Abstract
PurposeIn this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.Design/methodology/approachThe concepts of relative ρ-stability and weak ρ-stability are introduced, and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable are established for the newly introduced strong ρ-quasi-contractive-like class of maps.FindingsNoor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries in this study.Originality/valueThe results obtained in this work are complementary to those proved in normed and metric spaces in the literature.
Highlights
Introduction and preliminary definitionsModular function spaces are well-known generalizations of both function and sequence variants of many important spaces such as Calderon–Lozanovskii, Kothe, Lebesgue, Lorentz, Musielak–Orlicz, Orlicz and Orlicz–Lorentz spaces
It is worthy to mention that modulartype conditions are far more natural as their assumptions can be verified than their corresponding metrics or norms, especially when related to fixed-point results and applications to integral-type operators
Let D be a ρ − closed, ρ − bounded and convex subset of a ρ − complete modular space Lρ, and T : D → PρðDÞ be a multivalued mapping such that PρT is a ρ − quasicontractive-like mapping, satisfying contractive-like condition (1.8)
Summary
Introduction and preliminary definitionsModular function spaces are well-known generalizations of both function and sequence variants of many important spaces such as Calderon–Lozanovskii, Kothe, Lebesgue, Lorentz, Musielak–Orlicz, Orlicz and Orlicz–Lorentz spaces.
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