Some Fixed Point Results for Commuting Families of Mappings in Modular Spaces

Assume that ρ is a convex modular satisfying the -type condition and the modular space is either α-uniformly ρ-noncompact convex or it satisfies the strong Opial condition. We prove that the fixed point set of a commuting family of asymptotically non-expansive mappings defined from a ρ-convex weakly...

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Bibliographic Details
Published in:Numerical functional analysis and optimization 2021-10, Vol.42 (13), p.1608-1625
Main Authors: Domínguez Benavides, T., Lorenzo Ramírez, P.
Format: Article
Language:English
Subjects:
Asymptotic series
Asymptotically nonexpansive mapping
Commuting
fixed point
Function space
measure of noncompactness
modular vector space
nonexpansive mapping
Primary 47H10
Secondary 46E30
Topology
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Summary:Assume that ρ is a convex modular satisfying the -type condition and the modular space is either α-uniformly ρ-noncompact convex or it satisfies the strong Opial condition. We prove that the fixed point set of a commuting family of asymptotically non-expansive mappings defined from a ρ-convex weakly compact set C into C is a non-empty nonexpansive retract of C. We show that our results about existence of fixed point for commuting families of nonexpansive mappings also work if we replace the weak topology by some other topologies weaker than the ρ-topology. In particular, we can obtain fixed point results when C is ρ-a.e. compact in a modular function space.
ISSN:0163-0563
1532-2467
DOI:10.1080/01630563.2021.2001825