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 is an element of (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 < r(1) < . . . < 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 is an element of R such that x(t) = ct(R) + o(t(R)) as t --> infinity where R = max{n - 1, r(m)}.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 is an element of (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 < r(1) < . . . < 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 is an element of R such that x(t) = ct(R) + o(t(R)) as t --> infinity where R = max{n - 1, r(m)}.