Impulsive Evolution Equations with Causal Operators

: In this paper, we establish sufﬁcient conditions for the existence of mild solutions for certain impulsive evolution differential equations with causal operators in separable Banach spaces. We rely on the existence of mild solutions for the strongly continuous semigroups theory, the measure of noncompactness and the Schauder ﬁxed point theorem. We consider the impulsive integro-differential evolutions equation and impulsive reaction diffusion equations (which could include symmetric kernels) as applications to illustrate our main results.


Introduction
Let R be the set of real numbers and let R + be the set of non-negative real numbers. Let E be a real Banach space endowed with the norm · . We denote by C( The aim of this paper is to establish existence results for mild solutions of the following impulsive evolution equation with the causal operator      u (t) = Au(t) + (Cu)(t) for t ∈ [0, T]\{t 1 , ..., t N } where A : D(A) ⊂ E → E is the infinitesimal generator of a C 0 -semigroup {T (t); t ≥ 0} and C : PC([0, T], E) → L p ([0, T], E) is a continuous causal operator; here 1 ≤ p ≤ ∞, N ∈ N, 0 = t 0 < t 1 < t 2 < ... < t N < t N+1 = T and I k : PC([0, T], E) → PC([0, T], E) is a continuous causal operator for each k = 1, 2, ..., N.
The theory of differential equations involving causal operators allows a unified treatment for general classes of differential equations, such as: Ordinary differential equations, differential equations with delay, integro-differential equations, Volterra integral equations and so on. The term causal operator (or Volterra abstract operator) was introduced by Tonelli [1], and the theory of these classes of operators was developed by Tychonoff [2]. The class of causal operators is quite large and it includes a number of operators that are used in mathematical modeling of some phenomena in engineering and physics. An important class of causal operators is the class of superposition operators or Nemytskij operators (see [3]) C : L p ([0, T], E) → L p ([0, T], E) defined by (Cu)(t) := F(t, u(t)), is another example of a causal operator. A more general example of causal operators is the operator C : Several examples of causal operators and their applications can be found in the monograph [4]. Although it does not specifically study the theory of causal operators, several monographs, such as [5][6][7][8][9], address some aspects of differential equations involving causal operators. Detailed studies on differential equations with causal operators in finite dimensional spaces can be found in the monographs [4,[10][11][12]. Applications of differential equations with causal operators in optimal control, adaptive control or hysteresis phenomena can be found in the papers [13][14][15][16][17][18][19][20]. Theoretical aspects regarding existence, stability or periodicity of solutions of differential equations with causal operators in finite or infinite dimensional spaces were presented in a series of works, such as: [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39].
The study of evolution equations with causal operators was first presented in [40], where an existence result was obtained and some applications were given, but impulsive evolution differential equations with causal operators has not yet been studied. In this paper we study the class of impulsive evolution equations involving causal operators. In Section 2 we recall some results on C 0 -semigroups of linear operators and some properties of the Hausdorff measure of noncompactness. In Section 3 we obtain the existence of mild solutions for a class of impulsive evolution equations with causal operators. Also, we show that a mild solution can be obtained as the limit of a sequence of successive approximations. In the last section we give some applications.

Preliminaries
We denote the space of all bounded linear operators acting on a Banach space E by L(E). We recall that a family {T (t); t ≥ 0} ⊂ L(E) is called a C 0 -semigroup if the following three properties are satisfied: (a) T (0) = I, the identity operator on E; The generator is always a closed, densely defined operator. For further details on the theory of the C 0 -semigroups see [41,42].
We denote by χ(B) the Hausdorff measure of non-compactness of a nonempty bounded set B ⊂ E, and it is defined by [43]: B admits a finite cover by balls of radius ≤ ε}.
We recall some properties of χ (see [43]). If A, B are bounded subsets of E, then (χ7) Generalized Cantor's intersection property : If {B n } n≥1 is a decreasing sequence of bounded closed nonempty subsets of E and lim n→∞ χ(B n ) = 0, then ∞ n=1 B n is a nonempty and compact subset of E (see [44]).
A function u(·) ∈ PC([0, T], E) is called a mild solution of (4) if it satisfies Let us introduce the following conditions.
and Proof. First, we remark that there exists an r > 0 such that Indeed, from (5) it follows that there exists η 0 > 0 such that for every η > η 0 , so that for every η > η 0 . Consequently, we can choose a r > η 0 such that (9) holds. Now, let and define the operator F : for t ∈ [0, T]. Since ξ(t, ·) is increasing on R + for a.e. t ∈ [0, T] for every u(·) ∈ B 0 , using (5) we have Next, we will show that B k 1 is equicontinuous on J k for every k = 1, 2, ..., N. For this, we shall write the operator F as (Fu First, we show that [42], Corollary 2.3), then there exists δ 1 = δ 1 (ε/5) > 0 such that for every t ∈ [0, T] and h ∈ R with |h| < δ 1 and t + h ∈ [0, T]. On [0, T], the function t → t 0 ψ(s)ds is uniformly continuous and thus there exists for every t ∈ [0, T] and h ∈ R with |h| < δ 2 and t + h ∈ [0, T]. Then for t = 0 we have for each u ∈ B 0 and h ∈ (0, T] with h < min{δ 1 , δ 2 }. It follows that G 1 is equicontinuous at t = 0. Next, take t ∈ (0, T] and let us choose 0 Since for each u ∈ B 0 . By similar reasoning, we obtain and so, by (13), we conclude that for each u ∈ B 0 and h ∈ R with |h| < η and t + h ∈ [0, T]. Furthermore, we have that is, for each u ∈ B 0 and h ∈ R with |h| < min{η, δ 1 , δ 2 } and t + h ∈ [0, T]. Now, using (15), (16) and (18), from (14) it follows that From this it follows that G k 1 is equicontinuous on J k for every k = 1, 2, ..., N. Next, we show that, for a given ν ∈ {1, 2, ..., N}, the set G ν 2 is equicontinuous on J ν , where G 2 := F 2 B 0 . Since I k is a compact operator, I k B 0 is a relatively compact set in PC([0, T], E) and so, by Lemma 1 I k B 0 is a relatively compact set in C(J k , E) for each k = 1, 2, ..., N. Using the Ascoli-Arzela theorem, from the compactness of I k B 0 in C(J k , E), it follows that (I k B 0 )(t) is relatively compact in E for every t ∈ J k and k = 1, 2, ..., N.
In particular, (I k B 0 )(t k ) is relatively compact for every k = 1, 2, ..., N, and thus K : x is jointly continuous from [0, ∞) × K to E, it follows that there exists a δ > 0 such that for every t k , k = 1, 2, ..., N, t, s ∈ J k with |t − s| < δ. Next, for every u(·) ∈ B 0 , t, s ∈ J ν with |t − s| < δ, we have .., N, it follows that B k 1 is equicontinuous on J k for every k = 1, 2, ..., N. Next, for each n ≥ 1, we define B n = conv(FB n−1 ). Then, for every n ≥ 1, B n ⊂ PC([0, T], E) is a bounded, closed and convex set. Now, from FB 0 ⊂ B 0 , it follows that If we suppose that B ν ⊂ B ν−1 for a given ν > 1, then and thus, by induction it follows that B n ⊂ B n−1 for every n ≥ 1. Moreover, it is easy to see that B k n is equicontinuous on J k for each k = 1, 2, ..., N and for every n ≥ 1. Now, we will show that χ PC (B n ) → 0 as n → ∞. From Lemma 2, it follows that there exists a sequence {v m (·)} m≥1 in FB n−1 such that where V := {v m (·); m ≥ 1}. From the above inequality it follows that Since for each k = 1, 2, ..., N, the equicontinuity of V k n and Lemma 3 imply Let t ∈ [0, T] be fixed. Since for all m ≥ 1, and for a.e. s ∈ [0, t], by Lemma 3 it follows that Also, by the continuity of the operators T(t) and by the compactness of the operators I k , it follows that the set {T (t − t k )(I k V)(t k )} is relatively compact for every t ∈ [0, T]. Therefore, we have that From (19), (21), and (22), we obtain where ρ := 2MT T 0 γ V (s)ds < 1.
Since ε > 0 is arbitrary, it follows that Since the last inequality is true for every n ≥ 1 and 0 < ρ < 1, passing to the limit as n → ∞, we obtain lim n→∞ χ PC (B n ) = 0. Now, using the generalized Cantor's intersection property, it follows that the set B := ∞ n=1 B n is a nonempty and compact subset of PC([0, T], E). Since every set B n is a convex set, the set B is also a convex set. Next, we verify that FB ⊂ B. Indeed, for every n ≥ 1, we have that For this, let u n (·) → u(·) in B. If 1 ≤ p < ∞ and 1/p + 1/q = 1, then by Hölder's inequality we have Using the continuity of the operators C and I k it follows that for 1 ≤ p ≤ ∞ we have that (Fu n )(·) − (Fu)(·) PC → 0 as n → ∞, so that F : B → B is a continuous operator. Since B is a nonempty compact convex set, and F : B → B is a continuous operator, by Schauder's fixed point theorem it follows that there exists at least one u(·) ∈ B such that for all t ∈ [0, T]; that is, u(·) ∈ B is a mild solution for (4).

Remark 2.
It is easy to see that the conclusion Theorem 1 remains true if (6) is replaced by Next, suppose that f (·, ·) : [0, T] × E → E is a function which satisfies the following condition:  η ∈ R + . Also, it is easy to see that C satisfies (H2)(b ). We obtain the following result.
has at least one mild solution on [0, T].
In the next result we show that, under the conditions (H1) and (H2), we can construct a sequence of successive approximations which converges to a mild solution of (4). Proof. Let r > 0 be such that (9) holds, and let B 0 and F : B 0 → B 0 be given by (10) and (11), respectively. We construct a sequence {u n (·)} n≥1 of functions u n (·) ∈ PC([0, T], E) as follows. Let n ∈ N. For each i ∈ {1, 2, ..., n}, we define Let us suppose that ||u i n (t)|| ≤ r for t ∈ [0, iT/n] and i ∈ {1, 2, ..., ν} with ν ≤ n − 1. Then we have for all t ∈ [0, (i + 1)T/n]. Hence, by induction on i we have that ||u i n (t)|| < r for all i ∈ {1, 2, ..., n} and t ∈ [0, iT/n]. In the following, to simplify the notation, we put u n (·) = u n n (·), n ≥ 1. By the causality of C and I k , the sequence {u n (·)} n≥1 can be written as for every n ≥ 1. Moreover, u n (·) ∈ B 0 for all n ≥ 1. Next, if 0 ≤ t ≤ T/n, then it is easy to see that If T/n ≤ t ≤ T, then we have Therefore, we obtain that Let V = {u n (·); n ≥ 1}. Since by (28) and the equicontinuity of Then by the property of the measure of non-compactness we obtain Let t ∈ [0, T] be fixed and let ε > 0. The we can find n(ε) ≥ 1 such that t t−T/n ψ(s)ds < ε/2M for n ≥ n(ε). Since for a.e. s ∈ [0, t] and n ≥ 1, by Remark 1 we conclude that Using the last inequality and the fact that χ ∑ 0<t k <t−T/n T (t − t k )(I k V)(t k ) = 0, we obtain that Since V(t) is bounded, by Lemma 3 and (H2) it follows that Therefore, V is a relatively compact subset of B 0 . Then, by the Arzela-Ascoli theorem, and extracting a subsequence if necessary, we may assume that the sequence {u n (·)} n≥1 converges uniformly on [0, T] to a continuous function u(·) ∈ B 0 . Since by the continuity of F and (28), we get (Fu)(·) − u(·) PC = 0. It follows that for all t ∈ [0, T]; that is, u(·) is a mild solution of the causal evolution Equation (4).

Conclusions
The theory of impulsive evolution differential equations with causal operators is an important one because it covers a large class of different types of impulsive evolution differential equations. The study of these evolution equations hopefully will be continued with impulsive evolution equations with nonlinear operators or impulsive evolution differential inclusions involving causal operators. Another direction of investigation is to study fractional differential equations with causal operators and their applications.