Approximate Controllability of Fractional Sobolev-Type Evolution Equations in Banach Spaces

and Applied Analysis 3 To investigate the approximate controllability of system (9), we assume the following conditions. (H4) The functionf:[0, b]×X → X satisfies the following: (a) f(t, ⋅) : X → X is continuous for each t ∈ [0, b] and for each x ∈ X, f(⋅, x) : [0, b] → X is strongly measurable; (b) there is a positive integrable function n ∈ L 1 ([0, b], [0, +∞)) and a continuous nondecreasing function Λ f : [0,∞) → (0,∞) such that for every (t, x) ∈ [0, b] × X, we have 󵄩󵄩󵄩󵄩f (t, x) 󵄩󵄩󵄩󵄩 ≤ n (t) Λ f (‖x‖) , lim inf r→∞ Λ f (r) r = σ f < ∞. (10) (H5) The following relationship holds: (1 + 1 ε M 2 B M 2 T b 2α−1 2α − 1 ) M 󵄩󵄩󵄩󵄩 E −1 󵄩󵄩󵄩󵄩 Γ (α) b α α × σ f sup s∈[0,b] n (s) < 1, (11) hereM B := ‖B‖, MT := ‖TE‖. (H6) For every h ∈ X, z α (h) = ε(εI + Γ b 0 J) −1 (h) converges to zero as ε → 0+ in strong topology, where