Multi-term fractional differential equations in a nonreflexive banach space

Abstract

In this paper we establish an existence result for the multi- term fractional differential equation (D-alpha m - (m-1)Sigma(aiD alpha i)(i=1))u(t) = f(t, u(t)) for t epsilon [0, 1], u(0) = 0, (1) where D(p)(alpha m)y(.) and D(p)(alpha i)y(.) are fractional pseudo- derivatives of a weakly absolutely continuous and pseudo- differentiable function u(.) : T -> E of order alpha(m) and alpha(i), i = 1, 2,..., m - 1, respectively, the function f(t, .) : T x E -> E is weakly-weakly sequentially continuous for every t is an element of T and f (., y(.)) is Pettis integrable for every weakly absolutely continuous function y(.) : T -> E, T is a bounded interval of real numbers and E is a nonreflexive Banach space, 0 < alpha(1) < alpha(2) < ... <alpha(m) < 1 and a(1), a(2),..., a(m-1) are real numbers such that a := Sigma(m-1)(i=1) vertical bar a(j)vertical bar/Gamma(alpha(m)-alpha(j)+1) < 1.