# Existence of Solutions for Sturm-Liouville Boundary Value Problem of Impulsive Differential Equations

Authors: Hong-Rui Sun; Ya-Ning Li; Juan J. Nieto; Qing Tang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned with the existence of solutions for Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations, under different assumptions on the nonlinearity and impulsive functions, existence criteria of single and multiple solutions are established. The main tools are variational method and critical point theorems. Some examples are also given to illustrate the main results.


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## Body

## 1. Introduction

Impulsive differential equation is one of the main tools to study the dynamics of processes in which sudden changes occur. The theory of impulsive differential equation has recently received considerable attention, see [1–9]. Some classical tools such as fixed-point theorems in cones and the method of upper and lower solutions with monotone iterative technique have been widely used to study impulsive differential equations. In the last few years, variational method has been used to determine the existence of solutions for impulsive differential equation possessing a variational structure under certain boundary condition, one can refer to [10–26] and the references therein for detailed discussions.Especially, in [12], Nieto and O’Regan studied the nonlinear Dirichlet impulsive problem:
(1.1)-u′′(t)+u(t)=f(t,u(t)),t∈[0,T]∖{t1,t2,…,tp},(1.2)Δu′(tj)=Ij(u(tj-)),j=1,2,…,p,(1.3)u(0)=u(T)=0,
where f∈C([0,T]×ℝ,ℝ),Ij∈C(ℝ,ℝ),and j=1,2,…,p. By the least action principle, the existence of a solution was obtained by assuming sublinear growth on the nonlinearity and impulses.In [26], Zhou and Li discussed the problem:
(1.4)-u′′(t)+g(t)u(t)=f(t,u(t)),t∈[0,T]∖{t1,t2,…,tp},
with impulse conditions (1.2), and Dirichlet boundary condition (1.3), where f∈C([0,T]×ℝ,ℝ) and g∈L∞([0,T],ℝ). By using a symmetric Mountain Pass theorem, the existence result of an infinite number of solutions was obtained.Tian and Ge in [20] studied the existence of multiple solutions for the following equation with impulsive effect:
(1.5)-(p(t)u′)′+q(t)u=f(t,u),t∈[0,1]∖{t1,t2,…,tp},(1.6)-Δ(p(tj)u(tj))=Ij(u(tj)),j=1,2,…,p,(1.7)αu(0)-βu′(0)=0,γu(1)+σu′(1)=0.
By applying variational methods and upper and lower solutions methods, they obtained the existence of at least four solutions and gave some accurate characteristics of the solutions. In the special case of p(t)≡1,q(t)≡λ and Ij(u(tj))=dj, in [19] they studied the corresponding linear and nonlinear problem and established some existence results of positive solutions via critical point theory and variational methods. In [13], Sun and Chen considered the equation:
(1.8)-(p(t)u′)′+r(t)u′+q(t)u=f(t,u),t∈[0,1]∖{t1,t2,…,tp},
with impulsive condition (1.6) and Neumann boundary condition u′(0+)=u′(1-)=0, they got existence criteria of at least one solution, two solutions and infinitely many solutions under some different conditions.In the above-cited articles, analogous results are also given when the impulses are absent, so they cannot reflect the impact of the impulses on the existence of the solutions. In [24], Zhang and Li studied the periodic solutions generated by impulses for a second-order impulsive differential equation. As defined in [24], a solution is called a solution generated by impulses if this solution is nontrivial only if impulsive terms are not zero.Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Many systems involve the impulsive condition which depends not only on the derivative of a function but also the function itself. A few papers discussed the solutions of impulsive differential equations involving impulses both on the function and on its derivative by critical point theory; see for example [21, 22].Inspired by the above results, in this paper we consider the existence and multiplicity of solution to the following Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations:(1.9)-u′′(t)+u(t)=f(t,u(t)),t∈[0,T]∖{t1,t2,…,tp},(1.10)-Δu(tj)=Aju(tj-),j=1,2,…,p,(1.11)-Δu′(tj)=Ij(u(tj-))-Aju′(tj+),j=1,2,…,p,(1.12)αu(0)-βu′(0)=0,γu(T)+σu′(T)=0,
where f∈C([0,T]×ℝ,ℝ), α,β,γ,σ≥0, α+β>0, γ+σ>0. p∈ℕ, 0=t0<t1<t2<⋯<tp<tp+1=T, Δu(tj)=u(tj+)-u(tj-), u(tj+) (or u(tj-)) denotes the right limit (or left limit) of u(t) at t=tj,Δu′(tj)=u′(tj+)-u′(tj-),u′(tj+) (or u′(tj-)) denotes the right limit (or left limit) of u′(t) at t=tj, Aj∈ℝ, Ij∈C(ℝ,ℝ), j=1,2,…,p. For convenience, we denote (1.9)–(1.12) as problem (P).We begin by establishing the corresponding variational framework of problem (P) in an appropriate space of functions. Then under the assumption that the nonlinearity and the impulsive functions are superlinear, we get the existence of at least one nontrivial solution. Furthermore, when f is odd about the second variable and Ij are odd, the existence of an infinite number of solutions is obtained. Moreover, we study the existence of solutions generated by impulses. Under suitable hypothesis, existence criteria of at least one solution and infinitely many solutions generated by impulses of problem (P) are established. The main tools are the Mountain Pass theorem, symmetric Mountain Pass theorem, and the least action principle.Note that whenAj=0, the impulsive conditions (1.10) and (1.11) of problem (P) degenerate to the impulsive condition (1.2), and when β=σ=0, (1.12) is Dirichlet boundary condition; however, our assumptions on f are different from the assumptions on f in [12]. The construction of a suitable space of functions and corresponding energy functional become more complicated with the impulse effects of (1.10), (1.11), and Sturm-Liouville boundary conditions taken into consideration.The rest of this paper is organized as follows. In Section2, we list some preliminary results and several lemmas, which are important in proving the existence of solutions. In Section 3, we state and prove the main results and give some examples to illustrate them.


## 2. Preliminary

We define the space:(2.1)X={u∈L2(0,T)∣u∣(tj,tj+1)∈H1(tj,tj+1),j=0,1,…,p,-Δu(tj)=Aju(tj-),j=1,…,p,u(0)=0ifβ=0,u(T)=0ifσ=0}.
Its norm is induced by the inner product:
(2.2)(u,v)=∫0T[u′(t)v′(t)+u(t)v(t)]dt,u,v∈X,
that is,
(2.3)‖u‖=(∫0T[(u′(t))2+u2(t)]dt)1/2,u∈X.Ifu∈X, then u∣(tj,tj+1)∈H1(tj,tj+1), and thus the limits u(tj+) and u(tj+1-) exist, we define
(2.4)uj(t)={u(t),t∈(tj,tj+1),u(tj+),t=tj,u(tj+1-),t=tj+1,
then uj∈C[tj,tj+1].Now we state some lemmas, which are needed in the proof of the main results.Lemma 2.1.
X is a Hilbert space.Proof.
Clearly,X is a linear space equipped with the inner product (2.2), so X is an inner space. Next, we will show that X is complete.
Suppose{un}⊂X is an arbitrary Cauchy sequence, from the definition of X, for j=0,1,…,p, {unj} is also a Cauchy sequence in H1(tj,tj+1). Then, there exists uj∈H1(tj,tj+1) with limn→∞unj=uj in H1(tj,tj+1); therefore, {unj} converges to uj in C[tj,tj+1], that is, ∥unj-uj∥C[tj,tj+1]→0 as n→∞. So we have
(2.5)uj(tj)=lim⁡n→∞unj(tj)=lim⁡n→∞un(tj+)=lim⁡n→∞(1-Aj)un(tj-)=lim⁡n→∞(1-Aj)unj-1(tj)=(1-Aj)uj-1(tj),
that is,
(2.6)(1-Aj)uj-1(tj)=uj(tj).
Set
(2.7)u(t)={uj(t),t∈(tj,tj+1),j=0,…,p,u0(0+),t=0,up(T-),t=T.
Then we have
(2.8)(1-Aj)u(tj-)=(1-Aj)uj-1(tj)=uj(tj)=u(tj+).
So -Δu(tj)=Aju(tj-) and u∈X.
Observe that(2.9)‖un-u‖2=∑j=0p∫tjtj+1|(unj)′(t)-(uj)′(t)|2+|unj(t)-uj(t)|2dt=∑j=0p‖unj-uj‖H1(tj,tj+1)2,
since {unj} converge to uj in H1(tj,tj+1), then the Cauchy sequence {un} converge to u in X, that is, X is complete. So X is a Hilbert space.Foru∈X, we denote
(2.10)‖u‖∞=max⁡j∈{0,1,…,p}{max⁡t∈[tj,tj+1]|uj(t)|},
then we can get the following result.Lemma 2.2.
Letc=maxj∈{0,1,…,p}{(tj+1-tj)-1/2,(tj+1-tj)1/2}, then
(2.11)‖u‖∞≤c‖u‖,
for every u∈X.Proof.
Letu∈X, by mean value theorem, there exists τ∈[tj,tj+1], such that
(2.12)uj(τ)=1tj+1-tj∫tjtj+1uj(θ)dθ.
For t∈[tj,tj+1], from Hölder inequality, we have
(2.13)|uj(t)|=|uj(τ)+∫τt(uj)′(θ)dθ|≤1tj+1-tj∫tjtj+1|uj(θ)|dθ+∫tjtj+1|(uj)′(θ)|dθ≤(tj+1-tj)-1/2(∫tjtj+1|uj(θ)|2dθ)1/2+(tj+1-tj)1/2(∫tjtj+1|(uj)′(θ)|2dθ)1/2≤max⁡j∈{0,1,…,p}{(tj+1-tj)-1/2,(tj+1-tj)1/2}(∫tjtj+1(|uj(θ)|2+|(uj)′(θ)|2)dθ)1/2=c‖uj‖H1(tj,tj+1)≤c‖u‖.Definition 2.3.
A functionu∈L2(0,T) is said to be a classical solution of problem (P), if u|(tj,tj+1)∈C2(tj,tj+1) for j=0,1,2,…,p, and u satisfies (1.9) on [0,T]∖{t1,t2,…,tp}, the limits u′(tj+),u′(tj-),u(tj+),u(tj-) exist, and (1.10), (1.11), and (1.12) hold.For convenience, denote(2.14)α1={αβ,β≠0,0,β=0,α2={γσ,σ≠0,0,σ=0,F(t,u)=∫0uf(t,ξ)dξ.
For u∈X, set
(2.15)φ(u)=12∫0T[(u′(t))2+u2(t)]dt+12α1u2(0)+12α2u2(T)-∫0TF(t,u(t))dt-∑j=1p∫0u(tj-)Ij(s)ds.
From the assumption of the continuity of f and Ij(j=1,2,…,p), we can see that φ∈C1(X,ℝ) and
(2.16)(φ′(u),v)=∫0T[u′(t)v′(t)+u(t)v(t)]dt+α1u(0)v(0)+α2u(T)v(T)-∫0Tf(t,u(t))v(t)dt-∑j=1pIj(u(tj-))v(tj-),
for u,v∈X.Lemma 2.4.
If the functionu∈X is a critical point of φ, then u is a classical solution of problem (P).Proof.
Suppose thatu∈X is a critical point of φ, then (φ′(u),v)=0 for every v∈X. By (2.16), we have
(2.17)∫0T[u′v′+uv]dt+α1u(0)v(0)+α2u(T)v(T)-∫0Tf(t,u)vdt-∑j=1pIj(u(tj-))v(tj-)=0.
For j∈{0,1,2,…,p}, choose v=0 on [0,T]∖(tj,tj+1) and v|(tj,tj+1)∈C0∞(tj,tj+1), then v∈X and
(2.18)∫tjtj+1u′v′dt=∫tjtj+1(f(t,u)-u)vdt.
By the definition ofH1(tj,tj+1) and since u|(tj,tj+1)∈H1(tj,tj+1)⊂C[tj,tj+1],f∈C([0,T]×ℝ,ℝ), (2.18) gives us ((u|(tj,tj+1))′)′∈C[tj,tj+1], thus (u|(tj,tj+1))′∈C1[tj,tj+1] and u|(tj,tj+1)∈C2[tj,tj+1], so the limits u′(tj+),u′(tj-) exist, then integrating (2.18) by parts, we obtain
(2.19)∫tjtj+1(-u′′+u-f(t,u))vdt=0,v∈X,v|(tj,tj+1)∈C0∞(tj,tj+1).
So
(2.20)-u′′+u-f(t,u)=0,t∈(tj,tj+1).
Thus, we can get that u satisfies (1.9) for t∈(0,T)∖{t1,…,tp}.
In view of (2.17), (2.19), and -Δu(tj)=Aju(tj-), we get that
(2.21)0=∑j=0p∫tjtj+1u′v′dt+∫0T(u-f(t,u))vdt+α1u(0)v(0)+α2u(T)v(T)-∑j=1pIj(u(tj-))v(tj-)=∑j=0p(u′(tj+1-)v(tj+1-)-u′(tj+)v(tj+))+∫0T(-u′′+u-f(t,u))vdt+α1u(0)v(0)+α2u(T)v(T)-∑j=1pIj(u(tj-))v(tj-)=∑j=1p(u′(tj-)v(tj-)-u′(tj+)v(tj-)+u′(tj+)v(tj-)-u′(tj+)v(tj+))+u′(T)v(T)-u′(0)v(0)+α1u(0)v(0)+α2u(T)v(T)-∑j=1pIj(u(tj-))v(tj-)=-∑j=1p(Δu′(tj)+Ij(u(tj-))-Aju′(tj+))v(tj-)+(α1u(0)-u′(0))v(0)+(α2u(T)+u′(T))v(T).
If we choose v∈X, such that v(0)=v(T)=0, then it follows from the above equality that
(2.22)-Δu′(tj)=Ij(u(tj-))-Aju′(tj+),
and the impulsive condition (1.11) holds.
Ifβ=σ=0, then from the definition of X, we know that u(0)=u(T)=0, and the boundary condition (1.12) holds. If β≠0, σ≠0, in view of (2.21) and (2.22), we can conclude that
(2.23)(α1u(0)-u′(0))v(0)+(α2u(T)+u′(T))v(T)=0,v∈X.
Since v(0) and v(T) are arbitrary, we deduce that
(2.24)α1u(0)-u′(0)=0,α2u(T)+u′(T)=0,
and u satisfies the boundary condition (1.12). If one of β and σ is equal to 0, similarly we can obtain the conclusion.
Therefore,u is a classical solution of problem (P).Lemma 2.5 (Mountain Pass theorem [10, Theorem 2.2]).
LetE be a real Banach space and φ∈C1(E,ℝ) satisfying the (PS) condition. Suppose φ(0)=0 and(C1)
there are constantsρ,α>0 such that φ|∂Bρ≥α, where Bρ={x∈E:∥x∥<ρ};(C2)
there is ane∈E∖Bρ¯ such that φ(e)≤0.Then φ possesses a critical value c≥α. Moreover c can be characterized as
(2.25)c=inf⁡g∈Γmax⁡u∈g([0,1])φ(u),
where
(2.26)Γ={g∈C([0,1],E)∣g(0)=0,g(1)=e}.Lemma 2.6 (see [10, Theorem 9.12]).
LetE be an infinite dimensional Banach space, and let φ∈C1(E,ℝ) be even functional which satisfies the (PS) condition, and φ(0)=0. If E=V⊕X, where V is finite dimensional, and φ satisfies(C1′)
there are constantsρ,α>0 such that φ|∂Bρ⋂X≥α, where Bρ={x∈E:∥x∥<ρ};(C2′)
for each finite dimensional subspaceẼ⊂E, there is an R=R(Ẽ) such that φ≤0 on Ẽ∖BR(Ẽ).Then φ possesses an unbounded sequence of critical values.


## 3. Main Results

Lemma 3.1.
If there existμ>2 and δ>0 such that
(3.1)0<μF(t,x)≤xf(t,x),fort∈[0,T],|x|≥δ,(3.2)0<μ∫0xIj(s)ds≤xIj(x),for|x|≥δ,j=1,2,…,p,
then the functional φ satisfies the (PS) condition.Proof.
Let{un} be a sequence in X satisfying that {φ(un)} is bounded and φ′(un)→0 as n→∞. Now we shall prove that {un} is a bounded sequence in X. By (2.15), (2.16), (3.1), and (3.2), we have
(3.3)μφ(un)-(φ′(un),un)=μ2∫0T[(un′)2+(un)2]dt+μα12(un(0))2+μα22(un(T))2-μ∫0TF(t,un)dt-μ∑j=1p∫0un(tj-)Ij(s)ds-∫0T[(un′)2+(un)2]dt-α1(un(0))2-α2(un(T))2+∫0Tf(t,un)undt+∑j=1pIj(un(tj-))un(tj-)≥(μ2-1)(‖un‖2+α1(un(0))2+α2(un(T))2)+Tmin⁡t∈[0,T],|x|≤δ(f(t,x)x-μF(t,x))+∑j=1pmin⁡|x|≤δ(Ij(x)x-μ∫0xIj(s)ds)≥(μ2-1)‖un‖2+M,
where M=Tmint∈[0,T],|x|≤δ(f(t,x)x-μF(t,x))+∑j=1pmin|x|≤δ(Ij(x)x-μ∫0xIj(s)ds) is a constant. So {un} is bounded in X.
From the reflexivity ofX, we may extract a weakly convergent subsequence that, for simplicity, we call {un}, un⇀u. In what follows, we will prove that {un} strongly converges to u. By un⇀u, we see that, for j=0,1,…,p, unj⇀uj in H1(tj,tj+1), and {unj} converges to uj in C[tj,tj+1]. From the continuity of f and Ij(j=1,2,…,p), we know that
(3.4)∫0T(f(t,un)-f(t,u))(un-u)dt⟶0,asn⟶∞,∑j=1p(Ij(un(tj-))-Ij(u(tj-)))(un(tj-)-u(tj-))⟶0,asn⟶∞.
In view of (2.16), we get
(3.5)(φ′(un)-φ′(u),un-u)=‖un-u‖2+α1(un(0)-u(0))2+α2(un(T)-u(T))2+∫0T(f(t,un)-f(t,u))(un-u)dt+∑j=1p(Ij(un(tj-))-Ij(u(tj-)))(un(tj-)-u(tj-)).
By φ′(un)→0 and un⇀u, we obtain that
(3.6)(φ′(un)-φ′(u),un-u)⟶0,asn⟶∞.
From the previous discussions, it is easy to see that ∥un-u∥→0 as n→∞. Therefore, φ satisfies the (PS) condition.Theorem 3.2.
Suppose that (3.1), (3.2) hold and
(3.7)f(t,x)=o(x),x⟶0,t∈[0,T],(3.8)Ij(x)=o(x),x⟶0,j=1,2,…,p,
then problem (P) has at least one nontrivial solution.Proof.
In order to show that problem (P) has at least one nontrivial solution, that is, φ has at least one nontrivial critical point, it is sufficient to check the conditions in Lemma 2.5. Clearly φ∈C1(X,R), φ(0)=0, and φ satisfies the (PS) condition from Lemma 3.1.
First, it follows from (3.7) and (3.8) that there exists a constant ɛ>0, such that for |x|≤ɛ:
(3.9)|F(t,x)|≤18c2Tx2,∫0xIj(s)ds≤18c2px2,j=1,2,…,p,
where c is given in Lemma 2.2. Choose u∈X with ∥u∥≤ɛ/c, then ∥u∥∞≤ɛ, and by (3.9) and (2.11), we have
(3.10)φ(u)=12‖u‖2+12α1u2(0)+12α2u2(T)-∫0TF(t,u(t))dt-∑j=1p∫0u(tj-)Ij(s)ds≥12‖u‖2-∫0T18c2Tu2(t)dt-∑j=1p18c2p(u(tj-))2≥12‖u‖2-∫0T18c2T‖u‖∞2dt-∑j=1p18c2p‖u‖∞2=12‖u‖2-14c2‖u‖∞2≥12‖u‖2-14‖u‖2=14‖u‖2.
If we choose α=ɛ2/4c2, ρ=ɛ/c, then φ|∂Bρ≥α. Thus, condition (C1) of Lemma 2.5 holds.
By (3.1) and (3.2), we know that there are positive constants c1,c2,d1,andd2 such that
(3.11)F(t,x)≥c1|x|μ-c2,(t,x)∈[0,T]×R,(3.12)∫0xIj(s)ds≥d1|x|μ-d2,x∈R,j=1,2,…,p.
If we choose u0∈X∖{0}, then for any r>0, we have
(3.13)φ(ru0)=12‖ru0‖2+12α1(ru0(0))2+12α2(ru0(T))2-∫0TF(t,ru0)dt-∑j=1p∫0ru0(tj-)Ij(s)ds≤12‖ru0‖2+r22(α1(u0(0))2+α2(u0(T))2)-∫0T(c1|ru0|μ-c2)dt-∑j=1pd1|ru0(tj-)|μ+pd2=r22(‖u0‖2+α1(u0(0))2+α2(u0(T))2)-(c1∫0T|u0|μdt+d1∑j=1p|u0(tj-)|μ)rμ+c2T+pd2.
Since μ>2, we can find r>1 such that ∥ru0∥>ρ and φ(ru0)<0. Hence the condition (C2) of Lemma 2.5 is satisfied.Theorem 3.3.
Under the same assumptions as Theorem3.2, if f(t,x) is odd in x, and Ij(j=1,2,…,p) are odd, then problem (P) has infinitely many solutions.Proof.
The proof follows the analogous ideas as that we have developed for Theorem3.2. By the assumption we know that φ(u) is even, φ∈C1(X,R),φ(0)=0, and φ satisfies the (PS) condition from Lemma 3.1.
If we letV be any finite dimensional subspace of X, and W=V⊥, then for any u∈W with ∥u∥≤ɛ/c, it follows from (3.10) that
(3.14)φ(u)≥α,foru∈∂Bρ∩W,
where α=ɛ2/4c2,ρ=ɛ/c. So φ satisfies the condition (C1′). On the other hand, fix u0∈V∖{0}, the same as (3.13), we can find some r>1 such that
(3.15)‖ru0‖>ρ,φ(ru0)<0,
then (C2′) of Lemma 2.6 is true. Thus, φ possesses an infinite number of critical points, that is, problem (P) has infinitely many solutions.WhenAj≡0 and Ij≡0 for j=1,2,…,p, problem (P) degenerates to the following problem (P0):
(3.16)-u′′(t)+u(t)=f(t,u(t)),t∈[0,T],αu(0)-βu′(0)=0,γu(T)+σu′(T)=0.
Now, we show the nonexistence of solutions for problem (P0).Lemma 3.4.
Assume thatf satisfies
(3.17)xf(t,x)≤0,(t,x)∈[0,T]×R.
Then, (P0) has no nontrivial solution.Proof.
Ifu is a nontrivial solution of (P0), then we have
(3.18)0=-∫0Tu′′udt+∫0Tu2dt-∫0Tf(t,u)udt=∫0T[(u′)2+u2]dt+α1u2(0)+α2u2(T)-∫0Tf(t,u)udt=‖u‖2+α1u2(0)+α2u2(T)-∫0Tf(t,u)udt≥0.
Since α1≥0,α2≥0 and (3.17), we obtain u≡0.Remark 3.5.
By Lemma3.4, if we get a solution of problem (P) under the assumption of (3.17), then the solution is generated by impulses.
The next theorem gives some sufficient conditions that problem (P) has at least one solution and infinitely many solutions generated by impulses under the assumption of (3.17), compared with Theorem 3.2, we change the conditions on nonlinearity and make certain requirements about Aj,j=1,2,…,p.Theorem 3.6.
Suppose0≤Aj≤1 for j=1,2,…,p, (3.2), (3.8), and (3.17) hold, and there exist a>0 and η∈(0,1) such that
(3.19)f(t,x)≥-a|x|η,(t,x)∈[0,T]×R.
Then problem (P) has at least one nontrivial solution generated by impulses. Furthermore, if f(t,x) is odd in x, and Ij(j=1,2,…,p) are odd, then problem (P) has infinitely many solutions generated by impulses.Proof.
Firstly, we show thatφ satisfies the (PS) condition. Let {un} be a sequence in X satisfying that {φ(un)} is bounded and φ′(un)→0 as n→∞. Now, we shall prove that {un} is a bounded sequence in X.
Equation (3.17) implies that
(3.20)F(t,x)≤0,(t,x)∈[0,T]×R.
By (3.2), (3.19), and (3.20), we have
(3.21)μφ(un)-(φ′(un),un)=(μ2-1)∫0T[(un′)2+(un)2]dt+(μ2-1)α1(un(0))2+(μ2-1)α2(un(T))2-μ∫0TF(t,un)dt+∫0Tf(t,un)undt-μ∑j=1p∫0un(tj-)Ij(s)ds+∑j=1pIj(un(tj-))un(tj-)≥(μ2-1)‖un‖2+∫0Tf(t,un)undt-μ∑j=1p∫0un(tj-)Ij(s)ds+∑j=1pIj(un(tj-))un(tj-)≥(μ2-1)‖un‖2-aT‖un‖∞η+1+∑j=1pmin⁡|x|≤δ(Ij(x)x-μ∫0xIj(s)ds)≥(μ2-1)‖un‖2-aTcη+1‖un‖η+1+M1,
where M1=∑j=1pmin|x|≤δ(Ij(x)x-μ∫0xIj(s)ds) is a constant. So {un} is bounded in X, since η<1,μ>2. Similar to the proof in Lemma 3.1, we can see that {un} has a convergent subsequence in X. So, φ satisfies the (PS) condition.
From (3.8), there exists a constant ɛ1>0 such that for j=1,2,…,p, we have
(3.22)∫0xIj(s)ds≤14c2px2,|x|≤ɛ1,
where c is given in Lemma 2.2. Hence combining this with (3.20), for u∈X with ∥u∥≤ɛ1/c, we have
(3.23)φ(u)=12‖u‖2+12α1u2(0)+12α2u2(T)-∫0TF(t,u(t))dt-∑j=1p∫0u(tj-)Ij(s)ds≥12‖u‖2-∑j=1p∫0u(tj-)Ij(s)ds≥12‖u‖2-∑j=1p14c2pu2(tj-)≥12‖u‖2-14c2‖u‖∞2≥12‖u‖2-14‖u‖2=14‖u‖2.
If we choose α=ɛ12/4c2,ρ=ɛ1/c, then φ|∂Bρ≥α. Thus, condition (C1) in Lemma 2.5 holds.
For some constantk>0, let
(3.24)u¯(t)={kt,t∈[0,t1),kt1∏i=1j(1-Ai),t∈(tj,tj+1),j=1,…,p-1,kt1∏i=1p(1-Ai)tp-T(t-T),t∈(tp,T].
Clearly, u¯∈X and u¯(t)≥0 for t∈[0,T]. Then for any r>0, by (3.19), we have
(3.25)F(t,ru¯(t))=∫0ru¯(t)f(t,s)ds≥-∫0ru¯(t)a|s|ηds=-aη+1|ru¯(t)|η+1≥-aη+1‖ru¯‖∞η+1.
In view of (3.25) and (3.12), one can get
(3.26)φ(ru¯)=12‖ru¯‖2+12α1(ru¯(0))2+12α2(ru¯(T))2-∫0TF(t,ru¯)dt-∑j=1p∫0ru¯(tj-)Ij(s)ds≤r22(‖u¯‖2+α1(u¯(0))2+α2(u¯(T))2)+aTcη+1η+1‖u¯‖η+1rη+1-∑j=1pd1|u¯(tj-)|μrμ+pd2.
Since μ>2,η<1, we can find r>1 such that ∥ru¯∥>ρ and φ(ru¯)<0. Hence, the condition (C2) in Lemma 2.5 is satisfied.
Finally, by Remark3.5 and Lemma 2.5, the problem (P) has at least one nontrivial solution generated by impulses. Analogous to the proof of Theorem 3.3, we know that when f(t,x) is odd about x, and Ij(j=1,2,…,p) are odd, problem (P) has an infinite number of solutions which are generated by impulses.Theorem 3.7.
Suppose (3.17) holds, 0<Aj<1 for j=1,2,…,p, and there exist positive constants a0,aj,ãj,bj and η0,ηj,νj which satisfy ηj∈(0,1) and 0<νj<min{η0,1} such that(i)
Ij(x)≤aj|x|ηj+bj for x∈ℝ,j=1,2,…,p;(ii)
Ij(x)≥ãj|x|νj for |x|≤1,x≠0, and f(t,x)≥-a0|x|η0 for |x|≤1,t∈[0,T]. Then problem (P) has at least one nontrivial solution generated by impulses.Proof.
Firstly, we show thatφ is weakly lower semi continuous. Let {un} be a sequence which weakly converges to some u in X, then
(3.27)‖u‖≤lim⁡inf⁡n→∞‖un‖.
On the other hand, we have that {un|(tj,tj+1)} weakly converges to u|(tj,tj+1) in H1(tj,tj+1), then {un|(tj,tj+1)} uniformly converges to u|(tj,tj+1) on [tj,tj+1], then
(3.28)lim⁡n→∞(12α1un2(0)+12α2un2(T)-∫0TF(t,un(t))dt-∑j=1p∫0un(tj-)Ij(s)ds)=lim⁡n→∞(12α1un2(0)+12α2un2(T)-∑j=0p∫tjtj+1F(t,un(t))dt-∑j=1p∫0un(tj-)Ij(s)ds)=12α1u2(0)+12α2u2(T)-∑j=0p∫tjtj+1F(t,u(t))dt-∑j=1p∫0u(tj-)Ij(s)ds=12α1u2(0)+12α2u2(T)-∫0TF(t,u(t))dt-∑j=1p∫0u(tj-)Ij(s)ds.
From (3.27) and (3.28), we conclude that φ(u)≤liminfn→∞φ(un).
Next, we verify thatφ is coercive.
From (3.17), the hypothesis (i) and Lemma 2.2, we have
(3.29)φ(u)=12‖u‖2+12α1u2(0)+12α2u2(T)-∫0TF(t,u(t))dt-∑j=1p∫0u(tj-)Ij(s)ds≥12‖u‖2-∑j=1p∫0u(tj-)Ij(s)ds≥12‖u‖2-∑j=1p∫0‖u‖∞(aj|x|ηj+bj)dx≥12‖u‖2-∑j=1p(ajηj+1‖u‖∞ηj+1+bj‖u‖∞)≥12‖u‖2-∑j=1p(ajcηj+1ηj+1‖u‖ηj+1+bjc‖u‖),
which implies that φ is coercive since ηj∈(0,1) for j=1,2,…,p.
Therefore by the least action principle [27, Theorem 1.1], we obtain that c′=infXφ(u) is a critical value of φ, this means that there exists ũ∈X such that φ(ũ)=c′.
Finally, we show thatũ is nontrivial.
In (3.24), if we choose positive constant k such that k<1/t1, then u¯(t)∈[0,1] for t∈[0,1], and for any r∈(0,1), by the assumption (ii) and with same calculation of (3.25), we have
(3.30)F(t,ru¯)≥-a0η0+1‖ru¯‖∞η0+1,∫0ru¯(tj-)Ij(s)ds≥ãjνj+1|ru¯(tj-)|νj+1.
Then
(3.31)φ(ru¯)=12‖ru¯‖2+12α1(ru¯(0))2+12α2(ru¯(T))2-∫0TF(t,ru¯)dt-∑j=1p∫0ru¯(tj-)Ij(s)ds≤r22(‖u¯‖2+α1(u¯(0))2+α2(u¯(T))2)+a0Tcη0+1η0+1‖u¯‖η0+1rη0+1-∑j=1pãjνj+1|u¯(tj-)|νj+1rνj+1.
For r>0 small enough, we have that φ(ru¯)<0 since 0<νj<min{η0,1}. Hence, φ(ũ)=infXφ(u)<0, and by Remark 3.5, ũ is a nontrivial solution generated by impulses.Example 3.8.
LetT=1,t1∈(0,1), consider the system:
(3.32)-u′′(t)+u(t)=f(t,u(t)),t∈[0,1]∖{t1},u(t1-)-u(t1+)=13u(t1-),u′(t1-)-u′(t1+)=u3(t1-)-13u′(t1+),u(0)-u′(0)=0,u(1)+u′(1)=0.
Compare with problem (P), A1=1/3,I1(u)=u3.
If we choosef(t,u)=up/q, where p and q are odd numbers and p/q>1, then the conditions of Theorem 3.3 are satisfied. So, the impulsive boundary value problem (3.32) has an infinite number of nontrivial solutions.Example 3.9.
Consider (3.32) with f(t,u) defined by
(3.33)f(t,u)={-ueu,u≤1,-e,u>1.
If we choose a=e,η=1/2, the assumptions of Theorem 3.6 are satisfied, then the impulsive boundary value problem (3.32) has at least one nontrivial solution generated by impulses.Example 3.10.
Consider the following problem:(3.34)-u′′(t)+u(t)=f(t,u(t)),t∈[0,1],αu(0)-βu′(0)=0,γu(1)+σu′(1)=0.
If we choose f(t,u)=-tum/n-up/q, where m,n,p,andq are odd numbers, then uf(t,u)≤0, from Lemma 3.4, we know that (3.34) has no nontrivial solution.Example 3.11.
LetT=1,t1∈(0,1), we consider the following problem:
(3.35)-u′′(t)+u(t)=-u1/9(t),t∈[0,1]∖{t1},u(t1-)-u(t1+)=13u(t1-),u′(t1-)-u′(t1+)=u1/3(t1-)-13u′(t1+),u(0)-u′(0)=0,u(1)+u′(1)=0,
here f(t,u)=-u1/9,A1=1/3,I1(u)=u1/3. In view of uf(t,u)=-u10/9≤0 and Lemma 3.4, we get that the above problem without impulsive, that is, the problem:
(3.36)-u′′(t)+u(t)=-u1/9(t),t∈[0,1],u(0)-u′(0)=0,u(1)+u′(1)=0
has no nontrivial solution. In addition, it is easy to verify that the hypotheses of Theorem 3.7 are fulfilled. Thus, via Theorem 3.7, the impulsive boundary value problem (3.35) has at least one nontrivial solutions generated by impulses.



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*Source: 707163-2012-06-06.xml*