The aim of this paper is to employ variational techniques and critical point theory to prove some conditions for the existence of solutions to a nonlinear impulsive dynamic equation with homogeneous Dirichlet boundary conditions. Also, we are interested in the solutions of the impulsive nonlinear problem with linear derivative dependence satisfying an impulsive condition.
MSC: 34B37, 34N05.
Keywords:impulsive dynamic equations; second-order boundary value problem; variational techniques; critical point theory; time scales
This paper is concerned with the existence of solutions of second-order impulsive dynamic equations on time scales. More precisely, we consider the following boundary value problem:
It is well known that the theory of impulsive dynamic equations provides a natural framework for mathematical modeling of many real world phenomena. The impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time.
Applications of impulsive dynamic equations arise in biology (biological phenomena involving thresholds), medicine (bursting rhythm models), pharmacokinetics, mechanics, engineering, chaos theory, etc. As a consequence, there has been a significant development in impulse theory in recent years. We can see some general and recent works on the theory of impulsive differential equations; see [1-9] and the references therein.
For a second-order dynamic equation, we usually consider impulses in the position and velocity. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position. The impulses only on the velocity occur also in impulsive mechanics. An impulsive problem with impulses in the derivative is considered in .
Moreover, we are interested in the solutions of the impulsive nonlinear problem in time scale with derivative dependence satisfying an impulsive condition. We can see, for example, recent works on the theory of impulsive differential equations in [1,3,6,8,11].
There have been several approaches to studying the solutions of impulsive dynamic equations on time scales, such as the method of lower and upper solutions, fixed-point theory [12-14]. Sobolev spaces of functions on time scales, which were first introduced in , opened a very fruitful new approach in the study of dynamic equations on time scales: the use of variational methods in the context of boundary value problems on time scales (see [16,17]) or in second-order Hamiltonian systems . Moreover, the study of the existence and multiplicity of solutions for impulsive dynamic equations on time scales has also been done by means of the variational method (see, for example, [19,20]).
The aim of this paper is to use variational techniques and critical point theory to derive the existence of multiple solutions to (P); we refer the reader to [21-24] for a broad introduction to dynamic equations on time scales and to [25,26] for variational methods and critical point theory.
The goal of Section 3 is to exhibit the variational formulation for the impulsive Dirichlet problem. As we will see, all these problems can be understood and solved in terms of the minimization of a functional, usually related to the energy, in an appropriate space of functions. The results presented in the part where we address the linear problem are basic but crucial to revealing that a problem can be solved by finding the critical points of a functional. Moreover, we prove some sufficient conditions for the existence of at least one positive solution to (P).
To finish, in Section 4, we present an impulsive nonlinear problem with linear derivative dependence. We transform the problem into an equivalent one that has no dependence on the derivative, and then we prove that the problem has at least one solution. Also, with additional conditions in nonlinearities and impulse functions, we can show the existence of at least two solutions by using the mountain pass theorem.
Proposition 2.2 (Poincare’s inequality)
Proposition 2.3 (Corollary 3.3 in )
It is the consequence of Poincare’s inequality that
3 Variational formulation of (P) and existence results
Firstly, to show the variational structure underlying an impulsive dynamic equation, we consider the lineal problem
We can deduce the following regularity properties which allow us to assert that the solutions to (LP) are precisely the critical points of φ.
Lemma 3.1The following statements are valid.
We will use the following result in linear functional analysis, which ensures the existence of a critical point of φ.
Theorem 3.1 (Lax-Milgram theorem)
attains its minimum atu.
By the Lax-Milgram theorem, we obtain the following result.
3.1 Impulsive nonlinear problem
We consider the nonlinear Dirichlet problem
We now consider the functional
Proposition 3.1The functionalφdefined by (3.4) is continuous, differentiable, and weakly lower semi-continuous. Moreover, the critical points ofφare weak solutions of (P).
This implies that , and φ is coercive. Hence (Th. 1.1 of ), φ has a minimum, which is a critical point of φ. □
4 Impulsive nonlinear problem with linear derivative dependence
Consider the following problem:
We transform the problem (NP) into the following equivalent form:
and the norm induced
Hence, a weak solution of (NP) is a critical point of the following functional:
It is evident that A is bilinear, continuous and symmetric.
Lemma 4.1 (Theorem 38.A of )
(i) Xis a reflexive Banach space.
(ii) Mis bounded and weak sequentially closed.
(iii) φis sequentially lower semi-continuous onM.
Lemma 4.2 (Analogous to Lemma 2.2 of )
Proof The result is followed by the following inequalities:
Lemma 4.4The functionalψdefined by (4.1) is continuous, continuously differentiable and weakly lower semi-continuous.
We will apply the mountain pass theorem in order to obtain at least two critical points of ψ.
Suppose that X is a Banach space (in particular, a Hilbert space) and is differentiable and . We say that ϕ satisfies the Palais-Smale condition if every bounded sequence in the space X such that contains a convergent subsequence.
Theorem 4.2 (Mountain pass theorem)
Hence, we have
Since is a Hilbert space, it is easy to deduce that is bounded and weak sequentially closed. Lemma 4.4 has shown that ψ is weak lower semi-continuous on and, besides, is a reflexive Banach space. So, by Lemma 4.1 we can have this such that .
In fact, from (4.2) and (4.3), we obtain
We can choose
From (4.4) and (4.5), we have
The next step is to show that ψ satisfies the Palais-Smale condition.
By (4.6), we have
So, we have
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.
The authors are grateful to the referees for their valuable suggestions that led to the improvement of the original manuscript. The research of V Otero-Espinar has been partially supported by Ministerio de Educación y Ciencia (Spain) and FEDER, Project MTM2010-15314.
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