# An Existence Result for Second-Order Impulsive Differential Equations

Authors: Meili Li; Haiqiang Liu
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

The Leray-Schauder alternative is used to investigate the
existence of solutions for second-order impulsive differential equations with nonlocal conditions
in Banach spaces. The results improve some recent results.


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## Body

## 1. Introduction

The theory of impulsive differential equations is emerging as an important area of investigation since it is a lot richer than the corresponding theory of nonimpulsive differential equations.  Many evolutionary processes in nature are characterized by the fact that at certain moments in time an abrupt change of state is experienced. That is the reason for the rapid development of the theory of impulsive differential equations; see the monographs [1, 2].This paper is concerned with the study on existence of second-order impulsive differential equations with nonlocal conditions of the form(1.1)x′′(t)=f(t,x(t),x′(t)),t∈J=[0,b],t≠tk,Δx∣t=tk=Ik(x(tk)),k=1,…,m,Δx′∣t=tk=I¯k(x(tk),x′(tk)),k=1,…,m,x(0)+g(x)=x0,x′(0)=η,
where the state x(·) takes values in Banach space X with the norm ∥·∥,x0,η∈X,  0=t0<t1<⋯<tm<tm+1=b,Δx∣t=tk=x(tk+)-x(tk-),Δx′∣t=tk=x′(tk+)-x′(tk-).  f,g, and Ik,I¯k(k=1,2,…,m) are given functions to be specified later.The nonlocal condition is a generalization of the classical initial condition. The first results concerning the existence and uniqueness of mild solutions to Cauchy problems with nonlocal conditions were studied by Byszewski [3]. Recently, theorems about existence, uniqueness and continuous dependence of impulsive differential abstract evolution Cauchy problems with nonlocal conditions have been studied by Fu and Cao [4], Anguraj and Karthikeyan [5], Abada et al. [6], Li and Han [7], and in the references therein.Up to now there have been very few papers in this direction dealing with the existence of solutions for second-order impulsive differential equations with nonlocal conditions. Our purpose here is to extend the results of first-order impulsive differential equations to second-order impulsive differential equations with nonlocal conditions.Our main results are based on the following lemma [8].Lemma 1.1 (Leray-Schauder alternative).
LetS be a convex subset of a normed linear space E and assume that 0∈S. Let G:S→S be a completely continuous operator, and let
(1.2)ζ(G)={x∈S:x=λGx,forsome0<λ<1}.Then eitherζ(G) is unbounded or G has a fixed point.


## 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.DenoteJ0=[0,t1],Jk=(tk,tk+1],J′=J∖{tk},k=1,2,…,m. We define the following classes of functions:PC(J,X)={x:J→X:xk∈C(Jk,X),k=0,1,…,m, and there exist x(tk+),x(tk-),k=1,…,m with x(tk)=x(tk-)},PC1(J,X)={x∈PC(J,X):xk′∈C(Jk,X),k=0,1,…,m, and there exist x′(tk+),x′(tk-),k=1,…,m with x′(tk)=x′(tk-)}, where xk and xk′ represent the restriction of x and x′ to Jk, respectively (k=0,…,m), and ∥xk∥Jk=sups∈Jk∥xk(s)∥.Obviously,PC(J,X) is a Banach space with the norm ∥x∥PC=max{∥xk∥Jk,k=0,…,m}, and PC1(J,X) is also a Banach space with the norm ∥x∥PC1=max{∥x∥PC,∥x′∥PC}.Definition 2.1.
A mapf:J×X×X→X is said to be an L1-Carathéodory if(i)
f:(·,w,v):J→X is measurable for every w,v∈X,(ii)
f:(t,·,·):X×X→X is continuous for almost all t∈J,(iii)
for eachi>0, there exists αi∈L1(J,R+) such that for almost all t∈J(2.1)sup∥w∥,∥v∥≤i∥f(t,w,v)∥≤αi(t).Definition 2.2.
A functionx∈PC1(J,X)∩C2(J′,X) is said to be a solution of (1.1) if x satisfies the equation x′′(t)=f(t,x(t),x′(t)) a.e. on J′, the conditions Δx|t=tk=Ik(x(tk)),Δx′|t=tk=I¯k(x(tk),x′(tk)),k=1,…,m, and x(0)+g(x)=x0,x′(0)=η.Lemma 2.3.
Ifx∈PC1(J,X)∩C2(J′,X) satisfies
(2.2)x′′(t)=f(t,x(t),x′(t)),t≠tk(k=1,2,…,m),
then
(2.3)x′(t)=x′(0)+∫0tf(s,x(s),x′(s))ds+∑0<tk<t[x′(tk+)-x′(tk)],∀t∈J,(2.4)x(t)=x(0)+x′(0)t+∑0<tk<t[x(tk+)-x(tk)]+∑0<tk<t[x′(tk+)-x′(tk)](t-tk)+∫0t(t-s)f(s,x(s),x'(s))ds,∀t∈J.Proof.
Assume thattk<t≤tk+1 ( here t0=0,tm+1=b ). Then
(2.5)x′(t1)-x′(0)=∫0t1f(s,x(s),x′(s))ds,x′(t2)-x′(t1+)=∫t1t2f(s,x(s),x′(s))ds,⋮x′(tk)-x′(tk-1+)=∫tk-1tkf(s,x(s),x′(s))ds,x′(t)-x′(tk+)=∫tk+tf(s,x(s),x′(s))ds.
Adding these together, we get(2.6)x′(t)=x′(0)+∫0tf(s,x(s),x′(s))ds+∑0<tk<t[x′(tk+)-x′(tk)],
that is, (2.3) holds.
Similarly, we have(2.7)x(t)=x(0)+∫0tx′(s)ds+∑0<tk<t[x(tk+)-x(tk)].
Substitution of (2.3) in (2.7) gives
(2.8)x(t)=x(0)+x′(0)t+∑0<tk<t[x(tk+)-x(tk)]+∑0<tk<t[x′(tk+)-x′(tk)](t-tk)+∫0t(t-s)f(s,x(s),x′(s))ds,∀t∈J,
that is, (2.4) holds.We assume the following hypotheses:(H1)
f:J×X×X→X is an L1-Carathéodory map;(H2)
Ik∈C(,X),I¯k∈C(X×X,X), and there exist constants dk,d¯k such that ∥Ik(w)∥≤dk,∥I¯k(w,v)∥≤d¯k(k=1,…,m) for every w,v∈X;(H3)
g:PC(J,X)→X is a continuous function and there exists a constant M such that
(2.9)∥g(x)∥≤M,foreachx∈PC(J,X);(H4)
there exists a functionp∈L1(J,R+) such that
(2.10)∥f(t,w,v)∥≤p(t)ψ(∥w∥+∥v∥),fora.e.t∈Jandeveryw,v∈X,
where ψ:[0,∞)→(0,∞) is a continuous nondecreasing function with
(2.11)(b+1)∫0bp(s)ds<∫c∞dsψ(s),
where(2.12)c=∥x0∥+M+(b+1)∥η∥+∑k=1m[dk+(b+1-tk)dk¯];(H5)
for each boundedB⊆PC1(J,X) and t∈J the set
(2.13){x0-g(x)+tη+∑0<tk<tIk(x(tk))+∑0<tk<tI¯k(x(tk),x′(tk))(t-tk)+∫0t(t-s)f(s,x(s),x′(s))ds:x∈B}
is relatively compact in X.


## 3. Main Results

Theorem 3.1.
If the hypotheses(H1)–(H5) are satisfied, then the second-order impulsive nonlocal initial value problem (1.1) has at least one solution on J.Proof.
Consider the spaceB=PC1(J,X) with norm
(3.1)∥x∥PC1=max{∥x∥PC,∥x′∥PC}.
We will now show that the operatorG defined by
(3.2)Gx(t)=x0-g(x)+tη+∑0<tk<tIk(x(tk))+∑0<tk<tI¯k(x(tk),x′(tk))(t-tk)+∫0t(t-s)f(s,x(s),x′(s))ds,t∈J
has a fixed point. This fixed point is then a solution of (1.1).
First we obtain a priori bounds for the following equation:(3.3)x(t)=λ[x0-g(x)+tη]+λ∑0<tk<tIk(x(tk))+λ∑0<tk<tI¯k(x(tk),x′(tk))(t-tk)+λ∫0t(t-s)f(s,x(s),x′(s))ds,t∈J.
We have(3.4)∥x(t)∥≤∥x0∥+M+b∥η∥+∑k=1mdk+∑k=1m(b-tk)d¯k+b∫0tp(s)ψ(∥x(s)∥+∥x′(s)∥)ds,t∈J.
Denoting byμ(t) the right-hand side of the above inequality, we have
(3.5)μ(0)=∥x0∥+M+b∥η∥+∑k=1m[dk+(b-tk)d¯k],∥x(t)∥≤μ(t),t∈J,μ′(t)=bp(t)ψ(∥x(t)∥+∥x′(t)∥),t∈J.
But
(3.6)x′(t)=λη+λ∫0tf(s,x(s),x′(s))ds+λ∑0<tk<tI¯k(x(tk),x′(tk)),t∈J.
Thus we have
(3.7)∥x′(t)∥≤∥η∥+∫0tp(s)ψ(∥x(s)∥+∥x′(s)∥)ds+∑k=1md¯k,t∈J.
Denoting by r(t) the right-hand side of the above inequality, we have
(3.8)r(0)=∥η∥+∑k=1md¯k,∥x′(t)∥≤r(t),t∈J,r′(t)=p(t)ψ(∥x(t)∥+∥x′(t)∥),t∈J.
Let
(3.9)w(t)=μ(t)+r(t),t∈J.
Then
(3.10)w(0)=μ(0)+r(0)=c,w′(t)=μ′(t)+r′(t)≤bp(t)ψ(w(t))+p(t)ψ(w(t))=(b+1)p(t)ψ(w(t)),t∈J.
This implies that(3.11)∫w(0)w(t)dsψ(s)≤(b+1)∫0bp(s)ds<∫c∞dsψ(s),t∈J.
This inequality implies that there is a constantK such that
(3.12)w(t)=μ(t)+r(t)≤K,t∈J.
Then
(3.13)∥x(t)∥≤μ(t),t∈J,∥x′(t)∥≤r(t),t∈J,
and hence
(3.14)∥x∥PC1=max{∥x∥PC,∥x′∥PC}≤K.
Second, we must prove that the operatorG:B→B is a completely continuous operator.
LetBi={x∈B:∥x∥PC1≤i} for some i≥1. We first show that G maps Bi into an equicontinuous family. Let x∈Bi and t,t¯∈J. Then for 0<t<t¯≤b, we have
(3.15)∥(Gx)(t)-(Gx)(t¯)∥≤∥tη-t¯η∥+∥∑t≤tk<t¯Ik(x(tk))∥+∥∑0<tk<t[(t-tk)-(t¯-tk)]I¯k(x(tk),x′(tk))∥+∥∑t≤tk<t¯(t¯-tk)I¯k(x(tk),x′(tk))∥+∥∫0t[(t-s)-(t¯-s)]f(s,x(s),x′(s))ds∥+∥∫tt¯(t¯-s)f(s,x(s),x′(s))ds∥≤(t¯-t)∥η∥+∑t≤tk<t¯dk+∑0<tk<t(t¯-t)d¯k+∑t≤tk<t¯(t¯-tk)d¯k+∫0t(t¯-t)αi(s)ds+∫tt¯(t¯-s)αi(s)ds,
and similarly
(3.16)∥(Gx)′(t)-(Gx)′(t¯)∥≤∥∫0tf(s,x(s),x′(s))ds-∫0t¯f(s,x(s),x′(s))ds∥+∥∑0<tk<tI¯k(x(tk),x′(tk))-∑0<tk<t¯I¯k(x(tk),x′(tk))∥≤∫tt¯αi(s)ds+∑t≤tk<t¯d¯k.
The right-hand sides are independent ofx∈Bi and tend to zero as t¯-t→0. Thus G maps Bi into an equicontinuous family of functions. It is easy to see that the family GBi is uniformly bounded. And from (H5), we know that GBi¯ is compact. Then by Arzela-Ascoli theorem, we can conclude that the map G:B→B is compact.
Next, we show thatG:B→B is continuous. Let {un}0∞⊑B with un→u in B. Then there is an integer q such that ∥un(t)∥,∥un′(t)∥≤q for all n and t∈J, so un∈Bq and u∈Bq. By (H1), f(t,un(t),un′(t))→f(t,u(t),u'(t))(n→∞) for almost all t∈J, and since ∥f(t,un(t),un′(t))-f(t,u(t),u'(t))∥<2αq(t), we have by the dominated convergence theorem that
(3.17)∥Gun-Gu∥PC=supt∈J∥[g(un)-g(u)]+∫0t(t-s)[f(s,un(s),un′(s))-f(s,u(s),u′(s))]ds+∑0<tk<tIk(un(tk))-∑0<tk<tIk(u(tk))+∑0<tk<t(t-tk)I¯k(un(tk),un′(tk))-∑0<tk<t(t-tk)I¯k(u(tk),u′(tk))∥≤∥g(un)-g(u)∥+∫0b(b-s)∥f(s,un(s),un′(s))-f(s,u(s),u′(s))∥ds+∑k=1m∥Ik(un(tk))-Ik(u(tk))∥+∑k=1m(b-tk)∥I¯k(un(tk),un′(tk))-I¯k(u(tk),u′(tk))∥→0,(3.18)∥(Gun)′-(Gu)′∥PC=supt∈J∥∫0t[f(s,un(s),un′(s))-f(s,u(s),u′(s))]ds+∑0<tk<tI¯k(un(tk),un′(tk))-∑0<tk<tI¯k(u(tk),u′(tk))∥≤∫0b∥[f(s,un(s),un′(s))-f(s,u(s),u′(s))]∥ds+∑k=1m∥I¯k(un(tk),un′(tk))-I¯k(u(tk),u′(tk))∥→0.
Thus G is continuous. This completes the proof that G is completely continuous.
Finally, the setζ(G)={x∈B:x=λGx,λ∈(0,1)} is bounded, as we proved in the first step. As a consequence of Lemma 1.1, we deduce that G has a fixed point x∈B which is a solution of (1.1).



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*Source: 293846-2010-08-11.xml*