Existence Theorem for Noninstantaneous Impulsive Evolution Equations

In this note, the variational form of the classical Lax–Milgram theorem is used for the divulgence of variational structure of the first-order noninstantaneous impulsive linear evolution equation. The existence and uniqueness of the weak solution of the problem is obtained. In future, this constructive theory can be used for the corresponding semilinear problems.


Introduction
Evolution equations interpret the differential law of development with respect to time. A vast theory, in this regard, has been developed [1][2][3][4][5][6][7][8][9]. In [10], Hernándaz and O'Regan established the theory of noninstantaneous impulsive equations and showed the existence of corresponding mild solutions. Recently, in [11], JinRong Wang acquired ample conditions to guarantee the asymptotic stability of linear and semilinear noninstantaneous impulsive evolution equations. In [12], Tang and Nieto used the variational method to show existence of the solution to impulsive evolution equations. However, this approach, to the best of our knowledge, has not been used to show the existence (uniqueness) of the solution to noninstantaneous impulsive evolution equations.
In this paper, we intend to acquire the variational structure associated to the following first-order noninstantaneous impulsive linear evolution equations: where A: D(A) ⊆ V ⟶ V is the infinitesimal generator of a strongly continuous semigroup of linear operators Z(t) { } t≥0 on a Banach space V endowed with a norm, the impulsive jump operator B i ∈ B(V), i ∈ N, and the sequences t i i∈N 0 and s i i∈N 0 satisfy the relation t i < s i < t i+1 , i ∈ N 0 and set t 0 � s 0 � 0. Furthermore, I denotes the identity operator.

Preliminaries
Let Ω � (0, τ) ⊂ R be an open set and let 1 ≤ p ≤ ∞. e Sobolev space W 1,p (Ω) is the space of all functions u ∈ L p (Ω) whose distributional first-order derivatives belong to L p (Ω); that is, there exists a function ] ∈ L p (Ω) such that for all ϕ ∈ C ∞ c (Ω). e function ] is called the weak or distributional derivative of u. e space W 1,p (Ω) is a Banach space with respect to the following norm: A special case is when p � 2. en, it is often written as H 1 0 (Ω) which is a Hilbert space with the following inner product: where the corresponding norm is First, let us review that, for we have, for u ∈ H 1 0 (0, τ), where α � π(τ) −1/2 + τ 1/2 .
In [13], Drivaliaris and Yannakakis proved the following variant form of generalized Lax-Milgram eorem [14]. Theorem 1. Suppose X and Y are Banach Spaces, X is reflexive, and that B: X × Y ⟶ R is a bounded, nondegenerate (w.r.t the second variable), and bilinear functional.
Then, for every bounded linear functional S on Y * , there exists a unique x ∈ X with for all y ∈ Y.

Main Results
Using the technique of variational approach to impulsive differential equations of [15,16], we have following results.

Lemma 1. A: D(A) ⊆ V ⟶ V being the infinitesimal generator of a strongly continuous semigroup of linear operators Z(t)
{ } t≥0 on a Banach space V endowed with a norm. For each v ∈ H 1 0 (0, τ), problem (1) has the following equivalent form: Using the conditions of (1), 2

Journal of Mathematics
While, on the contrary, Comparing (14) and (16) yields Since For details, we refer [3,17]. Hence, (17) becomes Proof. Since a bounded bilinear form B(u, v) on a Hilbert space H is of the form (Fu, v), for a unique bounded operator F on H, in this case, if we choose F � u ′ , we have By using (8), the result follows. Define S: H 1 0 (0, τ) ⟶ R: Note that a function u ∈ H 1 0 (0, τ) is the weak solution of problem (1) if and only if It can be easily noted that H 1 0 (0, τ) is a reflexive Banach space, a(u, v) is a nondegenerate bilinear form, and S(v) is linear. Moreover, by using (9),

Example of the Main Result
Now, we present an example pertaining to the main result stated above. Let V � L 2 (0, 1), t 0 � s 0 � 0, t 1 � π, and s 1 � T � 2π. Now, we let the operator A be the second-order partial derivative operator, i.e., Au � z 2 u/zt 2 , for u ∈ D(A). en, A is the infinitesimal generator of a strongly continuous semigroup of linear operators Z(t) { } t≥0 and Z(·) is bounded with ‖Z(t)‖ ≤ e − t , for all t ≥ 0. Now, we look at the following periodic problem: