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arXiv · 2606.25799

Weak and Strong Convergence of Implicit Iterations for Lipschitzian Hemi-Contractive and {\alpha}-Hemi-Contractive Semigroups in Banach Spaces

Ferit Gurbuz
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Abstract

We investigate an implicit iterative scheme for approximating common fixed points of one-parameter semigroups generated by Lipschitzian hemi-contractive and {\alpha}-hemi-contractive mappings on closed convex subsets of Banach spaces. Under suitable conditions on the control sequences, we establish weak convergence of the proposed iteration in uniformly convex Banach spaces satisfying Opial's condition. Furthermore, strong convergence results are obtained in general Banach spaces without imposing uniform convexity assumptions. The analysis is based on a careful use of recursive inequality techniques and weak convergence principles, allowing us to extend several known results in the literature. In particular, the proposed framework provides a unified approach that encompasses important classes of nonlinear operators, including pseudocontractive and demicontractive mappings, as special cases of hemi-contractive semigroups. Finally, we apply our results to nonlinear evolution, variational inequality, and nonlinear integral equations to demonstrate their broad applicability.

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Ferit Gurbuz. 2026-06-24. Weak and Strong Convergence of Implicit Iterations for Lipschitzian Hemi-Contractive and {\alpha}-Hemi-Contractive Semigroups in Banach Spaces. https://arxiv.org/abs/2606.25799

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