In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the nonautonomous second-order system on time scales with impulsive effects
where , (), , is a symmetric matrix-valued function defined on with for all , (, ) are continuous and . Finally, two examples are presented to illustrate the feasibility and effectiveness of our results.
MSC: 34B37, 34N05.
Keywords:nonautonomous second-order systems; time scales; impulse; variational approach
Consider the nonautonomous second-order system on time scales with impulsive effects
In , the authors study the Sobolev’s spaces on time scales and their properties. As applications, they present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for (1.2).
In , the authors establish some sufficient conditions on the existence of solutions of (1.3) by means of some critical point theorems when . When , until now, it is unknown whether problem (1.1) has a variational structure or not.
Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. The theory of impulsive differential systems has been developed by numerous mathematicians (see [3-5]). Applications of impulsive differential equations with or without delays occur in biology, medicine, mechanics, engineering, chaos theory and so on (see [6-9]).
For a second-order differential equation , one usually considers impulses in the position u and the 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 (see ). The impulses only on the velocity occur also in impulsive mechanics (see ). An impulsive problem with impulses in the derivative only is considered in .
The study of dynamical systems on time scales is now an active area of research. One of the reasons for this is the fact that the study on time scales unifies the study of both discrete and continuous processes, besides many others. The pioneering works in this direction are Refs. [13-17]. The theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988, providing a rich theory that unifies and extends discrete and continuous analysis [18,19]. The time scales calculus has a tremendous potential for applications in some mathematical models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics, neural networks and social sciences (see ). For example, it can model insect populations that are continuous while in season (and may follow a difference scheme with variable step-size), die out in winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population.
There have been many approaches to study solutions of differential equations on time scales, such as the method of lower and upper solutions, fixed-point theory, coincidence degree theory and so on (see [1,20-29]). In , authors used the fixed point theorem of strict-set-contraction to study the existence of positive periodic solutions for functional differential equations with impulse effects on time scales. However, the study of the existence and multiplicity of solutions for differential equations on time scales using the variational method has received considerably less attention (see, for example, [1,29]). The variational method is, to the best of our knowledge, novel and it may open a new approach to deal with nonlinear problems, with some type of discontinuities such as impulses.
Motivated by the above, we research the existence of variational construction for problem (1.1) in an appropriate space of functions and study the existence of solutions for (1.1) by some critical point theorems in this paper. All these results are new.
2 Preliminaries and statements
In this section, we present some fundamental definitions and results from the calculus on time scales and Sobolev’s spaces on time scales that will be required below. These are a generalization to of definitions and results found in .
Definition 2.1 ([, Definition 1.1])
(supplemented by and , where ∅ denotes the empty set). A point is called right-scattered, left-scattered, if , hold, respectively. Points that are right-scattered and left-scattered at the same time are called isolated. Also, if and , then t is called right-dense, and if and , then t is called left-dense. Points that are right-dense and left-dense at the same time are called dense. The set which is derived from the time scale as follows. If has a left-scattered maximum m, then ; otherwise, .
Definition 2.2 ([, Definition 1.10])
Definition 2.3 ([, Definition 2.3])
and let . Then we define (provided it exists). We call the delta (or Hilger) derivative of f at t. The function f is delta (or Hilger) differentiable provided exists for all . The function is then called the delta derivative of f on .
Definition 2.4 ([, Definition 2.7])
Definition 2.5 ([, Definition 2.5])
The Δ-measure and Δ-integration are defined as those in .
Definition 2.6 ([, Definition 2.7])
Definition 2.7 ([, Definition 2.3])
with the norm
We have the following theorem.
Theorem 2.1 ([, Theorem 2.1])
Definition 2.8 ([, Definition 2.11])
Now, we recall the Sobolev space on defined in . For the sake of convenience, in the sequel we let .
Definition 2.9 ([, Definition 2.12])
It follows from Remark 2.2 in  that
Theorem 2.2 ([, Theorem 2.5])
are satisfied and
Theorem 2.3 ([, Theorem 2.21])
Theorem 2.4 ([, Theorem 2.23])
Theorem 2.5 ([, Theorem 2.25])
Theorem 2.6 ([, Theorem 2.27])
3 Variational setting
In this section, we recall some basic facts which will be used in the proofs of our main results. In order to apply the critical point theory, we make a variational structure. From this variational structure, we can reduce the problem of finding solutions of (1.1) to the one of seeking the critical points of a corresponding functional.
Combining (3.1), we have
Considering the above, we introduce the following concept solution for problem (1.1).
By Definition 3.1 and Lemma 3.1, the weak solutions of problem (1.1) correspond to the critical points of φ.
We see that
Remark 3.1K has only finitely many eigenvalues with since K is compact on . Hence is finite dimensional. Notice that is a compact perturbation of the self-adjoint operator I. By a well-known theorem, we know that 0 is not in the essential spectrum of . Hence, is a finite dimensional space too.
To prove our main results, we need the following definitions and theorems.
Definition 3.2 ([, ])
Firstly, we state the local linking theorem.
Definition 3.3 ([, Definition 2.2])
contains a subsequence which converges to a critical point of I.
Theorem 3.1 [, Theorem 2.2]
(I3) Imaps bounded sets into bounded sets.
ThenIhas at least two critical points.
Secondly, we state another three critical point theorems.
Theorem 3.2 ([, Theorem 5.29])
Theorem 3.3 ([, Theorem 9.12])
thenIpossesses an unbounded sequence of critical values.
In order to state another critical point theorem, we need the following notions. Let X and Y be Banach spaces with X being separable and reflexive, and set . Let be a dense subset. For each , there is a semi-norm on E defined by
For a functional , we write . Recall that is said to be weak sequentially continuous if, for any in E, one has for each , i.e., is sequentially continuous. For , we say that Φ satisfies the condition if any sequence such that and as contains a convergent subsequence.
Theorem 3.4 ()
4 Main results
First of all, we give two existence results.
Theorem 4.1Suppose that (A) and the following conditions are satisfied.
Then problem (1.1) has at least two weak solutions. The one is a nontrivial weak solution, the other is a trivial weak solution.
Then we have
We divide our proof into four parts in order to show Theorem 4.1.
From (F5) and (2.5), we have that
Thus, by (4.1), (4.10) and (4.11), we obtain
Secondly, we show that φ maps bounded sets into bounded sets.
It follows from (3.2), (4.4), (4.5) and (4.6) that
This implies that
On the other hand, it follows from (F6) that
This implies that
Thus, by Theorem 3.1, problem (1.1) has at least one nontrivial weak solution. The proof is complete. □
then all conditions of Theorem 4.1 hold. According to Theorem 4.1, problem (4.22) has at least one nontrivial weak solution. In fact,
is the solution of problem (4.22).
Theorem 4.2Assume that (A), (F5), (F6), (F7) and the following conditions are satisfied.
Then problem (1.1) has at least one nontrivial weak solution.
Firstly, we prove that φ satisfies the (PS) condition. Indeed, let be a sequence such that and as . As the proof of Theorem 4.1, it suffices to show that is bounded in . By (F9) there exist positive constants , such that
(see ). By (F9), (4.11) and (4.23), we have
Combining (3.2), (4.6), (4.11) and (4.25), we obtain
Moreover, we can prove that is compact (see [, p.1437]). It follows from (3.4), (4.29) and Lemma 4.1 that φ satisfies the conditions (I5), (I6) and (I7)(i) with of Theorem 3.2.
Thus, we have
Next, we given two multiplicity results.
Theorem 4.3Assume that (A), (F5), (F7), (F8), (F9) and the following conditions are satisfied.
Then problem (1.1) has an unbounded sequence of weak solutions.
Moreover, by (F10) and (F12), we know that φ is even and . In view of Theorem 3.3, φ has a sequence of critical points such that . If is bounded in E, then by the definition of φ, one knows that is also bounded, a contradiction. Hence, is unbounded in E. The proof is completed. □
Theorem 4.4Assume that (A), (F5), (F7), (F8), (F9), (F11) and the following condition are satisfied.
Then problem (1.1) has an infinite sequence of distinct weak solutions.
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183, the Natural Sciences Foundation of Yunnan Province (2011Y116, 2012FB111, IRTSTYN) and the third batch young skeleton teachers training plan of Yunnan University (XT412003).
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