Abstract
Inspired by Suzuki’s generalization for nonexpansive mappings, we define the ( C ) -property on modular spaces, and provide conditions concerning the fixed points of newly introduced class of mappings in this new framework. In addition, Kirk’s Lemma is extended to modular spaces. The main outcomes extend the classical results on Banach spaces. The major contribution consists of providing inspired arguments to compensate the absence of subadditivity in the case of modulars. The results herein are supported by illustrative examples.
Highlights
The first idea regarding the concept of modular space was initiated by Orlicz in [1] through a remarkable example
The notion was reactivated in the expanded framework of vector spaces and one important direction was settled by Khamsi [6] in connection with the fixed point theory
We extend the above result to modular spaces
Summary
The first idea regarding the concept of modular space was initiated by Orlicz in [1] through a remarkable example. The notion was reactivated in the expanded framework of vector spaces and one important direction was settled by Khamsi [6] in connection with the fixed point theory Nowadays, this approach is fructified in several works: Okeke et al [7], Khan [8], Abbas et al [9], Abdou and Khamsi [10], Alfuraidan et al [11] and the papers referenced there. Many of the fixed point theory outcomes on Banach spaces can be extended, as the above-mentioned references prove, to modular structures This paper continues this approach by extending Suzuki’s concept of generalized nonexpasive mapping to modular vector spaces and by analyzing the existence of fixed points. The Goebel and Kirk Lemma [15] are provided, bypassing, again, the subadditivity
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