Complete Controllability of Fractional Neutral Differential Systems in

and Applied Analysis 3 Definition 4. Let X be a Banach space and ΩX the bounded set ofX. The Kuratowski’s measure of noncompactness is the map α : ΩX → [0,∞) defined by α (D) = inf {d > 0 : D ⊆ n ⋃ i=1 Di, diam (Di) ≤ d} , (8) hereD ∈ ΩX. One will use the following basic properties of the α measure and Sadovskii’s fixed point theorem here (see [37– 39]). Lemma 5. Let D1 and D2 be two bounded sets of a Banach space X. Then (i) α(D1) = 0 if and only ifD1 is relatively compact; (ii) α(D1) ≤ α(D2) ifD1 ⊆ D2; (iii) α(D1 + D2) ≤ α(D1) + α(D2). Lemma 6 (sadovskii’s fixed point theorem). Let N be a condensing operator on a Banach space X, that is, N is continuous and takes bounded sets into bounded sets, and α(N(D)) < α(D) for every bounded setD ofX with α(D) > 0. IfN(S) ⊂ S for a convex closed and bounded set S ofX, thenN has a fixed point in S. 3. Complete Controllability Result To study the system (1), we assume the function xt represents the history of the state from −∞ up to the present time t and xt : (−∞, 0] → X, xt(θ) = x(t + θ) belongs to some abstract phase space B, which is defined axiomatically. In this article, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato [40] and follow the terminology used in [41]. Thus, B will be a linear space of functionsmapping (−∞, 0] intoX endowedwith a seminorm ‖ ⋅ ‖B. We will assume that B satisfies the following axioms: (A) If x ∈ (−∞, σ + a) → X, a > 0, is continuous on [σ, σ + a] and xσ ∈ B, then for every t ∈ [σ, σ + a] the following conditions hold: (i) xt is in B; (ii) ‖x(t)‖ ≤ H‖xt‖B; (iii) ‖xt‖B ≤ K(t − σ) sup{‖x(t)‖ : σ ≤ s ≤ t} + M(t − σ)‖xσ‖B. Here H ≥ 0 is a constant, K,M : [0, +∞) → [0, +∞), K is continuous and M is locally bounded, and H,K,M are independent of x(t). (B) For the function x(⋅) in (A), xt is a B-valued continuous function on [σ, σ + a]. (C) The space B is complete. Now we give the basic assumptions on the system (1). (H0) (i) A generates a uniformly bounded analytic semigroup {T(t), t ≥ 0} in X; (ii) for all bounded subsets D ⊂ X and x ∈ D, ‖T(tq 2 θ)x − T(t q 1 θ)x‖ → 0 as t2 → t1 for each fixed θ ∈ [0,∞]. (H1) F: [0, a] × B → X is continuous function, and there exists a constant β ∈ (0, 1) and L, L1 > 0 such that the function F isXβ-valued and satisfies the Lipschitz condition: 󵄩󵄩󵄩󵄩 A β F (s1, φ1) − A β F (s2, φ2) 󵄩󵄩󵄩󵄩 ≤ L ( 󵄨󵄨󵄨󵄨s1 − s2 󵄨󵄨󵄨󵄨 + 󵄩󵄩󵄩󵄩φ1 − φ2 󵄩󵄩󵄩󵄩B) , (9) for 0 ≤ s1, s2 ≤ a, φ1, φ2 ∈ B, and the inequality 󵄩󵄩󵄩󵄩 A β F (t, φ) 󵄩󵄩󵄩󵄩 ≤ L1 ( 󵄩󵄩󵄩󵄩φ 󵄩󵄩󵄩󵄩B + 1) (10) holds for t ∈ [0, a], φ ∈ B. (H2) The functionG : [0, a]×B → X satisfies the following conditions: (i) for each t ∈ [0, a], the function G(t, ⋅) : B → X is continuous and for each φ ∈ B the function G(⋅, φ) : [0, a] → X is strongly measureable; (ii) for each positive number k, there is a positive function gk ∈ L 1([0, a]), 0 < q1 < q such that sup ‖‖B 󵄩󵄩󵄩󵄩G (t, φ) 󵄩󵄩󵄩󵄩 ≤ gk (t) , lim inf 1 k 󵄩󵄩󵄩󵄩gk 󵄩󵄩󵄩󵄩L1/q1 [0,a] = γ < ∞. (11) (H3) The linear operator C is bounded, W from U into X is defined by