Abstract

This paper introduces convex contractions of order two in complete suprametric spaces and establishes a conditional fixed point theorem for such mappings. The suprametric setting produces a nonlinear Picard recurrence with a quadratic term, requiring explicit orbit-smallness and diameter conditions to ensure convergence. Under these hypotheses, we prove the existence and uniqueness of a fixed point and the geometric convergence of the Picard sequence, recovering Istrăţescu’s classical theorem when the suprametric parameter is zero. Examples are provided to illustrate both the role and applicability of the conditions. The result is further applied to fractional Volterra–Fredholm integro-differential equations and fractional discrete-time neural networks, yielding existence, uniqueness, iterative convergence, and Mittag-Leffler stability of solutions.

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