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:
where the impulsive points are right-dense points in an arbitrary time scale , with . Here and , , are continuous functions.
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.
Let be an arbitrary time scale. We assume that has the topology that it inherits from the standard topology on ℝ. Assume that are points in and define the time scale interval . We denote .
Definition 2.1 Let be such that and . We say that u belongs to if and only if , and there exists such that and
and is the set of all continuous functions on J such that they are Δ-differentiable on and their Δ-derivatives are rd-continuous on .
Theorem 2.1Assume that and . The set is a Banach space with the norm defined for every as
Moreover, the set is a Hilbert space with the inner product given for every by
Proposition 2.1Assume that with , then there exists a constant , only dependent on , such that the inequality
holds for all , and hence the immersion is continuous.
Definition 2.2 Let be such that , define the set as the closure of the set in . We define .
The spaces and are endowed with the norm induced by , defined in (2.1), and the inner product induced by , defined in (2.2). These spaces satisfy the following properties.
Proposition 2.2 (Poincare’s inequality)
Let be such that . Then there exists a constant , only dependent on , such that
Proposition 2.3 (Corollary 3.3 in )
If , then
holds, where is the smallest positive eigenvalue of problem ; and .
In the Sobolev space with and , consider the inner product
inducing the norm .
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
where we consider J with and and , , are fixed constants.
Suppose that is such that . Moreover, assume that for every , is such that .
Definition 3.1 We say that u is a classical solution of (LP) if the limits and exist for every and it satisfies the equation on (LP) for Δ-almost everywhere (Δ-a.e.) .
Take , multiply the equation by and integrate between 0 and T:
Taking into account that and integrating by parts, we get
We define the bilinear form by
and the linear operator by
Thus, the concept of weak solution for the impulsive problem (LP) is a function such that is valid for any .
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 defined by
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 and
2. If is a critical point ofφdefined by (3.3), thenuis a weak solution of the impulsive problem (LP).
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)
LetHbe a Hilbert space and let be a bounded bilinear form. Ifais coercive, i.e., there exists such that for every , then for any (the conjugate space ofH) there exists a unique such that
Moreover, ifais also symmetric, then the functional defined by
attains its minimum atu.
By the Lax-Milgram theorem, we obtain the following result.
Theorem 3.2If then the problem (LP) has a weak solution for any . Moreover, anduis a classical solution anduminimizes the functional (3.3), and hence it is a critical point ofφ.
3.1 Impulsive nonlinear problem
We consider the nonlinear Dirichlet problem
We assume that .
A weak solution of (P) is a function such that
for every .
We now consider the functional
One can deduce, from the properties of H, f and , the following regularity properties of φ.
Proposition 3.1The functionalφdefined by (3.4) is continuous, differentiable, and weakly lower semi-continuous. Moreover, the critical points ofφare weak solutions of (P).
Theorem 3.3Suppose thatfis bounded and that the impulsive functions are bounded. Then there is a critical point ofφ, and (P) has at least one solution.
Proof Take and , , such that
Using that , there exists such that for any
Thus, using Proposition 2.1, (2.3) and , we have
This implies that , and φ is coercive. Hence (Th. 1.1 of ), φ has a minimum, which is a critical point of φ. □
Theorem 3.4Suppose thatfis sublinear and the impulsive functions have sublinear growth. Then there is a critical point ofφand (P) has at least one solution.
Proof Let , and , , such that
Again, using that , Proposition 2.1, (2.3) and , , we have
where and .
Since , then for every . □
4 Impulsive nonlinear problem with linear derivative dependence
Consider the following problem:
where f and , are continuous and g is continuous and regressive.
We assume that . Here, , , where is an exponential function. Note that, as g is regressive, is the solution of the problem
We transform the problem (NP) into the following equivalent form:
Obviously, the solutions of (NPE) are solutions of (NP). Consider the Hilbert space with the inner product:
and the norm induced
A weak solution of (NPE) is a function such that
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 )
For the functional withMnot empty, has solutions in case the following hold:
(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 )
There exist constants such that
Proof In fact, by Poincare’s inequality, if , we can take , ; if , then we can take and . □
Lemma 4.3If , then there exists a constant such that , where
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.
Theorem 4.1Suppose that , fand are bounded, , then (NP) has at least one solution.
Proof Take and , , such that
For any , using Lemma 4.3 and Proposition 2.3, we have
This implies that , and ψ is coercive. Hence, ψ has a minimum, which is a critical point of ψ. □
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)
Let be such that it satisfies the Palais-Smale condition. Assume that there exist and a bounded neighborhood Ω of such that and
Then there exists a critical point ofϕ.
Theorem 4.3Suppose that , then the problem (NP) has at least two solutions if the following conditions hold:
( ) There exist constants and such that for all ,
( ) There exists a positive such that and uniformly for as , .
( ) and uniformly for as , .
Proof From ( ), ( ) and the continuities of f and , it is easy to see that for any and , there exist and such that
Hence, for any and , we have
where and .
From the condition ( ), the following hold:
Integrating the above two inequalities with respect to u on and , respectively (in this case, these are integrals on ℝ), we have
Thus there exists a constant such that for all .
From the continuity of , there exists a constant such that
Hence, we have
Similarly, there exist such that
Firstly, we apply Lemma 4.1 to show that there exists ρ such that ψ has a local minimum .
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 .
Now we will show that for some .
In fact, from (4.2) and (4.3), we obtain
We can choose
For any , , we have . Besides, . Then for any . So, . Hence, ψ has a local minimum .
Next, we will show that there exists with such that .
From (4.4) and (4.5), we have
For any with , we have
So, since . Then there exists such that .
Hence, for the above , there exists such that and .
Then, we have .
The next step is to show that ψ satisfies the Palais-Smale condition.
Let be a bounded sequence such that . Now we show that is bounded. By (4.1) we have
Note that , where , , and that there exists a constant c such that
So, by ( ) and (4.7), we have
where , and are constants (independent of k).
Analogously, there exists a constant (independent of k) such that .
Since is bounded, we have is a bounded sequence.
Hence, there exists a subsequence (for simplicity denoted again by ) such that weakly converges to some u in . Then the sequence converges uniformly to u in .
By (4.6), we have
So, we have
Then converges in . Since is a Hilbert space, and the sequence satisfies , then converges to u, i.e., . ψ satisfies the Palais-Smale condition.
Now, by Theorem 4.2, there exists a critical point . Therefore, and are two critical points of ψ, and they are classical solutions of (NPE). Hence, and are classical solutions of (NP). □
Example 4.1 Let for , and . Thus, and . Consider the following boundary value problem:
where is a constant.
We can see that is regressive and continuous. If we take and , by Theorem 4.3, Eq. (4.9) has at least two solutions.
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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