Weak and Strong Convergence of an Implicit Iteration for Hemi-Contractive Semigroups in Banach Spaces

Abstract

We study an implicit iteration for approximating common fixed points of strongly continuous semigroups on closed convex subsets of real Banach spaces. Under a stepwise Lipschitz restriction and hemi-contractivity relative to the common fixed point set, we establish well-definedness and Fej% ér monotonicity of the iteration. In reflexive Banach spaces satisfying Opial's condition, we obtain a sufficient condition for weak convergence based on a normalized residual estimate, uniform continuity at zero along the generated sequence, and demiclosedness at fixed positive times. In general real Banach spaces, we prove strong convergence under a time-dependent directional contraction condition and a divergence condition coupling the iteration weights with the contraction strength. A linear example shows that vanishing time steps and a divergent sum of iteration weights alone do not ensure convergence to a common fixed point. A nonlinear planar evolution semigroup illustrates the applicability of the strong convergence theorem beyond the nonexpansive class.

arXiv commentSubstantially revised version correcting errors in the previous convergence statements and proofs. Assumptions and convergence conditions reformulated, invalid examples removed, and a verified nonlinear evolution example added