Convergence of Implicit Iterative Processes for Semigroups of Nonlinear Operators Acting in Regular Modular Spaces

Wojciech M. Kozlowski ()
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Wojciech M. Kozlowski: School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia

Mathematics, 2024, vol. 12, issue 24, 1-14

Abstract: This paper focuses on one-parameter semigroups of ρ -nonexpansive mappings T t : C → C , where C is a subset of a modular space X ρ , the parameter t ranges over [ 0 , + ∞ ) , and ρ is a convex modular with the Fatou property. The common fixed points of such semigroups can be interpreted as stationary points of a dynamic system defined by the semigroup, meaning they remain unchanged during the transformation T t at any given time t . We demonstrate that, under specific conditions, the sequence { x k } generated by the implicit iterative process x k + 1 = c k T t k + 1 ( x k + 1 ) + ( 1 − c k ) x k is ρ -convergent to a common fixed point of the semigroup. Our findings extend existing convergence results for semigroups of operators, from Banach spaces to a broader class of regular modular spaces.

Keywords: common fixed point; modular space; semigroup of nonlinear operator; implicit iterative process; convergence of fixed point approximation process (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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