Periodic and homoclinic solutions generated by impulses
Introduction
In the present paper we are interested in the existence of periodic and homoclinic solutions of the following second order impulsive differential equations, where , , with , , , , for each , and there exist an and a such that , and for all (that is, is -periodic in ).
Impulsive differential equations arise from real world and are used to describe the dynamics of processes which possess sudden, discontinuous jumps. Such processes naturally occur in control theory, biology, optimization theory and some physics and mechanics problems. Due to their significance, a lot of efforts have been made in studying the qualitative properties of such systems; see, for instance, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].
Most of the results in the literature concerning the existence of solutions of impulsive differential equations are obtained via fixed point theory, topological degree theory (including continuation method and coincidence degree theory) and comparison method (including upper and lower solutions method and monotone iterative method). See [12], [2], [5], [6], [8], [13], [14], [10] and the references therein.
Recently some authors studied the existence of solutions for impulsive systems via variational methods [7], [15], [16], [17], [18]. In [15], Tian and Ge studied the existence of solutions for a second order impulsive system of a more general form than (0.1)–(0.2) with Sturm–Liouville boundary conditions. In [7], [18], the authors studied the existence of weak solutions for a system similar to (0.1)–(0.2) with Dirichlet boundary conditions. In [16], [17], Zhang and Li studied the existence of periodic solutions for similar impulsive systems. All the results of [7], [15], [16], [17], [18] can be seen as generalizations of corresponding ones for second order ordinary differential equations. In other words, these results can be applied to impulsive systems when the impulses are absent and still give the existence of solutions in this situation. This, in some sense, means that the nonlinear term plays a more important role than the impulsive terms do in guaranteeing the existence of solutions in these results.
However, in the present paper we are more interested in the existence of solutions of system (0.1)–(0.2) controlled by the impulses. More precisely, in this paper we will study the existence of periodic and homoclinic solutions of system (0.1)–(0.2) generated by impulses. Here, we say a solution of (0.1)–(0.2) is generated by impulses if this solution is non-trivial when for some , but it is trivial when for all . For example, if system (0.1)–(0.2) does not possess non-zero periodic solution when for all , then a non-zero periodic solution of system (0.1)–(0.2) with for some is called a periodic solution generated by impulses.
The results of this paper show that under appropriate conditions, system (0.1)–(0.2) possesses at least one non-zero periodic solutions and that this periodic solution is generated by impulses when . In particular, one of the results (Theorem 2) gives a lower bound of the number of -periodic solutions generated by impulses. This lower bound is completely determined by the number of impulses occur in periods of the system, that is, this bound is totally determined by the number of -s in the interval .
For the existence of homoclinic solutions of impulsive systems, to the best of the authors’ knowledge, there is still no result in the literature, but there are many papers studying the existence of homoclinic solutions for ordinary differential equations, for instance, [19], [20], [21], [22], [23], [24], [25], [26], [27]. Since 1980s, some mathematicians have began studying the existence of homoclinic solutions for Hamiltonian systems via variational methods and fruitful results were proved. For example, in [24], [26], [27] the authors proved the multiple existence of homoclinic solutions for a class of Hamiltonian systems via variational methods; in [19], [20], [22], the authors took another approach: they first proved the existence of -periodic solutions via variational methods, and then they showed that there exists a homoclinic solution as the limit of a sequence -periodic solutions when goes to .
Based on the facts mentioned above, in this paper we study the existence of homoclinic solutions of (0.1)–(0.2) via variational methods. Our results show that system (0.1)–(0.2) possesses at least one non-zero homoclinic solution and this homoclinic solution is generated by impulses when .
This paper is organized as follows: in Section 1, the main results of this paper are stated and the non-existence of non-zero periodic and homoclinic solutions is proved when and for , and two examples are also given to illustrate the results; in Section 2, the variational structure of (0.1)–(0.2) is studied, and the equivalence of periodic solutions of (0.1)–(0.2) and critical points of the action functional is proved; in Sections 3 Existence of periodic solutions, 4 Existence of homoclinic solutions, Theorem 1–2 and Theorem 3 are proved respectively.
Section snippets
Main results
To state the main results we need the following assumptions.
is continuous differentiable and -periodic, and there exist positive constants , such that
for all ;
for ;
there exists a such that
for all ,
Remark 1
By Fact 2.1 of [19], implies
Preliminaries
For each integer set Obviously, equipped with the inner product is a Hilbert space. Let denote the norm on induced by and denote the uniform norm on . Set . Consider the functional on defined as follows, Define by
Existence of periodic solutions
In this section Theorem 1, Theorem 2 will be proved, and the main tool used is the well-known Mountain Pass Lemma:
Mountain Pass Lemma Let be a real Banach space and satisfying the Palais–Smale (P.S.) condition. Suppose and there are constants such that , where , and there is an such that .
Then possesses a critical value . Moreover can be characterized aswhere
Lemma 2
Under the hypotheses of Theorem 1,
Existence of homoclinic solutions
In this section we prove Theorem 3 through two lemmas. We assume that the hypotheses of Theorem 3 hold in this section.
Set be a -periodic extension of on for each integer . Consider the system with the periodic boundary conditions Set and is defined on the Hilbert space
Acknowledgements
The authors would like to thank the anonymous referees for their careful reading and helpful comments.
References (29)
Basic theory for nonresonance impulsive periodic problems of first order
J. Math. Anal. Appl.
(1997)Periodic boundary value problems for first-order impulsive ordinary differential equations
Nonlinear Anal. TMA
(2002)- et al.
Variational approach to impulsive differential equations
Nonlinear Anal. RWA
(2009) - et al.
Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations
J. Math. Anal. Appl.
(2006) - et al.
Periodic boundary value problems for second-order impulsive integro-differential equations
J. Comput. Appl. Math.
(2007) - et al.
On the solvability of periodic boundary value problems with impulse
J. Math. Anal. Appl.
(2007) - et al.
Periodic solutions for ordinary differential equations with sublinear impulsive effects
J. Math. Anal. Appl.
(2005) - et al.
Variational approach to impulsive differential equation with periodic boundary conditions
Nonlinear Anal. RWA
(2010) - et al.
An application of variational methods to Dirichlet boundary value problem with impulses
Nonlinear Anal. RWA
(2010) - et al.
Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential
J. Math. Anal. Appl.
(2007)
Impulsive Differential Equations and Inclusions
Impulsive periodic solutions of first-order singular differential equations
Bull. Lond. Math. Soc.
Impulsive and Hybrid Dynamical Systems. Stability, Dissipativity, and Control
Theory of Impulsive Differential Equations
Cited by (40)
Mono/multi-periodicity generated by impulses control in time-delayed memristor-based neural networks
2020, Nonlinear Analysis: Hybrid SystemsCitation Excerpt :Throughout this paper, the following hypotheses are needed. This section summarizes definitions and lemmas on multivalued analysis, differential inclusion and fixed-point theory of multivalued map (or multimap) [4,35–52]. Gronwall’s Inequality [35]
Subharmonic solutions with prescribed minimal period of an impulsive forced pendulum equation
2016, Applied Mathematics LettersCitation Excerpt :The variational approach to the study of impulsive differential equations possibly is due to Nieto and O’Regan (2007) [11], who constructed a variational structure and converted the problem on the existence of solutions of a second-order impulsive equation to that on the existence of critical points of the corresponding variational functional. Since then, many important results have been obtained, such as boundary value problems [12–14], periodic solutions [15,16], homoclinic solutions [17]. Variational methods are employed in this paper and we get subharmonic solutions as critical points of a certain functional defined in a Hilbert space.
Existence of solution for impulsive differential equations with indefinite linear part
2016, Applied Mathematics LettersCitation Excerpt :In recent years, many researchers have extensively studied the theory and applications of impulsive differential equations, see [1–5] for details. Recently, variational methods have been widely used to study the existence of solutions for impulsive problems (see, [6–21] for details). The rest of the paper is organized as follows.
Three Solutions for Impulsive Fractional Boundary Value Problems with p -Laplacian
2022, Bulletin of the Iranian Mathematical Society