Existence of anti-periodic solutions with symmetry for some high-order ordinary differential equations

Boundary Value Problems20122012:108

DOI: 10.1186/1687-2770-2012-108

Received: 25 May 2012

Accepted: 24 September 2012

Published: 9 October 2012

Abstract

The existence of anti-periodic solutions with symmetry for high-order Duffing equations and a high-order Duffing type p-Laplacian equation has been studied by using degree theory. The results obtained enrich some known works to some extent.

MSC:34B15, 34C25.

Keywords

anti-periodic solution with symmetry high-order ordinary differential equation p-Laplacian operator Leray-Schauder degree theory

1 Introduction

Anti-periodic problems arise naturally from the mathematical models of various physical processes (see [1, 2]) and also appear in the study of partial differential equations and abstract differential equations (see [35]). For instance, electron beam focusing system in traveling-wave tube theories is an anti-periodic problem (see [6]).

In mechanics, the simplest model of oscillation equation is a single pendulum equation
x + ω 2 sin x = e ( t ) ( e ( t + 2 π ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equa_HTML.gif
whose anti-periodic solutions satisfy
x ( t + π ) = x ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equb_HTML.gif

During the past twenty years, anti-periodic problems have been studied extensively by numerous scholars. For example, for first-order ordinary differential equations, a Massera’s type criterion was presented in [7] and the validity of the monotone iterative technique was shown in [8]. Moreover, for higher-order ordinary differential equations, the existence of anti-periodic solutions was considered in [912]. Recently, existence results were extended to anti-periodic boundary value problems for impulsive differential equations (see [13]), and anti-periodic wavelets were discussed in [14].

It is well known that higher-order p-Laplacian equations are derived from many fields such as fluid mechanics and nonlinear elastic mechanics. In the past few decades, many important results on higher-order p-Laplacian equations with certain boundary conditions have been obtained. We refer the readers to [1519] and the references cited therein.

In [10], the authors considered the existence of anti-periodic solutions for the high-order Duffing equation as follows:
x ( n ) + i = 1 n 1 a i x ( i ) + g ( t , x ) = e ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ1_HTML.gif
(1.1)
Moreover, in [15] the authors discussed the existence of anti-periodic solutions for the following higher-order Liénard type p-Laplacian equation:
( ϕ p ( x ( n ) ) ) ( n ) + f ( x ) x + g ( t , x ) = e ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ2_HTML.gif
(1.2)

However, to the best of our knowledge, there exist relatively few results on the existence of anti-periodic solutions with symmetry for (1.1) and (1.2). Thus, it is worthwhile to continue to investigate the existence of anti-periodic solutions with symmetry for (1.1) and (1.2).

Motivated by the works mentioned previously, in this paper, we study the existence of anti-periodic solutions with symmetry for high-order Duffing equations of the forms:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ3_HTML.gif
(1.3)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ4_HTML.gif
(1.4)
and high-order Duffing type p-Laplacian equation of the form:
( ϕ p ( x ( m + 1 ) ) ) ( m + 1 ) + g ( t , x ) = e ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ5_HTML.gif
(1.5)

where p > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq1_HTML.gif is a constant, m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq2_HTML.gif is an integer, ϕ p ( s ) = | s | p 2 s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq3_HTML.gif; a i R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq4_HTML.gif, g C ( R 2 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq5_HTML.gif, e C ( R , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq6_HTML.gif with g ( t + π , x ) g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq7_HTML.gif, e ( t + π ) e ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq8_HTML.gif. Obviously, the inverse operator of ϕ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq9_HTML.gif is ϕ q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq10_HTML.gif, where q > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq11_HTML.gif is a constant such that 1 p + 1 q = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq12_HTML.gif.

Notice that, when p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq13_HTML.gif, the nonlinear operator ( ϕ p ( x ( m + 1 ) ) ) ( m + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq14_HTML.gif reduces to the linear operator x ( 2 m + 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq15_HTML.gif. On the other hand, x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif is also a 2π-periodic solution if x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif is a π-anti-periodic solution. Hence, from the arguments in this paper, we can also obtain the existence results on periodic solutions for the above equations.

The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries. In Section 3 and Section 4, basing on the Leray-Schauder principle, we establish some existence theorems on anti-periodic solutions with symmetry of (1.3), (1.4) and (1.5). Our results are different from those of bibliographies listed in the previous texts.

2 Preliminaries

For the sake of convenience, we set
C k , π = { x C k ( R , R ) : x ( t + π ) x ( t ) } , k { 0 , 1 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equc_HTML.gif
with the norm
x C k = max i { 0 , 1 , , k } { x ( i ) 0 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equd_HTML.gif
where x 0 = max t [ 0 , 2 π ] | x ( t ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq17_HTML.gif, and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Eque_HTML.gif

with the norm C k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq18_HTML.gif.

Notice that, x C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq19_HTML.gif may be written as Fourier series as follows:
x ( t ) = i = 0 a 2 i + 1 cos ( 2 i + 1 ) t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equf_HTML.gif
and x C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq20_HTML.gif may be written as the following Fourier series:
x ( t ) = i = 0 b 2 i + 1 sin ( 2 i + 1 ) t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equg_HTML.gif
where a 2 i + 1 , b 2 i + 1 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq21_HTML.gif. We define the mapping J 0 : C 0 0 , π C 1 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq22_HTML.gif by
( J 0 x ) ( t ) = 0 t x ( s ) d s = i = 0 a 2 i + 1 2 i + 1 sin ( 2 i + 1 ) t , t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equh_HTML.gif
and the mapping J 1 : C 1 0 , π C 0 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq23_HTML.gif by
( J 1 x ) ( t ) = 0 t x ( s ) d s i = 0 b 2 i + 1 2 i + 1 = i = 0 b 2 i + 1 2 i + 1 cos ( 2 i + 1 ) t , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equi_HTML.gif

It is easy to prove that the mappings J 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq24_HTML.gif, J 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq25_HTML.gif are completely continuous by using the Arzelà-Ascoli theorem.

Next, we introduce a continuation theorem (see [20]) as follows.

Lemma 2.1 (Continuation theorem)

Let Ω be open bounded in a linear normal space X. Suppose that f is a completely continuous field on Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq26_HTML.gif. Moreover, assume that the Leray-Schauder degree
deg ( f , Ω , p ) 0 , for p X f ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equj_HTML.gif

Then the equation f ( x ) = p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq27_HTML.gif has at least one solution in Ω.

3 Anti-periodic solutions with symmetry of (1.3) and (1.4)

In this section, some existence results on anti-periodic solutions with symmetry of (1.3) and (1.4) will be given.

Theorem 3.1 Assume that

(H1) the functions g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq28_HTML.gif and e ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq29_HTML.gif are odd in t, i.e.,
g ( t , ) = g ( t , ) , e ( t ) = e ( t ) , t R ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equk_HTML.gif
(H2) there exist non-negative functions α 1 , β 1 C ( R , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq30_HTML.gif such that
| g ( t , x ) | α 1 ( t ) | x | + β 1 ( t ) , t , x R ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equl_HTML.gif

(H3) i = 1 m | a i | + α 1 0 1 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq31_HTML.gif.

Then (1.3) has at least one even anti-periodic solution x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif, i.e., x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif satisfies
x ( t + π ) = x ( t ) , x ( t ) = x ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equm_HTML.gif
Proof For making use of the Leray-Schauder degree theory to prove the existence of even anti-periodic solutions for (1.3), we consider the following homotopic equation of (1.3):
x ( 2 m + 1 ) = λ i = 1 m a i x ( 2 i 1 ) λ g ( t , x ) + λ e ( t ) , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ6_HTML.gif
(3.1)
Define the operator D 01 : C 0 2 m + 1 , π C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq32_HTML.gif by
( D 01 x ) ( t ) = x ( 2 m + 1 ) ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equn_HTML.gif
Obviously, the operator D 01 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq33_HTML.gif is invertible. Let N 01 : C 0 2 m 1 , π C 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq34_HTML.gif be the Nemytskii operator
( N 01 x ) ( t ) = i = 1 m a i x ( 2 i 1 ) ( t ) g ( t , x ( t ) ) + e ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equo_HTML.gif
By hypothesis (H1), it is easy to see that
( N 01 x ) ( t + π ) ( N 01 x ) ( t ) , ( N 01 x ) ( t ) ( N 01 x ) ( t ) , x C 0 2 m 1 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equp_HTML.gif
Thus, the operator N 01 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq35_HTML.gif sends C 0 2 m 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq36_HTML.gif into C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq37_HTML.gif. Hence, the problem of even anti-periodic solutions for (3.1) is equivalent to the operator equation
D 01 x = λ N 01 x , x C 0 2 m + 1 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equq_HTML.gif
From hypotheses (H2), (H3) and (5) in [10], for the possible even anti-periodic solution x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif of (3.1), there exists a prior bounds in C 0 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq38_HTML.gif, i.e., x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif satisfies
x C 2 m + 1 T 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ7_HTML.gif
(3.2)

where T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq39_HTML.gif is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator F λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq40_HTML.gif, which sends C 0 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq38_HTML.gif into C 0 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq38_HTML.gif, such that the fixed points of operator F 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq41_HTML.gif in some open bounded set are the even anti-periodic solutions of (1.3).

With this in mind, let us define the set as follows:
Ω 01 = { x C 0 2 m + 1 , π : x C 2 m + 1 < T 1 + 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equr_HTML.gif
Obviously, the set Ω 01 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq42_HTML.gif is a open bounded set in C 0 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq43_HTML.gif and zero element θ Ω 01 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq44_HTML.gif. Define the completely continuous operator F λ : Ω 01 ¯ C 0 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq45_HTML.gif by
F λ x = J 1 J 0 J 0 J 1 2 m + 1 λ N 01 x = λ D 01 1 N 01 x , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equs_HTML.gif
Let us define the completely continuous field h λ ( x ) : Ω 01 ¯ × [ 0 , 1 ] C 0 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq46_HTML.gif by
h λ ( x ) = x F λ x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equt_HTML.gif
By (3.2), we get that zero element θ h λ ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq47_HTML.gif for all λ [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq48_HTML.gif. So, the following Leray-Schauder degrees are well defined and
deg ( i d F 1 , Ω , θ ) = deg ( h 1 , Ω , θ ) = deg ( h 0 , Ω , θ ) = deg ( i d , Ω , θ ) = 1 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equu_HTML.gif

Consequently, the operator F 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq41_HTML.gif has at least one fixed point in Ω 01 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq42_HTML.gif by using Lemma 2.1. Namely, (1.3) has at least one even anti-periodic solution. The proof is complete. □

Theorem 3.2 Assume that

(H4) the function g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq28_HTML.gif is even in t, x and e ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq29_HTML.gif is even in t, i.e.,
g ( t , x ) = g ( t , x ) , e ( t ) = e ( t ) , t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equv_HTML.gif

and the assumptions (H2), (H3) are true.

Then (1.3) has at least one odd anti-periodic solution x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif, i.e., x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif satisfies
x ( t + π ) = x ( t ) , x ( t ) = x ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equw_HTML.gif
Proof We consider the homotopic equation (3.1) of (1.3). Define the operator D 11 : C 1 2 m + 1 , π C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq49_HTML.gif by
( D 11 x ) ( t ) = x ( 2 m + 1 ) ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equx_HTML.gif
Let N 11 : C 1 2 m 1 , π C 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq50_HTML.gif be the Nemytskii operator
( N 11 x ) ( t ) = i = 1 m a i x ( 2 i 1 ) ( t ) g ( t , x ( t ) ) + e ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equy_HTML.gif
By hypothesis (H4), it is easy to see that
( N 11 x ) ( t ) ( N 11 x ) ( t ) , x C 1 2 m 1 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equz_HTML.gif
Thus, the operator N 11 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq51_HTML.gif sends C 1 2 m 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq52_HTML.gif into C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq53_HTML.gif. Hence, the problem of odd anti-periodic solutions for (3.1) is equivalent to the operator equation
D 11 x = λ N 11 x , x C 1 2 m + 1 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equaa_HTML.gif
Our problem is reduced to construct one completely continuous operator G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq54_HTML.gif, which sends C 1 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq55_HTML.gif into C 1 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq55_HTML.gif, such that the fixed points of operator G 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq56_HTML.gif in some open bounded set are the odd anti-periodic solutions of (1.3). With this in mind, let us define the following set:
Ω 11 = { x C 1 2 m + 1 , π : x C 2 m + 1 < T 1 + 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equab_HTML.gif
Define the completely continuous operator G λ : Ω 11 ¯ C 1 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq57_HTML.gif by
G λ x = J 0 J 1 J 1 J 0 2 m + 1 λ N 11 x = λ D 11 1 N 11 x , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equac_HTML.gif

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 3.3 Assume that

(H5) the functions g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq28_HTML.gif and e ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq29_HTML.gif are even in t, i.e.,
g ( t , ) = g ( t , ) , e ( t ) = e ( t ) , t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equad_HTML.gif

and the assumptions (H2), (H3) are true.

Then (1.4) has at least one even anti-periodic solution.

Proof We consider the homotopic equation of (1.4) as follows:
x ( 2 m + 2 ) = λ i = 1 m a i x ( 2 i ) λ g ( t , x ) + λ e ( t ) , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ8_HTML.gif
(3.3)
Define the operator D 02 : C 0 2 m + 2 , π C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq58_HTML.gif by
( D 02 x ) ( t ) = x ( 2 m + 2 ) ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equae_HTML.gif
Let N 02 : C 0 2 m , π C 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq59_HTML.gif be the Nemytskii operator
( N 02 x ) ( t ) = i = 1 m a i x ( 2 i ) ( t ) g ( t , x ( t ) ) + e ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equaf_HTML.gif
By hypothesis (H5), it is easy to see that
( N 02 x ) ( t ) ( N 02 x ) ( t ) , x C 0 2 m , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equag_HTML.gif
Thus, the operator N 02 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq60_HTML.gif sends C 0 2 m , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq61_HTML.gif into C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq53_HTML.gif. Hence, the problem of even anti-periodic solutions for (3.3) is equivalent to the operator equation
D 02 x = λ N 02 x , x C 0 2 m + 2 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equah_HTML.gif
Our problem is reduced to construct one completely continuous operator L λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq62_HTML.gif, which sends C 0 2 m + 2 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq63_HTML.gif into C 0 2 m + 2 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq63_HTML.gif, such that the fixed points of operator L 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq64_HTML.gif in some open bounded set are the even anti-periodic solutions of (1.4). With this in mind, let us define the following set:
Ω 02 = { x C 0 2 m + 2 , π : x C 2 m + 2 < T 2 + 1 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equai_HTML.gif
where T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq65_HTML.gif is a positive constant independent of λ. Define the completely continuous operator L λ : Ω 02 ¯ C 0 2 m + 2 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq66_HTML.gif by
L λ x = J 1 J 0 J 1 J 0 2 m + 2 λ N 02 x = λ D 02 1 N 02 x , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equaj_HTML.gif

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 3.4 Assume that

(H6) the function g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq28_HTML.gif is odd in t, x and e ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq29_HTML.gif is odd in t, i.e.,
g ( t , x ) = g ( t , x ) , e ( t ) = e ( t ) , t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equak_HTML.gif

and the assumptions (H2), (H3) are true.

Then (1.4) has at least one odd anti-periodic solution.

Proof We consider the homotopic equation (3.3) of (1.4). Define the operator D 12 : C 1 2 m + 2 , π C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq67_HTML.gif by
( D 12 x ) ( t ) = x ( 2 m + 2 ) ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equal_HTML.gif
Let N 12 : C 1 2 m , π C 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq68_HTML.gif be the Nemytskii operator
( N 12 x ) ( t ) = i = 1 m a i x ( 2 i ) ( t ) g ( t , x ( t ) ) + e ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equam_HTML.gif
By hypothesis (H6), it is easy to see that
( N 12 x ) ( t ) ( N 12 x ) ( t ) , x C 1 2 m , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equan_HTML.gif
Thus, the operator N 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq69_HTML.gif sends C 1 2 m , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq70_HTML.gif into C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq37_HTML.gif. Hence, the problem of odd anti-periodic solutions for (3.3) is equivalent to the operator equation
D 12 x = λ N 12 x , x C 1 2 m + 2 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equao_HTML.gif
Our problem is reduced to construct one completely continuous operator P λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq71_HTML.gif which sends C 1 2 m + 2 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq72_HTML.gif into C 1 2 m + 2 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq72_HTML.gif, such that the fixed points of operator P 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq73_HTML.gif in some open bounded set are the odd anti-periodic solutions of (1.4). With this in mind, let us define the set as follows:
Ω 12 = { x C 1 2 m + 2 , π : x C 2 m + 2 < T 2 + 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equap_HTML.gif
Define the completely continuous operator P λ : Ω 12 ¯ C 1 2 m + 2 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq74_HTML.gif by
P λ x = J 0 J 1 J 0 J 1 2 m + 2 λ N 12 x = λ D 12 1 N 12 x , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equaq_HTML.gif

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

When g ( t , x ) = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq75_HTML.gif, we can remove the assumption (H2) in Theorem 3.1, Theorem 3.2 and obtain the following results.

Theorem 3.5 Assume that

(H7) i = 1 m | a i | 1 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq76_HTML.gif and the assumption (H1) is true.

Then (1.3) ( g ( t , x ) = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq75_HTML.gif) has at least one even anti-periodic solution.

Theorem 3.6 Suppose that the assumptions (H4), (H7) are true. Then (1.3) ( g ( t , x ) = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq75_HTML.gif) has at least one odd anti-periodic solution.

Basing on the proof of Theorem 2 in [10], for the possible anti-periodic solution x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif of (3.1) ( g ( t , x ) = g ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq75_HTML.gif), the hypothesis (H7) yields that there exists a prior bounds in C 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq77_HTML.gif, i.e., x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif satisfies
x C 2 m + 1 T 3 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equar_HTML.gif

where T 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq78_HTML.gif is a positive constant independent of λ. The remainder of the proof work of Theorem 3.5 and Theorem 3.6 is quite similar to the proof of Theorem 3.1 and Theorem 3.2, so we omit the details.

4 Anti-periodic solutions with symmetry of (1.5)

In this section, we will give some existence results on anti-periodic solutions with symmetry of (1.5).

Theorem 4.1 Assume that

(H8) there exist non-negative functions α 2 , β 2 C ( R , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq79_HTML.gif such that
| g ( t , x ) | α 2 ( t ) | x | p 1 + β 2 ( t ) , t , x R ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equas_HTML.gif

(H9) α 2 0 λ 1 ( m + 1 ) 1 < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq80_HTML.gif and the assumption (H5) is true.

Then (1.5) has at least one even anti-periodic solution.

Proof We consider the following homotopic equation of (1.5):
( ϕ p ( x ( m + 1 ) ) ) ( m + 1 ) = λ g ( t , x ) + λ e ( t ) , λ [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equ9_HTML.gif
(4.1)
Define the operator D 03 : D ( D 03 ) C 0 0 , π L 1 ( [ 0 , 2 π ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq81_HTML.gif by
( D 03 x ) ( t ) = ( ϕ p ( x ( m + 1 ) ( t ) ) ) ( m + 1 ) , t R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equat_HTML.gif
where
D ( D 03 ) = { x C 0 2 m + 1 , π : ( ϕ p ( x ( m + 1 ) ( t ) ) ) ( m ) is absolutely continuous on  R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equau_HTML.gif
Let N 03 : C 0 0 , π L 1 ( [ 0 , 2 π ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq82_HTML.gif be the Nemytskii operator
( N 03 x ) ( t ) = g ( t , x ( t ) ) + e ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equav_HTML.gif
Obviously, the operator D 03 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq83_HTML.gif is invertible and the problem of even anti-periodic solutions for (4.1) is equivalent to the operator equation
D 03 x = λ N 03 x , x D ( D 03 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equaw_HTML.gif
From hypotheses (H8), (H9) and (3.8) in [15], for the possible even anti-periodic solution x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif of (4.1), there exists a prior bounds in C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq53_HTML.gif, i.e., x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif satisfies
x C 0 T 4 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equax_HTML.gif

where T 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq84_HTML.gif is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator Q λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq85_HTML.gif, which sends C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq53_HTML.gif into C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq53_HTML.gif, such that the fixed points of operator Q 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq86_HTML.gif in some open bounded set are the even anti-periodic solutions of (1.5).

With this in mind, let us define the set as follows:
Ω 03 = { x C 0 0 , π : x C 0 < T 4 + 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equay_HTML.gif
By hypothesis (H5), it is easy to see that
( N 03 x ) ( t ) ( N 03 x ) ( t ) , x C 0 0 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equaz_HTML.gif
Hence, the operator N 03 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq87_HTML.gif sends C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq53_HTML.gif into C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq53_HTML.gif. Define the completely continuous operator Q λ : Ω 03 ¯ C 0 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq88_HTML.gif by
Q λ x = J 1 J 0 J 0 J 1 m + 1 ϕ q J 0 J 1 J 1 J 0 m + 1 λ N 03 x = ϕ q ( λ ) D 03 1 N 03 x , λ [ 0 , 1 ] ( if  m = 2 n , n = 1 , 2 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equba_HTML.gif
or
Q λ x = J 1 J 0 J 1 J 0 m + 1 ϕ q J 1 J 0 J 1 J 0 m + 1 λ N 03 x = ϕ q ( λ ) D 03 1 N 03 x , λ [ 0 , 1 ] ( if  m = 2 n 1 , n = 1 , 2 , ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbb_HTML.gif

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 4.2 Suppose that the assumptions (H6), (H8), (H9) are true. Then (1.5) has at least one odd anti-periodic solution.

Proof We consider the homotopic equation (4.1) of (1.5). Define the operator D 13 : D ( D 13 ) C 1 0 , π L 1 ( [ 0 , 2 π ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq89_HTML.gif by
( D 13 x ) ( t ) = ( ϕ p ( x ( m + 1 ) ( t ) ) ) ( m + 1 ) , t R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbc_HTML.gif
where
D ( D 13 ) = { x C 1 2 m + 1 , π : ( ϕ p ( x ( m + 1 ) ( t ) ) ) ( m ) is absolutely continuous on  R } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbd_HTML.gif
Let N 13 : C 1 0 , π L 1 ( [ 0 , 2 π ] , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq90_HTML.gif be the Nemytskii operator
( N 13 x ) ( t ) = g ( t , x ( t ) ) + e ( t ) , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Eqube_HTML.gif
Thus, the problem of odd anti-periodic solutions for (4.1) is equivalent to the operator equation
D 13 x = λ N 13 x , x D ( D 13 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbf_HTML.gif
Our problem is reduced to construct one completely continuous operator W λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq91_HTML.gif, which sends C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq37_HTML.gif into C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq37_HTML.gif, such that the fixed points of operator W 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq92_HTML.gif in some open bounded set are the odd anti-periodic solutions of (1.5). With this in mind, let us define the following set:
Ω 13 = { x C 1 0 , π : x C 0 < T 4 + 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbg_HTML.gif
By hypothesis (H6), it is easy to see that
( N 13 x ) ( t ) ( N 13 x ) ( t ) , x C 1 0 , π . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbh_HTML.gif
Hence, the operator N 13 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq93_HTML.gif sends C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq37_HTML.gif into C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq37_HTML.gif. Define the completely continuous operator W λ : Ω 13 ¯ C 1 0 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq94_HTML.gif by
W λ x = J 0 J 1 J 1 J 0 m + 1 ϕ q J 1 J 0 J 0 J 1 m + 1 λ N 13 x = ϕ q ( λ ) D 13 1 N 13 x , λ [ 0 , 1 ] ( if  m = 2 n , n = 1 , 2 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbi_HTML.gif
or
W λ x = J 0 J 1 J 0 J 1 m + 1 ϕ q J 0 J 1 J 0 J 1 m + 1 λ N 13 x = ϕ q ( λ ) D 13 1 N 13 x , λ [ 0 , 1 ] ( if  m = 2 n 1 , n = 1 , 2 , ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbj_HTML.gif

The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □

Theorem 4.3 Assume that g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq28_HTML.gif has the decomposition
g ( t , x ) = u ( t , x ) + v ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbk_HTML.gif

such that

(H10) there exist non-negative constants γ, r with r > p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq95_HTML.gif, such that
( 1 ) m + 1 x u ( t , x ) γ | x | r , t , x R ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbl_HTML.gif
(H11) there are non-negative functions α 3 , β 3 C ( R , R + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq96_HTML.gif such that
| v ( t , x ) | α 3 ( t ) | x | r 1 + β 3 ( t ) , t , x R ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbm_HTML.gif

(H12) α 3 0 γ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq97_HTML.gif and the assumption (H5) is true.

Then (1.5) has at least one even anti-periodic solution.

Theorem 4.4 Suppose that the assumptions (H6), (H10), (H11), (H12) are true. Then (1.5) has at least one odd anti-periodic solution.

Basing on the proof of Theorem 3.2 in [15], for the possible anti-periodic solution x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif of (4.1), the hypotheses (H10), (H11), (H12) yield that there exists a prior bounds in C 2 m + 1 , π http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq77_HTML.gif, i.e., x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq16_HTML.gif satisfies
x C 0 T 5 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_Equbn_HTML.gif

where T 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq98_HTML.gif is a positive constant independent of λ. The remainder of the proof work of Theorem 4.3 and Theorem 4.4 is quite similar to the proof of Theorem 4.1 and Theorem 4.2, so we omit the details.

Remark Assumptions (H10), (H11), (H12) guarantee that the degree with respect to x of g ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq28_HTML.gif is allowed to be greater than p 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-108/MediaObjects/13661_2012_Article_207_IEq99_HTML.gif, which is different from the hypothesis (H8) of Theorem 4.1 and Theorem 4.2.

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundations of China (50904065) and the Program for New Century Excellent Talents in University (NCET-09-0728). As well, this work was sponsored by the Qing Lan Project and the Fundamental Research Funds for the Central Universities (China University of Mining and Technology).

Authors’ Affiliations

(1)
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology
(2)
School of Mechanics and Civil Engineering, China University of Mining and Technology
(3)
School of Mathematics and Physical Sciences, Xuzhou Institute of Technology

References

  1. Ahn C, Rim C: Boundary flows in general coset theories. J. Phys. A 1999, 32(13):2509-2525. 10.1088/0305-4470/32/13/004MathSciNetView Article
  2. Kleinert H, Chervyakov A: Functional determinants from Wronski Green function. J. Math. Phys. 1999, 40(11):6044-6051. 10.1063/1.533069MathSciNetView Article
  3. Aizicovici S, McKibben M, Reich S: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities. Nonlinear Anal. 2001, 43(2):233-251. 10.1016/S0362-546X(99)00192-3MathSciNetView Article
  4. Nakao M: Existence of an anti-periodic solution for the quasilinear wave equation with viscosity. J. Math. Anal. Appl. 1996, 204(3):754-764. 10.1006/jmaa.1996.0465MathSciNetView Article
  5. Souplet P: Optimal uniqueness condition for the antiperiodic solutions of some nonlinear parabolic equations. Nonlinear Anal. 1998, 32(2):279-286. 10.1016/S0362-546X(97)00477-XMathSciNetView Article
  6. Lu Z: Travelling Tube. Shanghai Sci. Technol., Shanghai; 1962.
  7. Chen Y: On Massera’s theorem for anti-periodic solution. Adv. Math. Sci. Appl. 1999, 9(1):125-128.MathSciNet
  8. Yin Y: Monotone iterative technique and quasilinearization for some anti-periodic problems. Nonlinear World 1996, 3(2):253-266.MathSciNet
  9. Aftabizadeh AR, Pavel NH, Huang Y: Anti-periodic oscillations of some second-order differential equations and optimal control problems. J. Comput. Appl. Math. 1994, 52(1-3):3-21. Oscillations in nonlinear systems: Applications and numerical aspects 10.1016/0377-0427(94)90345-XMathSciNetView Article
  10. Chen T, Liu W, Zhang J: The existence of anti-periodic solutions for high order Duffing equation. J. Appl. Math. Comput. 2008, 27(1-2):271-280. 10.1007/s12190-008-0056-1MathSciNetView Article
  11. Liu B: Anti-periodic solutions for forced Rayleigh-type equations. Nonlinear Anal., Real World Appl. 2009, 10(5):2850-2856. 10.1016/j.nonrwa.2008.08.011MathSciNetView Article
  12. Liu W, Zhang J, Chen T: Anti-symmetric periodic solutions for the third order differential systems. Appl. Math. Lett. 2009, 22(5):668-673. 10.1016/j.aml.2008.08.004MathSciNetView Article
  13. Luo Z, Shen J, Nieto JJ: Antiperiodic boundary value problem for first-order impulsive ordinary differential equation. Comput. Math. Appl. 2005, 49(2-3):253-261. 10.1016/j.camwa.2004.08.010MathSciNetView Article
  14. Chen H: Antiperiodic wavelets. J. Comput. Math. 1996, 14(1):32-39.MathSciNet
  15. Chen T, Liu W: Anti-periodic solutions for higher-order Liénard type differential equation with p -Laplacian operator. Bull. Korean Math. Soc. 2012, 49(3):455-463. 10.4134/BKMS.2012.49.3.455MathSciNetView Article
  16. Li X: Existence and uniqueness of periodic solutions for a kind of high-order p -Laplacian Duffing differential equation with sign-changing coefficient ahead of linear term. Nonlinear Anal. 2009, 71(7-8):2764-2770. 10.1016/j.na.2009.01.153MathSciNetView Article
  17. Pang H, Ge W, Tian M: Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a p -Laplacian. Comput. Math. Appl. 2008, 56(1):127-142. 10.1016/j.camwa.2007.11.039MathSciNetView Article
  18. Su H, Wang B, Wei Z, Zhang X: Positive solutions of four-point boundary value problems for higher-order p -Laplacian operator. J. Math. Anal. Appl. 2007, 330(2):836-851. 10.1016/j.jmaa.2006.07.017MathSciNetView Article
  19. Xu F, Liu L, Wu Y: Multiple positive solutions of four-point nonlinear boundary value problems for a higher-order p -Laplacian operator with all derivatives. Nonlinear Anal. 2009, 71(9):4309-4319. 10.1016/j.na.2009.02.118MathSciNetView Article
  20. Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.View Article

Copyright

© Pu and Yang; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.