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Existence and multiplicity of solutions for some second-order systems on time scales with impulsive effects
Boundary Value Problems volume 2012, Article number: 148 (2012)
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.
Consider 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 satisfies the following assumption:
is Δ-measurable in t for every and continuously differentiable in x for Δ-a.e. , and there exist , such that
for all and Δ-a.e. , where denotes the gradient of in x.
For the sake of convenience, in the sequel, we denote , .
When , , and is a zero matrix, (1.1) is the Hamiltonian system on time scales
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).
When , , and is not a zero matrix, until now the variational structure of (1.1) has not been studied.
Problem (1.1) covers the second-order Hamiltonian system with impulsive effects (when )
as well as the second-order discrete Hamiltonian system (when , , )
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])
Let be a time scale. For , the forward jump operator is defined by
while the backward jump operator is defined by
(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, .
When , , we denote the intervals , and in by
respectively. Note that if b is left-dense and if b is left-scattered. We denote , therefore if b is left-dense and if b is left-scattered.
Definition 2.2 ([, Definition 1.10])
Assume that is a function and let . Then we define to be the number (provided it exists) with the property that given any , there is a neighborhood U of t (i.e., for some ) such that
We call the delta (or Hilger) derivative of f at t. The function f is delta (or Hilger) differentiable on provided exists for all . The function is then called the delta derivative of f on .
Definition 2.3 ([, Definition 2.3])
Assume that is a function,
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])
For a function , we will talk about the second derivative provided is differentiable on with derivative .
Definition 2.5 ([, Definition 2.5])
For a function , we will talk about the second derivative provided is differentiable on with derivative .
The Δ-measure and Δ-integration are defined as those in .
Definition 2.6 ([, Definition 2.7])
Assume that is a function, and let A be a Δ-measurable subset of . f is integrable on A if and only if () are integrable on A, and .
Definition 2.7 ([, Definition 2.3])
Let . B is called a Δ-null set if . Say that a property P holds Δ-almost everywhere (Δ-a.e.) on B, or for Δ-almost all (Δ-a.a.) if there is a Δ-null set such that P holds for all .
For , , we set the space
with the norm
We have the following theorem.
Theorem 2.1 ([, Theorem 2.1])
Let be such that . Then the space is a Banach space together with the norm . Moreover, is a Hilbert space together with the inner product given for every by
where denotes the inner product in .
Definition 2.8 ([, Definition 2.11])
A function . We say that f is absolutely continuous on (i.e., ) if for every , there exists such that if is a finite pairwise disjoint family of subintervals of satisfying , then .
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])
Let be such that and . We say that if and only if and there exists such and
For , , we denote
It follows from Remark 2.2 in  that
is true for every with . These two sets are, as a class of functions, equivalent. It is the characterization of functions in in terms of functions in too. That is the following theorem.
Theorem 2.2 ([, Theorem 2.5])
Suppose that for some with , and that (2.1) holds for . Then there exists a unique function such that the equalities
are satisfied and
By identifying with its absolutely continuous representative for which (2.2) holds, the set can be endowed with the structure of a Banach space. That is the following theorem.
Theorem 2.3 ([, Theorem 2.21])
Assume and . The set is a Banach space together with the norm defined as
Moreover, the set is a Hilbert space together with the inner product
The Banach space has some important properties.
Theorem 2.4 ([, Theorem 2.23])
There exists such that the inequality
holds for all , where .
Moreover, if , then
Theorem 2.5 ([, Theorem 2.25])
If the sequence converges weakly to u in , then converges strongly in to u.
Theorem 2.6 ([, Theorem 2.27])
Let be Δ-measurable in t for each and continuously differentiable in for Δ-almost every . If there exist , and () such that for Δ-almost and every , one has
where , then the functional defined as
is continuously differentiable on and
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.
If , by identifying with its absolutely continuous representative for which (2.2) holds, then u is absolutely continuous and . In this case, may not hold for some . This leads to impulsive effects.
Take and multiply the two sides of the equality
by and integrate on , then we have
Moreover, combining , one has
Combining (3.1), we have
Considering the above, we introduce the following concept solution for problem (1.1).
Definition 3.1 We say that a function is a weak solution of problem (1.1) if the identity
holds for any .
Consider the functional defined by
Lemma 3.1 The functional φ is continuously differentiable on and
Proof Set for all and . Then satisfies all assumptions of Theorem 2.6. Hence, by Theorem 2.6, we know that the functional ψ is continuously differentiable on and
for all .
On the other hand, by the continuity of , , , one has that and
for all . Thus, φ is continuously differentiable on and (3.3) holds. □
By Definition 3.1 and Lemma 3.1, the weak solutions of problem (1.1) correspond to the critical points of φ.
Moreover, we need more preliminaries. For any , let
We see that
where is the bounded self-adjoint linear operator defined, using the Riesz representation theorem, by
and I denote an identity matrix and an identity operator, respectively. By (3.2), can be rewritten as
The compact imbedding of into implies that K is compact. By classical spectral theory, we can decompose into the orthogonal sum of invariant subspaces for
where and , are such that, for some ,
Remark 3.1 K 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 ([, ])
Let X be a real Banach space and . I is said to be satisfying (PS) condition on X if any sequence for which is bounded and as , possesses a convergent subsequence in X.
Firstly, we state the local linking theorem.
Let X be a real Banach space with a direct decomposition . Consider two sequences of a subspace
For every multi-index , we denote by the space . We say , . A sequence is admissible if, for every , there is such that .
Definition 3.3 ([, Definition 2.2])
Let . The functional I satisfies the condition if every sequence such that is admissible and
contains a subsequence which converges to a critical point of I.
Theorem 3.1 [, Theorem 2.2]
Suppose that satisfies the following assumptions:
(I1) and I has a local linking at 0 with respect to ; that is, for some ,
(I2) I satisfies condition.
(I3) I maps bounded sets into bounded sets.
(I4) For every , as , .
Then I has at least two critical points.
Remark 3.2 Since , by the condition (I1) of Theorem 3.1, 0 is the critical point of I. Thus, under the conditions of Theorem 3.1, I has at least one nontrivial critical point.
Secondly, we state another three critical point theorems.
Theorem 3.2 ([, Theorem 5.29])
Let E be a Hilbert space with and . Suppose , satisfies (PS) condition, and
(I5) , where and is bounded and self-adjoint, ,
(I6) is compact, and
(I7) there exist a subspace and sets , and constants such that
Q is bounded and ,
S and ∂Q link.
Then I possesses a critical value .
Theorem 3.3 ([, Theorem 9.12])
Let E be a Banach space. Let be an even functional which satisfies the (PS) condition and . If , where V is finite dimensional, and I satisfies
(I8) there are constants such that , where ,
(I9) for each finite dimensional subspace , there is an such that on ,
then I possesses 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
We denote by the topology on E induced by a semi-norm family , and let w and denote the weak-topology and weak*-topology, respectively.
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.
() for any , is -closed, and is continuous;
() there exists such that , where
() there exist a finite dimensional subspace and such that and , where
Theorem 3.4 ()
Assume that Φ is even and ()-() are satisfied. Then Φ has at least pairs of critical points with critical values less than or equal to provided Φ satisfies the condition for all .
Remark 3.3 In our applications, we take = so that is the product topology on given by the weak topology on X and the strong topology on Y.
4 Main results
Lemma 4.1 is compact on .
Proof Let be any bounded sequence. Since is a Hilbert space, we can assume that . Theorem 2.5 implies that . By (2.5), we have
The continuity of and this imply that in . The proof is complete. □
First of all, we give two existence results.
Theorem 4.1 Suppose that (A) and the following conditions are satisfied.
(F1) uniformly for Δ-a.e. ,
(F2) uniformly for Δ-a.e. ,
(F3) there exist and such that
(F4) there exists such that
(F5) there exist and such that
(F6) for every , , ,
(F7) there exists such that
Then problem (1.1) has at least two weak solutions. The one is a nontrivial weak solution, the other is a trivial weak solution.
Proof By Lemma 3.1, . Set with being its Hilbertian basis, and define
Then we have
We divide our proof into four parts in order to show Theorem 4.1.
Firstly, we show that φ satisfies the condition.
Let be a sequence in such that is admissible and
then there exists a constant such that
for all large n. On the other hand, by (F3), there are constants and such that
for all and Δ-a.e. . By (A) one has
for all and Δ-a.e. . It follows from (4.2) and (4.3) that
for all and Δ-a.e. . Since for all , there exists a constant such that
From (F5) and (2.5), we have that
for all , where , . Combining (4.4), (4.5), (4.6) and Hölder’s inequality, we have
for all large n, where . On the other hand, by (F3), there exist and such that
for all and Δ-a.e. . By (A),
for all and Δ-a.e. , where . Combining (4.8) and (4.9), one has
for all and Δ-a.e. . According to (F7), there exists such that
Thus, by (4.1), (4.10) and (4.11), we obtain
for all large n. From (4.12), is bounded. If , by Hölder’s inequality, we have
Since for all , , by (4.7) and (4.13), is bounded in . If , by (2.5), we obtain
Since , , by (4.7) and (4.14), is also bounded in . Hence, is also bounded in . Going if necessary to a subsequence, we can assume that in . From Theorem 2.5, we have and . Since
This implies , and hence . Therefore, in . Hence φ satisfies the condition.
Secondly, we show that φ maps bounded sets into bounded sets.
It follows from (3.2), (4.4), (4.5) and (4.6) that
for all . Thus, φ maps bounded sets into bounded sets.
Thirdly, we claim that φ has a local linking at 0 with respect to .
Applying (F2), for , there exists such that
for all and Δ-a.e. . By (F7), for , there exists such that
Let . For with , by (2.5), (3.2), (3.6), (4.15) and (4.16), we have
This implies that
On the other hand, it follows from (F6) that
for all . Let satisfy . Using (F4), (2.5), (3.2), (3.5) and (4.17), we obtain
This implies that
Let . Then φ satisfies the condition of Theorem 3.1.
Finally, we claim that for every ,
For given , since is a finite dimensional space, there exists such that
By (F1), there exists such that
for all and Δ-a.e. . From (A), we get
for all and Δ-a.e. . Equations (4.19) and (4.20) imply that
for all and Δ-a.e. , where . Using (3.2), (3.6), (4.5), (4.17), (4.18) and (4.21), we have, for ,
where . Hence, for every , as and .
Thus, by Theorem 3.1, problem (1.1) has at least one nontrivial weak solution. The proof is complete. □
Example 4.1 Let , , , . Consider the second-order Hamiltonian system with impulsive effects
for all and ,
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 4.2 Assume that (A), (F5), (F6), (F7) and the following conditions are satisfied.
(F8) uniformly for Δ-a.e. ,
(F9) there exist constants and such that for all and ,
(F10) for all and Δ-a.e. .
Then problem (1.1) has at least one nontrivial weak solution.
Proof Set , and . Then E is a real Hilbert space, , and .
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
for large k, where . Equation (4.24) implies that there exists such that
Combining (3.2), (4.6), (4.11) and (4.25), we obtain
for large k. Since , , by (4.26), is bounded in .
For any small , by (F8) we know that there is a such that
By (F7), for , there exists such that
Let . For with , by (2.5), (3.2), (3.6), (4.27) and (4.28), we have
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.
Set , , , and . Then S and ∂Q link, where . Set
By (F10), (3.4), (3.5) and (4.17), we know . For each , one has and . By the equivalence of a finite dimensional space and (4.23), there exists such that
Thus, we have
for large due to .
Moreover, for each , one has , and . By the equivalence of a finite dimensional space and (4.23), one has
for large .
Summing up the above, φ satisfies all conditions of Theorem 3.2. Hence, φ possesses a critical value , and hence problem (1.1) has at least one nontrivial weak solution. The proof is complete. □
Remark 4.1 There are a number of functions satisfying (A), (F8), (F9) and (F10), for example, .
Next, we given two multiplicity results.
Theorem 4.3 Assume that (A), (F5), (F7), (F8), (F9) and the following conditions are satisfied.
(F11) (, ) are odd.
(F12) is even in x and .
Then problem (1.1) has an unbounded sequence of weak solutions.
Proof Set , and . Then , and . From the proof of Theorem 4.2, we know that φ satisfies the (PS) condition, and there exist and such that
For each finite dimensional subspace , combining (3.2), (4.5), (4.6), (4.23) and the equivalence of a finite dimensional space, there exists such that
This implies that there is an such that on .
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. □
Example 4.2 Let , , , , . Consider the second-order Hamiltonian system with impulsive effects
where is the unit matrix and
for all , . All conditions of Theorem 4.3 hold. According to Theorem 4.3, problem (4.31) has an unbounded sequence of weak solutions.
Remark 4.2 In Theorem 4.3, if we delete the condition ‘’, we have the following theorem.
Theorem 4.4 Assume that (A), (F5), (F7), (F8), (F9), (F11) and the following condition are satisfied.
(F13) is even in x.
Then problem (1.1) has an infinite sequence of distinct weak solutions.
Proof Set , and in Theorem 3.4. Then, from the proof of Theorem 4.3, we know that , , φ is even, satisfies the (PS) condition, and there are constants such that and , where .
For each finite dimensional subspace , by (4.30), we know that
Consequently, for each finite dimensional subspace , the condition () holds. Moreover, by and , we know that () holds too. Therefore, the conclusion follows from Theorem 2.6. □
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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).
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.