# Existence of anti-periodic solutions for second-order ordinary differential equations involving the Fučík spectrum

- Xin Zhao
^{1}and - Xiaojun Chang
^{2}Email author

**2012**:149

**DOI: **10.1186/1687-2770-2012-149

© Zhao and Chang; licensee Springer 2012

**Received: **20 July 2012

**Accepted: **20 September 2012

**Published: **21 December 2012

## Abstract

In this paper, we study the existence of anti-periodic solutions for a second-order ordinary differential equation. Using the interaction of the nonlinearity with the Fučík spectrum related to the anti-periodic boundary conditions, we apply the Leray-Schauder degree theory and the Borsuk theorem to establish new results on the existence of anti-periodic solutions of second-order ordinary differential equations. Our nonlinearity may cross multiple consecutive branches of the Fučík spectrum curves, and recent results in the literature are complemented and generalized.

### Keywords

anti-periodic solutions Fučík spectrum Leray-Schauder degree theory Borsuk theorem## 1 Introduction and main results

*T*is a positive constant. A function $x(t)$ is called an anti-periodic solution of (1.1) if $x(t)$ satisfies (1.1) and $x(t+\frac{T}{2})=-x(t)$ for all $t\in \mathbb{R}$. Note that to obtain anti-periodic solutions of (1.1), it suffices to find solutions of the following anti-periodic boundary value problem:

In what follows, we will consider problem (1.2) directly.

we can see that the conditions on the ratio $\frac{2F(t,s)}{{s}^{2}}$ are more general than those on the ratio $\frac{f(t,s)}{s}$. In fact, by using the asymptotic interaction of the ratio $\frac{2F(t,s)}{{s}^{2}}$ with the spectrum of $-{x}^{\u2033}$, the ratio $\frac{f(t,s)}{s}$ can cross multiple spectrum curves of $-{x}^{\u2033}$. In this paper, we are interested in the nonresonance condition on the ratio $\frac{2F(t,s)}{{s}^{2}}$ for the solvability of (1.1) involving the Fučík spectrum of $-{x}^{\u2033}$ under the anti-periodic boundary condition.

Note that the study of anti-periodic solutions for nonlinear differential equations is closely related to the study of periodic solutions. In fact, since $f(t,s)=-f(t+\frac{T}{2},-s)=f(t+T,s)$, $x(t)$ is a *T*-periodic solution of (1.1) if $x(t)$ is a $\frac{T}{2}$-anti-periodic solution of (1.1). Many results on the periodic solutions of (1.1) have been worked out. For some recent work, one can see [2–5, 8–10, 17]. As special periodic solutions, the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations coming from some models in applied sciences. During the last thirty years, anti-periodic problems of nonlinear differential equations have been extensively studied since the pioneering work by Okochi [18]. For example, in [19], anti-periodic trigonometric polynomials are used to investigate the interpolation problems, and anti-periodic wavelets are studied in [20]. Also, some existence results of ordinary differential equations are presented in [17, 21–24]. Anti-periodic boundary conditions for partial differential equations and abstract differential equations are considered in [25–32]. For recent developments involving the existence of anti-periodic solutions, one can also see [33–35] and the references therein.

It is easily seen that the set Σ can be seen as a subset of the Fuc̆ík spectrum of $-{x}^{\u2033}$ under the corresponding Dirichlet boundary condition; one can see the definition of the set ${\mathrm{\Sigma}}_{2i+1}$, $i\in \mathbb{N}$, or Figure 1 in [12]. Without loss of generality, we assume that ${\phi}_{m}$ is an eigenfunction of (1.3) corresponding to $({\lambda}_{+},{\lambda}_{-})\in {\mathrm{\Sigma}}_{m}$ such that ${\phi}_{m}(0)=0$ and ${\phi}_{m}^{\prime}(0)=a\in \mathbb{R}\setminus \{0\}$. Denote ${\mathrm{\Sigma}}_{m,1}=\{({\lambda}_{+},{\lambda}_{-})\in {\mathbb{R}}^{2}:\frac{(m+1)\pi}{\sqrt{{\lambda}_{+}}}+\frac{m\pi}{\sqrt{{\lambda}_{-}}}=\frac{T}{2},m\in {\mathbb{Z}}^{+}\}$ and ${\mathrm{\Sigma}}_{m,2}=\{({\lambda}_{+},{\lambda}_{-})\in {\mathbb{R}}^{2}:\frac{m\pi}{\sqrt{{\lambda}_{+}}}+\frac{(m+1)\pi}{\sqrt{{\lambda}_{-}}}=\frac{T}{2},m\in {\mathbb{Z}}^{+}\}$. Then if $a>0$, we obtain only a one-dimensional function ${\phi}_{m}$, denoted by ${\phi}_{m,1}$, corresponding to the point $({\lambda}_{+},{\lambda}_{-})\in {\mathrm{\Sigma}}_{m,1}$, and if $a<0$, we obtain only a one-dimensional function ${\phi}_{m}$, denoted by ${\phi}_{m,2}$, corresponding to the point $({\lambda}_{+},{\lambda}_{-})\in {\mathrm{\Sigma}}_{m,2}$.

In this paper, together with the Leray-Schauder degree theory and the Borsuk theorem, we obtain new existence results of anti-periodic solutions of (1.1) when the nonlinearity $f(t,s)$ is asymptotically linear in *s* at infinity and the ratio $\frac{F(t,s)}{{s}^{2}}$ stays asymptotically at infinity in some rectangular domain between Fučík spectrum curves ${\mathrm{\Sigma}}_{m}$ and ${\mathrm{\Sigma}}_{m+1}$.

Our main result is as follows.

**Theorem 1.1**

*Assume that*$f\in C({\mathbb{R}}^{2},\mathbb{R})$, $f(t+\frac{T}{2},-s)=-f(t,s)$.

*If the following conditions*:

- (i)
*There exist positive constants**ρ*, ${C}_{1}$,*M**such that*$\rho \le \frac{f(t,s)}{s}\le {C}_{1},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in \mathbb{R},\mathrm{\forall}|s|\ge M;$(1.4) - (ii)
*There exist connect subset*$\mathrm{\Gamma}\subset {\mathbb{R}}^{2}\setminus \mathrm{\Sigma}$,*constants*${p}_{1},{q}_{1},{p}_{2},{q}_{2}>0$*and a point of the type*$(\lambda ,\lambda )\in {\mathbb{R}}^{2}$*such that*$(\lambda ,\lambda )\in [{p}_{1},{q}_{1}]\times [{p}_{2},{q}_{2}]\subset \mathrm{\Gamma}$(1.5)

*and*

*hold uniformly for all* $t\in \mathbb{R}$,

*then* (1.1) *admits a* $\frac{T}{2}$-*anti*-*periodic solution*.

Simple computation implies that the operator $-{x}^{\u2033}$ with the anti-periodic boundary condition has a sequence of eigenvalues ${\lambda}_{m}=\frac{4{(2m-1)}^{2}{\pi}^{2}}{{T}^{2}}$, $m\in {\mathbb{Z}}^{+}$, and the corresponding eigenspace is two-dimensional.

**Corollary 1.2**

*Assume that*$f\in C({\mathbb{R}}^{2},\mathbb{R})$, $f(t,s)=-f(t+\frac{T}{2},-s)$.

*If*(1.4)

*holds and there exist constants*

*p*,

*q*

*and*$m\in {\mathbb{Z}}^{+}$

*such that*

*holds uniformly for all* $t\in \mathbb{R}$, *then* (1.1) *admits a* $\frac{T}{2}$-*anti*-*periodic solution*.

**Remark**It is well known that (1.1) has a $\frac{T}{2}$-anti-periodic solution if

for some ${\sigma}_{1}>0$ (see Theorem 3.1 in [22]), which implies that the ratio $\frac{f(t,s)}{s}$ stays at infinity asymptotically below the first eigenvalue ${\lambda}_{1}$ of (1.6). In this paper, this requirement on the ratio $\frac{f(t,s)}{s}$ can be relaxed to (1.4), with some additional restrictions imposed on the ratio $\frac{2F(t,s)}{{s}^{2}}$. In fact, the conditions relative to the ratios $\frac{f(t,s)}{s}$ and $\frac{2F(t,s)}{{s}^{2}}$ as in Theorem 1.1 and Corollary 1.2 may lead to that the ratio $\frac{f(t,s)}{s}$ oscillates and crosses multiple consecutive eigenvalues or branches of the Fučík spectrum curves of the operator $-{x}^{\u2033}$. In what follows, we give an example to show this.

for all $t\in \mathbb{R}$. It is obvious that (1.7) implies that the assumption (i) of Theorem 1.1 holds. Take ${p}_{1}=\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}-{\sigma}_{1}$, ${p}_{2}=\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}+{\sigma}_{1}$, ${q}_{1}=\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}-{\sigma}_{2}$, ${q}_{2}=\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}-{\sigma}_{2}$ such that $[{p}_{1},{p}_{2}]\times [{q}_{1},{q}_{2}]\subset {\mathbb{R}}^{2}\setminus \mathrm{\Sigma}$. Then (1.8) implies that the assumption (ii) of Theorem 1.1 holds. Thus, by Theorem 1.1 we can obtain a $\frac{T}{2}$-anti-periodic solution of equation (1.1). Here the ratio $\frac{2F(t,s)}{{s}^{2}}$ stays at infinity in the rectangular domain $[{p}_{1},{p}_{2}]\times [{q}_{1},{q}_{2}]$ between Fučík spectrum curves ${\mathrm{\Sigma}}_{m}$ and ${\mathrm{\Sigma}}_{m+1}$, while the ratio $\frac{f(t,s)}{s}$ can cross at infinity multiple Fučík spectrum curves ${\mathrm{\Sigma}}_{1},{\mathrm{\Sigma}}_{2},\dots ,{\mathrm{\Sigma}}_{m+1}$.

This paper is organized as follows. In Section 2, some necessary preliminaries are presented. In Section 3, we give the proof of Theorem 1.1.

## 2 Preliminaries

which implies that the operator *J* is continuous. In view of the Arzela-Ascoli theorem, it is easy to see that *J* is completely continuous.

Denote by deg the Leray-Schauder degree. We need the following results.

**Lemma 2.1** ([[36], p.58])

*Let* Ω *be a bounded open region in a real Banach space* *X*. *Assume that* $K:\overline{\mathrm{\Omega}}\to \mathbb{R}$ *is completely continuous and* $p\notin (I-K)(\partial \mathrm{\Omega})$. *Then the equation* $(I-K)(x)=p$ *has a solution in* Ω *if* $deg(I-K,\mathrm{\Omega},p)\ne 0$.

**Lemma 2.2** ([[36], Borsuk theorem, p.58])

*Assume that* *X* *is a real Banach space*. *Let* Ω *be a symmetric bounded open region with* $\theta \in \mathrm{\Omega}$. *Assume that* $K:\overline{\mathrm{\Omega}}\to \mathbb{R}$ *is completely continuous and odd with* $\theta \notin (I-K)(\partial \mathrm{\Omega})$. *Then* $deg(I-K,\mathrm{\Omega},\theta )$ *is odd*.

## 3 Proof of Theorem 1.1

*Proof of Theorem 1.1*Consider the following homotopy problem:

where $(\lambda ,\lambda )\in [{p}_{1},{p}_{2}]\times [{q}_{1},{q}_{2}]$, $\mu \in [0,1]$.

*f*is continuous, there exist ${n}_{0}\in {\mathbb{Z}}^{+}$, ${C}_{1}>0$ such that

*n*, we get

*t*, we get

*λ*, if $z(t)>0$, we have

It is easy to see that ${z}^{\prime}({t}_{0})\ne 0$. In fact, if not, in view of (3.7)-(3.10), we get $z(t)=0$, ${z}^{\prime}(t)=0$, $\mathrm{\forall}t\in [0,T]$, which is contrary to ${\parallel z\parallel}_{{C}^{1}}=1$.

a contradiction.

If there is $\overline{t}\in (\u03f5,{t}_{1})$ such that $\overline{z}(\overline{t})={z}_{1}(t)$, then comparing (3.9) with (3.13), we can obtain that ${\overline{z}}^{\prime}(\overline{t})\ge {z}_{1}^{\prime}(\overline{t})$, which implies that if $t>\overline{t}$, we have $\overline{z}(\overline{t})\ge {z}_{1}(\overline{t})$. Then $\overline{z}(t)\ge {z}_{1}(t)$ for $t\in (0,{t}_{1}]$. Similarly, we have $\overline{z}(t)\le {z}_{2}(t)$, $\mathrm{\forall}t\in [0,{t}_{1}]$. Hence, (3.17) holds.

where ${t}_{2}$ is the first zero point on $({t}_{1},T)$.

which is contrary to (3.22).

If ${\overline{z}}^{\prime}(0)<0$, then by the assumption (ii), we can obtain a contradiction using similar arguments.

*μ*such that

*f*, we obtain

which implies that $\phi \in {C}_{\frac{T}{2}}^{0}$.

Now, using Lemma 2.1, we can see that (1.2) has a solution and hence (1.1) has a $\frac{T}{2}$-anti-periodic solution. The proof is complete. □

## Declarations

### Acknowledgements

The authors sincerely thank Prof. Yong Li for his instructions and many invaluable suggestions. This work was supported financially by NSFC Grant (11101178), NSFJP Grant (201215184), and the 985 Program of Jilin University.

## Authors’ Affiliations

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