First Order Impulsive Differential Inclusions with Periodic Conditions

In this paper, the authors consider the first-order nonresonance impulsive differential inclusion with periodic conditions y 0 (t) �y(t) 2 F(t,y(t)), a.e. t 2 J\{t1,...,tm}, y(t + ) y(t k ) = Ik(y(t k )), k = 1,2,... ,m, y(0) = y(b), where J = [0,b] and F : J × R n ! P(R n ) is a set-valued map. The functions Ik characterize the jump of the solutions at impulse points tk (k = 1,2,... ,m). The topological structure of solution sets as well as some of their geometric properties (contractibility and R�-sets) are studied. A continuous version of Filippov’s theorem is also proved.


Introduction
Many processes in engineering, physics, biology, population dynamics, medicine, and other areas are subject to abrupt changes such as shocks or perturbations (see for instance [1,34] and the references therein).These changes may be viewed as impulses.For example, in the treatment of some diseases, periodic impulses correspond to the administration of a drug.In environmental sciences such impulses correspond to seasonal changes of the water level of artificial reservoirs.Such models can be described by impulsive differential equations.Contributions to the study of the mathematical aspects of such equations can be found, for EJQTDE, 2009 No. 24, p. 1 example, in the works of Bainov and Simeonov [9], Lakshmikantham, Bainov, and Simeonov [35], Pandit and Deo [38], and Samoilenko and Perestyuk [41].
Impulsive ordinary differential inclusions and functional differential inclusions with different conditions have been intensely studied in the last several years, and we refer the reader to the monographs by Aubin [6] and Benchohra et al. [11], as well as the thesis of Ouahab [37], and the references therein.
We will consider the problem y (t) − λy(t) ∈ F (t, y(t)), a.e.t ∈ J\{t 1 , . . ., t m }, where λ = 0 is a parameter, J = [0, b], F : J × R n → P(R n ) is a multi-valued map, . ., m, t 0 = 0 < t 1 < . . .< t m < t m+1 = b, ∆y| 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).In 1923, Kneser proved that the Peano existence theorem can be formulated in such a way that the set of all solutions is not only nonempty but is also compact and connected (see [39,40]).Later, in 1942, Aronszajn [5] improved Kneser's theorem by showing that the set of all solutions is even an R δ -set.It should also be clear that the characterization of the set of fixed points for some operators implies the corresponding result for the solution sets.
Lasry and Robert [36] studied the topological properties of the sets of solutions for a large class of differential inclusions including differential difference inclusions.The present paper is a continuation of their work but for a general class of impulsive differential inclusions with periodic conditions.Aronszajn's results for differential inclusions with difference conditions was improved by several authors, for example, see [2,3,4,21,23,24].Very recently, properties of the solutions of impulsive differential inclusions with initial conditions were study by Djebali et al. [18].
The main goal of this paper is to examine some properties of solutions sets for impulsive differential inclusions with periodic conditions and to present a continuous version of Filippov's theorem.

Preliminaries
Here, we introduce notations, definitions, and facts from multi-valued analysis that will be needed throughout this paper.We let C(J, R) denote the Banach space of all continuous functions from J into R with the Tchebyshev norm and we let L 1 (J, R) be the Banach space of measurable functions x : J −→ R that are Lebesgue integrable with the norm By AC i (J, R n ), we mean the space of functions y : J → R n that are i-times differentiable and whose i th derivative, y (i) , is absolutely continuous.
For a metric space (X, d), the following notations will be used throughout this paper: • P(X) = {Y ⊂ X : Y = ∅}.
Let (X, .) be a separable 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.Lemma 2.1 ( [15,20]) The mapping F is measurable if and only if for each x ∈ X, the function ζ : J → [0, +∞) defined by 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 say that F has convex (closed) values if F (x) is convex (closed) for all x ∈ X, and that F is totally bounded if Let (X, d) and (Y, ρ) be two metric spaces and let F : X → P cl (Y ) be a multi-valued mapping.We say that F is upper semi-continuous (u.s.c. for short) 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  (P cl (X), H d ) is a generalized metric space (see [33]).Moreover, H d satisfies the triangle inequality.Note that if x 0 ∈ R n , then Additional details on multi-valued maps can be found in the works of Aubin and Cellina [7], Aubin and Frankowska [8], Deimling [17], Gorniewicz [20], Hu and Papageorgiou [30], Kamenskii [32], Kisielewicz [33], and Tolstonogov [42].EJQTDE, 2009 No. 24, p. 4

Background in Geometric Topology
We begin with some elementary notions from geometric topology.For additional details, we recommend [12,22,28,31].In what follows, we let (X, d) denote a metric space.A set A ∈ P(X) is contractible 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.The following concepts are needed in the sequel.Definition 2.4 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 f : Y → X of f over Y , i.e., f(x) = f (x) for every x ∈ B. Definition 2.5 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 It is well known that any contractible set is acyclic and that the class of acyclic sets is larger than that of contractible sets.The continuity of the Čech cohomology functor yields the following lemma.
Lemma 2.7 ( [22]) Let X be a compact metric space.If X is an R δ -set, then it is an acyclic space.

Space of Solutions
Let J k = (t k , t k+1 ], k = 0, 1, . . ., m, and let y k be the restriction of a function y to J k .In order to define mild solutions to the problem (1)-( 3), consider the space Endowed with the norm this is a Banach space.

Solutions Sets
In this section, we present results about the topological structure of the set of solutions of some nonlinear functional equations due to Aronszajn [5] and further developed by Browder and Gupta in [14].
Theorem 4.1 Let X be a space, let (E, • ) be a Banach space, and let f : X → E be a proper map, i.e., f is continuous and for every compact Assume further that for each > 0 a proper map f : X → E is given, and the following two conditions are satisfied: (ii) for every > 0 and u ∈ E such that u ≤ , the equation f (x) = u has exactly one solution.
Then the set ) is measurable for every x ∈ X, and for each x ∈ X there exists a neighborhood V x of x and an integrable function The following result is know as the Lasota-Yorke Approximation Theorem.20]) Let E be a normed space, X be a metric space, and let f : X → E be a continuous map.Then, for each > 0 there is a locally Lipschitz map f : We consider the impulsive problem where The following result of Graef and Ouahab will be used to prove our main existence theorems.

Lemma 4.4 ([26])
The function y is the unique solution of the problem ( 4)-( 6) if and only if where We denote by S(f, 0, b) the set of all solutions of the impulsive problem ( 4)- (6).Now, we are in a position to state and prove our first Aronsajn type result.For the study of this problem, we first list the following hypotheses.
(R 1 ) There exist functions p, q ∈ L 1 (J, R + ) and α ∈ [0, 1) such that Proof.Let G : P C → P C defined by Thus, F ixG = S(f, 0, b).From (R 1 ) and (R 2 ), we have S(f, 0, b) = ∅ (see [26]) and there exists M > 0 such that y P C ≤ M for every y ∈ S(f, 0, b). Define Since f is an L 1 −Carathédory function, f is Carathédory and integrably bounded.We consider the following modified problem We can easily prove that S(f, 0, b) = S( f , 0, b).Since f integrably bounded, there exists By inequality (11) and the continuity of the I k , we have where EJQTDE, 2009 No. 24, p. 8 By the same method used in [25,26,37], we can prove that G : P C → P C is a compact operator, and we define the vector filed associated with G by g = y − G(y).From the compactness of G and the Lasota-Yorke Approximation Theorem (Theorem 4.3 above), we can easily prove that all the conditions of Theorem 4.1 are satisfied, and so S( f , 0, b) is R δ .That it is acyclic follows from Lemma 2.7.
The following definition and lemma can be found in [20,29].
Let S(F, 0, b) denote the set of all solutions of (1)-( 3).We are now going to characterize the topological structure of S(F, 0, b).First, we prove the following result.
Theorem 4.8 Let F : J × R n → P cp,cv (R n ) be a Carathéodory map that is mLL-selectionable.In addition to conditions (R 1 )-(R 2 ), assume that: EJQTDE, 2009 No. 24, p. 9 By the Banach fixed point theorem, we can prove that the problem ( 12)-( 14 12)-( 14).Note that We shall show that h(y n , α n ) → h(y, α).We have Hence, h(y n , α n ) − h(y, α) P C → 0 as n → ∞.If α n = 0 and 0 < lim x(t), for t ∈ (αb, b]. Since y n ∈ S(F, 0, b), there exist v n ∈ S F,yn such that Since y n converges to y, there exists R > 0 such that Hence, from (R 1 ), we have From the Lebesgue Dominated Convergence Theorem, we have that v ∈ L 1 (J, R n ), so v ∈ S F,y .Using the continuity of I k , we have Thus, h(y n , α n ) − h(y, α) P C → 0 as n → ∞.

σ-selectionable multivalued maps
The next two definitions and the theorem that follows can be found in [20,29] (see also [7], p. 86).
Definition 4.9 We say that a map F : Moreover, if F is integrably bounded, then F is σ−mLL-selectionable.
We are now in a position to state and prove another characterization of the geometric structure of the set S(F, 0, b) of all solutions of the problem (1)-(3).Theorem 4.12 Let F : J × R n → P cp,cv (R n ) be a Carathéodory and mLL-selectionable multi-valued map and assume that conditions (R 1 )-(R 2 ) and (H 1 )-(H 2 ) hold with Proof.Since F is σ−Ca-selectionable, there exists a decreasing sequence of multivalued maps F k : J × R n → P(R n ) (k ∈ N) that have Carathéodory selections and satisfy F k+1 (t, u) ⊂ F k (t, x) for almost all t ∈ J and all x ∈ R n and From Theorems 5.1 and 5.2 in [27], the sets S(F k , 0, b) are compact for all k ∈ N. Using Theorem 4.8, the sets Alternately, we have the following result.

Filippov's Theorem
Let A be a subset of J × R n .We say that A is L ⊗ B measurable if A belongs to the σ-algebra generated by all sets of the form J ×R n where J is Lebesgue measurable in J and D is Borel measurable in R n .A subset A of L 1 (J, R) is decomposable if for all u, v ∈ A and measurable J ⊂ J, uχ J + vχ J−J ∈ A, where χ stands for the characteristic function.The family of all nonempty closed and decomposable subsets of L 1 (J, R n ) is denoted by D. Definition 5.1 Let Y be a separable metric space and let N : Y → P(L 1 (J, R n )) be a multivalued operator.We say N has property (BC) if 1) N is lower semi-continuous (l.s.c.), and 2) N has nonempty closed and decomposable values.
Let F : J × R n → P(R n ) be a multivalued map with nonempty compact values.Assign to F the multivalued operator The operator F is called the Niemytzki operator associated to F .
Definition 5.2 Let F : J × R n → P(R n ) be a multivalued function with nonempty compact values.We say F is of lower semi-continuous type (l.s.c.type) if its associated Niemytzki operator F is lower semi-continuous and has nonempty closed and decomposable values.
Next, we state a selection theorem due to Bressan and Colombo.

Theorem 5.3 ([13])
Let Y be separable metric space and let N : Y → P(L 1 (J, R n )) be a multivalued operator that has property (BC).Then N has a continuous selection, i.e., there exists a continuous (single-valued) function g : The following result is due to Colombo et al.

Proposition 5.4 ([16]
) Consider a l.s.c.multivalued map G : S → D and assume that φ : S → L 1 (J, R n ) and ψ : S → L 1 (J, R + ) are continuous maps such that for every s ∈ S, the set is nonempty.Then the map H : S → D is l.s.c. and admits a continuous selection.
Let us introduce the following hypotheses which are assumed hereafter.
(H 4 ) For each q > 0, there exists a function h q ∈ L 1 (J, R + ) such that F (t, y) ≤ h q (t) for a.e.t ∈ J and for y ∈ R n with y ≤ q.
The following lemma is crucial in the proof of our main theorem.
The following two lemmas are concerned with the measurability of multi-functions; they will be needed in this section.The first one is the well known Kuratowski-Ryll-Nardzewski selection theorem.Lemma 5.6 ([20,Theorem 19.7])Let E be a separable metric space and G a multi-valued map with nonempty closed values.Then G has a measurable selection.Lemma 5.7 ( [43]) 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 Proof Let v : [0, b] → R + be defined by v (t) = > 0.Then, from Lemma 5.7, there exist a measurable selection u of G such that We take = 1 n , n ∈ N, hence for every n ∈ N, we have Using the fact that G has compact values, we may pass to a subsequence if necessary to get that u n (•) converges to a measurable function u in E. Thus, the proof of the corollary.In the case of both differential equations and inclusions, existence results for problem (1)-( 3) can be found in [25,26,37].The main result in this section is contained in the following theorem.It is a Filippov type result for problem (1)-(3).EJQTDE, 2009 No. 24, p. 14 Theorem 5.9 In addition to (H 1 ), (H 3 ), and (H 4 ), assume that the following conditions hold.
(H 5 ) There exist a function p ∈ L 1 (J, R + ) such that (H 6 ) There exists continuous mappings g(•) : If and there exists ∈ (0, 1] such that d(g(y 0 )(t), F (t, y 0 (t)) < and then the problem (1)-( 3) has at least one solution y satisfying the estimates where Proof Let f 0 (y 0 )(t) = g(x)(t), t ∈ J, and EJQTDE, 2009 No. 24, p. 15 Let G 1 : P C → P(L 1 (J, R n )) be given by and G 1 : P C → P(L 1 (J, R n )) be defined by From [10,26], the problem ( 16)-( 18) has at least one solution which we denote by y 1 .Consider where y 1 is a solution of the problem ( 16)- (18).For every t ∈ J, we have Then, We then have EJQTDE, 2009 No. 24, p. 17 Thus, We have Hence, from the estimates above, we have Repeating the process for n = 1, 2, . . ., we arrive at the bound By induction, suppose that (21) holds for some n.Let From (24) and the Lebesgue Dominated Convergence Theorem, we conclude that f n (y n ) converges to f (y) in L 1 (J, R n ).Passing to the limit in (22), the function |x| 1 = b 0 |x(s)|ds.EJQTDE, 2009 No. 24, p. 2 a∈A d(a, B), sup b∈B d(A, b) where d(A, b) = inf a∈A d(a, b), d(a, B) = inf b∈B d(a, b).Then (P b,cl (R n ), H d ) is a metric space and

Theorem 4 .
13 Let F : J × R n → P cp,cv (R n ) be an upper-Scorza-Dragoni. Assume that all conditions of Theorem 4.12 are satisfied.Then the solution set S(F, 0, b) is an R δ -set.Proof Since F is upper-Scorza-Dragoni, then from Theorem 4.11, F is a σ −Ca-selection map.Therefore S(F, 0, b) is an R δ -set.EJQTDE, 2009 No.24, p. 12

Corollary 5 . 8
Let G : [0, b] → P cp (E) be a measurable multifunction and g : [0, b] → E be a measurable function.Then there exists a measurable selection u of G such that