# Existence of Solutions for Superlinear Second-Order System with Noninstantaneous Impulses

Authors: Yucheng Bu
Journal: Abstract and Applied Analysis
 (2021)

---

## Abstract

Variational methods are used in order to establish the existence of nontrivial weak solution for superlinear second-order system with noninstantaneous impulses. The main result is obtained when a kind of definition of the weak solution for this system is introduced. Meanwhile, an example is presented to illustrate the main result.


---

## Body

## 1. Introduction

In this paper, we consider the following superlinear system with noninstantaneous impulsive effects:(1)−u¨=λbit∇Fiu−uti+1,t∈si,ti+1,i∈M,u̇t=αi,t∈ti,si,i∈M\0,u̇si+=u̇si−,i∈M\0,u0=0=uT,u̇0=α0,where ut=u1t,u2t,⋯,uNt, λ>0, bi∈Lsi,ti+1, Fi∈F, αi∈RNi∈M, 0=s0<t1<s1<t2<s2<⋯<tm<sm<tm+1=T, the impulses start abruptly at points ti and keep the derivative constant on a finite time interval ti,si, u̇si±=lims⟶si±u̇si∈M\0. Here, m is a given positive integer and T>0. Denote
(2)M≔0,1,2,⋯,m,F≔F∈C1RN,R:F0=0,Ls,t≔b∈L1s,t\0:b≥0a.e.ins,t,where 0≤s<t≤T.This paper is mainly motivated by two facts. For the past thirty years, impulsive differential equations have been studied extensively in the literature. There are many approaches to investigate solutions for impulsive differential equations, such as the method of lower and upper solutions, fixed point theory, coincidence degree theory, and geometric approach (see for example [1–8]). In addition, variational methods have been also dealt with impulsive differential equations since papers [9, 10] appeared. Meanwhile, many existence and multiplicity results of solutions for various impulsive differential equations have been obtained. We refer the reader to a large quantity of the references for papers [9, 10]. It is worth noting that in all these references, most of impulsive effects are instantaneous. However, in [11, 12], the authors considered a kind of noninstantaneous impulsive effects, which is motivated by paper [13]. On the other hand, in recent years, some new critical point theorems are used to study nonlinear differential equations with no impulsive effects (see for example, [14–21]). Based on these two facts, we investigate the existence of nontrivial solution for (1).This paper is organized as follows. In Section2, we recall a critical point theorem in [16] and introduce a definition of the weak solution for (1), which generalizes Definition 2.3 in [12] to an N-dimensional case. In Section 3, we present our result and complete its proof via variational methods. The result is new in the field of studying noninstantaneous impulsive problem. In Section 4, we present an example in order to illustrate our result.


## 2. Preliminaries

The following critical point theorem is crucial in our arguments.Theorem 1 (see [16], Theorem 3.2).
LetX be a real Banach space and let Φ,Ψ:X⟶R be two continuously Gâteaux differentiable functions such that Φ is bounded from below and Φ0=Ψ0=0. Fix r>0 such that supu∈Φ−1−∞,rΨu<+∞ and assume that for each λ∈0,r/supu∈Φ−1−∞,rΨu, the functional Iλ=Φ−λΨ satisfies the (PS)-condition and it is unbounded from below. Then, for each λ∈0,r/supu∈Φ−1−∞,rΨu, the functional Iλ admits at least two distinct critical points.Consider the Hilbert space:(3)H01≔u∈H10,T;RN∣u0=0=uT,with the inner product
(4)u,vH01=∫0Tu̇t,v̇tdt+∫0Tut,vtdt,∀u,v∈H01,where ·,· is the inner product in RN. The corresponding norm is defined by
(5)uH01=∫0Tu̇t2dt+∫0Tut2dt1/2,∀u∈H01.Meanwhile, for everyu,v∈H01, we define
(6)u,v1=∫0Tu̇t,v̇tdt,which is also an inner product in H01, whose corresponding norm is that
(7)u≔u,u11/2.Poincare’s inequality [22] implies that
(8)∫0Tut2dt≤1λ1∫0Tu̇t2dt,where λ1=π2/T2 is the first eigenvalue of the Dirichlet problem
(9)−u¨t=λut,t∈0,T;u0=0=uT.Hence,(10)λ1λ1+1uH01≤u≤uH01.That is,· is equivalent to ·H01. It is well known that H01,·H01 is compactly embedded in C0,T,RN;·∞, where u∞=maxt∈0,Tut. Since the norms ·H01 and · are equivalent, there exists a positive number γ such that
(11)u∞≤γu,where γ≤k≔21+1/λ1maxT,1/T.Following the idea of the variational approach for impulsive differential equation of [11, 12] and combining with (1), one has that for every v∈H01,
(12)∫0T−u¨t,vtdt=∑i=0M∫siti+1−u¨t,vtdt+∑i=1m∫tisi−u¨t,vtdt=∫0Tu̇t,v̇tdt−∑i=1mu̇ti−−u̇ti+,vti−∑i=1mu̇si−−u̇si+,vsi=∫0Tu̇t,v̇tdt+∑i=1m−u̇ti−+αi,vti.In addition, combing with (1) again, one has
(13)∑i=0m−1∫siti+1λbitFiu−uti+1,vti+1dt=∑i=0m−1∫siti+1−u¨t,vti+1dt=∑i=1m∫si−1ti−u¨t,vtidt=∑i=1m−u̇ti−+αi−1,vti.(12) and (13) show that
(14)∫0T−u¨t,vtdt=∫0Tu̇t,v̇tdt+∑i=1mαi−αi−1,vti+∑i=0m−1∫siti+1λbitFiu−uti+1,vti+1dt.On the other hand,(15)∫0T−u¨t,vtdt=∑i=0m∫siti+1λbitFiu−uti+1,vtdt+∑i=1m∫tisiddtαi,vtdt=λ∑i=0m∫siti+1bitFiu−uti+1,vtdt.Hence, (14) and (15), together with vtm+1=vT=0, imply that
(16)∫0Tu̇t,v̇tdt=∑i=1mαi−1−αi,vti+λ∑i=0m∫siti+1bit∇Fiut−uti+1,vt−vti+1dt.Similar to Definition 2.3 in [12], we can introduce the following concept of a weak solution for problem (1).Definition 2.
Say thatu∈H01 is a weak solution for problem (1) if (16) holds for any v∈H01.Now, we consider the functionalsIλ,Φ,Ψ:H01⟶R defined by putting
(17)Φu≔12u2,(18)Ψu≔1λ∑i=1mαi−1−αi,uti+∑i=0m∫siti+1bitFiut−uti+1dt,(19)Iλu≔Φu−λΨu.Sincebi∈Lsi,ti+1, Fi∈F and αi∈RN, Φ and Ψ are two continuously Gâteaux differentiable with
(20)Φ′uv=∫0Tu̇t,v̇tdt,(21)Ψ′uv=1λ∑i=1mαi−1−αi,vti+∑i=0m∫siti+1bit∇Fiut−uti+1,vt−vti+1dt,for every u,v∈H01. Hence,
(22)Iλ′uv=Φ′uv−λΨ′uv.Namely,Iλ∈C1H01,R. Definition 2, together with (20)–(22), shows that a critical point of Iλ corresponds to a weak solution for (1).


## 3. Main Result and Its Proof

From now on, we refer toM as the range of i, unless specifically stated. Let bi1 be the usual norm in L1si,ti+1. The following theorem is our main result.Theorem 3.
Suppose that
(F) There exist ri>0 and μi>2 such that for ξ≥ri,
(23)0<μiFiξ≤∇Fiξ,ξ,hold. Then, for every bi∈Lsi,ti+1\0 and for every
(24)λ∈0,supc>0c2/2k2−c∑i=1mαi−1−αi∑i=1mbi1maxξ≤2cFiξ,
(1) admits at least one nontrivial weak solution.Proof.
We complete the proof in four steps.
Step 1. Clearly,H01 is a real Banach space. Φ,Ψ are two continuously Gâteaux differentiable functional. In addition, (17), (18), and Fi∈F imply that Φ is bounded from below and Φ0=Ψ0=0.
Step 2.Iλ satisfies the (PS)-condition. That is, every un such that Iλun is bounded and Iλ′un⟶0 as n⟶+∞ contains a convergent subsequence.
By (11), we have
(25)∑i=1mαi−1−αi,uti≤∑i=1mαi−1−αiu∞≤∑i=1mαi−1−αiγu.
Letμ≔minμi:i∈M, then μ>2. From F and (25), we deduce that
(26)μIλun−Iλ′unun=μ2−1un2−μ−1∑i=1mαi−1−αi,unti−λ∑i=0m∫siti+1bitμFiunt−unti+1−∇Fiunt−unti+1,unt−unti+1dt≥μ2−1un2−μ−1∑i=1mαi−1−αiγun,which implies that un is bounded in H01. Since H01 is a reflexive Banach space, passing to a subsequence if necessary, we may assume that there is a u0∈H01 such that unu0 in H01. Then, un converges uniformly to u0 on 0,T and un⟶u0 in L20,T;RN. In addition,
(27)Iλ′um−Iλ′un,um−un=um−un2−λ∑i=0m∫siti+1bit∇Fiumt−umti+1−∇Fiunt−unti+1,umt−umti+1−unt+unti+1dt,(28)umt−umti+1−unt+unti+1≤umt−unt+umti+1−unti+1≤2um−un∞⟶0,asm,n⟶∞.
Hence, the second term on the right hand of (27) converges to 0 as m,n⟶∞ because of (28) and Fi∈C1RN. Moreover, I′λun⟶0 as n⟶+∞ implies
(29)Iλ′um−Iλ′un,um−un≤I′λum−I′λunum−un⟶0,as m,n⟶∞. (27)–(29) show that um−un⟶0 as m,n⟶∞. By the completeness of H01, we see that un contains a convergent subsequence in H01.
Step 3.Iλ is unbounded from below.
By (F), there exist αi,βi>0 such that
(30)Fiξ≥αiξμi−βi.for every ξ∈RN.
Denoteα≔minαi, β≔maxβi, μ=minμi. Hence,
(31)Fiξ≥αξμ−β.for every ξ∈RN.
Letk>0 and v0∈H01 with ∥v0∥=1 and v0t is not a constant for a.e. 0,t1. In view of (31), we have
(32)∫sitj+1Fikv0−kv0ti+1dt≥αkμ∫sitj+1v0−v0ti+1μdt−βtj+1−si.
Denote(33)Ki≔∫sitj+1v0−v0ti+1μdt.
Then,(34)0≤Ki≤∫sitj+12μv0∞μdt≤∫sitj+12γμv0μdt=2γμti+1−si
Hence, by (17)–(19) and (32)–(34), we have
(35)Iλkv0=12kv02−∑i=1mαi−1−αi,kv0ti+∑i=0m∫siti+1bitFikv0t−kv0ti+1dt≤12k2+kγ∑i=1mαi−1−αi−α∑i=0mbi1ti+1−si2kγμ+βT=−α∑i=0mbi1ti+1−si2γμkμ+12k2+γ∑i=1mαi−1−αik+βT,which shows that Iλkv0⟶−∞ as k⟶∞ because the coefficient of kμ is negative.
Step 4.λ∈0,supc>0c2/2k2c2/2k2−c∑i=1m∣αi−1−αi∣/∑i=1mbi1maxξ≤2cFiξ.
Letc be such that
(36)0<λ<c2/2k2−c∑i=1mαi−1−αi∑i=1mbi1maxξ≤2cFiξ,and put
(37)r=c22k2.
By (18), one has
(38)supu∈Φ−1−∞,rΨu≤cλ∑i=1mαi−1−αi+∑i=1mbi1maxξ≤2cFiξ<+∞,0<λ<rc/λ∑i=1mαi−1−αi+∑i=1mbi1maxξ≤2cFiξ≤rsupu∈Φ−1−∞,rΨu.
Now, we apply Theorem1 to conclude that (1) admits at least one nontrivial weak solution.


## 4. Example

Let us consider the following problem:(39)−u¨=t+0.1u−uti+1u−uti+12,t∈si,ti+1,i=0,1,u̇t=0.1,0.11,0.12,t∈ti,si,i=0,1,u̇si+=u̇si−,i=0,1,u0=0=u1,u̇0=0,0,0,where 0=s0<t1=0.1<0.9=s1<t2=1.In this case,N=3, T=λ=1, bit=t+0.1,t∈si,ti+1,Fiu=1/4u4,i=0,1, k=21+1/π2. Choose c=0.5, one can verify that all the conditions in Theorem 3 are satisfied and (39) admits at least one nontrivial weak solution.



---

*Source: 5515575-2021-10-25.xml*