Asymptotic Integration of Fractional Differential Equations with Integrodifferential Right-Hand Side

Abstract

In this paper we deal with the problem of asymptotic integration of a class of fractional differential equations of the Caputo type. The left-hand side of such type of equation is the Caputo derivative of the fractional order r ∈ (n − 1, n) of the solution, and the right-hand side depends not only on ordinary derivatives up to order n − 1 but also on the Caputo derivatives of fractional orders 0 < r₁ < · · · < r_m < r, and the Riemann–Liouville fractional integrals of positive orders. We give some conditions under which for any global solution x(t) of the equation, there is a constant c ∈ R such that x(t) = ct^R + o(t^R) as t → ∞, where R = max{n − 1, r_m}.