0 Boundary Value Problems

a SpringerOpen Journal

RESEARCH

Open Access

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

Xin Zhao1 and Xiaojun Chang2

"Correspondence: changxj1982@hotmail.com 2College of Mathematics, Jilin University, Changchun, Jilin 130012, China

Fulllist of author information is available at the end of the article

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 FuCik 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 Fucik spectrum curves, and recent results in the literature are complemented and generalized.

Keywords: anti-periodic solutions; Fucik spectrum; Leray-Schauder degree theory; Borsuk theorem

1 Introduction and main results

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

-x" = f (t, x), (1.1)

wheref e C(R2, R), f (t + -|,-s) = -f (t, s), Vt, s e 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 + 2) = -x(t) for all t e R. Note that to obtain anti-periodic solutions of (1.1), it suffices to find solutions of the following anti-periodic boundary value problem:

ft Spri

ringer

x" = -f (t, x),

J (1.2)

x(i)(0) = -x(i)( 2), i = 0,1.

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 [113]). Usually, the asymptotic interaction of the ratio f(Ss) with the Fucik spectrum of -x" under various boundary conditions was required as a nonresonance condition to obtain the solvability of equation (1.1). Recall that the Fucik spectrum of -x" with an anti-periodic

© 2012 Zhao and Chang; licensee Springer. 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 originalwork is properly cited.

boundary condition is the set of real number pairs (h+, h-) e M2 such that the problem x — h+x h_x ,

x(i) (0) — -x(i) (2), i — 0,1

has nontrivial solutions, where x+ — max{0, x}, x- — max{0, -x}; while the concept of Fucik spectrum was firstly introduced in the 1970s by Fucik [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 ^S-A, where F(t, s) — /0s/(t, x) dx, with the spectrum of -x" under different boundary conditions; for instance, see [10] for the periodic boundary condition, [16] for the two-point boundary condition. Note that

/(t, s) . 2F (t, s) 2F (t, s) / (t, s) liminf-< liminf--— < limsup--— < limsup-,

s s2 s2 s

we can see that the conditions on the ratio ^Fr^ are more general than those on the ratio In fact, by using the asymptotic interaction of the ratio ^Fr^ with the spectrum of -x", the ratio ^^ can cross multiple spectrum curves of -x". In this paper, we are interested in the nonresonance condition on the ratio 2Fs(zt,s) for the solvability of (1.1) involving the Fucik spectrum of -x" 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 /(t, s) — -/(t + f,-s) — /(t + T, s), x(t) is a T-periodic solution of (1.1) ifx(t) is a 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 X the Fucik spectrum of the operator -x" under the anti-periodic boundary condition. Simple computation implies that X — Um—1 Xm, where

{s _ 9 (m + 1)n mn T mn (m + 1)n T 1

(h+,e+ vx= = t or + 2,meNJ.

It is easily seen that the set X can be seen as a subset of the Fucik spectrum of -x" under the corresponding Dirichlet boundary condition; one can see the definition of the set X2i+i, i e N, or Figure 1 in [12]. Without loss of generality, we assume that ym is an eigenfunction of (1.3) corresponding to (h+, h_) e Xm such that <pm(0) — 0 and y'm(0) — a e R\{0}.Denote Xm,1 — {(h+, h-) e M2: + ^ — 2, m e Z+} and Xm,2 — {(h+, kJ) e M2: ^ + —

■2, m e Z+}. Then if a > 0, we obtain only a one-dimensional function ym, denoted by pm>i, corresponding to the point (X+, X-) e Xm,i, and if a <0, we obtain only a one-dimensional function ym, denoted by ^m,2, corresponding to the point (X+, X-) e Xm,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 ^jr^ stays asymptotically at infinity in some rectangular domain between Fucik spectrum curves Xm and Xm+1.

Our main result is as follows.

Theorem 1.1 Assume thatf e C(R2, R), f (t + f ,-s) = -f (t, s). If the following conditions:

(i) There exist positive constants p, C1, M such that

p < /XM) < Ci, Vt e r, V|s|>M; (.4)

(ii) There exist connect subset T c R2 \ X, constants p1, q^ p2, q2 > 0 and a point of the type (X, X) e R2 such that

(X, X) e p1, qj x [p2, q2] C T (1.5)

. 2F (t, s) 2F (t, s) p1 < liminf--— < limsup--— < p2,

s^+rn s2 s2

2F (t, s) 2F (t, s) q1 < liminf--— < limsup--— < q2,

s^-TO s2 s^-ro s2

hold uniformly for all t e R, then (1.1) admits a ■ -anti-periodic solution.

In particular, if X+ = X-, then problem (1.3) becomes the following linear eigenvalue problem:

x — Xx,

x(i)(0) ——x(i)( f), i — 0,1.

Simple computation implies that the operator -x" with the anti-periodic boundary condition has a sequence of eigenvalues Xm — 4(2mr1 n , m e Z+, and the corresponding eigenspace is two-dimensional.

Corollary 1.2 Assume thatf e C(R2, R), f (t, s) — -— (t + f,—s). If (1.4) holds and there exist constants p, q and me Z+ such that

4(2m — 1)2n2 . . 2F (t, s) 2F (t, s) 4(2m + 1)2n2

<p < liminf--— < limsup--— < q <

T2 s- si^. +œ S- T-

holds uniformly for all t e R, then (1.1) admits a ■ -anti-periodic solution.

Remark It is well known that (1.1) has a T-anti-periodic solution if

f (t, s) 4n2 limsup^——— < a < — = ki, Vt e R,

Isl^- s T

for some a1 > 0 (see Theorem 3.1 in [22]), which implies that the ratio ^^ stays at infinity asymptotically below the first eigenvalue k1 of (1.6). In this paper, this requirement on the ratio ^^ can be relaxed to (1.4), with some additional restrictions imposed on the ratio ^SM. In fact, the conditions relative to the ratios ^^ and ^^^ as in Theorem 1.1

s2 s s2

and Corollary 1.2 may lead to that the ratio oscillates and crosses multiple consecutive eigenvalues or branches of the Fucik spectrum curves of the operator -x". In what follows, we give an example to show this. Denote km — 4(2m-1) n for some positive integer m > 1. Define

f{t, s) = cos('2n A + km +km+1 s +i km +km+1 - 5 )s cos s, Vt e R, s e R,

where 5 e (0, ko). Clearly,

f[t + -2,-s Ï = cos

— - t + - -

- ^ 2/.

= -f (t, s).

km + km+i i km + km+i „ >

-s +--5 s cos s

In addition,

/(t, s) COs(T" t) hm + hm+1 / hm + hm+1 A

- — -T- + - +--S cos s,

s s 2 2

2F(t, s) 2 cos(T t) km + km+1 / km + km+1 \s sin s + cos s — — ^^ + —+ ' ~--S

for all t e R, s e R, which imply that

S — liminf/(t' s) < limsup— hm + hm+1 - S, (1.7)

2F(t,s) hm + hm+1 ^ oX

lim -—-o--(1.8)

|s|^+— s2 2

for all t e R.It is obvious that (1.7) implies that the assumption (i) of Theorem 1.1 holds. Take p1 — hm+hm+1 - oi, P2 — hm+^m+1 + 01, q1 — - a2, q2 — hm+2~m+1 - 02 such that

[p1,p2] x [q1, q2] C R2 \ X. Then (1.8) implies that the assumption (ii) of Theorem 1.1 holds. Thus, by Theorem 1.1 we can obtain a 2-anti-periodic solution of equation (1.1). Here the ratio ^Fr^ stays at infinity in the rectangular domain [p1,p2] x [q1, q2] between Fucik spectrum curves Xm and Xm+1, while the ratio ^^ can cross at infinity multiple Fucik spectrum curves X1, X2,..., Xm+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

Assume that T >0. Define

CT = \x e Ck(R,R):xlt + T ) = -x(t),Vt e R[,

■k = max |x(t) I + ••• + max \x(k)(t) |, Vx e Ck([0, T], R), k = 0,1, -....

te[0,T ]

te[0,T ]

For x e CkT, we can write the Fourier series expansion as follows:

x(t) = J2

-n (-i + 1)t , -n (-i + 1)i'

a-i+i cos---+ b-i+1 sin

Define an operator J : CkT ^ CT1 by

pi T w U

(Jx)(t)= x(s) ds - — -i

j0 i=0

-i+1 -i+l

a-i+1 -n (-i + 1)t b-i+1 -n (-i + 1)t' sin-—--——- cos

-i + 1

-i + 1

Clearly,

dJx(t) dt

T œ h

--x(t), (Jx)(0) = - Tb^, -n -i + 1

which implies that d2(J2 x(t))

— x(t).

Furthermore, we obtain

|jx(t)| < l |x(s)| dS + T £ -+1 < T Mc + ^^ b-i+1) (îeit-t^

Note that

( ±—

(-i + 1)-/ -j-'

using the Parseval equality /0 |x(s)|2 ds — f Si——0 [a2i+i + ¿^J, we get T ~

|jx(t)| < THx^ck + [a-i+1 + b-i+1]

< THxHck +

T - 'T

x(s)\ ds

< — HxHck, Vt e [0, T],

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 fi be a bounded open region in a realBanach space X. Assume that K : fi ^ R is completely continuous and p é (I - K)(9fi). Then the equation (I -K)(x) = p has a solution in fi if deg(I - K, fi,p) = 0.

Lemma 2.2 ([36, Borsuk theorem, p.58]) Assume that X is a real Banach space. Let fi be a symmetric bounded open region with Séfi. Assume that K : fi ^ R is completely continuous and odd with 0 é (I - K)(9fi). Then deg(I - K, fi, 0) is odd.

3 Proof of Theorem 1.1

Proof of Theorem 1.1 Consider the following homotopy problem:

x" = -if (t,x) -(1 - i)Xx = <p(i, t,x(t)), (.1)

x»=-x(i)(f), ,= <», ,3,,

where (X, X) é [pi,p2] x [qi, q2], i é [0,1].

We first prove that the set of all possible solutions of problem (3.1)-(3.2) is bounded. Assume by contradiction that there exist a sequence of number {in} c [0,1] and corresponding solutions {xn} of (3.1)-(3.2) such that

||Xn ||C1 ^ (3.3)

Set zn = ||xX| t. Obviously, ||zn||C1 = 1 and zn satisfies

-zn = fxù +(1-in)Xzn, (3.4)

||Xn||c1

zni)(0) = -zni^ T), i = 0,1. (3.5)

By (1.4), (3.3) and the fact thatf is continuous, there exist n0 é Z+, C1 > 0 such that

|f (t, xn)\ \\Xn\\cl

f (t, Xn)

| xn |

< Cl for n > n0.

\Xn\cl

In view of ¡x„ e [0,1], together with the choice of (h, h), it follows that there exists M1 > 0 such that, for all n > n0,

K'(t)|< Ml Vte [0, T].

It is easily seen that {zn(t)} and {Zn(t)} are uniformly bounded and equicontinuous on [0, T]. Then, using the Arzela-Ascoli theorem, there exist uniformly convergent subsequences on [0, T] for {zn(t)} and {Zn(t)} respectively, which are still denoted as {zn(t)} and

{z'n (t)}, such that

lim Zn(t) = z(t), lim Zn(t) = Z(t). (3.6)

n—>to n—>TO

Clearly, UzHc1 = 1. Since xn(t) is a solution of (3.1)-(3.2), for each n, we get

j xn(t) dt = j~ xn(t) dt + j~ xn^t + = 0,

which implies that there exists tn e [0, T] such that xn(tn) = 0. Then

lim Zn(tn) = lim ^^ = 0. (3.7)

n—TO n—TO Uxn UC1

Owing to that the sequences {tn} and {¡n} are uniformly bounded, there exist t0 e [0, T] and ¡0 e [0,1] such that, passing to subsequences if possible,

lim tn = t0, lim ¡¡n = ¡¡0. (3.8)

n—TO n—TO

Multiplying both sides of (3.4) by z'n(t) and integrating from tn to t, we get

[4 (t„)]-- [z'n(t)Y

= ^ -F(t,xn(t)) x2n(t) -F(t,xn(tn))

xn(t) HxnHC1 Hxn Hc1

+ (1 - (zn(t^2 - (zn(tn))-].

Taking a superior limit as n — to, by (3.3) and (3.6)-(3.8), we obtain [z'(t0)]2 - [z'(t)]2 = ¡0 limsup 2(t2xn(t)) • Z2(t) + (1 - ¡0)^z2(t).

n—TO xn(t)

By the assumption (ii) and the choice of A, if z(t) > 0, we have

[z'(t0)]2 - [z'(t)]2 <P2z2(t).

Similarly, we obtain

[z'(t0)]2 - [z'(t)]2 >p1z2(t) for z(t) > 0, [z'(t0)]2 - [z'(t)]2 < q2z2(t) for z(t) < 0, [z'(t0)]2 - [z'(t)]2 > q1z2(t) for z(t) < 0.

Note that z(t) e C1 [0, T], the above inequalities can be rewritten as the following equivalent forms:

-P2 [z(t)]2 < [z'(t)]2 - [z'(tc)]2 < -P1 [z(t)]2, z(t) > 0, (3.9)

-q2[z(t)]2 < [z'(t)]2 - [z'(tc)]2 < -q:[z(t)]2, z(t) < 0. (3.10)

It is easy to see that z'(t0) =0. In fact, if not, in view of (3.7)-(3.10), we get z(t) = 0,Z(t) = 0, Vt e [0, T], which is contrary to ||z||C1 = 1.

We claim that z'(t) has only finite zero points on [0, T]. In fact, if not, we may assume that there are infinitely many zero points Z} c [0, T] ofz'(t). Without loss of generality, we assume that there exists Zo e [0, T] such that Zi = Zo. Letting t = Zi in (3.9)-(3.10)

and taking i ^ro, we can obtain that z(Z0) = 0. Without loss of generality, we assume that z(Z0) > 0. Since z(t) is continuous, there exist n, $ >0 such that z(t) > n > 0, Vt e [t0 - S, t0 + 5]. Then there exists n1 > 0 such that, if n > n1, we have

zn(t) > n, Vt e [t0- $, t0 + $]. (3.11)

Clearly, z-(t) = 0, Vt e [t0 - S, t0 + $]. Take Z*, Z* e [t0 - S, t0 + S] with Z* < Z* such that z'(Z*) = z'(Z*) = 0. Integrating (3.4) from Z* to Z*,

1 fZ* rZ*

z!n(Z*)-z'n(Z*) = Inz—— f(t,Xn(s)) ds +(1 - ^n) I hzn(s) ds. (3.12)

yxnyc1 Jz* JZ*

By (3.3), (3.11), we obtain

xn(t) = zn(t)||xn||C1 > nlxn|C1 ^ as n ^

holds uniformly for t e [Z*, Z*]. Thus, using (1.4), we get

f (t,xn(t)K p w c rZ Z*1

-mar > * Vte Z1

which implies that

f (t, Xn(t)) f (t, Xn(t))

Zn(t) > P ■ n >0, vte [z*,Z*].

\\Xn\cl Xn(t)

Then, together with (3.6), (3.8) and (3.12), we obtain

0 > ßo ■ P ■ n(Z* - Z*) + (1 - ^o) ■ k ■ n(Z* - Z*) > 0,

a contradiction.

Now, we show that (3.9)-(3.10) has only a trivial anti-periodic solution. In fact, if not, we assume that (3.9)-(3.10) has a nontrivial anti-periodic solution Z(t). Without loss of generality, we assume t0 = 0. Firstly, we consider the case that Z'(0) > 0. Assume that z1, z2 satisfy the following equations respectively:

Z(t)]2- [z1 (0)]2 = -^2[z1(t)]2, Z1(t) > 0, (3.13)

[z2(t)]2- [z2(0)]2 = -jP1[z2(t)]2, Z2(t) > 0 (.14)

Z(0)= Z1(0) = Z2(0), (3.15)

z1 (0) < Z'(0) < z2(0). (3.16)

Take ti as the first zero point of z(t) on (0, T]. Then by (3.13)-(3.16) it follows that

In fact, by (3.15)-(3.16) and the fact that z, z1, z2 are continuous differential, it is easy to see that there exists sufficiently small e e (0, t1) such that

z1(t) < z(t) < z2(t), t e (0, e), z1 (t) < z'(t) < z2(t), t e (0, e).

If there is i e (e, t1) such that z(i) = z1(t), then comparing (3.9) with (3.13), we can obtain that z'(i) > z1(i), which implies that if t > i, we have z(i) > z1(i). Then z(t) > z1(t) for t e (0, t1]. Similarly, we have z(t) < z2(t), Vt e [0, tj. Hence, (3.17) holds. Similarly, if z1, z2 satisfy

z(t1) = z1(t1) = z2(t1), z1 (t1) < z'(t1) < z2(t1),

then we obtain

z1(t) < z(t) < z2(t), Vt e [t1, t2],

where t2 is the first zero point on (t1, T).

Since z'(t) has finite zero points, (3.13), (3.14), (3.18), (3.19) can be transformed into the following equations respectively:

z"(t) = -p2z(t), z"(t) = -p1z(t), z(t) > 0, (3.20)

z"(t) = -q2z(t), z"(t) = -q1z(t), z(t) < 0. (3.21)

Then there exist A, B, C, D >0 such that

A sin vp2t < z(t) < B sin vp!t, Vt e [0, t1],

-C sin (t -11) < z(t) < -D sin VqT(t -11), Vt e [t1, t2].

It is easy to get

Z1(t) < z(t) < z2(t), Vt e [0, t1].

(3.17)

[z1 (t)]2- [z1 (0^2 = -?2 [Z1(t^2, Z1(t) < 0, [z2(t)]2 - [z2(0)]2 = -q1[z2(t)]2, Z2(t) < 0

(3.18) (.19)

Since z is anti-periodic and z'(0) > 0, there exists m e Z+ such that

(m + l)n mn T (m + l)n mn

-+-< tm = — <-+-,

s/ql vp2 2 vqr vPl

which implies that there exists a real number pair (p , q) e [pl,p2] x [ql, q2] such that (m + l)n mn T

— + TP*= t (

On the other hand, in view of the assumption (ii), by the definition of X and (p , q) e [pl,p2] x [ql,q2], it follows that

(m + l)n mn T -r^— + ~r=i = _, ^m e Z+,

Vp" Vq" 2

which is contrary to (3.22).

If z'(0) < 0, then by the assumption (ii), we can obtain a contradiction using similar arguments.

In a word, we can see that there exists C >0 independent of ¡x such that

Mb < C. (.23)

& = {x e C\ : ||x||Cl < C + l}. Clearly, & is a bounded open set in C\. Note that, for x e C\, using the assumption onf,

we obtain

/ T / T H ¡, t + -,xl t + -

= -f t + --,^t + -(l-x)Xx^t + T = ¡f (t, x) + (l - ¡i)Xx(t) = -^(x, t, x(t)),

which implies that h e CT.

Define Gx : & — C2T by

Gp( x(t))= J2 t, x(t)).

Clearly, Gx is completely continuous, and by (2.l) and (3.l) it follows that the fixed point of Gl in & is the anti-periodic solution of problem (l.l). Define the homotopy H: & x [0, l] — C\ as follows:

H(x, //) =x - Gx(x).

In view of (3.23), it follows that

H(x, j) = 0, V(x, j) e dQ x [0,1]. Hence,

deg(I - G1, Q,0) = deg(I - G0, Q,0). Note that the operator G0 is odd. By Lemma 2.2 it follows that deg(I - G0, Q,0) = 0. Thus, deg(I - G1, Q,0) =0.

Now, using Lemma 2.1, we can see that (1.2) has a solution and hence (1.1) has a ^-anti-periodic solution. The proof is complete. □

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors read and approved the finalmanuscript.

Author details

'College of Information Technology, Jilin AgriculturalUniversity, Changchun, 130118, P.R. China. 2College of Mathematics, Jilin University, Changchun, Jilin 130012, China.

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.

Received: 20 July 2012 Accepted: 20 September 2012 Published: 21 December 2012

References

1. Bravo, JL, Torres, PJ: Periodic solutions of a singular equation with indefinite weight. Adv. Nonlinear Stud. 10, 927-938 (2010)

2. Chu, J, Fan, N, Torres, PJ: Periodic solutions for second order singular damped differentialequations. J. Math. Anal. Appl. 388, 665-675 (2012)

3. Chu, J, Torres, PJ: Applications of Schauder's fixed point theorem to singular differentialequations. Bull. Lond. Math. Soc. 39, 653-660 (2007)

4. Chu, J, Torres, PJ, Zhang, M: Periodic solutions of second order non-autonomous singular dynamicalsystems. J. Differ. Equ. 239, 196-212 (2007)

5. Chu, J, Zhang, Z: Periodic solutions of singular differentialequations with sign-changing potential. Bull. Aust. Math. Soc. 82, 437-445 (2010)

6. Fonda, A: On the existence of periodic solutions for scalar second order differentialequations when only the asymptotic behaviour of the potentialis known. Proc. Am. Math. Soc. 119,439-445 (1993)

7. Habets, P, Omari, P, Zanolin, F: Nonresonance conditions on the potentialwith respect to the Fucik spectrum for the periodic boundary value problem. Rocky Mt. J. Math. 25,1305-1340 (1995)

8. Halk, R, Torres, PJ: On periodic solutions of second-order differentialequations with attractive-repulsive singularities. J. Differ. Equ. 248,111-126(2010)

9. Halk, R, Torres, PJ, Zamora, M: Periodic solutions of singular second order differentialequations: the repulsive case. Topol. Methods Nonlinear Anal. 39,199-220 (2012)

10. Liu, W, Li, Y: Existence of 2^-periodic solutions for the non-dissipative Duffing equation under asymptotic behaviors of potentialfunction. Z. Angew. Math. Phys. 57,1-11 (2006)

11. Omari, P, Zanolin, F: Nonresonance conditions on the potentialfor a second-order periodic boundary value problem. Proc. Am. Math. Soc. 117,125-135 (1993)

12. Tomiczek, P: PotentialLandesman-Lazer type conditions and the Fucik spectrum. Electron. J. Differ. Equ. 2005, Art. ID 94 (2005)

13. Zhang, M: Nonresonance conditions for asymptotically positively homogeneous differentialsystems: the Fucik spectrum and its generalization. J. Differ. Equ. 145, 332-366 (1998)

14. Fucik, S: Boundary value problems with jumping nonlinearities. (Cas. Pest. Mat. 101,69-87 (1976)

15. Dancer, EN: Boundary-value problems for weakly nonlinear ordinary differentialequations. Bull. Aust. Math. Soc. 15, 321-328(1976)

16. Marcos, A: Nonresonance conditions on the potentialfor a semilinear Dirichlet problem. Nonlinear Anal. 70, 335-351 (2009)

17. Gao, E, Song, S, Zhang, X: Solving singular second-order initial/boundary value problems in reproducing kernel Hilbert space. Bound. Value Probl. 2012, Art. ID 3 (2012)

18. Okochi, H: On the existence of periodic solutions to nonlinear abstract parabolic equations. J. Math. Soc. Jpn. 40, 541-553 (1988)

19. Delvos, FJ, Knoche, L: Lacunary interpolation by antiperiodic trigonometric polynomials. BIT Numer. Math. 39, 439-450(1999)

20. Chen, H: Antiperiodic wavelets. J. Comput. Math. 14, 32-39 (1996)

21. Chen, Y: Note on Massera's theorem on anti-periodic solution. Adv. Math. Sci. Appl. 9,125-128 (1999)

22. Chen, T, Liu, W, Yang, C: Antiperiodic solutions for Liénard-type differentialequation with p-Laplacian operator. Bound. Value Probl. 2010, Art. ID 194824 (2010)

23. Chen, T, Liu, W, Zhang, J, Zhang, H: Anti-periodic solutions for higher-order nonlinear ordinary differentialequations. J. Korean Math. Soc. 47, 573-583 (2010)

24. Chen, Y, Nieto, JJ, O'Regan, D: Anti-periodic solutions for fully nonlinear first-order differentialequations. Math. Comput. Model. 46, 1183-1190 (2007)

25. Aizicovici, S, McKibben, M, Reich, S: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities. Nonlinear Anal. 43, 233-251 (2001)

26. Chen, Y, Nieto, JJ, O'Regan, D: Anti-periodic solutions for evolution equations associated with maximalmonotone mappings. Appl. Math. Lett. 24, 302-307 (2011)

27. Chen, Y, O'Regan, D, Agarwal, RP: Anti-periodic solutions for semilinear evolution equations in Banach spaces. J. Appl. Math. Comput. 38,63-70 (2012)

28. Ji, S: Time-periodic solutions to a nonlinear wave equation with periodic or anti-periodic boundary conditions. Proc. R. Soc. Lond. Ser. A465, 895-913 (2009)

29. Ji, S, Li, Y: Time periodic solutions to one dimensionalwave equation with periodic or anti-periodic boundary conditions. Proc. R. Soc. Edinb., Sect. A, Math. 137, 349-371 (2007)

30. Liu, Z: Anti-periodic solutions to nonlinear evolution equations. J. Funct. Anal. 258, 2026-2033 (2010)

31. Nakao, M: Existence of anti-periodic solution for the quasilinear wave equation with viscosity. J. Math. Anal. Appl. 204, 754-764(1996)

32. N'Guérékata, GM, Valmorin, V: Antiperiodic solutions ofsemilinearintegrodifferential equations in Banach spaces. Appl. Math. Comput. 218, 11118-11124 (2012)

33. Anahtarci, B, Djakov, P: Refined asymptotics of the spectralgap for the Mathieu operator. J. Math. Anal. Appl. 396, 243-255 (2012). doi:10.1016/j.jmaa.2012.06.019

34. Ahmad, B, Nieto, JJ: Existence of solutions for impulsive anti-periodic boundary value problems of fractionalorder. Taiwan. J. Math. 15, 981-993 (2011)

35. Pan, L, Cao, J: Anti-periodic solution for delayed cellular neuralnetworks with impulsive effects. Nonlinear Anal. 12, 3014-3027(2011)

36. Deimling, K: Nonlinear FunctionalAnalysis. Springer, New York (1985)

doi:10.1186/1687-2770-2012-149

Cite this article as: Zhao and Chang: Existence of anti-periodic solutions for second-order ordinary differential equations involving the Fuak spectrum. Boundary Value Problems 2012 2012:149.

Submit your manuscript to a SpringerOpen journal and benefit from:

► Convenient online submission

► Rigorous peer review

► Immediate publication on acceptance

► Open access: articles freely available online

► High visibility within the field

► Retaining the copyright to your article

Submit your next manuscript at ► springeropen.com