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Abstract and Applied Analysis

Volume 2014, Article ID 382970, 11 pages

http://dx.doi.org/10.1155/2014/382970
Review Article

Variational Approach to Impulsive Problems: A Survey of Recent Results

1Nanjing College of Information Technology, Nanjing 210046, China

2School of Science, Shandong University of Technology, Zibo 255049, China

Received 21 April 2014; Accepted 5 May 2014; Published 20 May 2014

Academic Editor: Yonghui Xia

Copyright © 2014 Fang-fang Liao and Juntao Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a survey on the existence of nontrivial solutions to impulsive differential equations by using variational methods, including solutions to boundary value problems, periodic solutions, and homoclinic solutions.

1. Introduction

There are many processes and phenomena in the real world, which are subjected during their development to the short-term external influences. Their duration is negligible compared with the total duration of the studied phenomena and processes. Therefore, it can be assumed that these external effects are “instantaneous”; that is, they are in the form of impulses. The investigation of such “leaps and bounds” developing dynamical states is a subject of different sciences: mechanics, control theory, pharmacokinetics, epidemiology, population dynamics, economics, ecology, and so forth. We refer the reader to [16].

During the last two decades, the theory of the existence of solutions for impulsive differential equations has attracted much attention because of its important applications; see [3, 712]. Classical approaches to such problems include fixed point theory and the method of upper and lower solutions. In 2008, Tian and Ge first applied variational methods in the study of boundary value problems (BVPs) of impulsive differential equations. From then on, variational methods have been widely used to study impulsive problems, such as boundary value problems, periodic solutions, and homoclinic solutions. We refer the reader to [3, 1322].

The main aim of this survey is to present some recent existence results for impulsive differential equations via variational methods. More precisely, we will explore the variational framework of impulsive differential equations and study the existence and multiplicity of solutions of boundary value problems, periodic solutions, and homoclinic solutions.

The rest of this paper is organized as follows. In Section 2, we will state some important results for the second-order impulsive differential equations with two kinds of boundary conditions. Periodic solutions to impulsive problems will be considered in Section 3. In Section 4, homoclinic solutions to impulsive differential equations will be studied.

Finally in this section, we must note that besides the results presented in this survey many interesting and important results on impulsive differential equations via variational methods have been obtained by other researchers; see, for example, [2331] and the references cited therein.

2. Boundary Value Problems

In this section, we recall some results of boundary value problems for impulsive differential equations.

Firstly, we consider a class of impulsive problems with Dirichlet boundary condition where is a constant, is continuous, and , , are continuous.

From now on, we assume that and define Then we have the following.

Proposition 1 (see [21, Proposition 1.1]). If , then there exist constants such that

Definition 2. One says that a function is a weak solution to problem (1) if holds for any .

Theorem 3 (see [15, Theorem 4.2]). Suppose that , is bounded, and the impulsive functions are bounded. Then problem (1) has at least one weak solution to .

If is sublinear at infinity on and the impulsive functions have sublinear growth, then we have the following.

Theorem 4 (see [15, Theorem 4.3]). Assume that and the following conditions are satisfied. ( )There exist and such that ( )there exist and such that Then problem (1) has at least one weak solution to .

If is superlinear at infinity on , then we have the following.

Theorem 5 (see [21, Theorem 1.2]). Assume that holds and the following conditions are satisfied. ( )There is a constant such that for every and ;( ) satisfies the following inequality where and is defined in Proposition 1.

Then problem (1) has at least one weak solution to .

Furthermore, we have the following result of multiple solutions.

Theorem 6 (see [21, Theorem 1.3]). Let . Assume that the conditions , , and hold. Moreover, and the impulsive functions are odd about ; then problem (1) has infinitely many solutions.

Next, we consider another Dirichlet impulsive problem where , is continuous, and , , are continuous.

Theorem 7 (see [20, Theorem 3.1]). Let be fixed constants. If and ; then problem (8) has a unique solution if ess .

Similar to Theorem 4, we have the following.

Theorem 8 (see [20, Theorem 3.2]). Assume that conditions and hold. Then problem (8) has at least one weak solution to ess .

If condition is replaced by the following condition:( ) satisfy for all ,then we have the following.

Theorem 9 (see [30, Theorem 3.1]). Assume that conditions and hold. Then problem (8) has at least one weak solution to ess .

Now we study a class of impulsive Hamiltonian systems with periodic boundary conditions where is a continuous map from the interval to the set of -order symmetric matrices, , are two positive parameters, is a real positive number, , , are the instants where the impulses occur and , are continuous, and are measurable with respect to , for every , continuously differentiable in , for almost every , and satisfy the following standard summability condition:

If is a real Banach space, denote by the class of all functionals possessing the following property: if is a sequence in converging weakly to and , then has a subsequence converging strongly to .

For example, if is uniformly convex and is a continuous, strictly increasing function, then, by a classical result, the functional belongs to the class .

Theorem 10 (see [32]). Let be a separable and reflexive real Banach space; let be a coercive, sequentially weakly lower semicontinuous functional, belonging to , bounded on each bounded subset of , whose derivative admits a continuous inverse on ;   is a functional with compact derivative. Assume that has a strict local minimum with . Finally, setting assume that .

Then, for each compact interval (with the conventions , there exists with the following property: for every and every functional with compact derivative, there exists such that, for each , has at least three solutions in whose norms are less than .

The following two results of Ricceri guarantee the existence of three solutions for a given equation.

Theorem 11 (see [33]). Let be a reflexive real Banach space; is an interval; let be a sequentially weakly lower semicontinuous functional, bounded on each bounded subset of , whose derivative admits a continuous inverse on ; is a functional with compact derivative. Assume that for all and that there exists such that Then there exist a nonempty open set and a positive number with the following property: for every and every functional with compact derivative, there exists such that, for each , has at least three solutions in whose norms are less than .

Proposition 12 (see [34]). Let be a nonempty set and two real functions on . Assume that there are and such that Then, for each , satisfying one has

We always assume that the following conditions hold: is a symmetric matrix with for every ; there exists a positive constant such that for any , and , are nondecreasing in and Define the Sobolev space by with the inner product where denotes the inner product in . The corresponding norm is defined by For every , we define and observe that, using assumptions and , (24) defines an inner product in , whose corresponding norm is A simple computation shows that for every and . So, putting together and (26), one has where and ; that is, the norm is equivalent to (23). At this point, since is compactly embedded in (see [35]), due to (27) we claim that there exists a positive number such that where and .

If , then is absolutely continuous and . In this case, is not necessarily valid for every and the derivative may present some discontinuities. This leads to the impulsive effects.

Following the ideas of [15], take and multiply the two sides of the equality by and integrate from to ; we have Moreover, combining , one has Combining (30), we get Considering the above equality, we introduce the following concept of a weak solution to problem (9).

Definition 13. One says that a function is a weak solution to problem (9) if holds for any .

In order to study problem (9), we will use the functionals defined by putting respectively, for every ; here, By the continuity of , , , we get that functional is a continuous Gâteaux differential functional whose Gâteaux derivative is the functional , given by for any . On the other hand, condition (10) implies that and are well defined, continuously Gâteaux differentiable in . More precisely, their Gâteaux derivatives are respectively, for every . Put Then we have the following.

Theorem 14 (see [16, Theorem 3.1]). Suppose that - and hold. Moreover, there exist a constant and constant vector such that    ; .

Then, for each compact interval , there exists with the following property: for every and there exists such that, for each , problem (9) has at least three weak solutions whose norms are less than .

Theorem 15 (see [16, Theorem 3.2]). Suppose that - and hold and there exist , three positive constants with , and such that ; here, ; for each , ; , for all and a.e. .Then, there exist a nonempty open set and a positive number with the following property: for every and every there exists such that, for each , problem (9) has at least three weak solutions whose norms are less than .

As a special case of , let , where , . Then we get the following two corollaries.

Corollary 16 (see [16, Corollary 3.1]). Suppose that - and hold. Moreover, there exist a constant and constant vector such that    ; .Then, for each compact interval , there exists with the following property: for every and there exists such that, for each , problem (9) has at least three weak solutions whose norms are less than .

Corollary 17 (see [16, Corollary 3.2]). Suppose that - and hold. Furthermore, there exist five positive constants , , , , and with and such that    ; for any , ; for every .Then, there exist a nonempty open set and a positive number with the following property: for every and every there exists such that, for each , problem (9) has at least three weak solutions whose norms are less than .

Finally, we consider the existence of infinitely many solutions for a class of impulsive systems with periodic boundary conditions where is a continuous map from the interval to the set of -order symmetric matrices, is a real positive number, , , , are the instants where the impulses occur and , are continuous, and are measurable with respect to , for every , continuously differentiable in , for almost every , and satisfy the following standard summability condition:

Now we state our main results.

Theorem 18 (see [36, Theorem 1.1]). Assume that is even in and the following conditions hold: are odd and satisfy there exist constants and such that uniformly for ; , as uniformly for ; there exist constants , , , and such that for every and with , Then problem (39) has infinitely many weak solutions.

Theorem 19 (see [36, Theorem 1.2]). Assume that the following conditions hold: are odd and satisfy for any ; there exist constants and such that , where is a positive constant and not a spectrum point of , is even in , and ; there exist with , and such that Then problem (39) has infinitely many weak solutions.

Theorem 20 (see [36, Theorem 1.3]). Assume that is even in , - and the following conditions hold: for any ; there are constants and with such that there exist constants and such that Then problem (39) has infinitely many weak solutions.

3. Periodic Solutions

Firstly, we consider periodic solutions for second-order regular impulsive problems with a perturbation: where , , with , , , , for each , and there exist and such that , , and for all (i.e., is -periodic in ).

Theorem 21 (see [20, Theorem 1]). Assume that is continuous and -periodic and is continuous, -periodic in . Furthermore, assume that and satisfy the following conditions: is continuous differentiable and -periodic, and there exist positive constants such that for all ; there exists such that Then there exists such that, for any positive integer , if , then problem (48) possesses at least one nonzero -periodic solution.

Theorem 22 (see [20, Theorem 1]). Suppose that is even about and satisfies , , and is continuous, odd about , -periodic in , and satisfies the following: for all , Then there exists such that if then problem (48) possesses at least distinct pairs of -periodic solutions generated by impulses.

Theorem 23 (see [14, Theorem 1]). Assume that and is continuous, -periodic in satisfying . Furthermore, assume that satisfies the following condition: is continuous differentiable and -periodic, and for all and .Then problem (48) possesses one nonzero periodic solution generated by impulses.

Corollary 24 (see [14, Corollary 1]). Assume that and is continuous, -periodic in satisfying . Furthermore, assume that satisfies the following condition: for all and , where for some and for all and .Then problem (48) possesses one nonzero periodic solution generated by impulses.

Theorem 25 (see [14, Theorem 2]). Assume that and and satisfy the following conditions: for all ; , where and satisfies the fact that there exists such that Then problem (48) possesses one nonzero periodic solution generated by impulses.

Now we discuss -periodic solution to second-order impulsive problems with singularity as follows: where , is -periodic, with ; and ; and .

Theorem 26 (see [18, Theorem 1.2]). Assume that is -periodic and ; there exist two constants such that, for any , where Then problem (54) has at least a positive -periodic solution.

Set with the inner product The corresponding norm is defined by Then is a Banach space (in fact it is a Hilbert space).

In order to study problem (54), for any , we consider the following modified problem: Following the ideas of [15], we introduce the following concept of a weak solution to problem (60).

Definition 27. One says that a function is a weak solution to problem (60) if holds for any .

Let be defined by and consider the functional defined by Clearly, is well defined on , continuously Gâteaux differentiable functional whose Gâteaux derivative is the functional , given by for any . Moreover, it is easy to verify that is weakly lower semicontinuous. Indeed, if , , and , then converges uniformly to on and on , and combining the fact that , we have By the standard discussion, the critical points of are the weak solutions of problem (1); see [15, 16].

The following version of the mountain pass theorem will be used in our argument.

Theorem 28 (see Theorem 4.10, [35]). Let be a Banach space and let . Assume that there exist and a bounded open neighborhood of such that and Let If satisfies the (PS) condition, that is, a sequence in satisfying is bounded and as has a convergent subsequence, then is a critical value of and .

If ( ), then we have the following result of problem (54).

Theorem 29 (see [17, Theorem 1.2]). Assume that is -periodic. Then problem (54) has a positive -periodic weak solution if and only if .

Remark 30. From Theorem 29 we can see that if but , then problem (54) still admits a positive -periodic solution. This shows that the existence of positive -periodic solution to problem (54) depends on the forced term and impulsive functions together, not single one.

4. Homoclinic Solutions

In this section, we are interested in the existence of homoclinic solutions for problem (48).

First we consider the case of . When is asymptotically linear, we show that there exists a nonzero homoclinic solution as the limit of a sequence -periodic solutions as goes to , and when is sublinear, we attain a nonzero homoclinic solution in the same way by strengthening the effect of impulses. The results are as follows.

Theorem 31 (see [14, Theorem 3]). Supposing that , is -periodic in and satisfies condition , and satisfies condition , then problem (48) possesses at least one nonzero homoclinic solution generated by impulses.

Theorem 32 (see [14, Theorem 4]). Supposing that , is -periodic in and satisfies condition , and satisfies condition , then problem (48) possesses at least one nonzero homoclinic solution generated by impulses.

If and for all , then we have the following result for problem (48) when satisfies the superlinear condition at infinity on .

Theorem 33 (see [13, Theorem 1.1]). Assume that and the following conditions hold: there exists a positive number such that uniformly for ; there exists a such that there exist constants and such that there exists a constant , with , such that Then there exists a nontrivial weak homoclinic solution to problem (48).

The proof of Theorem 33 is based on the following version of Mountain pass theorem.

Theorem 34 (see [37]). Let be a Banach space and let , , be such that and . Let </