Approximate Controllability of Fractional Neutral Evolution Equations in Banach Spaces

and Applied Analysis 3 Lemma 5. The operators S α and A α have the following properties: (i) for any fixed t ≥ 0 and any x ∈ X β , one has the operators S α (t) and A α (t) which are linear and bounded operators; that is, for any x ∈ X β , 󵄩󵄩󵄩󵄩Sα (t) x 󵄩󵄩󵄩󵄩β ≤ M‖x‖β, 󵄩󵄩󵄩󵄩Aα (t) x 󵄩󵄩󵄩󵄩β ≤ M Γ (α) ‖x‖ β ; (12) (ii) the operatorsS α (t) andA α (t) are strongly continuous for all t ≥ 0; (iii) S α (t) andA α (t) are norm continuous inX for t > 0; (iv) S α (t) andA α (t) are compact operators inX for t > 0; (v) for every t > 0, the restriction of S α (t) to X β and the restriction ofA α (t) toX β are norm continuous; (vi) for every t > 0, the restriction of S α (t) to X β and the restriction ofA α (t) toX β are compact operators inX β ; (vii) for all x ∈ X and t ∈ [0, T], 󵄩󵄩󵄩󵄩 A β A α (t) x 󵄩󵄩󵄩󵄩 ≤ C β t −αβ ‖x‖ , C β := M β αΓ (2 − β) Γ (1 + α (1 − β)) . (13) In this paper, we adopt the following definition of mild solution of (1). Definition 6. A solution x(⋅; u) ∈ C([0, T], X) is said to be a mild solution of (1) if for any u ∈ L 2 ([0, T], U) the integral equation x (t) = S α (t) (φ (0) + g (0, φ)) − g (t, x t )