Multi-term fractional differential equations in a nonreflexive banach space

In this paper we establish an existence result for the multi-term fractional differential equation (Equation Presented) where D_p ^αmy(·) are fractional pseudo-derivatives of a weakly absolutely continuous and pseudo-differentiable function u(·) : T → E of order α_m and α_i, i = 1,2, . . ., m - 1, respectively, the function f (t, ·) : T x E → E is weakly-weakly sequentially continuous for every t ∈ 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 < α₁ < α₂ < ⋯ < α_m < 1 and a₁, a₂, . . . , a _m-1 are real numbers such that a := ∑_i=1^m-1 |a_i|/Γ(α_m-α_i+1) < 1.