graph. Some examples are presented to support the results proved herein. Our results generalize and extend various comparable results in the existing literature. MSC: 47H10; 54H25; 54E50

Some fixed point theorems on equivalent metric spaces

“…Hence, the fixed point theorems can be studied as homotopies, for details see [25,32]. In a similar way, a contractive condition of integral type can be rewrite with respect to the itself equivalent metric space, hence we can obtain new theorems related to results from [11,14,16,23,26,30,33].…”

“…[11]. In the last decade, the literature records many papers in which this type of contractive condition are studied, see for example [14,16,30,23,33] and reference therein.…”

Fixed Point Theorems for a New Class of Mappings in Modular Spaces Endowed with a Graph

“…In the last 10 years, many authors have considered the approach of Jachymski in the setting of modular spaces. Öztürk et al [14] presented some fixed point results for mappings satisfying the integral contraction condition in a vector modular space equipped with a graph. Also, Alfuraidan [15] proved a generalization of Banach's theorem in modular metric spaces endowed with a graph.…”

Fixed points of Ψ-contractive mappings in modular spaces

“…Oztürk et al ( [26] ) defined the notions of C ρ -graph and orbitally G ρ -continuity as follows: Definition 2.8. Let {T n x} be a sequence, there exists C > 0 such that ρ (C (T n x − x * )) → 0 for x * ∈ X ρ and T n x, T n+1 x ∈ E (G) for all n ∈ N. Then a graph G is called a C ρ -graph if there exists a subsequence {T n p x} of {T n x} such that (T n p x, x * ) ∈ E (G) for p ∈ N. Definition 2.9.…”