Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 31, 1-40; http://www.math.u-szeged.hu/ejqtde/

First Order Impulsive Differential Inclusions with Periodic Conditions

John R Graef^{1} and Abdelghani Ouahab^{2}

1Department of Mathematics, University of Tennessee at Chattanooga

Chattanooga, TN 37403-2504 USA e-mail: John-Graef@utc.edu 2 Laboratoire de Mathématiques, Université de Sidi Bel Abbès BP 89, 22000 Sidi Bel Abbès, Algérie e-mail: agh ouahab@yahoo.fr

Abstract

In this paper, we present an impulsive version of Filippov’s Theorem for the first-order nonresonance impulsive differential inclusion

y′(t) − λy(t) ∈ F (t, y(t)), a.e. t ∈ J\{t_{1}, …, t_{m}},
y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,
y(0) = y(b),

where J = [0, b] and F : J × ℝ^{n} → P(ℝ^{n}) is a set-valued map. The functions I_{k} characterize the jump of the solutions at impulse points t_{k} (k = 1, …, m.). Then the relaxed problem is considered and a Filippov-Wasewski result is obtained. We also consider periodic solutions of the first order impulsive differential inclusion

y′(t) ∈ φ(t, y(t)), a.e. t ∈ J\{t_{1}, …, t_{m}},
y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,
y(0) = y(b),

where φ : J × ℝ^{n} → P(ℝ^{n}) is a multi-valued map. The study of the above problems use an approach based on the topological degree combined with a Poincaré operator.

Key words and phrases: Impulsive differential inclusions, Filippov’s theorem, relaxation theorem, boundary value problem, compact sets, Poincaré operator, degree theory, contractible, R_{δ}–set, acyclic. AMS (MOS) Subject Classifications: 34A60, 34K45, 34B37.

1 Introduction

The dynamics of many processes in physics, population dynamics, biology, medicine,

and other areas may be subject to abrupt changes such as shocks or perturbations (see for instance [1, 30] and the references therein). These perturbations may be viewed 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 are described by impulsive differential equations. Important contributions to the study of the mathematical aspects of such equations can be found in the works by Bainov and Simeonov [7], Lakshmikantham, Bainov, and Simeonov [31], Pandit and Deo [36], and Samoilenko and Perestyuk [37] among others.

During the last couple of years, impulsive ordinary differential inclusions and functional differential inclusions with different conditions have been intensely studied; see, for example, the monographs by Aubin [4] and Benchohra et al. [10], as well as the thesis of Ouahab [35], and the references therein.

In this paper, we will consider the problem

y′(t) − λy(t) ∈ F (t, y(t)), a.e. t ∈ J := [0, b],  (1)
y(t_{k}^{+}) − y(t_{k}) = I_{k}(y(t_{k})), k = 1, …, m,  (2)
y(0) = y(b),  (3)

where λ ≠ 0 is a parameter, F : J × ℝ^{n} → P(ℝ^{n}) is a multi-valued map, I_{k} ∈ C(ℝ^{n}, ℝ^{n}), k = 1, …, m, t_{0} = 0 < t_{1} < … < t_{m} < t_{m+1} = b, Δy|_{t=t_{k}} = y(t_{k}^{+}) − y(t_{k}^{−}),

y(t_{k}^{+}) = lim_{h→0^{+}} y(t_{k} + h), and y(t_{k}^{−}) = lim_{h→0^{+}} y(t_{k} − h).

First, we shall be concerned with Filippov’s theorem for first order nonresonance impulsive differential inclusions. This is the aim of Section 3. Section 4 is devoted to the relaxed problem associated with problem (1)–(3), that is, the problem where we consider the convex hull of the right-hand side. The compactness of the solution sets is examined in Section 5. In Section 6, we study the existence of solutions of first order impulsive differential inclusions with periodic conditions.

EJQTDE, 2008 No. 31, p. 2

2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis that are used throughout this paper. Here, C(J, ℝ) will denote the Banach space of all continuous functions from J into ℝ with the Tchebyshev norm

∥x∥_{∞} = sup{|x(t)| : t ∈ J}.

In addition, we let L^{1}(J, ℝ) be the Banach space of measurable functions x : J ⟶ ℝ which are Lebesgue integrable with the norm

|x|_{1} = ∫_{0}^{b} |x(s)|ds.

If (X, d) is a metric space, the following notations will be used throughout this paper: • P(X) = {Y ⊂ X : Y ≠ ∅}.

• P_{p}(X) = {Y ∈ P(X) : Y has the property “p”} where p could be: cl=closed, b=bounded, cp=compact, cv=convex, etc. Thus,

• P_{cl}(X) = {Y ∈ P(X) : Y closed}.

• P_{b}(X) = {Y ∈ P(X) : Y bounded}.

• P_{cv}(X) = {Y ∈ P(X) : Y convex}.

• P_{cp}(X) = {Y ∈ P(X) : Y compact}.

• P_{cv,cp}(X) = P_{cv}(X) ∩ P_{cp}(X).

Let (X, ∥.∥) be a Banach space and F : J → P_{cl}(X) be a multi-valued map. We say that F is measurable provided for every open U ⊂ X, the set F^{+1}(U) = {t ∈ J : F (t) ⊂ U} is Lebesgue measurable in J. We will need the following lemma.

EJQTDE, 2008 No. 31, p. 3 Lemma 2.1 ([13, 17]) The mapping F is measurable if and only if for each x ∈ X, the function ζ : J → [0, +∞) defined by

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

is Lebesgue measurable.

Let (X, ∥·∥) be a Banach space and F : X → P(X) be a multi-valued map. We say that F has a fixed point if there exists x ∈ X such that x ∈ F (x). The set of fixed points of F will be denoted by F ix F. We will say that F has convex (closed) values if F (x) is convex (closed) for all x ∈ X, and that F is totally bounded if F (A) = ⋃_{x∈A}{F (x)} is bounded in X for each bounded set A of X, i.e.,

sup_{x∈A}{sup{∥y∥ : y ∈ F (x)}} < ∞.

Let (X, d) and (Y, ρ) be two metric spaces and F : X → P_{cl}(Y) be a multi-valued mapping. Then F is said to be lower semi-continuous (l.s.c.) if the inverse image of V by F

F^{−1}(V) = {x ∈ X : F (x) ∩ V ≠ ∅}

is open for any open set V in Y. Equivalently, F is l.s.c. if the core of V by F

F^{+1}(V) = {x ∈ X : F (x) ⊂ V }

is closed for any closed set V in Y.

Likewise, the map F is called upper semi-continuous (u.s.c.) on X if for each x_{0} ∈ X the set F (x_{0}) is a nonempty, closed subset of X, and if for each open set N of Y containing F (x_{0}), there exists an open neighborhood M of x_{0} such that F (M) ⊆ Y. That is, if the set F^{−1}(V) is closed for any closed set V in Y. Equivalently, F is u.s.c. if the set F^{+1}(V) is open for any open set V in Y.

EJQTDE, 2008 No. 31, p. 4

The mapping F is said to be completely continuous if it is u.s.c. and, for every bounded subset A ⊆ X, F (A) is relatively compact, i.e., there exists a relatively compact set K = K(A) ⊂ X such that

F (A) = ⋃{F (x) : x ∈ A} ⊂ K.

Also, F is compact if F (X) is relatively compact, and it is called locally compact if for each x ∈ X, there exists an open set U containing x such that F (U) is relatively compact.

We denote the graph of F to be the set Gr(F) = {(x, y) ∈ X × Y, y ∈ F (x)} and recall the following facts.

Lemma 2.2 ([12], [15, Proposition 1.2]) If F : X → P_{cl}(Y) is u.s.c., then Gr(F) is a closed subset of X × Y, i.e., for any sequences (x_{n})_{n∈ℕ} ⊂ X and (y_{n})_{n∈ℕ} ⊂ Y, if x_{n} → x_{∗} and y_{n} → y_{∗} as n → ∞, and y_{n} ∈ F (x_{n}), then y_{∗} ∈ F (x_{∗}). Conversely, if F has nonempty compact values, is locally compact, and has a closed graph, then it is u.s.c.

The following two lemmas are concerned with the measurability of multi-functions; they will be needed in this paper. The first one is the well known Kuratowski-Ryll-Nardzewski selection theorem.

Lemma 2.3 ([17, Theorem 19.7]) Let E be a separable metric space and G a multivalued map with nonempty closed values. Then G has a measurable selection.

Lemma 2.4 ([40]) Let G : J → P(E) be a measurable multifunction and let g : J → E be a measurable function. Then for any measurable v : J → ℝ_{+} there exists a measurable selection u of G such that

|u(t) − g(t)| ≤ d(g(t), G(t)) + v(t).

EJQTDE, 2008 No. 31, p. 5 For any multi-valued function G : J × ℝ^{n} → P(E), we define

∥G(t, z)∥_{P} := sup{|v| : v ∈ G(t, z)}.

Definition 2.5 The mapping G is called a multi-valued Carathéodory function if:

(a) The function t ↦ G(t, z) is measurable for each z ∈ D;

(b) For a.e. t ∈ J, the map z ↦ G(t, z) is upper semi-continuous. Furthermore, it is an L^{1}−Carathéodory if it is locally integrably bounded, i.e., for each positive r, there exists some h_{r} ∈ L^{1}(J, ℝ^{+}) such that

∥G(t, z)∥_{P} ≤ h_{r}(t) for a.e. t ∈ J and all ∥z∥ ≤ r.

Consider the Hausdorf pseudo-metric H_{d}: P(E) × P(E) ⟶ ℝ^{+} ∪ {∞} defined by

H_{d}(A, B) = max {sup_{a∈A} d(a, B), sup_{b∈B} d(A, b)} ,

where d(A, b) = inf_{a∈A} d(a, b) and d(a, B) = inf_{b∈B} d(a, b). Then (P_{b,cl}(E), H_{d}) is a metric space and (P_{cl}(X), H_{d}) is a generalized metric space (see [28]). In particular, H_{d} satisfies the triangle inequality. Also, notice that if x_{0} ∈ E, then

d(x_{0}, A) = inf_{x∈A} d(x_{0}, x) whereas H_{d}({x_{0}}, A) = sup_{x∈A} d(x_{0}, x).

Definition 2.6 A multi-valued operator N: E → P_{cl}(E) is called:

(a) γ-Lipschitz if there exists γ > 0 such that H_{d}(N(x), N(y)) ≤ γd(x, y), for each x, y ∈ E;

(b) a contraction if it is γ-Lipschitz with γ < 1. EJQTDE, 2008 No. 31, p. 6 Notice that if N is γ−Lipschitz, then

H_{d}(F (x), F (y)) ≤ kd(x, y) for all x, y ∈ E.

For more details on multi-valued maps, we refer the reader to the works of Aubin and Cellina [5], Aubin and Frankowska [6], Deimling [15], Gorniewicz [17], Hu and Papageorgiou [25], Kamenskii [27], Kisielewicz [28], and Tolstonogov [38].

3 Filippov’s Theorem

Let J_{k} = (t_{k}, t_{k+1}], k = 0, …, m, and let y_{k} be the restriction of a function y to J_{k}. In order to define mild solutions for problem (1)–(3), consider the space

P C = {y: J → ℝ^{n} | y_{k} ∈ C(J_{k}, ℝ^{n}), k = 0, …, m, and

y(t_{k}^{−}) and y(t_{k}^{+}) exist and satisfy y(t_{k}^{−}) = y(t_{k}) for k = 1, …, m}.

Endowed with the norm

∥y∥_{P C} = max{∥y_{k}∥_{∞} : k = 0, …, m},

this is a Banach space.

Definition 3.1 A function y ∈ P C ∩ ∪_{k=0}^{m}AC(J_{k}, ℝ^{n}) is said to be a solution of problem (1)–(3) if there exists v ∈ L^{1}(J, ℝ^{n}) such that v(t) ∈ F (t, y(t)) a.e. t ∈ J, y′(t) − λy(t) = v(t) for t ∈ J\{t_{1}, …, t_{m}}, y(t_{k}^{+}) − y(t_{k}) = I_{k}(y(t_{k})), k = 1, …, m, and y(0) = y(b).

We will need the following auxiliary result in order to prove our main existence theorems.

Lemma 3.2 ([21]) Let f : ℝ^{n} → ℝ^{n} be a continuous function. Then y is the unique solution of the problem

y′(t) − λy(t) = f(y(t)), t ∈ J, t ≠ t_{k}, k = 1, …, m,  (4)
y(t_{k}^{+}) − y(t_{k}) = I_{k}(y(t_{k})), k = 1, …, m,  (5)
y(0) = y(b),  (6)

if and only if

y(t) = ∫_{0}^{b} H(t, s)f(y(s))ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})),  (7)

where

H(t, s) = (e^{−λb} − 1)^{−1} \begin{cases}e^{−λ(b+s−t)}, & if 0 ≤ s ≤ t ≤ b, \\ e^{−λ(s−t)}, & if 0 ≤ t < s ≤ b. \end{cases}

In the case of both differential equations and inclusions, existence results for problem (1)–(3) can be found in [20, 21, 35]. The main result of this section is a Filippov type result for problem (1)–(3).

Let g ∈ L^{1}(J, ℝ^{n}) and let x ∈ P C be a solution to the linear impulsive problem

\{ \begin{matrix}x′(t) − λx(t) & = & g(t), & a.e. t ∈ J\\{t_{1}, …, t_{m}\}, \\ x(t_{k}^{+}) − x(t_{k}) & = & I_{k}(x(t_{k}^{−})), & k = 1, …, m, \\ x(0) & = & x(b). \end{matrix}  (8)

Our main result in this section is contained in the following theorem.

Theorem 3.3 Assume the following assumptions hold.

(H_{1}) The function F: J × ℝ^{n} → P_{cl}(ℝ^{n}) satisfies (a) for all y ∈ ℝ^{n}, the map t ↦ F (t, y) is measurable, and

(b) the map t ↦ γ(t) = d(g(t), F (t, x(t)) is integrable.

(H_{2}) There exist constants c_{k} ≥ 0 such that

|I_{k}(u) − I_{k}(z)| ≤ c_{k}|u − z| for each u, z ∈ ℝ^{n}.

EJQTDE, 2008 No. 31, p. 8 (H_{3}) There exist a function p ∈ L^{1}(J, ℝ^{+}) such that

H_{d}(F (t, z_{1}), F (t, z_{2})) ≤ p(t)|z_{1} − z_{2}| for all z_{1}, z_{2} ∈ ℝ^{n}.

If

\frac{H_{∗}∥p∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k}} < 1,

then the problem (1)–(3) has at least one solution y satisfying the estimates

∥y − x∥_{P C} ≤ \frac{∥γ∥_{L1}}{(1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}})}.

and

|y′(t) − λy(t) − g(t)| ≤ H̃p(t) + |γ(t)|,

where

H̃ = \frac{H_{∗}∥γ∥_{L^{1}}}{(1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}})}

and

H_{∗} = sup{H(t, s) | (t, s) ∈ J × J}.

Proof Let f_{0} = g and

y_{0}(t) = ∫_{0}^{b} H(t, s)f_{0}(s)ds + ∑_{k=0}^{m} H(t, t_{k})I_{k}(x(t_{k})), y_{0}(t_{k}) = x(t_{k}).

Let U_{1} : J → P(ℝ^{n}) be given by U_{1}(t) = F (t, y_{0}(t)) ∩ B(g(t), γ(t)). Since g and γ are measurable, Theorem III.4.1 in [13] implies that the ball B(g(t), γ(t)) is measurable. Moreover, F (t, y_{0}) is measurable and U_{1} is nonempty. Indeed, since v = 0 is a measurable function, from Lemma 2.4, there exists a function u which is a measurable selection of F (t, y_{0}) and such that

|u(t) − g(t)| ≤ d(g(t), F (t, y_{0})) = γ(t).

EJQTDE, 2008 No. 31, p. 9 Then u ∈ U_{1}(t), proving our claim. We conclude that the intersection multi-valued operator U_{1}(t) is measurable (see [6, 13, 17]). By the Kuratowski-Ryll-Nardzewski selection theorem (Lemma 2.3), there exists a function t → f_{1}(t) which is a measurable selection for U_{1}. Hence, U_{1}(t) = F (t, y_{0}(t)) ∩ B(g(t), γ(t)) ≠ ∅. Consider

y_{1}(t) = ∫_{0}^{b} H(t, s)f_{1}(s)ds + ∑_{k=0}^{m} H(t, t_{k})I_{k}(y_{1}(t_{k})), t ∈ J,

where y_{1} is a solution of the problem

\{ \begin{matrix}y′(t) − λy(t) & = & f_{1}(t), & a.e. t ∈ J\\{t_{1}, …, t_{m}\}, \\ y(t_{k}^{+}) − y(t_{k}) & = & I_{k}(y(t_{k}^{−})), & k = 1, …, m, \\ y(0) & = & y(b). \end{matrix}  (9)

For every t ∈ J, we have

|y_{1}(t) − y_{0}(t)| ≤ ∫_{0}^{b} |H(t, s)||f_{1}(s) − f_{0}(s)| ds

+ ∑_{k=1}^{m} |I_{k}(y_{0}(t_{k})) − I_{k}(y_{1}(t_{k}))|

≤ H_{∗}∥γ∥_{L^{1}} + H_{∗} ∑_{k=1}^{m} c_{k}|y_{1}(t_{k}) − y_{0}(t_{k})|.

Then,

∥y_{1} − y_{0}∥_{P C} ≤ \frac{H_{∗}}{1 − H_{∗} ∑_{k=1}^{m} c_{k}} ∥γ∥_{L^{1}}.

Define the set-valued map U_{2}(t) = F (t, y_{1}(t))∩B(f_{1}(t), p(t)|y_{1}(t)−y_{0}(t)|). The multifunction t → F (t, y_{1}) is measurable and the ball B(f_{1}(t), p(t)∥y_{1}−y_{0}∥_{D}) is measurable by Theorem III.4.1 in [13]. To see that the set U_{2}(t) = F (t, y_{1})∩B(f_{1}(t), p(t)∥y_{1}−y_{0}∥_{D}) is nonempty, observe that since f_{1} is a measurable function, Lemma 2.4 yields a measurable selection u of F (t, y_{1}) such that

|u(t) − f_{1}(t)| ≤ d(f_{1}(t), F (t, y_{1}))

EJQTDE, 2008 No. 31, p. 10 Moreover, ∥y_{1} − y_{0}∥_{D} ≤ η_{0}(t_{1}) ≤ β. Then, using (H_{2}), we have

|u(t) − f_{1}(t)| ≤ d(f_{1}(t), F (t, y_{1}))
≤ H_{d}(F (t, y_{0}), F (t, y_{1}))
≤ p(t)∥y_{0} − y_{1}∥_{D},

i.e., u ∈ U_{2}(t) ≠ ∅. Since the multi-valued operator U_{2} is measurable (see [6, 13, 17]), there exists a measurable function f_{2}(t) ∈ U_{2}(t). Then define

y_{2}(t) = ∫_{0}^{b} H(t, s)f_{2}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y_{2}(t_{k})), t ∈ J,

where y_{2} is a solution of the problem

\{ \begin{matrix}y′(t) − λy(t) & = & f_{2}(t), & a.e. t ∈ J\\{t_{1}, …, t_{m}\}, \\ y(t_{k}^{+}) − y(t_{k}) & = & I_{k}(y(t_{k}^{−})), & k = 1, …, m, \\ y(0) & = & y(b). \end{matrix}  (10)

We then have

|y_{2}(t) − y_{1}(t)| ≤ H_{∗} ∫_{0}^{b} |f_{2}(s) − f_{1}(s)| ds + H_{∗} ∑_{k=1}^{m} c_{k}|y_{2}(t_{k}) − y_{1}(t_{k})|
≤ H_{∗} ∫_{0}^{b} p(s)|y_{1}(s) − y_{0}(s)| ds + H_{∗} ∑_{k=1}^{m} c_{k}|y_{2}(t_{k}) − y_{1}(t_{k})|
≤ \frac{H_{∗}^{2}}{1 − H_{∗} ∑_{k=1}^{m} c_{k}} ∥γ∥_{L^{1}}∥p∥_{L^{1}} + H_{∗} ∑_{k=1}^{m} c_{k}|y_{2}(t_{k}) − y_{1}(t_{k})|.

Thus,

∥y_{2} − y_{1}∥_{P C} ≤ \frac{H_{∗}^{2}}{(1 − H_{∗} ∑_{k=1}^{m} c_{k})^{2}} ∥p∥_{L^{1}}∥γ∥_{L^{1}}.

Let U_{3}(t) = F (t, y_{2}(t)) ∩ B(f_{2}(t), p(t)|y_{2}(t) − y_{1}(t)|). Arguing as we did for U_{2} shows that U_{3} is a measurable multi-valued map with nonempty values, so there exists a measurable selection f_{3}(t) ∈ U_{3}(t). Consider

y_{3}(t) = ∫_{0}^{b} H(t, s)f_{3}(s)ds + ∑_{k=1}^{m} I_{k}(y_{3}(t_{k})), t ∈ J,

where y_{3} is a solution of the problem

\{ \begin{matrix}y′(t) − λy(t) & = & f_{3}(t), & a.e. t ∈ J\\{t_{1}, …, t_{m}\}, \\ y(t_{k}^{+}) − y(t_{k}) & = & I_{k}(y(t_{k}^{−})), & k = 1, …, m, \\ y(0) & = & y(b). \end{matrix}  (11)

We have

|y_{3}(t) − y_{2}(t)| ≤ H_{∗} ∫_{0}^{t} |f_{3}(s) − f_{2}(s)| ds + H_{∗} ∑_{k=1}^{m} c_{k}|y_{3}(t_{k}) − y_{2}(t_{k})|.

Hence, from the estimates above, we have

∥y_{3} − y_{2}∥_{P C} ≤ \frac{H_{∗}^{3}}{(1 − H_{∗} ∑_{k=1}^{m} c_{k})^{3}} ∥p∥_{L^{1}}^{2}∥γ∥_{L^{1}}.

Repeating the process for n = 1, 2, …, we arrive at the bound

∥y_{n} − y_{n−1}∥_{P C} ≤ \frac{H_{∗}^{n}}{(1 − H_{∗} ∑_{k=1}^{m} c_{k})^{n}} ∥p∥_{L^{1}}^{n−1}∥γ∥_{L^{1}}.  (12)

By induction, suppose that (12) holds for some n. Let

U_{n+1}(t) = F (t, y_{n}(t)) ∩ B(f_{n}(t), p(t)|y_{n}(t) − y_{n−1}(t)|).

Since again U_{n+1} is measurable (see [6, 13, 17]), there exists a measurable function f_{n+1}(t) ∈ U_{n+1}(t) which allows us to define

y_{n+1}(t) = ∫_{0}^{b} H(t, s)f_{n+1}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y_{n+1}(t_{k})), t ∈ J.  (13)

Therefore,

|y_{n+1}(t) − y_{n}(t)| ≤ H_{∗} ∫_{0}^{b} |f_{n+1}(s) − f_{n}(s)| ds + H_{∗} ∑_{k=1}^{m} c_{k}|y_{n}(t_{k}) − y_{n−1}(t_{k})|.

Thus, we arrive at

∥y_{n+1} − y_{n}∥_{P C} ≤ \frac{H_{∗}^{n+1}}{(1 − H_{∗} ∑_{k=1}^{m} c_{k})^{n+1}} ∥p∥_{L^{1}}^{n}∥γ∥_{L^{1}}.  (14)

Hence, (12) holds for all n ∈ ℕ, and so {y_{n}} is a Cauchy sequence in P C, converging uniformly to a function y ∈ P C. Moreover, from the definition of U_{n}, n ∈ ℕ,

|f_{n+1}(t) − f_{n}(t)| ≤ p(t)|y_{n}(t) − y_{n−1}(t)| for a.e. t ∈ J.

Therefore, for almost every t ∈ J, {f_{n}(t) : n ∈ ℕ} is also a Cauchy sequence in ℝ^{n} and so converges almost everywhere to some measurable function f(·) in ℝ^{n}. Moreover, since f_{0} = g, we have

|f_{n}(t)| ≤ |f_{n}(t) − f_{n−1}(t)| + |f_{n−1}(t) − f_{n−2}(t)| + … + |f_{2}(t) − f_{1}(t)|

+ |f_{1}(t) − f_{0}(t)| + |f_{0}(t)|

≤ ∑_{k=2}^{n} p(t)|y_{k−1}(t) − y_{k−2}(t)| + γ(t) + |f_{0}(t)|

≤ p(t) ∑_{k=2}^{∞} |y_{k−1}(t) − y_{k−2}(t)| + γ(t) + |g(t)|

≤ H̃p(t) + γ(t) + |g(t)|.

Then, for all n ∈ ℕ,

|f_{n}(t)| ≤ H̃p(t) + γ(t) + g(t) a.e. t ∈ J.  (15)

From (15) and the Lebesgue Dominated Convergence Theorem, we conclude that f_{n} converges to f in L^{1}(J, ℝ^{n}). Passing to the limit in (13), the function

y(t) = ∫_{0}^{b} H(t, s)f(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})), t ∈ J,

is a solution to the problem (1)–(3).

Next, we give estimates for |y′(t) − λy(t) − g(t)| and |x(t) − y(t)|. We have

|y′(t) − λy(t) − g(t)| = |f(t) − f_{0}(t)|
≤ |f(t) − f_{n}(t)| + |f_{n}(t) − f_{0}(t)|

EJQTDE, 2008 No. 31, p. 13

≤ |f(t) − f_{n}(t)| + ∑_{k=2}^{n} |f_{k}(t) − f_{k−1}(t)| + γ(t)

≤ |f(t) − f_{n}(t)| + ∑_{k=2}^{n} p(t)|y_{k−1}(t) − y_{k−2}(t)| + γ(t).

Using (14) and passing to the limit as n → +∞, we obtain

|y′(t) − λy(t) − g(t)| ≤ p(t) ∑_{k=2}^{∞} |y_{k−1}(t) − y_{k−2}(t)| + γ(t) + |f(t) − f_{n}(t)|

≤ p(t) ∑_{k=2}^{∞} \frac{H_{∗}^{k−1}∥p∥_{L^{1}}^{k−2}∥γ∥_{L^{1}}}{(1 − H_{∗} ∑_{i=1}^{m} c_{i})^{k−1}} + |γ|,

so

|y′ − λy − g| ≤ H̃p(t) + γ(t), t ∈ J.

Similarly,

|x(t) − y(t)| = |∫_{0}^{b} H(t, s)g(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(x(t_{k}))

− ∫_{0}^{b} H(t, s)f(s)ds − ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k}))|

≤ H_{∗} ∫_{0}^{b} |f(s) − f_{0}(s)|ds + H_{∗} ∑_{k=1}^{m} c_{k}|x(t_{k}) − y(t_{k})|

≤ H_{∗} ∫_{0}^{b} |f(s) − f_{n}(s)|ds + H_{∗} ∫_{0}^{b} |f_{n}(s) − f_{0}(s)|ds

+ H_{∗} ∑_{k=1}^{m} c_{k}|x(t_{k}) − y(t_{k})|.

As n → ∞, we arrive at

∥x − y∥_{P C} ≤ \frac{H_{∗}∥p∥_{L^{1}}∥γ∥_{L^{1}}}{(1 − H_{∗} ∑_{k=1}^{m} c_{k})(1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}})} + \frac{∥γ∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k}}

= \frac{∥γ∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}}} ,

completing the proof of the theorem.

4 Relaxation Theorem

In this section, we examine to what extent the convexification of the right-hand side of the inclusion introduces new solutions. More precisely, we want to find out if the solutions of the nonconvex problem are dense in those of the convex one. Such a result is known in the literature as a Relaxation theorem and has important implications in optimal control theory. It is well-known that in order to have optimal state-control pairs, the system has to satisfy certain convexity requirements. If these conditions are not present, then in order to guarantee existence of optimal solutions we need to pass to an augmented system with convex structure by introducing the so-called relaxed (generalized, chattering) controls. The resulting relaxed problem has a solution. The Relaxation theorems tell us that the relaxed optimal state can be approximated by original states, which are generated by a more economical set of controls that are much simpler to build. In particular, “strong relaxation” theorems imply that this approximation can be achieved using states generated by bang-bang controls. More precisely, we compare trajectories of (1)–(3) to those of the relaxation impulsive differential inclusion

x′(t) − λx(t) ∈ \overline{co}F (t, x(t)), a.e. t ∈ J\\{t_{1}, …, t_{m}\},  (16)
x(t_{k}^{+}) − x(t_{k}) = I_{k}(x(t_{k}^{−})), k = 1, …, m,  (17)
x(0) = x(b).  (18)

Theorem 4.1 ([24]) Let U : J → P_{cl}(E) be a measurable, integrably bounded setvalued map and t → d(0, U(t)) be an integrable map. Then, the integral ∫_{0}^{b} U(t)dt is

EJQTDE, 2008 No. 31, p. 15 convex and t → \overline{co}U(t) is measurable. Moreover, for every ϵ > 0 and every measurable selection of u of \overline{co}U(t), there exists a measurable selection \overline{u} of U such that

sup_{t∈J}|∫_{0}^{t} u(s)ds − ∫_{0}^{t} \overline{u}(s)ds| ≤ ϵ

and

\overline{∫_{0}^{b} \overline{co}U(t)dt} = \overline{∫_{0}^{b} U(t)dt} = ∫_{0}^{b} \overline{co}U(t)dt.

We will need the following lemma to prove our main result in this section.

Lemma 4.2 (Covitz and Nadler [14]) Let (X, d) be a complete metric space. If G : X → P_{cl}(X) is a contraction, then F ixG ≠ ∅.

We now present a relaxation theorem for the problem (1)–(3).

Theorem 4.3 Assume that (H_{2}) and (H_{3}) hold and

(\overline{H}_{1}) The function F: J × ℝ^{n} → P_{cl}(ℝ^{n}) satisfies that for all y ∈ ℝ^{n} the map t ↦ F (t, y) is measurable, and the map t ↦ γ(t) = d(g(t), F (t, 0)) is integrable.

Assume that

\frac{H_{∗}∥p∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k}} < 1,

and let x be a solution of (16)–(18). Then, for every ϵ > 0 there exists a solution y to (1)–(3) on J satisfying

∥x − y∥_{∞} ≤ ϵ.

This implies that S^{co} = \overline{S^{F}} , where

S^{co} = {x | x is a solution to (16)–(18)}

and

S^{F} = {y | y is a solution to (1)–(3)}.

EJQTDE, 2008 No. 31, p. 16

Proof. First, we prove that S^{co} ≠ ∅. We transform the problem (16)–(18) into a fixed point problem. Consider the operator N : P C(J, ℝ^{n}) → P(P C(J, ℝ^{n})) defined by

N(y) = \{h ∈ P C : h(t) = ∫_{0}^{b} H(t, s)g(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})), g ∈ S_{\overline{co}F,y}\}.

We shall show that N satisfies the assumptions of Lemma 4.2. The proof will be given in two steps.

Step 1: N(y) ∈ P_{cl}(P C) for each y ∈ P C.

Let (y_{n})_{n≥0} ∈ N(y) be such that y_{n} ⟶ ỹ in P C. Then ỹ ∈ P C and there exists g_{n} ∈ S_{\overline{co}F,y} such that

y_{n}(t) = ∫_{0}^{b} H(t, s)g_{n}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})).

From \overline{H}_{1} and H_{3}, we have

|g_{n}(t)| ≤ p(t)∥y∥_{P C} + d(0, F (t, 0)) := M(t),

and observe that

n ∈ ℕ implies g_{n}(t) ∈ M(t)B(0, 1) for t ∈ J.

Now B(0, 1) is compact in ℝ^{n}, so passing to a subsequence if necessary, we have that {g_{n}} converges to some function g. An application of the Lebesgue Dominated Convergence Theorem shows that

∥g_{n} − g∥_{L^{1}} → 0 as n → ∞.

Using the continuity of I_{k}, we have

y_{n}(t) ⟶ ỹ(t) = ∫_{0}^{b} H(t, s)g(s)ds + ∑_{k=1}^{m} H_{k}(t, t_{k})I_{k}(y(t_{k})),

and so ỹ ∈ N(y).

Step 2: There exists γ < 1 such that H(N(y), N(\overline{y})) ≤ γ∥y − \overline{y}∥_{P C} for each y, \overline{y} ∈ P C.

Let y, \overline{y} ∈ P C and h_{1} ∈ N(y). Then there exists g_{1}(t) ∈ F (t, y(t)) such that for each t ∈ J,

h_{1}(t) = ∫_{0}^{b} H(t, s)g_{1}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})).

From (H_{2}) and (H_{3}), it follows that

H(F (t, y(t)), F (t, \overline{y}(t)) ≤ p(t)|y(t) − \overline{y}(t)|.

Hence, there is w ∈ F (t, \overline{y}(t)) such that

|g_{1}(t) − w| ≤ p(t)|y(t) − \overline{y}(t)|, t ∈ J.

Consider U : J → P(ℝ^{n}) given by

U(t) = \{w ∈ ℝ^{n} : |g_{1}(t) − w| ≤ p(t)|y(t) − \overline{y}(t)|\}.

Since the multi-valued operator V (t) = U(t) ∩ F (t, \overline{y}(t)) is measurable (see [6, 13]), there exists a function g_{2}(t), which is a measurable selection for V. So, g_{2}(t) ∈ F (t, \overline{y}(t)) and

|g_{1}(t) − g_{2}(t)| ≤ p(t)|y(t) − \overline{y}(t)|, for each t ∈ J.

Let us define for each t ∈ J,

h_{2}(t) = ∫_{0}^{b} H(t, s)g_{2}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(\overline{y}(t_{k})).

Then, we have

|h_{1}(t) − h_{2}(t)| ≤ ∫_{0}^{b} H(t, s)|g_{1}(s) − g_{2}(s)|ds + ∑_{k=1}^{m} |H(t, t_{k})|c_{k}|y(t_{k}) − \overline{y}(t_{k})|

≤ H_{∗} ∫_{0}^{b} p(s)|y(s) − \overline{y}(s)|ds + H_{∗} ∑_{k=1}^{m} c_{k}∥y − \overline{y}∥_{P C}
≤ H_{∗} ∫_{0}^{b} p(s)ds∥y − \overline{y}∥_{P C} + H_{∗} ∑_{k=1}^{m} c_{k}∥y − \overline{y}∥_{P C}.

Thus,

∥h_{1} − h_{2}∥_{P C} ≤ \frac{H_{∗}∥p∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k}} ∥y − \overline{y}∥_{P C}.

By an analogous relation obtained by interchanging the roles of y and \overline{y}, it follows that

H_{d}(N(y), N(\overline{y})) ≤ \frac{H_{∗}∥p∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k}} ∥y − \overline{y}∥_{P C}.

Therefore, N is a contraction, and so by Lemma 4.2, N has a fixed point y that is solution to (16)–(18).

We next prove that S^{co} = \overline{S^{F}}. Let x be a solution of Problem (16)–(18); then there exists g ∈ S_{\overline{co}F,x} such that

x(t) = ∫_{0}^{b} H(t, s)g(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(x(t_{k})), t ∈ J.

Hence, x is a solution of the problem

\{ \begin{matrix}x′(t) − λx(t) & = & g(t), & a.e. t ∈ J\\{t_{1}, …, t_{m}\}, \\ x(t_{k}^{+}) − x(t_{k}) & = & I_{k}(x(t_{k}^{−})), & k = 1, …, m, \\ x(0) & = & x(b). \end{matrix}  (19)

From Theorem 4.1, we have that for every ϵ > 0 and g ∈ \overline{co}F (t, x(t)) there exists a measurable selection f_{∗} of t → F (t, x(t)) such that

|∫_{0}^{b} H(t, s)f_{∗}(s)ds − ∫_{0}^{b} H(t, s)g(s)ds| ≤ H_{∗}|∫_{0}^{t} f_{∗}(s)ds − ∫_{0}^{t} g(s)ds|
≤ H_{∗} sup_{t∈J}|∫_{0}^{t} f_{∗}(s)ds − ∫_{0}^{t} g(s)ds|
≤ \frac{H_{∗}ϵ (1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}})}{2∥p∥_{L^{1}}} := δ.

Let

z(t) = ∫_{0}^{b} H(t, s)f_{∗}(s)ds + ∑_{0<t_{k}<t} H_{k}(t, t_{k})I_{k}(x(t_{k})), t ∈ J.

Observing that z(t_{k}) = x(t_{k}), we see that for t ∈ J,

|x(t) − z(t)| ≤ δ.

It follows that for all u ∈ B(x(t), δ),

γ(t) := d(g(t), F (t, x(t)) ≤ d(g(t), u) + H_{d}(F (t, z(t)), F (t, x(t))),
≤ H_{d}(\overline{co}F (t, x(t)), \overline{co}F (t, z(t))) + H_{d}(F (t, z(t)), F (t, x(t)))
≤ 2p(t)|x(t) − z(t)| ≤ 2p(t)δ.

Since γ is measurable (see [6]), the above inequality also shows that γ ∈ L^{1}(J, ℝ^{n}). From Theorem 3.3, problem (1)–(3) has a solution y such that

∥x − y∥_{P C} ≤ \frac{∥γ∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}}} .

Since γ(t) ≤ 2δp(t), this becomes

∥x − y∥_{P C} ≤ \frac{2δ∥p∥_{L^{1}}}{1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}}} ,

so

∥x − y∥_{P C} ≤ H_{∗}ϵ.

Since ϵ > 0 is arbitrary, we have S^{co} = \overline{S^{F}}, which completes the proof of the theorem.

5 Compactness of the Solution Set

Let us introduce the following hypotheses. Notice that the first part of condition (A_{2}) below is actually condition (H_{3}) above, and condition (A_{3}) is the same as (H_{2}) above. We list them here in this form for the convenience of the reader.

EJQTDE, 2008 No. 31, p. 20 (A_{1}) F : J × ℝ^{n} ⟶ P_{cl,cv}(ℝ^{n}); t ⟼ F (t, x) is measurable for each x ∈ ℝ^{n}. (A_{2}) There exists a function p ∈ L^{1}(J, ℝ^{+}) such that, for a.e. t ∈ J and all x, y ∈ ℝ^{n},

H_{d}(F (t, x), F (t, y)) ≤ p(t)|x − y|

and

H_{d}(0, F (t, 0)) ≤ p(t) for a.e. t ∈ J.

(A_{3}) There exist constants c_{k} ≥ 0 such that

|I_{k}(u) − I_{k}(z)| ≤ c_{k}|u − z|, for each u, z ∈ ℝ^{n}.

Our first compactness result is the following. Theorem 5.1 Suppose that hypotheses (A_{1}) − (A_{3}) are satisfied. If

H_{∗}∥p∥_{L^{1}} + H_{∗} ∑_{k=1}^{m} c_{k} < 1,

then the solution set of the problem (1)–(3) is nonempty and compact.

Proof. Let N : P C(J, ℝ^{n}) → P(P C(J, ℝ^{n})) be defined by

N(y) = {h ∈ P C : h(t)) = ∫_{0}^{b} H(t, s)v(s)ds + ∑_{k=0}^{m} H(t, t_{k})I_{k}(x(t_{k})), v ∈ S_{F,y}},

where

S_{F,y} = {v ∈ L^{1}(J, ℝ^{n}) : v(t) ∈ F (t, y(t)) a.e. t ∈ J}.

First we show that N(y) ∈ P_{cl}(P C) for each y ∈ P C. To do this, let (y_{n})_{n≥1} ∈ N(y) be such that y_{n} ⟶ ỹ in P C. Then, there exists v_{n} ∈ S_{F,y}, n = 0, 1, …, such that for each t ∈ J,

y_{n}(t) = ∫_{0}^{b} H(t, s)v_{n}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y_{n}(t_{k})).

EJQTDE, 2008 No. 31, p. 21 From (A_{2}), we have v_{n}(t) ∈ \overline{B}(0, p(t)|y(t)| + p(t)), where

\overline{B}(0, p(t)|y(t)| + p(t)) = \{w ∈ ℝ^{n} : |w| ≤ p(t)|y(t)| + p(t)\} := φ(t).

It is clear that φ : J → P_{cp,cv}(ℝ^{n}) is a multi-valued map that is integrably bounded. Since {v_{n}(·) : n ≥ 1} ∈ φ(·), we may pass to a subsequence if necessary to get that v_{n} converges weakly to v in L_{w}^{1}(J, ℝ^{n}). From Mazur’s lemma, there exists

v ∈ \overline{conv}\{v_{n}(t) : n ≥ 1\},

so there exists a subsequence \{v̄_{n}(t) : n ≥ 1\} in \overline{conv}\{v_{n}(t) : n ≥ 1\}, such that v̄_{n} converges strongly to v ∈ L^{1}(J, ℝ^{n}). From (A_{2}), we have for every ϵ > 0 there exists n_{0}(ϵ) such that for every n ≥ n_{0}(ϵ), we have

v_{n}(t) ∈ F (t, y_{n}(t)) ⊆ F (t, ỹ(t)) + ϵp(t)B(0, 1).

This implies that v(t) ∈ F (t, y(t)), a.e. t ∈ J. Thus, we have

ỹ(t) = ∫_{0}^{b} H(t, s)v(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(ỹ(t_{k})).

Hence, ỹ ∈ N(y). By the same method used in [8, 20, 35], we can prove that N has at least one fixed point.

Now we prove that S^{F} ∈ P_{cp}(P C), where

S^{F} = {y ∈ P C | y is a solution of the problem (1)–(3)}.

Let (y_{n})_{n∈ℕ} ∈ S^{F}; then there exist v_{n} ∈ S_{F,y_{n}}, n ∈ ℕ, such that

y_{n}(t) = ∫_{0}^{b} H(t, s)v_{n}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})), t ∈ J.

From (A_{2}) and (A_{3}), we have

|y_{n}(t)| ≤ H_{∗} ∫_{0}^{b} p(s)|y_{n}(s)|ds + H_{∗}∥p∥_{L^{1}}

+H_{∗} ∑_{k=1}^{m} c_{k}|y_{n}(t_{k})| + H_{∗} ∑_{k=1}^{m} c_{k}|I_{k}(0)|.

Hence,

∥y_{n}∥_{P C} ≤ \frac{1}{1 − H_{∗} ∑_{k=1}^{m} c_{k} − H_{∗}∥p∥_{L^{1}}} (H_{∗}∥p∥_{L^{1}} + H_{∗} ∑_{k=1}^{m} |I_{k}(0)|) := M, for all n ∈ ℕ.

Next, we prove that {y_{n} : n ∈ ℕ} is equicontinuous in P C. Let 0 < τ_{1} < τ_{2} ≤ b; then we have

|y_{n}(τ_{2}) − y_{n}(τ_{1})| ≤ ∫_{0}^{b} |H(τ_{2}, s) − H(τ_{1}, s)||v_{n}(s)|ds

+ ∑_{k=1}^{m} |H(τ_{2}, t_{k}) − H(τ_{1}, t_{k})|[Mc_{k} + |I_{k}(0)|]

≤ (M + 1) ∫_{0}^{b} |H(τ_{2}, s) − H(τ_{1}, s)|p(s)ds

+ ∑_{k=1}^{m} |H(τ_{2}, t_{k}) − H(τ_{1}, t_{k})|[Mc_{k} + |I_{k}(0)|].

The right-hand side tends to zero as τ_{2} − τ_{1} → 0. This proves the equicontinuity for the case where t ≠ t_{i} i = 1, …, m. It remains to examine the equicontinuity at t = t_{i}.

Set

h_{1}(t) = ∑_{k=1}^{m} H(t, t_{k})I_{k}(y_{n}(t_{k}))

and

h_{2}(t) = ∫_{0}^{b} H(t, s)y_{n}(s)ds.

First, we prove equicontinuity at t = t_{i}^{−}. Fix δ_{1} > 0 such that {t_{k} : k ≠ i}∩[t_{i} −δ_{1}, t_{i} + δ_{1}] = ∅ and

h_{1}(t_{i}) = ∑_{k=1}^{m} H(t_{i}, t_{k})I_{k}(y(t_{k}))

For 0 < h < δ_{1}, we have

|h_{1}(t_{i} − h) − h_{1}(t_{i})| ≤ ∑_{k=1,k≠i}^{m} |[H(t_{i} − h, t_{k}) − H(t_{i}, t_{k})]I(y_{n}(t_{k}^{−}))|

≤ ∑_{k=1,k≠i}^{m} |H(t_{i} − h, t_{k}) − H(t_{i}, t_{k})|[Mc_{k} + |I_{k}(0)|].

The right-hand side tends to zero as h → 0. Moreover,

|h_{2}(t_{i} − h) − h_{2}(t_{i})| ≤ (M + 1) ∫_{0}^{b} |H(t_{i} − h, s) − H(t_{i}, s)|p(s)ds

which tends to zero as h → 0.

Next, we prove equicontinuity at t = t_{i}^{+}. Fix δ_{2} > 0 such that {t_{k} : k ≠ i} ∩ [t_{i} − δ_{2}, t_{i} + δ_{2}] = ∅. Then, for 0 < h < δ_{2}, we have

|h_{1}(t_{i} + h) − h_{1}(t_{i})| ≤ ∑_{k=1,k≠i}^{m} |[H(t_{i} + h, t_{k}) − H(t_{i}, t_{k})]I(y_{n}(t_{k}^{−}))|
≤ ∑_{k=1,k≠i}^{m} |H(t_{i} + h, t_{k}) − H(t_{i}, t_{k})|[Mc_{k} + |I_{k}(0)|]

Again, the right-hand side tends to zero as h → 0. Similarly,

|h_{2}(t_{i} + h) − h_{2}(t_{i})| ≤ (M + 1) ∫_{0}^{b} |H(t_{i} + h, s) − H(t_{i}, s)|p(s)ds

tends to zero as h → 0.

Thus, the set {y_{n} : n ∈ ℕ} is equicontinuous in P C. As a consequence of the Arzelá-Ascoli Theorem, we conclude that there exists a subsequence of {y_{n}} converging to y in P C. As we did above, we can easily prove that there exists v(·) ∈ F (·, y_{.}) such that

y(t) = ∫_{0}^{b} H(t, s)v(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})), t ∈ J.

Hence, S^{F} ∈ P_{cp}(P C). This completes the proof of the theorem.

Our next theorem yields the same conclusion under the somewhat different hypotheses.

EJQTDE, 2008 No. 31, p. 24 Theorem 5.2 Assume that the following conditions hold. (H_{4}) The multifunction F : J × ℝ^{n} → P_{cp,cv}(ℝ^{n}) is L^{1}-Carathéodory.

(H_{5}) There exist functions p̄, q̄ ∈ L^{1}(J, ℝ_{+}) and α ∈ [0, 1) such that

∥F (t, y)∥_{P} ≤ p̄(t)|y|^{α} + q̄(t) for each (t, y) ∈ J × ℝ^{n}.

In addition, suppose that there exist constants c_{k}^{∗}, b_{k}^{∗} ∈ ℝ_{+} and α_{k} ∈ [0, 1) such that

|I_{k}(y)| ≤ c_{k}^{∗} + b_{k}^{∗}|y|^{α_{k}}, y ∈ ℝ^{n}.

Then the solution set of the problem (1)–(3) is nonempty and compact.

Proof. Let S^{F} = {y ∈ P C | y is a solution of the problem (1)–(3)}. From results in [9, 20, 35], it follows that S^{F} ≠ ∅. Now, we prove that S^{F} is compact. Let (y_{n})_{n∈ℕ} ∈ S^{F}; then there exist v_{n} ∈ S_{F,y_{n}}, n ∈ ℕ, such that

y_{n}(t) = ∫_{0}^{b} H(t, s)v_{n}(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})), t ∈ J.

From (H_{4}), we can prove that there exists an M_{1} > 0 such that

∥y_{n}∥_{P C} ≤ M_{1}, for every n ≥ 1.

Similar to what we did in the proof of Theorem 5.1, we can use (H_{5}) to show that the set {y_{n} : n ≥ 1} is equicontinuous in P C. Hence, by the Arzelá-Ascoli Theorem, we can conclude that there exists a subsequence of {y_{n}} converging to y in P C. We shall show that there exist v(·) ∈ F (·, y(·)) such that

y(t) = ∫_{0}^{b} H(t, s)v(s)ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})), t ∈ J.

Since F (t, ·) is upper semicontinuous, for every ε > 0, there exist n_{0}(ϵ) ≥ 0 such that for every n ≥ n_{0}, we have

v_{n}(t) ∈ F (t, y_{n}(t)) ⊂ F (t, y(t)) + εB(0, 1), a.e. t ∈ J.

EJQTDE, 2008 No. 31, p. 25 Since F (·, ·) has compact values, there exists a subsequence v_{n_{m}}(·) such that

v_{n_{m}}(·) → v(·) as m → ∞

and

v(t) ∈ F (t, y(t)), a.e. t ∈ J, and for all m ∈ ℕ.

It is clear that

|v_{n_{m}}(t)| ≤ p̄(t), a.e. t ∈ J.

By the Lebesgue Dominated Convergence Theorem and the continuity of I_{k}, we conclude that v ∈ L^{1}(J, ℝ^{n}) so v ∈ S_{F,y}. Thus,

y(t) = ∫_{0}^{b} H(t, s)v(s) ds + ∑_{k=1}^{m} H(t, t_{k})I_{k}(y(t_{k})), t ∈ J.

Therefore, S^{F} ∈ P_{cp}(P C), and this completes the proof of the theorem.

6 Periodic Solutions

In this section, we consider the impulsive periodic problem

y′(t) ∈ φ(t, y(t)), a.e. t ∈ J\{t_{1}, …, t_{m}},  (20)
y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,  (21)
y(0) = y(b),  (22)

where φ : J × ℝ^{n} → P(ℝ^{n}) is a multifunction.

A number of papers have been devoted to the study of initial and boundary value problems for impulsive differential inclusions. Some basic results in the theory of periodic boundary value problems for first order impulsive differential equations and inclusions may be found in [21, 32, 33, 34, 35] and the references therein. Our goal in this section is to give an existence result for the above problem by using topological degree combined with a Pointcaré operator.

EJQTDE, 2008 No. 31, p. 26

6.1 Background in Geometric Topology

First, we begin with some elementary concepts from geometric topology. For additional details, we recommend [11, 19, 22, 26]. In what follows, (X, d) denotes a metric space. A set A ∈ P(X) is called a contractible set provided there exists a continuous homotopy h : A × [0, 1] → A such that

(i) h(x, 0) = x, for every x ∈ A, and

(ii) h(x, 1) = x_{0}, for every x ∈ A.

Note that if A ∈ P_{cv,cp}(X), then A is contractible. Clearly, the class of contractible sets is much larger than the class of all compact convex sets.

Definition 6.1 A space X is called an absolute retract (written as X ∈ AR) provided that for every space Y, a closed subset B ⊆ Y, and a continuous map f : B → X, there exists a continuous extension f̃: Y → X of f over Y, i.e., f̃(x) = f(x) for every x ∈ B.

Definition 6.2 A space X is called an absolute neighborhood retract (written as X ∈ ANR) if for every space Y, any closed subset B ⊆ Y, and any continuous map f : B → X, there exists a open neighborhood U of B and a continuous map f̃: U → X such that f̃(x) = f(x) for every x ∈ B.

Definition 6.3 A space X is called an R_{δ}−set provided there exists a sequence of nonempty compact contractible spaces {X_{n}} such that:

X_{n+1} ⊂ X_{n} for every n;

X = ⋂_{n=1}^{∞} X_{n}.

It is well known that any contractible set is acyclic and that the class of acyclic sets is larger then that of contractible sets. From the continuity of the Čech cohomology functor, we have the following lemma.

Lemma 6.4 ([19]) Let X be a compact metric space. If X is an R_{δ}–set, then it is an acyclic space.

Set

K^{n}(r) = K^{n}(x, r), S^{n−1}(r) = ∂K^{n}(r), and P^{n} = ℝ^{n}\{0},

where K^{n}(r) is a closed ball in ℝ^{n} with center x and radius r, and ∂K^{n}(r) stands for the boundary of K^{n}(r) in ℝ^{n}. For any X ∈ ANR−space X, we set

J(K^{n}(r), X) = {F : X → P(X) | F u.s.c with R_{δ}–values}.

Moreover, for any continuous f : X → ℝ^{n}, where X ∈ ANR, we set

J_{f}(K^{n}(r), X) = {φ :K^{n}(r) → P(X) | φ = f ∘ F for some
F ∈ J(K^{n}(r), X) and φ(S^{n−1}(r)) ⊂ P^{n}}.

Finally, we define

CJ(K^{n}(r), ℝ^{n}) = ∪{J_{f}(K^{n}(r), ℝ^{n}) | f : X → ℝ^{n} is continuous and X ∈ ANR}.

It is well known that (see [17]) that for the multi-valued maps in this class, the notion of topological degree is available. To define it, we need an appropriate concept of homotopy in CJ(K^{n}(r), ℝ^{n})).

Definition 6.5 Let ϕ_{1}, ϕ_{2} ∈ CJ(K^{n}(r), ℝ^{n}) be two maps of the form

ϕ_{1} = f_{1} ∘ F_{1} : K^{n}(r)⟶^{F_{1}}P(X)⟶^{f_{1}}ℝ^{n}

ϕ_{2} = f_{2} ∘ F_{2} : K^{n}(r)⟶^{F_{2}}P(X)⟶^{f_{2}}ℝ^{n}.

EJQTDE, 2008 No. 31, p. 28 We say that ϕ_{1} and ϕ_{2} are homotopic in CJ(K^{n}(r), ℝ^{n}) if there exist an u.s.c. R_{δ}– valued homotopy χ : [0, 1]×K^{n}(r) → P(X) and a continuous homotopy h : [0, 1]×X → ℝ^{n} satisfying

(i) χ(0, u) = F_{1}(u), χ(1, u) = F_{2}(u) for every u ∈ K^{n}(r),

(ii) h(0, x) = f_{1}(x), h(1, x) = f_{2}(x) for every x ∈ X,

(iii) for every (u, λ) ∈ [0, 1] × S^{n−1}(r) and x ∈ χ(λ, u), we have h(x, λ) ≠ 0.

The map H : [0, 1] × K^{n}(r) → P(ℝ^{n}) given by

H(λ, u) = h(λ, χ(λ, u))

is called a homotopy in CJ(K^{n}(r), ℝ^{n}) between ϕ_{1} and ϕ_{2}.

Theorem 6.6 ([17]) There exist a map Deg : CJ(K^{n}(r), ℝ^{n}) → ℤ, called the topological degree function, satisfying the following properties:

(C_{1}) If φ ∈ CJ(K^{n}(r), ℝ^{n}) is of the form φ = f ∘ F with F single valued and continuous, then Deg(φ) = deg(φ), where deg(φ) stands for the ordinary Brouwer degree of the single valued continuous map φ : K^{n}(r) → ℝ^{n}.

(C_{2}) If Deg(φ) = 0, where φ ∈ CJ(K^{n}(r), ℝ^{n}), then there exists u ∈ K^{n}(r) such that 0 ∈ φ(u).

(C_{3}) If φ ∈ CJ(K^{n}(r), ℝ^{n}) and {u ∈ K^{n}(r)|0 ∈ φ(u)} ⊂ IntK^{n}(r_{0}) for some 0 < r_{0} < r, then the restriction φ_{0} of φ to K^{n}(r_{0}) is in CJ(K^{n}(r), ℝ^{n}) and Deg(φ_{0}) = Deg(φ).

Let A ⊂ ℝ^{n} and B ⊂ ℝ^{m}; CJ_{0}(A, B) will denote the class of mappings

CJ_{0}(K^{n}(r), ℝ^{n}) = {φ : A →P(B) | φ = f ∘ F, F : A → P(X), F is u.s.c.

EJQTDE, 2008 No. 31, p. 29

with R_{δ}–values and f : X → B is continuous},

where X ∈ ANR. The next two definitions were introduced in [18]

Definition 6.7 A metric space X is called acyclically contractible if there exists an acyclic homotopy Π : X × [0, 1] → P(X) such that

(a) x_{0} ∈ Π(x, 1) for every x ∈ X and for some x_{0} ∈ X;

(b) x ∈ Π(x, 0) for every x ∈ X.

Notice that any contractible space and any acyclic, compact metric space are acyclically contractible (see [3], Theorem 19). Also, from [17], any acyclically contractible space is acyclic.

Definition 6.8 A metric space X is called R_{δ}−contractible if there exists a multivalued homotopy Π : X × [0, 1] → P(X) which is u.s.c. and satisfies:

(a) x ∈ Π(x, 1) for every x ∈ X;

(b) Π(x, 0) = B for every x ∈ X and for some B ⊂ X;

(c) Π(x, α) is an R_{δ}−set for every α ∈ [0, 1] and x ∈ X.

6.2 Poincaré translation operator

By Poincaré operators we mean the translation operator 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. The translation operator is sometimes also called the Poincaré-Andronov, or Levinson, or simply the T -operator. In the classical theory (see [29, 39] and the references therein), both these operators are defined to be single-valued, when assuming, among other things, the

EJQTDE, 2008 No. 31, p. 30 uniqueness of solutions of initial value problems. In the absence of uniqueness, it is often possible to approximate the right-hand sides of the given systems by locally lipschitzian ones (implying uniqueness already), and then apply a standard limiting argument. This might be, however, rather complicated and is impossible for discontinuous right-hand sides. On the other hand, set-valued analysis allows us to handle effectively such classically troublesome situations. For additional background details, see [2, 17].

Let φ : J × ℝ^{n} → P(ℝ^{n}) be a Carathédory map. We define a multi-valued map

S_{φ} : ℝ^{n} → P(P C)

by

S_{φ}(x) = {y | y(·, x) is a solution of the problem satisfying y(0, x) = x}.

Consider the operator P_{t} defined by P_{t} = Ψ ∘ S_{φ} where

P_{t} : ℝ^{n}⟶^{S_{φ}}P(P C)⟶^{Ψ_{t}}P(ℝ^{n})

and

Ψ_{t}(y) = y(0) − y(t).

Here, P_{t} is called the Poincaré translation map associated with the Cauchy problem

y′(t) ∈ φ(t, y(t)), a.e. t ∈ J\{t_{1}, …, t_{m}},  (23)
y(t_{k}^{+}) − y(t_{k}^{−}) = I_{k}(y(t_{k}^{−})), k = 1, …, m,  (24)
y(0) = y_{0} ∈ ℝ^{n}.  (25)

The following lemma is easily proved.

Lemma 6.9 Let φ : J × ℝ^{n} → P_{cv,cp}(ℝ^{n}) be a Carathédory multfunction. Then the periodic problem (20)–(22) has a solution if and only if for some y_{0} ∈ ℝ^{n} we have 0 ∈ P_{b}(y_{0}), where P_{b} is the Poincaré map associated with (23)–(25).

EJQTDE, 2008 No. 31, p. 31

Next, we define what is meant by an upper-Scorza-Dragoni map.

Definition 6.10 We say that a multi-valued map F : J ×ℝ^{n} → P_{cl}(ℝ^{n}) has the upper-Scorza-Dragoni property if, given δ > 0, there is a closed subset A_{δ} ⊂ J such that the measure μ(A_{δ}) ≤ δ and the restriction F̃ of F to A_{δ} × ℝ^{n} is u.s.c.

We also need the following two lemmas.

Lemma 6.11 ([16]) Let φ: J × ℝ^{n} → P_{cp,cv}(ℝ^{n}) be upper-Scorza-Dragoni. Assume that:

(R_{1}) There exist functions p ∈ L^{1}(J, ℝ_{+}) and ψ : ℝ^{n} → ℝ such that

∥φ(t, y)∥_{P} ≤ p(t)ψ(|y|) for each (t, y) ∈ J × ℝ^{n}.

(R_{2}) There exist constants c_{k}^{∗}, b_{k}^{∗} ∈ ℝ_{+} and α_{k} ∈ [0, 1) such that

|I_{k}(y)| ≤ c_{k}^{∗} + b_{k}^{∗}|y|^{α_{k}} y ∈ ℝ^{n}.

Then the set S_{φ} is R_{δ}−contractible.

Lemma 6.12 ([16]) Let φ : J × ℝ^{n} → P_{cv,cp}(ℝ^{n}) be upper-Scagoni-Dragoni. Let P_{b} : ℝ^{n} → P(ℝ^{n}) be the Poincaré map associated with the problem (23)–(25). Assume that there exists r > 0 such that

0 ∉ P_{b}(y_{0}) for every y_{0} ∈ S^{n−1}(r).

Then,

P_{b} ∈ CJ(K^{n}(r), ℝ^{n}).

Furthermore, if Deg(P_{b}) ≠ 0, then the impulsive periodic problem (20)–(22) has a solution.

EJQTDE, 2008 No. 31, p. 32

The following Theorem due to Gorniewicz [17] is critical in the proof of the main result in this section.

Theorem 6.13 (Nonlinear Alterntive). Assume that φ ∈ CJ_{0}(K^{n}(r), ℝ^{n}). Then φ has at least one of the following properties:

(i) F ix(φ) ≠ ∅,

(ii) there is an x ∈ S^{n−1}(r) with x ∈ λφ(x) for some 0 < λ < 1.

The following definition and lemma can be found in [17, 23].

Definition 6.14 A mapping F : X → P(Y) is LL-selectionable provided there exists a measurable, locally-Lipchitzian map f : X → Y such that f ⊂ F.

Lemma 6.15 If φ : X → P_{cp,cv}(ℝ^{n}) is an u.s.c. multi-valued map, then φ is σ−LLselectionable.

We are now ready to give our main result in this section.

Theorem 6.16 Let φ: ℝ^{n} → P_{cp,cv}(ℝ^{n}) be an u.s.c. multifunction. In addition to conditions (R_{1}) − (R_{2}), assume that (R_{3}) There exists r > 0 such that

\frac{r}{ψ(r)∥p∥_{L^{1}} + ∑_{k=1}^{m}[c_{k}^{∗} + b_{k}^{∗}r^{α_{k}}]} > 1.

Then the problem (20)–(22) has at least one solution.

Proof. From Lemma 6.15, φ is σ−LL-selectionable, so by a result of Djebali et al. [16], S_{φ} is R_{δ}−contractible. Set A = B = ℝ^{n} and X = P C ∈ ANR. We will prove that

Ψ : P C → ℝ^{n} defined by y → Ψ(y) = y(0) − y(·)

EJQTDE, 2008 No. 31, p. 33 is a continuous map. Let {y_{n}} be a sequence such that y_{n} → y in P C. Then,

|Ψ(y_{n})(t) − Ψ(y)(t)| ≤ 2∥y_{n} − y∥_{P C} → 0 as n → ∞.

Hence,

P_{b} ∈ CJ_{0}(K^{n}(r), ℝ^{n}).

Let a ∈ P_{t}(a) = λ(Ψ_{t} ∘ S_{φ})(a) for some λ ∈ (0, 1). Then, there exist y ∈ P C such that y ∈ S_{φ}(a). This implies y(0) = a and a = λ(a − y(t)), a ∈ S^{n−1}(r). For t ∈ J, we have

|a| ≤ ∥y(t)∥
≤ ∫_{0}^{t} p(s)ψ(|y(s)|)ds + ∑_{k=1}^{m}[c_{k}^{∗} + b_{k}^{∗}|y(t_{k})|^{α_{k}}]
≤ ψ(r) ∫_{0}^{b} p(s)ds + ∑_{k=1}^{m}[c_{k}^{∗} + b_{k}^{∗}r^{α_{k}}].

Hence,

\frac{|a|}{ψ(r)∥p∥_{L^{1}} + ∑_{k=1}^{m}[c_{k}^{∗} + b_{k}^{∗}r^{α_{k}}]} ≤ 1.

Next, we will show that S_{φ} is u.s.c. by proving that the graph

Γ_{φ} := {(x, y) | y ∈ S_{φ}(x)}

of S_{φ} is closed. Let (x_{n}, y_{n}) ∈ Γ_{φ}, i.e., y_{n} ∈ S_{φ}(x_{n}), and let (x_{n}, y_{n}) → (x, y) as n → ∞. Since y_{n} ∈ S_{φ}(x_{n}), there exists v_{n} ∈ L^{1}(J, ℝ^{n}) such that

y_{n}(t) = x_{n} + ∫_{0}^{t} v_{n}(s)ds + ∑_{0<t_{k}<t} I_{k}(y_{n}(t_{k})), t ∈ J.

Since (x_{n}, y_{n}) converge to (x, y), there exists M > 0 such that

|x_{n}| ≤ M for all n ∈ ℕ.

By using (R_{1}) − (R_{2}), we can easily prove that there exist \overline{M} > 0 such that

∥y_{n}∥_{P C} ≤ \overline{M} for all n ∈ ℕ.

EJQTDE, 2008 No. 31, p. 34 From the definition of y_{n}, we have y′_{n}(t) = v_{n}(t) a.e. t ∈ J, so

|v_{n}(t)| ≤ p(t)ψ(M), t ∈ J.

Thus, v_{n}(t) ∈ p(t)ψ(M)\overline{B}(0, 1) := χ(t) a.e. t ∈ J. It is clear that χ : J → P_{cp,cv}(ℝ^{n}) is a multivalued map that is integrably bounded. Since {v_{n}(·) : n ≥ 1} ∈ χ(·), we may pass to a subsequence if necessary to obtain that v_{n} converges weakly to v in L_{w}^{1}(J, ℝ^{n}). From Mazur’s lemma, there exists

v ∈ \overline{conv}\{v_{n}(t) : n ≥ 1\},

so there exists a subsequence \{v̄_{n}(t) : n ≥ 1\} in \overline{conv}\{v_{n}(t) : n ≥ 1\}, such that v̄_{n} converges strongly to v ∈ L^{1}(J, ℝ^{n}). Since F (t, .) is u.s.c., for every ϵ > 0 there exists n_{0}(ϵ) such that for every n ≥ n_{0}(ϵ), we have

v̄_{n}(t) ∈ F (t, y_{n}(t)) ⊆ F (t, ỹ(t)) + ϵB(0, 1).

This implies that v(t) ∈ F (t, y(t)), a.e. t ∈ J. Let

z(t) = x + ∫_{0}^{t} v(s)ds + ∑_{0<t_{k}<t} I_{k}(y(t_{k})), t ∈ J.

Since the functions I_{k}, k = 1, …, m are continuous, we obtain the estimates

||y_{n} − z∥_{P C} ≤ |x_{n} − x| + ∫_{0}^{b} |v̄_{n}(s) − v(s)|ds + ∑_{k=1}^{m} |I_{k}(y_{n}(t_{k})) − I_{k}(y(t_{k}))|.

The right-hand side of the above expression tends to 0 as n → +∞. Hence,

y(t) = x + ∫_{0}^{t} v(s)ds + ∑_{0<t_{k}<t} I_{k}(y(t_{k})), t ∈ J.

Thus, y ∈ S_{φ}(x). Now, we show that S_{φ} maps bounded sets into relatively compact sets of P C. Let B be a bounded set in ℝ^{n} and let {y_{n}} ⊂ S_{φ}(B). Then there exist {x_{n}} ⊂ B such that

y_{n}(t) = x_{n} + ∫_{0}^{t} v_{n}(s)ds + ∑_{0<t_{k}<t} I_{k}(y_{n}(t_{k})), t ∈ J,

where v_{n} ∈ S_{φ,y_{n}}, n ∈ ℕ. Since {x_{n}} is a bounded sequence, there exists a subsequence of {x_{n}} converging to x, so from (R_{1}) − (R_{2}), there exist M_{∗} > 0 such that

∥y_{n}∥_{P C} ≤ M_{∗}, n ∈ ℕ.

As in the proof of Theorem 5.1, we can show that {y_{n} : n ∈ ℕ} is equicontinuous in P C. As a consequence of the Arzelá-Ascoli Theorem, we conclude that there exists a subsequence of {y_{n}} converging to y in P C. By a similar argument to the one above, we can prove that

y(t) = x + ∫_{0}^{t} v(s)ds + ∑_{0<t_{k}<t} I_{k}(y(t_{k})), t ∈ J,

where v ∈ S_{F,y}. Thus, y ∈ S_{φ}(x). This implies that S_{φ} is u.s.c.

As a consequence of the nonlinear alternative of Leray Schauder type [17], we conclude that F ix P_{b} ≠ ∅. This completes the proof of the theorem.

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(Received July 11, 2008)