Abstract
In this paper, we discuss a class of mappings more general than ρ-nonexpansive mapping defined on a modular space endowed with a graph. In our investigation, we prove the existence of fixed point results of these mappings. Then, we also introduce an iterative scheme for which proves the convergence to a fixed point of such mapping in a modular space with a graph.
Highlights
The abstract definition of modular spaces was introduced by Nakano [1] in 1950
Using Mann’s [20] iteration, we prove the existence of fixed points of this large class in modular spaces endowed with an oriented graph
Let C be a nonempty subset of a modular space Xρ and Δ = fðx, xÞ: x ∈ Cg be the loop set
Summary
The abstract definition of modular spaces was introduced by Nakano [1] in 1950. Further, Musielak and Orlicz [2] redefined the modular spaces in such a way that a modular function space is seen as a vector space endowed with a modular function. Fixed point theorems for monotone mappings in metric spaces endowed with partial ordering are considered firstly by Ran and Reurings [3] in 2004. Afterwards, Alfuraidan examined in [16] the existence of fixed points for multivalued monotone G-contraction and G-nonexpansive mappings in modular function spaces. Alfuraidan [16] justified the existence and convergence for multivalued mapping in modular function spaces endowed with a graph. He gave some results on the approximation of fixed points of Ćirić quasicontraction mappings in metric modular spaces equipped with a graph (see [15]). Afterwards, we establish some convergence results for a new iterative process (see (55)) which can be considered an accelerated version of the AK iteration scheme
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