EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 9, No. 4, 2016, 383-401 ISSN 1307-5543 – www.ejpam.com Controllability of Mild Solutions for Evolution Equations with Infinite State-Dependent Delay Djillali Aoued, Selma Baghli-Bendimerad∗ Department of Mathematics, Exact Sciences Faculty, Djillali Liabes University, B. P. 89, Sidi Bel-Abbes 22000, Algeria Abstract. We consider in this paper the controllability of mild solutions defined on the semi-infinite positive real interval for two classes of first order partial functional and neutral functional evolution equations with infinite state-dependent delay using a nonlinear alternative due to Avramescu for sum of compact and contraction operators in Fréchet spaces, combined with the semigroup theory. 2010 Mathematics Subject Classifications: 93B05, 34G20, 34G25, 34K40 Key Words and Phrases: Controllability, Mild solution, Evolution equations, Neutral problems, Infinite delay, State-dependent delay, Fixed point, Nonlinear alternative, Semigroup theory, Fréchet spaces 1. Introduction Controllability of mild solutions is given in this paper over the semi-infinite real interval J := [0,+∞) for two classes of first order partial and neutral functional evolution equations with infinite state-dependent delay in a real separable Banach space (E, | · |). In Section 3, we study the following evolution equation y ′(t) =A(t)y(t) + Cu(t) + f (t, yρ(t,yt )), a.e. t ∈ J , y0 =φ ∈B (1) and in Section 4, we study the following neutral evolution equation d d t [y(t)− g(t, yρ(t,yt ))] =A(t)y(t) + Cu(t) + f (t, yρ(t,yt )), a.e. t ∈ J , y0 =φ ∈B , (2) whereB is an abstract phase space to be specified later, f , g : J ×B → E, ρ : J ×B → R and φ ∈ B are given functions, the control function u(·) is given in L2(R+; E), the Banach space ∗Corresponding author. Email addresses: ouadjillali@yahoo.fr (D. Aoued), selma_baghli@yahoo.fr (S. Baghli-Bendimerad) http://www.ejpam.com 383 c© 2016 EJPAM All rights reserved. D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 384 of admissible control function with E is a real separable Banach space with the norm | · |, C is a bounded linear operator from E into E and {A(t)}0≤t<+∞ is a family of linear closed (not necessarily bounded) operators from E into E that generates an evolution system of operators {U(t, s)}(t,s)∈J×J for s ≤ t. For any continuous function y and any t ≤ 0, we denote by yt the element of B defined by yt(θ ) = y(t + θ ) for θ ≤ 0: Here yt(·) represents the history of the state from time t ≤ 0 up to the present time t. Finally in Section 5, we illustrate by examples the previous abstract theory obtained. Controllability problem of linear and nonlinear systems represented by ODEs in finite di- mensional space has been extensively studied. Several authors have extended the controlla- bility concept to infinite dimensional systems in Banach space with unbounded operators (see [13, 28]) and developed more results in [26, 30, 34]. Carmichael and Quinn [12] have shown that the controllability problem can be converted into a fixed point problem. Then, interesting controllability results are given for neutral problems with impulses by Balachandran et al. in [4] and for integrodifferential equations by Machado et al. in [27] and for inclusions by Gu- nasekar et al. in [18, 19, 31]. Recently Baghli et al. have studied many classes of functional evolution equations and inclusions in [6, 7] and proposed some controllability results in [1] and [8] when the delay is finite and infinite. However, complicated situations in which the delay depends on the unknown functions have been proposed in modelling in recent years. These equations are frequently called equa- tions with state-dependent delay. Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case is called distributed delay; see for instance the books [21, 24, 32], and the papers [14, 20]. Existence results and among other things were derived recently for functional differential equations when the solution is depending on the delay on a bounded interval for impulsive problems. We refer the reader to the papers by Hernandez et al. [22] and Li et al. [25]. Very recently, Baghli et al. considered when the solution is depending in the delay for evolution equations in [9], for multivalued problems in [10] and for perturbed evolution equations in [3]. Our main purpose in this paper is to extend the controllability results obtained by Baghli et al. in [1] and [8] when ρ(t, yt) = t to the control problems (1) and (2) with infinite state- dependent delay as in [9]. We provide sufficient conditions for the existence of mild solutions using the nonlinear alternative of Avramescu [5] due to Burton and Kirk [11] for contractions maps in Fréchet spaces, combined with semigroup theory [2, 29]. 2. Preliminaries We introduce notations, definitions and theorems which are used in this paper. Let C(R+; E) be the space of continuous functions from R+ into E and B(E) be the space of all bounded linearoperators from E into E, with the usual supremum norm ‖N‖B(E) = sup { |N(y)| : |y|= 1 } for all N ∈ B(E). A measurable function y : R+ → E is Bochner integrable if and only if |y| is Lebesgue integrable (See the Bochner integral properties in Yosida [33]). D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 385 Let L1(R+, E) be the Banach space of measurable functions y : R+→ E which are Bochner integrable normed by ‖y‖L1 = ∫ +∞ 0 |y(t)| d t. In this paper, we will employ an axiomatic definition of the phase spaceB introduced by Hale and Kato in [20] and follow the terminology used by Hino et al. in [23] (More details and some examples of phase spaces could be found in [23]). Thus, (B ,‖·‖B)will be a seminormed linear space of functions mapping (−∞, 0] into E, and satisfying the following axioms : (A1) If y : (−∞, b)→ E, b > 0, is continuous on [0, b] and y0 ∈B , then for every t ∈ [0, b) the following conditions hold : (i) yt ∈B; (ii) There exists a positive constant H such that |y(t)| ≤ H‖yt‖B ; (iii) There exist two functions K(·), M(·) : R+→ R+ independent of y with K continuous and M locally bounded such that : ‖yt‖B ≤ K(t) sup s∈[0,t] |y(s)|+M(t)‖y0‖B . (A2) For the function y in (A1), yt is aB−valued continuous function on [0, b]. (A3) The spaceB is complete. Denote Kb = supt∈[0,b] K(t) and Mb = supt∈[0,b]M(t). Definition 1. A function f : J ×B → E is said to be an L1-Carathéodory function if it satisfies: (i) for each t ∈ J the function f (t, ·) :B → E is continuous; (ii) for each y ∈B the function f (·, y) : J → E is measurable; (iii) for every positive integer k there exists hk ∈ L1(J ;R+) such that | f (t, y)| ≤ hk(t) for all ‖y‖B ≤ k and almost every t ∈ J In what follows, for the family {A(t), t ≥ 0} of closed densely defined linear unbounded operators on the Banach space E we assume that it satisfies the following assumptions (see [2]). (P1) The domain D(A(t)) is independent of t and is dense in E. (P2) For t ≥ 0, the resolvent R(λ, A(t)) = (λI−A(t))−1 exists for all λ with Reλ≤ 0 and there is a constant M independent of λ and t such that ‖R(t, A(t))‖ ≤ M(1+ |λ|)−1, for Reλ≤ 0. (P3) There exist constants L > 0 and 0< α≤ 1 such that ‖(A(t)− A(θ ))A−1(τ)‖ ≤ L|t −τ|α, for t,θ ,τ ∈ J . D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 386 Lemma 1 ([2]). Under assumptions (P1)-(P3), the Cauchy problem y ′(t)− A(t)y(t) = 0, t ∈ J and y(0) = y0, has a unique evolution system U(t, s), (t, s) ∈∆ := {(t, s) ∈ J × J : 0≤ s ≤ t < +∞} satisfying the following properties : (i) U(t, t) = I where I is the identity operator in E, (ii) U(t, s)U(s,τ) = U(t,τ) for 0≤ τ≤ s ≤ t < +∞, (iii) U(t, s) ∈ B(E) the space of bounded linear operators on E where for every (t, s) ∈ ∆ and for each y ∈ E, the mapping (t, s)→ U(t, s) y is continuous. For more details on evolution systems and their properties, see [2, 15, 16, 29]. Let X be a Fréchet space with a family of semi-norms {‖ · ‖n}n∈N. We assume that the family of semi-norms {‖ · ‖n} verifies: ‖x‖1 ≤ ‖x‖2 ≤ ‖x‖3 ≤ . . . for every x ∈ X . Let Y ⊂ X , we say that Y is bounded if for every n ∈ N, there exists M n > 0 such that ‖y‖n ≤ M n for all y ∈ Y . To X we associate a sequence of Banach spaces {(X n,‖ · ‖n)} as follows: For every n ∈ N, we consider the equivalence relation ∼n defined by: x ∼n y if and only if ‖x − y‖n = 0 for x , y ∈ X . We denote X n = (X |∼n ,‖ · ‖n) the quotient space, the completion of X n with respect to ‖ · ‖n. To every Y ⊂ X , we associate a sequence {Y n} of subsets Y n ⊂ X n as follows: For every x ∈ X , we denote [x]n the equivalence class of x of subset X n and we defined Y n = {[x]n : x ∈ Y }. We denote Y n, intn(Y n) and ∂nY n, respectively, the closure, the interior and the boundary of Y n with respect to ‖ · ‖n in X n. We give now the definition of the appropriate concept of contraction in X then we state the corresponding nonlinear alternative result. Definition 2. [17] A function f : X → X is said to be a contraction if for each n ∈ N there exists kn ∈ (0, 1) such that: ‖ f (x)− f (y)‖n ≤ kn ‖x − y‖n for all x , y ∈ X . Theorem 1 (Avramescu’s Nonlinear Alternative [5]). Let X be a Fréchet space and let A, B : X → X be two operators satisfying: (i) A is a compact operator, (ii) B is a contraction. Then either one of the following statements holds: (C1) The operator A+ B has a fixed point; (C2) The set � x ∈ X , x = λA(x) +λB � x λ � is unbounded for some λ ∈ (0, 1). D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 387 3. Semilinear Evolution Equations Before stating and proving our first main result, we define firstly the corresponding mild solution then we define the concept of controllability for that problem and finally we expose the properties of state-dependent delay. Definition 3. We say that the function y : R→ E is a mild solution of (1) if y(t) = φ(t) for all t ≤ 0 and y satisfies for each t ≥ 0 the following integral equation y(t) = U(t, 0)φ(0) + ∫ t 0 U(t, s)Cu(s)ds+ ∫ t 0 U(t, s) f (s, yρ(s,ys))ds. (3) Definition 4. The evolution problem (1) is said to be controllable if for every initial function φ ∈ B , y∗ ∈ E and for some n ∈ N, there is some control u ∈ L2([0, n]; E) such that the mild solution y(·) of (1) satisfies the terminal condition y(n) = y∗. SetR(ρ−) = {ρ(s,φ) : (s,φ) ∈ J×B ,ρ(s,φ)≤ 0}. We always assume that ρ : J×B → R is continuous. Additionally, we introduce the following hypothesis: (Hφ) The function t → φt is continuous from R(ρ−) into B and there exists a continuous and bounded function Lφ :R(ρ−)→ (0,+∞) such that ‖φt‖B ≤ Lφ(t)‖φ‖B for every t ∈ R(ρ−). Remark 1. Continuous and bounded functions verified frequently the condition (Hφ), for more details, see for instance [23]. Lemma 2 ([22]). If y : (−∞, b]→ E is a function such that y0 = φ, then ‖ys‖B ≤ (Mb + Lφ)‖φ‖B + Kb sup{|y(θ )|;θ ∈ [0, max{0, s}]}, s ∈ R(ρ−)∪ J where Lφ = supt∈R(ρ−) Lφ(t). Proposition 1. From (Hφ), (A1) and Lemma 2, for all t ∈ [0, n] and n ∈ N we have ‖yρ(t,yt )‖B ≤ Kn|y(t)|+ (Mn + Lφ)‖φ‖B . We will need to introduce the following hypothesis which are assumed thereafter : (H0) U(t, s) is compact for t − s > 0. (H1) There exists a constant ÒM ≥ 1 such that ‖U(t, s)‖B(E) ≤ ÒM for every (t, s) ∈∆. (H2) There exists a function p ∈ L1 loc(J ;R+) and a continuous nondecreasing function ψ : R+→ (0,+∞) and such that: | f (t, u)| ≤ p(t) ψ(‖u‖B) for a.e. t ∈ J and each u ∈B . D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 388 (H3) For all R> 0, there exists lR ∈ L1 loc(J ;R+) such that: | f (t, u)− f (t, v)| ≤ lR(t) ‖u− v‖B for all u, v ∈B with ‖u‖B ≤ R and ‖v‖B ≤ R. (H4) For each n ∈ N, the linear operator W : L2([0, n]; E)→ E is defined by Wu= ∫ n 0 U(n, s)Cu(s)ds, has a pseudo invertible operator W̃−1 which takes values n L2([0, n]; E)/ker W and there exists positive constants eM and eM1 such that: ‖C‖ ≤ eM and ‖W̃−1‖ ≤ eM1. For the construction of W̃−1 see the paper of Carmichael et al. [12]. Consider the following space B+∞ = � y : R→ E : y|[0,T] continuous for T > 0 and y0 ∈B where y|[0,T] is the restriction of y to the real compact interval [0, T]. Let us fix τ > 1. For every n ∈ N, we define in B+∞ the semi-norms by: ‖y‖n := sup t∈[0,n] e−τ L∗n(t) |y(t)| where L∗n(t) = ∫ t 0 ln(s) ds, ln(t) = KnÒMln(t) and ln is the function from (H3). Then B+∞ is a Fréchet space with those family of semi-norms ‖ · ‖n∈N. Theorem 2. Assume that (Hφ) and (H0)-(H4) hold and moreover for each n ∈ N, there exists a constant M n ? > 0 such that M n ? αn + KnÒM(ÒM eM eM1n+ 1)ψ(M n ? ) ‖p‖L1 > 1, (4) with αn = KnÒM eM eM1n|y∗| + � Mn + Lφ + KnÒMH � ÒM eM eM1n+ 1 �� ‖φ‖B . Then the evolution problem (1) is controllable on R. Proof. We transform the problem (1) into a fixed-point problem. Consider the operator N : B+∞→ B+∞ defined by: N(y)(t) = ¨ φ(t) if t ≤ 0; U(t, 0)φ(0) + ∫ t 0 U(t, s) C uy(s)ds+ ∫ t 0 U(t, s) f (s, yρ(s,ys))ds if t ∈ J . Clearly, fixed points of the operator N are mild solutions of the problem (1). Using assumption (H4), for arbitrary function y(·), we define the control uy(t) = W̃−1 � y∗ − U(n, 0) φ(0)− ∫ n 0 U(n,τ) f (τ, yρ(τ,yτ)) dτ � (t). D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 389 Applying (H2), we get |uy(t)| ≤ eM1 � |y∗|+ ÒMH‖φ‖B + ÒM ∫ n 0 p(τ) ψ(‖yρ(τ,yτ)‖B)dτ � . (5) We shall show that using this control the operator N has a fixed point y(·). Then y(·) is a mild solution of the evolution system (1). For φ ∈ B , we will define the function x(·) : R → E by x(t) = φ(t) for t ≤ 0 and x(t) = U(t, 0)φ(0) for t ∈ J . Then x0 = φ. For each function z ∈ B+∞ with z(0) = 0, we denote by z the function defined by z(t) = 0 for t ≤ 0 and z(t) = z(t) for t ∈ J . If y(·) satisfies (3), we can decompose it as y(t) = z(t) + x(t), t ≥ 0, which implies yt = zt + x t , for every t ∈ J and the function z(·) satisfies for t ∈ J z(t) = ∫ t 0 U(t, s) C uz+x(s)ds+ ∫ t 0 U(t, s) f (s, zρ(s,zs+xs) + xρ(s,zs+xs))ds. Let B0 +∞ = � z ∈ B+∞ : z0 = 0 ∈B . For any z ∈ B0 +∞ we have ‖z‖+∞ = sups≥0 |z(s)|. Thus (B0 +∞,‖ · ‖+∞) is a Banach space. We define the operators F, G : B0 +∞ → B0 +∞ by F(z)(t) = ∫ t 0 U(t, s) C uz+x(s)ds and G(z)(t) = ∫ t 0 U(t, s) f (s, zρ(s,zs+xs) + xρ(s,zs+xs))ds. Obviously the operator N has a fixed point is equivalent to F + G has one, so it turns to prove that F +G has a fixed point. The proof will be given in several steps. First we show that F is continuous and compact. Step 1: F is continuous. Let (zn)n∈N be a sequence in B0 +∞ such that zn → z in B0 +∞. By (H1), (H4) and (5), we get for every t ∈ [0, n] |F(zn)(t)− F(z)(t)| ≤ÒM eM ∫ t 0 |uzn+x(s)− uz+x(s)|ds ≤ÒM2 eM eM1 ∫ t 0 ∫ n 0 | f (τ, znρ(τ,znτ+xτ) + xρ(τ,znτ+xτ)) − f (τ, zρ(τ,zτ+xτ) + xρ(τ,zτ+xτ))|dτds ≤ÒM2 eM eM1n ∫ n 0 | f (s, znρ(s,zns+xs) + xρ(s,zns+xs))− f (s, zρ(s,zs+xs) + xρ(s,zs+xs))|ds. Since f is continuous, we obtain by the Lebesgue Dominated Convergence theorem |F(zn)(t)− F(z)(t)| → 0 as n→ +∞. Thus F is continuous. Step 2: F maps bounded sets of B0 +∞ into bounded sets. For any d > 0, there exists a positive constant ` such that for each z ∈ Bd = {z ∈ B0 +∞ : ‖z‖n ≤ d} one has ‖F(z)‖n ≤ `. Let z ∈ Bd . By (H1), (H2) and (5), we have for each t ∈ [0, n] |F(z)(t)| ≤ÒM eM ∫ t 0 eM1 � |by|+ ÒMH‖φ‖B D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 390 +ÒM ∫ n 0 p(τ) ψ(‖zρ(τ,zτ+xτ) + xρ(τ,zτ+xτ)‖B) dτ � ds ≤ÒM eM eM1n � |by|+ ÒMH‖φ‖B + ÒM ∫ n 0 p(s) ψ(‖zρ(s,zs+xs) + xρ(s,zs+xs)‖B)ds � . Using Proposition 1, we get ‖zρ(s,zs+xs) + xρ(s,zs+xs)‖B ≤Kn|z(s)|+ (Mn +L φ)‖z0‖B + Kn|x(s)|+ (Mn +L φ)‖x0‖B ≤Kn|z(s)|+ Kn‖U(s, 0)‖B(E)|φ(0)|+ (Mn +L φ)‖φ‖B ≤Kn|z(s)|+ (Mn +L φ + KnÒMH)‖φ‖B . Set cn := (Mn +L φ + KnÒMH)‖φ‖B , then we obtain ‖zρ(s,zs+xs) + xρ(s,zs+xs)‖B ≤ Kn|z(s)|+ cn. (6) Since z ∈ Bd , then we have for δn := Knd + cn ‖zρ(s,zs+xs) + xρ(s,zs+xs)‖B ≤ Kn|z(s)|+ cn ≤ δn. (7) Using the nondecreasing character of ψ, we get for each t ∈ [0, n] |F(z)(t)| ≤ ÒM eM eM1n � |by|+ ÒMH‖φ‖B + ÒMψ(δn)‖p‖L1 � := %. Thus there exists a positive number % such that ‖F(z)‖n ≤ %. Hence F(Bd) ⊂ B%. Step 3: F maps bounded sets into equicontinuous sets of B0 +∞. We consider Bd as in Step 2 and we show that F(Bd) is equicontinuous. Let τ1,τ2 ∈ J with τ2 > τ1 and z ∈ Bd . Then |F(z)(τ2)− F(z)(τ1)| ≤ ∫ τ1 0 ‖U(τ2, s)− U(τ1, s)‖B(E) ‖C‖|uz+x(s)| ds + ∫ τ2 τ1 ‖U(τ2, s)‖B(E) ‖C‖|uz+x(s)| ds. By the inequalities (5) and (6) and using the nondecreasing character of ψ, we get |uz+x(t)| ≤ eM1 � |y∗|+ ÒMH‖φ‖B + ÒM ψ(δn) ‖p‖L1 � :=ω. (8) Then |F(z)(τ2)− F(z)(τ1)| ≤‖C‖B(E) ω ∫ τ1 0 ‖U(τ2, s)− U(τ1, s)‖B(E) ds + ‖C‖B(E) ω ∫ τ2 τ1 ‖U(τ2, s)‖B(E) ds. Noting that |F(z)(τ2)− F(z)(τ1)| tends to zero as τ2 − τ1 → 0 independently of z ∈ Bd . The right-hand side of the above inequality tends to zero as τ2 − τ1 → 0. Since U(t, s) is a D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 391 strongly continuous operator and the compactness of U(t, s) for t > s implies the continuity in the uniform operator topology (see [2, 29]). As a consequence of Steps 1 to 3 together with the Arzelá-Ascoli theorem it suffices to show that the operator F maps Bd into a precompact set in E. Let t ∈ J be fixed and let ε be such that 0< ε < t. For z ∈ Bd we define Fε(z)(t) = U(t, t − ε) ∫ t−ε 0 U(t − ε, s) C uz+x(s) ds. Since U(t, s) is a compact operator, the set Zε(t) = {Fε(z)(t) : z ∈ Bd} is pre-compact in E for every ε sufficiently small, 0< ε < t. Moreover using (8), we have |F(z)(t)− Fε(z)(t)| ≤ ∫ t t−ε ‖U(t, s)‖B(E) ‖C‖ |uz+x(s)| ds ≤‖C‖B(E) ω ∫ t t−ε ‖U(t, s)‖B(E) ds. Therefore there are precompact sets arbitrary close to the set {F(z)(t) : z ∈ Bd}. Hence the set {F(z)(t) : z ∈ Bd} is precompact in E. So we deduce from Steps 1, 2 and 3 that F is a continuous compact operator. Step 4: We shall show now that the operator G is a contraction. Indeed, consider z, z ∈ B0 +∞. By (H1), (H3) and (7), we get for each t ∈ [0, n] and n ∈ N |G(z)(t)− G(z)(t)| ≤ ∫ t 0 ÒM ln(s) ‖zρ(s,zs+xs) − zρ(s,zs+xs)‖Bds ≤ ∫ t 0 ÒM Kn ln(s) |z(s)− z(s)|ds ≤ ∫ t 0 � ln(s) eτL∗n(s) � � e−τL∗n(s) |z(s)− z(s)| � ds ≤ ∫ t 0 � eτ L∗n(s) τ �′ ds ‖z − z‖n ≤ 1 τ eτ L∗n(t) ‖z − z‖n. Therefore, ‖G(z)− G(z)‖n ≤ 1 τ ‖z − z‖n. So, the operator G is a contraction for all n ∈ N. Step 5: To apply Theorem 1, we must check (C2): i.e. it remains to show that the following set is bounded E = � z ∈ B0 +∞ : z = λ F(z) +λ G � z λ � for some 0< λ < 1 . Let z ∈ E . By (5), we have for each t ∈ [0, n] |z(t)| λ ≤ÒM eM eM1n � |by|+ ÒMH‖φ‖B � D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 392 + ÒM2 eM eM1n ∫ n 0 p(s) ψ � ‖zρ(s,zs+xs) + xρ(s,zs+xs)‖B � ds + ÒM ∫ t 0 p(s) ψ � zρ(s, zs λ +xs) λ + xρ(s, zs λ +xs) B � ds. Using the first inequality in (6), we get zρ(s, zs λ +xs) λ + xρ(s, zs λ +xs) B ≤ Kn|z(s)| λ + Mn +L φ λ ‖z0‖B + Kn|x(s)|+ � Mn +L φ � ‖x0‖B ≤ Kn|z(s)| λ + Kn‖U(s, 0)‖B(E)|φ(0)|+ � Mn +L φ � ‖φ‖B ≤ Kn|z(s)| λ + � KnÒMH +Mn +L φ � ‖φ‖B . Then, we get zρ(s, zs λ +xs) λ + xρ(s, zs λ +xs) B ≤ Kn|z(s)| λ + cn. (9) By the previous inequality and the nondecreasing character of ψ, we obtain |z(t)| λ ≤ÒM eM eM1n � |by|+ ÒMH‖φ‖B � + ÒM2 eM eM1n ∫ n 0 p(s) ψ(Kn|z(s)|+ cn)ds + ÒM ∫ t 0 p(s) ψ � Kn|z(s)| λ + cn � ds. Consider the function eu(t) := supθ∈[0,t] |z(θ )|. Then by the nondecreasing character of ψ, we get for λ < 1 and for t ∈ [0, n] eu(t) λ ≤ ÒM eM eM1n � |by|+ ÒMH‖φ‖B � + ÒM � ÒM eM eM1n+ 1 � ∫ n 0 p(s) ψ � Kneu(s) λ + cn � ds. We consider the function µ defined by µ(t) = sups∈[0,t] Kneu(s) λ + cn for t ∈ J . Let t? ∈ [0, t] be such that µ(t) = Knu(t?) λ +cn. If t? ∈ [0, n], by the previous inequality and the nondecreasing character of ψ, we have for αn := cn + KnÒM eM eM1n � |by|+ ÒMH‖φ‖B � µ(t)≤ αn + KnÒM � ÒM eM eM1n+ 1 � ∫ n 0 p(s) ψ(µ(s)) ds. Consequently, ‖z‖n αn + KnÒM � ÒM eM eM1n+ 1 � ψ(‖z‖n)‖p‖L1 ≤ 1. Then by the condition (4), there exists a constant M n ? such that µ(t)≤ M n ? . Since ‖z‖n ≤ µ(t), we have ‖z‖n ≤ M n ? . This shows that the set E is bounded, i.e. the statement D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 393 (C2) in Theorem 1 does not hold. Then the Avramescu’s nonlinear alternative [5] implies that (C1) holds: i.e. the operator F + G has a fixed-point z?. Then, there exists at least y?(t) = z?(t) + x(t), t ∈ R which is a fixed point of the operator N , which is a mild solution of the problem (1). Thus the evolution system (1) is controllable on R. Then, the proof is complete. 4. Semilinear Neutral Evolution Equations Before stating and proving our second main result, we define firstly the corresponding mild solution then we define the concept of controllability for that problem. Definition 5. We say that the function y(·) : R→ E is a mild solution of (2) if y(t) = φ(t) for all t ≤ 0 and y satisfies the following integral equation y(t) =U(t, 0)[φ(0)− g(0,φ)] + g(t, yρ(t,yt )) + ∫ t 0 U(t, s)A(s)g(s, yρ(s,ys))ds + ∫ t 0 U(t, s)Cu(s)ds+ ∫ t 0 U(t, s) f (s, yρ(s,ys))ds, (10) for each t ≥ 0. Definition 6. The neutral evolution problem (2) is said to be controllable if for every initial function φ ∈ B , y∗ ∈ E and n ∈ N, there is some control u ∈ L2([0, n]; E) such that the mild solution y(·) of (2) satisfies y(n) = y∗. We consider the function ρ : J ×B −→ R satisfies the hypothesis (Hφ) and the Lemma 1. We assume here that the hypotheses (H0)-(H4) hold and we will need the following assump- tions: (H5) There exists a constant M0 > 0 such that ‖A−1(t)‖B(E) ≤ M0 for all t ∈ J . (H6) There exists a constant 0< L < 1 M0Kn , such that |A(t) g(t,φ)| ≤ L (‖φ‖B + 1) for all t ∈ J and φ ∈B . (H7) There exists a constant L∗ > 0 such that |A(s) g(s,φ)− A(s) g(s,φ)| ≤ L∗ (|s− s|+ ‖φ −φ‖B) for all s, s ∈ J and φ,φ ∈B . (H8) The function g is completely continuous and for each bounded sub-set Q ⊂ B , the mapping {t −→ g(t, xρ(s,ys))} is equicontinous in C(J , E). D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 394 Theorem 3. Suppose that hypotheses (H0)-(H8) are satisfied and moreover M?? γn + KnÒM 1−M0 LKn � ÒM eM eM1n+ 1 � � M?? +ψ(M??) � ψ(‖z‖n)‖ζ‖L1 > 1, (11) where ζ(t) =max(L; p(t)) and γn = (Mn +L φ + KnÒMH)‖φ‖B + Knβn 1−M0 LKn with βn = � (ÒM + 1)M0 L + ÒM Ln � � ÒM eM eM1n+ 1 � + ÒM eM eM1n � 1+ KnM0 L � |by| + �� ÒM eM eM1n+ 1 � M0 L � ÒM +Mn +L φ � + ÒMH � ÒM eM eM1n+M0 LKn �� ‖φ‖B . Then the neutral evolution problem (2) is controllable on R. Proof. Consider the operator eN : B+∞→ B+∞ defined by: eN(y)(t) =      φ(t) if t ≤ 0; U(t, 0) [φ(0)− g(0,φ)] + g(t, yρ(t,yt )) + ∫ t 0 U(t, s)A(s)g(s, yρ(s,ys))ds + ∫ t 0 U(t, s)Cu(s)ds+ ∫ t 0 U(t, s) f (s, yρ(s,ys))ds if t ∈ J . Then, fixed points of the operator eN are mild solutions of the problem (2). Using assumption (H4), for arbitrary function y(·), we define the control uy(t) =W̃−1 � y∗ − U(n, 0) � φ(0)− g(0,φ) � − g(n, yρ(n,yn)) − ∫ n 0 U(n,τ)A(τ)g(τ, yρ(τ,yτ))dτ− ∫ n 0 U(n,τ) f (τ, yρ(τ,yτ))dτ � (t). Noting that by (H1), (H2), (H4), (H5) and (H7) we get |uy(t)| ≤ eM1 � |y∗|+ ÒM � H +M0 L � ‖φ‖B + � ÒM + 1 � M0 L + ÒM Ln � + eM1M0 L‖yρ(n,yn)‖B + eM1ÒM L ∫ n 0 ‖yρ(τ,yτ)‖Bdτ + eM1ÒM ∫ n 0 p(τ)ψ(‖yρ(τ,yτ)‖B)dτ. (12) Using this control the operator eN has a fixed point y(·). Then y(·) is a mild solution of the neutral evolution system (2). For φ ∈ B , we will define the function x(·) : R → E by x(t) = φ(t) for t ≤ 0 and x(t) = U(t, 0) φ(0) for t ∈ J . Then x0 = φ. For each function z ∈ B+∞ with z(0) = 0, we denote by z the function defined by z(t) = 0 for t ≤ 0 and z(t) = z(t) for t ∈ J . If y(·) satisfies (10), we decompose it as y(t) = z(t) + x(t), t ≥ 0, which implies yt = zt + x t , for every t ∈ J and the function z(·) satisfies z0 = 0 and for t ∈ J , we get z(t) =g(t, zρ(t,zt+x t ) + xρ(t,zt+x t ))− U(t, 0)g(0,φ D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 395 + ∫ t 0 U(t, s)A(s)g(s, zρ(s,zs+xs) + xρ(s,zs+xs))ds+ ∫ t 0 U(t, s)Cuz+x(s)ds + ∫ t 0 U(t, s) f (s, zρ(s,zs+xs) + xρ(s,zs+xs))ds. Let us define the operators eF , G : B0 +∞ −→ B0 +∞ by eF(z)(t) =g(t, zρ(t,zt+x t ) + xρ(t,zt+x t ))− U(t, 0)g(0,φ) + ∫ t 0 U(t, s)A(s)g(s, zρ(s,zs+xs) + xρ(s,zs+xs))ds+ ∫ t 0 U(t, s)Cuz+x(s)ds and G(z)(t) = ∫ t 0 U(t, s) f (s, zρ(s,zs+xs) + xρ(s,zs+xs))ds. Obviously the operator eN has a fixed point is equivalent to eF + G has one, so it turns to prove that eF + G has a fixed point. We can show as in Section 3 that the operator eF is continuous and compact and the operator G is a contraction. For applying Avramescu’s nonlinear alternative, we must check (C2) in Theorem 1: i.e. it remains to show that the following set eE = § z ∈ B0 +∞ : z = λeF(z) +λG � z λ � for some 0< λ < 1 ª is bounded. Let z ∈ eE . Then, using (H1)-(H6) and (12), we have for each t ∈ [0, n] |z(t)| λ ≤ � (ÒM + 1)M0 L + ÒM Ln � � ÒM eM eM1n+ 1 � + ÒM eM eM1n|by| + ÒM � M0 L � ÒM eM eM1n+ 1 � + ÒM eM eM1nH � ‖φ‖B + ÒM eM eM1M0 Ln‖zρ(n,zn+xn) + xρ(n,zn+xn)‖B +M0 L‖zρ(t,zt+x t ) + xρ(t,zt+x t )‖B + ÒM L ∫ t 0 ‖zρ(s,zs+xs) + xρ(s,zs+xs)‖B ds + ÒM2 eM eM1 Ln ∫ n 0 ‖ � zρ(τ,zτ+xτ) � + xρ(τ,zτ+xτ)‖B dτ + ÒM2 eM eM1n ∫ n 0 p(τ) ψ(‖ � zρ(τ,zτ+xτ) � + xρ(τ,zτ+xτ)‖B) dτ + ÒM ∫ t 0 p(s) ψ � zρ(s, zτ λ +xs) λ + xρ(s, zs λ +xs) B � ds. By Proposition 1, we obtain ‖zρ(n,zn+xn)+ xρ(n,zn+xn)‖B ≤ Kn|by|+(Mn+L φ)‖φ‖B . Using the inequalities (6) and (9), we have |z(t)| λ ≤ � (ÒM + 1)M0 L + ÒM Ln � � ÒM eM eM1n+ 1 � + ÒM eM eM1n|by| D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 396 + ÒM � M0 L � ÒM eM eM1n+ 1 � + ÒM eM eM1nH � ‖φ‖B + ÒM eM eM1M0 Ln � Kn|by|+ (Mn +L φ)‖φ‖B � +M0 L � Kn|z(t)|+ (Mn +L φ + KnÒMH)‖φ‖B � + ÒM L ∫ t 0 � Kn|z(s)|+ cn � ds+ ÒM2 eM eM1 Ln ∫ n 0 � Kn|z(τ)|+ cn � dτ + ÒM2 eM eM1n ∫ n 0 p(τ) ψ � Kn|z(τ)|+ cn � dτ + ÒM ∫ t 0 p(s) ψ � Kn|z(s)| λ + cn � ds. We consider the function eu(t) := supθ∈[0,t] |z(θ )| then by the nondecreasing character of ψ, we obtain for βn := � (ÒM + 1)M0 L + ÒM Ln � � ÒM eM eM1n+ 1 � + ÒM eM eM1n � 1+ KnM0 L � |by| + �� ÒM eM eM1n+ 1 � M0 L � ÒM +Mn +L φ � + ÒMH � ÒM eM eM1nM0 LKn �� ‖φ‖B and for λ < 1, eu(t) λ � 1−M0 LKn � ≤βn + ÒM L � ÒM eM eM1n+ 1 � ∫ n 0 � Kneu(s) λ + cn � ds + ÒM � ÒM eM eM1n+ 1 � ∫ n 0 p(s) ψ � Kneu(s) λ + cn � ds. We consider the function µ defined by µ(t) = sups∈[0,t] Kneu(s) λ + cn for t ∈ J . Let t? ∈ [0, t] be such that µ(t) = Knu(t?) λ + cn. By the previous inequality, we have for γn := cn + Knβn 1−M0 LKn , then we obtain for t ∈ [0, n] µ(t)≤ γn + KnÒM 1−M0 LKn � ÒM eM eM1n+ 1 � ∫ n 0 � Lµ(s) + p(s) ψ(µ(s)) � ds. Set ζ(t) :=max(L; p(t)) for t ∈ [0, n]. Consequently, we get ‖z‖n γn + KnÒM 1−M0 LKn � ÒM eM eM1n+ 1 � � ‖z‖n +ψ(‖z‖n) � ψ(‖z‖n)‖ζ‖L1 ≤ 1. Then by the condition (11), there exists a constant M n ?? such that µ(t) ≤ M n ??. Since ‖z‖n ≤ µ(t), we have ‖z‖n ≤ M n ??. This shows that the set eE is bounded, i.e. the statement (C2) in Theorem 1 does not hold. Then the nonlinear alternative due to Avramescu [5] implies that (C1) holds: i.e. the operator eF + G has a fixed-point z??. Then, there exists at least y??(t) = z??(t)+ x(t), t ∈ R which is a fixed point of the operator eN , which is a mild solution of the problem (2). Thus the neutral evolution system (2) is controllable on R. Then, the proof is complete. D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 397 5. Examples To illustrate the previous results, we give in this section two examples. Example 1 Consider the partial functional differential equation ∂ z ∂ t (t,ξ) = ∂ 2z(t,ξ) ∂ ξ2 + d(ξ)u(t) + a0(t,ξ)z(t,ξ) + ∫ 0 −∞ a1(s− t)z � s−ρ1(t)ρ2 �∫ π 0 a2(θ )|z(t,θ )|2dθ � ,ξ � ds, for t ≥ 0, ξ ∈ [0,π], z(t, 0) = z(t,π) = 0, for t ≥ 0, z(θ ,ξ) = z0(θ ,ξ), for −∞< θ ≤ 0, ξ ∈ [0,π], (13) where a : R+ × [0,π]→ R is a continuous function and is uniformly Hölder continuous in t; a0 : R+ × [0,π]→ R; a1 : R−→ R; a2 : [0,π]→ R; ρi : R+→ R for i = 1, 2; z0 : R− × [0,π] → R and d : R+ → E are continuous functions. u(·) : R+ → E is a given control. To study this system, we consider the space E = L2([0,π],R) and the operator A : D(A) ⊂ E → E given by Aw = w′′ with D(A) := {w ∈ E : w′′ ∈ E, w(0) = w(π) = 0}. It is well known that A is the infinitesimal generator of an analytic semigroup {T (t)}t≥0 on E. Furthermore, A has discrete spectrum with eigenvalues −n2, n ∈ N, and corresponding nor- malized eigenfunctions given by zn(ξ) = sin(nξ) p 2p π . In addition, {zn : n ∈ N} is an orthonormal basis of E and T (t)x = ∑+∞ n=1 e−n2 t(x , zn)zn for x ∈ E and t ≥ 0. It follows from this repre- sentation that T (t) is compact for every t > 0 and that ‖T (t)‖ ≤ e−t for every t ≥ 0. On the domain D(A), we define the operators A(t) : D(A) ⊂ E→ E by A(t)x(ξ) = Ax(ξ) + a0(t,ξ)x(ξ). By assuming that a0(·) is continuous and that a0(t,ξ) ≤ −δ0 (δ0 > 0) for every t ∈ R, ξ ∈ [0,π], it follows that the system u′(t) = A(t)u(t) t ≥ s, and u(s) = x ∈ E, has an associated evolution family given by U(t, s)x(ξ) = � T (t − s)ex p � ∫ t s a0(τ,ξ)dτ � x � (ξ). From this expression, it follows that U(t, s) is a compact linear operator and that ‖U(t, s)‖ ≤ e−(1+δ0)(t−s) for every (t, s) ∈∆. Let B = BUC(R−; E) the space of bounded uniformly continuous functions defined from R− to E endowed with the uniform norm ‖φ‖= supθ∈R− |φ(θ )|. Theorem 4. Let φ ∈ B . Assume that the condition (Hφ) holds and the functions d : R+ → E, ρi : R+ → R+, i = 1, 2, a1 : R− → R and a2 : [0,π] → R are continuous. Then the evolution system (13) is controllable on (−∞,+∞). D. Aoued, S. Baghli-Bendimerad / Eur. J. Pure Appl. Math, 9 (2016), 383-401 398 Proof. From the assumptions, we have that f (t,ψ)(ξ) = ∫ 0 −∞ a1(s)ψ(s,ξ)ds and ρ(s,ψ) = s−ρ1(s)ρ2 � ∫ π 0 a2(θ )|ψ(0,ξ)|2dθ � , are well defined functions and let C ∈ L(R; E) be defined as: Cu(t)(ξ) = d(ξ)u(t), t ≥ 0, ξ ≥ 0, u ∈ R, d(ξ) ∈ E, which permit to trans- form system (13) into the abstract system (1). Moreover, the function f is a bounded linear operator. Now, the controllability of mild solutions can be deduced from a direct application of Theorem 2. Thus, the conclusion of our theorem hold. From Remark 1, we have the following result. Corollary 1. Let φ ∈ B be continuous and bounded. Then the evolution problem (13) is con- trollable on R. Example 2 Consider the semilinear neutral evolution equation ∂ ∂ t � u(t,ξ)− ∫ 0 −∞ a3(s− t)u � s−ρ1(t)ρ2 �∫ π 0 a2(θ )|u(t,θ )|2dθ � ,ξ � ds � = ∂ 2u(t,ξ) ∂ ξ2 + a0(t,ξ)u(t,ξ) + ∫ 0 −∞ a1(s− t)u � s−ρ1(t)ρ2 �∫ π 0 a2(θ )|u(t,θ )|2dθ � ,ξ � ds, for t ≥ 0, ξ ∈ [0,π], v(t, 0) = v(t,π) = 0, for t ≥ 0, v(θ ,ξ) = v0(θ ,ξ), for −∞< θ ≤ 0, ξ ∈ [0,π], (14) where a3 : R− → R is a continuous function and a, ai for i = 0,1, 2, ρi for i = 1,2, z0, d and u(·) are defined as in (13). Theorem 5. Let B = BUC(R−; E) and φ ∈ B . Assume that the condition (Hφ) holds and the functions d : R+ → E, ρi : R+ → R+, i = 1, 2, a1, a3 : R− → R and a2 : [0,π] → R are continuous. Then the evolution system (14) is controllable on (−∞,+∞). Proof. 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