Variational approach to second-order impulsive dynamic equations on time scales
© Otero-Espinar and Pernas-Castaño; licensee Springer. 2013
Received: 27 February 2013
Accepted: 23 April 2013
Published: 9 May 2013
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.
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 .
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 .
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)
Proposition 2.3 (Corollary 3.3 in )
holds, where is the smallest positive eigenvalue of problem ; and .
inducing the norm .
3 Variational formulation of (P) and existence results
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.) .
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 .
We can deduce the following regularity properties which allow us to assert that the solutions to (LP) are precisely the critical points of φ.
- 1.φ is differentiable at any and
If is a critical point of φ defined by (3.3), then u is 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)
attains its minimum at u.
By the Lax-Milgram theorem, we obtain the following result.
Theorem 3.2 If then the problem (LP) has a weak solution for any . Moreover, and u is a classical solution and u minimizes the functional (3.3), and hence it is a critical point of φ.
3.1 Impulsive nonlinear problem
We assume that .
for every .
One can deduce, from the properties of H, f and , the following regularity properties of φ.
Proposition 3.1 The functional φ defined by (3.4) is continuous, differentiable, and weakly lower semi-continuous. Moreover, the critical points of φ are weak solutions of (P).
Theorem 3.3 Suppose that f is bounded and that the impulsive functions are bounded. Then there is a critical point of φ, and (P) has at least one solution.
This implies that , and φ is coercive. Hence (Th. 1.1 of ), φ has a minimum, which is a critical point of φ. □
Theorem 3.4 Suppose that f is sublinear and the impulsive functions have sublinear growth. Then there is a critical point of φ and (P) has at least one solution.
where and .
Since , then for every . □
4 Impulsive nonlinear problem with linear derivative dependence
where f and , are continuous and g is continuous and regressive.
It is evident that A is bilinear, continuous and symmetric.
Lemma 4.1 (Theorem 38.A of )
X is a reflexive Banach space.
M is bounded and weak sequentially closed.
φ is sequentially lower semi-continuous on M.
Lemma 4.2 (Analogous to Lemma 2.2 of )
Proof In fact, by Poincare’s inequality, if , we can take , ; if , then we can take and . □
Lemma 4.4 The functional ψ defined by (4.1) is continuous, continuously differentiable and weakly lower semi-continuous.
Theorem 4.1 Suppose that , f and are bounded, , then (NP) has at least one solution.
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)
Then there exists a critical point of ϕ.
Theorem 4.3 Suppose that , then the problem (NP) has at least two solutions if the following conditions hold:
() There exists a positive such that and uniformly for as , .
() and uniformly for as , .
where and .
Thus there exists a constant such that for all .
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 .
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 .
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.
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 .
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). □
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 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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