Abstract
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; secondorder boundary value problem; variational techniques; critical point theory; time scales1 Introduction
This paper is concerned with the existence of solutions of secondorder impulsive dynamic equations on time scales. More precisely, we consider the following boundary value problem:
where the impulsive points
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 [19] and the references therein.
For a secondorder 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 [10].
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, fixedpoint theory [1214]. Sobolev spaces of functions on time scales, which were first introduced in [15], 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 secondorder Hamiltonian systems [18]. 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 [2124] for a broad introduction to dynamic equations on time scales and to [25,26] for variational methods and critical point theory.
The paper is organized as follows. In Section 2 we gather together essential properties about Sobolev spaces on time scales proved in [15,27,28] which one needs to read this paper.
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.
2 Preliminaries
Let
Below we set out some results proved in [15,27] about Sobolev spaces on time scales.
Definition 2.1 Let
with
and
Theorem 2.1Assume that
Moreover, the set
Proposition 2.1Assume that
holds for all
Definition 2.2 Let
The spaces
Proposition 2.2 (Poincare’s inequality)
Let
Proposition 2.3 (Corollary 3.3 in [27])
If
holds, where
In the Sobolev space
inducing the norm
It is the consequence of Poincare’s inequality that
and
3 Variational formulation of (P) and existence results
Firstly, to show the variational structure underlying an impulsive dynamic equation, we consider the lineal problem
where we consider J with
Suppose that
Definition 3.1 We say that u is a classical solution of (LP) if the limits
Take
Taking into account that
Hence,
We define the bilinear form
and the linear operator
Thus, the concept of weak solution for the impulsive problem (LP) is a function
We can prove that a defined by (3.1) and l defined by (3.2) are continuous, and, from Proposition 2.3, that a is coercive if
Consider
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.
1. φis differentiable at any
2. If
We will use the following result in linear functional analysis, which ensures the existence of a critical point of φ.
Theorem 3.1 (LaxMilgram theorem)
LetHbe a Hilbert space and let
Moreover, ifais also symmetric, then the functional
attains its minimum atu.
By the LaxMilgram theorem, we obtain the following result.
Theorem 3.2If
3.1 Impulsive nonlinear problem
We consider the nonlinear Dirichlet problem
We assume that
A weak solution of (P) is a function
for every
We now consider the functional
where
One can deduce, from the properties of H, f and
Proposition 3.1The functionalφdefined by (3.4) is continuous, differentiable, and weakly lower semicontinuous. Moreover, the critical points ofφare weak solutions of (P).
Theorem 3.3Suppose thatfis bounded and that the impulsive functions
Proof Take
and
Using that
Thus, using Proposition 2.1, (2.3) and
where
This implies that
Theorem 3.4Suppose thatfis sublinear and the impulsive functions
Proof Let
Again, using that
where
Since
4 Impulsive nonlinear problem with linear derivative dependence
Consider the following problem:
where f and
We assume that
We transform the problem (NP) into the following equivalent form:
Obviously, the solutions of (NPE) are solutions of (NP). Consider the Hilbert space
and the norm induced
A weak solution of (NPE) is a function
Hence, a weak solution of (NP) is a critical point of the following functional:
where
and
It is evident that A is bilinear, continuous and symmetric.
Lemma 4.1 (Theorem 38.A of [29])
For the functional
(i) Xis a reflexive Banach space.
(ii) Mis bounded and weak sequentially closed.
(iii) φis sequentially lower semicontinuous onM.
Lemma 4.2 (Analogous to Lemma 2.2 of [5])
There exist constants
Proof In fact, by Poincare’s inequality, if
Lemma 4.3If
Proof The result is followed by the following inequalities:
□
Lemma 4.4The functionalψdefined by (4.1) is continuous, continuously differentiable and weakly lower semicontinuous.
Theorem 4.1Suppose that
Proof Take
For any
This implies that
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
Theorem 4.2 (Mountain pass theorem)
Let
Then there exists a critical point
Theorem 4.3Suppose that
(
where
(
(
Proof From (
Hence, for any
where
From the condition (
Integrating the above two inequalities with respect to u on
That is,
Thus there exists a constant
From the continuity of
Hence, we have
where
Similarly, there exist
Firstly, we apply Lemma 4.1 to show that there exists ρ such that ψ has a local minimum
Since
Now we will show that
In fact, from (4.2) and (4.3), we obtain
Hence,
We can choose
For any
Next, we will show that there exists
From (4.4) and (4.5), we have
Thus,
For any
So,
Hence, for the above
Then, we have
The next step is to show that ψ satisfies the PalaisSmale condition.
Let
Thus,
where
Note that
So, by (
where
Analogously, there exists a constant
Hence,
Since
Hence, there exists a subsequence
By (4.6), we have
So, we have
Then
Now, by Theorem 4.2, there exists a critical point
Example 4.1 Let
where
We can see that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in this article. They read and approved the final manuscript.
Acknowledgements
The authors are grateful to the referees for their valuable suggestions that led to the improvement of the original manuscript. The research of V OteroEspinar has been partially supported by Ministerio de Educación y Ciencia (Spain) and FEDER, Project MTM201015314.
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