First-order periodic impulsive semilinear differential inclusions: Existence and structure of solution sets

Mathematical and Computer Modelling 52 (2010) 683–714

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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm

First-order periodic impulsive semilinear differential inclusions: Existence and structure of solution sets Smaïl Djebali a , Lech Górniewicz b,c,∗ , Abdelghani Ouahab d a

Laboratory of EDP&HM, Department of Mathematics, E.N.S., P.B. 92, 16050 Kouba. Algiers, Algeria

b

Institute of Mathematics, Kazimierz Wielki University, Bydgoszcz, Poland

c

Schauder Center for Nonlinear Studies, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

d

Department of Mathematics, Sidi-Bel-Abbès University, P.B. 89, 22000. Sidi-Bel-Abbès, Algeria

article

info

Article history: Received 12 January 2010 Received in revised form 24 April 2010 Accepted 26 April 2010 Keywords: Impulsive semilinear differential inclusions Periodic solutions Filippov’s theorem Poincaré operator Contractibility Acyclicity

abstract In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions:

 0 (y − Ay)(t ) ∈ F (t , y(t )), y(t + ) − y(t − ) = Ik (y(tk− )), y(0k) = y(b)k

a.e. t ∈ J \ {t1 , . . . , tm }, k = 1, . . . , m,

where J = [0, b] and 0 = t0 < t1 < · · · < tm < tm+1 = b (m ∈ N∗ ) A is the infinitesimal generator of a C0 -semigroup T on a separable Banach space E and F is a multi-valued map. The functions Ik characterize the jump of the solutions at impulse points tk (k = 1, . . . , m). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of T (b). Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The dynamics of many processes in physics, population dynamics, biology, medicine may be subject to abrupt changes such that shocks, perturbations (see for instance [1,2] and the references therein). These perturbations may be seen as impulses. For instance, in the periodic treatment of some diseases, impulses correspond to the administration of a drug treatment or a missing product. In environmental sciences, impulses correspond to seasonal changes of the water level of artificial reservoirs. Their models may be described by impulsive differential equations. The mathematical study of boundary value problems for differential equations with impulses were considered in 1960 by Milman and Myshkis [3] and then followed by a period of active research which culminated in 1968 with the monograph by Halanay and Wexler [4]. So far, various mathematical results (existence, asymptotic behavior, etc.) have been obtained (see [5–10] and the references therein).

∗

Corresponding author at: Institute of Mathematics, Kazimierz Wielki University, Bydgoszcz, Poland. Tel.: +48 56 4775520. E-mail addresses: [email protected] (S. Djebali), [email protected] (L. Górniewicz), [email protected] (A. Ouahab).

0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.04.016

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Given a real separable Banach space E with norm | · |, consider the following problem

 0 (y − Ay)(t ) ∈ F (t , y(t )), ∆y tk = Ik (y(tk− )), y(0t)== y(b),

a.e. t ∈ J \ {t1 , . . . , tm }, k = 1, . . . , m,

(1.1)

where 0 = t0 < t1 < · · · < tm < tm+1 = b (m ∈ N∗ ). The nonlinearity F : J × E → P (E ) is a multi-valued map, A is the infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 , Ik ∈ C (E , E ) (k = 1, . . . , m), ∆y|t =tk = y(tk+ ) − y(tk− ), y(tk+ ) = limh→0+ y(tk + h) and y(tk− ) = limh→0+ y(tk − h) stand for the right and the left limits of y(t ) at t = tk , respectively. In case the space E is finite dimensional, some existence results of mild solutions for Problem (1.1) in the particular case Ay = λy (λ ∈ R) have been recently obtained in [11]. Our goal in this work is to complement and extend some of these results to the case of infinite-dimensional spaces; moreover the right-hand side multi-valued nonlinearity may be either convex or nonconvex. Some auxiliary results from multi-valued analysis, semigroup theory, . . . are gathered together in Section 2. In the first part of this work, we assume that 1 ∈ ρ(T (b)) and then prove some existence results based on the nonlinear alternative of Leray–Schauder type (in the convex case), on the Bressan–Colombo–Fryszkowski selection theorem and on the Covitz and Nadler fixed point theorem for contraction multi-valued maps in a generalized metric space (in the nonconvex case). Some topological ingredients including some notions of measure of noncompactness are recalled and employed to prove closedness and compactness of the solution set; some geometric properties are also provided. This is the content of Section 3. We will also discuss the question of dependence of F on parameters and present a Filippov’s theorem for Problem (1.1) in Sections 4 and 5 respectively. Making using of the Poincaré operator, we finally prove in Section 6 some existence results for Problem (1.1) in case 1 6∈ ρ(T (b)). A new nonlinear alternative is proved and employed and some ingredients from algebraic topology are used. We end the paper with some concluding remarks and a rich bibliography. 2. Preliminaries In this section, we recall from the literature some notations, definitions, and auxiliary results which will be used throughout this paper. Let (E , | · |) be a separable Banach space, J = [0, b] an interval in R and C (J , E ) the Banach space of all continuous functions from J into E with the norm

kyk∞ = sup{|y(t )|: 0 ≤ t ≤ b}. B(E ) refers to the Banach space of linear bounded operators from E into E with norm

kN kB(E ) = sup{|N (y)|: |y| = 1}. A function y: J → E is called measurable provided for every open subset U ⊂ E, the set y−1 (U ) = {t ∈ J: y(t ) ∈ U } is Lebesgue measurable. A measurable function y: J → E is Bochner integrable if |y| is Lebesgue integrable. For properties of the Bochner integral, we refer to [12]. In what follows, L1 (J , E ) denotes the Banach space of functions y: J −→ E which are Bochner integrable with norm b

Z

|y(t )|dt .

kyk1 = 0

Denote by P (E ) = {Y ⊂ E: Y 6= ∅}, Pcl (E ) = {Y ∈ P (E ): Y closed}, Pb (E ) = {Y ∈ P (E ): Y bounded}, Pc v (E ) = {Y ∈ P (E ): Y convex}, and Pcp (E ) = {Y ∈ P (E ): Y compact}. 2.1. Multi-valued analysis Let (X , d) and (Y , ρ) be two metric spaces and G : X → Pcl (Y ) be a multi-valued map. A single-valued map g : X → Y is said to be a selection of G and we write g ⊂ G whenever g (x) ∈ G(x) for every x ∈ X . G is called upper semi-continuous (u.s.c. for short) on X if for each x0 ∈ X , the set G(x0 ) is a nonempty, closed subset of X , and for each y0 ∈ G(x0 ) and for each open neighborhood N of y0 containing G(x0 ), there exists an open subset M of x0 such that G(M ) ⊆ N. That is if the set G−1 (V ) = {x ∈ X , G(x) ⊂ V } is open for any open set V in Y . Equivalently, F is u.s.c. if the −1 set G+ (V ) = {x ∈ X , G(x) ∩ V 6= ∅} is closed for any closed subset V in Y . G is called lower semi-continuous (u.s.c. for short) on X if for each x0 ∈ X , the set G(x0 ) is a nonempty, closed subset of X , and for each y0 ∈ G(x0 ) and for each open neighborhood N of y0 such that N ∩ G(x0 ) 6= ∅, there exists an open neighborhood 1 M of x0 such that N ∩ G(M ) 6= ∅. That is, if the set G− + (V ) = {x ∈ X , G(x) ∩ V 6= ∅} is open for any open set V in Y . −1 Equivalently, F is u.s.c. if the set G (V ) = {x ∈ X , G(x) ⊂ V } is closed for any closed set V in Y . The following two results are easily deduced from the limit properties. Lemma 2.1 (See e.g. [13], Theorem 1.4.13). If G : X −→ Pcp is u.s.c., then for any x0 ∈ X , lim sup G(x) = G(x0 ). x→ x0

Lemma 2.2 (See e.g. [13], Lemma 1.1.9). Let (Kn )n∈N ⊂ K ⊂ X be a sequence of subsets where K is compact in the separable Banach space X . Then

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685

! co (lim sup Kn ) = n→∞

\

[

co

N >0

Kn

,

n≥N

where co A refers to the closure of the convex hull of A. G is said to be completely continuous if it is u.s.c. and, for every bounded subset A ⊆ X , G(A) is relatively compact, i.e. there S exists a relatively compact set K = K (A) ⊂ X such that G(A) = {G(x), x ∈ A} ⊂ K .G is compact if G(X ) is relatively compact. It is called locally compact if, for each x ∈ X , there exists U ∈ V (x) such that G(U ) is relatively compact. G is quasicompact if, for each subset A ⊂ X , G(A) is relatively compact. 2.1.1. Measurable selections Definition 2.1. A multi-valued map F : J = [0, b] → Pcl (Y ) is said measurable provided for every open U ⊂ Y , the set F +1 (U ) is Lebesgue measurable. We have Lemma 2.3 ([14,15]). The mapping F is measurable if and only if for each x ∈ Y , the function ζ : J → [0, +∞) defined by

ζ (t ) = dist(x, F (t )) = inf{kx − yk : y ∈ F (t )},

t ∈ J,

is Lebesgue measurable. The following two lemmas are needed in this paper. The first one is the celebrated Kuratowski–Ryll-Nardzewski selection theorem. Lemma 2.4 ([15, Theorem 19.7]). Let Y be a separable metric space and F : [a, b] → P (Y ) a measurable multi-valued map with nonempty closed values. Then F has a measurable selection. Lemma 2.5 ([16, Lemma 3.2]). Let F : [0, b] → P (Y ) be a measurable multi-valued map and u : [a, b] → Y a measurable function. Then for any measurable v : [a, b] → (0, +∞), there exists a measurable selection fv of F such that for a.e. t ∈ [a, b],

|u(t ) − fv (t )| ≤ d(u(t ), F (t )) + v(t ). Corollary 2.1. Let F : [0, b] → Pcp (Y ) be a measurable multi-valued map and u : [0, b] → E a measurable function. Then there exists a measurable selection f of F such that for a.e. t ∈ [0, b],

|u(t ) − f (t )| ≤ d(u(t ), F (t )). Proof. Taking v(t ) = vn (t ) =

1 n

in Lemma 2.5, we get a measurable selection fn of F such that

|u(t ) − fn (t )| ≤ d(u(t ), F (t )) + 1/n. Using the fact that F has compact values, we may pass to a subsequence if necessary to get that (fn (·)) converges to a measurable function f , yielding our claim.  2.1.2. Closed graphs We denote the graph of G to be the set Gr (G) = {(x, y) ∈ X × Y , y ∈ G(x)}. Definition 2.2. G is closed if Gr (G) is a closed subset of X × Y , i.e. for every sequences (xn )n∈N ⊂ X and (yn )n∈N ⊂ Y , if xn → x∗ , yn → y∗ as n → ∞ with yn ∈ F (xn ), then y∗ ∈ G(x∗ ). We recall the following two results; the first one is classical. Lemma 2.6 ([17, Proposition 1.2]). If G : X → Pcl (Y ) is u.s.c., then Gr (G) is a closed subset of X × Y . Conversely, if G is locally compact and has nonempty compact values and a closed graph, then it is u.s.c. Lemma 2.7. If G : X → Pcp (Y ) is quasicompact and has a closed graph, then G is u.s.c. Proof. Assume that G is not u.s.c. at some point x. Then there exists an open neighborhood U of G(x) in Y , a sequence (xn ) which converges to x, and for every l ∈ N there exists nl ∈ N such that G(xnl ) 6⊂ U. Then for each l = 1, 2, . . . , there are ynl such that ynl ∈ G(xnl ) and ynl 6∈ U ; this implies that ynl ∈ Y \ U. Moreover {ynl : l ∈ N} ⊂ G({xn : n ≥ 1}). Since G is compact, there exists a subsequence of {ynl : l ∈ N} which converges to y. G closed implies that y ∈ G(x) ⊂ U ; but this is a contradiction to the assumption that ynl 6∈ U for each nl .  Lemma 2.8 ([18]). Given a Banach space X , let F : [a, b] × X −→ Pcp,c v (X ) be an L1 -Carathéodory multi-valued map such that for each y ∈ C ([a, b], X ), SF ,y 6= ∅ and let Γ be a linear continuous mapping from L1 ([a, b], X ) into C ([a, b], X ). Then the

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operator

Γ ◦ SF : C ([a, b], X ) −→ Pcp,c v (C ([a, b], X )), y 7−→ (Γ ◦ SF )(y) := Γ (SF ,y ) has a closed graph in C ([a, b], X ) × C ([a, b], X ). Given a separable Banach space (E , | · |) and a multi-valued map F : J × E → P (E ), denote

kF (t , x)kP := sup{|v| : v ∈ F (t , x)}. Definition 2.3. F is said (a) integrable if it has a summable selection f ∈ L1 (J , E ), (b) integrably bounded, if there exists q ∈ L1 (J , R+ ) such that

kF (t , z )kP ≤ q(t ), for a.e. t ∈ J and every z ∈ E . Definition 2.4. A multi-valued map F is called a Carathéodory function if (a) the function t 7→ F (t , x) is measurable for each x ∈ E ; (b) for a.e. t ∈ J, the map x 7→ F (t , x) is upper semi-continuous. Furthermore, F is L1 -Carathéodory if it is locally integrably bounded, i.e., for each positive r, there exists hr ∈ L1 (J , R+ ) such that

kF (t , x)kP ≤ hr (t ),

for a.e. t ∈ J and all |x| ≤ r .

For each x ∈ C (J , E ), the set SF ,x = f ∈ L1 (J , E ) : f (t ) ∈ F (t , x(t )) for a.e. t ∈ [0, b]





is known as the set of selection functions. Remark 2.1. (a) For each x ∈ C (J , E ), the set SF ,x is closed whenever F has closed values. It is convex if and only if F (t , x(t )) is convex for a.e. t ∈ J . (b) From [19], Theorem 5.10 (see also [18] when E is finite dimensional), we know that SF ,x is nonempty if and only if the mapping t 7→ inf{kvk : v ∈ F (t , x(t ))} belongs to L1 (J ). It is bounded if and only if the mapping t 7→ kF (t , x(t ))kP = sup{kvk: v ∈ F (t , x(t ))} belongs to L1 (J ); this particularly holds true when F is L1 -Carathéodory. For the sake of completeness, we also refer to Theorem 1.3.5 in [20] which states that SF ,x contains a measurable selection whenever x is measurable and F is a Carathéodory function. For further readings and details on multi-valued analysis, we refer to the books by Andres and Górniewicz [21], Aubin and Celina [22], Aubin and Frankowska [13], Deimling [17], Górniewicz [15], Hu and Papageorgiou [23,24], Kamenskii et al. [20], and Tolstonogov [25]. 2.2. Semi-compactness in L1 ([0, b], E ) Definition 2.5. A sequence (vn )n∈N ⊂ L1 ([0, b], E ) is said to be semi-compact if (a) it is integrably bounded, i.e. there exists q ∈ L1 (J , R+ ) such that

|vn (t )|E ≤ q(t ),

for a.e. t ∈ J and every n ∈ N,

(b) the image sequence (vn (t ))n∈N is relatively compact in E for a.e. t ∈ [0, b]. We recall two fundamental results. The first one follows from the Dunford–Pettis theorem (see [20, Proposition 4.2.1]). In the case dim E < ∞, a proof may be found in [26], Proposition 3.6. This result is of particular importance if E is reflexive in which case (a) implies (b) in Definition 2.5. Lemma 2.9. Every semi-compact sequence L1 ([0, b], E ) is weakly compact in L1 ([0, b], E ). The second one is due to Mazur, 1933: Lemma 2.10 (Mazur’s Lemma, [27], Theorem 21.4). Let E be a normed space {xk }k∈N ⊂ E be a sequence weakly converging Pand m to a limit x ∈ E. Then there exists a sequence of convex combinations ym = k=1 αmk xk with αmk > 0 for k = 1, 2, . . . , m and Pm k=1 αmk = 1, which converges strongly to x. 2.3. C0 -Semigroups Definition 2.6. A semigroup is a one-parameter family {T (t ): t ≥ 0} ⊂ B(E ) satisfying the conditions: (a) T (t ) ◦ T (s) = T (t + s), for t , s ≥ 0, (b) T (0) = I where I denotes the identity operator in E.

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Definition 2.7. A semigroup T (t ) is uniformly continuous if lim kT (t ) − I kB(E ) = 0

t → 0+

that is if lim kT (t ) − T (s)kB(E ) = 0.

|t −s|→0

Definition 2.8. We say that the semigroup {T (t )t ≥0 } is strongly continuous (or a C0 -semigroup) if the map t → T (t )(x) is strongly continuous, for each x ∈ E, i.e. lim T (t )x = T (0)x,

∀ x ∈ E.

t → 0+

Definition 2.9. Let T (t ) be a C0 -semigroup defined on E. The infinitesimal generator A ∈ B(E ) of T (t ) is the linear operator defined by A(x) = lim

T (t )(x) − T (0)x

t → 0+

t

where D(A) = {x ∈ E: limt → 0+

,

for x ∈ D(A),

T (t )(x)−x t

exists in E }.

The following properties are classical (see Engel and Nagel [28,29] or [30]). Proposition 2.1. A linear operator A : D(A) ⊂ E −→ E is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator. In this case, the semigroup can be defined by T (t ) = eAt , t ≥ 0. Proposition 2.2. (a) If {T (t )}t ≥0 is a C0 -semigroup of bounded linear operators, then there exist constants ω ≥ 0 and M ≥ 1 such that

kT (t )kB(E ) ≤ M exp(ωt ),

for t ≥ 0.

(b) If A is the infinitesimal generator of a C0 -semigroup{T (t )}t ≥0 , then D(A), the domain of A, is dense in X and A is a closed linear operator. Proposition 2.3. Let {T (t )}t ≥0 be a uniformly continuous semigroup of a bounded linear operator. Then there exists some constant ω ≥ 0 such that

kT (t )kB(E ) ≤ exp(ωt ),

for t ≥ 0.

Proposition 2.4. If {T (t )}t ≥0 is a compact C0 -semigroup for t > 0, then it is uniformly continuous, for t > 0. Let A : E −→ E be a linear operator. Definition 2.10. The resolvent set Λ(A) of A consists of all complex numbers λ for which the linear operator λI − A is invertible, i.e. (λI − A)−1 is a bounded linear operator in E . The family R(λ, A) = (λI − A)−1 , λ ∈ Λ(A) is called the resolvent of A. All complex numbers λ not in Λ(A) form a set called the spectrum of A. 3. Existence results: 1 ∈ ρ(T (b)) 3.1. Mild Solutions Let Jk = (tk , tk+1 ], k = 0, . . . , m and let yk be the restriction of a function y to Jk . In order to define mild solutions for Problem (1.1), consider the space of piece-wise continuous functions PC = y: [0, b] → E , yk ∈ C (Jk , E ), k = 0, . . . , m, such that



y(tk− ) and y(tk+ ) exist and satisfy y(tk ) = y(tk− ) for k = 1, . . . , m .



Endowed with the norm

kykPC = max{kyk k∞ , k = 0, . . . , m}, PC is a Banach space. Let A be the infinitesimal generator of a C0 -semigroup {T (t )}t ≥0 such that 1 ∈ ρ(T (b)) and let f : J −→ E be a continuous function.

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Lemma 3.1. If y ∈ PC is a solution of the problem

 0 y (t ) − Ay(t ) = f (t ), y(t − ) − y(tk ) = Ik (y(tk )), y(0k) = y(b),

t ∈ J \ {t1 , . . . , tm }, t 6= tk , k = 1, . . . , m,

(3.1)

then it is given by y(t ) = T (t )(I − T (b))

m X

−1

T (b − tk )Ik (y(tk )) + −

X

T (b − s)f (s)ds

T (t − tk )Ik (y(tk− )),

t

Z

T (t − s)f (s)ds

+ 0

0

k=1

+

!

b

Z

for t ∈ J .

(3.2)

0
Proof. Let y be a solution of Problem (3.1) and L(s) = T (t − s)y(s) for fixed t ∈ J. We have L0 (s) = −T 0 (t − s)y(s) + T (t − s)y0 (s)

= −AT (t − s)y(s) + T (t − s)y0 (s) = T (t − s)[y0 (s) − Ay(s)] = T (t − s)f (s).

(3.3)

Let 0 < t < t1 . Integrating the previous equation, we get for k = 1 L(t ) − L(0) =

t

Z

T (t − s)f (s)ds.

0

Hence y(t ) = T (t )y(0) +

t

Z

T (t − s)f (s)ds. 0

More generally, for tk < t < tk+1 t1

Z

L0 (s)ds +

Z

0

t2

L0 (s)ds + · · · +

Z

t1

t

L0 (s)ds = tk

t

Z

T (t − s)f (s)ds ⇔ 0

L(t1− ) − L(0) + L(t2− ) − L(t1+ ) + · · · + L(t ) − L(tk+ ) =

t

Z

T (t − s)f (s)ds. 0

Therefore y(t ) = T (t )y(0) +

X

[L(tk+ ) − L(tk− )] +

0
t

Z

T (t − s)f (s)ds. 0

Since y(0) = y(b) and 1 ∈ ρ(T (b)), then I − T (b) is invertible. Hence we obtain after substitution y(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y(tk )) +

Z

!

b

T (b − s)f (s)ds

+

T (t − tk )Ik (y(tk ))

0
0

k=1

X

t

Z

T (t − s)f (s)ds,

+

for t ∈ J ,

0

proving the lemma.



This lemma leads to the definition of a mild solution. Definition 3.1. A function y ∈ Ω is said to be a mild solution of Problem (1.1) if there exists f ∈ L1 (J , E ) such that f (t ) ∈ F (t , y(t )) a.e. on J, y(0) = y(b) and y(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y(tk )) +

0

k=1

+

X

T (t − tk )Ik (y(tk− )),

Z

!

b

T (b − s)f (s)ds

Z +

t

T (t − s)f (s)ds

0

for t ∈ J .

0
In this section, we assume again that 1 ∈ ρ(T (b)), prove some existence results and describe the structure of the solution sets.

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3.2. The convex case: a direct approach Consider the following assumptions: (B1 ) F : J × E −→ Pcp,c v (E ) is an integrably bounded multi-valued map, i.e. there exists p ∈ L1 (J , E ) such that

kF (t , x)kP ≤ p(t ), for every x ∈ E and a.e. t ∈ J . (B2 ) There exist constants ak , bk > 0 and α ∈ [0, 1) such that |Ik (x)| ≤ ak |x|α + bk , for every x ∈ E , k = 1, . . . , m. (B3 ) For every t > 0, T (t ) is uniformly continuous. Remark 3.1. One can relax Assumption (B1 ) by a sublinear growth condition: (B10 ) there exist p, q ∈ L1 (J , R+ ) and β ∈ [0, 1 − α) such that

kF (t , x)kP ≤ q(t ) + p(t )|x|β ,

for every x ∈ E and a.e. t ∈ J .

Our first main existence result is: Theorem 3.1. Assume that F : J × E → Pcp,c v (E ) is a Carathéodory map satisfying (B1 )–(B3 ). Then Problem (1.1) has at least one solution. If further E is a reflexive space, then the solution set is compact in PC . The following so-called nonlinear alternatives of Leray and Schauder will be needed in the proof (see [31,15]). Lemma 3.2. Let (X , k · k) be a normed space and F : X → Pcl,c v (X ) a compact, u.s.c. multi-valued map. Then either one of the following conditions holds: (a) F has at least one fixed point, (b) the set M := {x ∈ X , x ∈ λF (x), λ ∈ (0, 1)} is unbounded. The single-valued version may be stated as follows: Lemma 3.3. Let X be a Banach space and C ⊂ X a nonempty bounded, closed, convex subset. Assume U is an open subset of C with 0 ∈ U and let G : U → C be a continuous compact map. Then (a) either there is a point u ∈ ∂ U and λ ∈ (0, 1) with u = λG(u), (b) or G has a fixed point in U. Proof of Theorem 3.1. Part 1. Existence of solutions. It is clear that all solutions of Problem (1.1) are fixed points of the multi-valued operator N : PC → P (PC ) defined by

!  Z b m X  − −1   T (b − tk )I (y(tk )) + T (b − s)f (s)ds T (t )(I − T (b)) 0 k = 1 Z t N (y) := h ∈ PC : h(t ) = X      + T (t − s)f (s)ds + T (t − tk )Ik (y(tk− )), for t ∈ J       

0 < tk < t

0

        

where f ∈ SF ,y = f ∈ L1 ([0, b], E ) : f (t ) ∈ F (t , y(t )),



for a.e. t ∈ [0, b] .



Notice that the set SF ,y is nonempty (see Remark 2.1, (b)). Since, for each y ∈ PC , the nonlinearity F takes convex values, the selection set SF ,y is convex and therefore N has convex values. Step 1. N is completely continuous. (a) N sends bounded sets into bounded sets in PC . Let q > 0, Bq := {y ∈ PC : kykPC ≤ q} be a bounded set in PC and y ∈ Bq . Then for each h ∈ N (y), there exists f ∈ SF ,y such that h(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )I (y(tk )) +

X

T (b − s)f (s)ds 0

k=1

+

!

b

Z

T (t − tk )Ik (y(tk− )),

t

Z

T (t − s)f (s)ds

+ 0

for t ∈ J .

0
Thus for each t ∈ J, we have the estimates

khkPC ≤ e

2ωb

k(I − T (b)) kB(E ) −1

m X (ak qα + bk ) + kpkL1 k=1

Notice that we have used Proposition 2.3 both with B3 .

! + eωb kpkL1 + eωb

m X (ak qα + bk ). k=1

690

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

(b) N maps bounded sets into equicontinuous sets of PC . Let τ1 , τ2 ∈ J , 0 < τ1 < τ2 and Bq be a bounded set of PC as in Step 1. Let y ∈ Bq ; then for each t ∈ J ωb

|h(τ2 ) − h(τ1 )| ≤ e kT (τ2 ) − T (τ1 )kB(E ) k(I − T (b)) kB(E ) −1

m X (ak qα + bk ) + kpkL1

!

k=1

τ2

Z +

τ1

Z

kT (τ1 − s) − T (τ2 − s)kB(E ) p(s)ds kT (τ2 − s)kB(E ) p(s)ds + 0 X kT (τ1 − tk ) − T (τ2 − tk )kB(E ) (ak qα + bk ).

τ1

+

τ1
Since T (t ) is uniformly continuous, the right-hand side tends to zero as τ2 − τ1 → 0. This proves the equicontinuity for the case where t 6= ti i = 1, . . . , m. It remains to examine the equicontinuity at t = ti . Set h1 (t ) = T (t )(I − T (b))

m X

−1

T (b − tk )I (y(tk )) +

!

b

Z

T (b − s)f (s)ds

T (t − tk )Ik (y(tk− )).

0
0

k=1

X

+

First we prove equicontinuity at t = ti− . Let δ1 > 0 be such that {tk : k 6= i} ∩ [ti − δ1 , ti + δ1 ] = ∅. Then h1 (ti ) = T (ti )(I − T (b))

m X

−1

T (b − tk )I (y(tk )) +

= T (ti )(I − T (b))

m X

T (b − s)f (s)ds

T (b − tk )I (y(tk )) +

! T (b − s)f (s)ds

+

0

k=1

T (ti − tk )Ik (y(tk ))

0
Z

X

+

0

k=1

−1

!

b

Z

i−1 X

T (ti − tk )Ik (y(tk )).

k=1

For 0 < h < δ1 , we have

|h1 (ti − h) − h1 (ti )| ≤ |(T (ti − h) − T (ti ))|k(I − T (b)) kB(E ) e −1

ωb

m X

! α

ωb

(ak q + bk ) + e kpkL1

k=1

+

i −1 X

|(T (ti − h − tk ) − T (ti − tk ))|(ck qα + bk ).

k=1

By the uniform continuity of T (t ), the right-hand side tends to zero as h → 0. Rt If h2 (t ) = 0 T (t − s)f (s)ds, then

|h2 (ti − h) − h2 (ti )| ≤

ti − h

Z

|[T (ti − h − s) − T (ti − s)]p(s)|ds + eωb

Z

ti

p(s)ds,

ti − h

0

which also tends to zero as h → 0. Next we prove the equicontinuity at t = ti+ . Define hˆ 0 (t ) = h(t ) t ∈ [0, t1 ] and

 ˆhi (t ) = h(t ), h(ti+ ),

t ∈ (ti , ti+1 ], t = ti .

Let δ2 > 0 be such that {tk : k 6= i} ∩ [ti − δ2 , ti + δ2 ] = ∅. Then hˆ (ti ) = T (ti )

m X

T (b − tk )I (y(tk )) +

Z

!

b

T (b − s)f (s)ds

0

k =1

Z +

ti

T (ti − s)f (s)ds +

0

i X

T (ti − tk )Ik (y(tk )).

k=1

For 0 < h < δ2 , we have

|hˆ (ti + h) − hˆ (ti )| ≤ eωb |(T (ti + h) − T (ti ))|

m X (ak qα + bk ) + kpkL1 k=1

+e

ωb

Z ti

which tends to zero as h → 0.

ti + h

p(s)ds +

i X k=1

!

Z +

ti

|[T (ti + h − s) − T (ti − s)]|p(s)ds

0

[T (ti + h − tk ) − T (ti − tk )](ak qα + bk ),

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

691

(c) As a consequence of parts (a), (b) together with the Arzelá–Ascoli theorem, it suffices to show that N maps Bq into a precompact set in E. Let 0 < t ≤ b and let 0 < ε < t. For y ∈ Bq , define hε (t ) = T (ε)T (t − ε)(I − T (b))

−1

m X

T (b − tk )I (y(tk )) +

+ T (ε)

T (b − s)f (s)ds 0

k=1 t −ε

Z

!

b

Z

X

T (t − s − ε)f (s)ds + T (ε)

T (t − ε − tk )Ik (y(tk )).

0
0

Then

|h(t ) − hε (t )| ≤

Z

t

kT (t − s)kB(E ) p(s)ds +

t −ε

kT (t − tk )kB(E ) (ak qα + bk ).

X t −ε
The right-hand side tends to 0, as ε → 0. Therefore, there are precompact sets arbitrarily close to the set H (t ) = {h(t ) : h ∈ N (y)}. This set is then precompact in E . Step 2. N has a closed graph. Let hn ∈ N (yn ) be such that hn −→ h∗ and yn −→ y∗ . We shall prove that h∗ ∈ N (y∗ ). hn ∈ N (yn ) means that there exists fn ∈ SF ,yn such that for each t ∈ J hn (t ) = T (t )(I − T (b))

m X

−1

T (b − tk )Ik (yn (tk )) +

!

b

Z

T (b − s)fn (s)ds

+

T (t − tk )Ik (yn (tk ))

0 < tk < t

0

k=1

X

t

Z

T (t − s)fn (s)ds.

+ 0

First, notice that, as n → ∞

k(hn − T (t )(I − T (b))−1 )

m X

T (t − tk )Ik (yn (tk )) − (h∗ − T (t )(I − T (b))−1 )

k=1

m X

T (t − tk )Ik (y∗ (tk ))k∞ −→ 0.

k=1

Now, consider the continuous linear operator Γ : L1 (J , E ) −→ PC (J , E ) defined by

(Γ v)(t ) =

t

Z

T (t − s)v(s)ds + T (t )(I − T (b))−1

b

Z

0

T (t − s)v(s)ds. 0

From the definition of Γ , we know that m X

(hn (t ) − T (t )(I − T (b))−1 )

T (t − tk )Ik (yn (tk )) ∈ Γ (SF ,yn ).

k=1

Since yn −→ y∗ and Γ ◦ SF is a closed graph operator by Lemma 2.8, then there exists f∗ ∈ SF ,y∗ such that h∗ (t ) = T (t )(I − T (b))

m X

−1

T (b − tk )Ik (y∗ (tk )) +

!

b

Z

T (b − s)f∗ (s)ds

+

T (t − tk )Ik (y∗ (tk ))

0 < tk < t

0

k=1

X

t

Z

T (t − s)f∗ (s)ds.

+ 0

Hence h∗ ∈ N (y∗ ), proving our claim. Lemma 2.7 yields that N is u.s.c. Step 3. A priori bounds on solutions. Let y ∈ PC be such that y ∈ N (y). Then there exists f ∈ SF ,y such that y(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )I (y(tk )) +

X

T (b − s)f (s)ds 0

k=1

+

!

b

Z

T (t − tk )Ik (y(tk− )),

for t ∈ J .

By Proposition 2.3, there exists a constant ω ≥ 0 such that

|y(t )| ≤ e

k(I − T (b)) kB(E ) −1

Z m X α (ak |y(tk )| + bk ) + k=1

+e ω b

t

Z 0

|f (s)|ds + eωb

m X (ak |y(tk− )|α + bk ). k=1

T (t − s)f (s)ds

+ 0

0
2ωb

t

Z

!

b

|f (s)|ds 0

692

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

Thus

kykPC ≤ e

2ωb

m X (ak kykαPC + bk ) + kpkL1

k(I − T (b)) kB(E ) −1

! + eωb kpkL1 + eωb

k=1

m X (ak kykαPC + bk ). k=1

If kykPC > 1, then since 0 ≤ α < 1, we have

kyk

1−α PC

≤e

2ωb

k(I − T (b)) kB(E ) −1

m X (ak + bk ) + kpkL1

! + eωb kpkL1 + eωb

k=1

m X (ak + bk ). k=1

Hence

kykPC ≤ e

2ωb

k(I − T (b)) kB(E ) −1

m X (ak + bk ) + kpkL1

! ωb

+ e kpkL1 + e

ωb

m X (ak + bk )

k=1

1 ! 1−α

:= M .

k=1

Therefore

e. kykPC ≤ max(1, M ) := M Let

e + 1}, U := {y ∈ PC : kykPC < M and consider the operator N : U → Pc v,cp (PC ). From the choice of U, there is no y ∈ ∂ U such that y ∈ γ N (y) for some γ ∈ (0, 1). As a consequence of the Leray and Schauder nonlinear alternative (Lemma 3.2), we deduce that N has a fixed point y in U, solution of Problem (1.1). Part 2. Compactness of the solution set. Let SF = {y ∈ PC : yis a solution of Problem (1.1)}.

e such that for every y ∈ SF , kykPC ≤ M. e Since N is completely continuous, then N (SF ) From Part 1, SF 6= ∅ and there exists M is relatively compact in PC . Let y ∈ SF ; then y ∈ N (y) and SF ⊂ N (SF ). It remains to prove that SF is a closed set in PC . Let yn ∈ SF such that (yn ) converge to y. For every n ∈ N, there exists vn (t ) ∈ F (t , yn (t )), a.e. t ∈ J such that yn (t ) = T (t )(I − T (b))

m X

−1

T (b − tk )Ik (yn (tk )) +

!

b

Z

T (b − s)vn (s)ds

X

T (t − tk )Ik (yn (tk ))

0
0

k=1

+

t

Z

T (t − s)vn (s)ds.

+

(3.4)

0

(B1 ) implies that for a.e. t ∈ J vn (t ) ∈ p(t )B(0, 1), hence (vn )n∈N is integrably bounded. Note that this still remains true when (B1 )0 holds for SF is a bounded set. Since E is reflexive, (vn )n∈N is semi-compact. By Lemma 2.9, there exists a subsequence, still denoted by (vn )n∈N , which converges weakly to some limit v ∈ L1 (J , E ). Moreover, the mapping Γ : L1 (J , E ) → PC (J , E ) defined by

Γ (g )(t ) =

t

Z

T (t − s)g (s)ds 0

is a continuous linear operator. Then it remains continuous if these spaces are endowed with their weak topologies [32]. Therefore for a.e. t ∈ J, the sequence (yn (t )) converge to y(t ) and by the continuity of Ik , it follows that y(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y(tk )) +

k=1

Z 0

!

b

T (b − s)v(s)ds

+

X

T (t − tk )Ik (y(tk ))

0
t

Z

T (t − s)v(s)ds.

+ 0

It remains to prove that v(t ) ∈ F (t , y(t )), for a.e. t ∈ J. Lemma 2.10 yields the existence of constants αin ≥ 0, i =

Pk(n)

n, . . . , k(n) such that i=1 αin = 1 and the sequence of convex combinations gn (·) = some limit v in L1 . Since F takes convex values, using Lemma 2.2, we obtain that

v(t ) ∈

\

{gn (t )},

a.e. t ∈ J

n≥1

⊂

\ n≥1

co{vk (t ), k ≥ n}

Pk(n) i=1

αin vi (·) converges strongly to

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

⊂

\ n ≥1

co

( [

693

) F (t , yk (t ))

k≥n

= co(lim sup F (t , yk (t ))).

(3.5)

k→∞

Since F is u.s.c. and has compact values, then by Lemma 2.1, we have lim sup F (t , yn (t )) = F (t , y(t )), n→∞

for a.e. t ∈ J .

This with (3.5) imply that v(t ) ∈ co F (t , y(t )). Since F (., .) has closed, convex values, we deduce that v(t ) ∈ F (t , y(t )), for a.e. t ∈ J, as claimed. Hence y ∈ SF which proves that SF is closed, hence compact in PC .  3.3. The convex case: an MNC approach 3.3.1. Reminders First, we gather together some material on the measure of noncompactness. For more details, we refer the reader to [33,34,20] and the references therein. Definition 3.2. Let E be a Banach space and (A, ≥) a partially ordered set. A map β : P (E ) → A is called a measure of noncompactness on E, MNC for short, if for every Ω ∈ P (E ), β(co Ω ) = β(Ω ). Notice that if D is dense in Ω , then co Ω = co D and hence β(Ω ) = β(D). Definition 3.3. A measure of noncompactness β is called (a) monotone if Ω0 , Ω1 ∈ P (E ), Ω0 ⊂ Ω1 implies β(Ω0 ) ≤ β(Ω1 ), (b) nonsingular if β({a} ∪ Ω ) = β(Ω ) for every a ∈ E , Ω ∈ P (E ), (c) invariant with respect to the union with compact sets if β(K ∪ Ω ) = β(Ω ) for every relatively compact set K ⊂ E, and Ω ∈ P (E ), (d) real if A = R+ = [0, ∞] and β(Ω ) < ∞ for every bounded Ω , (e) semi-additive if β(Ω0 ∪ Ω1 ) = max(β(Ω0 ), β(Ω1 )) for every Ω0 , Ω1 ∈ P (E ), (f) regular if the condition β(Ω ) = 0 is equivalent to the relative compactness of Ω . As example of an MNC, one may consider the Hausdorff MNC

χ (Ω ) = inf{ε > 0: Ω has a finite ε -net }. Recall that a bounded set A ⊂ E has a finite ε -net if there exits a finite subset S ⊂ E such that A ⊂ S + ε B where B is a closed ball in E . Other examples are given by the following measures of noncompactness defined on the space of continuous functions C ([0, b], E ) with values in a Banach space E: (i) the modulus of fiber noncompactness

ϕ(Ω ) = sup χE (Ω (t )), t ∈[0,b]

where χE is the Hausdorff MNC in E and Ω (t ) = {y(t ) : y ∈ Ω }; (ii) the modulus of equicontinuity modC (Ω ) = lim sup max ky(τ1 ) − y(τ2 )k. δ→0 y∈Ω |τ1 −τ2 |≤δ

It should be mentioned that these MNCs satisfy all above-mentioned properties except regularity. Definition 3.4. Let M be a closed subset of a Banach space E and β : P (E ) → (A, ≥) an MNC on E. A multi-valued map F : M → Pcp (E ) is said to be β -condensing if for every Ω ⊂ M, the relation β(Ω ) ≤ β(F (Ω )) implies the relative compactness of Ω . Some important results on fixed point theory with MNCs are recalled hereafter (see e.g., [20] for the proofs and further details). The first one is a compactness criterion. Lemma 3.4 ([20, Theorem 5.1.1]). Let N: L1 ([a, b], E ) → C ([a, b], E ) be an abstract operator satisfying the following conditions:

(S1 ) N is ξ -Lipschitz: there exists ξ > 0 such that for every f , g ∈ L1 ([a, b], E ) Z b |Nf (t ) − Ng (t )| ≤ ξ |f (s) − g (s)|ds, for all t ∈ [a, b]. a

(S2 ) N is weakly–strongly sequentially continuous on compact subsets: for any compact K ⊂ E and any sequence {fn }∞ n=1 ⊂ L1 ([a, b], E ) such that {fn (t )}∞ n=1 ⊂ K for a.e. t ∈ [a, b], the weak convergence fn * f0 implies the strong convergence N (fn ) → N (f0 ) as n → +∞.

694

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

Then for every semi-compact sequence {fn }∞ n =1 C ([a, b], E ).

⊂ L1 ([0, b], E ), the image sequence N ({fn }∞ n=1 ) is relatively compact in

Lemma 3.5 ([20, Theorem 5.2.2]). Let an operator N: L1 ([a, b], E ) → C ([a, b], E ) satisfy conditions (S1 )–(S2 ) together with

(S3 ) There exits η ∈ L1 ([a, b]) such that for every integrably bounded sequence {fn }∞ n=1 , we have χ ({fn (t )}∞ n=1 ) ≤ η(t ), for a.e. t ∈ [a, b]. Then

χ ({N (fn )(t )}∞ n =1 ) ≤ 2 ξ

b

Z

η(s)ds,

for all t ∈ [a, b],

a

where ξ is the constant in (S1 ) and χ is the Hausdorff MNC. The next result is concerned with the nonlinear alternative for β -condensing u.s.c. multi-valued maps. Lemma 3.6 ([20]). Let V ⊂ E be a bounded open neighborhood of zero and N : V → Pcp,c v (E ) a β -condensing u.s.c. multivalued map, where β is a nonsingular measure of noncompactness defined on subsets of E, satisfying the boundary condition x 6∈ λN (x) for all x ∈ ∂ V and 0 < λ < 1. Then FixN 6= ∅. Lemma 3.7 ([20]). Let W be a closed subset of a Banach space E and F : W → Pcp (E ) is a closed β -condensing multi-valued map where β is a monotone MNC on E. If the fixed point set Fix F is bounded, then it is compact. 3.3.2. Main results In all this part, we assume that there exists M > 0 such that

kT (t )kB(E ) ≤ M ,

for every t ∈ [0, b].

(3.6)

Let F : J × E → Pcp,c v (E ) be a Carathéodory multi-valued map which satisfies Lipschitz conditions with respect to the Hausdorff MNC: (B4 ) There exists p ∈ L1 ([0, b], R+ ) such that for every bounded D in E

χ (F (t , D)) ≤ p(t )χ (D). (B5 ) There exist ck > 0 such that for every bounded D in E .

χ (Ik (D)) ≤ ck χ (D),

k = 1, 2, . . . , m.

Lemma 3.8. Under conditions (B1 ) and (B4 ), the operator N is closed and N (y) ∈ Pcp,c v (PC ), for every y ∈ PC where N is as defined in the proof of Theorem 3.1. Proof. Step 1. N is closed. Let hn −→ h∗ , hn ∈ N (yn ) and yn −→ y∗ . We shall prove that h∗ ∈ N (y∗ ). hn ∈ N (yn ) means that there exists fn ∈ SF ,yn such that for a.e. t ∈ J hn (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (yn (tk )) +

!

b

Z

T (b − s)fn (s)ds

+

T (t − tk )Ik (yn (tk ))

0
0

k=1

X

t

Z

T (t − s)fn (s)ds.

+

(3.7)

0

Since {fn (t ) : n ∈ N} ⊆ F (t , yn (t )), Assumption (B1 ) implies that (fn )n∈N is integrably bounded. In addition, the set {fn (t ) : n ∈ N} is relatively compact for a.e. t ∈ J because Assumption (B4 ) both with the convergence of (yn )n∈N imply that

χ ({fn (t ) : n ∈ N}) ≤ χ (F (t , yn (t ))) ≤ p(t )χ ({yn (t ) : n ∈ N}) = 0. Then the sequence {fn : n ∈ N} is semi-compact, whence weakly compact in L1 ([0, b]; E ) to some limit f∗ by Lemma 2.9. Arguing as in the proof of Theorem 3.1, Part 2, and passing to the limit in (3.7), we obtain that f∗ ∈ SF ,y∗ and for each t ∈ J h∗ (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y∗ (tk )) +

k =1

b

Z

! T (b − s)f∗ (s)ds

0

+

X

T (t − tk )Ik (y∗ (tk ))

0
t

Z

T (t − s)f∗ (s)ds.

+ 0

As a consequence, h∗ ∈ N (y∗ ), as claimed.

(3.8)

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

695

Step 2. N has compact, convex values. The convexity of N (y) follows immediately from the convexity of the values of F . To prove the compactness of the values of F , let N (y) ∈ P (E ) for some y ∈ PC and hn ∈ N (y). Then there exists fn ∈ SF ,y satisfying (3.7). Arguing again as in Step 1, we prove that (fn ) is semi-compact and converges weakly to some limit f∗ ∈ F (t , y(t )), a.e. t ∈ [0, b] hence passing to the limit in (3.7), hn tends to some limit h∗ in the closed set N (y) with h∗ satisfying (3.8). Therefore the set N (y) is sequentially compact, hence compact [35].  Lemma 3.9. Under the conditions (B4 )–(B5 ), the operator N is u.s.c. Proof. Using Lemmas 2.7 and 3.8, we only prove that N is quasicompact. Let K be a compact set in PC and hn ∈ N (yn ) such that yn ∈ K . Then there exists fn ∈ SF ,yn such that hn (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (yn (tk )) +

!

b

Z

T (b − s)fn (s)ds

+

T (t − tk )Ik (yn (tk ))

0 < tk < t

0

k=1

X

t

Z

T (t − s)fn (s)ds.

+ 0

Since K is compact, we may pass to a subsequence, if necessary, to get that (yn ) converges to some limit y∗ in PC . Arguing as in the proof of Theorem 3.1, Step 1, we can prove the existence of a subsequence (fn ) which converges weakly to some limit f∗ and hence (hn ) converges to h∗ , where h∗ (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y∗ (tk )) +

!

b

Z

T (b − s)f∗ (s)ds

+

T (t − tk )Ik (y∗ (tk ))

0 < tk < t

0

k=1

X

t

Z

T (t − s)f∗ (s)ds.

+ 0

As a consequence, N is u.s.c.



We are now in position to prove our second existence result in the convex case. Theorem 3.2. Assume that F satisfies Assumptions (B1 ) and (B4 )–(B5 ). If q := M (kM (I − T (b))−1 kB(E ) + 1)

m X

p(s)ds

ck + 2

k=1

!

b

Z

< 1,

0

then the set of solutions for Problem (1.1) is nonempty and compact. Proof. It is clear that all solutions of Problem (1.1) are fixed points of the multi-valued operator N defined in Theorem 3.1. By Lemmas 3.8 and 3.9, N (.) ∈ Pc v,cp (PC ) and it is u.s.c. Next, we prove that N is a β -condensing operator for a suitable MNC β . Given a bounded subset D ⊂ K0 , let modC (D) denotes the modulus of equicontinuity of the set of functions D: modC (D) = lim sup max |x(τ1 ) − x(τ2 )|. δ→0 x∈D |τ2 −τ1 |≤δ

It is well known (see e.g. Example 2.1.2 in [20]) that modC (D) defines an MNC in PC which satisfies all of the properties in the Definition 3.3 except regularity. Given the Hausdorff MNC χ , let γ be the real MNC defined on bounded subsets on PC by

γ (D) = sup χ (D(t )). t ∈[0,b]

Finally, define the following MNC on bounded subsets of PC by

β(D) = max (γ (D), modC (D)), D∈∆(PC )

where ∆(PC ) is the collection of all denumerable subsets of B. Then the MNC β is monotone, regular and nonsingular (see Example 2.1.4 in [20]). This measure is also used in [36–40] in the discussion of semilinear evolution differential inclusions. To show that N is β -condensing, let B ⊂ PC be a bounded set in PC such that

β(B) ≤ β(N (B)).

(3.9)

We will show that B is relatively compact. Let {yn : n ∈ N} ⊂ B and let N = L1 + L2 ◦ Γ1 ◦ SF + Γ ◦ SF where L1 : PC → PC is defined by

(L1 y)(t ) = T (t )(I − T (b))−1

m X k=1

L2 : R+ → B(E ) is defined by L2 (y) = T (t )(I − T (b))−1 y

T (b − tk )Ik (y(tk )) +

X 0
T (t − tk )Ik (y(tk ))

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SF : PC → L1 (J , E ) is defined by SF (y) = SF ,y = {v ∈ L1 (J , E ) : v(t ) ∈ F (t , y(t )) a.e t ∈ [0, b]},

Γ1 : L1 (J , E ) → PC is defined by Z b T (t − s)f (s)ds, Γ1 (f )(t ) =

t ∈ [0, b]

0

and

Γ (f )(t ) =

t

Z

T (t − s)f (s)ds,

t ∈ [0, b].

0

Then t

Z

kT (t − s)k · |f1 (s) − f2 (s)| ds Z t |f1 (s) − f2 (s)| ds. ≤M

|Γ f1 (t ) − Γ f2 (t )| ≤

0

0

Moreover, each element hn in N (yn ) can be represented as hn = L1 (yn ) + T (.)(I − T (b))−1 Γ1 (fn ) + Γ (fn ),

with fn ∈ SF (yn ).

(3.10)

Moreover (3.9) yields

β({hn : n ∈ N}) ≥ β({yn : n ∈ N}).

(3.11)

From Assumption (B4 ), it holds that for a.e. t ∈ [0, b],

χ ({fn (t ) : n ∈ N}) ≤ χ (F (t , {yn (t )})∞ n=1 ) ∞ ≤ p(t )χ ({yn (t )}n=1 ) ≤ p(t ) sup χ ({yn (s)}∞ n =1 ) 0≤s≤t

≤ p(t )γ ({yn }∞ n=1 ).

(3.12)

Lemmas 3.4 and 3.5 imply that ∞ χ ({Γ (fn )(t )}∞ n=1 ) ≤ γ ({yn }n=1 )2M

t

Z

p(s)ds, 0

−1 χ ({T (.)(I − T (b))−1 Γ1 (fn )(t )}∞ kB(E ) {Γ1 (fn )(t )}∞ n=1 ) ≤ M k(I − T (b)) n=1 Z 2 −1 ∞ ≤ 2M k(I − T (b)) kB(E ) γ ({yn }n=1 )

b

p(s)ds, 0

and 2 −1 χ (L1 {yn (t )}∞ kB(E ) n=1 ) ≤ M k(I − T (b))

m X

χ (Ik {yn (tk )}∞ n=1 ) + M

k =1

≤ M 2 k(I − T (b))−1 kB(E )

m X

m X

χ (Ik {yn (tk )}∞ n =1 )

k=1

ck χ ({yn (tk )}∞ n =1 ) + M

k=1

≤ (M 2 k(I − T (b))−1 kB(E ) + M )

m X

ck χ ({yn (tk )}∞ n=1 )

k=1 m X

ck γ ({yn }∞ n=1 ).

k=1

Hence

γ (hn ) ≤ M k(I − T (b)) kB(E ) + M 2

−1

m
X

! ck + 2kpkL1

γ ({yn }+∞ n=1 ).

k=1

Therefore ∞ ∞ ∞ γ ({yn }∞ n=1 ) ≤ γ ({hn }n=1 ) = sup χ ({hn (t )}n=1 ) ≤ qγ ({yn }n=1 ). t ∈[0,b]

(3.13)

Since 0 < q < 1, we infer that

γ ({yn }∞ n=1 ) = 0.

(3.14)

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697

γ (yn ) = 0 implies that χ ({yn (t )}) = 0, for a.e. t ∈ [0, b]. In turn, (3.12) yields that χ ({fn (t )}) = 0,

for a.e. t ∈ [0, b].

Hence (fn (·))n∈N is semi-compact. Note that the Cauchy operator Γ satisfies (S1 ) and (S2 ) in Lemma 3.4. The latter lemma then implies that (fn (·))n∈N is relatively compact. Therefore γ ({hn }) = 0 and modC ({hn }) = 0. As a consequence β(N (B)) = 0 and hence β(B) = 0 follows from (3.9). We have proved that B is relatively compact. Hence N : U → Pcp,c v (PC ) is u.s.c and β -condensing, where U is as in the proof of Theorem 3.1 (see also [41,42,8] for the existence of such U). From the choice of U, there is no y ∈ ∂ U such that y ∈ λN (y) for some λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray–Schauder type for condensing maps (Lemma 3.6), we deduce that N has a fixed point y in U, a solution to Problem 1.1. Finally, since FixN is bounded, by Lemma 3.7, FixN is further compact.  3.4. The nonconvex case In this section, we present a second existence result for Problem (1.1) when the multi-valued nonlinearity is not necessarily convex. In the proof, we will make use of the nonlinear alternative of Leray–Schauder type [31] combined with a selection theorem due to [43–45] for lower semi-continuous multi-valued maps with decomposable values. The main ingredients are presented hereafter. We first start with some definitions (see e.g. [13]). Consider a topological space E and a family A of subsets of E . Definition 3.5. A is called a σ -algebra if it verifies the following properties: (a) ∅ ∈ A. (b) O ∈ A ⇒ E \ O ∈ A. S (c) On ∈ A, n = 1, 2, . . . ⇒ n≥1 On ∈ A. Definition 3.6. A is called L ⊗ B measurable if A belongs to the σ -algebra generated by all sets of the form I × D where I is Lebesgue measurable in J and D is Borel measurable in E. Definition 3.7. A subset A ⊂ L1 (J , E ) is decomposable if for all u, v ∈ A and for every Lebesgue measurable set I ⊂ J, we have: uχI + vχJ \I ∈ A, where χA stands for the characteristic function of the set A. Let F : J × E → P (E ) be a multi-valued map with nonempty closed values. Assign to F the multi-valued operator F : C (J , E ) → P (L1 (J , E )) defined by F (y) = SF ,y . The operator F is called the Nemyts’ki˘ı operator associated to F . Definition 3.8. Let F : J × E → P (E ) be a multi-valued map with nonempty compact values. We say that F is of lower semi-continuous type (l.s.c. type) if its associated Nemyts’ki˘ı operator F is lower semi-continuous and has nonempty closed and decomposable values. Next, we state a classical selection theorem due to Bressan and Colombo. Lemma 3.10 (see [22,43,17,23]). Let X be a separable metric space and let E be a Banach space. Then every l.s.c. multi-valued operator N : X → Pcl (L1 (J , E )) with closed decomposable values has a continuous selection, i.e. there exists a continuous singlevalued function f : X → L1 (J , E ) such that f (x) ∈ N (x) for every x ∈ X . Let us introduce the following hypothesis.

(H1 ) F : [0, b] × E −→ P (E ) has nonempty compact values and (a) the mapping (t , y) 7→ F (t , y) is L ⊗ B measurable, (b) the mapping y 7→ F (t , y) is lower semi-continuous for a.e. t ∈ [0, b]. The following lemma is crucial in the proof of our existence theorem. Lemma 3.11 (see e.g. [17,46]). Let F : J × E → Pcp (E ) be an integrably bounded multi-valued map satisfying (H1 ). Then F is of lower semi-continuous type. Theorem 3.3. Suppose that the hypotheses (B1 )–(B3 ) and (H1 ) are satisfied. Then Problem (1.1) has at least one solution. Proof. (B1 ) and (H1 ) imply, by Lemma 3.11, that F is of lower semi-continuous type. From Lemma 3.10, there is a continuous selection f : PC → L1 ([0, b], E ) such that f (y) ∈ F (y) for all y ∈ PC . Consider the problem

 0 y (t ) − Ay(t ) = f (y)(t ), ∆y| = Ik (y(tk− )), y(0)t ==tk y(b),

t ∈ [0, b], t 6= tk , k = 1, . . . , m, k = 1, . . . , m,

(3.15)

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and the operator G : PC → PC defined by G(y)(t ) = T (t )(I − T (b))

m X

−1

T (b − tk )I (y(tk )) +

X

T (b − s)f (y)(s)ds

T (t − tk )Ik (y(tk− )),

t

Z

T (t − s)f (y)(s)ds

+ 0

0

k=1

+

!

b

Z

for t ∈ [0, b].

0
As in Theorem 3.1, we can prove that the single-valued operator G is compact and there exists M∗ > 0 such that for all possible solutions y, we have kykPC < M∗ . Now, we only check that G is continuous. Let (yn ) be a sequence such that yn −→ y in PC (J , E ), as n → +∞. Then

|G(yn (t )) − G(y(t ))| ≤ eωb (eωb k(I − T (b))−1 kB(E ) )

b

Z

|f (yn (s)) − f (y(s))|ds 0

+ eωb (eωb k(I − T (b))−1 kB(E ) )

m X

|Ik (yn (tk− )) − Ik (y(tk− ))|.

k=1

Since the functions f and Ik , k = 1, . . . , m are continuous, we have

kG(yn ) − G(y)kPC ≤ e2ωb (k(I − T (b))−1 kB(E ) )kf (yn ) − f (y)kL1 m X + e2ωb (k(I − T (b))−1 kB(E ) ) |Ik (yn (tk− )) − Ik (y(tk− ))|, k=1

which, by continuity of f and Ik (1 ≤ k ≤ n), tends to 0, as n → +∞. Let U = {y ∈ PC : kykPC < M∗ }. From the choice of U, there is no y ∈ ∂ U such that y = λNy for in λ ∈ (0, 1). As a consequence of the nonlinear alternative of Leray–Schauder (Lemma 3.3), we deduce that G has a fixed point y ∈ U which is a solution of Problem (3.15), hence a solution to the problem (1.1).  3.5. A further result In this part, we present a second existence result to Problem (1.1) with a nonconvex valued right-hand side. First, consider the Hausdorff pseudo-metric distance Hd : P (E ) × P (E ) −→ R+ ∪ {∞} defined by





Hd (A, B) = max sup d(a, B), sup d(A, b) a∈A

b∈B

where d(A, b) = infa∈A d(a, b) and d(a, B) = infb∈B d(a, b). Then (Pb,cl (E ), Hd ) is a metric space and (Pcl (X ), Hd ) is a generalized metric space (see [47]). In particular, Hd satisfies the triangle inequality. Definition 3.9. A multi-valued operator N: E → Pcl (E ) is called (a) γ -Lipschitz if there exists γ > 0 such that Hd (N (x), N (y)) ≤ γ d(x, y),

for each x, y ∈ E ,

(b) a contraction if it is γ -Lipschitz with γ < 1. Our proofs are based on the following classical fixed point theorem for contraction multi-valued operators proved by Covitz and Nadler in 1970 [48] (see also [17, Theorem 11.1]). Lemma 3.12. Let (X , d) be a complete metric space. If G : X → Pcl (X ) is a contraction, then FixN 6= ∅. Let us introduce the following hypotheses:

(A1 ) F : J × E −→ Pcp (E ); t 7−→ F (t , x) is measurable for each x ∈ E . (A2 ) There exist constants ck , such that |Ik (x) − Ik (y)| ≤ ck |x − y|, for all x, y ∈ E (k = 1, . . . , m). (A3 ) There exists a function l ∈ L1 (J , R+ ) such that Hd (F (t , x), F (t , y)) ≤ l(t )|x − y|, with d(0, F (t , 0)) ≤ l(t ),

for a.e. t ∈ J .

for a.e. t ∈ J and all x, y ∈ E ,

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699

Theorem 3.4. Let Assumptions (A1 )–(A3 ) be satisfied. If Me

ωb

k=m X

! 1 + Meωb k(I − T (b))−1 kB(E ) < 1,




ck + klkL1

k=1

then Problem (1.1) has at least one solution. Proof. In order to transform the problem (1.1) into a fixed point problem, let the multi-valued operator N : PC (J , E ) → P (PC (J , E )) be as defined in Theorem 3.1. We shall show that N satisfies the assumptions of Lemma 3.12. Note that (A3 ) implies that F has at most linear growth, i.e.

kF (t , x)kP ≤ l(t )|x|,

for a.e. t ∈ J and all x ∈ E .

(a) N (y) ∈ Pcl (PC (J , E )) for each y ∈ PC (J , E ). The proof is similar to that in Theorem 3.1, Part 1, Step 2 and is omitted. (b) There exists γ < 1, such that Hd (N (y), N (y)) ≤ γ ky − ykPC , ∀ y, y ∈ PC (J , E ). Let y, y ∈ PC (J , E ) and h ∈ N (y). Then there exists g (t ) ∈ F (t , y(t )) (g is a measurable selection) such that for each t ∈ J h(t ) = T (t )(I − T (b))

m X

−1

T (b − tk )I (y(tk )) +

X

T (b − s)g (s)ds

t

Z

T (t − s)g (s)ds

+

0

k=1

+

!

b

Z

0

T (t − tk )Ik (y(tk− )).

0
(A3 ) tells us that Hd (F (t , y(t )), F (t , y(t ))) ≤ l(t )|y(t ) − y(t )|,

a.e. t ∈ J .

Hence there is w ∈ F (t , y(t )) such that

|g (t ) − w| ≤ l(t )|y(t ) − y(t )|,

t ∈ J.

Then consider the mapping U : J → P (E ), given by U (t ) = {w ∈ E : |g (t ) − w| ≤ l(t )|y(t ) − y(t )|},

t ∈ J,

that is U (t ) = B (g (t ), l(t )|y(t ) − y(t )|). Since g , l, y, y are measurable, Theorem III.4.1 in [14] tells us that the closed ball U is measurable. In addition (A1 ) and (A3 ) imply that for each y ∈ PC (J , E ), F (t , y(t )) is measurable. Finally the set V (t ) = U (t ) ∩ F (t , y(t )) is nonempty since it contains w . Therefore the intersection multi-valued operator V is measurable with nonempty, closed values (see [13–15]). By Lemma 2.4, there exists a function g (t ) which is a measurable selection for V . Thus g (t ) ∈ F (t , y(t )) and

|g (t ) − g (t )| ≤ l(t )|y(t ) − y(t )|,

for a.e. t ∈ J .

Let us define for a.e. t ∈ J h(t ) = T (t )(I − T (b))

m X

−1

T (b − tk )I (y(tk )) +

X

T (b − s)g (s)ds

t

Z

T (t − s)g (s)ds

+

0

k=1

+

!

b

Z

0

T (t − tk )Ik (y(tk )).

0
Then

|h(t ) − h(t )| ≤ M 2 e2ωb k(I − T (b))−1 kB(E )

m X

ck |y(tk ) − y(tk )|

k=1

+ M 2 e2ωb k(I − T (b))−1 kB(E )

b

Z

l(s)|y(s) − y(s)|ds + Meωb

0

+ Meωb

m X k=1

b

Z

l(s)|y(s) − y(s)|ds. 0

Hence

kh − hk∞ ≤ Me

ωb

k =m X k=1

! ck + klkL1

1 + Meωb k(I − T (b))−1 kB(E ) ky − ykPC .




ck |y(tk ) − y(tk )|

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S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

By an analogous relation, obtained by interchanging the roles of y and y, we finally arrive at Hd (N (y), N (y)) ≤ Me

ωb

k=m X

! ck + klkL1

1 + Meωb k(I − T (b))−1 kB(E ) ky − ykPC .




k=1

So, N is a contraction and thus, by Lemma 3.12, N has a fixed point y, which is a mild solution to (1.1).



Arguing as in Theorem 3.1, we can also prove the following result the proof of which is omitted. Theorem 3.5. Let (E , k · k) be a reflexive Banach space and F : J × E → Pcp (E ) satisfies all conditions of Theorem 3.4. Then the solution set of Problem (1.1) is nonempty and compact. 4. The parameter-dependent case In this section, we consider a parameter-dependent impulsive problem:

 0 (y − Ay)(t ) ∈ F (t , y(t ), λ), ∆y tk = Ik (y(tk− ), λ), y(0t)== y(b)

a.e. t ∈ J \ {t1 , . . . , tm }, k = 1, . . . , m,

(4.1)

where F : J × H × Λ → P (H ) is a multi-valued map with compact values, Ik (., .) : H × Λ → H is a continuous functions, (Λ, dΛ ) is a complete metric space and H is a separable Hilbert space. In the case of a problem with no impulses, some existence results and properties of solutions for semilinear and evolutions differential inclusions with parameters were studied by Hu et al. [49], Papageorgiou and Yannakakis [50] and Tolstonogov [51,52]; see also [53] for a parameterdependent first-order Cauchy problem. In this section, we assume again that 1 ∈ ρ(T (b)) and consider separately the convex and the nonconvex cases. 4.1. The convex case We will assume the following assumptions.

e1 ) The multi-valued map F (., x, λ) : [0, b] → Pcp,c v (H ) is measurable for all x ∈ E and λ ∈ Λ. (B e2 ) The multi-valued map F (t , ., .) : H × Λ → Pcp,c v (H ) is u.s.c. for a.e. t ∈ [0, b]. (B e3 ) There exist α ∈ [0, 1) and p, q ∈ L1 (J , R+ ) such that (B kF (t , x, λ)kP ≤ p(t ) + q(t )kxkα ,

for a.e. t ∈ J and for all x ∈ E , λ ∈ Λ. e (B4 ) There exist constants ak , bk > 0 and β ∈ [0, 1) such that

|Ik (x)| ≤ ak |x|β + bk , for each x ∈ E , λ ∈ Λ (k = 1, . . . , m). e5 ) For every t > 0, T (t ) is uniformly continuous; hence there exists M > 0 such that (B kT (t )kB(E ) ≤ M ,

for every t ≥ 0.

e1 )–(B e3 ). Then for every fixed λ ∈ Λ, there exists y(., λ) ∈ PC solution of Problem (4.1). Theorem 4.1. Assume that F satisfies (B For fixed λ ∈ Λ, let Fλ (t , y) = F (t , y(t ), λ), (t , y) ∈ [0, b] × H and let Ikλ (y) = Ik (y, λ), k = 1, . . . , m. It is clear that Fλ (., u) is a measurable multi-valued map for all u ∈ E , Fλ (t , .) is u.s.c and

kFλ (t , x)kP ≤ q(t ) + p(t )|x|α ,

for a.e. t ∈ J and each x ∈ H ,

e3 ). To transform Problem (4.1) into a fixed point problem, consider the where p, q ∈ L (J , R ) and α are as defined in (B operator N1 : PC → P (PC ) defined by 1

+

!  Z b m X  −1   T (b − tk )I (y(tk ), λ) + T (b − s)v(s)ds T (t )(I − T (b)) 0 k=1 Z N1 (y) := h ∈ PC : h(t ) = t X      + T (t − s)v(s)ds + T (t − tk )Ik (y(tk− ), λ), for t ∈ J       

0

0
where

v ∈ SFλ ,y = {v ∈ L1 (J , H ) : v(t ) ∈ Fλ (t , y(t )), a.e. t ∈ J }. Define the mapping S : Λ → Pcp (H )

        

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

701

by S (λ) = {y ∈ PC : y is a solution of Problem (4.1)}. From Theorem 3.1, S (λ) 6= ∅ so that S is well defined. Next, we prove the upper semi-continuity of solutions with respect to the parameter λ.

e1 )–(B e3 ) hold and E is reflexive, then S is u.s.c. Proposition 4.1. If hypotheses (B Proof. Step 1. S (.) ∈ Pcp (H ). Let λ ∈ Λ and yn ∈ S (λ), n ∈ N. Then there exists vn ∈ SFλ ,yn such that yn (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (yn (tk ), λ) +

!

b

Z

T (b − s)vn (s)ds

+

T (t − tk )Ik (yn (tk ), λ)

0 < tk < t

0

k=1

X

t

Z

T (t − s)vn (s)ds,

+

t ∈ J.

0

e3 ) and the continuity of Ik , k = 1, . . . , m, we can prove that there exists M > 0 such that kyn kPC ≤ M , n ∈ N. As From (B in the proof of Theorem 3.1, Part 2, we can easily prove that the set {yn : n ≥ 1} is compact in PC ; hence there exists a subsequence of (yn ) which converges to y in PC . Since {vn }(·) is integrably bounded, then arguing as in the proof of Theorem 3.1, Step 1, there exists a subsequence which converges weakly to v and then we obtain at the limit: y(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y(tk ), λ) + −

!

b

Z

T (b − s)v(s)ds

T (t − tk )Ik (y(tk− ), λ)

0
0

k=1

X

+

t

Z

T (t − s)v(s)ds,

+

for t ∈ J .

(4.2)

0

Hence S (.) ∈ Pcp (H ). Step 2. S (.) is quasicompact. Let K be a compact set in Λ. To show that S (K ) is compact, let yn ∈ S (λn ), λn ∈ K . Then there exists vn ∈ SF (.,.,λn ,yn ) , n ∈ N such that yn (t ) = T (t )(I − T (b))−1

m X

T (b − tk )Ik (yn (tk− ), λn ) +

!

b

Z

T (b − s)vn (s)ds 0

k=1

+

X

T (t − tk )Ik (yn (tk− ), λn )

0 < tk < t

t

Z

T (t − s)vn (s)ds,

+

t ∈ J.

(4.3)

0

As mentioned in Step 1, {yn : n ≥ 1} is compact in PC ; then there exists a subsequence of (yn ) which converges to y in PC . Since K is compact, there exists a subsequence still denoted by {λn : n ≥ 1} ⊂ K such that (λn ) converges to λ ∈ Λ. As we did above, we can easily prove that there exists v(·) ∈ F (·, y(.), λ) such that y satisfies (4.2). Step 3. S (.) is closed. For this, let λn ∈ Λ be such that (λn ) converge to λ and let yn ∈ S (λn ), n ∈ N be a sequence which

e3 ) to show that the set {yn : n ≥ 1} converges to some limit y in PC . Then yn satisfies (4.3) and as we did above, we can use (B is equicontinuous in PC . By the Arzelá–Ascoli theorem, we conclude that there exists a subsequence of (yn ) converging to some limit y in PC and there exists a subsequence of (vn ) which converges to v(.) ∈ F (., y(.), λ) such that y satisfies (4.2). Therefore S (.) has a closed graph, hence u.s.c. by Lemma 2.6.  4.2. The nonconvex case The following hypotheses will be assumed in this sub-section:

e1 ) F : J × H × Λ → Pcp (H ) is such that for all (x, λ) ∈ H × Λ, the multi-valued map t 7→ F (t , x, λ) is measurable. (H e2 ) For every compact B ⊂ Λ, there exists a function pB ∈ L1 (J , H ) such that (H Hd (F (t , x, λ), F (t , y, λ)) ≤ pB (t )|x − y|, e (H3 ) There exist ck > 0, d ≥ 0 such that

|Ik (x, λ) − Ik (y, λ)| ≤ ck |x − y|,

a.e. t ∈ J , for each x, y ∈ H , λ ∈ B.

for every x, y ∈ H , λ ∈ Λ.

e1 )–(H e3 ) and that for every compact subset B ⊂ Λ, we have Theorem 4.2. Assume that F satisfies (H M k(I − T (t )) 2

−1

kB(H ) kpB kL1 +

m X k=1

! ck

< 1.

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e5 ) holds Then for every λ ∈ Λ, there exists at least one solution of Problem (4.1). Assume further that F (., ., .) ∈ Pcp,c v (H ) and (B together with e4 ) λ 7−→ F (t , x, λ) is lower semi-continuous, for a.e. t ∈ J and each x ∈ H . (H e5 ) For every compact B ⊂ Λ, there exist a function pB ∈ L1 (J , H ) and a continuous nondecreasing function ψ : [0, ∞) → (H [0, ∞) such that kF (t , x, λ)kP ≤ pB (t )ψ(|x|),

for a.e. t ∈ J each x ∈ H , ∈ and λ ∈ B.

Then the multi-valued map λ ( S (λ) is l.s.c. from Λ into Pcp (PC ).

e1 )–(H e3 ), we have that Proof. Arguing as in Theorem 3.3, we can prove that for every λ ∈ Λ, S (λ) 6= ∅ and that under (H S (.) ∈ Pcp (PC ). To show that S (.) is l.s.c., let λn → λ in Λ. We need to show that S (λ) ⊂ limn→∞ S (λn ) = {y ∈ PC : y = lim yn , yn ∈ S (λn ), n > 0}. n→∞

Let y ∈ S (λ); then there exists v ∈ SF (.,y(.),λ) such that y satisfies (4.2). Let h(t , λn ) = Proj(v(t ), F (t , y(t ), λn )) and g (t , u, λn ) = Proj(h(t , λn ), F (t , u, λn )) where

|v(t ) − h(t , λn )| = d(v(t ), F (t , y(t ), λn )) and

|h(t , λn ) − g (t , u, λn )| = d(h(t , λn ), F (t , u, λn )). It is clear that h(., λn ) and g (., u, λn ) are measurable functions. Furthermore, the mapping u → g (t , u, λn ) is continuous (more details can be found in Theorem 3.33 of Attouch [54] or Proposition 3 in Hu and Papageorgiou [23]). Consider the following impulsive problem y0 (t ) − Ay(t ) = g (t , y, λn ), a.e. t ∈ [0, b], ∆y|t =tk = Ik (y(tk ), λn ), k = 1, . . . , m, y(0) = y(b).

(

(4.4)

e5 ), there exists pB ∈ L1 (J , H ) such that Since (λn ) converges to λ, the set B = {λn : n ≥ 1} is compact in Λ. Hence from (H |g (t , y, λn )| ≤ pB (t )ψ(|y|),

∀ n ∈ N, y ∈ H .

By the Leray–Schauder nonlinear alternative (Lemma 3.3), we can easily prove that for every n ∈ N, Problem (4.4) has at least one solution denoted by yn , that is yn (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (yn (tk ), λ), λ + −

+

T (b − s)g (s, yn (s), λn )ds 0

k=1

X

!

b

Z

T (t − tk )Ik (yn (tk ), λn ) + −

0
t

Z

T (t − s)g (s, yn (s), λn )ds,

t ∈ J.

0

Next we prove that (yn ) converges to y. We have the estimates:

|yn (t ) − y(t )| ≤ [M k(I − T (t )) kB(H ) + M ] 2

−1

b

Z

|h(s, λn ) − v(s)|ds 0

+ [M 2 k(I − T (t ))−1 kB(H ) + M ]

b

Z

|g (s, yn (s), λn ) − h(s, λn )|ds 0 m

X

+ [M 2 k(I − T (t ))−1 kB(H ) + M ]

|Ik (yn (tk ), λn ) − Ik (y(tk ), λ)|

0
≤ [M k(I − T (t )) kB(H ) + M ] 2

−1

b

Z

d(v(s), F (s, y(s), λn ))ds 0

+ [M 2 k(I − T (t ))−1 kB(H ) + M ]

Z

b

|d(h(s, λn ), F (s, y(s), λn ))|ds

0 m

+ [M 2 k(I − T (t ))−1 kB(H ) + M ]

X k=1

ck |yn (tk ) − y(tk )|.

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

703

Then sup |yn (t ) − y(t )| ≤

[M 2 k(I − T (t ))−1 kB(H ) + M ]

Rb

|d(v(s), F (s, y(s), λn ))|ds  ·  m P 1 − [M 2 k(I − T (t ))−1 kB(H ) + M ] kpB kL1 + ck

t ∈[0,b]

0

k =1

e4 ) and the fact that the mapping λ → F (t , y(t ), λ) is l.s.c., we deduce that the mapping λ → d(v(s), F (t , y(t ), λ)) Using (H is u.s.c. Since v(t ) ∈ F (t , y(t ), λ), by Fatou’s Lemma, and the convergence of (λn ) to λ, we obtain that b

Z

|d(v(s), F (s, y(s), λn ))|ds → 0,

as n → ∞.

0

Hence kyn − ykPC −→ 0, as n → +∞. Noting that yn ∈ S (λn ), n ∈ N, we infer that S (λ) ⊂ lim infn→+∞ S (λn ), as claimed.  5. Filippov’s theorem The family of all nonempty closed and decomposable subsets of L1 (J , E ) is denoted by D . Let S 6= ∅ be a nonempty set. The following result is due to [55]. Lemma 5.1. Consider a l.s.c. multi-valued map G : S → D and assume that φ, ψ : S → L1 (J , Rn ) are continuous maps such that for every s ∈ S, the set H (s) = {u ∈ G(s) : |u(t ) − φ(s)(t )| < ψ(s)(t ), a.e. t ∈ J } is nonempty. Then the map H : S → D is l.s.c. and admits a continuous selection. The main result in this section is a Filippov type result for Problem (1.1). Filippov’s Theorem yields an estimate of the distance of a given solution to the solution set of a problem providing a kind of Gronwall’s inequality (see also [56] Thm. 4.5, p. 91). The problem was investigated by Filippov [57] in 1967 for finite-dimensional first-order differential inclusions and later by Frankowska [58] in 1990 for first-order semilinear differential inclusions; see e.g. Aubin and Cellina [22] (Thm. 1, p. 120), Aubin and Frankowska [13] (Thm. 10.4.1, p. 401), and also [59] for second-order differential inclusions. Theorem 5.1. In addition to (H1 ), (A2 ), (A3 ), assume that F : J × E −→ Pcp (E ) satisfies the following condition:

(H2 ) there exist a continuous mapping g (·) : PC → L1 (J , E ) and x ∈ PC a mild solution of the problem:  0 x (t ) − Ax(t ) = g (x)(t ), a.e. t ∈ J \ {t1 , . . . , tm }, x(tk+ ) − x(tk ) = Ik (x(tk− )), k = 1, . . . , m, x(0) = x(b) such that there exists r ∈ L1 (J , R+ ) with d(g (x)(t )), F (t , x(t )) < r (t ),

a.e. t ∈ J .

If R 2klk1 +

m X

! ck

< 1,

k=1

where R := Meωb M ωb k(I − T (b))−1 kB(E ) + 1 ,




then Problem (1.1) has at least one solution y satisfying the estimates

ky − xkPC ≤

2R(e H + kg (x)k1 ) 1−R

m P

ck

k=1

and

|y0 (t ) − Ay(t ) − g (x)(t )| ≤ 2e H kr k1 p(t ) + |l(t )|, where



m P

1 − R klk1 +

e H =

 ck

k=1

 1 − R 2klk1 +

m P k=1

· ck

a.e. t ∈ J ,

(5.1)

704

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

Remark 5.1. Assumption (H2 ) is satisfied for instance if g is an L1 -selection of the multi-valued map F . For this, see the proof in Step 1 with f1 as a nonlinear right-hand side. Proof. Step 1. Let y0 := x and f0 := g, that is y0 (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y0 (tk )) +

!

b

Z

T (b − s)f0 (y0 )(s)ds

T (t − tk )Ik (y0 (tk ))

0 < tk < t

0

k=1

X

+

t

Z

T (t − s)f0 (y0 )(s)ds.

+ 0

Let G1 : PC → P (L1 (J , E )) be defined by G1 (y) = {v ∈ L1 (J , E ) : v(t ) ∈ F (t , y(t )), a.e. t ∈ J } and e G1 : PC → P (L1 (J , E )) be given by

e G1 (y) = {v ∈ G1 (y) : |v(t ) − g (y0 )(t )| < p(t )|y(t ) − y0 (t )| + p(t )}. Since, from Assumption (H2 ), the mapping t → F (t , y(t )) is, for fixed y ∈ PC , a measurable multi-valued map, then by Corollary 2.1, there exists a measurable selection v1 (t ) ∈ F (t , y(t )) a.e. t ∈ J such that, using (A3 )

|v1 (t ) − g (y0 )(t )| ≤ d(g (y0 )(t ), F (t , y(t ))) ≤ d(g (y0 )(t )), F (t , y0 (t )) + Hd (F (t , y0 (t )), F (t , y(t ))) < r (t ) + l(t )|y0 (t ) − y(t )|. e Hence v1 ∈ G1 (y) 6= ∅ and thus e G1 is well defined. By Lemma 3.11, F is of lower semi-continuous type. Then G1 is l.s.c. and has decomposable values. Therefore e G1 is l.s.c. with decomposable values. By Lemma 3.10, there exists a continuous selection f1 : PC → L1 (J , E ) such that f1 (y) ∈ e G1 (y) for all y ∈ PC . Let the problem  0 y (t ) − Ay(t ) = f1 (y)(t ), t ∈ J , t 6= tk , k = 1, . . . , m, (5.2) ∆y| = I (y(tk− )), k = 1, . . . , m, y(0)t ==tk y(b).k Define an operator N 1 : PC → PC by N 1 (y)(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )I (y(tk )) +

X

T (b − s)f1 (y)(s)ds

T (t − tk )Ik (y(tk− )),

t

Z

T (t − s)f1 (y)(s)ds

+

0

k =1

+

!

b

Z

0

for t ∈ [0, b].

0
As in Theorem 3.1, we can prove that N 1 is completely continuous. To establish a priori estimates on all possible solutions, let y ∈ PC be such that y = λN 1 (y) for some λ ∈ (0, 1). By Proposition 2.2, there exist constants ω ≥ 0 and M ≥ 1 such that 2 2ωb

|y(t )| ≤ M e

k(I − T (b)) kB(E ) −1

Z m X (ck |y(tk )| + |Ik (0)|) +

! |f1 (y)(s)|ds + Meωb

0

k=1

+ Meωb

b

m X [ck |y(tk− )| + |Ik (0)|] k=1

kykPC

! m X ≤ Me (Me k(I − T (b)) kB(E ) + 1) (ck kykPC + |Ik (0)|) + klk1 . ωb

−1

k=1

Hence



Meωb (Meωb k(I − T (b))−1 kB(E ) + 1) klk1 +

kykPC ≤

m P

 |Ik (0)|

k=1 m

1 − Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

P k=1

Set U := {y ∈ PC : kykPC < M + 1}

:= M . ck

|f1 (y)(s)|ds 0

and so ωb

t

Z

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

705

and consider the operator N 1 : U → PC . From the choice of U, there is no y ∈ ∂ U such that y = λN 1 (y) for some λ ∈ (0, 1). As a consequence of the Leray–Schauder nonlinear alternative (Lemma 3.3), we deduce that N 1 has a fixed point y in U which is a solution of Problem (5.2). This solution, denoted by y1 , satisfies for t ∈ J y1 (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y1 (tk )) +

!

b

Z

T (b − s)f1 (y1 )(s)ds

T (t − tk )Ik (y1 (tk ))

0
0

k=1

X

+

t

Z

T (t − s)f1 (y1 )(s)ds.

+ 0

Writing

|f1 (y1 ) − f0 (y0 )| ≤ |f1 (y1 ) − f1 (y0 )| + |f1 (y0 ) − f0 (y0 )|, and using the fact that f1 (y1 )(t ) ∈ F (t , y1 (t )),

f1 (y0 )(t ) ∈ F (t , y0 (t )),

we get, for every t ∈ J, the following estimates

|y1 (t ) − y0 (t )| ≤ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

b

Z

|f1 (y1 )(s) − f0 (y0 )(s)| ds 0

+ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

m X

|Ik (y1 (tk )) − Ik (y0 (tk ))|

k=1

ωb

ωb

≤ Me (Me k(I − T (b)) kB(E ) + 1) −1

b

Z

l(s)|y1 (s) − y0 (s)| ds 0

+ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)kr k1 m X + Meωb (Meωb k(I − T (b))−1 kB(E ) + 1) ck |y1 (tk ) − y0 (tk )|. k=1

Passing to the supremum over J, we get the bound: Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)kr k1

ky1 − y0 kPC ≤ 1−

Meωb (Meωb k(I

−

T (b))−1 k



B(E )

+ 1) klk1 +

m P

· ck

k=1

Step 2. Define the set-valued map G2 : PC → P (L1 (J , E )) by G2 = G1 and

e G2 (y) = {v ∈ G2 (y) : |v(t ) − f1 (y1 )(t )| < l(t )|y(t ) − y1 (t )| + l(t )|y1 (t ) − y0 (t )|}. Since, for fixed y ∈ PC , the map t → F (t , y(t )) is measurable, then there exists, by Corollary 2.1, a function v2 ∈ e G2 which is a measurable selection of F (·, y(·)) such that

|v2 (t ) − f1 (y1 )(t )| ≤ ≤ ≤ <

d(f1 (y1 )(t ), F (t , y(t ))) Hd (F (t , y1 (t )), F (t , y(t ))) l(t )|y1 (t ) − y(t )| l(t )|y1 (t ) − y(t )| + l(t )|y1 (t ) − y0 (t )|.

Then v2 ∈ e G2 (y) 6= ∅. Arguing as in Step 1, we can prove that e G2 has at least one continuous selection denoted by f2 . Hence, there exists y2 ∈ PC solution of the problem  0 y (t ) − Ay(t ) = f2 (y)(t ), a.e. t ∈ J \ {t1 , . . . , tm }, (5.3) y(tk+ ) − y(tk ) = Ik (y(tk− )), k = 1, . . . , m, y(0) = y(b), that is y2 (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y2 (tk )) +

t

Z

T (t − s)f2 (y2 )(s)ds,

+ 0

T (b − s)f2 (y2 )(s)ds 0

k=1

t ∈ J.

!

b

Z

+

X 0
T (t − tk )Ik (y2 (tk ))

706

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

f2 ∈ e G2 implies that |f2 (y2 )(s) − f1 (y1 )(s)| ≤ l(s)|y2 (s) − y(s)| + l(s)|y1 − y0 |. Therefore

|y2 (t ) − y1 (t )| ≤ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

b

Z

|f2 (y2 )(s) − f1 (y1 )(s)| ds 0 m X

+ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

ck |y2 (tk ) − y1 (tk )|

k=1

ωb

ωb

≤ Me (Me k(I − T (b)) kB(E ) + 1) −1

b

Z

l(s)|y2 (s) − y1 (s)| ds 0

+ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1) +Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

Z

b

l(s)|y1 (s) − y0 (s)| ds

0 m

X

ck |y2 (tk ) − y1 (tk )|.

k=1

Setting R := Meωb M ωb k(I − T (b))−1 kB(E ) + 1 ,




we get the bound Rklk1 ky1 − y0 kPC

ky2 − y1 kPC ≤

 1 − R klk1 +

m P

 ck

k=1

R2 klk1 kr k1

≤ 

 1 − R klk1 +

m P

2 · ck

k=1

Step 3. Define the set-valued map G3 : PC → P (L1 (J , E )) by G3 = G2 = G1 and

e G3 (y) = {v ∈ G3 (y) : |v(t ) − f2 (y2 )(t )| < l(t )|y(t ) − y2 (t )| + l(t )|y2 (t ) − y1 (t )|}· Arguing as we did for e G2 , we can prove that e G3 is an l.s.c. type multi-valued map with nonempty decomposable values. Thus there exists a continuous selection f3 (y) ∈ e G3 (y) for all y ∈ PC . Hence we can prove existence of a solution y3 such that y3 (t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y3 (tk )) +

!

b

Z

T (b − s)f3 (y3 )(s)ds

X

T (t − tk )Ik (y3 (tk ))

0 < tk < t

0

k=1

+

t

Z

T (t − s)f3 (y3 )(s)ds,

+

t ∈J

0

solution of the problem

 0 y (t ) − Ay(t ) = f3 (y)(t ), y(t + ) − y(t ) = Ik (y(tk− )), y(0k) = y(b).k

a.e. t ∈ J \ {t1 , . . . , tm }, k = 1, . . . , m,

(5.4)

Since f3 (y) ∈ e G3 (y), we obtain the estimates


|y3 (t ) − y2 (t )| ≤ Meωb M ωb k(I − T (b))−1 kB(E ) + 1 | + Meωb M ωb k(I − T (b))−1 kB(E ) + 1

Z

t

|f3 (y3 (s)) − f2 (y2 (s))| ds

0 m


X

ck |y3 (tk ) − y2 (tk )|

k=1

≤ Me

ωb

M

ωb

k(I − T (b)) kB(E ) + 1

t

Z

|l(s)(y3 (s)) − (y2 (s))| ds Z t
M ωb k(I − T (b))−1 kB(E ) + 1 |l(s)(y2 (s)) − (y1 (s))| ds −1




0

+ Meωb

0

+ Me

ωb

M

ωb

k(I − T (b)) kB(E ) + 1 −1

m
X k=1

ck |y3 (tk ) − y2 (tk )|.

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

707

From the estimates of |y2 (t ) − y1 (t )| and |y1 (t ) − y0 (t )| above, we infer the bound R3 klk2 kr k

ky3 − y2 kPC ≤ 

 1 − R klk1 +

m P

3 ·

k=1

Step 4. Repeating the process for n = 1, 2, . . . , we finally arrive at Rn klkn−1 kr k

kyn − yn−1 kPC ≤ 



1 − R klk1 +

m P

n ·

(5.5)

k=1

By induction, suppose that (5.5) holds for some n and let e Gn+1 be a multi-valued map defined by

e Gn+1 (y) = {v ∈ Gn+1 (y) : |v(t ) − fn (yn )(t )| < p(t )|y(t ) − yn (t )| + p(t )|yn (t ) − yn−1 (t )|}. e Since Gn+1 is a l.s.c type multi-valued map, there exists a continuous function fn+1 (y) ∈ e Gn+1 (y) which allows us to define ! Z m b X −1 T (b − tk )Ik (yn+1 (tk )) + T (b − s)fn+1 (yn+1 )(s)ds yn+1 (t ) = T (t )(I − T (b)) 0

k=1

+

X

T (t − tk )Ik (yn+1 (tk )) +

t

Z

0 < tk < t

T (t − s)fn+1 (yn+1 )(s)ds,

t ∈ J.

(5.6)

0

We have

|yn+1 (t ) − yn (t )| ≤ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

b

Z

|fn+1 (yn+1 )(s) − fn (yn )(s)| ds 0

+ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

m X

ck |yn (tk ) − yn+1 (tk )|

k =1

≤ Meωb M ωb k(I − T (b))−1 kB(E ) + 1




b

Z

|l(s)(yn+1 (s)) − (yn (s))| ds 0

+ Meωb M ωb k(I − T (b))−1 kB(E ) + 1 + Meωb M ωb k(I − T (b))−1 kB(E ) + 1




Z

b

|l(s)(yn (s)) − (yn−1 (s))| ds

0 m


X

ck |yn+1 (tk ) − yn (tk )|.

k=1

Hence

kyn+1 − yn kPC ≤

Rklk1 kyn+1 − yn kPC

 1 − R klk1 +

m P

·

ck

k=1

Using (5.5), we obtain that

kyn+1 − yn kPC ≤ 

Rn klkn1+1 kr k1



1 − R klk1 +

m P

n ck

k=1

 =

Rklk1

1 − R klk1 +

m X

!!!n ck

klk1 kr k1 ·

(5.7)

k=1

Hence, (5.5) holds for all n ∈ N, and so (yn )n∈N is a Cauchy sequence in PC , converging uniformly to a function y ∈ PC , as n → +∞. Moreover, from the definition of e Gn (y), n ∈ N, we deduce

|fn+1 (yn+1 )(t ) − fn (yn )(t )| ≤ l(t )|yn (t ) − yn−1 (t )| + l(t )|yn+1 (t ) − yn (t )| a.e. t ∈ J . Therefore, for almost every t ∈ J, {fn (yn )(t ) : n ∈ N} is also a Cauchy sequence in E and thus converges almost everywhere to some measurable function f (·) in E. Moreover, since f0 = g, we have successively the estimates:

|fn (yn )(t )| ≤ |fn (yn )(t ) − fn−1 (yn−1 )(t )| + |fn−1 (yn−1 )(t ) − fn−2 (yn−2 )(t )| + · · ·

708

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

+ |f2 (y2 )(t ) − f1 (y1 )(t )| + |f1 (y1 )(t ) − f0 (y0 )(t )| + |f0 (y0 )(t )| n X ≤2 l(t )|yk (t ) − yk−1 (t )| + |f0 (y0 )(t )| k=1

≤ 2l(t )

∞ X

|yk (t ) − yk−1 (t )| + |g (x)(t )|

k=1

≤ 2l(t )

∞ X

 Rklk1

1 − R(klk1 +

k=1

m X

!!k ck )

+ |g (x)(t )|.

k=0

Hence

|fn (yn )(t )| ≤ 2l(t )e H + |g (x)(t )|,

a.e. t ∈ J ,

(5.8)

where



m P

1 − R klk1 +

 ck

k=1

e H :=



m P

1 − R 2klk1 +

· ck

k=1

From (5.8) and the Lebesgue Dominated Convergence Theorem, we conclude that (fn (yn )) converges to f (y) in L1 (J , E ). Passing to the limit in (5.6), we obtain a solution to Problem (1.1), namely y(t ) = T (t )(I − T (b))

−1

m X

T (b − tk )Ik (y(tk )) +

!

b

Z

T (b − s)f (y)(s)ds

T (t − tk )Ik (y(tk ))

0
0

k=1

X

+

t

Z

T (t − s)f (y)(s)ds,

+

for t ∈ J .

0

Next, we give estimates for |y0 (t ) − Ay(t ) − g (x)(t )| and |x(t ) − y(t )|, t ∈ J. We have

|y0 (t ) − Ay(t ) − g (x)(t )| = |f (y)(t ) − f0 (x)(t )| ≤ |f (y)(t ) − fn (yn )(t )| + |fn (yn )(t ) − f0 (x)(t )| n X ≤ |f (y)(t ) − fn (yn )(t )| + |fk (yk )(t ) − fk−1 (yk−1 )(t )| k=1

≤ |f (y)(t ) − fn (yn )(t )| + 2

n X

l(t )|yk (t ) − yk−1 (t )| + |l(t )|.

k=1

Using (5.7) and passing to the limit as n → +∞, we get

|y0 (t ) − Ay(t ) − g (x)(t )| ≤ 2l(t )

∞ X

|yk (t ) − yk−1 (t )| + |l(t )|

k =1

≤ 2l(t )

∞ X

 Rklk1

1 − R klk1 +

k =0

m X

!!!k ck

klk1 kr k1 + |l(t )|,

k=1

hence

|y0 (t ) − Ay(t ) − g (x)(t )| ≤ (2klk1 kr k1e H + 1)l(t ),

a.e. t ∈ J .

Similarly, ωb

ωb

|x(t ) − y(t )| ≤ Me (Me k(I − T (b)) kB(E ) + 1) −1

b

Z

|f (y)(s) − f0 (y0 )(s)| ds 0

+ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

m X

ck |y(tk ) − x(tk )|

k=1

≤ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

b

Z

|f (y)(s) − fn (yn )(s)|ds 0

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

+ Meωb (Meωb k(I − T (b))−1 kB(E ) + 1) + Meωb (Meωb k(I − T (b))−1 kB(E ) + 1)

Z

709

b

|fn (yn )(s) − g (x)(s)|ds

0 m

X

ck |x(tk ) − y(tk )|.

k=1

As n → ∞, we finally arrive at 2R(e H kpk1 + |g (x)|)

kx − ykPC ≤ 1−

Meωb (Meωb k(I

−

T (b))−1 k

B(E )

+ 1)

m P

, ck

k=1

ending the proof of the theorem.



6. Existence of solutions: 1 6∈ ρ(T (b)) In the particular case where Ay = λy, some basic results in the theory of periodic boundary value problems for first-order impulsive differential equations and inclusions may be found in [60,11,61,62,42] and in the references therein. Our goal in this section is to give an existence result when 1 6∈ ρ(T (b)) using the topological degree theory combined with the Poincaré operator properties (see [63]). Consider the problem

 0 (y − Ay)(t ) ∈ ϕ(t , y(t )), y(t + ) − y(t − ) = I (y(tk− )), y(0k) = y(b)k ∈ E , k

a.e. t ∈ J \ {t1 , . . . , tm }, k = 1, . . . , m,

(6.1)

where ϕ : J × E −→ P (E ) is a multi-valued map. First, we begin by recalling the definition of retracts, contractible and Rδ -sets. More details can be found in [64,65,31,66]. In what follows (X , d) and (Y , d0 ) stand for two metric spaces. 6.1. Rδ -sets Definition 6.1. (a) A subset A ⊂ X is a retract of the space X if there exists a retraction r : X −→ A, i.e. r (x) = x, for every x ∈ A. In other words, A is a retract of X if and only if the identity map Id|A over A possesses a continuous extension onto X . (b) A is a neighborhood retract of X if there exists an open subset U ⊂ X such that A ⊂ U and A is a retract of U . (c) We say that X ∈ ANR (absolute neighborhood retract) if for any space Y and for any embedding h : X −→ Y , the set h(X ) is a retract of Y . By an embedding of a space X into Y , it is meant any homeomorphism h : X −→ Y such that h(X ) is a closed subset of Y . Definition 6.2. A space X is called absolute retract (in short X ∈ AR) provided that for every space Y , every closed subset B ⊆ Y and any continuous map f : B → X , there exists a continuous extension e f : Y → X of f over Y , i.e. e f (x) = f (x) for every x ∈ B. In other words, for every space Y and for any embedding f : X −→ Y , the set f (X ) is a retract of Y . Definition 6.3. A set A ∈ P (X ) is called contractible space provided there exist a continuous homotopy h : A × [0, 1] → A and x0 ∈ A such that (a) h(x, 0) = x, for every x ∈ A, (b) h(x, 1) = x0 , for every x ∈ A, i.e. if the identity map A −→ A is homotopic to a constant map (A is homotopically equivalent to a point). Remark 6.1. It is clear that if A is a retract of X , then A is a closed subset of X . Note that if A ∈ Pc v,cl (X ), then A is contractible. Still the class of contractible sets is much larger than the class of closed convex sets. Finally from [21], Proposition 2.15, we know that if X ∈ AR, then it is a contractible space. Definition 6.4. A compact nonempty space X is T called Rδ -set provided there exists a decreasing sequence of compact ∞ nonempty contractible spaces {Xn } such that X = n=1 Xn . 6.2. A nonlinear alternative Set K (r ) = {x ∈ X : kxk ≤ r },

S (r ) = ∂ K (r ),

and X ∗ = X \ {0},

710

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

where K (r ) is the closed ball in X with center x and radius r and ∂ K (r ) stands for the boundary of K (r ) in X . For any ANR-space X , let J (K (r ), X ) = {F : K (r ) → P (X ) : F is u.s.c. and has Rδ -set values}. Moreover, for any continuous function f : X → X where X ∈ ANR, define the sets

 Jf (K (r ), X ) = φ : K (r ) → P (X ), φ = f ◦ F with F ∈ J (K (r ), X ), φ(S (r )) ⊂ E ∗ . CJ (K (r ), X ) = ∪{Jf (K (r ), X ), f : X −→ X }. CJC (K (r ), X ) = {Φ : K (r ) → P (X ) : Φ = f ◦ F with F ∈ J (K (r ), X ), Fix Φ ∩ S (r ) = ∅ and Φ (K (r )) is compact}. In what follows, for given Φ ∈ CJ (K (r ), X ), we shall associate the vector field φ : K (r ) → X , φ = j − Φ defined by

φ(x) = j(x) − Φ (x),

∀ x ∈ K (r ),

where j : K (r ) → X , j(x) = x is the inclusion map. Note if Φ ∈ CJC (K (r ), X ), then φ ∈ CJ (K (r ), X ) (Proposition 28.1 in [15]), hence φ(S (r )) ⊂ X ∗ . Let CJCV (K (r ), X ) = {φ ∈ CJ (K (r ), X ) : φ is a compact field associated with some Φ ∈ CJC (K (r ), X )}. It is well known (see e.g., [15]) that for multi-valued maps in this class, one can define a notion of topological degree. To this end, we need an appropriate concept of homotopy in CJCV (K (r ), X ). We omit the details. The following result is due to Górniewicz. Proposition 6.1 ([15, Proposition 28.9]). Let X ∈ AR and Φ : X → P (X ) a compact u.s.c. map of the form: f

F

Φ = f ◦ F : X −−−−−−→P (Y )−−−−−−→X , where F is u.s.c. with Rδ -set values, Y ∈ ANR and f is continuous. Then Fix(Φ ) 6= ∅. In what follows, our main ingredient tool is the following nonlinear alternative. The finite-dimensional version is given in [15], Proposition 26.8. Proposition 6.2 (Nonlinear Alternative). Let φ : X → P (X ) be a multi-valued map associated with Φ and let M > 0 be such that φ ∈ CJCV (K (M ), X ). Then (a) either there exist λ ∈ (0, 1) and x ∈ S (M ) such that x ∈ λΦ (x). (b) or Fix(Φ ) 6= ∅ and hence 0 ∈ φ.

ˆ defined by Proof. Since φ ∈ CJCV (K (r ), X ), then Φ ∈ CJC (K (r ), X ). Consider the multi-valued map Φ ˆ = f ◦ F ◦ r : X −−−−r−−→K (M )−−−−F−−→P (X )−−−−f−−→X , Φ where r is the radial retraction r (x) =

 x,

kx k ≤ M

Mx



kx k

,

kx k > M .

We know that X ∈ AR and the composition F ◦ r is u.s.c. with Rδ values. Furthermore X ∈ ANR, f is a continuous function and

ˆ (X ) = (f ◦ F ◦ r )(X ) = (f ◦ F )(r (X )) = (f ◦ F )(K (M )) = Φ (K (M )). Φ ˆ ˆ This implies that  Φ is compact. From Proposition 6.1, there exists x ∈ X such that x ∈ Φ (x). Assume that kxk ≥ M; then x ∈ (f ◦ F ) kMx xk Mx

kx k

∈

M

kx k

which implies that

Φ

Mx

!

kx k

,

leading to a contradiction with Fix(λΦ ) ∩ S (r ) = ∅. Then Fix Φ 6= ∅ and 0 ∈ ϕ(x).



S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

711

6.3. A Poincaré translation operator By a Poincaré operator for a differential system, we mean the translation operator (or the Poincaré–Andronov or Levinson operator, or simply the T -operator [67]) along the trajectories of the associated differential system, and the first return (or section) map defined on the cross section of the torus by means of the flow generated by the vector field. Both of these operators are single valued when the uniqueness of solutions of initial value problems is assumed. In the absence of uniqueness, it is often possible to approximate the right-hand sides of the given system by locally Lipschitzian ones (implying uniqueness), and then apply a standard limiting argument which may be rather complicated for discontinuous right-hand sides. However, set-valued analysis allows us to handle effectively such problems. For further details, we refer to the monographs [21,15]. Given a Carathédory map G : J × X → P (X ), define the multi-valued map SG : X → P (PC ) by SG (x) = {y : yis a solution of Problem (6.2)}. For some positive real number b, consider the operator Pb defined by Pb = Ψb ◦ SG where Ψb

SG

Pb : X −−−−−−→P (PC )−−−−−−→P (X ) and

Ψb (y) = y(0) − y(b). Pb is called the Poincaré translation map associated with the impulsive Cauchy problem

 0 (y − Ay) ∈ G(t , y(t )), y(t + ) − y(t − ) = Ik (y(tk− )), y(0k) = x ∈ kX .

a.e. t ∈ J \ {t1 , . . . , tm }, k = 1, . . . , m,

(6.2)

The following lemma is easily proved. Lemma 6.1. Let G : J × X → Pc v,cp (X ) be a Carathédory multi-valued map. Then the periodic problem (6.2) has a solution if and only if for some x ∈ X , 0 ∈ Pb (x), where Pb is the Poincaré map associated with (6.2). We define upper Scorza–Dragoni maps (see [15,68,23] or [22], p. 86). Definition 6.5. We say that a multi-valued map F : J × X → Pcl (X ) has the upper-Scorza–Dragoni property if, given δ > 0, there is a closed subset Aδ ⊂ J such that the measure µ(J \ Aδ ) ≤ δ and the restriction of F to Aδ × X is u.s.c. Definition 6.6. We say that a multi-valued map F : X → P (Y ) is σ -selectionable if there exists a decreasing sequence of compact valued u.s.c. multi-valued maps Fn : X → Y satisfying: (a) Fn has aT selection, for all n ≥ 0, (b) F (x) = n≥0 Fn (x), for all x ∈ X . F : X → P (Y ) is σ -Ca-selectionable if the multi-valued maps Fn has a Carathédory selection, for all n ≥ 0 (Fn are called Ca-selectionable). Definition 6.7. A mapping F : X → P (Y ) is mLL-selectionable provided there exists a measurable, locally Lipschitzian map f : X → Y such that f ⊂ F . It is σ -mLL-selectionable if Fn in Definition 6.6 are mLL-selectionable. The next two lemmas are crucial for the sequel. Lemma 6.2 (See [15, Theorem 19.19]). Let E , E1 be two separable Banach spaces and let F : [a, b] × E → Pcp,c v (E1 ) be an upper Scorza–Dragoni map. Then F is σ -Ca-selectionable, the maps Fn : [a, b] × E → P (E1 ) (n ∈ N) are upper Scorza–Dragoni and we have

! Fn (t , e) ⊂ conv

[

Fn (t , x) .

x∈E

Moreover, if F is integrably bounded, then F is σ -mLL-selectionable. Lemma 6.3 ([39, Theorem 6.18]). Let G: J × X → Pcp,c v (X ) be a Carathéodory σ -Ca-selectionable multi-valued map. Assume Pk=m that (A2 ) holds together with k=1 ck < 1 and

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S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

(R1 ) there exist a function p∗ ∈ L1 (J , R+ ) and a continuous nondecreasing function ρ : [0, ∞) → [0, ∞) such that kG(t , y)k ≤ p∗ (t )ρ(kyk),

for each (t , y) ∈ J × X

with

Z

b

p∗ (s)ds <

∞

Z 1

0

du

ρ(u)

·

(R2 ) There exist constants c k > 0 and continuous functions ψk : R+ → R+ such that |Ik (x)| ≤ c k ψk (|x|) for each x ∈ X , k = 1, . . . , m. Then for every x ∈ X , the set SG (x) is an Rδ -set. 6.4. Existence result Given an ANR space X , we are in position to state our main existence result. T (t ) is assumed uniformly continuous. Theorem 6.1. Let G : J × X → Pcp,c v (X ) be a Carathéodory multi-valued map with the upper-Scorza–Dragoni property. In Pk=m addition to conditions (R1 )–(R2 ) and (A2 ) with k=1 ck < 1, assume that

(R3 ) here exists p ∈ L1 ([0, b], R+ ) such that for every bounded subset D in E χ (F (t , D)) ≤ p(t )χ (D) and there exist Lk > 0, k = 0, . . . , m such that qk : = 2Me

tk+1

Z

ωtk+1

sup

t ∈[tk ,tk+1 ]

e−Lk (t −s) p(s)ds < 1,

k = 0, . . . , m.

tk

Here χ is the Hausdorff MNC. Then Problem (6.1) has at least one solution. Proof. (a) From Theorem 5.11 in [39], we know that the problem (6.2) has at least one solution, and that for each x ∈ X the set SG (x) 6= ∅ is compact. From Lemma 6.2, G is σ -Ca-selectionable and so, by Lemma 6.3, for every x ∈ X , SG (x) is an Rδ -set. In addition, the mapping

Ψ : PC → X defined by y 7→ Ψ (y) = y(0) − y(·) is continuous. Indeed, let (yn ) be a sequence such that yn → y in PC . Then,

|Ψ (yn )(t ) − Ψ (y)(t )| ≤ 2kyn − ykPC → 0,

as n → ∞.

(b) Using the conditions (R1 )–(R2 ), we can prove, as mentioned at the end of the proof of Theorem 3.2, that there exists M∗ > 0 independent of x such that, for every y solution of the problem (6.2), we have kykPC ≤ M∗ . Let K = {x ∈ X : |x| ≤ 2M∗ + 1}. We have to show that Pb ∈ CJCV (K , X ). Let x ∈ Pt (x) = λ(Ψt ◦ SG )(x) for some λ ∈ (0, 1). Then, there exists y ∈ PC such that y ∈ SG (x). This yields y(0) = x and x = λ(x − y(t )), x ∈ S (2M∗ + 1). For t ∈ J, we have the estimates

|x| ≤ |y(0)| + |y(t )| ≤ 2kykPC ≤ 2M∗ which is contradiction to |x| = 2M∗ + 1. (c) Making use of Lemma 2.7, we will show that SG is u.s.c. by proving that the graph of SG

ΓG := {(x, y) : y ∈ SG (x)} is closed. Let (xn , yn ) ∈ ΓG , i.e., yn ∈ SG (xn ) and suppose that (xn , yn ) → (x, y), as n → ∞. Since yn ∈ SG (xn ), there exists vn ∈ SG,yn such that yn (t ) = T (t )xn +

t

Z

T (t − s)vn (s)ds +

X

T (t − tk )Ik (yn (tk− )),

0
0

Since (xn , yn ) converges to (x, y), there exists M1 > 0 such that

|xn | ≤ M1 for every n ∈ N. Moreover

kyn kPC ≤ M∗ ,

for every n ∈ N.

So

|vn (t )| ≤ p(t )ρ(M∗ ),

t ∈ J.

t ∈ J.

(6.3)

S. Djebali et al. / Mathematical and Computer Modelling 52 (2010) 683–714

713

Hence xn ∈ M1 B(0, 1) and vn (t ) ∈ p∗ (t )ρ(M∗ )B(0, 1) := H (t ) a.e. t ∈ J. The multi-valued map χ : J → Pcp,c v (X ) is integrably bounded. Since {vn (·) : n ≥ 1} ∈ H (·) is bounded, we may pass to a subsequence if necessary to obtain that (vn ) converges weakly to v in L1w (J , X ). Arguing as in the proof of Lemma 3.9, we deduce that for each t ∈ J, the sequence (yn ) converges to y with y(t ) = T (t )x +

t

Z

T (t − s)v(s)ds + 0

X

T (t − tk )Ik (y(tk− )),

t ∈ J.

(6.4)

0
Thus, y ∈ SG (x). (c) It remains to show that SG (.) maps compact sets into relatively compact sets of PC . Let K be a compact set in X and let (yn ) ⊂ SG (K ). Then there exists a sequence {xn : n ∈ N} ⊂ K satisfying (6.3). Since (xn ) is a compact sequence, there exists a subsequence of (xn ) converging to some limit x. From the boundedness of the sequence (yn ) and arguing as in the proof of Theorem 5.11 in [39], we can show that (yn ) is equicontinuous in PC . The Arzelá–Ascoli Theorem then implies that there exists a subsequence of (yn ) converging to some limit y in PC . By a similar argument to the one above, we can prove that y satisfies (6.4) and y ∈ SG (x). Hence Pb ∈ CJCV (K , PC ). By the nonlinear alternative (Proposition 6.2), we conclude that Fix (j − Pb ) 6= ∅, i.e. 0 ∈ Pb , ending the proof of the theorem.  7. Concluding remarks In this work, we have obtained some existence results of mild solutions for Problem (1.1) in the convex and nonconvex cases when 1 ∈ ρ(T (b)). Indeed, in this case, the periodic problem may be formulated as a fixed point problem for a multi-valued operator. Then, we have mainly appealed to some ingredients from multi-valued analysis, topological fixed point theory, and measure of noncompactness to get existence of mild solutions. The topological properties of the solution set (closedness and compactness) are also studied (see [69,70] for the case of differential inclusions); and a result of Filippov’s theorem type is presented. The dependence upon a parameter is also investigated. The nonlinearity satisfies Lipschitz conditions with respect to the Hausdorff distance in generalized metric spaces or Nagumo–Bernstein type growth conditions. 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