# Variational approach to impulsive differential system

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## 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

impulsive problem Dirichlet condition variational methods## MSC

34A08 34B15## 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, 4, 5, 6, 7, 8, 9, 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, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 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, 29, 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.

*f*and \(I_{j}\) satisfy some conditions, problem (1.2) has at least two positive solutions via variational method.

In this paper we consider the impulsive nonlinear coupled differential system (1.1) motivated by the results [32, 33, 34, 35]. Our main result extends the studies made in [32, 33, 34, 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$$\bigl\vert f_{x}(x,y)\bigr\vert \leq a+b|x|^{\gamma_{1}}\quad \mbox{for every } (x,y)\in{\mathbb{R}}^{2} $$$$\bigl\vert f_{y}(x,y)\bigr\vert \leq a+b|y|^{\gamma_{2}}\quad \mbox{for every } (x,y)\in{\mathbb{R}}^{2}. $$
- (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

*p*-times integrable on \([0,T]\) under the norm

*X*, for any \((u,v)\in X\), we set the norm

_{1}), we also introduce the norm

### 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

*u*,

*v*, \(u'\) and \(v'\) are both absolutely continuous. Meanwhile, \(u'',v''\in L^{2}(0,T)\). Hence, \(u'(t^{+})=u'(t^{-})\) and \(v'(t^{+})=v'(t^{-})\) for any \(t\in[0,T]\). If \(u,v\in H_{0}^{1}(0,T)\), then

*u*,

*v*are absolutely continuous and \(u',v'\in L^{2}(0,T)\). In this case, the one-sided derivatives \(u'(t^{+})\), \(u'(t^{-})\), \(v'(t^{+})\) and \(v'(t^{-})\) may not exist. Thus, we need to introduce a concept of solution which is different from a classical solution. We say that \((u,v)\) is a classical solution of problem (1.1) if it satisfies the corresponding equations a.e. on \([0,T]\), the limits \(u'(t_{k}^{+})\), \(u'(t_{k}^{-})\), \(v'(t_{k}^{+})\) and \(v'(t_{k}^{-})\), \(k=1,2,\ldots,m\), exist and (2.2), (2.3) hold.

*φ*and

*ψ*respectively, then integrating from 0 to

*T*, we have

### Lemma 2.3

[36]

*Let* *X* *be a reflexive Banach space and* \(F:X\rightarrow{\mathbb{R}}\) *be continuously Fréchet*-*differentiable*. *If* *F* *is weakly lower semi*-*continuous 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 Fréchet*-*differentiable and weakly lower semi*-*continuous*.

### 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).

*u*and

*v*on \([0,T]\) respectively. In connection with the fact that \(\liminf_{i\rightarrow\infty}\|(u_{i},v_{i})\|_{X} \geq\|(u,v)\|_{X}\), one has

### Proof of Theorem 1.1

_{2}), (H

_{3}) and Lemma 2.2, we have

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

_{1}) holds. Next, taking \(a=\pi^{2}\), \(b=1\), \(\gamma _{1}=\frac{1}{2}\), and \(\gamma_{2}=\frac{1}{3}\), (H

_{2}) holds. Finally, taking \(a_{1}=2\), \(b=1\), and \(\beta_{1}=\frac{1}{3}\), (H

_{3}) holds. Then, by Theorem 1.1, the impulsive problem (4.1) has at least one nontrivial solution.

## Notes

### 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).

## References

- 1.Haddad, WM, Chellaboina, C, Nersesov, SG, Stability, GS: Dissipativity and Control. Princeton University Press, Princeton (2006) MATHGoogle Scholar
- 2.Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) MATHCrossRefGoogle Scholar
- 3.Agarwal, RP, Franco, D, O’Regan, D: Singular boundary value problem for first and second order impulsive differential equations. Aequ. Math.
**69**, 83-96 (2005) MATHMathSciNetCrossRefGoogle Scholar - 4.Ahmad, B, Nieto, JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. TMA
**69**, 3291-3298 (2008) MATHMathSciNetCrossRefGoogle Scholar - 5.Hernandez, E, Henriquez, HR, McKibben, MA: Existence results for abstract impulsive second-order neutral functional differential equations. Nonlinear Anal. TMA
**70**, 2736-2751 (2009) MATHCrossRefGoogle Scholar - 6.Li, J, Nieto, JJ, Shen, J: Impulsive periodic boundary value problems of first-order differential equations. J. Math. Anal. Appl.
**325**, 226-236 (2007) MATHMathSciNetCrossRefGoogle Scholar - 7.Luo, Z, Nieto, JJ: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. TMA
**70**, 2248-2260 (2009) MATHMathSciNetCrossRefGoogle Scholar - 8.Nieto, JJ: Basic theory for nonresonance impulsive periodic problems of first order. J. Math. Anal. Appl.
**205**, 423-433 (1997) MATHMathSciNetCrossRefGoogle Scholar - 9.Nieto, JJ, Rodríguez-López, R: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J. Math. Anal. Appl.
**318**, 593-610 (2006) MATHMathSciNetCrossRefGoogle Scholar - 10.Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) MATHGoogle Scholar
- 11.Braverman, E, Zhukovskiy, S: The problem of a lazy tester, or exponential dichotomy for impulsive differential equations revisited. Nonlinear Anal. Hybrid Syst.
**2**, 971-979 (2008) MATHMathSciNetCrossRefGoogle Scholar - 12.Dai, B, Su, H, Hu, D: Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse. Nonlinear Anal. TMA
**70**, 126-134 (2009) MATHMathSciNetCrossRefGoogle Scholar - 13.Del Pino, M, Elgueta, M, Manasevich, R: A homotopic deformation along
*p*of a Leray-Schauder degree result and existence for \((|u'|^{p-2}u')'+f(t,u)=0\), \(u(0)=u(T)=0\), \(p>1^{\ast}\). J. Differ. Equ.**80**, 1-13 (1989) MATHCrossRefGoogle Scholar - 14.Guo, H, Chen, L: Time-limited pest control of a Lotka-Volterra model with impulsive harvest. Nonlinear Anal., Real World Appl.
**10**, 840-848 (2009) MATHMathSciNetCrossRefGoogle Scholar - 15.Jiang, G, Lu, Q, Qian, L: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fractals
**31**, 448-461 (2007) MATHMathSciNetCrossRefGoogle Scholar - 16.Jiang, G, Lu, Q, Qian, L: Chaos and its control in an impulsive differential system. Chaos Solitons Fractals
**34**, 1135-1147 (2007) MATHMathSciNetCrossRefGoogle Scholar - 17.Mohamad, S, Gopalsamy, K, Akca, H: Exponential stability of artificial neural networks with distributed delays and large impulses. Nonlinear Anal., Real World Appl.
**9**, 872-888 (2008) MATHMathSciNetCrossRefGoogle Scholar - 18.Pei, Y, Li, C, Chen, L, Wang, C: Complex dynamics of one-prey multi-predator system with defensive ability of prey and impulsive biological control on predators. Adv. Complex Syst.
**8**, 483-495 (2005) MATHMathSciNetCrossRefGoogle Scholar - 19.Shen, J, Li, J: Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays. Nonlinear Anal., Real World Appl.
**10**, 227-243 (2009) MATHMathSciNetCrossRefGoogle Scholar - 20.Wang, W, Shen, J, Nieto, JJ: Permanence and periodic solution of predator-prey system with Holling type functional response and impulse. Discrete Dyn. Nat. Soc.
**2007**, Article ID 81756 (2007) MathSciNetCrossRefGoogle Scholar - 21.Wei, C, Chen, L: A delayed epidemic model with pulse vaccination. Discrete Dyn. Nat. Soc.
**2008**, Article ID 746951 (2008) MathSciNetCrossRefGoogle Scholar - 22.Xia, Y: Positive periodic solutions for a neutral impulsive delayed Lotka-Volterra competition system with the effect of toxic substance. Nonlinear Anal., Real World Appl.
**8**, 204-221 (2007) MATHMathSciNetCrossRefGoogle Scholar - 23.Yan, J, Zhao, A, Nieto, JJ: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems. Math. Comput. Model.
**40**, 509-518 (2004) MATHMathSciNetCrossRefGoogle Scholar - 24.Zeng, G, Wang, F, Nieto, JJ: Complexity of delayed predator-prey model with impulsive harvest and Holling type II functional response. Adv. Complex Syst.
**11**, 77-97 (2008) MATHMathSciNetCrossRefGoogle Scholar - 25.Zhang, H, Chen, L, Nieto, JJ: A delayed epidemic model with stage structure and pulses for management strategy. Nonlinear Anal., Real World Appl.
**9**, 1714-1726 (2008) MATHMathSciNetCrossRefGoogle Scholar - 26.Zhang, H, Xu, W, Chen, L: An impulsive infective transmission SI model for pest control. Math. Methods Appl. Sci.
**30**, 1169-1184 (2007) MATHMathSciNetCrossRefGoogle Scholar - 27.Zhou, J, Xiang, L, Liu, Z: Synchronization in complex delayed dynamical networks with impulsive effects. Physica A
**384**, 684-692 (2007) MathSciNetCrossRefGoogle Scholar - 28.Carter, TE: Necessary and sufficient conditions for optional impulsive rendezvous with linear equations of motions. Dyn. Control
**10**, 219-227 (2000) MATHCrossRefGoogle Scholar - 29.Liu, X, Willms, AR: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng.
**2**, 277-299 (1996) MATHCrossRefGoogle Scholar - 30.Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol. 65. Am. Math. Soc., Providence (1986) Google Scholar
- 31.Tian, Y, Ge, W: Applications of variational methods to boundary value problem for impulsive ordinary differential equations. Proc. Edinb. Math. Soc.
**51**, 509-527 (2008) MATHMathSciNetCrossRefGoogle Scholar - 32.Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.
**10**, 680-690 (2009) MATHMathSciNetCrossRefGoogle Scholar - 33.Zhang, H, Li, ZX: Variational approach to impulsive differential equations with periodic boundary conditions. Nonlinear Anal., Real World Appl.
**11**, 67-78 (2010) MATHMathSciNetCrossRefGoogle Scholar - 34.Zhang, ZH, Yuan, R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl.
**11**, 155-162 (2010) MATHMathSciNetCrossRefGoogle Scholar - 35.Zhou, JW, Li, YK: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. TMA
**71**, 2856-2865 (2009) MATHCrossRefGoogle Scholar - 36.Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989) MATHCrossRefGoogle Scholar

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