Existence results of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces

Mathematical and Computer Modelling 49 (2009) 516–526

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Existence results of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces Junhao Hu a,b , Xinzhi Liu c,∗ a Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China b College of Computer Science, South Central University for Nationalities, Wuhan, Hubei 430074, People’s Republic of China c Department of Applied Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontano, Canada N2L 3G1

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Article history: Received 21 September 2007 Accepted 15 February 2008

a b s t r a c t This paper investigates a class of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces. The existence of mild solutions of these inclusions is determined by using the theory of cosine families of bounded linear operators and a recent fixed point theorem for multivalued maps due to Dhage. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The theory of impulsive differential equations has become an important area of investigation in the past two decades because of their applications to various problems arising in communications, control technology, impact mechanics, electrical engineering, medicine, and biology, see the monograph of Laskshmikantham et al. [18] and the papers of Erbe et al. [1], Rogochenko [3], Kirane et al. [4], Liu [5] and the survey papers of Rogochenko [2], Bainov [6] and the references therein. Recently, the problems of existence of solutions and controllability of differential equation and differential inclusions have been extensively studied [7–16]. Benchohra et al. [7], Balachandran et al. [12,13] and George et al. [14] proved the existence of mild solutions and controllability for impulsive neutral second-order differential inclusions with delay. But the authors imposed some strict compactness assumptions on the cosine function which implies that the underlying space is of finite dimension. In [8–10], the authors considered existence results for second-order impulsive functional differential inclusions for the cases A = 0. Benchohra et al. [16] discussed the controllability for an infinite time horizon of secondorder differential inclusions in Banach spaces with nonlocal conditions using a fixed point theorem due to Ma which is an extension to the multivalued maps of the Schaefer theorem on locally convex topological spaces. In this paper, we shall study a class of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces described in the form   Z t  d 0   [ x (t) − g(t, xt , x0 (t))] ∈ Ax + F t, xt , x0 (t), h(t, s, xs , x0 (s))ds , t ∈ J, t 6= tk ,    dt 0 ∆x(tk ) = Ik1 (xtk , x0 (tk )), k = 1, 2, . . . , m, (1)  0 2 0   ∆x (tk ) = Ik (xtk , x (tk )), k = 1, 2, . . . , m,  x0 = φ ∈ B , x0 (0) = z, where J = [0, b], A is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators (C (t))t∈R on a Banach space X . For t ∈ J, xt represents the function xt : (−∞, 0] → X defined by xt (θ) = x(t + θ), −∞ < θ ≤ 0 which belongs to some abstract phase space B defined axiomatically, g : J × B × X → X , F : J × B × X × X → P (X ), and ∗ Corresponding author. E-mail address: [email protected] (X. Liu). 0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.02.005

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h : J × J ×B × X → X are appropriate functions and will be specified later, P (X ) denotes the class of all nonempty subsets of X . The impulsive moments {tk } are given such that 0 = t0 < t1 < · · · < tm < tm+1 = b, Iki : B× X → X (k = 1, 2, . . . , m, i = 1, 2), 1ξ(t) represents the jump of a function ξ at t, which is defined by 1ξ(t) = ξ(t+ ) − ξ(t− ), where ξ(t+ ) and ξ(t− ) are respectively the right and the left limits of ξ at t.

Motivated by the above-mentioned works, our goal in this paper is to establish the existence results on mild solutions of system (1) by using cosine families of bounded linear operators and a recent fixed point theorem for multivalued maps due to Dhage. The rest of this paper is organized as follows. In Section 2, we introduce some notations and necessary preliminaries. In Section 3, we prove the existence results of mild solutions of system (1). 2. Preliminaries In this section, we introduce some definitions, assumptions and preliminary results which will be used throughout this paper. Let X denote a separable Banach space with norm k · k, P (X ) denote the class of all nonempty subsets of X . Let Pbd,cl (X ), Pcp,cv (X ), Pbd,cl,cv (X ) denote respectively the family of all nonempty bounded-closed, compact-convex, boundedclosed-convex subsets of X . For x ∈ X and Y, Z ∈ Pbd,cl (X ), we define by D(x, Y ) = inf {kx − yk : y ∈ Y }, ρ(Y, Z ) = supa∈Y D(a, Z ) and the Hausdorff metric H : Pbd,cl (X ) × Pbd,cl (X ) → R+ by H(A, B) = max{ρ(A, B), ρ(B, A)}. Br [x, X ] denotes the closed ball in X with center at x and radius r. L(X1 , X2 ) stands for the Banach space of bounded linear operator from Banach space X1 to Banach space X2 . A multivalued map F : X → P (X ) is convex(closed) valued, if F (x) is convex(closed) for all x ∈ X . F is bounded on S bounded set if F (V ) = x∈V F (x) is bounded in X, for any bounded set V of X . F is called upper semicontinuous(shortly u.s.c.) on X , if for each x∗ ∈ X , the set F (x∗ ) is a nonempty, closed subset of X , and if for each open set of V of X containing F (x∗ ), there exists an open neighborhood N of x∗ such that F (N) ⊆ V . F is said to be completely continuous if F (V ) is relatively compact, for every bounded subset V ⊆ X . If the multivalued map F is completely continuous with nonempty compact values, then F is u.s.c. if and only if F has a closed graph, (i.e. xn → x∗ , yn → y∗ , yn ∈ F (xn ) imply y∗ ∈ F (x∗ )). A point x0 ∈ X is called a fixed point of the multivalued map F if x0 ∈ F (x0 ). A multivalued map F : J → Pbd,cl,cv (X ) is said to be measurable if for each x ∈ X , the function t 7→ D(x, F (t)) is a measurable function on J. For more details on the multivalued maps, see the books of Deimling [20]. Definition 2.1. Let F : X → Pbd,cl (X ) be a multivalued map. Then F is called a multivalued contraction if there exists a constant k ∈ (0, 1) such that for each x, y ∈ X we have H(F (x), F (y)) ≤ kkx − yk.

The constant k is called a contraction constant of F . In this paper, A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators

((C (t)))t∈R on a Banach space X . We refer the reader to [21] for the necessary concepts about cosine functions. Next we only mention a few results and notations about this matter R t needed to establish our results. We denote by S(t) the sine function associated with C (t) which is defined by S(t)x := 0 C (s)xds, x ∈ X , t ∈ R. and we always assume that M1 and M2 are positive constants such that kC (t)k ≤ M1 and kS(t)k ≤ M2 for every t ∈ J. As usual we denote by [D(A)] the domain of operator A endowed with the graph norm kxkA = kxk + kAxk, x ∈ D(A). Moreover, the notation E stands for the space formed by the vectors x ∈ X for which C (·)x is of class C 1 on R. It was proved by Kisińsky [24] that E endowed with the norm kxkE = kxk + sup0≤t≤1 kAS(t)xk, x ∈ E, is a Banach space. The operator-valued function G(t) =



C (t) AS(t)

 S(t) C (t )

is a strongly continuous group of linear operators on the space E × X generated by the operator   0 I A= A 0 defined on D(A) × E. From this, it follows that AS(t) : E → X is a bounded linear operator R t and that AS(t)x → 0 as t → 0, for each x ∈ E. Furthermore, if x : [0, ∞) → X is a locally integrable function then z(t) = 0 S(t − s)x(s)ds, defines an E-valued continuous function which is a consequence of the fact that  Z t   Z t S ( t − s ) x ( s ) d s   0 , Z0 G(t − s) ds =   t  x(s) 0 C (t − s)x(s)ds 0

defines an E × X -valued continuous function.

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The existence of solutions for the second-order abstract Cauchy problem x00 (t) = Ax(t) + h(t), x(σ) = z0 ,

σ ≤ t ≤ µ,

(2)

x (σ) = z1 , 0

(3)

where h : [σ, µ] → X is an integrable function, has been discussed in [25]. Similarly, the existence of solutions for the semilinear second-order abstract Cauchy problem has been treated in [26]. We only mention here that the function x(·) given by Z t x(t) = C (t − σ)z0 + S(t − σ)z1 + S(t − s)h(s)ds, σ ≤ t ≤ µ, (4) σ

is called a mild solution of (2)-(3) and that when z1 ∈ E, x(·) is continuously differentiable and Z t x0 (t) = AS(t − σ)z0 + C (t − σ)z1 + C (t − s)h(s)ds, σ ≤ t ≤ µ.

(5)

σ

Let

P C ([µ, τ ]; X ) = ϕ : [µ, τ ] → X : ϕ(·) is continuous at t 6= tk , ϕ(tk+ ) = ϕ(tk ) and ϕ(tk− ) exist for k = 1, 2, . . . , m . 



In this paper, the notation P C stands for the space formed by all functions u ∈ P C ([0, b]; X ). The norm k · kP C of the space P C is defined by kϕkP C = sup0≤s≤b kϕ(s)k. It is clear that (P C , k · kP C ) is a Banach space. Similarly, P C 1 will be the space of the functions ϕ ∈ P C such that ϕ is continuously differentiable on J \ {tk : k = 1, 2, . . . , m} and the lateral derivatives − + ϕ0R (t) = lims→0+ ϕ(t+s)−s ϕ(t ) , ϕ0L (t) = lims→0− ϕ(t+s)−s ϕ(t ) are continuous functions on [0, b) and (0, b] respectively. Next, for ϕ ∈ P C 1 we represent by ϕ0 (t) the right derivative at t ∈ (0, b] and by ϕ0 (0) the right derivative at zero. It is easy to see that P C 1 provided with the norm

kϕkP C 1 = kϕkP C + kϕ0 kP C is a Banach space. We will also employ an axiomatic definition of the phase space B which is similar to that used in [27]. B will be a linear space of functions mapping (−∞, 0] to X endowed with a seminorm k · kB . We will assume that B satisfies the following axioms: (A) If x : (−∞, µ + σ ] → X , σ > 0 is such that xµ ∈ B and x|[µ,µ+σ] ∈ P C 1 ([µ, µ + σ ]; X ), then for every t ∈ [µ, µ + σ ] the following conditions hold: (1) xt is in B , (2) kx(t)k ≤ Lkxt kB , (3) kxt kB ≤ K (t − µ) sup{kx(s)k : µ ≤ s ≤ t} + M(t − µ)kxµ kB , where L > 0 is a constant, K , M : [0, ∞) → [0, ∞), K is continuous, M is locally bounded, and L, K , M are independent of x(·). (B) The space B is complete.

˜ k , k = 0, 1, . . . , m, the unique continuous function ϕ˜ k ∈ C ([tk , tk+1 ]; X ) such that For ϕ ∈ P C we denote by ϕ  ϕ(t), for t ∈ [tk , tk+1 ), e ϕk (t) = ϕ(tk− ), for t = tk . Moreover, for B ⊆ P C we employ the notation e Bk , k = 0, 1, . . . , m, for the sets e ˜ k : ϕ ∈ B}. Bk = {ϕ Lemma 2.2. A set B ⊆ P C is relatively compact in P C if and only if each set e Bk is relatively compact in C ([tk , tk+1 ]; X ). Furthermore, we need the following extension of ([23], Lemma 1.1). Rt Lemma 2.3. Let h : [0, b] → X be an integrable function such that h ∈ P C . Then the function v(t) = 0 C (t − s)h(s)ds R t belongsR to P C 1 , the function s → AS(t − s)h(s) is integrable on [0, t], t ∈ [0, b], and v0 (t) = h(t) + A 0 S(t − s)h(s)ds = t h(t) + 0 AS(t − s)h(s)ds, t ∈ J.

The consideration in this paper is based on the recent fixed point theorem for multivalued maps by Dhage [17] (Theorem 3.3, P365). Theorem 2.4. Let X be a Banach space, C : X → Pbd,cl,cv (X ) and D : X → Pcp,cv (X ) be two multivalued maps satisfying: (a) C is a contraction with a contraction constant k, and (b) D is u.s.c. and completely continuous. Then either (i) The operator inclusion λx ∈ C x + D x has a solution for λ = 1, or (ii) The set E = {u ∈ X : λu ∈ C u + D u, λ > 1} is unbounded.

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3. Existence results In this section, we state and prove our main results. Let us begin by introducing the following definition. Definition 3.1. A function x : (−∞, b] → X is called a mild solution of system (1) if x0 = φ ∈ B , x0 (0) = z, x|J ∈ P C 1 , the impulsive conditions ∆x(tk ) = Ik1 (xtk , x0 (tk )), ∆x0 (tk ) = Ik2 (xtk , x0 (tk )), k = 1, 2, . . . , m, are satisfied and Z t Z t x(t) = C (t)φ(0) + S(t)[z − g(0, φ, z)] + C (t − s)g(s, xs , x0 (s))ds + S(t − s)f (s)ds 0

+

X 0
C (t − tk )Ik (xtk , x (tk )) + 1

0

0

0
where f ∈ SF,x = {f ∈ L1 (J, X ) : f (t) ∈ F (t, xt , x0 (t),

S(t − tk )Ik (xtk , x (tk )), 2

X

Rt 0

0

t∈J

(6)

h(t, s, xs , x0 (s))ds), for a.e. t ∈ J}.

Remark. In what follows, it is convenient to introduce the function defined by  φ(t), if t ∈ (−∞, 0], y(t) = C (t)φ(0) + S(t)[z − g(0, φ, z)], if t ∈ J, and M3 = supθ∈J kAS(θ)kL(E:X) . We shall need the following assumptions in what follows. (H0) A is the infinitesimal generator of a strongly continuous cosine function of bounded linear operators ((C (t)))t∈R on a Banach space X and S(t) is the sine function associated with C (t). There exist positive constants M1 and M2 such that kC (t)k ≤ M1 and kS(t)k ≤ M2 for every t ∈ J. (H1) The multivalued map F : J × B × X × X → Pbd,cl,cv (X ) satisfies the following conditions: (i) For every t ∈ J, the map F (t, ·) : B × X × X → Pbd,cl,cv (X ) is u.s.c. (ii) For each (ψ, x, w) ∈ B × X × X , the map F (·, ψ, x, w) : J → Pbd,cl,cv (X ) is measurable and the set     Z t SF,ψ = f ∈ L1 (J, X ) : f (t) ∈ F t, ψt , ψ0 (t), h(t, s, ψs , ψ0 (s))ds , for a.e. t ∈ J 0

is nonempty. (H2) The function h : J × J × B × X → X satisfies the following conditions: (i) For every (t, s) ∈ J × J the function h(t, s, ·) : B × X → X is continuous, (ii) For every (ψ, w) ∈ B × X the function h(·, ψ, w) : J × J → X is strongly measurable. T (iii) There exists a continuous positive function α ∈ C (J, R+ ) L2 (J, R+ ) such that

Z t



h(t, s, ψ, w)ds ≤ α(t)(kψkB + kwk)

0

for each t ∈ J and every (ψ, w) ∈ B × X . (H3) The impulsive functions satisfies the following conditions: j (i) The function Ik1 : B × X → E is completely continuous, there exist constants ck , j = 1, 2, k = 1, 2, . . . , m such that

kIk1 (ψ, w)kE ≤ ck1 (kψkB + kwk) + ck2 for every (ψ, w) ∈ B × X . (ii) The function Ik2 : B × X → E is continuous and for each s ∈ J and all r > 0, the set V 1 (s, r) = {S(s)Ik2 (ψ, w) : (ψ, w) ∈ j Br [0, B] × Br [0, X ]} is relatively compact in X . Moreover,there exist constants dk , j = 1, 2, k = 1, 2, . . . , m such that 2 1 2 kIk (ψ, w)kE ≤ dk (kψkB + kwk) + dk for every (ψ, w) ∈ B × X . (H4) There exist continuous functions g1 : J × X → E and g2 : J × B → E such that g(t, ψ, x) = g1 (t, x) + g2 (t, ψ) and the following conditions hold: (i) The function g2 : J × B → E is completely continuous, (ii) There exist constants L0 , L1 , L2 and a continuous nondecreasing function q : [0, ∞) → (0, ∞) such that

kg1 (t, x)kE ≤ L0 kxk + L1 and kg1 (t, x) − g1 (t, y)kE ≤ L2 kx − yk, kg2 (t, ψ)kE ≤ L3 kψkB + L4 ,

(t, ψ) ∈ J × B

and L2 (1 + bM3 + bM1 ) < 1. (iii) Let T = {x : (−∞, b] → X : x0 = 0, x|J ∈ P C} endowed with the norm of uniform convergence. and the set of functions {t 7→ g(t,^ xt + yt ) : x ∈ Br (0, P C )} be uniformly equicontinuous on [tk , tk+1 ] for every k = 0, 1, 2, . . . , m. k

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(H5) There exists a positive function p ∈ L2 (J, [0, ∞)) such that

kF (t, ψ, x, w)k := sup{kf k : f ∈ F (t, ψ, x, w)} ≤ p(t)Θ (kψkB + kxk + kwk) a.e. t ∈ J, ψ ∈ B , (x, w) ∈ X 2 , where Θ : [0, ∞) → (0, ∞) a continuous nondecreasing function with Θ (β(t)(kψkB + kxk)) ≤ β(t)Θ (kψkB + kxk)

for each t ∈ J and (ψ, x) ∈ B × X , β(t) is a positive function. (H6) For every t ∈ J, 0 < s ≤ t, and all r > 0, the set V 2 (s, t, r) = {S(s)f (t) : f (t) ∈ F (J × Br (0, B ) × Br (0, X )2 )} is relatively compact in X . Remark 3.1. (i) If dim X < ∞, then for each ψ ∈ B , SF,ψ 6= ∅ (see [22]), (ii) SF,ψ is nonempty if only if the function Y : J → R defined by Y (t) = inf {kf k : f ∈ F (t, ψ, x, w)} belongs to L1 (J, R). We need the following result due to Lasota and Opial [22]. Theorem 3.2. Let X be a Banach space, let F be a multivalued map satisfying (H1), and K be a linear continuous map from L1 (J, X ) to C (J, X ). Then the operator K ◦ SF,φ : C (J, X ) → Pcp,cv (C (J, X )) is a closed graph operator in C (J, X ) × C (J, X ). Now we are ready to state and prove the following existence result. Theorem 3.3. Let φ ∈ B , If the assumptions (H0)–(H6) are satisfied, then there is a mild solution of system (1) provided that ς < 1 and Z ∞ Z b ds N∗ ds < N 3 s + Θ (2s) 0 1−ς o n where N∗ (t) = max 1N−1ς , 1N−2ς p(t)(1 + α(t)) , and N1 = (Kb M1 + M3 )(L0 + L3 ), N2 = M1 + Kb M2 , N3 = (Kb M1 b + M3 b + 1)(L1 + Pm Pm Pm Pm L4 ) + (M3 + Kb M1 ) k=1 ck2 + (M1 + Kb M2 ) k=1 d2k , ς = L0 + L3 + 2(M3 + Kb M1 ) k=1 ck1 + 2(M1 + Kb M2 ) k=1 d1k . Proof. Let Bb1 be the space of all functions x : (−∞, b] → X such that x0 ∈ B and the restriction x|J ∈ P C 1 . For each x(·) ∈ Bb1 , let k · kb be the seminorm in Bb defined by

kxkb = kx0 kB + kx|J kP C + kx0 |J kP C . The multivalued map Φ : Bb1 → P (Bb1 ) is defined by Φ x the set of ρ ∈ Bb1 such that  φ(t), if t ∈ (−∞, 0],   Z t Z t    C (t − s)g(s, xs , x0 (s))ds + S(t − s)f (s)ds C (t)φ(0) + S(t)[z − g(0, φ, z)] + ρ(t) = 0X 0 X  1 0 2 0  +  C (t − tk )Ik (xtk , x (tk )) + S(t − tk )Ik (xtk , x (tk )), t ∈ J,   0
(7)

0
where f ∈ SF,x . We shall show that Φ has a fixed point which is consequently a mild solution of system (1). Set x(t) = u(t) + y(t), −∞ < t ≤ b, it is clear that x satisfies (6) if and only if u satisfies u0 = 0 and Z t X C (t − s)g(s, us + ys , u0 (s) + y0 (s))ds + C (t − tk )Ik1 (utk + ytk , u0 (tk ) + y0 (tk )) u(t) = 0

0
t

Z

+ 0

S(t − s)f (s)ds +

X 0
S(t − tk )Ik (utk + ytk , u0 (tk ) + y0 (tk )), 2

t ∈ J,

n o Rt where f ∈ SF,u = f ∈ L1 (J, X ) : f (t) ∈ F (t, ut + yt , u0 + y0 , 0 h(t, s, us + ys , u0 (s) + y0 (s))ds) .

Let Bb0 = {u : (−∞, b] → X : u0 = 0, u|J ∈ P C 1 , u0 (0) = g(0, φ, z)} endowed with the norm of P C 1 . thus (Bb0 , k · kP C 1 ) is a Banach space. Set Br = {u ∈ Bb0 : kukP C 1 ≤ r} for some r ≥ 0, then Br ⊆ Bb0 is uniformly bounded and for u ∈ Br , we have

kut + yt kB ≤ kut kB + kyt kB ≤ Kb [sup{ku(s)kP C 1 : 0 ≤ s ≤ t} + sup{ky(s)k : 0 ≤ s ≤ t}] + Mb kφkB ≤ Kb r + Mb kφkB + Kb [M1 kφ(0)k + M2 (kzk + kg(0, φ, z)k)] := r0 . Let the operator Φ1 : Bb0 → P (Bb0 ) be defined by Φ1 u the set of ρ¯ ∈ Bb0 such that  0, if t ∈ (−∞, 0],   Z t Z t Z t   0 0 C ( t − s ) g ( s , u ( s ) + y ( s )) d s + C ( t − s ) g ( s , u + y ) d s + S(t − s)f (s)ds 1 2 s s ¯ t) = 0 ρ( 0 0 X X  1 0 0 2   + C (t − tk )Ik (utk + ytk , u (tk ) + y (tk )) + S(t − tk )Ik (utk + ytk , u0 (tk ) + y0 (tk )),   0
0
t ∈ J.

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We see that if Φ1 has a fixed point in Bb0 , then Φ has a fixed point in Bb1 which is a solution of system (1). Now, we define the two operators as follows. The map C : Bb0 → Bb0 is defined by Z t  C (u) = C (t − s)g1 (s, u0 (s) + y0 (s))ds 0

and D : Bb → P (Bb0 ) is defined by  Z t 0

D (u) =



+

0

C (t − s)g2 (s, us + ys )ds +

t

Z 0

S(t − s)f (s)ds +

X 0
C (t − tk )Ik1 (utk + ytk , u0 (tk ) + y0 (tk ))

 

X 0
S(t − tk )Ik2 (utk + ytk , u0 (tk ) + y0 (tk )), f ∈ SF,u . 

Then Φ1 = C + D . We divide the proof into several steps. Step 1. Obviously, the operator C has closed-convex values. Next, we shall show that C maps bounded sets into bounded sets in Bb0 . Indeed, it is enough to show that there exists a positive constant ` such that for each C u, u ∈ Br = {u ∈ Bb0 : k · kP C 1 ≤ r}, one has kC ukP C 1 ≤ `. Z b kC u(t)k ≤ M1 (L0 ku0 (s) + y0 (s)k + L1 )ds 0

≤ M1 (L1 b + L0 r) + L0 M1

b

Z

ky0 (s)kds := `1 .

0

By Lemma 2.3, we have

C 0 u(t) = g1 (s, u0 (s) + y0 (s)) +

Z

t

0

AS(t − s)g1 (s, u0 (s) + y0 (s))ds

so

kC 0 u(t)k ≤ kg1 (s, u0 (s) + y0 (s))k +

Z 0

t

kAS(t − s)g1 (s, u0 (s) + y0 (s))kds

≤ L0 r + L0 sup ky0 (s)k + M3 b(L0 r + L1 ) + M3 L0 0≤s≤b

Z

b

0

ky0 (s)kds := `2

then kC ukP C 1 ≤ `1 + `2 := `. Step 2. C is a contraction operator. Let u1 , u2 ∈ Bb0 , by hypothesis (H4), we have Z b

kC u1 (t) − C u2 (t)k ≤ M1

0

kg1 (s, u01 (s) + y0 (s)) − g1 (s, u02 (s) + y0 (s))kds

≤ M1 bL2 ku01 − u02 k ≤ M1 bL2 ku1 − u2 kP C 1 . By Lemma 2.3, we have

kC 0 u1 − C 0 u2 k = kg1 (s, u01 (s) + y0 (s)) − g1 (s, u02 (s) + y0 (s))k Z

+ 0

t

kAS(t − s)kkg1 (s, u01 (s) + y0 (s)) − g1 (s, u02 (s) + y0 (s))kds

≤ L2 ku01 − u02 k + bM3 L2 ku01 − u02 k ≤ (L2 + bM3 L2 )ku1 − u2 kP C 1 so

kC u1 − C u2 kP C 1 ≤ L2 (1 + bM3 + bM1 )ku1 − u2 kP C 1 by hypothesis (H4), we obtain

kC u1 − C u2 kP C 1 < ku1 − u2 kP C 1 then C is a contraction map. Step 3. D u is convex for u ∈ Bb0 , D maps bounded sets into bounded sets in Bb0 . First, since F (·) is convex valued and SF,u is convex, D u has convex values. Secondly, D maps bounded sets into bounded sets in Bb0 . Indeed, it is enough to show that there exists a positive `¯ such that each ρ¯ 1 (t) ∈ D u, u ∈ Br = {u ∈ Bb0 : kukP C 1 ≤ r}, one has kρ¯ 1 kP C 1 ≤ `¯ .

J. Hu, X. Liu / Mathematical and Computer Modelling 49 (2009) 516–526

522

If ρ¯ 1 (t) ∈ D u, then there exists f ∈ SF,u , such that Z t X ρ¯ 1 (t) = C (t − s)g2 (s, us + ys )ds + C (t − tk )Ik1 (utk + ytk , u0 (tk ) + y0 (tk )) 0

0
t

Z

+ 0

S(t − s)f (s)ds +

X 0
S(t − tk )Ik2 (utk + ytk , u0 (tk ) + y0 (tk )).

By hypotheses (H2), (H3) and (H5), we observe that, for each t ∈ J

kρ¯ 1 (t)k ≤ M1 b(L3 r0 + L4 ) + M1

m X

(ck1 (r0 + r) + ck2 ) + M2

k=1 b

Z

+ M2 Θ ( r 0 + r )

0

m X

(d1k (r0 + r) + d2k )

k=1

p(s)(α(s) + 1)ds := `¯ 1 .

By Lemma 2.3, we have

ρ¯ 01 (t) = g2 (t, ut + yt ) + t

Z

+ 0

t

Z

AS(t − s)g2 (s, us + ys )ds +

0

C (t − s)f (s)ds +

X 0
X 0
AS(t − tk )Ik1 (utk + ytk , u0 (tk ) + y0 (tk ))

C (t − tk )Ik2 (utk + ytk , u0 (tk ) + y0 (tk )).

By hypotheses (H2), (H3) and (H5), we also observe that, for each t ∈ J

kρ¯ 01 (t)k ≤ (L3 r0 + L4 ) + M3 b(L3 r0 + L4 ) + M3

m X

(ck1 (r0 + r) + ck2 ) + M1

Z 0

b

(d1k (r0 + r) + d2k )

k=1

k =1

+ M1 Θ ( r 0 + r )

m X

p(s)(α(s) + 1)ds := `¯ 2 .

Then, for each ρ¯ 1 (t) ∈ D u, u ∈ B0r , we have kρ¯ 1 kP C 1 ≤ `¯ 1 + `¯ 2 := `¯ . Next, we prove that D is completely continuous. Let Br be, as above, a bounded set. For each u ∈ Br , ρ¯ 1 (t) ∈ D u, there exists f ∈ SF,u , such that, Z t X ρ¯ 1 (t) = C (t − s)g2 (s, us + ys )ds + C (t − tk )Ik1 (utk + ytk , u0 (tk ) + y0 (tk )) 0

0
t

Z

+ 0

S(t − s)f (s)ds +

X 0
S(t − tk )Ik2 (utk + ytk , u0 (tk ) + y0 (tk )).

We introduce the decomposition D (Br ) = D2 (Br ) + D3 (Br ) where D2 is defined by D2 u, u ∈ Br the set of ρ¯ 2 such that Z t Z t ρ¯ 2 (t) = C (t − s)g2 (s, us + ys )ds + S(t − s)f (s)ds 0

0

and D3 is defined by D3 u, u ∈ Br the set of ρ¯ 3 such that X 1 0 0

ρ¯ 3 (t) =

0
C (t − tk )Ik (utk + ytk , u (tk ) + y (tk )) +

X 0
S(t − tk )Ik2 (utk + ytk , u0 (tk ) + y0 (tk )).

Step 4. D2 (Br ) is completely continuous. We begin by showing D2 (Br ) is equicontinuous. For any ε > 0, since the function g2 : J ×B → E is completely continuous, there exists δ > 0 such that

k(C (t + h) − C (t))g2 (t, ψ)k ≤ ε,

kψkB ≤ r0

when |h| < δ. By the hypotheses (H2), (H4) and (H5), we get Z t

kρ¯ 2 (t + h) − ρ¯ 2 (t)k ≤ Z

+ 0

t

0

k(C (t + h − s) − C (t − s))g2 (s, us + ys )kds + M1 hq(r0 )

k(S(t + h − s) − S(t − s))f (s)kds + M2 Θ (r0 + r)

≤ εt + M1 h(L3 r0 + L4 ) + M1 hΘ (r0 + r)

Z 0

t

Z t

t +h

p(s)(α(s) + 1)ds

p(s)(α(s) + 1)ds + M2 Θ (r0 + r)

Z t

t +h

p(s)(α(s) + 1)ds.

As h → 0 and for ε sufficiently small, the right-hand side of the inequality above tends to zero independent of u, so D2 maps bounded sets into equicontinuous sets.

J. Hu, X. Liu / Mathematical and Computer Modelling 49 (2009) 516–526

523

Now we prove that D2 (Br )(t) = {ρ¯ 2 (t) : ρ¯ 2 (t) ∈ D2 (Br )} is relatively compact for every t ∈ J. Let ε > 0 and choose 0 = s1 < s2 < · · · < sk = t such that | si −si+1 |< ε for every i = 1, 2, . . . , k−1. Using the mean value theorem for the Bochner integral, see [19, Lemma 2.1.3], and the set V = {C (t)g2 (s, ψ) : t, s ∈ J, kψkB ≤ r0 } is relatively compact in X , we find that Z t k−1 Z si+1 k−1 Z si+1 X X ρ¯ 2 (t) = C (t − s)g2 (s, us + ys )ds + S(si )f (t − s)ds + (S(s) − S(si ))f (t − s)ds 0

i =1

∈ tco(V ) +

k−1 X

si

i=1

si

(si+1 − si )co(V 2 (si , t − s, max{r, r0 })) + Br000 [0, X ]

i =1

Rb where co(·) denotes the convex hull and r000 = εM1 Θ (r0 + r) 0 p(s)(α(s) + 1)ds. This implies that D2 (Br )(t) is totally bounded and then relatively compact. As a consequence of the above proof together with the Ascoli–Arzela theorem, we can conclude D2 is completely continuous. Step 5. D3 (Br ) is completely continuous. We begin by showing D3 (Br ) is equicontinuous. For any ε > 0, since the function Ik1 , k = 1, 2, . . . , m is completely continuous, we can choose δ > 0 such that

k(C (t + h) − C (t))Ik1 (ψ, w)k ≤

ε

m

,

kψkB ≤ r0 , kwk ≤ r

when |h| < δ. For each u ∈ Br , 0 < t ≤ b be fixed, t ∈ [ti , ti+1 ] and t + δ ∈ [ti , ti+1 ], ρ¯ 3 (t) ∈ D3 u, such that X X 0 1 0 0 2 0

ρ¯ 3 (t) =

0
C (t − tk )Ik (utk + ytk , u (tk ) + y (tk )) +

0
S(t − tk )Ik (utk + ytk , u (tk ) + y (tk )),

then we get m

X

f

f ¯ 3 ]i (t + h) − [ρ ¯ 3 ]i (t)

≤

[ρ

(C (t + h − tk ) − C (t − tk ))Ik1 (utk + ytk , u0 (tk ) + y0 (tk )) k=1

+

m

X

(S(t + h − tk ) − S(t − tk ))Ik2 (utk + ytk , u0 (tk ) + y0 (tk )) k =1

≤ ε + hM1

m X

(d1k (r0 + r) + d2k ).

k =1

As h → 0 and for ε sufficiently small, the right-hand side of the inequality above tends to zero independent of u, so [ρ¯^ 3 (Br )]i , i = 1, 2, . . . , m, are equicontinuous. Now we prove that [ρ¯^ 3 (Br )]i (t), i = 1, 2, . . . , m, is relatively compact for every t ∈ J . From the following relations X X f ¯ 3 ]i (t) = [ρ C (t − tk )Ik1 (utk + ytk , u0 (tk ) + y0 (tk )) + S(t − tk )Ik2 (utk + ytk , u0 (tk ) + y0 (tk )) 0
∈

0
C (t − tk )Ik1 (Br0 [0, B], Br [0, X ]) +

k=1

i X

V 1 (t − tk , max{r, r0 })

k=1

we infer that [ρ¯^ 3 (Br )]i (t), i = 1, 2, . . . , m, is relatively compact for every t ∈ [ti , ti+1 ]. By Lemma 2.2, we conclude that D3 (Br ) is relatively compact. As an application of the Ascoli–Arzela theorem, D3 (Br ) is completely continuous. From the step of 4 and 5, D is completely continuous. Step 6. D has a close graph. Let un → u∗ , ρ¯ n1 ∈ D un and ρ¯ n1 → ρ¯ ∗ , we shall prove that ρ¯ ∗ ∈ D u∗ . Indeed, if ρ¯ n1 ∈ D un means that there exists fn ∈ SF,un , such that Z t X C (t − tk )Ik1 (untk + ytk , (un )0 (tk ) + y0 (tk )) ρ¯ n1 (t) = C (t − s)g2 (s, uns + ys )ds + 0

0
Z

t

+ 0

S(t − s)fn (s)ds +

X 0
S(t − tk )Ik2 (untk + ytk , (un )0 (tk ) + y0 (tk )).

We must prove that there exists f∗ ∈ SF,u∗ , such that Z t X ρ¯ ∗1 (t) = C (t − s)g2 (s, u∗s + ys )ds + C (t − tk )Ik1 (u∗tk + ytk , (u∗ )0 (tk ) + y0 (tk )) 0

0
Z

+ 0

t

S(t − s)f∗ (s)ds +

X 0
S(t − tk )Ik2 (u∗tk + ytk , (u∗ )0 (tk ) + y0 (tk )).

J. Hu, X. Liu / Mathematical and Computer Modelling 49 (2009) 516–526

524

Consider the linear continuous operator

K : L1 (J, X ) → C (J, X ) t

Z

f 7→ K (f )(t) =

0

S(t − s)f (s)ds.

Now

Z t X

n

ρ ¯ ( t ) − C (t − s)g2 (s, uns + ys )ds − C (t − tk )Ik1 (untk + ytk , (un )0 (tk ) + y0 (tk ))

1 0

0
0
tk

1

k

s

0



S(t − tk )Ik2 (u∗tk + ytk , (u∗ )0 (tk ) + y0 (tk )) − C (t − tk )Ik1 (u∗tk + ytk , (u∗ )0 (tk ) + y0 (tk )) −

→ 0,

0
X

as n → ∞. From Theorem 3.2, it follows that K ◦ SF is a closed graph operator, and Z t X ρ¯ n1 (t) − C (t − s)g2 (s, uns + ys )ds − C (t − tk )Ik1 (untk + ytk , (un )0 (tk ) + y0 (tk )) 0

−

0
S(t − tk )Ik2 (untk + ytk , (un )0 (tk ) + y0 (tk )) ∈ K (SF,un ).

X 0
Since un → z∗ and ρ¯ n1 → ρ¯ ∗ , there exists a f∗ ∈ SF,u∗ such that Z t X ρ¯ ∗1 (t) − C (t − s)g2 (s, u∗s + ys )ds − C (t − tk )Ik1 (u∗tk + ytk , (u∗ )0 (tk ) + y0 (tk )) 0

−

X 0
0
S(t − tk )Ik2 (u∗tk + ytk , (u∗ )0 (tk ) + y0 (tk )) =

t

Z 0

S(t − s)f∗ (s)ds.

So we can conclude that D is a completely continuous multivalued map, which is u.s.c. and convex-closed valued. Step 7. The operator Φ1 has a solution in Bb0 . In fact, we show only that the second assertion of Theorem 2.4 is not true. Let u ∈ Bb0 be any solution of λu ∈ Φ1 u = C u + D u for some λ > 1, we get Z t Z t Z t u(t) = λ−1 C (t − s)g1 (s, u0 (s) + y0 (s))ds + λ−1 C (t − s)g2 (s, us + ys )ds + λ−1 S(t − s)f (s)ds 0 0 0 X X −1 1 0 0 −1 2 0 0

+λ

0
C (t − tk )Ik (utk + ytk , u (tk ) + y (tk )) + λ

0
S(t − tk )Ik (utk + ytk , u (tk ) + y (tk )),

t ∈ J, f ∈ SF,u .

It also follows from Lemma 2.3 that u0 (t) =

λ−1 g1 (t, u0 (t) + y0 (t)) + λ−1 + λ−1 + λ−1

t

Z 0

t

Z 0

AS(t − s)g1 (s, u0 (s) + y0 (s))ds + λ−1 g2 (t, ut + yt )

AS(t − s)g2 (s, us + ys )ds + λ−1

X 0
Z 0

t

C (t − s)f (s)ds + λ−1

C (t − tk )Ik2 (utk + ytk , u0 (tk ) + y0 (tk )),

X 0
AS(t − tk )Ik1 (utk + ytk , u0 (tk ) + y0 (tk ))

t ∈ J, f ∈ SF,u .

Let v(s) = Kb sup{ku(τ)k : 0 ≤

τ ≤ s} + sup{kys kB : 0 ≤ s ≤ b} + sup{u0 (τ) : 0 ≤ τ ≤ s} + sup{ky0 (s)k : 0 ≤ s ≤ b}

where Kb sup{ku(τ)k : 0 ≤

τ ≤ s} + sup{kys kB : 0 ≤ s ≤ b} ≥ kus kB + kys kB ≥ kus + ys kB .

Using the hypotheses (H0), (H2), (H3), (H4) and (H5), we obtain, for all t ∈ J, Z t Z t ku(t)k ≤ M1 (L0 v(s) + L1 )ds + M2 p(s)(1 + α(s))Θ (2v(s))ds 0

0

Z

+ M1

0

t

(L3 (v(s)) + L4 )ds + M1

m X

m X

k=1

k=1

(ck1 (2v(t)) + ck2 ) + M2

(d1k (2v(t)) + d2k )

(8)

J. Hu, X. Liu / Mathematical and Computer Modelling 49 (2009) 516–526

525

and

ku0 (t)k ≤ (L0 + L3 )v(t) + M3 Z

+ M1

0

t

t

Z 0

(L0 v(s) + L1 )ds + M3

p(s)(1 + α(s))Θ (2v(s))ds + M3

Z

m X

0

t

(L3 v(s) + L4 )ds + L1

(ck1 (2v(t)) + ck2 ) + L4 + M1

k =1

m X

(d1k (2v(t)) + d2k ).

(9)

k =1

Using (8) and (9), along with the fact that ς < 1, after some simplification, we get Z t Z t N1 N2 N3 v(t) ≤ v(s)ds + p(s)(1 + α(s))Θ (2v(s))ds + 1−ς 0 1−ς 0 1−ς where N1 = (Kb M1 + M3 )(L0 + L3 ), N2 = M1 + Kb M2 , N3 = (Kb M1 b + M3 b + 1)(L1 + L4 ) + (M3 + Kb M1 ) Pm Pm Pm Kb M2 ) k=1 d2k , ς = L0 + L3 + 2(M3 + Kb M1 ) k=1 ck1 + 2(M1 + Kb M2 ) k=1 d1k . Denote the right hand side of the last inequality by π(t), Computing π0 (t), and v(t) ≤ π0 (t), we obtain

π0 (t) ≤

N1

1−ς

π(t) +

N2

1−ς

t ∈ J, o where N∗ (t) = max 1−ς , 1−ς p(t)(1 + α(t)) . Integrating (10) over J, then yields Z π(t) Z b Z ∞ ds ds ≤ N∗ ds < N 3 s + Θ ( 2 s ) s + Θ (2s) π(0) 0 1−ς

where π(0) =

N3

1−ς

N1

k=1

ck2 + (M1 +

p(t)(1 + α(t))Θ (2π(t))

≤ N∗ (t)(π(t) + Θ (2π(t))), n

Pm

(10)

N2

. Therefore, there exists a constant N4 such that v(t) ≤ π(t) ≤ N4 , t ∈ J, and kukP C 1 is bounded. This shows

that the set E is bounded in Bb0 . As a result the second assertion of Theorem 2.4 does not hold. Hence the first assertion holds and the multivalued map Φ has a solution x on (−∞, b]. So, system (1) has a mild solution on (−∞, b]. This completes the proof.  Acknowledgement This research was supported by NSERC Canada. References [1] L.H. Erbe, H.I. Freedman, X.Z. Liu, J.H. Wu, Comparison principles for impulsive parabolic equations with applications to models of single species growths, J. Austral. Math. Soc. Ser. B 32 (1991) 382–400. [2] Y.V. Rogovchenko, Impulsive evolution systems: Main results and new trends, Dyn. Contin. Discrete Impuls. Syst. 3 (1) (1997) 57–88. [3] Y.V. Rogovchenko, Nonlinear impulse evolution systems and applications to population models, J. Math. Anal. Appl. 207 (2) (1997) 300–315. [4] M. Kirane, Y.V. Rogovchenko, Comparison results for systems of impulse parabolic equations with applications to populations dynamics, Nonlinear Anal. TMA 28 (2) (1997) 263–276. [5] J.H. Liu, Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impuls. Syst. 6 (1) (1999) 77–85. [6] D.D. Bainov, E. Minchev, Trends in the theory of impulsive partial differential equations, Nonlinear Word 3 (1996) 357–384. [7] M. Benchohra, J. Henderson, S.K. Ntouyas, Existence results for impulsive multivaued semilinear neutral functional differential inclusions in Banach spaces, J. Math. Anal. Appl. 263 (2001) 763–780. [8] M. Benchohra, J. Henderson, S.K. Ntouyas, On second-order multivalued impulsive functional differential inclusions in Banach spaces, Abstract Appl. Anal. 6 (2001) 369–380. [9] Yong-Kui Chang, Wan-Tong Li, Existence results for second order impulsive functional differential inclusions, J. Math. Anal. Appl. 301 (2005) 477–490. [10] M. Benchohra, A. Ouahab, Initial and boundary value problems for second order impulsive functional differential inclusions, Electron. J. Qualitative Theory Differen. Equations 3 (2003) 1–10. [11] K. Balachandran, S.M. Anthoni, Existence of solutions of second order neutral functional-differential equations, Tamkang J. Math. 30 (1999) 299–309. [12] K. Balachandran, D.G. Park, S.M. Anthoni, Existence of results of abstract-nonlinear second-order neutral functional integrodifferential equations, Comput. Math. Appl. 46 (2003) 1313–1324. [13] K. Balachandran, S.M. Anthoni, Controllbility of second-order semilinear neutral functional differential systems in Banach spaces, Comput. Math. Appl. 41 (2001) 1223–1235. [14] R.K. George, D.N. Chalishajar, A.K. Nandakumaran, Controllbility of second order semi-linear neutral functional differential inclusions in Banach spaces, Mediterr. J. Math. 1 (2004) 463–477. [15] J.R. Kang, Y.C. Kwun, J.Y. Park, Controllbility of the second-order differential inclusions in Banach spaces, J. Math. Anal. Appl. 285 (2003) 537–550. [16] M. Benchohra, S.K. Ntouyas, Controllability for an infinite-time horizon of second-order differential inclusions in Banach spaces with nonlocal conditons, J. Optim. Theory Appl. 109 (2001) 85–98. [17] B.C. Dhage, Multi-valued mappings and fixed points I, Nonlinear Funct. Anal. Appl. 10 (2005) 359–378. [18] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Pub. Co., Singapore, 1989. [19] R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Robert E. Krieger Publ. Co., Florida, 1987. [20] K. Deimling, Multivalued Differential Equations, de Gruyter, Berlin, New York, 1992. [21] H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, in: North-Holland Math. Stud., vol. 108, North-Holland, Amsterdam, 1985. [22] A. Lasota, Z. Opial, An application of the Kakutani–Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965) 781–786.

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Further reading [1] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, in: Lecture Notes in Mathematics, vol. 580, Springer, Berlin, 1977. [2] B.N. Sadovskii, On a fixed point principle, Funct. Anal. Appl. 1 (1967) 74–76. [3] N. Carmichael, M.D. Quinn, An approach to nonlinear control problems using fixed-point methods, degree theory, and pseodo-inverses, Numer. Funct. Anal. Optim. 7 (1984–1985) 197–219. [4] N. Papageorgiou, Boundary value problems for evolution inclusions, Comment. Math. Univ. Carol. 29 (1988) 355–363. [5] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in: Applied Methematical Sciences, vol. 44, Springer Verlag, New York, 1983. [6] J.K. Hale, J. Kato, Phase space for retarded equations with infinite delay, Funkcial. Ekvac. 21 (1978) 11–41. [7] K. Yosida, Functional Analysis, sixth ed., Spinger, Berlin, 1980.