Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

Nonlinear Analysis 74 (2011) 2141–2169

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Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces Smaïl Djebali a , Lech Górniewicz b,c,∗ , Abdelghani Ouahab d a

Laboratory of EDP & HM, Department of Mathematics, E.N.S., PB 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

Laboratory of Mathematics, Sidi-Bel-Abbès University, PB 89, 22000 Sidi-Bel-Abbès, Algeria

article

info

Article history: Received 15 June 2010 Accepted 8 November 2010 MSC: 34A37 34A60 34K30 34K45 47H10 54C60 54C65 55M15

abstract In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, AR and Rδ ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals. © 2010 Elsevier Ltd. All rights reserved.

Keywords: Impulsive differential inclusions Solution set Compactness Terminal problem Limit inverse systems Fréchet spaces Contractible Acyclic AR Rδ

1. Introduction Differential equations with impulses were considered for the first time by Milman and Myshkis [1] and this was then followed by a period of active research which culminated in the monograph by Halanay and Wexler [2]. Many phenomena and evolution processes in physics, chemical technology, population dynamics, and natural sciences may change state abruptly or be subject to short-term perturbations (see for instance [3–5] and the references therein). These perturbations may be seen as impulses. Impulsive problems arise also in various applications in communications, chemical technology,

∗

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

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.11.020

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mechanics (jump discontinuities in velocity), electrical engineering, medicine, and biology. A comprehensive introduction to the basic theory is well developed in the monographs by Bainov and Simeonov [6], Laskshmikantham et al. [7], Samoilenko and Perestyuk [8] and the survey paper by Rogovchenko [9]. For instance, in the periodic treatment of some diseases, impulses correspond to the administration of a drug treatment. In environmental sciences, impulses correspond to seasonal changes of the water level of artificial reservoirs. Their models are described by impulsive differential equations and inclusions. More recently, the questions of the existence of solutions and other mathematical properties of solutions to differential equations and inclusions have been extensively studied and have attracted much attention; important contributions have been obtained already (see the monographs and the recent papers [10,6,11–15,7,16] among others). In the case of problems posed on the half-line, the model problem reads

 ′ y (t ) ∈ F (t , y(t )), 1y|t =tk = Ik (y(tk− )),  y(0) = a,

a.e. t ∈ J := [0, ∞) \ {t1 , t2 , . . . , }, k = 1, . . . ,

(1.1)

where F : J × Rn → P (Rn ) is a multivalued map, P (Rn ) is the family of all nonempty subsets of Rn , 0 = t0 < t1 < · · · < tk < · · · , limk→∞ tk = ∞. 1y|t =tk = y(tk+ ) − y(tk− ) where 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 at t = tk , respectively, and a ∈ Rn . Throughout this paper Ik ∈ C (Rn , Rn ) for k = 1, . . . . In 1890, Peano [17] proved that the Cauchy problem for ordinary differential equations has local solutions although the uniqueness property does not hold in general. For the case where the uniqueness does not hold, Kneser [18] proved in 1923 that the solution set is a continuum, i.e. closed and connected. In 1942, Aronszajn [19] improved this result for differential inclusions in the sense that he showed that the solution set is compact and acyclic, and he specified this continuum to be an Rδ -set. An analogous result has been obtained for differential inclusions with u.s.c. convex valued nonlinearities by De Belasi and Myjak in 1985 (see [20]). Very recently, the topological and geometric structures of solution sets for impulsive differential inclusions on compact intervals were investigated in [21–24] where contractibility, AR, acyclicity, Rδ -sets properties are given. However, the topological structures of solution sets for some Cauchy problems without impulses and posed on noncompact intervals were studied by various techniques in [25–27,20,28,29]. In the case of differential inclusions on unbounded domains, some existence results together with topological structures of solution sets have also been obtained in [30–32,26,27,14,33–38]. The goal of this paper is to study the geometric and topological structures of solution sets of problem (1.1) in some Fréchet spaces. Some auxiliary results on set-valued analysis are first gathered together in Section 2. In Section 3, we present three existence theorems for problem (1.1). Different types of growth of the nonlinearity F are considered in the case where F is u.s.c., l.s.c., Lipschitz or satisfies a Nagumo-type condition. Section 4 is devoted to investigating the topological structures of solution sets (AR, Rδ or acyclicity) in both convex and nonconvex cases. The projective limit approach is employed. This section starts with some background material from homology and algebraic topology, with some general properties of inverse systems with their limits as well as with some definitions of selectionable maps. In Section 5, we study the question of existence and the topological structure of the solution set for the corresponding terminal problem. We end the paper with some concluding remarks and a rich bibliography. 2. Preliminaries In this section, we recall some auxiliary results which are needed in this paper. Let [a, b] be an interval in R and C ([a, b], Rn ) be the Banach space of all continuous functions from [a, b] into Rn with the norm

‖y‖∞ = sup{‖y(t )‖: a ≤ t ≤ b}. Throughout this paper, ‖ · ‖ refers to the Euclidean norm in Rn . In what follows, L1 ([a, b], Rn ) denotes the Banach space of all functions y: [a, b] −→ Rn which are Lebesgue integrable with norm b

∫

‖y(t )‖dt .

‖y‖L1 = a

Let E be a metric space. Define P (E ) = {Y ⊂ E: Y ̸= ∅}, Pcl (E ) = {Y ∈ P (E ): Y closed}, Pb (E ) = {Y ∈ P (E ): Y bounded}, Pcv (E ) = {Y ∈ P (E ): Y convex}, and Pcp (E ) = {Y ∈ P (E ): Y compact}. 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.

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Definition 2.1. A multivalued map N: E ( E is called (a) γ -Lipschitz if there exists γ > 0 such that Hd (N (x), N (y)) ≤ γ d(x, y),

∀ x, y ∈ E ,

(b) a contraction if it is γ -Lipschitz with γ < 1. Notice that if N is γ -Lipschitz and E is a Banach space, then for every γ ′ > γ , N (x) ⊂ N (y) + γ ′ d(x, y)B(0, 1),

∀ x, y ∈ A ,

where B(0, 1) refers to the unit ball in E. Let (X , d) and (Y , ρ) be two metric spaces and G : X → Pcl (Y ) a multivalued 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 ̸= ∅} is closed for any closed subset V in Y . G is called lower semi-continuous (l.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 ) ̸= ∅, there exists an open neighborhood 1 M of x0 such that N ∩ G(M ) ̸= ∅, that is, if the set G− + (V ) = {x ∈ X , G(x) ∩ V ̸= ∅} is open for any open set V in Y . −1 Equivalently, F is l.s.c. if the set G (V ) = {x ∈ X , G(x) ⊂ V } is closed for any closed set V in Y . 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  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 an open neighborhood U of x such that G(U ) is relatively compact. We define the graph of G to be the set Gr (G) = {(x, y) ∈ X × Y , y ∈ G(x)} and recall a useful result regarding connection between closed graphs and upper semi-continuity. Lemma 2.1 ([39], Proposition 1.2). If G : X → Pcl (Y ) is u.s.c., then Gr (G) is a closed subset of X × Y , i.e. for every sequence (xn )n∈N ⊂ X and (yn )n∈N ⊂ Y , if when n → ∞, xn → x∗ , yn → y∗ and yn ∈ G(xn ), then y∗ ∈ G(x∗ ). Conversely, if G is completely continuous and has a closed graph, then it is u.s.c. Finally, the following results are easily deduced from the theoretical limit set properties. Lemma 2.2 (See e.g. [40], Lemma 1.1.9). Let (Kn )n∈N ⊂ K ⊂ X be a sequence of subsets where K is a compact subset of a separable Banach space X . Then



 co (lim sup Kn ) = n→∞

 N >0

co



Kn

,

n ≥N

where co A refers to the closure of the convex hull of A. Lemma 2.3 (See e.g. [40], Theorem 1.4.13). Let X , Y be two metric spaces. If G : X −→ Pcp (Y ) is u.s.c., then for each x0 ∈ X , lim sup G(x) = G(x0 ). x→x0

We end these ingredients of multivalued analysis with some definitions and a result regarding the measurability of multivalued maps. Definition 2.2. Let

∑

be a nonempty set and A ⊂ P (

∑ ). A is called a σ -algebra if it satisfies the following properties:

(a) ∅ ∈ A. ∑ (b) O ∈ A ⇒ \O ∈ A.  (c) On ∈ A, n = 1, 2, . . . ⇒ n≥1 On ∈ A. Then (

∑ , A) is called a measurable space.

Definition 2.3. Let I be an interval of R and D ⊂ E, with E a metric space. A ⊂ P (I × D) 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. Lemma 2.4 ([Kuratowski–Ryll–Nardzewski Selection Theorem][See ∑ e.g. [41], Thm. 19.7]). Let ( , A) be a measurable space, (E , d) a separable, complete metric space (Polish space) and G : ( E a multivalued map with nonempty closed values. If G is measurable, then it has a measurable selection.

∑

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Finally recall: Definition 2.4. G is called a Carathéodory function if: (a) The multivalued map t ( G(t , z ) is measurable for each z ∈ E. (b) For a.e. t ∈ J, the multivalued map z ( G(t , z ) is upper semi-continuous. It is further a L1 -Carathéodory if it is locally integrably bounded, i.e. for each positive real number r, there exists some hr ∈ L1 (J , R+ ) such that

‖G(t , z )‖P ≤ hr (t ),

for a.e. t ∈ J and all ‖z ‖ ≤ r ,

where

‖G(t , z )‖P := sup{|v|E : v ∈ G(t , z )}. For further reading and details on multivalued analysis, we refer the reader to the books by Andres and Górniewicz [42], Aubin and Cellina [43], Aubin and Frankowska [40], Deimling [39], Górniewicz [41], Hu and Papageorgiou [44,45], Kamenskii et al. [46], Smirnov [47], and Tolstonogov [48]. 3. Existence results Many properties of solutions for differential equations, such as stability or oscillation, require global properties of solutions. This is the main motivation for searching for sufficient conditions that ensure global existence of solutions for impulsive differential equations and inclusions. In this direction, some questions have already been discussed by Graef and Ouahab [14,15], Guo [35,34], Guo and Liu [49], Henderson and Ouahab [36,37,50], Marino et al. [38], Ouahab [51], Stamov and Stamova [52], Weng [53], and Yan [54,55]. Consider the Banach space PCb = {y ∈ PC (R+ , Rn ) : y is bounded}, where PC = PC (R+ , Rn ) = {y: [0, ∞) → Rn , yk ∈ C ((tk , tk+1 ), Rn ), k = 0, . . . , m, y(tk− ) and y(tk+ ) exist and satisfy y(tk ) = y(tk− ) for k = 1, . . .} and yk := y|(tk ,tk+1 ) . Endowed with the norm

‖y‖b = sup{‖y(t )‖ : t ∈ [0, ∞)}, PCb is a Banach space. Next we define what we mean by a solution to problem (1.1). Definition 3.1. A function y ∈ PC is said to be a mild solution of problem (1.1) if there exists v ∈ L1 (J , Rn ) such that v(t ) ∈ F (t , y(t )) a.e. on J , y(0) = a and y(t ) = a +

t

∫

−

v(s)ds +

Ik (y(tk− )),

a.e. t ∈ J .

0
0

3.1. The upper semi-continuous case In this subsection, we present a global existence result and prove the compactness of the solution set for problem (1.1) by using a nonlinear alternative for multivalued maps combined with a compactness argument. The nonlinearity is u.s.c. with respect to the second variable and satisfies a Nagumo growth condition. Theorem 3.1. The impulsive functions Ik ∈ C (Rn , Rn ) satisfy:

(H1 ) There exist ck , dk > 0 such that ‖Ik (x)‖ ≤ ck ‖x‖ + dk ,

for every x ∈ Rn , k = 1, 2, . . .

with ∞ −

∞ −

ck < 1 and

k=1

dk < ∞.

k =1

The Carathéodory multivalued map F : J × Rn → P (Rn ) has compact, convex values and satisfies: (H2 ) There exist a continuous nondecreasing function ψ : [0, ∞) −→ (0, ∞) and p ∈ L1 (J , R+ ) such that

‖F (t , x)‖P ≤ p(t )ψ(‖x‖), with ∞

∫

m(s)ds < 0

∞

∫ c

du

ψ(u)

for a.e. t ∈ J and each x ∈ Rn ,

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

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where

‖ a‖ +

p(s)

m(s) =

1−

∞ ∑

1−

ck

dk

k=1

c=

and

∞ ∑

k=1

∞ ∑

.

ck

k=1

Then problem (1.1) has at least one solution. Moreover, the solution set SF (a) is compact and the multivalued map SF : a ( SF (a) is u.s.c. First, recall the well-known nonlinear alternative of Leray–Schauder for multivalued maps (see, e.g., [56,41]). Lemma 3.1. Let X be a normed space and N: X → Pcl,c v (X ) be a completely continuous, u.s.c. multivalued map. Then one of the following conditions holds: (a) N has at least one fixed point in X , (b) the set M := {x ∈ X , x ∈ λN (x), λ ∈ (0, 1)} is unbounded. Definition 3.2. Let E be a Banach space. A sequence (vn )n∈N ⊂ L1 ([a, b], E ) is said to be semi-compact if (a) it is integrably bounded, i.e. there exists q ∈ L1 ([a, b], R+ ) such that

|vn (t )|E ≤ q(t ),

for a.e. t ∈ [a, b] and every n ∈ N,

(b) the image sequence (vn (t ))n∈N is relatively compact in E for a.e. t ∈ [a, b]. The following important result follows from the Dunford–Pettis theorem (see [46], Proposition 4.2.1). Lemma 3.2. Every semi-compact sequence L1 ([a, b], E ) is weakly compact in L1 ([a, b], E ). When the nonlinearity takes convex values, Mazur’s Lemma, 1933, may be useful: Lemma 3.3 ([57], Theorem 21.4). Let E be a normed space to a limit x ∈ E. Then ∑m and (xk )k∈N ⊂ E a sequence weakly converging∑ m there exists a sequence of convex combinations ym = k=1 αmk xk with αmk > 0 for k = 1, 2, . . . , m and k=1 αmk = 1 which converges strongly to x. The following compactness criterion on unbounded domains is a simple extension of a compactness criterion in Cb (R+ , Rn ) (see [58], p. 62). Lemma 3.4. Let M ⊂ PCℓ = {x ∈ PC (R+ , Rn ) : limt →+∞ x(t ) exists}. Then M is relatively compact if it satisfies the following conditions: (a) M is uniformly bounded in PCℓ (R+ , Rn ). (b) The functions belonging to M are almost equicontinuous on R+ , i.e. equicontinuous on every compact interval of R+ . (c) The functions from M are equiconvergent, that is, given ε > 0, there corresponds T (ε) > 0 such that |x(t ) − x(+∞)| < ε for any t ≥ T (ε) and x ∈ M. Proof of Theorem 3.1. Step 1. Existence of solutions. Consider the operator N: PC → P (PC ) defined for y ∈ PC by

 N (y) =

h ∈ PC : h(t ) = a +



t

∫

v(s)ds + 0

−

Ik (y(tk )), a.e. t ∈ J

,

(3.1)

0
where v ∈ SF ,y = {v ∈ L1 (J , Rn ): v(t ) ∈ F (t , y(t )), a.e. t ∈ J }. Note that, from [59], Theorem 5.10 or [60], the set SF ,y is nonempty if and only if the mapping t → inf{‖v‖ : v ∈ F (t , y(t ))} belongs to L1 (J ). It is further bounded if and only if the mapping t → ‖F (t , y(t ))‖P = sup{‖v‖ : v ∈ F (t , y(t ))} belongs to L1 (J ); this particularly holds true when F satisfies (H2 ). Moreover, fixed points of the operator N are mild solutions of problem (1.1). We shall show that N satisfies the assumptions of Lemma 3.1. First notice that since SF ,y is convex (because F has convex values), then N takes convex values. Claim 1. N (PCb ) ⊂ PCℓ . Indeed, if y ∈ PCb and h ∈ N (y) then there exists v ∈ SF ,y such that h( t ) = a +

t

∫

v(s)ds + 0

−

Ik (y(tk )),

0 < tk < t

Since v ∈ L1 (J ) and

− 0
Ik (y(tk )) ≤ ‖y‖PCb +

∞ − k=1

dk ,

a.e. t ∈ J .

(3.2)

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

then clearly h ∈ PCℓ and h(∞) = a +

‖h(t )‖ ≤ ‖a‖ +

 +∞ 0

v(s)ds +

t

∫

−

‖F (s, y(s))‖P ds +

0
(y(tk )). Moreover

‖Ik (y(tk ))‖

0 < tk < t

0 t

∫

∑

−

p(s)ψ(‖y(s)‖)ds +

≤ ‖a‖ +

(ck ‖y(tk )‖ + dk ).

0 < tk < t

0

Hence

‖h‖PCb ≤ ‖a‖ + ψ(‖y‖PCb )

∞

∫

p(s)ds + ‖y‖PCb +

0

∞ −

dk .

k=1

This shows that N sends bounded sets in PCb into bounded sets of PCℓ . Claim 2. N sends bounded sets in PCb into almost equicontinuous sets of PCℓ . Let r > 0, Br := {y ∈ PCb : ‖y‖∞ ≤ r } be a bounded set in PCb , τ1 , τ2 ∈ J , τ1 < τ2 , and y ∈ Br . For each h ∈ N (y), we have τ2

∫

‖h(τ2 ) − h(τ1 )‖ ≤

−

‖v(s)‖ds +

τ1

‖Ik (y(tk ))‖

τ1
≤ ψ(r )

τ2

∫

τ1

−

p(s)ds +

(ck r + dk ).

τ1
1 + Since k=1 ck < ∞, k=1 dk < ∞ and p ∈ L (J , R ), the right-hand term tends to zero as |τ1 − τ2 | → 0, proving equi− continuity for the case where t ̸= ti , i = 1, . . . . To prove equicontinuity at t = ti for some i ∈ N∗ , fix ε0 > 0 such that {tj : j ̸= i} ∩ [ti − ε0 , ti + ε0 ] = ∅. Then for each 0 < ε < ε0 , we have the estimates

∑∞

∑∞

‖h(ti ) − h(ti − ε)‖ ≤

ti

∫

‖v(s)‖ds ≤ ψ(r )

∫

ti −ε

ti

p(s)ds.

ti −ε

Since p ∈ L1 (J , R+ ), the right-hand term tends to 0 as ε → 0. The equicontinuity at ti+ (i = 1, . . .) is proved in the same way. Claim 3. Let B(0, r ) be the closed ball centered at the origin with radius r > 0. We show that the set N (B(0, r )) is equiconvergent at ∞, i.e. for every ε > 0, there exists T (ε) > 0 such that ‖h(t ) − h(∞)‖ ≤ ε for every t ≥ T and each h ∈ N (B(0, r )). If h ∈ N (y) for some y ∈ B(0, r ), then there exists v ∈ SF ,y such that h satisfies (3.2). Then

‖h(t ) − h(∞)‖ ≤

∞

∫

‖v(s)‖ds +

≤ ψ(r )

∞

∫

p(s)ds +

k=1

∞ −

ck < ∞,

∑∞

k=1

(ck r + dk ) ≤

k=k0

−

(ck r + dk ).

t ≤tk <∞

t

∑∞

‖Ik (y(tk ))‖

t ≤tk <∞

t

Since

−

dk < ∞ and p ∈ L1 (J , R+ ), then there exist k0 and T (ε) > 0 such that

ε 2

and

ε

∞

∫

p(s) < t

2ψ(r )

,

∀ t ≥ T (ε).

Hence

‖h(t ) − h(∞)‖ ≤ ε,

∀ t ≥ max(k0 , T (ε)).

Then N (B(0, r )) is equiconvergent. With Lemma 3.4 and Claims 1–3, we conclude that N is completely continuous. Claim 4. N is u.s.c. To this end, it is sufficient to show that N has a closed graph. Let hn ∈ N (yn ) be such that hn −→ h and yn −→ y, as n → +∞. Then there exists M > 0 such that ‖yn ‖ ≤ M. We shall prove that h ∈ N (y). hn ∈ N (yn ) means that there exists vn ∈ SF ,yn such that, for a.e. t ∈ J, we have hn (t ) = a +

t

∫

vn (s)ds + 0

− 0 < tk < t

Ik (yn (tk )).

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

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(H2 ) implies that vn (t ) ∈ p(t )ψ(M )B(0, 1). Then (vn )n∈N is integrably bounded in L1 (J , Rn ). Since F has compact values, we deduce that (vn )n is semi-compact. By Lemma 3.2, there exists a subsequence, still denoted (vn )n∈N , which converges weakly to some limit v ∈ L1 (J , Rn ). Moreover, the mapping Γ : L1 (J , Rn ) → PCb (J , Rn ) defined by ∫ t g (s)ds Γ (g )(t ) = 0

is a continuous linear operator. Then it remains continuous if these spaces are endowed with their weak topologies [61,57]. Moreover for a.e. t ∈ J , yn (t ) converges to y(t ) and Ik is continuous for k = 1, . . . . Then we have h( t ) = a +

t

∫

v(s)ds +

−

Ik (y(tk )).

0 < tk < t

0

It remains to prove that v ∈ F (t , y(t )), a.e. t ∈ J. Mazur’s Lemma 3.3 yields the existence of αin ≥ 0, i = n, . . . , k(n) such that

∑k(n) i =1

αin = 1 and the sequence of convex combinations gn (·) =

∑k(n) i =1

αin vi (·) converges strongly to v in L1 . Using Lemma 2.2,

we obtain that

v(t ) ∈



{gn (t )},

a.e. t ∈ J

n ≥1

⊂



co{vk (t ), k ≥ n}

n ≥1

⊂





 

co

F (t , yk (t ))

k≥n

n ≥1

  = co lim sup F (t , yk (t )) .

(3.3)

k→∞

However, the fact that the multivalued x ( F (., x) is u.s.c. and has compact values, together with Lemma 2.3, implies that lim sup F (t , yn (t )) = F (t , y(t )), n→∞

a.e. t ∈ J .

This with (3.3) yields that v(t ) ∈ co F (t , y(t )). Finally F (., .) has closed, convex values; hence v(t ) ∈ F (t , y(t )), a.e. t ∈ J. Thus h ∈ N (y), proving that N has a closed graph. Finally, with Lemma 2.1 and the compactness of N, we conclude that N is u.s.c. Claim 5. A priori bounds on solutions. Let y ∈ PCb be such that y ∈ λN (y) and λ ∈ (0, 1). Then there exists v ∈ SF ,y such that

 y(t ) = λ a +



t

∫

−

v(s)ds +

Ik (y(tk )) ,

a.e. t ∈ J .

−

(3.4)

0 < tk < t

0

Arguing as in Claim 1, we get the estimates

‖y(t )‖ ≤ ‖a‖ +

t

∫

−

p(s)ψ(‖y(s)‖)ds +

(ck ‖y(tk )‖ + dk ),

a.e. t ∈ J .

0
0

Letting α(t ) = sup{‖y(s)‖ : s ∈ [0, t ]} and using the increasing character of ψ , we obtain that

α(t ) ≤ ‖a‖ +

t

∫

p(s)ψ(α(s))ds +

−

(ck α(t ) + dk ).

0
0

Hence



1

α(t ) ≤ 1−

t

∫

p(s)ψ(α(s))ds +

‖ a‖ +

∞ ∑

0

ck

∞ −

 dk

.

k=1

k=1

Denoting the right-hand side by β(t ), we have

‖y(t )‖ ≤ α(t ) ≤ β(t ),

t ∈J

as well as

‖ a‖ +

∞ ∑

dk

k=1

β(0) = 1−

∞ ∑ k=1

ck

and β ′ (t ) =

p(t )ψ(α(t )) 1−

∞ ∑ k=1

ck

≤

p(t )ψ(β(t )) 1−

∞ ∑ k=1

ck

.

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From (H2 ), this implies that for t ∈ J

Γ (β(t )) =

β(t )

∫

β(0)

ds

ψ(s)

∞ ∑

1−

∞

∫

1

≤

ck

p(s)ds <

∞

∫

0

β(0)

ds

ψ(s)

= Γ (+∞).

k=1

Thus





 ‖p‖L1  , β(t ) ≤ Γ −1  ∞   ∑ 1− ck

for every t ∈ J ,

k =1

where Γ (z ) =

z

du β(0) ψ(u) .

As a consequence,





 ‖p‖L1   := M . ‖y‖PCb ≤ Γ −1  ∞   ∑ 1− ck k=1

Finally, let

 + 1} U := {y ∈ PCb : ‖y‖PCb < M and consider the operator N : U → Pc v,cp (PCb ). 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 multivalued version of the nonlinear alternative of Leray–Schauder (Lemma 3.1), N has a fixed point y in U which is a solution of problem (1.1). Step 2: Compactness of the solution set. For each a ∈ Rn , let SF (a) = {y ∈ PCb : y is a solution of problem (1.1)}.

 such that for every y ∈ SF (a), ‖y‖PC ≤ M.  Since N is completely continuous, N (SF (a)) is From Step 1, there exists M b relatively compact in PCb . Let y ∈ SF (a); then y ∈ N (y) and hence SF (a) ⊂ N (SF (a)). It remains to prove that SF (a) is a closed subset in PCb . Let {yn : n ∈ N} ⊂ SF (a) be such that the sequence (yn )n∈N converges to y. For every n ∈ N, there exists vn such that vn (t ) ∈ F (t , yn (t )), a.e. t ∈ J, and yn (t ) = a +

t

∫

−

vn (s)ds +

Ik (yn (tk )).

(3.5)

0
0

Arguing as in Claim 4, we can prove that there exists v such that v(t ) ∈ F (t , y(t )) and y(t ) = a +

t

∫

v(s)ds +

−

Ik (y(tk )).

(3.6)

0
0

Therefore y ∈ SF (a) which yields that SF (a) is closed, and hence compact in PCb . Step 3: SF (.) is u.s.c. For this, we prove that the graph of SF

ΓSF := {(a, y) : y ∈ SF (a)} is closed. Let (an , yn ) ∈ ΓSF be such that (an , yn ) → (a, y) as n → ∞. Since yn ∈ SF (an ), there exists vn ∈ L1 (J , Rn ) such that yn (t ) = an +

t

∫

vn (s)ds + 0

−

Ik (yn (tk )),

t ∈ J.

(3.7)

0 < tk < t

Arguing as in Claim 4, we can prove that there exists v ∈ SF ,y such that y satisfies (3.6). Thus, y ∈ SF (a). Now, we show that SF maps bounded sets into relatively compact sets of PC . Let B be a bounded set in Rn and let {yn } ⊂ SF (B). Then there exist {an } ⊂ B and vn ∈ SF ,yn , n ∈ N such that (3.7) is satisfied. Since {an } is a bounded sequence, there exists a subsequence of {an } converging to a. As in Claims 2 and 3, we can show that {yn : n ∈ N} is equicontinuous on every compact of J and is equiconvergent at ∞. As a consequence of Lemma 3.4, we conclude that there exists a subsequence of {yn } converging to y in PC . By an argument similar to Claim 4, we can prove that y satisfies (3.6) for some v ∈ SF ,y . Thus y ∈ S (B). This implies that SF (.) is u.s.c., ending the proof of Theorem 3.1. 

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

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3.2. The Lipschitz case In this subsection, we prove the existence of solutions under Hausdorff–Lipschitz conditions. Let E be a Fréchet space with the topology generated by a family of semi-norms | · |n and corresponding distances dn (x, y) = |x − y|n (n ∈ N). First, we start with: Definition 3.3. A multivalued map F : E ( E is called an admissible contraction with constant {kn }n∈N if for each n ∈ N, there exists kn ∈ (0, 1) such that (a) Hdn (F (x), F (y)) ≤ kn |x − y|n for all x, y ∈ E, where Hd is the Hausdorff distance, (b) for every x ∈ E and every ε > 0, there exists y ∈ F (x) such that

|x − y|n ≤ dn (x, F (x)) + ε,

for every n ∈ N.

A subset A ⊂ E is bounded if for every n ∈ N, there exists Mn > 0 such that |x|n ≤ Mn , for every x ∈ A. Our main tool will be the following nonlinear alternative of Frigon for multivalued contractions [62]: Lemma 3.5. Let E be a Fréchet space, U ⊂ E an open neighborhood of the origin, and let N : U ( E be a bounded admissible multivalued contraction. Then one of the following statements holds: (a) N has a fixed point, (b) there exists λ ∈ [0, 1) and x ∈ ∂ U such that x ∈ λN (x). We now present our second existence result for problem (1.1). Here and hereafter Jk = [0, tk ] \ {tj , 0 < j < k}. Theorem 3.2. Suppose the multivalued map F : J × Rn → Pcp (Rn ) is such that t ( F (t , .) is measurable and

(H3 ) for each k = 1, 2, . . . , there exist lk ∈ L1 ([0, tk ], R+ ) such that Hd (F (t , x), F (t , y)) ≤ lk (t )‖x − y‖,

for x, y ∈ Rn and a.e. t ∈ Jk

and F (t , 0) ⊂ lk (t )B(0, 1),

(H4 )

∑∞

k=1

for a.e. t ∈ Jk .

‖Ik (0)‖ < ∞ and there exist constants ck ≥ 0 such that ‖Ik (x) − Ik (y)‖ ≤ ck ‖x − y‖,

∑∞

k=1

ck < 1 and

for each x, y ∈ R . n

Then problem (1.1) has at least one mild solution. Remark 3.1. (a) Note that (H4 ) implies (H1 ) with dk = ‖Ik (0)‖. (b) (H3 ) implies that the nonlinearity F has at most linear growth:

‖F (t , x)‖P ≤ lk (t )(1 + ‖x‖),

lk ∈ L1 (Jk , R+ ), a.e. t ∈ Jk , x ∈ Rn

and thus (H2 ) is satisfied locally. However, F is not Carathéodory and may take nonconvex values. Proof. We begin by defining a family of semi-norms on PC , thus rendering PC a Fréchet space. Let τ be a sufficiently large real parameter, say 1

τ

+

∞ −

ck < 1.

k=1

For each n ∈ N, define in PC the family of weighted semi-norms

|y|n = sup{e−τ Ln (t ) ‖y(t )‖ : 0 ≤ t ≤ tn } where t

∫

Ln (t ) =

ln (s)ds. 0

Thus PC =



n ≥1

PCn where PCn = {y : Jn → Rn such that y is continuous everywhere except for some tk at which y(tk− )

and y(tk ) exist and y(tk− ) = y(tk ) (k = 1, 2, . . . , n − 1)}. Then PC is a Fréchet space with the family of semi-norms {| · |n }. In order to transform problem (1.1) into a fixed point problem, define the operator N : PC → P (PC ) by (3.1). Since the fixed points of the operator N are solutions of problem (1.1), we first show that N : U → Pcl (PC ) is an admissible multivalued contraction, where U ⊂ PC is some open subset to be defined later on. +

Step 1. We claim that there exists γ < 1 such that Hd (N (y), N (y)) ≤ γ ‖y − y‖n ,

for each y, y ∈ PCn .

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

Let y, y ∈ PCn and h ∈ N (y). Then there exists v ∈ SF ,y such that h(t ) = a +

t

∫

−

v(s) ds +

Ik (y(tk− )),

a.e. t ∈ Jn .

0 < tk < t

0

(H3 ) implies that Hd (F (t , y(t )), F (t , y(t ))) ≤ ln (t )‖y(t ) − y(t )‖,

a.e. t ∈ Jn .

Hence, there is some w0 ∈ F (t , y(t )) such that

‖v(t ) − w0 ‖ ≤ ln (t )‖y(t ) − y(t )‖,

t ∈ Jn .

Consider the multivalued map Un : Jn → P (Rn ) defined by Un (t ) = {w ∈ F (t , y(t )) : ‖v(t ) − w‖ ≤ ln (t )‖y(t ) − y(t )‖, a.e. t ∈ Jn }. Then Un (t ) is a nonempty set for it contains w0 and Theorem III.4.1 in [63] tells us that Un is measurable. Moreover, the multivalued intersection operator Vn (.) = Un (.) ∩ F (., y(.)) is also measurable. Therefore, by Lemma 2.4, there exists a function t → v n (t ) which is a measurable selection for Vn , that is v n (t ) ∈ F (t , y(t )) and

‖v(t ) − v n (t )‖ ≤ ln (t )‖y(t ) − y(t )‖,

a.e. t ∈ Jn .

Define h by h(t ) = a +

t

∫

−

v n (s) ds +

Ik (y(tk− )),

a.e. t ∈ Jn .

0
0

Then, for a.e. t ∈ Jn , we have

‖h(t ) − h(t )‖ ≤

t

∫

‖v(s) − v n (s)‖ ds +

−

t

∫

−

ln (s)‖y(s) − y(s)‖ds +

≤

ck ‖y(tk ) − y(tk )‖

0
0 t

∫

‖Ik (y(tk− )) − Ik (y(tk− ))‖

0
0

≤

ln (s)eτ Ln (s) e−τ Ln (s) ‖y(s) − y(s)‖ds +

t

≤

ln (s)eτ Ln (s) ds‖y − y‖n +

−

ck eτ L(t ) ‖y − y‖n

0
0 t

∫

ck eτ Ln (t ) e−τ Ln (t ) ‖y(tk ) − y(tk )‖

0 < tk < t

0

∫

−

≤ 0

1

τ

(eτ Ln (s) )′ ds‖y − y‖n +

n −

ck eτ Ln (t ) ‖y − y‖n

k=1

 ≤ eτ Ln (t )

1

τ



n −

+

ck

‖y − y ‖n .

k=1

It follows that

 e

−τ Ln (t )

‖h(t ) − h(t )‖ ≤

1

τ

+

n −

 ck

‖y − y‖n .

k=1

By an analogous relation, obtained by interchanging the roles of y and y, we finally arrive at

 Hdn (N (y), N (y)) ≤

1

τ

+

n −

 ck

‖y − y‖n .

k=1

Moreover, since F is compact valued, we can prove that N has compact values too. Let x ∈ U and ε > 0. If x ̸∈ N (x), then dn (x, N (x)) ̸= 0. Since N (x) is compact, then there exists y ∈ N (x) such that dn (x, N (x)) = |x − y|n and we have

|x − y|n ≤ dn (x, N (x)) + ε. In the case where x ∈ N (x), we may take y = x. Therefore N is an admissible operator contraction.

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

2151

Step 2. A priori estimates. Given t ∈ Jn , let y ∈ λN (y) for some λ ∈ (0, 1]. Then there exists v ∈ SF ,y such that (3.4) is satisfied. Then we have

‖y(t )‖ ≤ ‖a‖ +

t

∫

‖v(s)‖ds +

−

‖Ik (y(tk ))‖

0 < tk < t

0

n −

t

∫

ln (s)(1 + ‖y(s)‖)ds +

≤ ‖a‖ + 0

ck ‖y(tk− )‖ +

k=1

n −

‖Ik (0)‖.

k=1

Consider the function µ defined on Jn by

µ(t ) = sup{‖y(s)‖ : 0 ≤ s ≤ t }. By the previous inequality, we have for t ∈ Jn



1

µ(t ) ≤ 1−

‖ a‖ +

n ∑

n −

‖Ik (0)‖ +

ln (s)(1 + µ(s))ds .

0

k=1

ck



t

∫

k=1

Let us take the right-hand side of the above inequality as β(t ). Then we have ∞ ∑

‖ a‖ +

‖Ik (0)‖

k=1 n

β(0) =

1−

= c,

∑

ck

k=1

µ(t ) ≤ β(t ),

t ∈ Jn

and

β ′ (t ) =

ln (t )(1 + µ(t )) 1−

∞ ∑

ln (t )(1 + β(t ))

≤

1−

ck

k=1

∞ ∑

,

t ∈ Jn .

ck

k=1

Integrating over t ∈ Jn yields

∫

β(t ) β(0)

ds 1+s

∫

1

≤ 1−

∞ ∑

ck

tn

ln (s)ds =: Mn .

0

k=1

Hence β(t ) ≤ Kn := (1 + β(0))eMn and as a consequence

‖y(t )‖ ≤ µ(t ) ≤ β(t ) ≤ Kn ,

t ∈ Jn .

Therefore

|y|n ≤ Kn ,

∀n ∈ N∗ .

Let U = {y ∈ PC : |y|n < Kn + 1, for all n ∈ N}. Clearly, U is a open subset of PC and there is no y ∈ ∂ U such that y ∈ λN (y) and λ ∈ (0, 1). By Lemma 3.5 and Steps 1, 2, N has at least one fixed point y solution to problem (1.1).  3.3. The lower semi-continuous case Our third existence result for problem (1.1) deals with the case where the nonlinearity is lower semi-continuous with respect to the second argument and does not necessarily have convex values. In the proof, we will make use of the nonlinear alternative of Leray–Schauder type (Lemma 3.1) combined with a selection theorem for lower semi-continuous multivalued maps with decomposable values. Consider a Banach space E and I = [a, b] an interval of the real line. Definition 3.4. A subset A ⊂ L1 (I , E ) is decomposable if for all u, v ∈ A and for every Lebesgue measurable subset I ′ ⊂ I, we have uχI ′ + vχI \I ′ ∈ A, where χA stands for the characteristic function of the set A. Definition 3.5. Let F : I × E → P (E ) be a multivalued 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 : C (I , E ) → P (L1 (I , E )) defined by F (y) = SF ,y is lower semi-continuous and has nonempty, closed, and decomposable values.

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

Next, we state the celebrated selection theorem of Bressan and Colombo [64], Fryszkowski [65,66]. Lemma 3.6. Let X be a separable metric space and let E be a Banach space. Then every l.s.c. multivalued operator N : X → Pcl (L1 (I , E )) with nonempty closed decomposable values has a continuous selection, i.e. there exists a continuous single-valued function f : X → L1 (I , E ) such that f (x) ∈ N (x) for every x ∈ X . Theorem 3.3. Suppose that:

1 ) There exist ck , dk > 0 such that (H ‖Ik (x)‖ ≤ ck ‖x‖ + dk ,

for every x ∈ Rn , k = 1, 2, . . .

with ∞ −

∞ −

ck < ∞ and

k=1

dk < ∞.

k=1

2 ) There exist p ∈ L1loc ([0, ∞), Rn ) and a continuous nondecreasing function ψ : [0, ∞) → [0, ∞) such that (H ‖F (t , x)‖P ≤ p(t )ψ(‖x‖),

for a.e. t ∈ J and each x ∈ Rn

with

∫

∞ ‖a‖

du

ψ(u)

= ∞.

(H5 ) F : [0, ∞) × Rn −→ P (Rn ) is a nonempty compact valued multimap such that (a) the mapping (t , y) ( F (t , y) is L ⊗ B measurable, (b) the mapping y ( F (t , y) is lower semi-continuous for a.e. t ∈ [0, ∞). Then problem (1.1) has at least one solution. For the proof, we need some auxiliary lemmas. Lemma 3.7 ([67]). Let F : I × E → Pcp (E ) be a locally integrably bounded multivalued map satisfying (H5 ). Then F is of lower semi-continuous type. The following result is known as the Gronwall–Bihari Theorem. Lemma 3.8 ([68]). Let u, g: I → R be positive real continuous functions. Assume there exist c > 0 and a continuous nondecreasing function h: R → (0, +∞) such that u(t ) ≤ c +

t

∫

g (s)h(u(s)) ds,

∀ t ∈ I.

a

Then u(t ) ≤ H −1

t

∫



g (s) ds ,

∀t∈I

a

provided +∞

∫ c

dy h(y)

>

b

∫

g (s) ds. a

Here H −1 refers to the inverse of the function H (u) =

u c

dy h(y)

for u ≥ c.

Also, recall the Schauder–Tikhonov fixed point theorem: Lemma 3.9 ([42]). Let E be a locally convex space, C a convex closed subset of E and N : C → C a continuous, compact map. Then N has at least one fixed point in C .

2 ) and (H5 ) imply, by Lemma 3.7, that F is of lower semi-continuous Proof of Theorem 3.3. Let F : Jm × Rn → P (Rn ). (H type. From Lemma 3.6, there is a continuous selection fm : PC (Jm , Rn ) → L1 (Jm , Rn ) such that fm (y) ∈ Fm (y) for every y ∈ PC (Jm , Rn ) where Fm is the Nemyts’ki˘ı operator associated with F on Jm : Fm (y) := {v ∈ L1 (Jm , Rn ) : v(t ) ∈ F (t , y(t )), a.e. t ∈ Jm }. Let f : PC → L1loc ([0, ∞), Rn ) be defined by f (y)(t ) = fm (y)(t ),

a.e. t ∈ Jm .

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

Then PC =



m≥1

2153

PCm is a Fréchet space with family of semi-norms {‖ · ‖m } where

‖y‖m = sup{‖y(t )‖ : t ∈ Jm }. Consider the problem

 ′ y (t ) = f (y)(t ), = I (y(tk− )), 1y| y(0)t ==tk a k

a.e. t ∈ J , k = 1, . . . ,

(3.8)

and the operator L : PC → PC defined by t

∫

L(y)(t ) = a +

−

f (y)(s)ds +

Ik (y(tk− )),

a.e. t ∈ J .

0
0

Clearly, the fixed points of the operator L are mild solutions of problem (1.1). Step 1. A priori estimates. Let y be a possible solution of problem (1.1). For t ∈ [0, t1 ], we have y(t ) = a +

t

∫

f (y)(s)ds. 0

Then

‖y(t )‖ ≤ ‖a‖ +

t

∫

p(s)ψ(‖y(s)‖)ds. 0

By Lemma 3.8 and ( H2 ), we have t

∫

‖y(t )‖ ≤ Γ1−1



p(s)ds ,

t ∈ [0, t1 ],

0

z

du . For t ∈ (t1 , t2 ], we have where Γ1 (z ) = ‖a‖ ψ( u)

y(t ) = a +

t

∫

f (y)(s)ds + I1 (y(t1 )). 0

Then

∫ t ‖y(t )‖ ≤ ‖a‖ + ‖I1 (y(t1 ))‖ + p(s)ψ(‖y(s)‖)ds 0 ∫ t ≤ ‖a‖ + K1 + p(s)ψ(‖y(s)‖)ds, 0

where K1 = sup{‖I1 (z )‖ : z ∈ B(0, M0 )}

and M0 = Γ1

−1

t1

∫



p(s)ds .

0

By Lemma 3.8, we again have

‖y(t )‖ ≤ Γ2−1

t

∫



p(s)ds ,

t ∈ (t1 , t2 ],

0

z

du where Γ2 (z ) = ‖a‖+K ψ( · We continue this process until we obtain, for every t ∈ (tm−1 , tm ], the estimate u) 1

‖y(t )‖ ≤ Γm−1

t

∫



p(s)ds ,

t ∈ (tm−1 , tm ],

0

where

Γm ( z ) =

∫

z

du

‖a‖+Km−1

ψ(u)

Km−1 = sup{‖Im−1 (z )‖ : z ∈ B(0, Mm−2 )} Mm−2 = Γm−−11

tm−1

∫ 0



p(s)ds .

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

Let

 C =

y ∈ PC : ‖y(t )‖ ≤ Γm−1

t

∫





p(s)ds , t ∈ (tm−1 , tm ], m = 1, 2, . . . . 0

It is clear that C is a convex closed and bounded subset in PC . Step 2. L(C ) ⊂ C . Given y ∈ C , we have for t ∈ [0, t1 ]

‖L(y)(t )‖ ≤ ‖a‖ +

t

∫

‖f (y)(s)‖ds

∫0 t ≤ ‖a‖ +

p(s)ψ(‖y(s)‖)ds

0

 ∫ s p(r )dr ds Γ1−1  ∫ s0 ∫0 t p(r )dr ds p(s)(Γ1−1 )′ = ‖a‖ + 0 0 ′ ∫ s ∫ t ds. = ‖a‖ + p(r )dr Γ1−1 t

∫

p(s)ψ

≤ ‖a‖ +



0

0

We have used the fact that Γ1 (0) = ‖a‖ and −1

ψ(z ) =

1

(Γ1 )′ (z )

= (Γ1−1 )′ (Γ1 )(z ).

Lemma 3.8 implies that

‖L(y)(t )‖ ≤ Γ1−1

t

∫



p(s)ds ,

a.e. t ∈ [0, t1 ].

(3.9)

0

Also for t ∈ (t1 , t2 ], we have

∫ t ‖L(y)(t )‖ ≤ ‖a‖ + ‖I1 (y(t1 ))‖ + p(s)ψ(‖y(s)‖)ds 0 ∫ t ≤ ‖a‖ + K1 + p(s)ψ(‖y(s)‖)ds. 0

Arguing as above, we obtain

‖L(y)(t )‖ ≤ Γ2

−1

t

∫



p(s)ds ,

a.e. t ∈ (t1 , t2 ].

(3.10)

0

We continue this process until we arrive at the estimate

‖L(y)(t )‖ ≤ Γm−1

t

∫



p(s)ds ,

a.e. t ∈ (tm−1 , tm ],

(3.11)

0

proving that L(C ) ⊂ C ; this implies that L(C ) is a bounded set in the Fréchet space PC . As in Claims 2 and 3, Step 1 of the proof of Theorem 3.1, we can prove that for every m ∈ N, the operator L : PCm → PCm is completely continuous; hence L : PC → PC is continuous and L(C ) is relatively compact. By Lemma 3.9, we conclude that L has at least one fixed point, a solution of problem (3.8), and hence a solution of problem (1.1).  4. Topological structure of solution sets: the projective limit approach In the second part of this work, we prove some properties such as AR, Rδ as well as acyclicity of the solution set. For this purpose, we shall use the approach of limits of inverse systems. 4.1. Selectionable maps The following definitions and the result can be found in [41] or [43], p. 86. Let (X , d) and (Y , d′ ) be two metric spaces. Definition 4.1. We say that a multivalued map F : X ( Y is σ -Ca-selectionable if there exists a decreasing sequence of compact valued u.s.c. maps Fn : X → Y satisfying: (a) Fn has a Carathédory selection, for all n ≥ 0 (the Fn are called Ca-selectionable), (b) F (x) = n≥0 Fn (x), for all x ∈ X .

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Definition 4.2. A single-valued map f : [0, a]× X → Y is said to be measurable locally Lipschitz (mLL) if f (·, x) is measurable for every x ∈ X and for every x ∈ X , there exists a neighborhood Vx of x ∈ X and an integrable function Lx : [0, a] → [0, ∞) such that d′ (f (t , x1 ), f (t , x2 )) ≤ Lx (t )d(x1 , x2 ) for a.e. t ∈ [0, a] and x1 , x2 ∈ Vx . Definition 4.3. A multivalued mapping F : [0, a] × X ( Y is mLL-selectionable if it has an mLL-selection. Definition 4.4. We say that a multivalued map φ : [a, b] × Rn ( Rn with closed values is upper Scorza–Dragoni if, given δ > 0, there exists a closed subset Aδ ⊂ [a, b] such that the measure µ([a, b] \ Aδ ) ≤ δ and the restriction φδ of φ to Aδ × Rn is u.s.c. Lemma 4.1 (See [41], Thm. 19.19). Let E , E1 be two separable Banach spaces and let F : [a, b] × E ( E1 be an upper Scorza– Dragoni multivalued map with compact, convex values. Then F is σ -Ca-selectionable, the maps Fn : [a, b] × E → P (E1 )(n ∈ N) are almost u.s.c. and Fn (t , e) ⊂ conv (∪x∈E Fn (t , x)). Moreover, if F is integrably bounded, then F is σ -mLL-selectionable. 4.2. Background in algebraic topology First, we start with some notions from geometric topology. For details, we recommend [69,56,70,41,71–75]. In what follows (X , d) and (Y , d′ ) stand for two metric spaces. Definition 4.5. A set A ⊂ X is called a contractible space provided that there exists 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). Note that any closed convex subset of X is contractible. Definition 4.6. A compact nonempty metric space X is called an Rδ -set provided that there exists a decreasing sequence of ∞ compact nonempty contractible metric spaces (Xn )n∈N∗ such that X = n=1 Xn . Definition 4.7. A space X is called an absolute retract (for 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  f : Y → X of f over Y , i.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 . Lemma 4.2. Let X ∈ AR and  X be such that X ≃  X . Then  X ∈ AR. Proof. Let Y be a space and B ⊂ Y a closed subset. Let f : B −→  X be a continuous map and ϕ :  X −→ X be a homeomorphism. Since X ∈ AR there exists  g : Y −→ X which is a continuous extension of ϕ ◦ f . Then  f = ϕ −1 ◦ g : Y −→  X is a continuous extension of f .  By an embedding, we mean any homeomorphism h : X −→ Y such that h(X ) is a closed subset of Y . From [42], Proposition 2.15, if X ∈ AR, then it is a contractible space. According to Hyman [73], a compact, metric space is Rδ if it can be expressed as the intersection of compact, absolute retracts. Definition 4.8. A space A is closed acyclic if (a) H0 (A) = Q, (b) Hn (A) = 0, for every n > 0, where H∗ = {Hn }n≥0 is the Čech-homology functor with compact carriers and coefficients in the field of rationals Q. In other words, a space A is acyclic if the map j : {p} → X , j(p) = x0 ∈ A, induces an isomorphism j∗ : H∗ ({p}) → H∗ (A). Definition 4.9. A u.s.c. map F : X → P (Y ) is called acyclic if for each x ∈ X , F (x) is compact acyclic. From the continuity of Čech-homology functors, if X is a contractible compact space, then it is acyclic. To sum up, for any compact space, we have convex ⊂ AR ⊂ contractible ⊂ Rδ ⊂ acyclic. 4.3. Limits of inverse systems Let us recall that an inverse system of Hausdorff topological spaces is a family S = (Xα , παβ , J ), where J is a poset directed by the relation ≤, Xα is a Hausdorff topological space, for every α ∈ J, and παβ : Xβ → Xα is a continuous mapping, for each

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γ

γ

γ

two elements α, β ∈ J, such that α ≤ β . Moreover, for each α ≤ β ≤ γ , παβ satisfies παα = IdXα and παβ πβ = πα . By παβ πβ ∏ γ we mean the composite παβ ◦ πβ . The following subset of the product α∈J Xα :





(xα ) ∈

lim S = ←

∏

Xα :

α∈J

παβ (xβ )

= xα , for all α ≤ β

is called a limit (or projective limit) of the inverse system S. The inverse limit of the corresponding inverse system is just the product. The limit projective lim S is also called the generalized intersection ∩α∈J Xα (see, e.g., [76] or [61], p. 439). An ←

element of lim S is called a thread or fibre of the system S. One can see that if we denote by πα : lim S → Xα a restriction ←

←

β of the projection pα : α∈J Xα → Xα onto the α th axis, then we obtain that πα πβ = πα , for each α ≤ β . Let us give an important example of inverse systems.

∏

Example 4.1. For every m ∈ N, let Cm = C ([0, m], Rn ) be the Banach space of all continuous functions on the closed interval [0, m] into Rn and C = C ([0, ∞), Rn ) an analogous Fréchet space of continuous functions. Consider the restriction maps p πmp : Cp → Cm defined  by πm (x) = x|[0,m] . It is easy to see that C is isometrically homeomorphic to the limit of the inverse p system Cm , πm , N . The maps defined by πm : C → Cm , φm (x) = x|[0,m] correspond to suitable projections. Some useful properties of limits of inverse systems are summarized in the following: Proposition 4.1. Let S = Xα , παβ , J be an inverse system.





(1) The limit lim S is a closed subset of

∏

←

α

Xα .

(2) If, for every α ∈ J , Xα is (a) compact, then lim S is compact, ←

(b) compact and nonempty, then lim S is compact and nonempty, ←

(c) compact and acyclic and lim S is nonempty, then lim S is compact and acyclic, ←

←

(d) metrizable and J is countable, then lim S is metrizable. ←

Part (c) is due to Gabor [28]. Proofs of Proposition 4.1 can be found in [30,31,42,77,28]. In the case where J is countable, we have (see [28], Prop. 3.2): p

Proposition 4.2. Let S = {Xn , πn , N} be an inverse system such that Xn is an Rδ -set. Then lim S is Rδ too. ←

Next, we introduce the notation of multivalued maps of inverse systems. Suppose that two systems S = Xα , παβ , J and



β′







S ′ = Yα ′ , πα ′ , J ′ are given. Definition 4.10. By a multivalued map from the system S into the system S ′ , we mean a family {σ , ϕσ (α ′ ) } consisting of a monotone function σ : J ′ → J, that is σ (α ′ ) ≤ σ (β ′ ), for α ′ ≤ β ′ , and of multivalued maps ϕσ (α ′ ) : Xσ (α ′ ) ( Yα ′ with nonempty values, defined for every α ′ ∈ J ′ and such that for each α ′ ≤ β ′ σ (β ′ )

β′

πα′ ϕσ (β ′ ) = ϕσ (α′ ) πσ (α′ ) . A map of systems {σ , ϕσ (α ′ ) } induces a limit map ϕ : lim S ( lim S ′ defined by ←

ϕ(x) =

∏

←

ϕσ (α′ ) (xσ (α′ ) ) ∩ lim S ′ . ←

α ′ ∈J ′

′ In other words, ∏∞ a limit map is a′ map such that πα′ ϕ = ϕσ (α′ ) πσ (α′ ) , for every α ∈ J. In terms of countable inverse systems, ϕ((xn )) = n=1 ϕn (xn ) ∩ lim S . ←

Since a topology of the limit of an inverse system is the one generated by the base consisting of all sets of the form

πα (Uα ), where α runs over an arbitrary set cofinal in J and Uα are open subsets of the space Xα , it is easy to prove the following continuity property for limit maps. β′

Proposition 4.3 ([31], Prop. 2.7). Let S = {Xα , παβ , J } and S ′ = {Yα ′ , πα ′ , J ′ } be two inverse systems and let ϕ : lim S ( lim S ′ be a limit map induced by the map {σ , ϕσ (α ′ ) }. If, for every α ′ ∈ J ′ , ϕσ (α ′ ) is (a) u.s.c. with compact values, then ϕ is u.s.c., (b) l.s.c., then ϕ is l.s.c.,

←

←

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(c) continuous, then ϕ is continuous. Regarding the structure of the fixed point sets of limit maps, we have (see [31], Thm. 2.8). Proposition 4.4. Let S = {Xα , παβ , J } be an inverse system and ϕ : lim S ( lim S be a limit map induced by the map {id, ϕα } ←

←

where ϕα : Xα ( Xα . Then the fixed point set of ϕ is a limit of the inverse system generated by the set Fix (ϕα ). In particular if the fixed point set of ϕα is compact acyclic (resp. Rδ ), then ϕ is compact acyclic (resp. Rδ ) as well. In order to define an inverse system for problem (1.1), consider, for every m ∈ N, the problems

 ′ y (t ) ∈ F (t , y(t )), = I (y(tk− )), 1y| y(0)t ==tk a. k

a.e. t ∈ Jm k = 1, . . . , m − 1

(4.1)

Under the assumptions of Theorem 3.1 (see Step 1, Claim 5 of the proof) and Theorem 3.2 (see Step 2 of the proof), we know that there exists Mm > 0 such that for all y, solutions of problem (4.1), we have ‖y‖∞ ≤ Mm . Consider the problem

 ′ y (t ) ∈ G(t , y(t )), 1y| = I (y(tk− )), y(0)t ==tk a, k

a.e. t ∈ Jm k = 1, . . . , m − 1

(4.2)

where Gm : [0, tm ] × Rn → P (Rn ) is the multivalued map defined by Gm (t , y) =

 F (t , y), F t ,

yMm

‖y ‖

‖y‖ ≤ Mm 

,

‖y‖ ≥ Mm .

(4.3)

The following result is a fundamental result proved by Aronszajn in 1942 [19] and later improved by Browder and Gupta in 1969 [78]. Proposition 4.5. Let X be a space, (E , ‖ · ‖) be a Banach space and f : X → E be a proper map, i.e. f is continuous and for every compact K ⊂ E, the set f −1 (K ) is compact. Assume further that, for each ε > 0, a continuous map fε : X → E is given and the following two conditions are satisfied: (a) ‖fε (x) − f (x)‖ < ε , for every x ∈ X , (b) for every ε > 0 and u ∈ E in a neighborhood of the origin such that ‖u‖ ≤ ε , the equation fε (x) = u has exactly one solution x. Then the set S = f −1 (0) is an Rδ -set. A useful tool for getting approximate functions which satisfy condition (b) is given by the following result (see [72], Thm. 3.1 or [79]). Proposition 4.6. Let E = C ([0, a], Rm ) be the Banach space of continuous maps with the usual max-norm and let X = B(0, r ) be the closed ball in E. Let F : X −→ E be a compact map and f : X −→ E be the associated compact vector field such that: (a) There exists x0 ∈ Rm such that for all u ∈ B(0, r ), F (u)(0) = x0 . (b) For every 0 < ε ≤ a and every u, v ∈ X , if u(t ) = v(t ) for each t ∈ [0, ε], then F (u)(t ) = F (v)(t ) for each t ∈ [0, ε]. Then there exists a sequence fn : X −→ E of continuous proper mappings satisfying conditions (a)–(b) in Proposition 4.5. 4.4. The nonconvex case We need the following result due to Bressan, Cellina and Fryszkowski. Lemma 4.3 ([80], Thm. 2). Let E be a Banach space, X = L1 (T , E ), for some measure space T , and N : E → P (X ) a contraction map with decomposable values. Then Fix (N ) is an absolute retract. Regarding the geometric structure of the solution set of problem (1.1), we state and prove our main result. Theorem 4.1. Under assumptions of Theorem 3.2, the solution set for problem (1.1) can be obtained as the limit of an inverse system of AR-spaces, for every a ∈ Rn .

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Proof. Step 1. A fixed point formulation. Since F is Hd -Lipschitz, the multivalued map G defined by (4.3) is also Hd -Lipschitz. Since F has further at most linear growth, there exists l ∈ L1loc (J , R+ ) such that for a.e. t ∈ Jm and y ∈ Rn , we have the estimates



(a) (b)

‖G(t , y)‖P ≤ l(t )(1 + Mm ) ‖G(t , y)‖P ≤ l(t )(1 + ‖y‖).

Now, if y ∈ SF then ‖y‖ ≤ Mm which implies from the definition of G that F (t , y(t )) = G(t , y(t )); hence y ∈ SG . Conversely, let y ∈ SG ; (b) means that G has a linear growth and thus Gronwall’s lemma can also be applied and implies that ‖y‖ ≤ Mm ; hence again F (t , y(t )) = G(t , y(t )) and y ∈ SF . This proves that F and G yield the same solution set. For m ∈ N∗ , define the space PACm := {y : Jm → Rn : y ∈ AC ((tk , tk+1 ), Rn ), k = 0, . . . , m − 1} and

Ωm = {y ∈ PACm : y(0) = a}.

(4.4)

Then (Ωm , ‖ · ‖Ωm ) is a Banach space with norm

‖y‖Ωm = ‖y‖∞ + ‖y′ ‖L1 .  Note that PAC = PACm is the appropriate space for mild solutions. Consider the multivalued Nemyts’ki˘ı operator associated with G: K m : Ωm ( L1 (Jm , Rn ) defined by K m (y) = {v ∈ L1 (Jm , Rn ) : v(t ) ∈ G(t , y(t )), a.e. t ∈ Jm } and Lm : L1 (Jm , Rn ) → Ωm defined by

 L (v)(t ),   0 L1 (v)(t ), Lm (v)(t ) =  · · · Lm−1 (v)(t ),

if t ∈ [0, t1 ], if t ∈ (t1 , t2 ],

(4.5)

··· if t ∈ (tm−1 , tm ],

where L0 (v)(t ) = a +

t

∫

v(s)ds,

t ∈ [0, t1 ]

0

L1 (v)(t ) = L0 (v)(t1 ) + I1 (L0 (v)(t1 )) +

t

∫

v(s)ds,

t ∈ (t1 , t2 ]

v(s)ds,

t ∈ (t2 , t3 ]

t1

L2 (v)(t ) = L1 (v)(t2 ) + I2 (L1 (v)(t2 )) +

t

∫

t2

··· Lm−1 (v)(t ) = Lm−2 (v)(tm−1 ) + Im−1 (Lm−2 (v)(tm−1 )) +

∫

t

v(s)ds,

t ∈ (tm−1 , tm ].

tm−1

From (4.5), we easily deduce that Lm (v)(t ) = a +

−

Ik (Lk−1 (v)(tk )) +

0
t

∫

v(s)ds,

a.e. t ∈ Jm .

0

Now if SGm (a) is the solution set of problem (4.2), then SGm (a) = Fix (Lm ◦ K m ), where Lm ◦ K m : Ωm ( Ωm . Step 2. Lm is a homeomorphism.

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2159

• Lm is injective. Let v1 , v2 ∈ L1 (Jm , Rn ) be such that Lm (v1 ) = Lm (v2 ). Then ∫ t ∫ t v2 (s)ds, t ∈ [0, t1 ] v1 (s)ds = 0 0 ∫ t ∫ t v2 (s)ds, v1 (s)ds = L0 (v2 )(t1 ) + I1 (L0 (v2 )(t1 )) + L0 (v1 )(t1 ) + I1 (L0 (v1 )(t1 )) + L1 (v1 )(t2 ) + I2 (L1 (v1 )(t2 )) +

∫

t ∈ (t1 , t2 ]

t1

t1 t

v1 (s)ds = L1 (v2 )(t2 ) + I2 (L1 (v2 )(t2 )) +

∫

t

v2 (s)ds,

t ∈ (t2 , t3 ]

t2

t2

··· Lm−1 (v1 )(tm1 ) + Im−1 (Lm−1 (v1 )(tm−1 )) +

∫

t

v1 (s)ds tm−1

= Lm−1 (v2 )(tm1 ) + Im−1 (Lm−1 (v2 )(tm−1 )) +

t

∫

v2 (s)ds,

t ∈ (tm−1 , tm ].

tm−1

This implies that v1 = v2 . t • Lm is surjective. Let y ∈ Ωm , then y(t ) = a + 0 y′ (s)ds, t ∈ [0, t1 ] and y′ ∈ L1 ([0, t1 ], Rn ), i.e. y(t ) = L0 (y′ )(t ),

t ∈ [0, t1 ].

Since y ∈ AC ((t1 , t2 ), R ), then n

y(t ) = L0 (y′ )(t1 ) + I1 (L0 (y′ (t1 ))) +

t

∫

y′ (s)ds,

t ∈ (t1 , t2 ].

t1

We continue the process until we arrive at y(t ) = Lm−2 (y )(tm−1 ) + Im−1 (Lm−2 (y (tm−1 ))) + ′

′

t

∫

y′ (s)ds,

t ∈ (tm−1 , tm ].

tm−1

Hence there exists y′ ∈ L1 (Jm , Rn ) such that Lm (y′ ) = y, proving our claim. • Lm is continuous. Let vn ∈ L1 (Jm , Rn ) be such that vn converges to a limit v in L1 (Jm , Rn ), as n → +∞. To show that Lm (vn ) converges to Lm (v), we use the estimates m−1

‖Lm (vn )(t ) − Lm (v)(t )‖ ≤

−

‖Ik (Lk−1 (vn )(tk )) − Ik (Lk−1 (v)(tk ))‖ + (2m − 1)

tm

∫

‖vn (s) − v(s)‖ds.

0

k=1

Then m−1

‖Lm (vn ) − Lm (v)‖Ωm ≤

−

‖Ik (Lk−1 (vn )(tk )) − Ik (Lk−1 (v)(tk ))‖ + (2m − 1)‖vn − v‖L1 .

k=1

Since

‖L0 (vn ) − L0 (v)‖∞ ≤ ‖vn − v‖L1 → 0,

as n → ∞

and using the fact that I1 is a continuous function, we get

‖I1 (L0 (vn )(t1 )) − I1 (L0 (v)(t1 ))‖ → 0,

as n → ∞.

More generally, for each k ∈ {2, . . . , m − 1}, we can prove that

‖Ik (Lk−1 (vn )(tk )) − Ik (Lk−1 (v)(tk ))‖ → 0,

as n → ∞.

Then

‖Lm (vn ) − Lm (v)‖Ωm → 0, To prove that Lm we have −1

−1

as n → ∞.

is continuous, let yn ∈ Ωm be a sequence converging to some limit y, as n → +∞. Since Lm −1

‖Lm (yn ) − Lm (y)‖L1 ≤ ‖y′n − y′ ‖L1 → 0, Hence Lm is a homeomorphism. Step 3. The set

Gm = {y′ : y ∈ SGm (a)}

as n → ∞.

−1

(y) = y′ ,

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

is an absolute retract. Let  Nm = K m ◦ Lm : L1 (Jm , Rn ) ( L1 (Jm , Rn ) be defined by

( Nm )(v) = {u ∈ L1 (Jm , Rn ) : u(t ) ∈ F (t , Lm (v)(t )), a.e. t ∈ Jm }. Then Gm = Fix  Nm . Moreover, since G is Hd -Lipschitz, then K m is Hd -Lipschitz. To prove that  Nm is a contraction, let v1 , v2 ∈ L1 (Jm , Rn ). Then there exist hi ∈ ( Nm )(vi ), i = 1, 2, such that hi (t ) ∈ G(t , Lm (vi )(t )),

a.e. t ∈ Jm .

We have Hd (G(t , Lm (v1 )(t ))), G(t , Lm (v2 )(t )) ≤ lm (t )‖Lm (v1 )(t ) − Lm (v2 )(t )‖,

a.e. t ∈ Jm .

Hence, there exists some w ∈ G(t , Lm (v2 )(t )) such that

‖h1 (t ) − w‖ ≤ lm (t )‖Lm (v1 )(t ) − Lm (v2 )(t )‖,

a.e. t ∈ Jm .

Consider the multivalued map U : Jm → P (R ) defined by n

U (t ) = {w ∈ Rn : ‖h1 (t ) − w‖ ≤ lm (t )‖Lm (v1 )(t ) − Lm (v2 )(t )‖}. Since the operator V defined by V (t ) = U (t ) ∩ G(t , Lm (v2 )(t )) is a measurable multivalued map, then there exists a function t → h2 (t ) which is a measurable selection for V (see the proof of Theorem 3.2). Hence h2 (t ) ∈ G(t , Lm (v2 )(t )) and

‖h1 (t ) − h2 (t )‖ ≤ lm (t )‖Lm (v1 )(t ) − Lm (v2 )(t )‖,

a.e. t ∈ Jm .

(4.6)

From the definition of Lm , we can prove that there exists Dm > 0 such that

‖Lm (v1 )(t ) − Lm (v2 )(t )‖ ≤ Dm

∫

t

‖v1 (s) − v2 (s)‖ds.

(4.7)

0

Let τm > max(Dm , 1), Lm (t ) =

 tm 0

‖v(t )‖e

−τm Lm (t )

∫ ‖v‖m =

tm

t

l 0 m

(s)ds, and let L1m be the weighted space of Lebesgue measurable functions such that

dt < ∞. Endowed with norm

‖v(t )‖e−τm Lm (t ) dt ,

for v ∈ L1m ,

0

this is a Banach space. From the inequalities (4.6) and (4.7), we obtain the successive estimates after performing an integration by parts: tm

∫ ‖h1 − h2 ‖m =

e−τm Lm (t ) ‖h1 (t ) − h2 (t )‖dt

0 tm

∫ ≤

lm (t )e−τm Lm (t ) Dm dt

t

∫

0

‖v1 (s) − v2 (s)‖ds 0

t −Dm tm −τm Lm (t ) ′ [e ] dt ‖v1 (s) − v2 (s)‖ds τm 0 0   ∫ tm ∫ tm −Dm −τm Lm (tm ) = e ‖v1 (t ) − v2 (t )‖dt − e−τm Lm (t ) ‖v1 (t ) − v2 (t )‖dt . τm 0 0

∫

∫

=

Hence

‖h1 − h2 ‖m ≤

Dm

τm

‖v1 − v2 ‖m .

By an analogous relation obtained by interchanging the roles of h1 and h2 , we finally obtain that for every v1 , v2 ∈ L1 (Jm , Rn ), Hd ( Nm (v1 ),  Nm (v2 )) ≤

Dm

τm

‖v1 − v2 ‖m .

Now the multivalued map t ( G(t , .) is measurable and x ( G(., x) is Hd -continuous. In addition G(., .) has compact values; hence G(., .) is measurable, continuous. Since the measurable multifunction G is integrably bounded, Lemma 3.7 implies that the Nemyts’ki˘ı operator K m has decomposable values; hence  Nm has decomposable values too. Since  Nm is contractive and ′  Fix (Nm ) = {y : y ∈ SGm (a)} = Gm , by Lemma 4.3, we conclude that the set Gm is an absolute retract. Step 4. Conclusion. Using the fact that Lm is a homeomorphism and that homeomorphic spaces have the same AR topological structure (Lemma 4.2), we deduce from Step 3 that the set Fix (Lm ◦ K m ) = SGm (a) is also an absolute retract. Hence, it is

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2161

p

p

acyclic. It remains to show that {Lm ◦ K m } is a map of the inverse system {Ωm , πm , N} where πm (y) = y|[0,tm ] , for every y ∈ Ωp and p ≥ m. This follows from the equalities

 (Lm ◦ K m )π (y)(t ) = a + p m

 π (Lm ◦ K m )(y)(t ) = p m

a+

−

Ik ((Lk−1 v)(tk )) +

0 < tk < t

−

t

∫

v(s)ds : v ∈ L (Jm ) and v(t ) ∈ G(t , y(t )), a.e. t ∈ Jm



1

0

Ik ((Lk−1 v)(tk )) +

0 < tk < t



t

∫ 0

v(s)ds : v ∈ L1 (Jp ) and v(t ) ∈ G(t , y(t )), a.e. t ∈ Jp .

Finally every Fréchet space is isometrically homeomorphic to a limit of some countable inverse system of Banach spaces and PAC is a limit of the Banach spaces PACm . Hence the maps {Lm ◦ K m } induce the limit one N : Ω → Ω with N (y)|[0,tm ] = {(Lm ◦ K m )(y|[0,tm ] )} and

Ω = {y ∈ PAC : y(0) = a}.

(4.8)

Since Fix N = SF (a), we conclude the proof of the theorem.



4.5. The convex case In this section, we consider the case where F has convex compact values and prove further more precise results. Theorem 4.2. Under the assumptions of Theorem 3.1, the solution set is homeomorphic to the intersection of absolute retracts. If further F is an upper Scorza–Dragoni map, then the solution set is an Rδ -set. Proof. Step 1. The solution set is homeomorphic to an intersection of AR spaces. From the assumptions and using the fact that F is a Carathéodory function with convex and compact values, we can apply the results of Benassi and Gavioli (see [81], Thm. 5.1) to get a sequence of multivalued maps Fk : J × Rn → P (Rn ) such that, for every k ≥ 1, (a) (b) (c) (d) (e)

Fk has nonempty, closed and connected values with the same growth as F , Fk (., y) is measurable, for every y ∈ Rn , Fk (., y) is locally Lipschitzian, for a.e. t ∈ J, F (t , y) ⊂ Fk+1 (t , y) ⊂ Fk (t , y), for a.e. t ∈ J and all y ∈ Rn , limk→∞ Hd (F (t , y), Fk (t , y)) = 0, for a.e. t ∈ J and all y ∈ Rn .

For every k ≥ 1, consider the sequence of impulsive problems

 ′ y (t ) ∈ Fk (t , y(t )), 1y| = I (y(ti− )), y(0)t ==ti a i

a.e. t ∈ Jm , i = 1, . . . , m − 1

(4.9)

k and denote by Sm (a) the solution set of problem (4.9). Since for a.e. t ∈ Jm , Fk (t , .) is Lipschitzian on bounded subsets of Rn , k using (Thm. 2, [80]), we can prove that Sm (a) is an absolute retract. Consider the multivalued operators Nm and Nmk defined by

 Nm (y) :=

h ∈ Ωm : h(t ) = a +



t

∫

−

v(s)ds +

Ik (y(ti )), a.e. t ∈ Jm

0 < ti < t

0



where v ∈ SF ,y = v ∈ L1 ([0, tm ], Rn ) : v(t ) ∈ F (t , y(t )), a.e. t ∈ Jm

 k Nm (y) :=

h ∈ Ωm : h(t ) = a +



t

∫



−

vk (s)ds +

Ik (y(ti )), a.e. t ∈ Jm

,

0 < ti < t

0





where vk ∈ SFk ,y = vk ∈ L1 (Jm , Rn ) : vk (t ) ∈ Fk (t , y(t )), a.e. t ∈ Jm . k k Note that Fix (Nm ) = Smk (a) ∈ AR and Sm (a) = ∩∞ k=1 Sm (a), where Sm is the solution set for problem (4.1). Moreover the map p Nm is a map of the inverse system (Ωm , πm , N) where Ωm is defined in (4.4). It is clear that Fix (N ) = SF (a). Since PAC is a limit of the Banach spaces PACm , then the Nm induce the limit map N : Ω → Ω where Ω is defined in (4.8) and N is given by

 N (y) :=

h ∈ Ω : h(t ) = a +



t

∫

v(s)ds + 0

− 0 < ti < t

Ik (y(ti )), a.e. t ∈ J

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

where

  v ∈ SF ,y = v ∈ L1loc ([0, ∞), Rn ) : v(t ) ∈ F (t , y(t )) a.e. t ∈ J . From Proposition 4.4, we infer that SF (a) = lim Sm (a). Let ←−

Zm = {(yi ) ∈

∞ ∏

Si (a) : yi = ym {[0, ti ], for i ≤ m}},

i =1 m

Z k = {(yi ) ∈

∞ ∏

Sik (a) : yi = ym {[0, ti ], for i ≤ m}} ∼ =

∞ ∏

Sik (a) ∈ AR.

i=m

i =1 m

m

m ∞ ∞ Therefore SF (a) = lim Sm (a) = ∩∞ m=1 Zm = ∩m=1 ∩k=1 Z k = ∩m=1 Z m , proving our claim. ←−

Step 2. The solution set is an Rδ -set. By Theorem 3.1, the solution set is nonempty. Moreover SF (a) = ∩∞ m=1 Sm (a), where Sm (a) is a solution set of problem (4.1). Since F is an upper Scorza–Dragoni map, then the function G defined by (4.3) is also an upper Scorza–Dragoni map. This implies that G is σ -Ca-selectionable and m-LL-selectionable. Then there exists a sequence of multivalued maps Gm such that Gm (., .) is Carathéodory and G(t , x) = ∩∞ m=1 Gm (t , x). Using the same method as in [21], Thm. 6.11 or 6.18, and making use of the Aronszajn–Browder–Gupta results (cf. Propositions 4.5 and 4.6), we can prove that the solution set SGm (a) of problem (4.2) is an Rδ -set. Hence the set Sm (a) is Rδ . Moreover, it can be easily seen that {Lm ◦ K m } is a map of the inverse system {PCm , πmp , N} where πmp (y) = y|[0,tm ] , for every y ∈ Ωp and p ≥ m. Indeed, since the Fréchet space PC is a limit of PCm , the mappings {Lm ◦ K m }∞ m=1 induce the limit mapping N : PC → PC defined by

 N (y) :=

h ∈ PC : h(t ) = a +



t

∫

f (s)ds + 0

−

Ik (y(tk )), for a.e. t ∈ J −

0
where f ∈ SF ,y = f ∈ L1loc ([0, ∞), Rn ) : f (t ) ∈ F (t , y(t )), a.e. t ∈ J .





Note that the fixed point set of the mapping N is the solution set of problem (1.1). Applying Proposition 4.4, the solution set of problem (1.1) is an Rδ -set, as claimed.  5. The terminal problem Our final existence theorem is concerned with the terminal (or target) problem, that is problem (1.1) in which the initial condition is replaced by a limit condition at positive infinity. We also prove compactness and acyclicity of the solution set, extending similar results obtained in [31] for differential inclusions. Consider the following problem:

 ′ y (t ) ∈ F (t , y(t )), 1y|t =tk = Ik (y(tk− )),  lim y(t ) = y∞ ∈ Rn .

a.e. t ∈ J k = 1, . . . ,

(5.1)

t →∞

For the existence problem, our arguments are based on the following fixed point theorem for contraction multivalued operators proved in 1970 by Covitz and Nadler (see e.g. [39], Theorem 11.1). Lemma 5.1 ([82]). Let (X , d) be a complete metric space. If N : X → Pcl (X ) is a contraction, then Fix N ̸= ∅. We are in position to state our final existence result. Theorem 5.1. Assume that Ik satisfies (H4 ) and the multivalued map F : J × Rn → Pcp (Rn ) is such that t ( F (t , .) is measurable;

3 ) there exists l ∈ L1 (J , R+ ) such that F (t , 0) ⊂ l(t )B(0, 1), a.e. t ∈ J, and (H Hd (F (t , x), F (t , y)) ≤ l(t )‖x − y‖,

for every x, y ∈ Rn and a.e. t ∈ J .

If

‖l‖L1 +

∞ −

ck < 1,

(5.2)

k=1

then the boundary value problem (5.1) has at least one solution. If further F : R+ × Rn ( Rn is a Carathéodory multivalued map with compact convex values, then the solution set is a contractible compact set, and hence acyclic.

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

2163

Proof. Part 1. Existence of a solution. Consider the operator N : PCb → P (PCb ) defined by N (y) = {h ∈ PCb } such that for t ∈J h( t ) = y ∞ −

∞

∫

v(s)ds − t

−

Ik (y(tk )),

tk ≥t

where v ∈ SF ,y = {v ∈ L1 (J , Rn ) : v(t ) ∈ F (t , y(t )), a.e. t ∈ J }. Clearly, the fixed points of the operator N are solutions of problem (5.1). Step 1. N (y) ∈ Pcl (PC ) for each y ∈ PCb . Indeed, let (yn )n≥0 ∈ N (y) be such that yn −→  y in PCb . Then there exists vn ∈ SF ,y such that for a.e. t ∈ J yn (t ) = y∞ −

∞

∫

−

vn (s)ds − t

Ik (y(tk )).

(5.3)

tk ≥ t

3 ), we get Since F has compact values, then there exists a subsequence (vnm (.)) which converges to v(.) in Rn . From (H ‖vnm (t )‖ ≤ l(t )‖y‖PC + l(t ),

a.e. t ∈ J .

The Lebesgue dominated convergence theorem then implies that (5.3) yields that  y ∈ N (y), as claimed.

∞ t

vn (s)ds converges to

∞ t

v(s)ds. Passing to the limit in

Step 2. N is a contraction. Let y, y ∈ PCb and h ∈ N (y). Then there exists v(t ) ∈ F (t , y(t )) such that for a.e. t ∈ J h( t ) = y ∞ −

∞

∫

v(s)ds − t

−

Ik (y(tk )).

tk ≥t

3 ) implies that there is some u ∈ F (t , y(t )) such that (H ‖v(t ) − u‖ ≤ l(t )‖y(t ) − y(t )‖,

a.e. t ∈ J .

Consider the multivalued map defined by V (t ) = U (t ) ∩ F (t , y(t )) where U : [0, ∞) → Pcl (Rn ) is given by U (t ) = {u ∈ Rn : ‖v(t ) − u‖ ≤ l(t )‖y(t ) − y(t )‖}. Arguing as in the proof of Theorem 3.2, we can find a function v(t ), which is a measurable selection for V . Thus, v(t ) ∈ F (t , y(t )) and

‖v(t ) − v(t )‖ ≤ l(t )‖y(t ) − y(t )‖,

a.e. t ∈ J .

Let us define for a.e. t ∈ J h( t ) = y ∞ −

∞

∫

v(s)ds − t

−

Ik (y(tk )).

tk ≥ t

Then h ∈ N (y) and

‖h(t ) − h(t )‖ ≤

∞

∫

l(s)‖y(s) − y(s)‖ds + 0

ck ‖y(tk ) − y(tk )‖

k=1

∞

∫

∞ −

l(s)‖y − y‖PCb ds +

≤ 0

∞ −

ck ‖y − y‖PCb .

k=1

Hence

 ‖h − h‖PCb ≤ ‖l‖L1 +

∞ −

 ck

‖y − y‖PCb .

k=1

Since y ∈ PCb and h ∈ N (y) are arbitrary, it follows that

 sup d(h, N (y)) ≤

h∈N (y)

‖l‖L1 +

∞ −

 ck

‖y − y‖PCb .

k=1

By an analogous relation, obtained by interchanging the roles of y and y, we have



Hd (N (y), N (y)) ≤ ‖l‖L1 +

∞  − ck ‖y − y‖PCb . k=1

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

By (5.2), N is a contraction on PCb and thus Lemma 5.1 implies that N has a fixed point in PCb , a solution of problem (5.1). For each y∞ ∈ Rn , let S (y∞ ) be the solution set of problem (5.1). Part 2. Structure of the solution set. First, we prove that for each y∞ ∈ Rn , the set S (y∞ ) is acyclic. Let y ∈ S (y∞ ); then there exists v ∈ L1 (J , R+ ) such that v(t ) ∈ F (t , y(t )) for a.e. t ∈ J and y(t ) = y∞ −

∞

∫

v(s)ds − t

−

Ik (y(tk )).

tk ≥t

Consequently ∞

∫

‖y(t )‖ ≤ ‖y∞ ‖ +

l(s)‖y(s)‖ds + ‖l‖L1 + 0

∞ −

ck ‖y(tk )‖ +

k=1

∞ −

‖Ik (0)‖,

a.e. t ∈ J .

k=1

Then ∞

∫

l(s)‖y‖PCb ds + ‖l‖L1 +

‖y‖PCb ≤ ‖y∞ ‖ + 0

∞ −

ck ‖y‖PCb +

k=1

∞ −

‖Ik (0)‖.

k =1

Hence



1

‖y‖PCb ≤ M :=

1 − ‖l‖L1 −

‖y∞ ‖ + ‖l‖L1 +

∞ ∑

∞ −

 ‖Ik (0)‖ .

(5.4)

k=1

ck

k=1

Step 1. Definition of a homotopy. Consider the modified problem

 ′ y (t ) ∈ G(t , y(t )), 1y|t =tk = Ik (y(tk− )),  lim y(t ) = y∞ ∈ Rn ,

a.e. t ∈ J k = 1, . . . ,

(5.5)

t →∞

where G : R+ × Rn → Pcp,c v (Rn ) is the multivalued map defined by G(t , y) =

 F (t , y), F t ,

‖y‖ ≤ M∗

yM∗

‖y‖



,

‖y‖ ≥ M∗ ,

where M∗ > M. Now, we show that S (y∞ ) = G(y∞ ) where G(y∞ ) is the solution set of problem (5.5). Let y ∈ S (y∞ ); then y(tk+ ) − y(tk− ) = Ik (y(tk )) for every k ∈ {1, 2, . . .}, limt →∞ y(t ) = y∞ and y′ (t ) ∈ F (t , y(t )), for a.e. t ∈ J. Thus

‖y(t )‖ ≤ M ,

a.e. t ∈ J ⇒ F (t , y(t )) = G(t , y(t )), a.e. t ∈ J .

This implies that y ∈ G(y∞ ), whence S (y∞ ) ⊂ G(y∞ ). Conversely, let y ∈ G(y∞ ); then y(t ) = y∞ −

∞

∫

v(s)ds − t

−

Ik (y(tk )),

a.e. t ∈ J ,

tk ≥t

where v ∈ SG,y . Setting W = {t ∈ [0, ∞) : ‖y(t )‖ > M∗ }, we show that mes(W ) = 0 where mes is the Lebesgue measure. Assume that mes(W ) ̸= 0. Then y(t ) = y∞ −

∫

∞

v(s)ds − W

∫

∞ [t ,∞)\W

v(s)ds −

−

Ik (y(tk )).

tk ≥ t

As a consequence,

‖y(t )‖ ≤ ‖y∞ ‖ +

∫

‖v(s)‖ds +

∫

W

[0,∞)\W

‖v(s)‖ds +

∞ −

ck ‖y(tk )‖ +

k =1

∞ −

‖Ik (0)‖.

k=1

3 ), we obtain the successive estimates From the definition of G, (H4 ) and (H ‖y(t )‖ ≤ ‖y∞ ‖ +

∫

‖v(s)‖ds + W

∫

l(s)ds +

≤ ‖y∞ ‖ + W

∫

∫ [0,∞)\W

‖v(s)‖ds +

M∗ l(s)ds + W

∞ −

ck ‖y(tk )‖ +

k=1

∫ [0,∞)\W

l(s)‖y(s)‖ds

∞ − k =1

‖Ik (0)‖

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

∫ + [0,∞)\W

l(s)ds +

∞ − k=1

∞

∫

ck ‖y(tk )‖ +

l(s)ds +

≤ ‖y∞ ‖ +

∞ −

‖Ik (0)‖

k=1

∞

∫

l(s)‖y(s)‖ds +

0

0

2165

∞ −

ck ‖y(tk )‖ +

k=1

∞ −

‖Ik (0)‖.

k=1

Then

‖y‖PCb ≤ ‖y∞ ‖ + ‖l‖L1 + ‖l‖L1 ‖y‖PCb +

∞ −

ck ‖y‖PCb +

∞ −

‖Ik (0)‖.

k =1

k=1

Therefore



1

‖y‖PCb ≤

1 − ‖l‖L1 −

∞ ∑

‖y∞ ‖ + ‖l‖L1 +

∞ −

 ‖Ik (0)‖ = M .

k=1

ck

k=1

This is contradiction to M < M∗ ; then meas(W ) = 0 which implies that G(t , y(t )) = F (t , y(t )),

a.e. t ∈ J .

Thus S (y∞ ) = G(y∞ ). Since G(., .) ∈ Pcp,c v (Rn ) because F (., .) ∈ Pcp,c v (Rn ) and is u.s.c., then there exists a Carathéodory Lipschitzian selection (see [43]) g : R+ × Rn → Rn such that g (t , x) ∈ G(t , x), Since ‖l‖L1 + (see [43]) and

∑∞

k=1

for a.e. t ∈ J and x ∈ Rn .

ck < 1, then we can choose a Lipschitzian function γ ∈ L1 (J , R+ ) such that

‖g (t , x) − g (t , y)‖ ≤ γ (t )‖x − y‖,

∞ 0

γ (s)ds +

∑∞

k=1

ck < 1

for a.e. t ∈ J and each x, y ∈ Rn .

Consider the single-valued problem

 ′ x (t ) = g (t , x(t )), 1x|t =tk = Ik (x(tk− )),  lim x(t ) = y∞ ∈ Rn .

a.e. t ∈ J k = 1, . . . ,

(5.6)

t →∞

Since g is a Carathéodory Lipschitz function, we can prove that problem (5.6) has a unique global solution given by x (see e.g. [51], Thms. 3.3 and 3.5). Furthermore x ∈ PC and limt →∞ x(t ) = y∞ . Since g (t , x) ∈ G(t , x), then

‖x‖PCb ≤ M∗ which implies that x ∈ PCb . Define the homotopy h : S (y∞ ) × [0, 1] → S (y∞ ) by

h(y, α)(t ) =

  y(t ),    z (t ),     x(t ),

for t ≥

1

α

− α, α ̸= 0

for 0 ≤ t < for α = 0,

1

− α, α ̸= 0

α

where, for each α ∈ (0, 1), z is the unique solution of the backward Cauchy problem:

  z ′ (t ) = g (t , z (t )),    1 z |t =tk =Ik (z (tk−)),     z 1 − α = x 1 − α . α α

[

a.e. t ∈ 0, k = 1, . . . ,

1

α

] − α \ {t1 , t2 , . . .} (5.7)

Step 2. h is a continuous function. Let (yn , αn ) ∈ S (y∞ ) × [0, 1] be such that (yn , αn ) → (y, α), as n → +∞. We shall prove that h(yn , αn ) → h(y, α). We have

h(yn , αn )(t ) =

   yn (t ),   z (t ),      x(t ),

for t ≥

1

αn

− αn , αn ̸= 0

for 0 ≤ t < for αn = 0.

We distinguish between different cases:

1

αn

− αn , αn ̸= 0

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S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

• If limn→∞ αn = 0, then ‖h(yn , αn )(t ) − h(y, 0)(t )‖ = ‖z (t ) − x(t )‖, for n large enough. Indeed when limn→∞ αn = 0, [ ] 1 1 − αn > t ∀ t ∈ 0, − α , ∃ n0 ∈ N : n ≥ n0 H⇒ α αn and thus h(yn , αn ) = x(t ). Moreover

 lim z

1

αn

n→+∞

− αn



 = lim x

1

− αn

αn

n→+∞



= y∞ ,

and hence limn→∞ ‖h(y, αn )(t ) − h(y, 0)(t )‖ = 0. • Let α  n ̸= 0 and0 < limn→∞ αn = α ≤ 1. Since z n is a solution of problem (5.7), then for every n ≥ 1 and for a.e. t ∈ 0, α1 − αn , we have n  ∫ 1/αn −αn  − 1 g (s, z (s))ds − − αn − z (t ) = x Ik (z (tk )). αn t t
Without loss of generality, we may assume αn ≥ α ; hence α1 − αn ≤ α1 − α . Consider the function n

   z (t ),    zn (t ) =  1  x − αn , αn

[

for t ∈ 0, for

1

αn

] − αn ,

1

αn

− αn ≤ t ≤

1

− α.

α

Since g and Ik for k = 1, . . . are Lipschitzian functions and x is bounded, then there exists a positive real number M independent of n ∈ N such that



[

sup ‖ zn (t )‖ : t ∈ 0,

1

α 

−α

]

Moreover the set  zn (t ) : t ∈ 0, α1 − α



‖zn (τ1 ) − zn (τ2 )‖ ≤

τ1

∫

τ1

≤ M, 

for every n ∈ N.

is equicontinuous in PCb

−

‖g (s, z (s))‖ds +





0, α1 − α , Rn . If τ1 , τ2 ∈ 0, α1 − αn , then n







‖Ik (z (tk ))‖.

τ1
Using the fact that x is bounded, we can easily prove that there exists K∗ > 0 such that ‖z ‖PC ≤ K∗ (for the proof, we refer the reader to [51], Thm. 3.3). Then

‖ zn (τ1 ) − zn (τ2 )‖ ≤ If τ1 , τ2 ∈



1

αn

τ2

∫

τ1

τ2

∫

K∗ γ (s)ds +

−

l(s)ds +

τ1

sup{‖Ik (x)‖ : x ∈ B(0, K∗ )}.

τ1
 − αn , α1 − α , then

‖ zn (τ1 ) − zn (τ2 )‖ = 0.     In the case where τ1 ∈ 0, α1 − αn and τ2 ∈ α1 − αn , α1 − α , we have the estimates n n ‖ zn (τ1 ) − zn (τ2 )‖ ≤

1

αn −αn

∫

τ1 τ2

∫ ≤

τ1

K∗ γ (s)ds +

K∗ γ (s)ds +

∫

τ2

τ1

τ1

 z (t ) = x

1

α



−α −

1/α−α

∫

−

sup{‖Ik (x)‖ : x ∈ B(0, K∗ )}.

τ1


0, α1 − α , Rn such that ( zn )n∈N converges to  z and



−



Ik ( z (tk )).

1 t < tk < α −α

Hence  z is a solution of problem (5.7). By uniqueness,  z = z. Thus



[

sup ‖ zn (t ) − z (t )‖ : t ∈ 0,

1

α

−α

]

sup{‖Ik (x)‖ : x ∈ B(0, K∗ )}

τ1
g (s, z (s))ds −

t

−

l(s)ds +

l(s)ds +

By the Ascoli–Arzéla Lemma, there exists  z ∈ PCb



τ2

∫

→ 0,

as n → ∞.

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

• Finally, let t ≥

2167

− αn and αn ̸= 0. Since (yn ) converges to y in PCb , we have

1

αn

sup{‖yn (t ) − y(t )‖ : t ∈ [0, ∞)} → 0,

as n → ∞.

To sum up, sup{‖h(yn , αn )(t ) − h(y, α)(t )‖ : t ∈ [0, ∞)} → 0,

as n → ∞,

proving our claim. Step 3. S (y∞ ) is closed in PCb . Let {yn : n ∈ N} ⊂ S (y∞ ) be such that (yn ) converges to y in PCb as n → +∞. Then there exists vn ∈ SG,yn such that ∞

∫

yn (t ) = y∞ −

vn (s)ds − t

−

Ik (yn (tk )),

a.e. t ∈ J .

tk ≥ t

From (5.4), there exists M > 0 such that sup{‖yn (t )‖ : t ∈ R+ } ≤ M

(5.8)

and

‖vn (t )‖ ≤ l(t )M + l(t ),

a.e. t ∈ J .

Since B(0, 1) is compact in R , then there exists a subsequence (vnm (.)) which converges to some limit v(.). By the Lebesgue dominated convergence theorem, we conclude that v ∈ L1 (R+ , Rn ). Using the fact that G(., .) ∈ Pcp,c v (Rn ) and that G(t , .) is u.s.c., the map G(t , .) has a closed graph; hence v ∈ SG,y . Therefore n

y(t ) = y∞ −

∞

∫

v(s)ds − t

−

Ik (y(tk )),

a.e. t ∈ J ,

tk ≥ t

proving that y ∈ S (y∞ ). From Steps 1–3, h is a continuous homotopy with h(y, 0) = x and h(y, 1) = y (see also [83] for the case of differential inclusions). Therefore the set S (.) is contractible. Let (yn )n∈N ⊂ S (y∞ ). (5.4) shows that (yn )n∈N is uniformly bounded. Also, we can easily prove that for every compact interval J∗ ⊂ [0, ∞), {yn : n ∈ N} is almost equicontinuous. It only remains to prove that {yn : n ∈ N} is equiconvergent at ∞, i.e. for every ε > 0, there exists T (ε) > 0 such that ‖yn (t ) − yn (∞)‖ ≤ ε for every t ≥ T and each n ∈ N. Let yn ∈ {yn : n ∈ N}; then there exists vn ∈ SF ,yn such that ∞

∫

yn (t ) = y∞ −

vn (s)ds − t

−

Ik (yn (tk )),

a.e. t ∈ J .

tk ≥ t

Then

‖yn (t ) − yn (∞)‖ ≤

∫

∞

−

‖vn (s)‖ds +

≤ (1 + M )

∞

∫

l(s)ds + t

Since

∑∞

k=1

∞ −

ck < ∞,

∑∞

k=1

‖Ik (yn (tk ))‖

t ≤tk <∞

t

−

(ck M + ‖Ik (0)‖).

t ≤tk <∞

‖Ik (0)‖ < ∞ and p ∈ L1 (J , R+ ), there exist k0 and T (ε) > 0 such that

(ck M + ‖Ik (0)‖) ≤

k=k0

ε 2

and ∞

∫

l(s)ds < t

ε 2(M + 1)

,

∀ t ≥ T (ε).

Hence

‖yn (t ) − yn (∞)‖ ≤ ε,

a.e. t ≥ max(k0 , T (ε)).

Then {yn : n ∈ N} is equiconvergent. This with Lemma 3.4 yields that S (y∞ ) is compact, and hence acyclic. The proof of Theorem 5.1 is complete. 

2168

S. Djebali et al. / Nonlinear Analysis 74 (2011) 2141–2169

6. Concluding remarks In 1942, Aronszajn [19] proved that the solution set of a Cauchy problem associated with a first-order differential equation in a finite-dimensional space is Rδ . Then Aronszajn’s result was improved by several authors. We cite Browder and Gupta [78] who gave in 1969 a way to obtain this result. For first-order differential inclusions, De Belasi and Myjak [20] proved in 1986 that for initial value problems for differential inclusions on bounded intervals, the solution set is Rδ whenever the nonlinearity F is a Carathéodory multivalued map which is integrably bounded and compact convex valued. In [72], Górniewicz noticed that the result remains true if F has sublinear growth and proved the previous results. In this work, we have extended these results to the initial problem for impulsive differential inclusions on unbounded intervals. The results of Theorem 4.2 were obtained by Andres et al. [25] in the case of differential inclusions and the solution set was proved to have compact Rδ structure in [30] under convex compact right-hand sides. Some existence results for first-order and second-order differential inclusions on the half-line can be found in the literature (see e.g. [31,32,83] and the references therein). Also some abstract differential equations and inclusions on Fréchet spaces are considered in [26] where the projective limit approach is employed. Our work complements these works since it deals with impulsive initial problems on the half-line. In 1976, Lasry and Robert [74] proved that, if the nonlinearity F is compact, convex valued, u.s.c. and bounded, then the set of all solutions for first-order differential inclusions with right-hand side F is a compact and acyclic set. In 1986, Górniewicz [71] discussed the topological structure of the set of solutions (contractibility and acyclic contractibility) when F is σ -selectionable. 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