Multiple Periodic Solutions for a Class of Second-Order Neutral Impulsive Functional Differential Equations

Abstract

In this paper, we study a class of second-order neutral impulsive functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of critical point theory and variational methods. We propose an example to illustrate the applicability of our result.

1. Introduction and Main Results

In this paper we consider a class of second-order neutral impulsive functional differential equationswhere , , and . The operator is defined as , where denotes the right-hand (left-hand) limit of at . , is a constant with and is a given positive integer.

The necessity to study delay differential equations is due to the fact that these equations are useful mathematical tools in modeling many real processes and phenomena studied in biology, medicine, chemistry, physics, engineering, economics, and so forth [1, 2].

On the other hand, impulsive differential equation not only is richer than the corresponding theory of differential equations but also represents a more natural framework for mathematical modeling of real world phenomena. People generally consider impulses in positions and for the second-order differential equation . However it is well known that in the motion of spacecraft instantaneous impulses depend on the position which result in jump discontinuities in velocity, with no change in position.

Thus, it is more realistic to consider the case of combined effects: impulses and time delays. This motivates us to consider neutral impulsive functional differential system (1).

The existence of periodic solutions of delay differential equations has been focused on by many researchers [3–6]. Several available approaches to tackle them include Lyapunov method, Fourier analysis method, fixed point theory, and coincidence degree theory [7–10]. Recently, some researchers have studied the existence of solutions for delay differential equations via variational methods [11–13]. In recent years, some researchers, by using critical point theory, have studied the existence of solutions for boundary value problems, periodic solutions, and homoclinic orbits of impulsive differential systems [14–19].

In this paper, we aim to establish existence of multiple periodic solutions for the second-order neutral impulsive functional differential equation (1) by using critical point theory and variational methods.

For (1) with , Shu and Xu [20] obtained the following periodic solutions result.

Theorem A. Assume that the following conditions are satisfied. (H1).(H2) There exists a function such that (H3) is -periodic in .(H4) satisfies and .(H5) if and only if , .(H6), where , .(H7) There exists a constant such that when , , .Moreover, if there exists an integer such that satisfying then the system possesses at least nonzero solutions with the period .

Our main result is stated as follows.

Theorem 1. Assume that (H1)–(H7) and the following condition are satisfied. (H8) is odd about , and there exists a constant such that , where .Moreover, if there exists an integer such thatthen system (1) admits at least nonzero solutions with the period .

Clearly, when , Theorem 1 generalizes Theorem  A.

Note that the first equation of system (1) is equivalent to the following equation:where and .

The rest of this paper is organized as follows. In Section 2, we present some preliminaries, which will be used to prove our main result. In Section 3 we prove our main result and provide an example to illustrate the applicability of our results.

2. Some Preliminaries

Let Then is a separable and reflexive Banach space and the inner product induces the norm

Definition 2. A function is a solution of system (1) if the function satisfies system (1).

Define a functional asThen is Fréchet differentiable at any . For any , by a simple calculation, we have From (H3), we get

Therefore, the corresponding Euler equation of functional is

Note that (6) is equivalent to system (13) and critical points of the functional are classical -periodic solutions of system (1).

Definition 3 (see [21]). Let be a real reflexive Banach space, and Define as follows: Then we say is the genus of .

Denote and , where .

Lemma 4 (see [22]). Let be a real Banach space and with even functional and satisfying the Palais-Smale (PS) condition. Suppose and (i)if there exist an -dimensional subspace of and a constant such that where is an open ball of radius in centered at , then we have ;(ii)if there exists -dimensional subspace of such that then we have . Moreover, if , then possesses at least distinct critical points.

3. Proof of Theorem 1 and an Example

We apply Lemma 4 to finish the proof. Under assumption (H4), it is easy to see that if function is a solution of system (1), then function is also a solution of system (1). Therefore, the solutions of system (1) are a set which is symmetric with respect to the origin in . It follows directly from (10), (H5), and (H8) that is even in and . The rest of the proof is divided into three steps.

Step 1. We show that the functional satisfies assumption (ii) of Lemma 4.
It follows from (H7) that there exists a constant such thatwhere . Combining (10) and (18), we getwhich implies that is bounded from below. By condition (ii) of Lemma 4, we have .

Step 2. We show that the functional satisfies the PS condition.
For any given sequence such that is bounded and , there exists a constant such thatwhere is the dual space of .
Combining (19) and (20), we haveIt follows that is bounded.
Since is a reflexive Banach space, so we may extract a weakly convergent subsequence, for simplicity, we also note again by , in . So we haveTherefore, by (22), we have . Hence the functional satisfies the PS condition.

Step 3. We show that the functional satisfies assumption (i) of Lemma 4.
Let , . By calculations, we obtain Define the -dimensional linear subspace as follows: It is clear to see that is a symmetric set. Take , when , where denotes boundary of , has expansion , , andBy (H6), for given with , there exists such that when , we have Combining (10), (25), and (26), when , we have Therefore . Consequently, system (1) admits at least nonzero -periodic solutions.

We conclude this section with the following example.

Example 5. Consider (1) with It is easy to see that and when , ; then (H1) and (H5) hold. Set , , , and then . By a simple computation, we have , , and . So conditions (H2)–(H4) hold. Clearly, the conditions (H6)–(H8) hold. Therefore system (1) admits at least nonzero solutions with the period .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by Hunan Provincial Natural Science Foundation of China (no. 2016JJ6122), National Natural Science Foundation of China (nos. 11661037 and 11471109), and Jishou University Doctor Science Foundation (no. jsdxxcfxbskyxm201504).