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Nonlinear boundary value conditions and ordinary differential systems with impulsive effects
Boundary Value Problems volume 2017, Article number: 45 (2017)
Abstract
We investigate solutions to nonlinear operator equations which are difficult to investigate with variational methods and obtain some abstract existence results by topology degree methods. These results apply to ordinary differential systems with impulsive effects satisfying nonlinear boundary value conditions, and we obtain some new results.
1 Introduction
We are interested in the problem
where \(V\in C^{1}([0,1]\times \mathbf {R}^{n},\mathbf {R})\), \(V'\) denotes the gradient of V with respect to x and \(M_{0},M_{1}:\mathbf {R}^{4n}\to \mathbf {R}^{n}\). When \(M_{0}\equiv x_{0}\), \(M_{1}\equiv x_{1}\) are constants, Ekeland et al. [1] investigated the problem in 1996. Setting \(x=y+(1-t)x_{0}+tx_{1}\), then (1.1)-(1.3) is equivalent to the problem
and its solutions are the critical points of the functional
defined on some suitable function space. However, if one of \(M_{0}\) and \(M_{1}\) is not constant, (1.1)-(1.3) cannot be solved by variational methods generally. Note that the problem is equivalent to the integral equation
where \(G(t,s)=t(1-s)\) as \(0\leq t\leq s\leq1\) and \(G(t,s)=s(1-t)\) as \(0\leq s\leq t\leq1\), \(M_{i}=M_{i}(x(0),x(1),x'(0),x'(1))\) (\(i=0,1\)).
Let \(X=L^{2}([0,1],\mathbf {R}^{n})\), \(D(A)=H_{0}^{2}([0,1],\mathbf {R}^{n})=\{x\in H^{2}([0,1],\mathbf {R}^{n})\vert x(0)=0=x(1)\}\), \(A:D(A)\rightarrow L^{2}([0,1],\mathbf {R}^{n})\) by \((Ax)(t)=-\ddot{x}(t)\), \(N:C^{1}([0,1],\mathbf {R}^{n})\rightarrow L^{2}([0,1],\mathbf {R}^{n})\) by \((Nx)(t)=V'(t,x(t))\), \(Y=C^{1}([0,1],\mathbf {R}^{n})\), \(M:C^{1}([0,1],\mathbf {R}^{n})\to C^{1}([0,1],\mathbf {R}^{n})\) by \((Mx)(t)=(1-t)M_{0}(x(0),x'(0),x(1),x'(1))+tM_{1}(x(0),x'(0),x(1),x'(1))\). Then A is an unbounded self-adjoint invertible operator in X with \(\sigma(A)=\{k^{2}\pi^{2}\}_{i=1}^{\infty}=\sigma_{d}(A)\), and (1.1)-(1.3) turn to the following operator equation:
In this paper we also denote \(N(x)\) and \(M(x)\) by Nx and Mx, respectively, when there is no confusion. We will first investigate (1.5), and then as applications we investigate ordinary differential systems satisfying nonlinear boundary value conditions including (1.1)-(1.3). In particular, we will investigate differential systems with impulsive effects.
Let X be a real infinite-dimensional separable Hilbert space with norm \(\Vert \cdot \Vert \) and inner product \((\cdot, \cdot)\). Let \(A:D(A)\subset X\rightarrow X\) be an unbounded self-adjoint and invertible operator satisfying \(\sigma(A)=\sigma_{d}(A)\). Assume that Y is a Banach space with the norm \(\Vert \cdot \Vert _{Y}\) satisfying \(D(A)\subset Y\subset X\), the inclusion map from \(D(A)\) to Y is compact and the inclusion from Y to X is continuous. Assume \(N:Y\rightarrow X\) is continuous, \(M:Y\rightarrow Y\) is compact and satisfies \(\Vert M(x) \Vert _{Y}\leq\rho\) for all \(x\in Y \) and some \(\rho>0\).
We will also use the following assumptions:
- (N1):
-
There exists \(B:Y\rightarrow\mathcal{L}_{s}(X)\), \(B_{1},B_{2}\in \mathcal{L}_{s}(X) \) with \(i_{A}(B_{1})=i_{A}(B_{2})\), \(\nu_{A}(B_{2})=0\) and there is an \(\varepsilon>0\) such that \(B_{1}\leq B(x)\leq B_{2}\), \(B_{1}\geq \varepsilon \mathit{Id}\) and \(Nx=B(x)x+C(x)\), \(\Vert C(x) \Vert \leq \rho\) for all \(x\in Y\) and some \(\rho>0\).
- (N2):
-
There exists \(B_{0}:Y\rightarrow\mathcal {L}_{s}(X)\), \(B_{01},B_{02}\in\mathcal{L}_{s}(X) \) with \(i_{A}(B_{01})=i_{A}(B_{02})\), \(\nu_{A}(B_{02})=0\) and there is an \(\epsilon>0\) and some \(r>0\) such that \(B_{01}\geq\varepsilon \mathit{Id}\), \(B_{01}\leq B_{0}(x)\leq B_{02}\) and \(Nx=B_{0}(x)x\) for all \(x\in Y\) with \(\Vert x \Vert _{Y}\leq r\).
- (M):
-
\(M(x)=o( \Vert x \Vert _{Y})\) as \(\Vert x \Vert _{Y}\rightarrow0\).
Theorem 1.1
Assume N satisfies (N1). Then (1.5) has one solution. If further (N2) and (M) hold, then (1.5) has a nontrivial solution provided \(i_{A}(B_{01})-i_{A}(B_{1})\) is odd.
We will give the proof in the next section, and now we return to a discussion of the problem at the beginning of the paper. Let \(\vert \cdot \vert \) denote the usual norm in \(\mathbf {R}^{m}\) for positive integer m. We need the following assumptions:
- (V1):
-
There is a \(\bar{B}:[0,1]\times \mathbf {R}^{n} \rightarrow\mathcal {L}_{s}(\mathbf {R}^{n})\) with \(\bar{B}(\cdot,x(\cdot))\in L^{\infty }([0,1],\mathcal{L}_{s}(\mathbf {R}^{n}))\) for all \(x\in C([0,1],\mathbf {R}^{n})\) and there exists \(\bar{B}_{1}\), \(\bar{B}_{2} \in L^{\infty}([0,1],\mathcal {L}_{s}(\mathbf {R}^{n}))\) such that
$$V'(t,x)=\bar{B}(t,x)x+h(t,x),\qquad \bar{B}_{1}(t)\leq \bar{B}(t,x)\leq\bar{B}_{2}(t) $$for all \((t,x)\in[0,1]\times \mathbf {R}^{n}\), \(h:[0,1]\times \mathbf {R}^{n}\to \mathbf {R}^{n}\) is bounded.
- (V2):
-
There exists \(\bar{B}_{0}:[0,1]\times \mathbf {R}^{n}\to\mathcal {L}^{\infty}(\mathbf {R}^{n})\) with \(\bar{B}_{0}(\cdot,x(\cdot))\in L^{\infty }([0,1],\mathcal{L}_{s}(\mathbf {R}^{n}))\) for all \(x\in C([0,1],\mathbf {R}^{n})\) and there exists \(\bar{B}_{01},\bar{B}_{02} \in L^{\infty}([0,1],\mathcal {L}_{s}(\mathbf {R}^{n}))\) such that
$$V'(t,x)=\bar{B}_{0}(t,x)x,\qquad \bar{B}_{01}(t)\leq \bar{B}_{0}(t,x)\leq\bar {B}_{02}(t) $$for all \((t,x)\in[0,1]\times \mathbf {R}^{n}\) with \(\vert x \vert \leq r\) for some \(r>0\).
- (M1):
-
\(M_{i}(\xi)=o( \vert \xi \vert )\) as \(\vert \xi \vert \rightarrow0\), \(M_{i}\) (\(i=0,1\)) are continuous and bounded.
We will also use the index \((\nu_{0,\pi}^{s}(\bar{B}),i_{0,\pi }^{s}(\bar{B}))\) concerning the following systems:
where \(\bar{B}\in L^{\infty}([0,1],\mathcal{L}_{s}(\mathbf {R}^{n}))\).
Definition 1.1
See Definition A.4
For any \(\bar{B}\in L^{\infty }([0,1],\mathcal{L}_{s}(\mathbf {R}^{n}))\), we define
Note that from Definition 1.1 for \(c\in \mathbf {R}\), \(\nu_{0,\pi}^{s}(cI_{n})=0\) as \(c\neq k^{2}\pi^{2}\) and \(\nu_{0,\pi}^{s}(cI_{n})=n\) as \(c =k^{2}\pi^{2}\) for \(k=1,2,\ldots\) ; and \(i_{0,\pi}^{s}(cI_{n})=0\) as \(c\leq\pi^{2}\) and \(i_{0,\pi}^{s}(cI_{n})=kn\) as \(k^{2}\pi^{2}< c\leq(k+1)^{2}\pi^{2}\) for \(k=1,2,\ldots\) .
Theorem 1.2
If V satisfies (V1) with \(i_{0,\pi}^{s}(\bar {B}_{1})=i_{0,\pi}^{s}(\bar{B}_{2})\), \(\nu_{0,\pi}^{s}(\bar{B}_{2})=0\), then (1.1)-(1.3) has one solution. Furthermore, if (V2) and (M1) hold, then (1.1)-(1.3) has one nontrivial solution provided \(i_{0,\pi}^{s}(\bar{B}_{01})=i_{0,\pi}^{s}(\bar{B}_{02})\), \(\nu _{0,\pi}^{s}(\bar{B}_{02})=0\) and \(i_{0,\pi}^{s}(\bar{B}_{01})-i_{0,\pi }^{s}(\bar{B}_{1})\) is odd.
Proof
We only give the proof for the case that there exists \(\epsilon>0\) such that \(B_{1}\geq\epsilon I_{n}\), \(B_{01}\geq\epsilon I_{n}\). The complete proof will be given in Section 4 as a special case of a more general result. Let \(X=L^{2}([0,1],\mathbf {R}^{n})\), \(D(A)=\{x\in H^{2}([0,1],\mathbf {R}^{n})\vert x(0)=0=x(1)\}\) and \(Y=C^{1}([0,1],\mathbf {R}^{n})\). The inclusion maps \(D(A)\to Y\), \(Y\to X\) are compact and continuous, respectively. Define \(A:D(A)\rightarrow L^{2}([0,1],\mathbf {R}^{n})\) by \((Ax)(t)=-\ddot{x}(t)\), then A is invertible. Define \(N:Y\rightarrow X\) and \(M: Y\to Y\) by \((Nx)(t)=V'(t,x(t))\) and \((Mx)(t)=tM_{1}(x(0),x(1),x'(0),x'(1))+(1-t)M_{0}(x(0),x(1),x'(0),x'(1))\), respectively. Then (1.1)-(1.3) is equivalent to (1.4) or (1.5). Because \(M_{i}\) is bounded, there exists \(c>0\) such that \(\vert M_{i}(\xi ) \vert \leq c\) for all \(\xi\in \mathbf {R}^{4n}\), \(i=0,1\). Assume \(\{x_{j}\} \subset Y\) is bounded. Then \(\Vert Mx_{j} \Vert _{Y}\leq3c\), and \(\vert (Mx_{j})(t)-(Mx_{j})(s) \vert \leq2c \vert t-s \vert \), \(\vert (Mx_{j})'(t)-(Mx_{j})'(s) \vert =0\) for all \(t,s\in[0,1]\). By Ascoli-Arzela’s theorem, \(\{Mx_{j}\}\) has a convergent subsequence in Y. Moreover, \(M:Y\to Y\) is continuous via the continuity of \(M_{i}\) (\(i=0,1\)). So M is compact. Because assumptions (V1), (V2), (M1) imply (N1), (N2), (M), Theorem 1.2 follows Theorem 1.1 directly. □
Remark
-
1.
As in [2], p.69, if we assume \(V\in C^{2}([0,1]\times \mathbf {R}^{n})\) and \(\bar{B}_{1}(t)\leq V''(t,x)\leq\bar{B}_{2}(t)\) for all \((t,x)\in[0,1]\times \mathbf {R}^{n}\) with \(\vert x \vert \geq r>0\), then (V1) (with \(\bar{B}_{1}\), \(\bar{B}_{2}\) replaced by \(\bar {B}_{1}-\epsilon I_{n}\), \(\bar{B}_{2}+\epsilon I_{n}\) for small \(\epsilon>0\)) holds. In fact, for any \(\epsilon>0\), there exists \(\delta\in(0,1)\) such that
$$\begin{gathered} \bar{B}_{1}-\frac{1}{2}\epsilon I_{n}\leq(1-\delta)\bar{B}_{1}\leq (1-\delta) \bar{B}_{2}\leq\bar{B}_{2}+\frac{1}{2}\epsilon I_{n}, \\ \frac{1}{2}\epsilon I_{n}\leq \int_{0}^{\delta}V''(t, \theta x)\,d\theta \leq\frac{1}{2}\epsilon I_{n}. \end{gathered} $$Set
$$\begin{aligned} \bar{B}(t,x)&= \int_{0}^{1}V''(t,\theta x)\,d\theta,\quad \vert x \vert \geq r\delta^{-1} \\ &=\bar{B}_{2}(t),\quad \vert x \vert \leq r\delta^{-1}. \end{aligned}$$It follows that
$$\begin{aligned}& \bar{B}_{1}(t)-\epsilon I_{n}\leq\bar{B}(t,x)\leq \bar{B}_{2}(t)+\epsilon I_{n} \end{aligned}$$for all \((t,x)\in[0,1]\times \mathbf {R}^{n}\). And \(h(t,x)= V'(t,x)-\bar {B}(t,x)=V'(t,0)\) (as \(\vert x \vert >r\delta^{-1}\)) is bounded. If \(i_{0,\pi}^{s}(\bar{B}_{1})=i_{0,\pi}^{s}(\bar{B}_{2})\), \(\nu_{0,\pi }^{s}(\bar{B}_{2})=0\), then there exists \(\epsilon>0\) such that \(i_{0,\pi }^{s}(\bar{B}_{1}-\epsilon I_{n})=i_{0,\pi}^{s}(\bar{B}_{2}+\epsilon I_{n})\), \(\nu _{0,\pi}^{s}(\bar{B}_{2}+\epsilon I_{n})=0\) via Proposition A.2(ii).
-
2.
In (1.1) and (\(V_{1}-V_{2}\)) if we replace \(V'(t,x)\) by \(F\in C([0,1]\times \mathbf {R}^{n},\mathbf {R}^{n})\), the results in Theorem 1.2 are also valid.
-
3.
Condition (N1) is called the asymptotically linear condition; concerning other conditions like superlinear or sublinear conditions for operator equations we refer to [3].
The proof of Theorem 1.1 will be given in Section 2 and in Sections 3-6 we will investigate its other applications. Especially we will investigate differential systems with impulsive effects [4–16], which is not easy to investigate by variational methods. In the Appendix we recall some useful results concerning the index theory for linear self-adjoint operator equations in [2] which will be used in other sections.
2 Proof of Theorem 1.1
In this section we will prove Theorem 1.1. We need two lemmas about the Leray-Schauder degree. Suppose X is a Banach space and \(\Omega\subset X\) is a bounded open set. \(T:\overline{\Omega}\rightarrow X\) is compact and \(x-Tx\) is not zero for all \(x\in\partial\Omega\), so the Leray-Schauder degree \(\deg(\mathit{Id}-T,\Omega)\in \mathbf {Z}\) is defined. We have the following well-known lemmas.
Lemma 2.1
-
(i)
If \(\deg(\mathit{Id}-T,\Omega)\) is not zero, then there exists \(x\in \Omega\) such that \(x-Tx=0\),
-
(ii)
If K is linear compact, \(\ker(\mathit{Id}-K)={0}\) and \(0\in\Omega\), then \(\deg(\mathit{Id}-K,\Omega)\neq0\),
-
(iii)
\(\deg(\mathit{Id}-T_{\lambda},\Omega)\) is constant for \(\lambda\in [0,1]\) provided \(x-T_{\lambda}x\) is not zero for all \(x\in\partial \Omega\) and \(T_{\lambda}x=(1-\lambda)T_{0}x+\lambda T_{1}x\) and \(T_{0},T_{1}:\overline{\Omega}\to X\) are compact.
Lemma 2.2
Assume \(K:X\rightarrow X\) is a linear compact operator, \(1\notin\sigma (K)\) the spectral of K. Let Ω be an open bounded subset of X with \(0\in\Omega\). Then \(\deg(\mathit{Id}-K,\Omega)=(-1)^{\beta}\) where \(\beta=\sum_{\lambda _{j}>1,\lambda_{j}\in\sigma(K)}\beta_{j}\) and \(\beta_{j}=\dim\ker \bigcup_{m=1}^{\infty}(K-\lambda_{j})^{m}\).
Proof of Theorem 1.1
Since (N1) holds, \(A^{-1}N+M\) is a compact operator on Y. Now we want to prove \(\deg (\mathit{Id}-(A^{-1}N+M),U_{R})\neq0\) for some open ball \(U_{R}\) in Y with center 0 and radius \(R>0\). It suffices to show that the possible solutions of the following equations are a priori bounded for \(\lambda\in(0,1)\) with respect to the norm \(\Vert \cdot \Vert _{Y}\):
If not, there exist \(\{x_{j}\}_{j=1}^{\infty}\subset Y\) with \(\Vert x_{j} \Vert _{Y}\to+\infty\), and \(\{\lambda_{j}\}_{j=1}^{\infty}\subset(0,1)\) such that
Set \(y_{j}=x_{j}/ \Vert x_{j} \Vert _{Y}\). Then (2.2) turns to
Because \(\Vert y_{j} \Vert _{Y}=1\), \(\{y_{j}\}\) is bounded in X. We may assume \(y_{j}\rightharpoonup y_{0}\) in Y and \(y_{j}\to y_{0}\) in X for some \(y_{0}\in Y\) by going to subsequences if necessary. Further we claim
in X for any given \(y\in X\) and some \(D_{1}\in{\mathcal {L}}_{s}(X)\). In fact, by (N1), \(\{ \Vert B({x_{j}}) \Vert \}\) is bounded, so it follows that
in X. Because X is separable, there exists a countably orthonormal basis \(\{e_{j}\}_{j=1}^{\infty}\). Since \(\{B(x_{j})e_{1}\}\) is bounded in X, we have \(B(x_{j_{1}(i)})e_{1}\rightharpoonup\xi_{1}\) in X, where \(j_{1}(i)\) is a subsequence of the positive integer sequence. Now \(\{ B(x_{j_{1}(i)})e_{2}\}\) is also bounded, again there exist a subsequence \(j_{2}(i)\) of \(j_{1}(i)\) and \(\xi_{2}\in X\) such that \(B(x_{j_{2}(i)})e_{2}\rightharpoonup\xi_{2}\). Repeating this process and using the standard diagonal process, there exists a subsequence \(j_{k}=j_{k}(k)\) such that \(B(x_{j_{k}})e_{l}\rightharpoonup\xi_{l}\) for any given l. Define a linear operator \(D_{1}\) on X by \(D_{1}e_{j}=\xi_{j}\). Then \(B(x_{j_{k}})x\rightharpoonup D_{1}x\) in X for any given \(x\in X\). So (2.4) holds. By assumptions, \(A^{-1}: X\to X\) is compact, thus \(A^{-1}B(x_{j})y_{j}\to A^{-1}D_{1}y_{0}\), \(A^{-1}B_{2}y_{j}\to A^{-1}B_{2}y_{0}\) in X via (2.4) and (2.5). By (N1), \(\frac{A^{-}C(x_{j})}{ \Vert x_{j} \Vert _{Y}}+\frac{M(x_{j})}{ \Vert x_{j} \Vert _{Y}} \rightarrow0\) in X. And from (2.3),
for any \(y\in X\). Further we assume \(\lambda_{j}\rightarrow\lambda_{0}\). Taking the limit in (2.6) and considering (2.4) and (2.5) yield
for all \(y\in X\) and
where \(B_{3}=\lambda_{0}D_{1}+(1-\lambda_{0})B_{2}\) satisfying \(B_{1}\leq B_{3}\leq B_{2}\). By Proposition A.1(ii), \(\nu_{A}(B_{3})=0\). By the above argument and (2.3), \(\{y_{j}\}\) is convergent in Y by going to subsequence if necessary. So \(\Vert y_{0} \Vert _{Y}=1\) and \(y=y_{0}\) is a nontrivial solution of \(Ay-B_{3}y=0\), a contradiction. Thus, there is \(R>0\) such that as \(\Vert x \Vert _{Y}\geq R\), \(x-\lambda(A^{-1}N(x)+M(x))-(1-\lambda )A^{-1}B_{2}x\neq0\) for all \(\lambda\in(0,1)\). So \(\deg(\mathit{Id}-T_{\lambda},U_{R})\) is well defined where \(T_{\lambda }=\lambda(A^{-1}N(x)+M(x))+(1-\lambda)A^{-1}B_{2}x\). By Lemma 2.1(ii)-(iii), \(\deg(\mathit{Id}-T_{1},U_{R})=\deg(\mathit{Id}-T_{0},U_{R})\neq0\) because of \(0\in U_{R}\) and \(\ker\{\mathit{Id}-A^{-1}B_{2}\}=\{0\}\) since \(\nu_{A}(B_{2})\). Hence, (1.5) has one solution.
Further assume (N2) and (M) hold. To obtain a nontrivial solution of (1.5), we claim that the following problem:
has no solution x satisfying \(0< \Vert x \Vert _{Y}\leq r\).
If not, there exist \(\{x_{k}\}_{k=1}^{\infty}\subset Y\) such that \(\Vert x_{k} \Vert _{Y}\rightarrow0\) and \(\{\lambda_{k}\} _{k=1}^{\infty}\subset(0,1)\) such that
We have
where \(\tilde{B}_{k}=\lambda_{k} B_{0}(x_{k})+(1-\lambda_{k})B_{01}\). Set \(y_{k}=\frac{x_{k}}{ \Vert x_{k} \Vert _{Y}}\). Then \(\Vert y_{k} \Vert _{Y}=1\), \(y_{k}\rightharpoonup y_{0}\) in X and (2.7) turns to
By (M), \(\frac{M(x_{k})}{ \Vert x_{k} \Vert }\rightarrow0\); and as before there exists a \(D_{0}\in\mathcal{L}_{s}(X)\) satisfying \(B_{01}\leq D_{0}\leq B_{02}\) such that \(A^{-1}\tilde {B}_{k}(y_{k})\rightarrow A^{-1}D_{0}y_{0}\) in Y. Taking the limit in (2.8) yields
where \(B_{01}\leq D_{0}\leq B_{02}\), so \(\nu_{A}(D_{0})=0\). As above we have \(\Vert y_{0} \Vert _{Y}=1\), \(y=y_{0}\) is a nontrivial solution of \(Ay_{0}-D_{0}y_{0}=0\), a contradiction. Now we prove
By Proposition A.1, setting \(K=A^{-1}B_{01}\) yields
By Lemma 2.2, in order to prove (2.9) we need only to show that \(\ker (K-\lambda)=\ker(K-\lambda)^{2}\). In fact, assume \(\ker(K-\lambda)^{2}x=0\). Then \(\bar{x}\equiv (K-\lambda)x=(A^{-1}-\lambda B_{01}^{-1})B_{01}x\in R(A^{-1}-\lambda B_{01}^{-1})\) and \(0=(K-\lambda)\bar{x}=(A^{-1}-\lambda B_{01}^{-1})B_{01}\bar{x}\), so \(B_{01}\bar{x}\in\ker(A^{-1}-\lambda B_{01}^{-1})\). Because \(A^{-1}-\lambda B_{01}^{-1}\) is self-adjoint, \((B_{01}\bar{x},\bar{c})=0\), and \(\bar{x}=0\).
Similarly,
Hence
since \(I_{A}(0, B_{1})-I_{A}(0,B_{01})=i_{A}(B_{1})-i_{A}(B_{01})\) (via Proposition A.1(ii)) is odd. Therefore (1.5) has one solution x with \(\Vert x \Vert _{Y}\in(r,R]\). □
Remark
As \(M(x)=0\), (1.5) reduces to the equation
When \(Y=D( \vert A \vert ^{\frac{1}{2}})\), Theorem 1.1 reduces to [2], Theorem 7.3.1, as \(\sigma(A)=\sigma_{d}(A)\) is bounded from below, and to [2], Theorem 8.4.1, as \(\sigma(A)=\sigma_{d}(A)\) is unbounded both from above and below.
3 Applications to first order Hamiltonian systems
Consider the following problem:
where \(H\in C^{1}([0,1]\times \mathbf {R}^{2n},\mathbf {R}^{2n})\) and \(H'(t,x)\) is the gradient of H with respect to x, \(x=(x_{1},x_{2})\), \(x_{1},x_{2}\in \mathbf {R}^{n}\), \(\alpha\in[0,\pi)\), \(\beta\in(0,\pi]\), J is the standard symplectic matrix and \(M_{i}\in C(\mathbf {R}^{2n}\times \mathbf {R}^{n},\mathbf {R}^{2n})\) are bounded (\(i=0,1\)). \(x:[0,1]\to \mathbf {R}^{2n}\) is said to be a solution of (3.1)-(3.3) if \(x\in C^{1}([0,1],\mathbf {R}^{2n})\) and \(x=x(t)\) satisfies (3.1)-(3.3).
We also make the following assumptions:
- (H1):
-
There exists \({\bar{B}}:[0,1]\times \mathbf {R}^{2n}\to{\mathcal {L}}_{s}(\mathbf {R}^{2n})\) with \({\bar{B}}(\cdot, x(\cdot))\in L^{\infty }([0,1],{\mathcal {L}}_{s}(\mathbf {R}^{2n}))\) for all \(x\in C([0,1],\mathbf {R}^{2n})\), \({\bar{B}}_{1}, {\bar{B}}_{2}\in L^{\infty}([0,1],\mathcal {L}_{s}(\mathbf {R}^{2n}))\) such that
$$H'(t,x)={\bar{B}}(t,x)x+h(t,x),\qquad {\bar{B}}_{1}(t)\leq{\bar{B}}(t,x)\leq {\bar{B}}_{2}(t) $$for all \((t,x)\in[0,1]\times \mathbf {R}^{2n}\), and \(h(t,x):[0,1]\times \mathbf {R}^{2n}\to \mathbf {R}^{2n}\) is bounded.
- (H2):
-
There exists \({\bar{B}}_{0}:[0,1]\times \mathbf {R}^{2n}\rightarrow \mathcal {L}_{s}(\mathbf {R}^{2n})\) with \({\bar{B}}_{0}(\cdot, x(\cdot))\in L^{\infty}([0,1],{\mathcal {L}}_{s}(\mathbf {R}^{2n}))\) for all \(x\in C([0,1],\mathbf {R}^{2n})\), \({\bar{B}}_{01},{\bar{B}}_{02}\in L^{\infty}([0,1],\mathcal {L}_{s}(\mathbf {R}^{2n}))\) such that
$$H'(t,x)=\bar{B}_{0}(t,x)x,\qquad {\bar{B}}_{01}(t) \leq{\bar{B}}_{0}(t,x)\leq {\bar{B}}_{02}(t) $$for all \((t,x)\in[0,1]\times \mathbf {R}^{2n}\) with \(\vert x \vert \leq r\) for some constant \(r>0\).
Theorem 3.1
If H satisfies (H1) with \(i_{\alpha,\beta}^{f}(\bar{B}_{1})=i_{\alpha,\beta}^{f}(\bar{B}_{2})\), \(\nu_{\alpha,\beta}^{f}(\bar{B}_{2})=0\), then (3.1)-(3.3) has one solution. Furthermore, if (H2) and (M1) hold, then (3.1)-(3.3) has one nontrivial solution provided \(i_{\alpha,\beta }^{f}(\bar{B}_{01})=i_{\alpha,\beta}^{f}(\bar{B}_{02})\), \(\nu_{\alpha ,\beta}^{f}(\bar{B}_{02})=0\) and \(i_{\alpha,\beta}^{f}(\bar {B}_{01})-i_{\alpha,\beta}^{f}(\bar{B}_{1})\) is odd.
Proof
Let \(X=L^{2}([0,1],\mathbf {R}^{2n})\), \(Y=C([0,1],\mathbf {R}^{2n})\), \(D(A_{1})=\{x\in H^{1}([0,1],\mathbf {R}^{2n})\vert x_{1}(0)\cos\alpha +x_{2}(0)\sin\alpha=0, x_{1}(1)\cos\beta+x_{2}(1)\sin\beta=0\}\), \(A_{1}:D(A_{1})\subset Y\rightarrow X\) by \((A_{1}x)(t)=-J\dot{x}(t)-\mu_{1} x(t)\) where \(\mu_{1}<0\), \(\mu _{1}\neq\beta-\alpha+k\pi\), \(k\in \mathbf {Z}\) and \(B_{1}-\mu_{1}I_{2n}\geq I_{2n}\), \(B_{01}-\mu_{1}I_{2n}\geq I_{2n}\). Then \(A_{1}\) is an unbounded self-adjoint and invertible operator in X with \(\sigma(A_{1})=\sigma_{d}(A_{1})=\{\beta-\alpha-\mu_{1}+k\pi \vert k\in \mathbf {Z}\}\). \(N_{1}:Y\rightarrow Y\) by \((N_{1}x)(t)=H'(t,x(t))-\mu_{1}x(t)\), \((B(x)y)(t)=\bar{B}(t,x(t))y(t)-\mu_{1}y(t)\). Hence (H1), (H2) imply (N1), (N2), respectively. Set \((Ax)(t)=-J\dot{x}(t)\), \((\widetilde{B}_{i}x)(t)={\bar{B}}_{i}(t)-\mu_{1}x(t)\), \((\widetilde{B}_{0i}x)(t)=\bar{B}_{0i}(t)-\mu_{1}x(t)\) and \((B_{i}x)(t)={\bar{B}}_{i}(t)\), \(( B_{0i}x)(t)={\bar{B}}_{0i}(t)\); then \(A_{1}=A-\mu_{1}\mathit{Id}\), \({\tilde{B}}_{i}=B_{i}-\mu_{1}\mathit{Id}\), \({\tilde{B}}_{0i}=B_{0i}-\mu_{1}\mathit{Id}\) (\(i=1,2\)). By the definition in the Appendix, \(\nu _{\alpha,\beta}^{f}(\bar{B}_{2})=\nu_{A}(B_{2})\), and
Hence, \(i_{\alpha,\beta}^{f}(\bar{B}_{2})=i_{\alpha,\beta}^{f}(\bar {B}_{1})\) implies \(i_{A}(B_{2})=i_{A}(B_{1})\) and \(i_{\alpha,\beta}^{f}(\bar {B}_{01})-i_{\alpha,\beta}^{f}(\bar{B}_{1})\) is odd means that \(i_{A}(B_{01})-i_{A}(B_{1})\) is odd. Therefore, in order to finish the proof we need only to show that (3.1)-(3.3) can be written in the form of (1.5). Noticing that (3.1) is equivalent to
Multiplying the equation with the integral factor \(e^{-J\mu_{1}t}\) and integrating over \([0,t]\), we can get
Considering (3.2)-(3.3) yields
where \(\Delta_{1}=\sin(\mu_{1}-\beta+\alpha)\). Then (3.1)-(3.3) is equivalent to
where, as \(0\leq s\leq t\leq1\),
as \(0\leq t\leq s\leq1\),
and
It is easy to see that \(M^{1}(x)\) is a compact operator satisfying \(\Vert M^{1}(x) \Vert _{Y}\leq\rho\) for all \(x\in Y\)and some \(\rho>0\) and (M1) implies (M). Hence Theorem 3.1 follows from Theorem 1.1. □
As an application of Theorem 3.1 we investigate the following second order Hamiltonian systems:
where \(V\in C^{1}([0,1]\times \mathbf {R}^{n},\mathbf {R})\), \(V'\) denotes the gradient of V with respect to x, \(\alpha\in[0,\pi)\), \(\beta\in(0,\pi ]\), \(M_{0},M_{1}:\mathbf {R}^{4n}\to \mathbf {R}^{n}\) are continuous and bounded. \(x:[0,1]\to \mathbf {R}^{n}\) is said to be a solution of (3.5)-(3.7) if \(x\in C^{2}([0,1],\mathbf {R}^{n})\) and \(x=x(t)\) satisfies (3.5)-(3.7).
Corollary 3.1
If V satisfies (V1) with \(i_{\alpha,\beta }^{s}(\bar{B}_{1})=i_{\alpha,\beta}^{s}(\bar{B}_{2})\), \(\nu_{\alpha,\beta }^{s}(\bar{B}_{2})=0\), then (3.5)-(3.7) has one solution. Furthermore, if (V2) and (M1) hold, then (3.5)-(3.7) have one nontrivial solution provided \(i_{\alpha,\beta}^{s}(\bar{B}_{01})=i_{\alpha,\beta}^{s}(\bar {B}_{02})\), \(\nu_{\alpha,\beta}^{s}(\bar{B}_{02})=0\) and \(i_{\alpha ,\beta}^{s}(\bar{B}_{01})-i_{\alpha,\beta}^{s}(\bar{B}_{1})\) is odd.
Proof
Define \(y=-\dot{x}\), \(z=(x,y)\), \(H(t,z)=\frac{1}{2} \vert y \vert ^{2}+V(t,x)\). Then (3.5)-(3.7) are equivalent to (3.1)-(3.3). If (V1) holds, then
and if (V2) holds, then
for all \((t,z)\in[0,1]\times \mathbf {R}^{2n}\) with \(\vert z \vert \leq r\). By Proposition A.2, \(\nu_{\alpha,\beta}^{s}(\bar {B}_{01})=\nu_{\alpha,\beta}^{f}(\operatorname{diag}\{\bar{B}_{01},I_{n}\})\), \(\nu _{\alpha,\beta}^{s}(\bar{B}_{1})=\nu_{\alpha,\beta}^{f}(\operatorname{diag}\{\bar {B}_{1},I_{n}\})\), and \(i_{\alpha,\beta}^{s}(\bar{B}_{0i})=i_{\alpha,\beta}^{f}(\operatorname{diag}\{\bar {B}_{0i},I_{n}\})\), \(i_{\alpha,\beta}^{s}(\bar{B}_{i})=i_{\alpha,\beta }^{f}(\operatorname{diag}\{\bar{B}_{i},I_{n}\})\) (\(i=1,2\)). Hence, the results follow from Theorem 3.1. □
Remark
-
1.
When \(\alpha=0\), \(\beta=\pi\), (3.6)-(3.7) reduce to (1.2)-(1.3), so that Corollary 3.1 contains Theorem 1.2 as a special case.
-
2.
When \(M_{0}(\xi)=0\), \(M_{1}(\xi)=0\) for \(\xi\in \mathbf {R}^{4n}\), the first part of Theorem 3.1 reduces [17], Theorem 3.4.3.
Next we discuss the problem
where \(P\in S_{p}(\mathbf {R}^{2n})\), \(M_{2}:\mathbf {R}^{2n}\times \mathbf {R}^{2n}\to \mathbf {R}^{2n}\) is continuous and bounded. \(x:[0,1]\to \mathbf {R}^{2n}\) is said to be a solution of (3.1) and (3.8) if \(x\in C^{1}([0,1],\mathbf {R}^{2n})\) and \(x=x(t)\) satisfies (3.1) and (3.8). We will use the following assumption:
- (M2):
-
\(M_{2}(\xi)=o( \vert \xi \vert )\) as \(\vert \xi \vert \to0\).
Theorem 3.2
If H satisfies (H1) with \(i_{P}^{f}(\bar {B}_{1})=i_{P}^{f}(\bar{B}_{2})\), \(\nu_{P}^{f}(\bar{B}_{2})=0\), then the problem (3.1) and (3.8) has one solution. Furthermore, if (H2) and (M2) hold, then the problem (3.1) and (3.8) has one nontrivial solution provided \(i_{P}^{f}(\bar{B}_{01})=i_{P}^{f}(\bar{B}_{02})\), \(\nu_{P}^{f}(\bar {B}_{02})=0\) and \(i_{P}^{f}(\bar{B}_{01})-i_{P}^{f}(\bar{B}_{1})\) is odd.
Proof
Let \(X=L^{2}([0,1],\mathbf {R}^{2n})\), \(Y=C([0,1],\mathbf {R}^{2n})\). Define \(D(A_{2})=\{x\in H^{1}([0,1],\mathbf {R}^{2n})\vert x(1)=Px(0)\}\), and \(A_{2}:D(A_{2})\subset Y\rightarrow X\) by \((A_{2}x)(t)=-J\dot {x}(t)-\mu_{2}x(t)\) where we choose \(\mu_{2}<0\) such that the operator \(A_{2}\) is invertible, the matrix \((e^{J\mu_{2}}-P)\) is also invertible and \(B_{1}-\mu_{2}I_{2n}\geq I_{2n}\), \(B_{01}-\mu_{2}I_{2n}\geq I_{2n}\). Then \(A_{2}\) is an unbounded self-adjoint and invertible operator in X with \(\sigma(A_{2})=\sigma_{d}(A_{2})\). \(N_{2}:Y\rightarrow Y\) by \((N_{2}x)(t)=H'(t,x(t))-\mu_{2} x(t)\equiv f_{2}(t)\).
Similar to the proof of Theorem 3.1, if \(x=x(t)\) is a solution of (3.1) and (3.8), then
Considering the boundary value condition (3.8) yields
Then the problem (3.1) and (3.8) is equivalent to
where
for \(0\leq s\leq t\leq1\);
for \(0\leq t\leq s\leq1\); and
\(M^{2}(x)\) is a compact operator and satisfies \(\Vert M^{2}(x) \Vert _{Y}\leq\rho\) for some \(\rho>0\). Hence (H1), (H2), (M1) imply (N1), (N2), (M), respectively. Hence, Theorem 3.1 follows from Theorem 1.1. □
Remark
When \(M_{2}(\xi)=0\) for \(\xi\in \mathbf {R}^{4n}\), the first part of Theorem 3.2 reduces to [17], Theorem 3.5.3.
4 Applications to second order Hamiltonian systems
We discuss the problem
where \(M_{i}:\mathbf {R}^{4n}\to \mathbf {R}^{n}\) (\(i=0,1\)) is continuous and bounded, \(G,H\in GL(n)\), \(G^{T}H=I_{n}\). \(x:[0,1]\to \mathbf {R}^{n}\) is said to be a solution of (4.1)-(4.3) if \(x\in C^{2}([0,1],\mathbf {R}^{n})\) and \(x=x(t)\) satisfies (4.1)-(4.3).
Theorem 4.1
If V satisfies (V1) with \(i_{M}^{s}(\bar {B}_{1})=i_{M}^{s}(\bar{B}_{2})\), \(\nu_{M}^{s}(\bar{B}_{2})=0\), then (4.1)-(4.3) have one solution. Furthermore, if (V2) and (M1) hold, then (4.1)-(4.3) have one nontrivial solution provided \(i_{M}^{s}(\bar {B}_{01})=i_{M}^{s}(\bar{B}_{02})\), \(\nu_{M}^{s}(\bar{B}_{02})=0\) and \(i_{M}^{s}(\bar{B}_{01})-i_{M}^{s}(\bar{B}_{1})\) is odd.
Proof
Let \(X=L^{2}([0,1],\mathbf {R}^{n})\), \(D(A_{3})=\{x\in H^{2}([0,1],\mathbf {R}^{n})\vert x(1)=Gx(0),x'(1)=Hx'(0)\}\), \(Y=C^{1}([0,1],\mathbf {R}^{n})\). The inclusion maps \(D(A_{3})\to Y\), \(Y\to X\) are compact. Define \(A_{3}:D(A_{3})\rightarrow L^{2}([0,1],\mathbf {R}^{n})\) by \((A_{3}x)(t)=-\ddot {x}(t)+ x(t)\). So \(A_{3}\) is an unbounded self-adjoint operator in X with \(\sigma (A_{3})=\sigma_{d}(A_{3})\). Define \(N_{3}:C^{1}([0,1],\mathbf {R}^{n})\rightarrow L^{2}([0,1],\mathbf {R}^{n})\) by \((N_{3}x)(t)=V'(t,x(t))+x(t)\equiv f_{3}(t)\). Then (4.1) is equivalent to
Multiplying the integral factor \(e^{t}\) and integrating over \([0,t]\), we can get
Multiplying the integral factor \(e^{-t}\) and integrating over \([0,t]\) again yields
Considering (4.2)-(4.3), we get the following system:
The system is equivalent to
where \(K_{1}=-I_{n}+\operatorname{ch}1(H+G)-HG\), \(K_{2}=I_{n}+\operatorname{ch}1(H+G)-GH\). Then
Then (4.1)-(4.3) are equivalent to
where
for \(0\leq s\leq t\leq1\);
for \(0\leq t\leq s\leq1\), and
It is easy to check that \(M^{3}(x)\) is a compact operator and satisfies \(\Vert M^{3}(x) \Vert _{Y}\leq\rho\) for some \(\rho>0\). Because (V1), (V2), (M1) imply (N1), (N2), (M), Theorem 4.1 follows from Theorem 1.1. □
5 Applications to first order Hamiltonian system with impulses
We first consider the following first order Hamiltonian system with impulses:
where \(\Delta x(t_{i})=x(t_{i}+0)-x({t_{i}-0})\), \(x=(x_{1},x_{2})\), \(x_{1},x_{2}\in \mathbf {R}^{n}\) and \(I_{i}:\mathbf {R}^{2n}\rightarrow \mathbf {R}^{2n}\), \(M_{0},M_{1}:\mathbf {R}^{2n}\times \mathbf {R}^{2n}\to \mathbf {R}^{n}\) are continuous and bounded. \(x:[0,1]\to \mathbf {R}^{2n}\) is said to be a solution of (5.1)-(5.4) if \(x\in C^{1}([0,1]\setminus \{t_{i}\}_{i=1}^{p},\mathbf {R}^{2n})\), \(x(t_{i}+0)\), \(x(t_{i}-0)\) exist and \(x=x(t)\) satisfies (5.1)-(5.4). We need the following assumption:n
- (I):
-
\(I_{i}(\xi)=o( \vert \xi \vert )\) as \(\vert \xi \vert \to0\) (\(i=1,2,\ldots,p\)).
Theorem 5.1
If H satisfies (H1) with \(i_{\alpha,\beta }^{f}(\bar{B}_{1})=i_{\alpha,\beta}^{f}(\bar{B}_{2})\), \(\nu_{\alpha ,\beta}^{f}(\bar{B}_{2})=0\), then (5.1)-(5.4) have one solution. Furthermore, if (H2), (M1) and (I) hold, then (5.1)-(5.4) have one nontrivial solution provided \(i_{\alpha,\beta}^{f}(\bar{B}_{01})=i_{\alpha,\beta }^{f}(\bar{B}_{02})\), \(\nu_{\alpha,\beta}^{f}(\bar{B}_{02})=0\) and \(i_{\alpha,\beta}^{f}(\bar{B}_{01})-i_{\alpha,\beta}^{f}(\bar {B}_{1})\) is odd.
Proof
Let \(X=L^{2}([0,1],\mathbf {R}^{2n})\), \(Y=C(0,1,t_{i};\mathbf {R}^{2n})=\{ x:[0,1]\rightarrow \mathbf {R}^{2n}\vert x(t) \mbox{ is continuous for } t\in [0,1]\setminus\{t_{i}\}_{i=1}^{p}, x(t_{i}+0), x(t_{i}-0)\mbox{ exist}, x(t_{i})=x(t_{i}-0),i=1,2,\ldots,p\}\), As in the proof of Theorem 3.1, (5.1)-(5.4) are equivalent to
where \(A_{1}\), \(N_{1}\) are defined as in Theorem 3.1 and
Hence Theorem 5.1 follows from Theorem 1.1. □
As an application of Theorem 5.1 we investigate the following second order Hamiltonian systems with impulses:
where \(\Delta x'(t_{i})=x'(t_{i}+0)-x'(t_{i}-0)\) and \(M_{0},M_{1}:\mathbf {R}^{4n}\to \mathbf {R}^{n}\), \(I_{i},J_{i}:\mathbf {R}^{n}\to \mathbf {R}^{n}\) (\(i=1,2,\ldots,p\)) are continuous and bounded. \(x:[0,1]\to \mathbf {R}^{n}\) is said to be a solution of (5.5)-(5.8) if \(x\in C^{2}([0,1]\setminus \{t_{i}\}_{i=1}^{p},\mathbf {R}^{n})\), \(x(t_{i}+0)\), \(x(t_{i}-0)\), \(x'(t_{i}+0)\), \(x'(t_{i}-0)\) exist, \(x(t_{i})=x(t_{i}-0)\) and \(x=x(t)\) satisfies (5.5)-(5.8). We need the following assumption:
- (J):
-
\(J_{i}(\xi)=o( \vert \xi \vert )\) as \(\vert \xi \vert \to0\) (\(i=1,2,\ldots,p\)).
Corollary 5.1
If V satisfies (V1) with \(i_{\alpha,\beta }^{s}(\bar{B}_{1})=i_{\alpha,\beta}^{s}(\bar{B}_{2})\), \(\nu_{\alpha,\beta }^{s}(\bar{B}_{2})=0\), then (5.5)-(5.8) have one solution. Furthermore, if (V2), (M1), (I) and (J) hold, then (5.5)-(5.8) have one nontrivial solution provided \(i_{\alpha,\beta}^{s}(\bar{B}_{01})=i_{\alpha,\beta }^{s}(\bar{B}_{02})\), \(\nu_{\alpha,\beta}^{s}(\bar{B}_{02})=0\) and \(i_{\alpha,\beta}^{s}(\bar{B}_{01})-i_{\alpha,\beta}^{s}(\bar{B}_{1})\) is odd.
Proof
Similar to the proof of Corollary 3.1. Then we consider the problem
where \(I_{i}:\mathbf {R}^{2n}\to \mathbf {R}^{2n}\) (\(i=1,2,\ldots,p\)), \(M_{2}:\mathbf {R}^{2n}\times \mathbf {R}^{2n}\to \mathbf {R}^{2n}\) is continuous and bounded. \(x:[0,1]\to \mathbf {R}^{n}\) is said to be a solution of (5.1), (5.2) and (5.9) if \(x\in C^{1}([0,1]\setminus \{t_{i}\}_{i=1}^{p},\mathbf {R}^{2n})\), \(x(t_{i}+0)\), \(x(t_{i}-0)\) exist, \(x(t_{i})=x(t_{i}-0)\) and \(x=x(t)\) satisfies (5.1), (5.2) and (5.9). □
Theorem 5.2
If H satisfies (H1) with \(i_{P}^{f}(\bar{B}_{1})=i_{P}^{f}(\bar {B}_{2})\), \(\nu_{P}^{f}(\bar{B}_{2})=0\), then the system (5.1), (5.2) and (5.9) has one solution. Furthermore, if (H2), (M2) and (I) hold, then the system (5.1), (5.2) and (5.9) has one nontrivial solution provided \(i_{P}^{f}(\bar{B}_{01})=i_{P}^{f}(\bar{B}_{02})\), \(\nu _{P}^{f}(\bar{B}_{02})=0\) and \(i_{P}^{f}(\bar{B}_{01})-i_{P}^{f}(\bar {B}_{1})\) is odd.
Proof
Let X, Y be defined in the proof of Theorem 5.1, and let \(D(A_{2})\) and \(A_{2}\) be defined in the proof of Theorem 3.2. Then (5.1), (5.2) and (5.9) are equivalent to
where \(A_{2}\), \(N_{2}\) are defined as in Theorem 3.2 and
It is easy to check that \(M^{5}(x):Y\rightarrow Y\) is a compact operator and satisfies \(\Vert M^{5}(x) \Vert _{Y}\leq\rho\) for some \(\rho>0\). □
6 Applications to second order Hamiltonian system with impulses
Consider the second order Hamiltonian system with impulses
where \(\Delta x(t_{i})=x(t_{i}+0)-x({t_{i}-0})\), \(\Delta x'(t_{i})=x'(t_{i}+0)-x'(t_{i}-0)\), \(I_{i},J_{i}:\mathbf {R}^{n}\to \mathbf {R}^{n}\) (\(i=1,2,\ldots,p\)), \(M_{i}:\mathbf {R}^{4n}\rightarrow \mathbf {R}^{n}\) (\(i=0,1\)) are continuous and bounded and \(G,H\in GL(n)\), \(G^{T}H=I_{n}\). \(x:[0,1]\to \mathbf {R}^{n}\) is said to be a solution of (6.1)-(6.4) if \(x\in C^{2}([0,1]\setminus \{ t_{i}\}_{i=1}^{p},\mathbf {R}^{n})\), \(x(t_{i}+0)\), \(x(t_{i}-0)\), \(x'(t_{i}+0)\), \(x'(t_{i}-0)\) exist, \(x(t_{i})=x(t_{i}-0)\) and \(x=x(t)\) satisfies (6.1)-(6.4).
Theorem 6.1
If V satisfies (V 1) with \(i_{M}^{s}(\bar{B}_{1})=i_{M}^{s}(\bar{B}_{2})\), \(\nu _{M}^{s}(\bar{B}_{2})=0\), then (6.1)-(6.4) has one solution. Furthermore, if (V2), (M1), (I) and (J) hold, then (6.1)-(6.4) has one nontrivial solution provided \(i_{M}^{s}(\bar{B}_{01})=i_{M}^{s}(\bar{B}_{02})\), \(\nu _{M}^{s}(\bar{B}_{02})=0\) and \(i_{M}^{s}(\bar{B}_{01})-i_{M}^{s}(\bar{B}_{1})\) is odd.
Proof
Let \(X=L^{2}([0,1],\mathbf {R}^{n})\), \(Y=C^{1}(0,1,t_{i};\mathbf {R}^{n})=\{ x:[0,1]\rightarrow \mathbf {R}^{n}\vert x'(t)\mbox{ is continuous for }t\in [0,1]\setminus\{t_{i}\}_{i=1}^{p}, x'(t_{i}+0), x'(t_{i}-0)\mbox{ exist}, x(t_{i})=x(t_{i}-0), x'(t_{i})=x'(t_{i}-0),i=1,\ldots,p\}\), and let \(D(A_{3})\), \(A_{3}\) be defined in the proof of Theorem 4.1. Then (6.1)-(6.4) are equivalent to
where \(G_{3}(t,x)\), \(f_{3}(s)\) are defined in the proof of Theorem 4.1. We have
and
References
Ekeland, I, Ghoussoub, N, Tehrani, H: Multiple solutions for a classical problem in the calculus of variations. J. Differ. Equ. 131, 229-243 (1996)
Dong, Y: Index Theory for Hamiltonian Systems and Multiple Solution Problems. Science Press, Beijing (2014)
Chen, Y, Dong, Y, Shan, Y: Existence of solutions for sub-linear or super-linear operator equations. Sci. China Math. 58, 1653-1664 (2015)
Hu, S, Lakshmikantham, V: PBVP for second order impulsive differential systems. Nonlinear Anal. 13, 75-85 (1989)
Li, Y, Zhou, Q: Periodic solutions to ordinary differential equations with impulses. Sci. China Ser. A 36, 778-790 (1993)
Dong, Y: Sublinear impulsive effects and solvability of boundary value problems for differential equations with impulses. J. Math. Anal. Appl. 264, 32-48 (2001)
Liu, B, Yu, J: Existence of solution for m-point boundary value problems of second-order differential systems with impulses. Appl. Math. Comput. 125, 155-175 (2002)
Zhang, F, Ma, Z, Yan, J: Boundary value problems for first order impulsive delay differential equations with a parameter. J. Math. Anal. Appl. 290, 213-223 (2004)
Qian, D, Li, X: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 303, 288-303 (2005)
Tian, Y, Ge, W: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. (2) 51 509-527 (2008)
Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 10, 680-690 (2009)
Yuan, X, Xia, Y, O’Regan, D: Nonautonomous impulsive systems with unbounded nonlinear terms. Appl. Math. Comput. 245, 391-403 (2014)
Sun, J, Chen, H, Nieto, JJ, Otero-Novoa, M: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 72, 4575-4586 (2010)
Zhou, J, Li, Y: Existence of solutions for a class of second order Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 72, 1594-1603 (2010)
Sun, J, Chen, H, Nieto, JJ: Infinitely many solutions for second-order Hamiltonian system with impulsive effects. Math. Comput. Model. 54, 544-555 (2011)
Qian, D, Chen, L, Sun, X: Periodic solutions of superlinear impulsive differential equations: a geometric approach. J. Differ. Equ. 258, 3088-3106 (2015)
Dong, Y: Index theory for linear self-adjoint operator equations and nontrivial solutions of asymptotically linear operator equations. Calc. Var. 38, 75-109 (2010)
Ekeland, I: Convexity Methods in Hamiltonian Mechanics. Springer, Berlin (1990)
Long, Y: Index Theory for Symplectic Paths with Applications. Progress in Math., vol. 207. Birkhäuser, Basel (2002)
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No.11171157) and the Jiangsu Planned Projects for Postdoctoral Research Funds. The author wants to express her sincere thanks to the referee for the valuable comments.
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Appendix
Appendix
In this section we will recall some results concerning index theory for self-adjoint operator equations from Dong [2, 17]. For index theories for Hamiltonian systems and symplectic paths we refer to [18, 19]. Let X be an infinite-dimensional Hilbert space, and let A be an unbounded self-adjoint invertible operator satisfying \(\sigma (A)=\sigma_{d}(A)\). For any \(B_{1},B_{2}\in\mathcal{L}_{s}(X)\), we write \(B_{1} < B_{2}\) w.r.t. \(X_{1}\) (a subspace of X) if and only if \((B_{1}x,x)<(B_{2}x,x)\) for all \(x\in X_{1}\setminus \{0\}\); and write \(B_{1}\leq B_{2}\) w.r.t. \(X_{1}\) if and only if \((B_{1}x,x)\leq(B_{2}x,x)\) for all \(x\in X_{1}\). If \(X=X_{1}\) we just write \(B_{1}< B_{2}\) or \(B_{1}\leq B_{2}\).
Definition A.1
-
(i)
For any \(B\in\mathcal{L}_{s}(X)\), the space of bounded self-adjoint operators on X, we define \(\nu_{A}(B)=\dim\ker(A-B)\), \(\nu_{A}(B)\) is called the nullity of B.
-
(ii)
For any \(B_{1},B_{2}\in\mathcal{L}_{s}(X)\) with \(B_{1}< B_{2}\), we define
$$I_{A}(B_{1},B_{2})=\sum _{\lambda\in[0,1)}\nu_{A}\bigl((1-\lambda )B_{1}+ \lambda B_{2}\bigr) $$and for any \(B_{1},B_{2}\in\mathcal{L}_{s}(X)\) we define
$$I_{A}(B_{1},B_{2})=I_{A}(B_{1},k\mathit{Id})-I_{A}(B_{2},k\mathit{Id}), $$where \(\mathit{Id}:X\rightarrow X\) is the identity map and \(k\mathit{Id}>B_{1}\), \(k\mathit{Id}>B_{2}\) for some real number \(k>0\).
-
(iii)
For any \(B\in\mathcal{L}_{s}(X)\), we define
$$i_{A}(B)=i_{A}(B_{0})+I_{A}(B_{0},B), $$where \(B_{0}\in\mathcal{L}_{s}(X)\) is fixed and \(i_{A}(B_{0})\) is a prescribed integer.
Proposition A.1
-
(i)
For any \(B\in{\mathcal {L}}_{s}(X)\), \((\nu_{A}(B),i_{A}(B))\in \mathbf {N}\times \mathbf {Z}\).
-
(ii)
For any \(B_{1},B_{2}\in{\mathcal {L}}_{s}(X)\), if \(B_{1}\leq B_{2}\), then \(i_{A}(B_{1})\leq i_{A}(B_{2})\), \(\nu_{A}(B_{1})+i_{A}(B_{1})\leq\nu_{A}(B_{2})+i_{A}(B_{2})\); if \(B_{1}\leq B_{2}\), and \(B_{1}< B_{2}\) with respect to \(\ker(A-B_{1})\), then \(\nu_{A}(B_{1})+i_{A}(B_{1})\leq i_{A}(B_{2})\).
-
(iii)
If \(\inf\sigma(A)\geq\lambda_{0}\) for some \(\lambda_{0}\in \mathbf {R}\), we can choose \(B_{0}=\lambda_{0}\mathit{Id}\) and \(i_{A}(B_{0})=0\), then the index defined by Definition A.1 satisfies
$$i_{A}(B)=\sum_{\lambda< 0}\nu_{A}(B+ \lambda \mathit{Id}). $$Define \(X_{1}=L^{2}([0,1],\mathbf {R}^{2n})\), \(D(A_{1})=\{x\in H^{2}([0,1],\mathbf {R}^{2n})\vert x_{1}(0)\cos\alpha +x_{2}(0)\sin\alpha=0,x_{1}(1)\cos\beta+x_{2}(1)\sin\beta=0\}\) and \((A_{1}x)(t)=-J\dot{x}(t)\) for all \(x\in D(A_{1})\). For any \(\bar {B}_{1},\bar{B}_{2}\in L^{\infty}([0,1],{\mathcal {L}}_{s}(\mathbf {R}^{n}))\), we define \(\bar{B}_{1}\leq\bar{B}_{2}\) if and only if \(\bar{B}_{1}(t)\leq\bar {B}_{2}(t)\) for a.e. \(t\in[0,1]\); and define \(\bar{B}_{1}<\bar{B}_{2}\) if and only if \(\bar{B}_{1}\leq\bar{B}_{2}\) and \(\bar{B}_{1}(t)\leq\bar {B}_{2}(t)\) on a subset of \((0,1)\) with positive measure. For \(\bar {B}\in L^{\infty}([0,1],{\mathcal {L}}_{s}(\mathbf {R}^{n}))\) we define \((Bx)(t)=\bar{B}(t)x(t)\) for all \(x\in X_{1}\). It is easy to check that \(\bar{B}_{1}\leq\bar{B}_{2}\) means that \(B_{1}< B_{2}\) w.r.t. \(\ker(A_{1}-B_{1})\).
Definition A.2
For any \(\bar{B}\in L^{\infty}([0,1],\mathcal{L}_{s}(\mathbf {R}^{2n}))\), we define
where \(i_{\alpha,\beta}^{s}(I_{n})\) will be defined in Definition A.4, and as \(\bar{B}_{1}<\bar{B}_{2}\) and
and for any \(\bar{B}_{1},\bar{B}_{2}\in L^{\infty}([0,1],{\mathcal {L}}_{s}(\mathbf {R}^{2n}))\), we define
where \(k\in \mathbf {R}\), \(kI_{2n}>\bar{B}_{1}\), \(kI_{2n}>\bar{B}_{2}\).
Define \(X_{2}=L^{2}([0,1],\mathbf {R}^{2n})\), \(D(A_{2})=\{x\in H^{2}([0,1],\mathbf {R}^{2n})\vert x(1)=Px(0)\}\), \(P\in S_{p}(2n)\) and \((A_{2}x)(t)=-J\dot{x}(t)\) for all \(x\in D(A_{2})\).
Definition A.3
-
(i)
For any \(\bar{B}\in L^{\infty}([0,1],\mathcal{L}_{s}(\mathbf {R}^{2n}))\), we define
$$\nu_{P}^{f}(\bar{B})=\dim\ker(A_{2}-B). $$ -
(ii)
For any \(\bar{B}_{1},\bar{B}_{2}\in L^{\infty}([0,1],\mathcal {L}_{s}(\mathbf {R}^{2n}))\) with \(\bar{B}_{1}<\bar{B}_{2}\), we define
$$I_{P}^{f}(\bar{B}_{1},\bar{B}_{2})= \sum_{s\in[0,1)}\nu _{P}^{f} \bigl((1-s)\bar{B}_{1}+s\bar{B}_{2}\bigr), $$and if \(\bar{B}_{1}<\bar{B}_{2}\) does not hold, we define
$$I_{P}^{f}(\bar{B}_{1},\bar{B}_{2})=I_{P}^{f}( \bar {B}_{1},cI_{2n})-I_{P}^{f}( \bar{B}_{2},cI_{2n}), $$where \(c\in \mathbf {R}\) such that \(cI_{2n}>\bar{B}_{1}\) and \(cI_{2n}>\bar{B}_{2}\).
-
(iii)
For any \(\bar{B}\in L^{\infty}([0,1],\mathcal{L}_{s}(\mathbf {R}^{2n}))\), we define
$$i_{P}^{f}(\bar{B})=i_{P}^{f}(0)+I_{P}^{f}(0, \bar{B}), $$where \(i_{P}^{f}(0)\in \mathbf {Z}\) is prescribed and depends only on P.
Define \(X_{3}=L^{2}([0,1],\mathbf {R}^{n})\), \(D(A_{3})=\{x\in H^{2}([0,1],\mathbf {R}^{n})\vert x(0)\cos\alpha-x'(0)\sin\alpha=0, x(1)\cos\beta-x'(0)\sin\beta=0\}\) for some constants \(\alpha\in [0,\pi)\), \(\beta\in(0,\pi]\) and \((A_{3}x)(t)=\ddot{x} (t)\) for all \(x\in D(A_{3})\).
Definition A.4
For any \(\bar{B}\in L^{\infty}([0,1],\mathcal{L}_{s}(\mathbf {R}^{n}))\), we define
where \((Bx)(t)=\bar{B}(t)x(t)\) for all \(x\in X\).
Define \(X_{4}=L^{2}([0,1],\mathbf {R}^{n})\), \(D(A_{4})=\{x\in H^{2}([0,1],\mathbf {R}^{n})\vert x(1)=Mx(0),x'(1)=Nx'(0)\}\) where \(M,N\in GL(n)\), \(M^{T}N=I_{n}\), and define \((A_{4}x)(t)=-\ddot{x} (t)\).
Definition A.5
For any \(\bar{B}\in L^{\infty}([0,1],\mathcal {L}_{s}(\mathbf {R}^{n}))\), we define
Proposition A.2
-
(i)
For any \(\bar{B}_{1},\bar{B}_{2}\in L^{\infty }([0,1],{\mathcal {L}}_{s}(\mathbf {R}^{n}))\), if \(\bar{B}_{1}\leq\bar{B}_{2}\), then \(i_{\alpha,\beta}^{s}(\bar{B}_{1})\leq i_{\alpha,\beta}^{s}(\bar{B}_{2})\), \(i_{\alpha,\beta}^{s}(\bar{B}_{1})+\nu_{\alpha,\beta}^{s}(\bar {B}_{1})\leq i_{\alpha,\beta}^{s}(\bar{B}_{2})+\nu_{\alpha,\beta }^{s}(\bar{B}_{2})\); if \(\bar{B_{1}}<\bar{B_{2}}\), then \(i_{\alpha,\beta }^{s}(\bar{B}_{1})+\nu_{\alpha,\beta}^{s}(\bar{B}_{1})\leq i_{\alpha ,\beta}^{s}(\bar{B}_{2})\).
-
(ii)
For any \(\bar{B}\in L^{\infty}([0,1],{\mathcal {L}}_{s}(\mathbf {R}^{n}))\),
$$\begin{gathered} \nu_{\alpha,\beta}^{s}(\bar{B})= \nu_{\alpha,\beta}^{f}\bigl(\operatorname{diag}\{\bar {B},I_{n}\}\bigr), \\ i_{\alpha,\beta}^{s}(\bar{B})=i_{\alpha,\beta}^{f}\bigl(\operatorname{diag}\{\bar{B},I_{n}\}\bigr). \end{gathered} $$
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Hu, T. Nonlinear boundary value conditions and ordinary differential systems with impulsive effects. Bound Value Probl 2017, 45 (2017). https://doi.org/10.1186/s13661-017-0777-x
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DOI: https://doi.org/10.1186/s13661-017-0777-x