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