In this paper, we study the existence of anti-periodic solutions for the following second-order ordinary differential equation:

$-{x}^{\u2033}=f(t,x),$

(1.1)

where

$f\in C({\mathbb{R}}^{2},\mathbb{R})$,

$f(t+\frac{T}{2},-s)=-f(t,s)$,

$\mathrm{\forall}t,s\in \mathbb{R}$ and

*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:

$\{\begin{array}{c}{x}^{\u2033}=-f(t,x),\hfill \\ {x}^{(i)}(0)=-{x}^{(i)}(\frac{T}{2}),\phantom{\rule{1em}{0ex}}i=0,1.\hfill \end{array}$

(1.2)

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

The problem of the existence of solutions of (1.1) under various boundary conditions has been widely investigated in the literature and many results have been obtained (see [

1–

13]). Usually, the asymptotic interaction of the ratio

$\frac{f(t,s)}{s}$ with the Fučík spectrum of

$-{x}^{\u2033}$ under various boundary conditions was required as a nonresonance condition to obtain the solvability of equation (

1.1). Recall that the Fučík spectrum of

$-{x}^{\u2033}$ with an anti-periodic boundary condition is the set of real number pairs

$({\lambda}_{+},{\lambda}_{-})\in {\mathbb{R}}^{2}$ such that the problem

$\{\begin{array}{c}-{x}^{\u2033}={\lambda}_{+}{x}^{+}-{\lambda}_{-}{x}^{-},\hfill \\ {x}^{(i)}(0)=-{x}^{(i)}(\frac{T}{2}),\phantom{\rule{1em}{0ex}}i=0,1\hfill \end{array}$

(1.3)

has nontrivial solutions, where

${x}^{+}=max\{0,x\}$,

${x}^{-}=max\{0,-x\}$; while the concept of Fučík spectrum was firstly introduced in the 1970s by Fučík [

14] and Dancer [

15] independently under the periodic boundary condition. Since the work of Fonda [

6], some investigation has been devoted to the nonresonance condition of (1.1) by studying the asymptotic interaction of the ratio

$\frac{2F(t,s)}{{s}^{2}}$, where

$F(t,s)={\int}_{0}^{s}f(t,\tau )\phantom{\rule{0.2em}{0ex}}d\tau $, with the spectrum of

$-{x}^{\u2033}$ under different boundary conditions; for instance, see [

10] for the periodic boundary condition, [

16] for the two-point boundary condition. Note that

$\underset{s\to \pm \mathrm{\infty}}{lim\hspace{0.17em}inf}\frac{f(t,s)}{s}\le \underset{s\to \pm \mathrm{\infty}}{lim\hspace{0.17em}inf}\frac{2F(t,s)}{{s}^{2}}\le \underset{s\to \pm \mathrm{\infty}}{lim\hspace{0.17em}sup}\frac{2F(t,s)}{{s}^{2}}\le \underset{s\to \pm \mathrm{\infty}}{lim\hspace{0.17em}sup}\frac{f(t,s)}{s},$

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.

Denote by Σ the Fuc̆ík spectrum of the operator

$-{x}^{\u2033}$ under the anti-periodic boundary condition. Simple computation implies that

$\mathrm{\Sigma}={\bigcup}_{m=1}^{+\mathrm{\infty}}{\mathrm{\Sigma}}_{m}$, where

${\mathrm{\Sigma}}_{m}=\{({\lambda}_{+},{\lambda}_{-})\in {\mathbb{R}}^{2}:\frac{(m+1)\pi}{\sqrt{{\lambda}_{+}}}+\frac{m\pi}{\sqrt{{\lambda}_{-}}}=\frac{T}{2}\text{or}\frac{m\pi}{\sqrt{{\lambda}_{+}}}+\frac{(m+1)\pi}{\sqrt{{\lambda}_{-}}}=\frac{T}{2},m\in \mathbb{N}\}.$

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)

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

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

In particular, if

${\lambda}_{+}={\lambda}_{-}$, then problem (1.3) becomes the following linear eigenvalue problem:

$\{\begin{array}{c}-{x}^{\u2033}=\lambda x,\hfill \\ {x}^{(i)}(0)=-{x}^{(i)}(\frac{T}{2}),\phantom{\rule{1em}{0ex}}i=0,1.\hfill \end{array}$

(1.6)

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* $\frac{4{(2m-1)}^{2}{\pi}^{2}}{{T}^{2}}<p\le \underset{|s|\to +\mathrm{\infty}}{lim\hspace{0.17em}inf}\frac{2F(t,s)}{{s}^{2}}\le \underset{|s|\to +\mathrm{\infty}}{lim\hspace{0.17em}sup}\frac{2F(t,s)}{{s}^{2}}\le q<\frac{4{(2m+1)}^{2}{\pi}^{2}}{{T}^{2}}$

*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

$\underset{|s|\to +\mathrm{\infty}}{lim\hspace{0.17em}sup}\frac{f(t,s)}{s}\le {\sigma}_{1}<\frac{4{\pi}^{2}}{{T}^{2}}={\lambda}_{1},\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in \mathbb{R},$

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.

Denote

${\lambda}_{m}=\frac{4{(2m-1)}^{2}{\pi}^{2}}{{T}^{2}}$ for some positive integer

$m\ge 1$. Define

$f(t,s)=cos\left(\frac{2\pi}{T}t\right)+\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}s+(\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}-\delta )scoss,\phantom{\rule{1em}{0ex}}\mathrm{\forall}t\in \mathbb{R},s\in \mathbb{R},$

where

$\delta \in (0,\frac{{\lambda}_{1}}{100})$. Clearly,

$\begin{array}{rcl}f(t+\frac{T}{2},-s)& =& cos\left[\frac{2\pi}{T}(t+\frac{T}{2})\right]-[\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}s+(\frac{{\lambda}_{m}+{\lambda}_{m+1}}{2}-\delta )scoss]\\ =& -f(t,s).\end{array}$

for all

$t\in \mathbb{R}$,

$s\in \mathbb{R}$, which imply that

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.