# Existence Theorem for Noninstantaneous Impulsive Evolution Equations

Authors: Gul I Hina Aslam; Amjad Ali; Maimona Rafiq
Journal: Journal of Mathematics
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

Evolution equations interpret the differential law of development with respect to time. A vast theory, in this regard, has been developed [1–9]. 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:(1)u′t=Aut,t∈si,ti+1,i∈ℕ0≔0,1,2,…,uti+=I+Biuti−,i∈ℕ≔1,2,…,ut=I+Biuti−,t∈ti,si,i∈ℕ,usi+=usi−,i∈ℕ,where A:DA⊆V⟶V is the infinitesimal generator of a strongly continuous semigroup of linear operators Ztt≥0 on a Banach space V endowed with a norm, the impulsive jump operator Bi∈BV,i∈ℕ, and the sequences tii∈ℕ0 and sii∈ℕ0 satisfy the relation ti<si<ti+1,i∈ℕ0 and set t0=s0=0. Furthermore, I denotes the identity operator.


## 2. Preliminaries

LetΩ=0,τ⊂ℝ be an open set and let 1≤p≤∞. The Sobolev space W1,pΩ is the space of all functions u∈LpΩ whose distributional first-order derivatives belong to LpΩ; that is, there exists a function ν∈LpΩ such that(2)∫Ωutϕ′dt=−∫Ωνtϕtdt,for all ϕ∈Cc∞Ω. The function ν is called the weak or distributional derivative of u. The space W1,pΩ is a Banach space with respect to the following norm:(3)uW1,pΩ≔uLpΩp+u′LpΩp.The spaceW01,pΩ is defined as the closure of the space Cc∞Ω in W1,pΩ (with respect to the topology of W1,pΩ). Precisely, this space is(4)W01,pΩ≔u∈W1,pΩ:u|∂Ω=0.A special case is whenp=2. Then, it is often written as H01Ω which is a Hilbert space with the following inner product:(5)u,v=∫0τutvtdt,where the corresponding norm is(6)u=∫0τut2dt1/2.First, let us review that, for(7)u∞=supt∈0,τut,uL2≔∫0τut2dt1/2,we have, for u∈H010,τ, (8)uL2≤τπu′tL2, (9)u∞≤αu,where α=πτ−1/2+τ1/2.Definition 1.
LetX and Y be Banach spaces. ℬ:X×Y⟶ℝ is nondegenerate with respect to the second variable if, for each y∈Y/0, there exists x∈X such that ℬx,y≠0.
In [13], Drivaliaris and Yannakakis proved the following variant form of generalized Lax–Milgram Theorem [14].Theorem 1.
SupposeX 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. There exists an m>0 such that, for each x∈X, supy=1ℬx,y≥mx.
Then, for every bounded linear functionalS on Y∗, there exists a unique x∈X with(10)ℬx,y=Sy,for all y∈Y.


## 3. Main Results

Using the technique of variational approach to impulsive differential equations of [15, 16], we have following results.Lemma 1.
A:DA⊆V⟶V being the infinitesimal generator of a strongly continuous semigroup of linear operators Ztt≥0 on a Banach space V endowed with a norm. For each v∈H010,τ, problem (1) has the following equivalent form:(11)∫0τu′tvtdt=∑i=1NBiZtivti+∑i=0N∫siti+1AsiZtvtdt.Proof.
(12)∫0∞u′tvtdt=∫0t1u′tvtdt+∑i=1N∫tisiu′tvtdt+∑i=1N−1∫siti+1u′tvtdt+∫sNτu′tvtdt+∫τ∞u′tvtdt=−∫0τutv′tdt+∑i=1Nusi−vsi−∑i=1Nusi+vsi+∑i=1Nuti−vti−∑i=1Nuti+vtior(13)∫0∞u′tvtdt=∑i=1Nuti−−uti+vti−∫0τutv′tdt.
Using the conditions of (1),(14)∫0∞u′tvtdt=−∑i=1NBiuti−vti−∫0τutv′tdt.
While, on the contrary,(15)∫0∞u′tvtdt=∑i=0N∫siti+1u′tvtdt+∑i=1N∫tisiu′tvtdt+∫τ∞u′tvtdt,∑i=0N∫siti+1u′tvtdt=∑i=0N∫siti+1Auttvtdt,∑i=1N∫tisiu′tvtdt=0.
Therefore,(16)∫0∞u′tvtdt=∑i=0N∫siti+1Autvtdt.
Comparing (14) and (16) yields(17)∫0τutv′tdt=−∑i=1NBiuti−vti+∑i=0N∫siti+1Autvtdt.
Since, fort∈si,ti+1, u′t=Aut, where A is an infinitesimal generator of a C0-semigroups Ztt≥0, therefore, ut=siZt, for t∈si,ti+1. For details, we refer [3, 17]. Hence, (17) becomes(18)∫0τutv′tdt=−∑i=1NBiZtivti+∑i=0N∫siti+1AsiZtvtdtor(19)∫0τu′tvtdt=∑i=1NBiZtivti+∑i=0N∫siti+1AsiZtvtdt.Lemma 2.
For a bilinear formℬu,v=∫0τutvtdt on H010,τ, there exists a k>0 such that, for each u∈H010,τ, supv=1ℬu,v≥ku.Proof.
Since a bounded bilinear formℬu,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(20)Fu,v=∫0τu′tvtdt=ℬu,v.
By using (8), the result follows.
DefineS:H010,τ⟶ℝ:(21)Sv=∑i=1NBiZtivti+∑i=0N∫siti+1AsiZtvtdt.
Note that a functionu∈H010,τ is the weak solution of problem (1) if and only if(22)ℬu,v=Sv,∀v∈H010,τ.
It can be easily noted thatH010,τ is a reflexive Banach space, au,v is a nondegenerate bilinear form, and Sv is linear. Moreover, by using (9),(23)Sv≤∑i=1NBiZtivti+∑i=0N∫siti+1AsiZtvtdt≤∑i=1NBiZtiv∞+ZtAv∞∑i=0Nsiti+1−si≤β∑i=1NBiZti+ZtA∑i=0Nsiti+1−siv.
Therefore, by using Theorem1, we have the following.Theorem 2.
Noninstantaneous impulsive evolution equation (1) has a unique weak solution u∈H010,τ if A is the generator of a uniformly continuous semigroup Ztt≥0.Theorem 3.
Noninstantaneous impulsive evolution equation (1) has a unique weak solution u∈H010,τ if A generates an exponentially stable C0-semigroup Ztt≥0.


## 4. Example of the Main Result

Now, we present an example pertaining to the main result stated above.LetV=L20,1,t0=s0=0,t1=π,ands1=T=2π. Now, we let the operator A be the second-order partial derivative operator, i.e., Au=∂2u/∂t2, for u∈DA. Then, A is the infinitesimal generator of a strongly continuous semigroup of linear operators Ztt≥0 and Z⋅ is bounded with Zt≤e−t, for all t≥0. Now, we look at the following periodic problem:(24)∂ut,y∂t=∂2ut,y∂y2,t∈0,π,y∈0,1,∂ut,0∂y=∂ut,1∂y=0,t∈0,π,uπ+,y=−yuπ−,y2eπ+uπ−,y+uπ−,y,ut=ycostuπ−,y2eπ+uπ−,y+uπ−,y,t∈π,2π,u0,y=u2π,y.Let us denoteBi=ycost/2eπ+uπ−,y, where Bi∈BV, the jump operator, and if I∈BV is the identity operator, then Iuπ−,y=uπ−,y. Then, the problem (24) can be written as(25)u′t=Aut,y,t∈0,π,uπ+=I+Biuπ−,y,ut=I+Biuπ−,y,t∈π,2π,u0=u2π.This problem has the variational form given by(26)∫02π∂ut,y∂tvtdt=B1Zπvπ+∫0π∂2ut,y∂y2vtdt.Since the right-hand side is bounded above andA is the generator of the uniformly continuous semigroup, so, by (8), it has a unique weak solution u∈H010,2π.



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