A SECOND-ORDER IMPULSIVE CAUCHY PROBLEM

We study the existence of mild and classical solutions for an abstract second-order impulsive Cauchy problem modeled in the form ü(t) = Au(t)+f(t,u(t),u̇(t)), t ∈ (−T0,T1), t ≠ ti; u(0) = x0, u̇(0) = y0; u(ti) = I1 i (u(ti)), u̇(ti) = I2 i (u̇(t+ i )), where A is the infinitesimal generator of a strongly continuous cosine family of linear operators on a Banach space X and f , I1 i , I 2 i are appropriate continuous functions.


Introduction. This paper is concerned with the second-order impulsive Cauchy
(1.1) In problem (1.1), A is the infinitesimal generator of a strongly continuous cosine family of linear operators (C(t)) t∈R on a Banach space X; −T 0 < 0 < t 1 < t 2 < ··· < t n < T 1 ; , and I j i : X → X, f : R ×X ×X → X are appropriate continuous functions.
The theory of impulsive differential equations has become an important area of investigation in recent years. Relative to this theory, we only refer the reader to the works of Rogovchenko [6,7], Liu [5], Sun [3], and Cabada [1].
Motivated for numerous applications, recently, Liu [5] studied the first-order impulsive evolution problemu (t) = Au(t) + f t, u(t) , (1.2) where A is the infinitesimal generator of a C 0 -semigroup of linear operators on a Banach space X. In the cited paper, Liu apply the semigroup theory to prove the existence of mild, strong and classical solutions for system (1.2) using usual assumptions on the function f .
Our goal is to give some existence results of mild and classical solutions for the second-order impulsive Cauchy problem (1.1) using the cosine functions theory. It is well known that, in general, the second-order abstract Cauchy problem u(t) = Au(t), cannot be studied by reducing this system to a first-order linear equation, therefore, the existence results in this work are not consequence of those in Liu [5]. Throughout this paper, X denotes an abstract Banach space endowed with a norm · and C(t) denotes a strongly continuous operator cosine function defined on X with infinitesimal generator A. We refer the reader to [2,10] for the necessary concepts about cosine functions. Next, we only mention a few results and notations needed to establish our results. We denote by S(t) the sine function associated with C(t) which is defined by For a closed operator B : D(B) → X, we denote by [D(B)] the space D(B) endowed with the graph norm · B , that is, Moreover, in this paper the notation E stands for the space formed by the vectors x ∈ X for which the function C(·)x is of class C 1 . It was proved by Kisyński [4] that E endowed with the norm is a Banach space. The operator-valued function G(t) = C(t) S(t) AS(t) C(t) is a strongly continuous group of linear operators on the space E ×X generated by the operator Ꮽ = 0 I A 0 defined on D(A) × E. From this, it follows that AS(t) : E → X is a bounded linear operator and that AS(t)x → 0, t → 0, for each x ∈ E. Furthermore, if x : [0, ∞) → X is a locally integrable function, then defines an E-valued continuous function. This is an immediate consequence of the fact that defines an (E × X)-valued continuous function.
The existence of solutions of the second-order abstract Cauchy problem where h : [0,a] → X is an integrable function, has been discussed in [8]. Similarly, the existence of solutions of the semilinear second-order abstract Cauchy problem treated in [9]. We only mention here that the function x(·) given by is called a mild solution of (1.10). In the case in which The properties in the next result are well known (see [9]). Lemma 1.1. In the previous condition, the following properties hold: In Section 2, we discuss the existence of solution for the impulsive problem (1.1). Firstly, we introduce the concept of mild and classical solution for the impulsive problem (1.1) and subsequently, employing the contraction mapping principle and the ideas in Travis [10], we prove the existence of mild and classical solutions.
The terminology and notations are those generally used in operator theory. In particular, if Z and Y are Banach spaces, we indicate by ᏸ(Z : Y ) the Banach space of the bounded linear operators from Z into Y and we abbreviate to ᏸ(Z) whenever Y = Z. In addition, B r (x : Z) will denote the closed ball in Z with center at x and radius r .

Existence results.
In this section, we discuss the existence of mild and classical solutions for the impulsive initial value problem (1.1). By comparison with the secondorder abstract Cauchy problem, we introduce the following definitions.
is a mild solution of the impulsive problem (1.1) if the impulsive conditions in (1.1) are satisfied and ..,t n } : X) and (1.1) is satisfied.

Mild solutions.
In order to establish the existence of mild solutions, we introduce the following technical assumptions: (A1) f : R × X × X → X is a continuous function and there exist positive constants for every x, x ,y and y ∈ X; (A2) the functions I j i : X → X are continuous and there exist positive constants K(I j i ) such that then there exists a unique mild solution of the impulsive problem (1.1). (2.8) Similarly, (2.9) Inequalities (2.8) and (2.9) and the assumption max{Λ 1 , Λ 2 } < 1 imply that Φ is a contraction. Clearly, the unique fixed point of Φ, is the unique mild solution of (1.1). The proof is completed.  (2) The cosine family (C(t)) t∈R verifies the condition F (see Travis [10]

d/dt)C(t)I(x) = AS(t)I(x) = B 2 S(t)I(x) = BS(t)BI(x) for every x ∈ X.
(3) Let r ∈ R and g : X → X a Lipschitz function. If I(x) = S(r )g(x), from Travis [10, for each x ∈ X.

Classical solutions.
Next, we establish the existence of classical solution for the impulsive initial value problem (1.1), under the assumption that f is continuously differentiable. For this purpose, we need the following lemmas. (2.12) We know that S(t)g(0) ∈ E, thus we only need to proof that the second term in the right-hand side of (2.12) is in E. In relation with this property, for ρ ∈ R, we have (2.14) Clearly, the right-hand side of equation (2.14) is a continuous function of t, therefore t 0 S(t − s)g (s)ds ∈ E. The proof is finished.

Lemma 2.6. Letx 0 ∈ D(A),ỹ 0 ∈ E and assume that the assumption in Theorem 2.3 holds. Suppose, furthermore, that f is continuously differentiable and that there exists
If t i and t i+1 , are two consecutive impulse instants, then there exists a unique classical solution u(·) ofẍ Proof. Let w : (−T 0 ,T 1 ) → X be the unique mild solution of On the other hand, (2.18) The proof is completed. Now we establish the principal result of this paper. Proof. Let w : (−T 0 ,t 1 ) → X be the unique classical solution of . From Lemma 2.6, we know that (u 1 (t 1 ), v 1 (t 1 )) ∈ D(A)×E; thus, there exists a unique classical solution w(·) of the abstract Cauchy problem (2.20) Similarly to the previous case, In general, if w is the classical solution of the second-order Cauchy problem

(2.22)
It is easy to see that (ũ,ṽ) is the unique classical solution of the impulsive problem (2.23) Next, we show that (ũ,ṽ) = (u, v). In order to reduce the proof, we introduce the group of linear operators function (ũ,ṽ) is a solution of the first-order impulsive Cauchy problemẆ from assumption (A1) and (2.27), we obtain that where C 1 is a constant independent of t ∈ (−T 0 ,T 1 ). The Gronwall's inequality implies that u =ũ. The proof is completed.

Consequences.
Next, we briefly consider the impulsive system for each t ∈ (−T 0 ,T 1 ).
In order to establish our next result, we introduce one more simple assumption: (A4) the functions f : R × X → X, I j i : X → X are continuous and exist positive constants K(f ) and K(I 1 i ) such that for every u, u ∈ X. Theorem 3.3. Let x 0 ,y 0 ∈ X and assume that assumptions (A2), (A4) hold. If then there exists a unique mild solution of (3.1).
Proof. The proof of this theorem is similar to the proof of Theorem 2.3. We omit the details.
Using the ideas in the proofs of [10, Lemma 3 and Proposition 2.4], we can prove the following lemma.
The proof of Theorem 3.5 follows from the steps in the proof of Theorem 2.7.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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