Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains

Nonlinear Analysis 69 (2008) 73–84 www.elsevier.com/locate/na

Impulsive semilinear differential inclusions: Topological structure of the solution set and solutions on non-compact domains Tiziana Cardinali a,∗ , Paola Rubbioni a,b a Department of Mathematics and Informatics, University of Perugia, via L. Vanvitelli, 1 - 06123 Perugia, Italy b INFM, University of Perugia, via L. Vanvitelli, 1 - 06123 Perugia, Italy

Received 5 October 2006; accepted 18 May 2007

Abstract This paper deals with an impulsive Cauchy problem governed by the semilinear evolution differential inclusion x 0 (t) ∈ A(t)x(t) + F(t, x(t)), where {A(t)}t∈[0,b] is a family of linear operators (not necessarily bounded) in a Banach space E generating an evolution operator and F is a Carath´eodory type multifunction. First a theorem on the compactness of the set of all mild solutions for the problem is given. Then this result is applied to obtain the existence of mild solutions for the impulsive Cauchy problem defined on non-compact domains. c 2007 Elsevier Ltd. All rights reserved.

MSC: 34G25; 34A37; 34A60; 34B15 Keywords: Impulsive semilinear evolution differential inclusion; Evolution system; Mild solution; Measure of non-compactness; Upper semicontinuous multifunction

1. Introduction The theory of impulsive differential equations or inclusions has in recent years been an object of increasing interest because of its wide applicability in biology, in medicine and in more and more fields. The reason for this applicability arises from the fact that impulsive differential problems are an appropriate model for describing processes which at certain moments change their state rapidly (in a mathematical simulation it is opportune to assume that the changes of the state are instantaneous and given by jumps) and which cannot be described using classical differential problems. But this theory is interesting in itself since it exhibits several new phenomena such as rhythmical beating, merging of solutions and non-continuability of solutions. For a wide bibliography and exposition on this subject see for instance the monographs [1,2,23,30] and the papers [3,4,11,17,18,24,26,28,31]; for applications of the theory of impulsive differential equations see [10,12,19, 22,29,32,33,35,36]; for applications to control differential inclusions see [8,13,20]. ∗ Corresponding author. Tel.: +39 075 5855042; fax: +39 075 5855024.

E-mail addresses: [email protected] (T. Cardinali), [email protected] (P. Rubbioni). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.05.001

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In 1999 Liu [21] obtained the existence and uniqueness of mild solutions for an impulsive Cauchy problem (with Lipschitz impulse functions) governed by a semilinear evolution equation x 0 (t) = A(t)x(t) + f (t, x(t)),

a.e. t ∈ [0, b]

where A is an unbounded operator generating a strongly continuous semigroup and f is a continuous function, Lipschitz with respect to the second variable. For non-Lipschitzian impulsive equations, see [25]. Since semilinear differential inclusions appear as the natural framework for describing hybrid systems with dry friction, processes of controlled heat transfer, obstacle problems and others (see e.g. [15] and references therein), during the last few years some existence theorems for impulsive initial value problems involving semilinear evolution inclusions have been proved (see e.g. [4]). In particular, in [6] the authors have proved the existence of mild solutions for the following impulsive Cauchy problem controlled by a semilinear evolution differential inclusion:  0 x (t) ∈ A(t)x(t) + F(t, x(t)), a.e. t ∈ [0, b], t 6= tk , k = 1, . . . , m (IP) x(tk+ ) = x(tk ) + Ik (x(tk )), k = 1, . . . , m  x(0) = a0 ∈ E where {A(t)}t∈[0,b] is a family of linear operators (not necessarily bounded) in a Banach space E generating an evolution operator; F is a Carath´eodory type multifunction; 0 = t0 < t1 < · · · < tm < tm+1 = b; Ik : E → E, k = 1, . . . , m, are impulse functions and x(t + ) = lims→t + x(s). In this paper we first provide (see Theorem 1) the compactness of the set of all mild solutions for the problem (IP). Then, we apply this result to deal with the delicate question of finding mild solutions for the impulsive Cauchy problem defined on an unbounded domain [0, +∞[, i.e.  0 x (t) ∈ A(t)x(t) + F(t, x(t)), a.e. t ∈ [0, +∞[, t 6= tk , k ≥ 1 (IP)∞ x(tk+ ) = x(tk ) + Ik (x(tk )), k≥1  x(0) = a0 ∈ E. Note that, in this case, the jump points are an increasing sequence of times (tk )+∞ k=0 such that t0 = 0 and limk→+∞ tk = +∞. Moreover we claim that, by means of the method that we use for the case of unbounded domains, one obtains the existence of mild solutions for impulsive Cauchy problems defined on non-closed intervals [0, b[, 0 < b < +∞, where the jump points are an increasing sequence (tk )+∞ k=0 such that t0 = 0 and limk→+∞ tk = b (obviously, by suitably adapting the assumptions). Finally, we present an example illustrating the applicability of the first abstract result. 2. Preliminaries Let X and Y be topological spaces and let us denote by P(Y ) the collection of all nonempty subsets of Y . A multimap F : X → P(Y ) is said to be upper semicontinuous (u.s.c.) in x ∈ X if, for every open set A in Y such that F(x) ⊂ A, there exists a neighborhood U of x such that F(x 0 ) ⊂ A for all x 0 ∈ U . Obviously, F is u.s.c. in X if F −1 (V ) = {x ∈ X : F(x) ⊂ V } is an open subset of X for every open V ⊂ Y . If Y is a metric space (Y, d), for every A ⊂ Y we define Wε (A) = {y ∈ Y : ρ(y, A) < ε}, where ρ(y, A) = infx∈A d(x, y). In this framework, if the multimap F has compact values then F is u.s.c. at a point x if and only if for every ε > 0 there exists a neighborhood V (x) such that F(x 0 ) ⊂ Wε (F(x)) for every x 0 ∈ V (x) (see e.g. [15], Theorem 1.1.8). Let (E, k · k) be a Banach space and (A, ≥) be a (partially) ordered set. We recall (see e.g. [15]) that a function β : P(E) → A is called a measure of non-compactness (MNC) in E if β(coΩ ) = β(Ω ),

Ω ∈ P(E).

Moreover, a MNC is said to be: monotone if Ω0 , Ω1 ∈ P(E), Ω0 ⊂ Ω1 implies β(Ω0 ) ≤ β(Ω1 );

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nonsingular if β({a} ∪ Ω ) = β(Ω ) for every a ∈ E, Ω ∈ P(E); real if A = [0, +∞] with the natural ordering and β(Ω ) < +∞ for every bounded Ω . If A is a cone in a Banach space we say that the MNC β is regular if the equality β(Ω ) = 0 is equivalent to the relative compactness of Ω . In the following we call χ the Hausdorff MNC: χ (Ω ) = inf{ε > 0 : Ω has a finite ε-net} which is a MNC possessing all the properties described above. In the sequel we also use the following notions from the literature (see e.g. [7,9,14,15]). Let [a, b] be a closed interval of the real line and let K (E)[K v(E)] denote the collection of all nonempty compact [nonempty compact convex] subsets of E. A multifunction G : [a, b] → K (E) is strongly measurable if there exists a sequence (Gn )+∞ n=1 of step multifunctions such that lim H (Gn (t), G(t)) = 0

n→+∞

for a.e. t ∈ [a, b] (on the interval [a, b] we consider the Lebesgue measure and H is the Hausdorff metric on K (E)). Every strongly measurable multifunction G admits a strongly measurable selection, i.e. there exists a strongly measurable function g : [a, b] → E such that g(t) ∈ G(t), a.e. t ∈ [a, b]. Let the symbol L 1 ([a, b]; E) denote the space of all Bochner summable functions and, for simplicity of notation, we write L 1+ ([a, b]) instead of L 1 ([a, b]; IR + ). Moreover, we call C([a, b]; E) the set of all the continuous functions defined on [a, b] which take values in E. A multifunction G : [a, b] → K (E) is said to be: integrable if it has a summable selection g ∈ L 1 ([a, b]; E); integrably bounded if there exists a summable function ω ∈ L 1+ ([a, b]) such that kG(t)k := sup{kgk : g ∈ G(t)} ≤ ω(t),

a.e. t ∈ [a, b].

By the symbol SG1 ,[a,b] we denote the set of all summable selections of multifunction G on the interval [a, b]. 1 Finally, a countable set { f n }+∞ n=1 ⊂ L ([a, b]; E) is said to be semicompact if: (i) it is integrably bounded: k f n (t)k ≤ ω(t) for a.e. t ∈ [a, b] and every n ≥ 1, where ω ∈ L 1+ ([a, b]); (ii) the set { f n (t)}+∞ n=1 is relatively compact for a.e. t ∈ [a, b]. 3. Compactness of the set of all mild solutions for the problem (IP) Let [0, b], b > 0, be a fixed interval of the real line and let ∆ = {(t, s) ∈ [0, b] × [0, b] : 0 ≤ s ≤ t ≤ b}. We recall (see e.g. [27]) that a two-parameter family {T (t, s)}(t,s)∈∆ , T (t, s) : E → E a bounded linear operator, (t, s) ∈ ∆, is called an evolution system if the following conditions are satisfied: 1. T (s, s) = I, 0 ≤ s ≤ b; T (t, r )T (r, s) = T (t, s), 0 ≤ s ≤ r ≤ t ≤ b; 2. (t, s) 7→ T (t, s) is strongly continuous on ∆ (see e.g. [16]). For every evolution system, we can consider the respective evolution operator T : ∆ → L(E), where L(E) is the space of all bounded linear operators in E. Note that, the evolution operator T being strongly continuous on the compact set ∆, there exists a constant D = D∆ > 0 such that kT (t, s)kL(E) ≤ D,

(t, s) ∈ ∆.

(1)

Now, we take into consideration the impulsive Cauchy problem (IP). On the linear part of the inclusion of this problem we assume the following hypothesis: (A) {A(t)}t∈[0,b] is a family of linear not necessarily bounded operators (A(t) : D(A) ⊂ E → E, t ∈ [0, b], D(A) is a dense subset of E not depending on t) generating an evolution operator T : ∆ → L(E), i.e. there exists an

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evolution system {T (t, s)}(t,s)∈∆ such that, on the region D(A), each operator T (t, s) is strongly differentiable (see e.g. [16]) relative to t and s, while ∂ T (t, s) ∂ T (t, s) = A(t)T (t, s) and = −T (t, s)A(s), (t, s) ∈ ∆. ∂t ∂s Besides we require that the nonlinear multimap F : [0, b]×E → K v(E) satisfies the following set of assumptions: (F) (F1) for every x ∈ E, the multifunction F(·, x) : [0, b] → K v(E) admits a strongly measurable selector; (F2) for a.e. t ∈ [0, b], the multifunction F(t, ·) : E → K v(E) is u.s.c. on E; (F3) there exists a function α ∈ L 1+ ([0, b]) such that, for every x ∈ E, we have kF(t, x)k ≤ α(t)(1 + kxk),

a.e. t ∈ [0, b] ;

(F4) there exists a function k ∈ L 1+ ([0, b]) such that, for every bounded B ⊂ E, we have χ (F(t, B)) ≤ k(t)χ (B),

a.e. t ∈ [0, b]

where χ is the Hausdorff MNC (see Section 2). 1 Let us note that under condition (F) the set S F(·,q(·)),[0,b] is nonempty for every strongly measurable function q : [0, b] → E. From now on we use the notation J0 = [0, t1 ]; Jk =]tk , tk+1 ], k = 1, . . . , m (remember that tm+1 = b). For every k ∈ {0, . . . , m}, we consider the generalized Cauchy operator G k : L 1 (Jk ; E) → C(Jk ; E) defined, for every f ∈ L 1 (Jk ; E), by Z t T (t, s) f (s)ds, t ∈ Jk G k f (t) = (2) tk

which satisfies the properties (cf. [5], Theorem 2): (G1) there exists ζk ≥ 0 such that kG k f (t) − G k g(t)k ≤ ζk

Z

t

k f (s) − g(s)k ds,

t ∈ Jk

tk

for every f, g ∈ L 1 (Jk ; E); +∞ 1 (G2) for every compact K ⊂ E and sequence ( f n )+∞ n=1 , f n ∈ L (Jk ; E), such that { f n (t)}n=1 ⊂ K for a.e. t ∈ Jk , the weak convergence f n * f 0 implies the convergence G k f n → G k f 0 . In particular, let us note that we may assume ζk = D for every k = 0, . . . , m, where D is from (1). Furthermore, let us also observe that G k satisfies the following property too (cf. [15], Theorem 4.2.2): (G3) for every set { f n }+∞ n=1 integrably bounded such that χ ({ f n (t)}+∞ n=1 ) ≤ η(t),

a.e. t ∈ Jk

where η ∈ L 1+ (Jk ), the following estimate holds: Z t +∞ χ ({G k f n (t)}n=1 ) ≤ 2D η(s)ds, t ∈ Jk . tk

Moreover, since G k satisfies the following Lipschitz condition (weaker than (G1)): 0

(G1 ) kG k f − G k gkC(Jk ;E) ≤ Dk f − gk L 1 (Jk ;E) , f, g ∈ L 1 (Jk ; E), we have that (cf. [15], Theorem 5.1.1) G k also verifies the property: +∞ 1 (G4) for every semicompact set { f n }+∞ n=1 ⊂ L (Jk ; E), the set {G k f n }n=1 is relatively compact in C(Jk ; E); moreover, if f n * f 0 then G k f n → G k f 0 .

In order to define a mild solution of the impulsive Cauchy problem (IP), we introduce the set Λ = {x : [0, b] → E : x|Jk ∈ C(Jk , E), k = 0, . . . , m, and there exists x(tk+ ) ∈ E, k = 1, . . . , m}.

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It is easy to check that (Λ, k · kΛ ) is a Banach space, endowed with the norm kxkΛ = max{kxk kC(Jk ;E) , k = 0 · · · , m} where x0 = x|J0 and, for k ∈ {1, . . . , m}, xk is the function defined as xk (t) =

n

x(t), x(tk+ ),

t ∈ Jk t = tk .

Definition 1. A function x continuous in the closed interval J0 and in every interval Jk , k = 1, . . . , m, is a mild solution for the impulse Cauchy problem (IP) if Rt P 1 (i) x(t) = T (t, 0)a0 + 0
where every set Σx10 is compact in C(J1 ; E) (see again [5], Theorem 4). We define the multifunction H 1 : Σ 0 → K (C(J1 ; E)) by H 1 (x 0 ) = Σx10 ,

x 0 ∈ Σ 0.

Since H 1 is defined on a compact set and it has nonempty compact values, if we prove that it is u.s.c. on Σ 0 we can deduce that Σ|J1 = H 1 (Σ 0 ) is compact (see e.g. [15], Theorem 1.1.7). 0 0 If H 1 is not u.s.c. in x¯ 0 for some x¯ 0 ∈ Σ 0 , then there exist ε¯ > 0 and two sequences (xn0 )+∞ n=1 , x n → x¯ in C(J0 ; E), +∞ 1 1 1 and (xn )n=1 , xn ∈ Σx 0 , such that n

  xn1 6∈ B Σx¯10 , ε¯ ,

n ≥ 1.

(3)

Of course, for every n ≥ 1, xn1 ∈ Σx10 means that n

xn1 (t) = T (t, t1 )a1,n +

Z

t t1

T (t, s) f n1 (s)ds,

t ∈ J1

(4)

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1 where f n1 ∈ S F(·,x 1 (·)),J and 1

n

a1,n = xn0 (t1 ) + I1 (xn0 (t1 )).

(5)

We observe that the sequence (xn1 )+∞ n=1 is bounded in C(J1 ; E). In fact, with the same arguments as in the proof of Theorem 4 in [5], for every n ≥ 1 and for all t ∈ J1 we can write   Z t kxn1 (t)k ≤ D ka1,n k + kαk L 1 ([0,b]) + α(s)kxn1 (s)k ds , +

t1

where D is from (1) and α is the summable positive function of assumption (F3). By applying the Gronwall–Bellmann inequality, we have   Dkαk 1 L + ([0,b]) kxn1 (t)k ≤ D ka1,n k + kαk L 1 ([0,b]) e := N1,n , t ∈ J1 , n ≥ 1. +

(6)


+∞ Now, since xn0 (t1 ) n=1 is a converging sequence and the function I1 is continuous, we have that there exists M1 > 0 such that ka1,n k = kxn0 (t1 ) + I1 (xn0 (t1 ))k ≤ M1 for every n ≥ 1. Hence (6) yields   Dkαk 1 L + ([0,b]) kxn1 (t)k ≤ D M1 + kαk L 1 ([0,b]) e (7) := K 1 , t ∈ J1 , n ≥ 1. +

{ f n1 (t)}+∞ n=1

Moreover, the set is relatively compact in E for a.e. t ∈ J1 . In fact, by using condition (F4), for a.e. t ∈ J1 and a.e. s ∈ [t1 , t], we have       +∞ +∞ 1 1 χ { f n1 (s)}+∞ (8) n=1 ≤ χ F(s, {x n (s)}n=1 ) ≤ k(s)χ {x n (s)}n=1 . If we put γ1 as the (non-regular) real MNC defined by γ1 (Ω ) = sup e−L 1 t χ (Ω (t)),

Ω ⊂ C(J1 ; E),

(9)

t∈J1

where Ω (t) = {x(t) : x ∈ Ω }, t ∈ J1 , and L 1 is a positive constant chosen so that Z t q1 := 2D sup e−L 1 (t−s) k(s)ds < 1

(10)

t∈J1 t1

(here D is the constant of condition (1) and k is the summable function of assumption (F4)), then for (8) we have the estimate       +∞ L1s −L 1 ξ 1 L1s 1 +∞ ≤ e χ { f n1 (s)}+∞ e k(s) sup χ {x (ξ )} = e k(s)γ {x } 1 n n n=1 . n=1 n=1 ξ ∈J1

So, since the generalized Cauchy operator G 1 satisfies (G3), we obtain    Z t 1 +∞ χ {G 1 f n1 (t)}+∞ ≤ 2Dγ {x } e L 1 s k(s)ds. 1 n n=1 n=1 t1

Therefore, from (11) and (10), we get       +∞ 1 +∞ −L 1 t 1 γ1 {xn1 }+∞ e = γ {G f } = sup χ {G f (t)} 1 1 1 n n n=1 n=1 n=1 t∈J1

Z t     1 +∞ e−L 1 (t−s) k(s)ds = γ1 {xn1 }+∞ ≤ 2Dγ1 {xn }n=1 sup n=1 q1 . t∈J1 t1

By using (10) again, we have   γ1 {xn1 }+∞ n=1 = 0

(11)

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and so, bearing in mind (9), we can write   χ {xn1 (t)}+∞ t ∈ J1 . n=1 = 0, From the last equality, by proceeding in the same way as to obtain (8), we have   χ { f n1 (t)}+∞ a.e. t ∈ J1 , n=1 = 0, i.e. the set { f n1 (t)}+∞ n=1 is relatively compact for a.e. t ∈ J1 . Then, by applying now (F3) and (7), we get k f n1 (t)k ≤ kF(t, xn1 (t))k ≤ α(t)(1 + kxn1 (t)k) ≤ α(t)(1 + K 1 ),

a.e. t ∈ J1 , n ≥ 1.

1 +∞ Therefore, the set { f n1 }+∞ n=1 is integrably bounded and we can conclude that the set { f n }n=1 is semicompact. Now, by means of the definition of G 1 (see (2)), for every n ≥ 1 the function xn1 given in (4) may be rewritten as

xn1 (t) = T (t, t1 )a1,n + G 1 f n1 (t), Since the sequence

(xn0 (t1 ))+∞ n=1

t ∈ J1 .

converges to x¯ 0 (t1 ) and the function I1 is continuous, from (5) we have

lim a1,n = x¯ 0 (t1 ) + I1 (x¯ 0 (t1 )).

(12)

n→+∞

Then, taking into account that the generalized Cauchy operator G 1 satisfies condition (G4), we can conclude that the set {xn1 }+∞ n=1 is relatively compact in C(J1 ; E). Therefore, w.l.o.g. we can assume that there exists x¯ 1 ∈ C(J1 ; E) such that xn1 → x¯ 1 in C(J1 ; E). Now, we prove that x¯ 1 ∈ H 1 (x¯ 0 ). From (4) and (5), for all n ≥ 1, we have Z t (13) xn1 (t) = T (t, t1 )[xn0 (t1 ) + I1 (xn0 (t1 ))] + T (t, s) f n1 (s)ds, t ∈ J1 t1

1 where f n1 ∈ S F(·,x 1 (·)),J . 1

n

Since we have already proved that the set { f n1 }+∞ n=1 is semicompact, we also have that it is weakly compact in 1 L (J1 ; E) (cf. [15], Proposition 4.2.1). Hence, w.l.o.g. we may assume that there exists f¯1 ∈ L 1 (J1 ; E) such that f n1 * f¯1 in L 1 (J1 ; E). Now, by using Lemma 5.1.1 in [15], we have f¯1 (t) ∈ F(t, x¯ 1 (t)),

a.e. t ∈ J1

and so we can say that 1 . f¯1 ∈ S F(·, x¯ 1 (·)),J 1

Since G 1 verifies property (G4), by considering the limit as n → +∞ in both sides of (13), by using (12) we get Z t x¯ 1 (t) = T (t, t1 )[x¯ 0 (t1 ) + I1 (x¯ 0 (t1 ))] + T (t, s) f¯1 (s)ds, t ∈ J1 t1

with

f¯1

∈

1 S F(·, , x¯ 1 (·)),J1

that is

x¯ 1 ∈ Σx¯10 = H 1 (x¯ 0 ). The fact that xn1 → x¯ 1 ∈ H 1 (x¯ 0 ) leads to a contradiction with (3). Therefore multimap H 1 is u.s.c. on Σ 0 . Now, with the same arguments as above, we can claim that every set Σ|Jk is compact in C(Jk ; E) for every k = 2, . . . , m. So, the set Σ is compact in Λ. 

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4. Existence of mild solutions on non-compact domains The main result of this section is the extension of the existence theorem provided by the authors in [6]; more precisely, we prove that the impulsive Cauchy problem (IP)∞ admits mild solutions on the whole interval [0, +∞[. Let us consider the set ∆∞ = {(t, s) ∈ IR0+ × IR0+ : 0 ≤ s ≤ t} and the evolution system {T (t, s)}(t,s)∈∆∞ , T (t, s) : E → E a bounded linear operator, (t, s) ∈ ∆∞ , such that the following conditions hold: 1’. T (s, s) = I, s ≥ 0; T (t, r )T (r, s) = T (t, s), 0 ≤ s ≤ r ≤ t; 2’. (t, s) 7→ T (t, s) is strongly continuous on ∆∞ . In this framework, for every natural number k ≥ 1 there exists a constant Dk = D∆k > 0 such that kT (t, s)kL(E) ≤ Dk ,

(t, s) ∈ ∆k = {(t, s) ∈ IR0+ × IR0+ : 0 ≤ s ≤ t ≤ tk+1 }.

Here we require the following assumption on the linear part of the inclusion: (A)∞ {A(t)}t∈[0,+∞[ is a family of linear not necessarily bounded operators (A(t) : D(A) ⊂ E → E, t ∈ [0, +∞[, D(A) is a dense subset of E not depending on t) generating an evolution operator T : ∆∞ → L(E). Let us denote by L 1loc ([0, +∞[) the set of all Bochner summable functions on the compact subsets of [0, +∞[ and 1,loc by S F(·,x(·)),[0,+∞[ the set of all L 1loc ([0, +∞[)-selections of the multifunction F(·, x(·)). Next, for the multivalued nonlinearity F : [0, +∞[×E → K v(E) we suppose that it satisfies the following set of hypotheses: (F)∞ (F1)∞ for every x ∈ E, the multifunction F(·, x) : [0, +∞[→ K v(E) admits a strongly measurable selector; (F2)∞ for a.e. t ∈ [0, +∞[, the multifunction F(t, ·) : E → K v(E) is u.s.c. on E; (F3)∞ there exists a function α ∈ L 1loc ([0, +∞[) such that, for every x ∈ E, we have kF(t, x)k ≤ α(t)(1 + kxk),

a.e. t ∈ [0, +∞[;

(F4)∞ there exists a function k ∈ L 1loc ([0, +∞[) such that, for every bounded B ⊂ E, we have χ (F(t, B)) ≤ k(t)χ (B),

a.e. t ∈ [0, +∞[.

Definition 2. A function x : [0, +∞[→ E continuous in the closed interval J0 and in every interval Jk , k ≥ 1, is a mild solution for the impulse Cauchy problem (IP)∞ if Rt P 1,loc (i) x(t) = T (t, 0)a0 + 0
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T. Cardinali, P. Rubbioni / Nonlinear Analysis 69 (2008) 73–84 1 where f 0 ∈ S F(·,z . 0 (·)),J0


+∞ Now, we consider the sequence ykn |J0 ∪J1 n=1 of mild solutions for the impulsive Cauchy problem (IP)1 . Since the set of all mild solutions of (IP)1 is compact (cf. Theorem 1), there exists a subsequence (ykn h |J0 ∪J1 )+∞ h=1 converging in the Banach space Λ|J0 ∪J1 to a mild solution of problem (IP)1 , Z t T (t, s) f 1 (s)ds, t ∈ J0 ∪ J1 , (15) z 1 (t) = T (t, 0)a0 + T (t, t1 )I1 (z 1 (t1 )) + 0 1 S F(·,z . 1 (·)),J0 ∪J1

where f 1 ∈ By iterating this process we get, for every natural number k ≥ 0, a mild solution for problem (IP)k : Z t X T (t, s) f k (s)ds, t ∈ J0 ∪ · · · ∪ Jk , z k (t) = T (t, 0)a0 + T (t, t j )I j (z k (t j )) + 0

0
1 where f k ∈ S F(·,z . k (·)),J0 ∪···∪Jk Moreover, we note that each function z k , k ≥ 1, has the property

z k|J0 ∪···∪Jk−1 = z k−1 .

(16)

Now, let z : [0, +∞[→ E be the function defined by z(t) =

+∞ X

X Jk (t)z k (t),

(17)

t ∈ [0, +∞[,

k=0

where X Jk is the characteristic function of interval Jk , k ≥ 0. We prove that z is a mild solution for the impulsive Cauchy problem (IP)∞ (cf. Definition 2). To this end, we consider the locally summable function f : [0, +∞[→ E defined as f (t) =

+∞ X

X Jk (t) f k (t),

(18)

t ∈ [0, +∞[.

k=0

Let t ∗ ∈ [0, +∞[ be fixed. Then, there exists a unique natural number k such that t ∗ ∈ Jk . If k = 0, from (17), (14) and (18), we have Z t∗ z(t ∗ ) = z 0 (t ∗ ) = T (t ∗ , 0)a0 + T (t ∗ , s) f (s)ds 0 1 where, again from (18) and (14), f |J0 = f 0 ∈ S F(·,z(·)),J . 0 If k = 1, by using (17), (15) and (18), we get

z(t ∗ ) = z 1 (t ∗ ) = T (t ∗ , 0)a0 + T (t ∗ , t1 )I1 (z(t1 )) + T (t ∗ , t1 )

t1

Z

T (t1 , s) f 1 (s)ds +

0

Now, we observe that by (15), (16) and (14) the following identity holds: Z t1 Z t1 T (t1 , s) f 1 (s)ds = T (t1 , s) f 0 (s)ds, 0

0

so, taking into account (18) and (19), this can be rewritten as Z t∗ ∗ ∗ ∗ z(t ) = T (t , 0)a0 + T (t , t1 )I1 (z(t1 )) + T (t ∗ , s) f (s)ds, 0 1 S F(·,z(·)),J . 0 ∪J1

where f |J0 ∪J1 ∈ If k = 2, first of all we note that the following two identities hold:

Z

t∗

T (t ∗ , s) f (s)ds.

(19)

t1

(20)

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1. from the evaluation of the mild solution z 2 in t1 , by means of (16) and (20), we have Z t1 Z t1 T (t1 , s) f 0 (s)ds; T (t1 , s) f 2 (s)ds = 0

0

2. from the evaluation of the mild solution z 2 in t2 and (16), we can write Z Z t2 T (t2 , s) f 2 (s)ds = z 1 (t2 ) − T (t2 , 0)a0 − T (t2 , t1 )I1 (z 1 (t1 )) −

t1

T (t2 , s) f 2 (s)ds.

0

t1

By virtue of the evaluation of z 2 in t1 and (16), the last term of this equality can be written as Z t1 Z t1 Z t1 T (t1 , s) f 1 (s)ds. T (t1 , s) f 2 (s)ds = T (t2 , t1 ) T (t2 , s) f 2 (s)ds = T (t2 , t1 ) 0

0

0

Then, from the value z 1 (t2 ), we achieve the identity Z t2 Z t2 T (t2 , s) f 2 (s)ds = T (t2 , s) f 1 (s)ds. t1

t1

By proceeding as in the case k = 1, taking into account the identities above and of (18), we can conclude that Z t1 z(t ∗ ) = T (t ∗ , 0)a0 + T (t ∗ , t1 )I1 (z(t1 )) + T (t ∗ , t2 )I2 (z(t2 )) + T (t ∗ , t1 ) T (t1 , s) f 2 (s)ds 0

+ T (t ∗ , t2 )

Z

t2

t1

T (t2 , s) f 2 (s)ds +

Z

t∗

T (t ∗ , s) f 2 (s)ds

t2

= T (t ∗ , 0)a0 + T (t ∗ , t1 )I1 (z(t1 )) + T (t ∗ , t2 )I2 (z(t2 )) +

t∗

Z

T (t ∗ , s) f (s)ds,

0 1 where f |J0 ∪J1 ∪J2 ∈ S F(·,z(·)),J . 0 ∪J1 ∪J2 The iteration of this process leads to claim that z satisfies condition (i) of Definition 2. Of course (ii) and (iii) are verified. 

Finally, we consider the impulsive Cauchy problem defined on a non-closed interval [0, b[, 0 < b < +∞:  0 x (t) ∈ A(t)x(t) + F(t, x(t)), a.e. t ∈ [0, b[, t 6= tk , k ≥ 1 (IP)∗ x(tk+ ) = x(tk ) + Ik (x(tk )), k≥1  x(0) = a0 ∈ E. Here the jump points are an increasing sequence (tk )+∞ k=0 such that t0 = 0 and limk→+∞ tk = b. Definition 3. A function x : [0, b[→ E continuous in the closed interval J0 and in every interval Jk , k ≥ 1, is a mild solution for the impulse Cauchy problem (IP)∗ if Rt P 1,loc (i) x(t) = T (t, 0)a0 + 0
T. Cardinali, P. Rubbioni / Nonlinear Analysis 69 (2008) 73–84

83

5. Example We consider the initial–boundary value problem  ∂x kb   (t, z) − ∆x(t, z) = α(t, z)x(t, z), a.e. on [0, b] × G, t 6= , k = 1, . . . , m   ∂t m   x(t, z) = 0, ! on [0, b] × ∂G    + kb kb   ,z = x , z + 1, k = 1, . . . , m x   m m    x(0, z) = a(z), on G.

(21)

Here G is a bounded open interval in IR and a ∈ W0 1,2 (G), where W0 1,2 (G) denotes the closure of C0∞ (G) in the Sobolev space W 1,2 (G). Moreover, the function α : [0, b] × G → IR verifies the following properties: (α1) for every t ∈ [0, b], α(t, ·) ∈ C01 (G); (α2) the function αˆ : [0, b] → W0 1,2 (G), defined as α(t) ˆ = α(t, ·),

t ∈ [0, b],

is strongly measurable. Let E = W0 1,2 (G). Let us consider the operator AF : D(AF ) ⊆ E → E which is the Friedrichs extension of −∆, with D(−∆) = C0∞ (G) (see [34], Section 19.9). Then the operator A = −AF is the infinitesimal generator of a C0 -semigroup (see [34], Example 19.42). Therefore, we can write the problem in the following abstract form:  0 x (t) ∈ Ax(t) + F(t, x(t)), a.e. on [0, b], t 6= tk , k = 1, . . . , m x(t + ) = x(tk ) + Ik (x(tk )), k = 1, . . . , m  k x(0) = a where F : [0, b] × E → K v(E) is defined by F(t, x) = {α(t)x} ˆ and, for every k = 1, . . . , m, tk = kb m and Ik ≡ 1. 1,2 Hence, by applying Theorem 1 the set of all mild solutions x : [0, b] → W0 (G) for the problem (21) is nonempty and compact. Acknowledgment The authors wish to express their gratitude to the referee for his/her very constructive remarks. References [1] D.D. Bainov, V. Covachev, Impulsive Differential Equations with a Small Parameter, in: Series on Advances in Math. for Applied Sciences, vol. 24, World Scientific, 1994. [2] D.D. Bainov, V. Lakshmikantham, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [3] G. Ballinger, X. Liu, Boundness for impulsive delay differential equations and applications to population growth models, Nonlinear Anal. 53 (2003) 1041–1062. [4] M. Benchohra, J. Henderson, S.K. Ntouyas, Existence results for first order impulsive semilinear evolution inclusions, Electron. J. Qual. Theory Differ. Equ. 1 (2001) 12 (electronic only). [5] T. Cardinali, P. Rubbioni, On the existence of mild solutions of semilinear evolution differential inclusions, J. Math. Anal. Appl. 308 (2005) 620–635. [6] T. Cardinali, P. Rubbioni, Mild solutions for impulsive semilinear evolution differential inclusions, J. Appl. Funct. Anal. 1 (3) (2006) 303–325. [7] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, in: Lect. Notes in Math., vol. 580, Springer-Verlag, Berlin, Heidelberg, New York, 1977. [8] Y.K. Chang, W.T. Li, J.J. Nieto, Controllability of evolution differential inclusions in Banach spaces, Nonlinear Anal. 67 (2) (2007) 623–632. [9] K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992. [10] A. D’Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett. 18 (2005) 729–732. [11] D. Franco, J.J. Nieto, First-order impulsive ordinary differential equations with anti-periodic and nonlinear boundary conditions, Nonlinear Anal. 42 (2000) 163–173. [12] S. Gao, L. Chen, J.J. Nieto, A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine 24 (2006) 6037–6045.

84

T. Cardinali, P. Rubbioni / Nonlinear Analysis 69 (2008) 73–84

[13] M. Guo, X. Xue, R. Li, Controllability of impulsive evolution inclusions with nonlocal conditions, J. Optim. Theory Appl. 120 (2004) 355–374. [14] S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Acad. Publ., Dordrecht, Boston, London, 1997. [15] M. Kamenskii, V. Obukhovskii, P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, in: De Gruyter Ser. Nonlinear Anal. Appl., vol. 7, Walter de Gruyter, Berlin, New York, 2001. [16] S.G. Krein, Linear Differential Equations in Banach Spaces, Amer. Math. Soc., Providence, 1971. [17] G. Jiang, Q. Lu, Impulsive state feedback control of a predator–prey model, J. Comput. Appl. Math. 200 (2007) 193–207. [18] J. Li, J.J. Nieto, J. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl. 325 (2007) 226–236. [19] W. Li, H. Huo, Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics, J. Comput. Appl. Math. 174 (2005) 227–238. [20] B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal. 60 (2005) 1533–1552. [21] J.H. Liu, Nonlinear impulsive evolution equations, Dyn. Contin. Discrete Impuls. Syst. 6 (1) (1999) 77–85. [22] X.Y. Lou, B.T. Cui, Global asymptotic stability of delay BAM neural networks with impulses, Chaos Solitons Fractals 29 (2006) 1023–1031. [23] V.D. Mil’man, A.D. Myshkis, On the solvability of motion in the presence of impulses, Siberian Math. J. 1 (2) (1960) 233–237. [24] J.J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Anal. 51 (2002) 1223–1232. [25] J.J. Nieto, R. Rodriguez-Lopez, Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. Math. Anal. Appl. 318 (2006) 593–610. [26] J.J. Nieto, R. Rodriguez-Lopez, New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl. 328 (2) (2007) 1343–1368. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. [28] D. Qian, X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl. 303 (2005) 288–303. [29] S.H. Saker, Oscillation and global attractivity of impulsive periodic delay respiratory dynamics model, Chinese Ann. Math. Ser. B 26 (2005) 511–522. [30] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995, Visca Skola, Kiev, 1987 (in Russian). [31] A.N. Sesekin, S.T. Zavalishchin, Dynamic Impulse Systems. Theory and Applications, in: Mathematics and its Applications, vol. 394, Kluwer Academic Publishers Group, Dordrecht, 1997. [32] S. Tang, L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol. 44 (2002) 185–199. [33] J. Yan, A. Zhao, J.J. Nieto, Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka–Volterra systems, Math. Comput. Modelling 40 (2004) 509–518. [34] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer-Verlag, New York, Berlin, Heidelberg, 1990. [35] W. Zhang, M. Fan, Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays, Math. Comput. Modelling 39 (2004) 479–493. [36] X. Zhang, Z. Shuai, K. Wang, Optimal impulsive harvesting policy for single population, Nonlinear Anal. RWA 4 (2003) 639–651.