Lacunary Cesaro-Kirk iteration for fixed point problems

Abstract

We propose a lacunary Cesaro–Kirk iterative process for fixed point approximation of asymptotically nonexpansive mappings in Banach spaces. The scheme is built by replacing classical Cesaro averages with lacunary block means and embedding them into a multi–step Kirk-type iteration via convex combinations of shifted iterates. We establish boundedness of the generated sequence under a summability condition on the lacunary asymptotic deviations induced by the asymptotic parameters of the mapping, and show that the distance to any fixed point admits a limit. Assuming an appropriate asymptotic regularity, we obtain strong convergence in the compact constraint case, and weak convergence in uniformly convex Banach spaces satisfying Opial's condition under a demiclosedness hypothesis. We also investigate the lacunary statistical convergence behavior of the proposed iteration and establish a connection between strong convergence and lacunary statistical convergence to fixed points. Our results yield lacunary analogues of several Cesàro and ergodic fixed point approximation schemes.

1. Introduction

Fixed point theory is one of the most active and influential areas in nonlinear analysis. One of the most fundamental results in this field is the Banach contraction principle [2]. This theorem serves as a cornerstone for the development of fixed point theory and has inspired extensive research over the past century. Numerous extensions and applications have been established in various branches of mathematics and applied sciences. In parallel with the development of fixed point theory, several classes of mappings and iterative approximation techniques have been introduced and investigated.

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Nuray, F. (2026). Lacunary Cesaro-Kirk iteration for fixed point problems. Asia Mathematika, 10(1), 49–63. https://doi.org/10.5281/zenodo.20996569

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