Averaging of Multivalued Integral Equations

Integral equations are extensively used in various fields of knowledge, such as the theory of elasticity, heat and mass transfer, the theory of oscillations, fluid dynamics, the theory of filtration, electrostatics, electrodynamics, biomechanics, game theory, control theory, voting theory, electrical engineering, economics, and medicine. The investigations of actual processes based on idealized mathematical models often lead to equations with small parameters. For their investigation, various asymptotic methods are extensively used. The choice of a specific asymptotic method depends on the structure of the equation used to describe the dynamics of an object. In recent years, averaging methods are rapidly developed in the nonlinear mechanics and in the theory of oscillations. The mathematical substantiation of the averaging method for ordinary differential equations was originated in 1937 by Krylov–Bogolyubov in their fundamental work [1]. The works by Grebenikov, Mitropol’skii, Moiseev, Perestyuk, Plotnikov, Samoilenko, Filatov, and other researchers [2–4, 7–11, 14] played a great role in the development of the averaging method for different classes of differential equations. Later, the ideas of the averaging method were generalized for differential equations with multivalued and fuzzy right-hand sides (see [5, 6, 9, 12, 13] and the references therein). In the present paper, we substantiate the averaging method for a multivalued integral equation. Let conv .R/ be a metric space of nonempty compact convex subsets R with the Hausdorff metric h.F;G/: Consider a multivalued integral equation