Extended unified interpolative and hybrid (, , F) contractions in metric spaces and applications to integral equations

This paper develops a unified and extended framework for contraction-type mappings in metric spaces by introducing extended unified interpolative Ćirić–Reich–Rus type (, , F) -contractions together with their r-order counterparts. Within this general setting, rigorous existence and uniqueness results for fixed points of self-mappings are established. The proposed approach not only subsumes numerous classical and contemporary contraction principles as particular cases, but also elucidates the structural relationships among them. The scope and effectiveness of the new framework are further demonstrated through applications to nonlinear integral equations, confirming its relevance and potential for broader analytical investigations.