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

1. Introduction and Main Results

In this paper we consider a class of second-order neutral impulsive functional differential equations

[mathematical expression not reproducible], (1)

where f [member of] C([R.sup.4], R), [I.sub.j] [member of] C(R, R), and 0 = [t.sub.0] < [t.sub.1] < [t.sub.2] < ... < [t.sub.l] < [t.sub.l+1] = 2k[tau]. The operator [DELTA] is defined as [DELTA]u' ([t.sub.j]) = u'([t.sup.+.sub.j])-u([t.sup.-.sub.j]), where u'([t.sup.+.sub.j])(u'([t.sup.-.sub.j])) denotes the right-hand (left-hand) limit of u' at [t.sub.j]. [lambda] [member of] R, [tau] is a constant with [tau] > 0 and k 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 u and u' for the second-order differential equation u" = f(t, u, u). 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 homo clinic 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 [I.sub.j] = 0, Shu and Xu [20] obtained the following periodic solutions result.

Theorem A. Assume that the following conditions are satisfied.

(H1) [partial derivative]f (t, [u.sub.1], [u.sub.2], [u.sub.3])/[partial derivative]t [not equal to] 0.

(H2) There exists a function F(t, [u.sub.1], [u.sub.2]) [member of] [C.sup.1]([R.sup.3], R) such that

[partial derivative]F(t, [u.sub.1], [u.sub.2])/[partial derivative][u.sub.2] + [partial derivative]F (t, [u.sub.1], [u.sub.2], [u.sub.3]). (2)

(H3) F(t, [u.sub.1], [u.sub.2]) is [tau]-periodic in t.

(H4) F satisfies F(t, -[u.sub.1], -[u.sub.2]) = F(t, [u.sub.1], [u.sub.2]) and f(t, -[u.sub.1], -[u.sub.2], -[u.sub.3]) = -f(t, [u.sub.1], [u.sub.2], [u.sub.3]).

(H5) F(t, [u.sub.1],[u.sub.2]) = 0 if and only if ([u.sub.1], [u.sub.2]) = 0, [for all]t [member of] [0, [tau]].

(H6) [lim.sub.[absolute value of u][right arrow]0] (F(t, [u.sub.1], [u.sub.2])/[[absolute value of u].sup.2]) = 1, where [absolute value of u] = [([[absolute value of [u.sub.1]].sup.2] + [[absolute value of [u.sub.2].sup.2]).sup.1/2], t [member of] [0, [tau]].

(H7) There exists a constant [alpha] > 0 such that when [[absolute value of [u.sub.1]].sup.2] + [[absolute value of [u.sub.2]].sup.2] > [[alpha].sub.2], F(t, [u.sub.1], [u.sub.2]) < 0, t [member of] [0, [tau]].

Moreover, if there exists an integer m > 0 such that [lambda] satisfying

[lambda] > [m.sup.2] ([[pi].sup.2] + [k.sup.2] [[tau].sup.2])/4k[[tau].sup.2], (3)

then the system

[mathematical expression not reproducible] (4)

possesses at least 2m nonzero solutions with the period 2k[tau].

Our main result is stated as follows.

Theorem 1. Assume that (H1)-(H7) and the following condition are satisfied.

(H8) [I.sub.j] is odd about u, and there exists a constant 0 [less than or equal to] D < 1 such that [absolute value of [I.sub.j](u)] [less than or equal to] D[absolute value of u], where j = 1, 2, ..., 1.

Moreover, if there exists an integer m > 0 such that

[lambda] > [m.sup.2] ([[pi].sup.2] + (1 + D) [k.sup.2][[tau].sup.2])/4[k.sup.2][[tau].sup.2], (5)

then system (1) admits at least 2m nonzero solutions with the period 2k[tau].

Clearly, when [I.sub.j] = 0, Theorem 1 generalizes Theorem A.

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

u" (t - [tau]) - u(t - [tau]) + [lambda] ([F'.sub.1] (t, u(t - [tau]), u(t - 2[tau])) + [F'.sub.2] (t, u(t), u(t - [tau]))) = 0, (6)

where [F'.sub.1] (t, u(t-[tau]), u(t-2[tau])) = [partial derivative]F(t, u(t-[tau]), u(t-2[tau]))/[partial derivative]u(t-[tau]) and [F'.sub.2] (t, u(t), u(t-[tau])) = [partial derivative]F(t, u(t), u(t-[tau]))/[partial derivative]u(t-[tau]).

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

[H.sup.1.sub.2k[tau]] = {u : R [right arrow] R | u, u' [member of] [L.sup.2] (([0, 2k[tau]]), R), u (0) = u (2k[tau]), u'(0) = u (2k[tau])}.

Then [H.sup.1.sub.2k[tau]] is a separable and reflexive Banach space and the inner product

[mathematical expression not reproducible] (8)

induces the norm

[mathematical expression not reproducible]. (9)

Definition 2. A function u [member of] [H.sup.1.sub.2k[tau]] is a solution of system (1) if the function u satisfies system (1).

Define a functional [phi] as

[mathematical expression not reproducible]. (10)

Then [phi] is Frechet differentiable at any u [member of] [H.sup.1.sub.2k[tau]]. For any v [member of] [H.sup.1.sub.2k[tau]], by a simple calculation, we have

[mathematical expression not reproducible]. (11)

From (H3), we get

[mathematical expression not reproducible]. (12)

Therefore, the corresponding Euler equation of functional [phi] is

u" (t) - u (t) + [lambda] ([F'.sub.1], (t, u(t), u(t - [tau])) + [F'.sub.2] (t, u(t + [tau]), u(t))) (13) = 0. (13)

Note that (6) is equivalent to system (13) and critical points of the functional [phi] are classical 2k[tau]-periodic solutions of system (1).

Definition 3 (see [21]). Let E be a real reflexive Banach space, and

[summation] = {A | A [subset] E \ {0} is closed, symmetric set}. (14)

Define [gamma] : [summation] [right arrow] [Z.sup.+] [union] {+[infinity]} as follows:

[mathematical expression not reproducible]. (15)

Then we say [gamma] is the genus of [summation].

Denote [i.sub.1]([phi]) = [lim.sub.a[right arrow]-0] [gamma]([[phi].sub.a]) and [i.sub.2]([phi]) = [lim.sub.a[right arrow]-[infinity]] [gamma]([[phi].sub.a]), where [[phi].sub.a] = {u [member of] E | [phi](u) [less than or equal to] a}.

Lemma 4 (see [22]). Let E be a real Banach space and [phi] [member of] [C.sup.1](E, R) with p even functional and satisfying the Palais-Smale (PS) condition. Suppose [phi](0) = 0 and

(i) if there exist an m-dimensional subspace X of E and a constant r > 0 such that

[mathematical expression not reproducible], (16)

where [B.sub.r] is an open ball of radius r in E centered at 0, then we have [i.sub.1] ([phi]) [greater than or equal to] m;

(ii) if there exists j-dimensional subspace V of E such that

[mathematical expression not reproducible], (17)

then we have [i.sub.2]([phi]) [less than or equal to] j.

Moreover, if m [greater than or equal to] j, then p possesses at least 2(m-j) 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 u is a solution of system (1), then function -u 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 [H.sup.1.sub.2k[tau]]. It follows directly from (10), (H5), and (H8) that [phi] is even in u and [phi](0) = 0. The rest of the proof is divided into three steps.

Step 1. We show that the functional p satisfies assumption (ii) of Lemma 4.

It follows from (H7) that there exists a constant M > 0 such that

[mathematical expression not reproducible], (18)

where [omega] = [0, [tau]] x [-[alpha], [alpha]] x [-[alpha], [alpha]]. Combining (10) and (18), we get

[mathematical expression not reproducible], (19)

which implies that [phi] is bounded from below. By condition (ii) of Lemma 4, we have [i.sub.2]([phi]) = 0.

Step 2. We show that the functional [phi] satisfies the PS condition.

For any given sequence {[u.sub.n]} [member of] [H.sup.1.sub.2k[tau]] such that {[phi]([u.sub.n])} is bounded and [lim.sub.n[right arrow][infinity]] [phi]' ([u.sub.n]) = 0, there exists a constant [C.sub.1] such that

[mathematical expression not reproducible], (20)

where [([H.sup.1.sub.2k[tau]]).sup.*] is the dual space of [H.sup.1.sub.2k[tau]].

Combining (19) and (20), we have

[mathematical expression not reproducible]. (21)

It follows that [mathematical expression not reproducible] is bounded.

Since [H.sup.1.sub.2k[tau]] is a reflexive Banach space, so we may extract a weakly convergent subsequence, for simplicity, we also note again by {[u.sub.n]}, [u.sub.n] [??] u in [H.sup.1.sub.2k[tau]]. So we have

[mathematical expression not reproducible]. (22)

Therefore, by (22), we have [mathematical expression not reproducible]. Hence the functional [phi] satisfies the PS condition.

Step 3. We show that the functional <p satisfies assumption (i) of Lemma 4.

Let [[beta].sub.j](t) = (k[tau]/j[pi])sin(j[pi]/[kappa][tau])t, j = 1, 2, ..., m. By calculations, we obtain

[mathematical expression not reproducible]. (23)

Define the m-dimensional linear subspace as follows:

[E.sub.m] = span {[beta.sub.1](t), [[beta].sub.2](t), ..., [[beta].sub.m](t)}. (24)

It is clear to see that [E.sub.m] is a symmetric set. Take r > 0, when u(t) [member of] [E.sub.m] [intersection] [S.sub.r], where [S.sub.r] denotes boundary of [B.sub.r], u(t) has expansion u(t) = [[summation].sup.m.sub.j=1] [b.sub.j][[beta].sub.j](t), [b.sub.j] [member of] R, and

[mathematical expression not reproducible]. (25)

By (H6), for given [member of] with 0 < [member of] < ([lambda][m.sup.2]/4[k.sup.2][[tau].sup.2])(4[k.sup.2][[tau].sup.2]/ [m.sup.2]-([[pi].sup.2] + (1 + D)[k.sup.2][[tau].sup.2])/[lambda]), there exists 0 < [delta] < 1 such that when [([[absolute value of u(t)].sup.2] + [[absolute value of u(t - [tau])].sup.2]).sup.1/2] < [delta], we have

[lambda]F (t, u(t), u(t - [tau])) > {[lambda] - e)([[absolute value of u(t)].sup.2] + [[absolute value of u(t - [tau])].sup.2]}. (26)

Combining (10), (25), and (26), when u(t) [member of] [E.sub.m] [intersection] [S.sub.r], we have

[mathematical expression not reproducible]. (27)

Therefore [i.sub.1] ([phi]) [greater than or equal to] m. Consequently, system (1) admits at least 2m nonzero 2k[tau]-periodic solutions.

We conclude this section with the following example.

Example 5. Consider (1) with

[mathematical expression not reproducible]. (28)

It is easy to see that

[partial derivative]f(t, [u.sub.1], [u.sub.2], [u.sub.3])/[partial derivative]t [not equal to] 0 and when ([u.sub.1], [u.sub.2]) = 0, F(t, [u.sub.1], [u.sub.2]) = 0; then (H1) and (H5) hold. Set [mathematical expression not reproducible]. By a simple computation, we have F(t + [tau], [u.sub.1], [u.sub.2]) = F(t, [u.sub.1], [u.sub.2]), F(t, -[u.sub.1], -[u.sub.2]) = F(t, [u.sub.1], [u.sub.2]), and f(t, -[u.sub.1], -[u.sub.2], -[u.sub.3]) = -f(t, [u.sub.1], [u.sub.2], [u.sub.3]). So conditions (H2)-(H4) hold. Clearly, the conditions (H6)-(H8) hold. Therefore system (1) admits at least 2m nonzero solutions with the period 2k[tau].

http://dx.doi.org/10.1155/2017/5041783

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).

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Jingli Xie, (1) Zhiguo Luo, (2) and Yuhua Zeng (3)

(1) College of Mathematics and Statistics, Jishou University, Jishou, Hunan 416000, China

(2) Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

(3) Department of Mathematics, Hunan First Normal University, Changsha, Hunan 410205, China

Correspondence should be addressed to Jingli Xie; xiejingli721124@163.com

Received 29 August 2016; Revised 1 December 2016; Accepted 20 December 2016; Published 12 January 2017

Academic Editor: Olfa Boubaker
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Title Annotation:Research Article
Author:Xie, Jingli; Luo, Zhiguo; Zeng, Yuhua
Publication:Mathematical Problems in Engineering
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