Mittag–Leffler and Finite-Time Stability in Quadratic Fractional Integral Equations

Abstract

This paper presents a comprehensive study of the stability of a quadratic fractional integral equation (QFIE) whose kernel is described by a generalized Q-function, a six-parameter extension that unifies several forms of the Mittag–Leffler function. Two different stability aspects are examined. First, Mittag–Leffler stability is established by constructing a suitable Lyapunov-type functional, which shows that solutions starting close to the equilibrium gradually approach zero following a Mittag–Leffler decay pattern. Second, finite-time stability is analyzed using a generalized Gronwall inequality associated with the Q-function kernel, leading to practical conditions that guarantee the boundedness of solutions within a given finite time interval. These findings broaden earlier results on existence and extremal solutions and are further demonstrated through several numerical examples accompanied by graphical illustrations.