Fixed-Point Theorem via a Composite Matkowski Comparison Function for a Generalized Ćirić–Reich–Rus Interpolative Contraction with Iterates and Applications to Fractional and Nonlinear Integral Equations

This paper establishes existence and uniqueness results for a generalized Ćirić–Reich–Rus interpolative contraction involving distinct forward horizons p,q>1 and a delay parameter r>1. By employing a Matkowski comparison function φ and introducing a composite condition Ψ(t)=φ(pβqγtα+β+γ)<t, we overcome the geometric expansion induced by multi-step triangle inequalities. The framework guarantees asymptotic regularity and the Cauchy property of Picard sequences, yielding a unique fixed point under mild continuity and right-continuity assumptions. We further demonstrate generalized Ulam–Hyers stability with explicit error bounds under a summability condition on the iterates of Ψ; apply the abstract result to fractional integral, Volterra, Hammerstein, and perturbed integral equations; and provide a sharp counterexample illustrating the necessity of the composite hypothesis. The analysis reveals how asymmetric iteration parameters optimize expansion absorption, offering a tunable framework for nonlinear discrete and integral dynamics.