Variational approach to impulsive differential system
 Yanqiang Wu^{1}Email author and
 Wenbin Liu^{1}
https://doi.org/10.1186/s1366201506411
© Wu and Liu 2015
Received: 29 June 2015
Accepted: 18 September 2015
Published: 30 September 2015
Abstract
In this work, we consider a nonlinear Dirichlet problem with impulses and obtain the existence of solutions to an impulsive problem by means of variational methods.
Keywords
MSC
1 Introduction
We point out that many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. Based on the significance, a lot of developments have been made in the theory and applications of impulsive differential systems by numerous mathematicians. We refer the reader to the classical monograph (see [1, 2]), the general works on the theory (see [3–10]) and applications of impulsive differential equations which occur in biology, control theory, optimization theory, population dynamics, medicine, mechanics, engineering and chaos theory, etc. (see [11–27]). These classical techniques contain fixed point theory, topological degree theory and comparison method (including monotone iterative method and upper and lower solutions methods).
For a second order differential equation \(u''=f(t,u,u')\), one usually considers, as impulsive, the position u and the velocity \(u'\). However, in the motion of spacecraft one has to deal with instantaneous impulses depending on the position that results in jump discontinuities in velocity, but no change in position (see [12, 28–30]). The impulses only on the velocity occur also in impulsive mechanics.
Many problems can be solved in terms of the minimization of a functional, usually related to the energy, in an appropriate space of functions. The purpose of this work is to investigate the variational structure under the impulsive differential system (1.1). Based on variational method, we introduce a different concept of solution, that is, a weak solution to problem (1.1). The critical points of the corresponding functional are indeed weak solutions of the impulsive problem (1.1). For the impulsive Dirichlet boundary value problems, the known results obtained by variational approach and critical point theory are as follows.
In this paper we consider the impulsive nonlinear coupled differential system (1.1) motivated by the results [32–35]. Our main result extends the studies made in [32–35] in the sense that we are concerned with a class of problems that is not considered in the papers.
 (H_{1}):

Assume that \(\alpha>\lambda_{1}\), where \(\alpha=\min\{ \operatorname{ess}\inf_{t\in[0,T]}g(t), \operatorname{ess}\inf_{t\in[0,T]}h(t)\}\) and \(\lambda_{1}=\frac{\pi^{2}}{T^{2}}\) is the first eigenvalue of the problem$$ \left \{ \textstyle\begin{array}{l} u''(t)=\lambda u(t), \quad t\in[0,T], \\ u(0)=u(T)=0. \end{array}\displaystyle \right . $$
 (H_{2}):

There exist \(a,b>0\) and \(\gamma_{1},\gamma_{2} \in[0,1)\) such thatand$$\bigl\vert f_{x}(x,y)\bigr\vert \leq a+bx^{\gamma_{1}}\quad \mbox{for every } (x,y)\in{\mathbb{R}}^{2} $$$$\bigl\vert f_{y}(x,y)\bigr\vert \leq a+by^{\gamma_{2}}\quad \mbox{for every } (x,y)\in{\mathbb{R}}^{2}. $$
 (H_{3}):

There exist \(a_{k},b_{k}>0\) and \(\beta_{k}\in[0,1)\) (\(k=1,2,\ldots,m\)) such thatand$$\bigl\vert I_{k}(u)\bigr\vert \leq a_{k}+b_{k}u^{\beta_{k}} \quad \mbox{for every } u\in{\mathbb{R}} $$$$\bigl\vert J_{k}(v)\bigr\vert \leq a_{k}+b_{k}v^{\beta_{k}} \quad \mbox{for every } v\in{\mathbb{R}}. $$
The main result of this paper is the following.
Theorem 1.1
Let assumptions (H_{1})(H_{3}) be satisfied. Then problem (1.1) has at least one nontrivial solution.
Obviously, Theorem 3.2 in [35] is a special case of Theorem 1.1 in this paper.
This paper is organized as follows. In Section 2, we introduce a Hilbert space \(X=H_{0}^{1}(0,T)\times H_{0}^{1}(0,T)\), on which the corresponding functional Φ of problem (1.1) is defined. Furthermore, we give some necessary notations and preliminaries. In Section 3, we prove the main result via variational approach.
2 Variational structure
Lemma 2.1
Assume that assumption (H_{1}) holds, then, for the Sobolev space X, the norm \(\\cdot\\) and the norm \(\\cdot\_{X}\) are equivalent.
Proof
Lemma 2.2
For any \((u,v)\in X\), there exists \(c_{2}>0\) such that \(\u\_{\infty},\v\_{\infty}\leq c_{2} \(u,v)\_{X}\).
Proof
Lemma 2.3
[36]
Let X be a reflexive Banach space and \(F:X\rightarrow{\mathbb{R}}\) be continuously Fréchetdifferentiable. If F is weakly lower semicontinuous and has a bounded minimizing sequence, then F has a minimum on X.
3 Main result
Lemma 3.1
Assume that conditions (H_{1})(H_{3}) are satisfied. Then the functional Φ defined by (2.9) is continuously Fréchetdifferentiable and weakly lower semicontinuous.
Proof
First, using the continuity of \(f_{u}\), \(f_{v}\), \(I_{k}\) and \(J_{k}\), \(k=1,2,\ldots,m\), we easily obtain the continuity and differentiability of Φ and \(\Phi':X=H_{0}^{1}(0,T)\times H_{0}^{1}(0,T)\rightarrow{\mathbb{R}}\) defined by (2.10).
Proof of Theorem 1.1
In connection with \(\gamma_{1},\gamma_{2},\beta_{k}\in [0,1)\), \(k=1,2,\ldots,m\), it follows that the functional Φ is coercive on X. Furthermore, by Lemma 3.1 and Lemma 2.3, we have that Φ has a minimum point on X. Hence, problem (1.1) has at least one nontrivial solution. □
Corollary 3.1
Assume that \(f_{u}\), \(f_{v}\), \(I_{k}\) and \(J_{k}\), \(k=1,2,\ldots,m\), are bounded. Then problem (1.1) has at least one solution.
4 Example
Declarations
Acknowledgements
The author would like to thank the referees very much for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (11271364).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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