Fixed point theorems for Reich, Hardy–Rogers, and Ćirić contractions in cone modular spaces via λ-iteration and applications to nonlinear integral equations

This paper presents a comprehensive study of fixed point theorems for Reich, Hardy–Rogers, and Ćirić-type contractions in the framework of cone modular spaces. We introduce a novel iteration scheme called [Formula: see text]-iteration, defined by [Formula: see text], with [Formula: see text], both Picard and Mann iterations are generalized. Complete convergence proofs are provided for each contractive condition, establishing the existence and uniqueness of fixed points. These results significantly extend existing work in cone metric spaces and modular metric spaces by considering more general contractive conditions in the unified framework of cone modular spaces. Explicit examples with full numerical computations demonstrate the convergence behavior. The paper concludes with discussions on potential applications and open problems for future research.