Nonlocal Problem for Fractional Evolution Equations of Mixed Type with the Measure of Noncompactness

and Applied Analysis 3 Definition 2 (see [3]). The Caputo fractional derivative of order q > 0 with the lower limit zero for a function u is defined as C D q t u (t) = 1 Γ (n − q) ∫ t 0 (t − s) n−q−1 u (n) (s) ds, t > 0, 0 ≤ n − 1 < q < n, (8) where the function u(t) has absolutely continuous derivatives up to order n − 1. If u is an abstract function with values in E, then the integrals which appear in Definitions 1 and 2 are taken in Bochner’s sense. For u ∈ E, define two operators Tq(t) (t ≥ 0) and S q (t) (t ≥ 0) by