Impulsive Evolution Inclusions with State-Dependent Delay and Multivalued Jumps

In this paper we prove the existence of a mild solution for a class of impulsive semilinear evolution dierential inclusions with state-dependent delay and multivalued jumps in a Banach space. We consider the cases when the multivalued nonlinear term takes convex values as well as nonconvex values.


Introduction
In this paper, we are concerned by the existence of mild solution of impulsive semilinear functional differential inclusions with state-dependent delay and multivalued jumps in a Banach space E.More precisely, we consider the following class of semilinear impulsive differential inclusions: x(t) = φ(t), t ∈ (−∞, 0], (1.3) where {A(t) : t ∈ J} is a family of linear operators in Banach space E generating an evolution operator, F be a Carathéodory type multifunction from J × B to the collection of all nonempty compact convex subsets of E, B is the phase space defined axiomatically (see section 2) which contains the mapping from (−∞, 0] into E, φ ∈ B, 0 = t 0 < t 1 < . . .< t m < t m+1 = b, I k : E → P(E), k = 1, . . ., m are multivalued maps with closed, bounded and convex values, x(t + k ) = lim h→0 + x(t k + h) and x(t − k ) = lim h→0 + x(t k −h) represent the right and left limits of x(t) at t = t k .Finally P(E) denotes the family of nonempty subsets of E, ρ : J × B → (−∞, b]. The theory of impulsive differential equations has become an important area of investigation in recent years, stimulated by the numerous applications to problems arising in mechanics, electrical engineering, medicine, biology, ecology, population dynamics, etc.During the last few decades there have been significant developments in impulse theory, especially in the area of impulsive differential equations and inclusions with fixed moments; see the monographs of Bainov and Simeonov [8], Benchohra et al. [11], Lakshmikantham et al. [29], Samoilenko and Perestyuk [33], and the references therein.For the case where the impulses are absent (i.e.I k = 0, k = 1, . . ., m) and F is a single-valued or multivalued map and A is a densely defined linear operator generating a C 0 -semigroup of bounded linear operators and the state space is C([−r, 0], E) or E, the problem (1.1)-(1.3)has been investigated in, for instance, the monographs by Ahmed [4,5], Hale and Verduyn Lunel [21], Hu and Papageorgiou [26], Kamenskii et al. [27] and Wu [34] and the papers by Benchohra and Ntouyas [12], Cardinali and Rubbioni [14], Gory et al. [18].Benedetti [13] considered the existence result in the autonomous case (A(t) ≡ A) and finite delay.Cardinali and Rubbioni [15] considered the non autonomous case.In [32] Obukhovskii and Yao considered local and global existence results for semilinear functional differential inclusions with infinite delay and impulse characteristics in a Banach space.Recently some existence results were obtained for certain classes of functional differential equations and inclusions in Banach spaces under assumption that the linear part generates an compact semigroup (see, e.g., [1,2,3]).
On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type of equations has received a significant amount of attention in the past several years (we refer to [7,16,22,23,24] and the references therein).The literature related to functional differential inclusions with state-dependent delay remains limited [1,3].
Our goal here is to give existence results for the problem (1.1)-(1.3)without any compactness assumption.In Section 2, we will recall briefly some basic definitions and preliminary facts which will be used throughout the following sections.In Section 3, we prove existence and compactness of solutions set for problem (1.1)- (1.3).In Section 4, we provide a condition which guarantee the existence of a solution of (1.1)-(1.3)by using a fixed point theorem due to Mönch [31].EJQTDE, 2013 No. 42, p. 2 We mention that the model with multivalued jump sizes may arise in a control problem where we want to control the jump sizes in order to achieve given objectives.To our knowledge, there are very few results for impulsive evolution inclusions with multivalued jump operators; see [3,6,10,13,30].The results of the present paper extend and complement those obtained in the absence of the impulse functions I k , and those with single-valued impulse functions I k .

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Let J := [0, b], b > 0 and (E, .) be a real separable Banach space.C(J, E) the space of E-valued continuous functions on J with the uniform norm L 1 (J, E) the space of E−valued Bochner integrable functions on J with the norm To define the solution of problem (1.1)-(1.3), it is convenient to introduce some additional concepts and notations.Consider the following spaces where y k is the restriction of y to J k = (t k , t k+1 ], k = 0, . . ., m.Let the space with the semi-norm defined by In this work, we will employ an axiomatic definition for the phase space B which is similar to those introduced in [25].Specifically, B will be a linear space of functions mapping (−∞, 0] into E endowed with a semi norm .In what follows we use the following notations K b = sup{K(t), t ∈ J} and M b = sup{M (t), t ∈ J}.A multifunction G : X → P(Y ) is said to be upper semicontinuous (u.s.c.) if The operator F is called the Niemytzki operator associated with F .We say F is the lower semi-continuous type if its associated Niemytzki operator F is lower semi-continuous and has nonempty closed and decomposable values.For details and equivalent definitions see [19,27,28].EJQTDE, 2013 No. 42, p. 4 Let us recall the following result that will be used in the sequel.
Lemma 2.3.[9] Let E be a separable metric space and let G : E → P(L 1 ([0, b]; E)) be a multi-valued operator which is lower semi-continuous and has nonempty closed and decomposable values.Then G has a continuous selection, i.e. there exists a continuous function f : As an example of the measure of noncompactness possessing all these properties is the Hausdorff of MNC which is defined by For more information about the measure of noncompactness we refer the reader to [27].
Now, let for every t ∈ J , A(t) : E → E be a linear operator such that EJQTDE, 2013 No. 42, p. 5 (i) For all t ∈ J, D(A(t)) = D(A) ⊆ E is dense and independent of t.
(ii) For each s ∈ I and each x ∈ E there is a unique solution v : [s, b] → E for the evolution equation In this case an operator T can be defined as where v is the unique solution of (2.1) and L(E) is the family of linear bounded operators on E.
Definition 2.8.The operator T is called the evolution operator generated by the family {A(t) : t ∈ J}.
3. (t, s) → T (t, s) is strongly continuous on ∆ and Definition 2.9.The operator G : L 1 (J, E) → C(J, E) defined by is called the generalized Cauchy operator, where T (., .) is the evolution operator generated by the family of operators {A(t) : t ∈ J}.
In the sequel we will need the following results.
Then for every semicompact set Lemma 2.13.[27] Let S : L 1 (J, E) → C(J, E) be an operator satisfying conditions (G1), (G2) and let the set {f n } ∞ n=1 be integrably bounded with the property χ({f n (t) : n ≥ 1}) ≤ η(t), for a.e.t ∈ J, where η(.) ∈ L 1 (J, R + ) and χ is the Hausdorff MNC.Then where ζ ≥ 0 is the constant in condition (G1).Lemma 2.14.[27] If U is a closed convex subset of a Banach space E and R : U → P cv,k (E) is a closed β-condensing multifunction, where β is a nonsingular MNC defined on the subsets of U .Then R has a fixed point.Lemma 2.15.[27] Let W be a closed subset of a Banach space E and R : W → P cv,k (E) be a closed multifunction which is β-condensing on every bounded subset of W , where β is a monotone measure of noncompactness.If the fixed points set FixR is bounded, then it is compact.In this section we prove the existence of mild solutions for the impulsive semilinear functional differential inclusions (1.1)-(1.3).We will always assume that ρ : J × B → (−∞, b] is continuous.In addition, we introduce the following hypotheses.(Hφ) The function t → φ t is continuous from R(ρ − ) = {ρ(s, ϕ) : (s, ϕ) ∈ J × B, ρ(s, ϕ) ≤ 0} into B and there exists a continuous and bounded function (H1) The multifunction F (., x) has a strongly measurable selection for every x ∈ B.
(H5) There exist constants a k , c k > 0, k = 1, . . ., m such that The next result is a consequence of the phase space axioms.
For any x ∈ Ω b we have Thus (Ω b , .b ) is a Banach space.We note that from assumptions (H1) and (H3) it follows that the superposition multioperator S 1 F : Ω b → P(L 1 (J, E)) defined by is nonempty set (see [27]) and is weakly closed in the following sense.
Lemma 3.5.If we consider the sequence where S 1 F and I k ∈ I k (x).It is clear that the integral multioperator N is well defined and the set of all mild solution for the problem (1.1)-(1.3)on J is the set FixN = {x : x ∈ N (x)}.
We shall prove that the integral multioperator N satisfies all the hypotheses of Lemma 2.14.The proof will be given in several steps.
Step 1.Using the fact that the maps F and I has a convex values it easy to check that N has convex values.
Step 2. N has closed graph.
Hypothesis (H3) implies that the set {f n } +∞ n=1 integrably bounded and for a.e.t ∈ J the set {f n (t)} +∞ n=1 relatively compact, we can say that {f n } +∞ n=1 is semicompact sequence.Consequently {f n } +∞ n=1 is weakly compact in L 1 (J; E), so we can assume that f n f * .From lemma 2.11 we know that the generalized Cauchy operator on the interval J, G : L 1 (J; E) → Ω b , defined by satisfies properties (G1) and (G2) on J.
Note that set {f n } +∞ n=1 is also semicompact and sequence (f n ) +∞ n=1 weakly converges to f * in L 1 (J; E).Therefore, by applying Lemma 2.12 for the generalized Cauchy operator G of (3.2) we have the convergence Gf n → Gf .By means of EJQTDE, 2013 No. 42, p. 10 (3.2) and (3.1), for all t ∈ J we can write where S 1 F , and I k ∈ I k (x).By applying Lemma 2.11, we deduce in Ω b and by using in fact that the operator With the same technique, we obtain that N has compact values.
Step 3. We consider the measure of noncompactness defined in the following way.
For every bounded subset where ∆(Ω) is the collection of all the denumerable subsets of Ω; where mod C (Ω) is the modulus of equicontinuity of the set of functions Ω given by the formula and L > 0 is a positive real number chosen such that From the Arzela-Ascoli theorem, the measure ν 1 give a nonsingular and regular measure of noncompactness, (see [27]).
We will demonstrate that the solution set is a priori bounded.Indeed, let x ∈ N .Then there exists f ∈ S 1 F and I k ∈ I k (x) such that for every t ∈ J we have Using Lemma 3.1, we have Since the last expression is a nondecreasing function of t, we have that Invoking Gronwall's inequality, we get EJQTDE, 2013 No. 42, p. 13 where which completes the proof.

The nonconvex case
This section is devoted to proving the existence of solutions for (1.1)-(1.3)with a nonconvex valued right-hand side.Our result is based on Mönch's fixed point theorem combined with a selection theorem due to Bressan and Colombo (see [9]).We will assume the following hypothesis: Let F be a multifunction defined from J × B to the family of nonempty closed convex subsets of E such that (H7) The multifunction F : (t, .)→ P k (E) is lower semicontinuous for a.e.t ∈ J.
Suppose that Ω ⊆ B r is countable and Ω ⊆ co({0} ∪ N (Ω)) We will prove that Ω is relatively compact.We consider the measure of noncompactness defined in (3.3) and L > 0 is a positive real number chosen such that where M = sup (t,s)∈∆ T (t, s) .
Let {y n } +∞ n=1 be the denumerable set which achieves that maximum ν 1 (N (Ω)), i,e; ν 1 (N (Ω)) = (γ 1 ({y n } +∞ n=1 ), mod C ({y n } +∞ n=1 )).Then there exists a set {x n } +∞ n=1 ⊂ Ω such that y n ∈ N (x n ), n ≥ 1.Then where f ∈ S 1 F and I k ∈ I k (x n ), so that We give an upper estimate for γ 1 ({y n } +∞ n=1 ).Fixed t ∈ J by using condition (H9), for all s ∈ [0, t] we have EJQTDE, 2013 No. 42, p. 15 By using condition (H8), the set {f n } +∞ n=1 is integrably bounded.In fact, for every t ∈ J, we have By applying Lemma 2.13, it follows that Thus, we get ) Therefore, we have that From (3.6), we obtain that Coming back to the definition of γ 1 , we can see By using the last equality and hypotheses (H8) and (H9) we can prove that set {f n } +∞ n=1 is semicompact.Now, by applying Lemma 2.11 and Lemma 2.12, we can conclude that set {Gf n } +∞ n=1 is relatively compact.The representation of y n given by (4.2) yields that set {y n } +∞ n=1 is also relatively compact in Ω b , since ν 1 is a monotone, nonsingular, regular MNC, we have that Step 2. It is clear that the superposition multioperator S 1 F has closed and decomposable values.Following the lines of [27], we may verify that S We consider a map N : Ω b → Ω b defined as Since the Cauchy operator is continuous, the map N is also continuous, therefore, it is a continuous selection of the integral multioperator.
We will demonstrate that the solution set is a priori bounded.Indeed, let x ∈ λN 1 and λ ∈ (0, 1).There exists f ∈ S 1 F and I k ∈ I k (x) such that for every t ∈ J we have x(t) = λT (t, 0)φ(0) + λ t 0 T (t, s)f (s)ds + λ From the choice of U there is no x ∈ ∂U such that x = λN x for some λ ∈ (0, 1).Thus, we get a fixed point of N 1 in Ū due to the Mönch Theorem.
We can show that problem (5.

Theorem 2 .
16.[31] Let E be a Banach space, U an open subset of E and 0 ∈ U .Suppose that N : U → E is a continuous map which satisfies Mönch's condition (that is, if D ⊆ U is countable and D ⊆ co({0} ∪ N (D)), then D is compact) and assume that x = λN (x), for x ∈ ∂U and λ ∈ (0, 1)holds.Then N has a fixed point in U .EJQTDE, 2013 No. 42, p. 7

(
A) {A(t) : t ∈ J} be a family of linear (not necessarily bounded) operators, A(t) : D(A) ⊂ E → E, D(A) not depending on t and dense subset of E and T : ∆ = {(t, s) : 0 ≤ s ≤ t ≤ b} → L(E) be the evolution operator generated by the family {A(t) : t ∈ J}.