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

Hai Pu; Jinyun Yang

doi:10.1186/1687-2770-2012-108

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

Hai Pu^{1, 2}Email author and
Jinyun Yang³

Boundary Value Problems20122012:108

https://doi.org/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 [3–5]). 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) (\equiv e (t + 2 π)),

whose anti-periodic solutions satisfy

x (t + π) = - x (t), \forall t \in R .

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 [9–12]. 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 [15–19] 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)} + \sum_{i = 1}^{n - 1} a_{i} x^{(i)} + g (t, x) = e (t) .

(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) .

(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:

(1.3)

(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),

(1.5)

where $p > 1$ is a constant, $m \geq 1$ is an integer, $ϕ_{p} (s) = {| s |}^{p - 2} s$ ; $a_{i} \in R$ , $g \in C (R^{2}, R)$ , $e \in C (R, R)$ with $g (t + π, - x) \equiv - g (t, x)$ , $e (t + π) \equiv - e (t)$ . Obviously, the inverse operator of $ϕ_{p}$ is $ϕ_{q}$ , where $q > 1$ is a constant such that $\frac{1}{p} + \frac{1}{q} = 1$ .

Notice that, when $p = 2$ , the nonlinear operator ${(ϕ_{p} (x^{(m + 1)}))}^{(m + 1)}$ reduces to the linear operator $x^{(2 m + 2)}$ . On the other hand, $x (t)$ is also a 2π-periodic solution if $x (t)$ 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 \in C^{k} (R, R) : x (t + π) \equiv - x (t)}, k \in {0, 1, \dots}

with the norm

{∥ x ∥}_{C^{k}} = max_{i \in {0, 1, \dots, k}} {{∥ x^{(i)} ∥}_{0}},

where

{∥ x ∥}_{0} = {max}_{t \in [0, 2 π]} | x (t) |

, and

with the norm ${∥ \cdot ∥}_{C^{k}}$ .

Notice that,

x \in C_{0}^{0, π}

may be written as Fourier series as follows:

x (t) = \sum_{i = 0}^{\infty} a_{2 i + 1} cos (2 i + 1) t,

and

x \in C_{1}^{0, π}

may be written as the following Fourier series:

x (t) = \sum_{i = 0}^{\infty} b_{2 i + 1} sin (2 i + 1) t,

where

a_{2 i + 1}, b_{2 i + 1} \in R

. We define the mapping

J_{0} : C_{0}^{0, π} ⟶ C_{1}^{1, π}

by

(J_{0} x) (t) = \int_{0}^{t} x (s) d s = \sum_{i = 0}^{\infty} \frac{a_{2 i + 1}}{2 i + 1} sin (2 i + 1) t, \forall t \in R

and the mapping

J_{1} : C_{1}^{0, π} ⟶ C_{0}^{1, π}

by

\begin{array}{rcl} (J_{1} x) (t) & = & \int_{0}^{t} x (s) d s - \sum_{i = 0}^{\infty} \frac{b_{2 i + 1}}{2 i + 1} \\ = & - \sum_{i = 0}^{\infty} \frac{b_{2 i + 1}}{2 i + 1} cos (2 i + 1) t, \forall t \in R . \end{array}

It is easy to prove that the mappings $J_{0}$ , $J_{1}$ 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

\bar{Ω}

. Moreover, assume that the Leray-Schauder degree

deg (f, Ω, p) \neq 0, for p \in X ∖ f (\partial Ω) .

Then the equation $f (x) = p$ 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)

and

e (t)

are odd in t, i.e.,

g (- t, \cdot) = - g (t, \cdot), e (- t) = - e (t), \forall t \in R;

(H₂) there exist non-negative functions

α_{1}, β_{1} \in C (R, R^{+})

such that

| g (t, x) | \leq α_{1} (t) | x | + β_{1} (t), \forall t, x \in R;

(H₃) $\sum_{i = 1}^{m} | a_{i} | + {∥ α_{1} ∥}_{0} - 1 < 0$ .

Then (1.3) has at least one even anti-periodic solution

x (t)

, i.e.,

x (t)

satisfies

x (t + π) = - x (t), x (- t) = x (t), \forall t \in R .

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 \sum_{i = 1}^{m} a_{i} x^{(2 i - 1)} - λ g (t, x) + λ e (t), λ \in [0, 1] .

(3.1)

Define the operator

D_{01} : C_{0}^{2 m + 1, π} ⟶ C_{1}^{0, π}

by

(D_{01} x) (t) = x^{(2 m + 1)} (t), \forall t \in R .

Obviously, the operator

D_{01}

is invertible. Let

N_{01} : C_{0}^{2 m - 1, π} ⟶ C^{0}

be the Nemytskii operator

(N_{01} x) (t) = - \sum_{i = 1}^{m} a_{i} x^{(2 i - 1)} (t) - g (t, x (t)) + e (t), \forall t \in R .

By hypothesis (H₁), it is easy to see that

(N_{01} x) (t + π) \equiv - (N_{01} x) (t), (N_{01} x) (- t) \equiv - (N_{01} x) (t), \forall x \in C_{0}^{2 m - 1, π} .

Thus, the operator

N_{01}

sends

C_{0}^{2 m - 1, π}

into

C_{1}^{0, π}

. 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 \in C_{0}^{2 m + 1, π} .

From hypotheses (H₂), (H₃) and (5) in [10], for the possible even anti-periodic solution

x (t)

of (3.1), there exists a prior bounds in

C_{0}^{2 m + 1, π}

, i.e.,

x (t)

satisfies

{∥ x ∥}_{C^{2 m + 1}} \leq T_{1},

(3.2)

where $T_{1}$ is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator $F_{λ}$ , which sends $C_{0}^{2 m + 1, π}$ into $C_{0}^{2 m + 1, π}$ , such that the fixed points of operator $F_{1}$ 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 \in C_{0}^{2 m + 1, π} : {∥ x ∥}_{C^{2 m + 1}} < T_{1} + 1} .

Obviously, the set

Ω_{01}

is a open bounded set in

C_{0}^{2 m + 1, π}

and zero element

θ \in Ω_{01}

. Define the completely continuous operator

F_{λ} : \bar{Ω_{01}} ⟶ C_{0}^{2 m + 1, π}

by

F_{λ} x = \underset{2 m + 1}{\underset{⏟}{J_{1} J_{0} \dots J_{0} J_{1}}} λ N_{01} x = λ D_{01}^{- 1} N_{01} x, λ \in [0, 1] .

Let us define the completely continuous field

h_{λ} (x) : \bar{Ω_{01}} \times [0, 1] ⟶ C_{0}^{2 m + 1, π}

by

h_{λ} (x) = x - F_{λ} x .

By (3.2), we get that zero element

θ \notin h_{λ} (\partial Ω)

for all

λ \in [0, 1]

. So, the following Leray-Schauder degrees are well defined and

\begin{array}{rcl} deg (i d - F_{1}, Ω, θ) & = & deg (h_{1}, Ω, θ) \\ = & deg (h_{0}, Ω, θ) = deg (i d, Ω, θ) = 1 \neq 0 . \end{array}

Consequently, the operator $F_{1}$ has at least one fixed point in $Ω_{01}$ 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)

is even in t, x and

e (t)

is even in t, i.e.,

g (- t, - x) = g (t, x), e (- t) = e (t), \forall t \in R

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

Then (1.3) has at least one odd anti-periodic solution

x (t)

, i.e.,

x (t)

satisfies

x (t + π) = - x (t), x (- t) = - x (t), \forall t \in R .

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}

by

(D_{11} x) (t) = x^{(2 m + 1)} (t), \forall t \in R .

Let

N_{11} : C_{1}^{2 m - 1, π} ⟶ C^{0, π}

be the Nemytskii operator

(N_{11} x) (t) = - \sum_{i = 1}^{m} a_{i} x^{(2 i - 1)} (t) - g (t, x (t)) + e (t), \forall t \in R .

By hypothesis (H₄), it is easy to see that

(N_{11} x) (- t) \equiv (N_{11} x) (t), \forall x \in C_{1}^{2 m - 1, π} .

Thus, the operator

N_{11}

sends

C_{1}^{2 m - 1, π}

into

C_{0}^{0, π}

. 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 \in C_{1}^{2 m + 1, π} .

Our problem is reduced to construct one completely continuous operator

G_{λ}

, which sends

C_{1}^{2 m + 1, π}

into

C_{1}^{2 m + 1, π}

, such that the fixed points of operator

G_{1}

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 \in C_{1}^{2 m + 1, π} : {∥ x ∥}_{C^{2 m + 1}} < T_{1} + 1} .

Define the completely continuous operator

G_{λ} : \bar{Ω_{11}} ⟶ C_{1}^{2 m + 1, π}

by

G_{λ} x = \underset{2 m + 1}{\underset{⏟}{J_{0} J_{1} \dots J_{1} J_{0}}} λ N_{11} x = λ D_{11}^{- 1} N_{11} x, λ \in [0, 1] .

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)

and

e (t)

are even in t, i.e.,

g (- t, \cdot) = g (t, \cdot), e (- t) = e (t), \forall t \in R

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 \sum_{i = 1}^{m} a_{i} x^{(2 i)} - λ g (t, x) + λ e (t), λ \in [0, 1] .

(3.3)

Define the operator

D_{02} : C_{0}^{2 m + 2, π} ⟶ C_{0}^{0, π}

by

(D_{02} x) (t) = x^{(2 m + 2)} (t), \forall t \in R .

Let

N_{02} : C_{0}^{2 m, π} ⟶ C^{0, π}

be the Nemytskii operator

(N_{02} x) (t) = - \sum_{i = 1}^{m} a_{i} x^{(2 i)} (t) - g (t, x (t)) + e (t), \forall t \in R .

By hypothesis (H₅), it is easy to see that

(N_{02} x) (- t) \equiv (N_{02} x) (t), \forall x \in C_{0}^{2 m, π} .

Thus, the operator

N_{02}

sends

C_{0}^{2 m, π}

into

C_{0}^{0, π}

. 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 \in C_{0}^{2 m + 2, π} .

Our problem is reduced to construct one completely continuous operator

L_{λ}

, which sends

C_{0}^{2 m + 2, π}

into

C_{0}^{2 m + 2, π}

, such that the fixed points of operator

L_{1}

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 \in C_{0}^{2 m + 2, π} : {∥ x ∥}_{C^{2 m + 2}} < T_{2} + 1},

where

T_{2}

is a positive constant independent of λ. Define the completely continuous operator

L_{λ} : \bar{Ω_{02}} ⟶ C_{0}^{2 m + 2, π}

by

L_{λ} x = \underset{2 m + 2}{\underset{⏟}{J_{1} J_{0} \dots J_{1} J_{0}}} λ N_{02} x = λ D_{02}^{- 1} N_{02} x, λ \in [0, 1] .

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)

is odd in t, x and

e (t)

is odd in t, i.e.,

g (- t, - x) = - g (t, x), e (- t) = - e (t), \forall t \in R

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}

by

(D_{12} x) (t) = x^{(2 m + 2)} (t), \forall t \in R .

Let

N_{12} : C_{1}^{2 m, π} ⟶ C^{0, π}

be the Nemytskii operator

(N_{12} x) (t) = - \sum_{i = 1}^{m} a_{i} x^{(2 i)} (t) - g (t, x (t)) + e (t), \forall t \in R .

By hypothesis (H₆), it is easy to see that

(N_{12} x) (- t) \equiv - (N_{12} x) (t), \forall x \in C_{1}^{2 m, π} .

Thus, the operator

N_{12}

sends

C_{1}^{2 m, π}

into

C_{1}^{0, π}

. 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 \in C_{1}^{2 m + 2, π} .

Our problem is reduced to construct one completely continuous operator

P_{λ}

which sends

C_{1}^{2 m + 2, π}

into

C_{1}^{2 m + 2, π}

, such that the fixed points of operator

P_{1}

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 \in C_{1}^{2 m + 2, π} : {∥ x ∥}_{C^{2 m + 2}} < T_{2} + 1} .

Define the completely continuous operator

P_{λ} : \bar{Ω_{12}} ⟶ C_{1}^{2 m + 2, π}

by

P_{λ} x = \underset{2 m + 2}{\underset{⏟}{J_{0} J_{1} \dots J_{0} J_{1}}} λ N_{12} x = λ D_{12}^{- 1} N_{12} x, λ \in [0, 1] .

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 $g (t, x) = g (x)$ , we can remove the assumption (H₂) in Theorem 3.1, Theorem 3.2 and obtain the following results.

Theorem 3.5 Assume that

(H₇) $\sum_{i = 1}^{m} | a_{i} | - 1 < 0$ and the assumption (H₁) is true.

Then (1.3) ( $g (t, x) = g (x)$ ) has at least one even anti-periodic solution.

Theorem 3.6 Suppose that the assumptions (H₄), (H₇) are true. Then (1.3) ( $g (t, x) = g (x)$ ) 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)

of (3.1) (

g (t, x) = g (x)

), the hypothesis (H₇) yields that there exists a prior bounds in

C^{2 m + 1, π}

, i.e.,

x (t)

satisfies

{∥ x ∥}_{C^{2 m + 1}} \leq T_{3},

where $T_{3}$ 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} \in C (R, R^{+})

such that

| g (t, x) | \leq α_{2} (t) {| x |}^{p - 1} + β_{2} (t), \forall t, x \in R;

(H₉) ${∥ α_{2} ∥}_{0} λ_{1}^{- (m + 1)} - 1 < 0$ 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 \in [0, 1] .

(4.1)

Define the operator

D_{03} : D (D_{03}) \subset C_{0}^{0, π} ⟶ L^{1} ([0, 2 π], R)

by

(D_{03} x) (t) = {(ϕ_{p} (x^{(m + 1)} (t)))}^{(m + 1)}, \forall t \in R,

where

\begin{array}{rcl} D (D_{03}) & = & {x \in C_{0}^{2 m + 1, π} : {(ϕ_{p} (x^{(m + 1)} (t)))}^{(m)} \\ is absolutely continuous on R} . \end{array}

Let

N_{03} : C_{0}^{0, π} ⟶ L^{1} ([0, 2 π], R)

be the Nemytskii operator

(N_{03} x) (t) = - g (t, x (t)) + e (t), \forall t \in R .

Obviously, the operator

D_{03}

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 \in D (D_{03}) .

From hypotheses (H₈), (H₉) and (3.8) in [15], for the possible even anti-periodic solution

x (t)

of (4.1), there exists a prior bounds in

C_{0}^{0, π}

, i.e.,

x (t)

satisfies

{∥ x ∥}_{C^{0}} \leq T_{4},

where $T_{4}$ is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator $Q_{λ}$ , which sends $C_{0}^{0, π}$ into $C_{0}^{0, π}$ , such that the fixed points of operator $Q_{1}$ 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 \in C_{0}^{0, π} : {∥ x ∥}_{C^{0}} < T_{4} + 1} .

By hypothesis (H₅), it is easy to see that

(N_{03} x) (- t) \equiv (N_{03} x) (t), \forall x \in C_{0}^{0, π} .

Hence, the operator

N_{03}

sends

C_{0}^{0, π}

into

C_{0}^{0, π}

. Define the completely continuous operator

Q_{λ} : \bar{Ω_{03}} ⟶ C_{0}^{0, π}

by

\begin{array}{rcl} Q_{λ} x & = & \underset{m + 1}{\underset{⏟}{J_{1} J_{0} \dots J_{0} J_{1}}} ϕ_{q} \underset{m + 1}{\underset{⏟}{J_{0} J_{1} \dots J_{1} J_{0}}} λ N_{03} x \\ = & ϕ_{q} (λ) D_{03}^{- 1} N_{03} x, λ \in [0, 1] (if m = 2 n, n = 1, 2, \dots) \end{array}

or

\begin{array}{rcl} Q_{λ} x & = & \underset{m + 1}{\underset{⏟}{J_{1} J_{0} \dots J_{1} J_{0}}} ϕ_{q} \underset{m + 1}{\underset{⏟}{J_{1} J_{0} \dots J_{1} J_{0}}} λ N_{03} x \\ = & ϕ_{q} (λ) D_{03}^{- 1} N_{03} x, λ \in [0, 1] (if m = 2 n - 1, n = 1, 2, \dots) . \end{array}

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}) \subset C_{1}^{0, π} ⟶ L^{1} ([0, 2 π], R)

by

(D_{13} x) (t) = {(ϕ_{p} (x^{(m + 1)} (t)))}^{(m + 1)}, \forall t \in R,

where

\begin{array}{rcl} D (D_{13}) & = & {x \in C_{1}^{2 m + 1, π} : {(ϕ_{p} (x^{(m + 1)} (t)))}^{(m)} \\ is absolutely continuous on R} . \end{array}

Let

N_{13} : C_{1}^{0, π} ⟶ L^{1} ([0, 2 π], R)

be the Nemytskii operator

(N_{13} x) (t) = - g (t, x (t)) + e (t), \forall t \in R .

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 \in D (D_{13}) .

Our problem is reduced to construct one completely continuous operator

W_{λ}

, which sends

C_{1}^{0, π}

into

C_{1}^{0, π}

, such that the fixed points of operator

W_{1}

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 \in C_{1}^{0, π} : {∥ x ∥}_{C^{0}} < T_{4} + 1} .

By hypothesis (H₆), it is easy to see that

(N_{13} x) (- t) \equiv - (N_{13} x) (t), \forall x \in C_{1}^{0, π} .

Hence, the operator

N_{13}

sends

C_{1}^{0, π}

into

C_{1}^{0, π}

. Define the completely continuous operator

W_{λ} : \bar{Ω_{13}} ⟶ C_{1}^{0, π}

by

\begin{array}{rcl} W_{λ} x & = & \underset{m + 1}{\underset{⏟}{J_{0} J_{1} \dots J_{1} J_{0}}} ϕ_{q} \underset{m + 1}{\underset{⏟}{J_{1} J_{0} \dots J_{0} J_{1}}} λ N_{13} x \\ = & ϕ_{q} (λ) D_{13}^{- 1} N_{13} x, λ \in [0, 1] (if m = 2 n, n = 1, 2, \dots) \end{array}

or

\begin{array}{rcl} W_{λ} x & = & \underset{m + 1}{\underset{⏟}{J_{0} J_{1} \dots J_{0} J_{1}}} ϕ_{q} \underset{m + 1}{\underset{⏟}{J_{0} J_{1} \dots J_{0} J_{1}}} λ N_{13} x \\ = & ϕ_{q} (λ) D_{13}^{- 1} N_{13} x, λ \in [0, 1] (if m = 2 n - 1, n = 1, 2, \dots) . \end{array}

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)

has the decomposition

g (t, x) = u (t, x) + v (t, x)

such that

(H₁₀) there exist non-negative constants γ, r with

r > p

, such that

{(- 1)}^{m + 1} x u (t, x) \geq γ {| x |}^{r}, \forall t, x \in R;

(H₁₁) there are non-negative functions

α_{3}, β_{3} \in C (R, R^{+})

such that

| v (t, x) | \leq α_{3} (t) {| x |}^{r - 1} + β_{3} (t), \forall t, x \in R;

(H₁₂) ${∥ α_{3} ∥}_{0} - γ \leq 0$ 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)

of (4.1), the hypotheses (H₁₀), (H₁₁), (H₁₂) yield that there exists a prior bounds in

C^{2 m + 1, π}

, i.e.,

x (t)

satisfies

{∥ x ∥}_{C^{0}} \leq T_{5},

where $T_{5}$ 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 $g (t, x)$ is allowed to be greater than $p - 1$ , 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

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 ArticleGoogle Scholar
Kleinert H, Chervyakov A: Functional determinants from Wronski Green function. J. Math. Phys. 1999, 40(11):6044-6051. 10.1063/1.533069MathSciNetView ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
Lu Z: Travelling Tube. Shanghai Sci. Technol., Shanghai; 1962.Google Scholar
Chen Y: On Massera’s theorem for anti-periodic solution. Adv. Math. Sci. Appl. 1999, 9(1):125-128.MathSciNetGoogle Scholar
Yin Y: Monotone iterative technique and quasilinearization for some anti-periodic problems. Nonlinear World 1996, 3(2):253-266.MathSciNetGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
Chen H: Antiperiodic wavelets. J. Comput. Math. 1996, 14(1):32-39.MathSciNetGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
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 ArticleGoogle Scholar
Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.View ArticleGoogle Scholar

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