Asymmetric Cross-Iterate Ćirić–Reich–Rus Contraction: Existence, Uniqueness, and Comparative Analysis

We introduce a novel generalization of the classical Ćirić–Reich–Rus (CRR) contraction, termed the Asymmetric Cross-Iterate CRR (ACI-CRR). The contraction condition involves distinct iterate orders p≠q in an asymmetric cross-coupled form. We establish the existence and uniqueness of a fixed point under the assumption of continuity. We then analyze the symmetric case p=q, demonstrating that continuity is indispensable for p>1, and contrast this with the classical p=q=1 case where continuity is unnecessary. We conclude with a structural example and a conjecture regarding the necessity of continuity in higher-iterate asymmetric contractions.