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

  • Xin Zhao1 and

    Affiliated with

    • Xiaojun Chang2Email author

      Affiliated with

      Boundary Value Problems20122012:149

      DOI: 10.1186/1687-2770-2012-149

      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

      In this paper, we study the existence of anti-periodic solutions for the following second-order ordinary differential equation:
      x = f ( t , x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ1_HTML.gif
      (1.1)
      where f C ( R 2 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq1_HTML.gif, f ( t + T 2 , s ) = f ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq2_HTML.gif, t , s R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq3_HTML.gif and T is a positive constant. A function x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq4_HTML.gif is called an anti-periodic solution of (1.1) if x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq4_HTML.gif satisfies (1.1) and x ( t + T 2 ) = x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq5_HTML.gif for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq6_HTML.gif. Note that to obtain anti-periodic solutions of (1.1), it suffices to find solutions of the following anti-periodic boundary value problem:
      { x = f ( t , x ) , x ( i ) ( 0 ) = x ( i ) ( T 2 ) , i = 0 , 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ2_HTML.gif
      (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 [113]). Usually, the asymptotic interaction of the ratio f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq7_HTML.gif with the Fučík spectrum of x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif with an anti-periodic boundary condition is the set of real number pairs ( λ + , λ ) R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq9_HTML.gif such that the problem
      { x = λ + x + λ x , x ( i ) ( 0 ) = x ( i ) ( T 2 ) , i = 0 , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ3_HTML.gif
      (1.3)
      has nontrivial solutions, where x + = max { 0 , x } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq10_HTML.gif, x = max { 0 , x } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq11_HTML.gif; 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 2 F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq12_HTML.gif, where F ( t , s ) = 0 s f ( t , τ ) d τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq13_HTML.gif, with the spectrum of x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif under different boundary conditions; for instance, see [10] for the periodic boundary condition, [16] for the two-point boundary condition. Note that
      lim inf s ± f ( t , s ) s lim inf s ± 2 F ( t , s ) s 2 lim sup s ± 2 F ( t , s ) s 2 lim sup s ± f ( t , s ) s , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equa_HTML.gif

      we can see that the conditions on the ratio 2 F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq12_HTML.gif are more general than those on the ratio f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq14_HTML.gif. In fact, by using the asymptotic interaction of the ratio 2 F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq12_HTML.gif with the spectrum of x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif, the ratio f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq14_HTML.gif can cross multiple spectrum curves of x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif. In this paper, we are interested in the nonresonance condition on the ratio 2 F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq12_HTML.gif for the solvability of (1.1) involving the Fučík spectrum of x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif 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 + T 2 , s ) = f ( t + T , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq15_HTML.gif, x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq4_HTML.gif is a T-periodic solution of (1.1) if x ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq4_HTML.gif is a T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq16_HTML.gif-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 [25, 810, 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, 2124]. Anti-periodic boundary conditions for partial differential equations and abstract differential equations are considered in [2532]. For recent developments involving the existence of anti-periodic solutions, one can also see [3335] and the references therein.

      Denote by Σ the Fuc̆ík spectrum of the operator x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif under the anti-periodic boundary condition. Simple computation implies that Σ = m = 1 + Σ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq17_HTML.gif, where
      Σ m = { ( λ + , λ ) R 2 : ( m + 1 ) π λ + + m π λ = T 2  or  m π λ + + ( m + 1 ) π λ = T 2 , m N } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equb_HTML.gif

      It is easily seen that the set Σ can be seen as a subset of the Fuc̆ík spectrum of x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif under the corresponding Dirichlet boundary condition; one can see the definition of the set Σ 2 i + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq18_HTML.gif, i N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq19_HTML.gif, or Figure 1 in [12]. Without loss of generality, we assume that φ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq20_HTML.gif is an eigenfunction of (1.3) corresponding to ( λ + , λ ) Σ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq21_HTML.gif such that φ m ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq22_HTML.gif and φ m ( 0 ) = a R { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq23_HTML.gif. Denote Σ m , 1 = { ( λ + , λ ) R 2 : ( m + 1 ) π λ + + m π λ = T 2 , m Z + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq24_HTML.gif and Σ m , 2 = { ( λ + , λ ) R 2 : m π λ + + ( m + 1 ) π λ = T 2 , m Z + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq25_HTML.gif. Then if a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq26_HTML.gif, we obtain only a one-dimensional function φ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq20_HTML.gif, denoted by φ m , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq27_HTML.gif, corresponding to the point ( λ + , λ ) Σ m , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq28_HTML.gif, and if a < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq29_HTML.gif, we obtain only a one-dimensional function φ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq20_HTML.gif, denoted by φ m , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq30_HTML.gif, corresponding to the point ( λ + , λ ) Σ m , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq31_HTML.gif.

      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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq32_HTML.gif is asymptotically linear in s at infinity and the ratio F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq33_HTML.gif stays asymptotically at infinity in some rectangular domain between Fučík spectrum curves Σ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq34_HTML.gif and Σ m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq35_HTML.gif.

      Our main result is as follows.

      Theorem 1.1 Assume that f C ( R 2 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq1_HTML.gif, f ( t + T 2 , s ) = f ( t , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq2_HTML.gif. If the following conditions:
      1. (i)
        There exist positive constants ρ, C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq36_HTML.gif, M such that
        ρ f ( t , s ) s C 1 , t R , | s | M ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ4_HTML.gif
        (1.4)
         
      2. (ii)
        There exist connect subset Γ R 2 Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq37_HTML.gif, constants p 1 , q 1 , p 2 , q 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq38_HTML.gif and a point of the type ( λ , λ ) R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq39_HTML.gif such that
        ( λ , λ ) [ p 1 , q 1 ] × [ p 2 , q 2 ] Γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ5_HTML.gif
        (1.5)
         
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equc_HTML.gif

      hold uniformly for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq6_HTML.gif,

      then (1.1) admits a T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq16_HTML.gif-anti-periodic solution.

      In particular, if λ + = λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq40_HTML.gif, then problem (1.3) becomes the following linear eigenvalue problem:
      { x = λ x , x ( i ) ( 0 ) = x ( i ) ( T 2 ) , i = 0 , 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ6_HTML.gif
      (1.6)

      Simple computation implies that the operator x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif with the anti-periodic boundary condition has a sequence of eigenvalues λ m = 4 ( 2 m 1 ) 2 π 2 T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq41_HTML.gif, m Z + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq42_HTML.gif, and the corresponding eigenspace is two-dimensional.

      Corollary 1.2 Assume that f C ( R 2 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq1_HTML.gif, f ( t , s ) = f ( t + T 2 , s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq43_HTML.gif. If (1.4) holds and there exist constants p, q and m Z + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq42_HTML.gif such that
      4 ( 2 m 1 ) 2 π 2 T 2 < p lim inf | s | + 2 F ( t , s ) s 2 lim sup | s | + 2 F ( t , s ) s 2 q < 4 ( 2 m + 1 ) 2 π 2 T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equd_HTML.gif

      holds uniformly for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq6_HTML.gif, then (1.1) admits a T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq16_HTML.gif-anti-periodic solution.

      Remark It is well known that (1.1) has a T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq16_HTML.gif-anti-periodic solution if
      lim sup | s | + f ( t , s ) s σ 1 < 4 π 2 T 2 = λ 1 , t R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Eque_HTML.gif

      for some σ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq44_HTML.gif (see Theorem 3.1 in [22]), which implies that the ratio f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq7_HTML.gif stays at infinity asymptotically below the first eigenvalue λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq45_HTML.gif of (1.6). In this paper, this requirement on the ratio f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq46_HTML.gif can be relaxed to (1.4), with some additional restrictions imposed on the ratio 2 F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq47_HTML.gif. In fact, the conditions relative to the ratios f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq7_HTML.gif and 2 F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq47_HTML.gif as in Theorem 1.1 and Corollary 1.2 may lead to that the ratio f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq7_HTML.gif oscillates and crosses multiple consecutive eigenvalues or branches of the Fučík spectrum curves of the operator x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq8_HTML.gif. In what follows, we give an example to show this.

      Denote λ m = 4 ( 2 m 1 ) 2 π 2 T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq41_HTML.gif for some positive integer m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq48_HTML.gif. Define
      f ( t , s ) = cos ( 2 π T t ) + λ m + λ m + 1 2 s + ( λ m + λ m + 1 2 δ ) s cos s , t R , s R , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equf_HTML.gif
      where δ ( 0 , λ 1 100 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq49_HTML.gif. Clearly,
      f ( t + T 2 , s ) = cos [ 2 π T ( t + T 2 ) ] [ λ m + λ m + 1 2 s + ( λ m + λ m + 1 2 δ ) s cos s ] = f ( t , s ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equg_HTML.gif
      In addition,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equh_HTML.gif
      for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq6_HTML.gif, s R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq50_HTML.gif, which imply that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ7_HTML.gif
      (1.7)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ8_HTML.gif
      (1.8)

      for all t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq6_HTML.gif. It is obvious that (1.7) implies that the assumption (i) of Theorem 1.1 holds. Take p 1 = λ m + λ m + 1 2 σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq51_HTML.gif, p 2 = λ m + λ m + 1 2 + σ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq52_HTML.gif, q 1 = λ m + λ m + 1 2 σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq53_HTML.gif, q 2 = λ m + λ m + 1 2 σ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq54_HTML.gif such that [ p 1 , p 2 ] × [ q 1 , q 2 ] R 2 Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq55_HTML.gif. Then (1.8) implies that the assumption (ii) of Theorem 1.1 holds. Thus, by Theorem 1.1 we can obtain a T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq16_HTML.gif-anti-periodic solution of equation (1.1). Here the ratio 2 F ( t , s ) s 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq12_HTML.gif stays at infinity in the rectangular domain [ p 1 , p 2 ] × [ q 1 , q 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq56_HTML.gif between Fučík spectrum curves Σ m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq34_HTML.gif and Σ m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq35_HTML.gif, while the ratio f ( t , s ) s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq14_HTML.gif can cross at infinity multiple Fučík spectrum curves Σ 1 , Σ 2 , , Σ m + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq57_HTML.gif.

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq58_HTML.gif. Define
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equi_HTML.gif
      For x C T 2 k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq59_HTML.gif, we can write the Fourier series expansion as follows:
      x ( t ) = i = 0 [ a 2 i + 1 cos 2 π ( 2 i + 1 ) t T + b 2 i + 1 sin 2 π ( 2 i + 1 ) t T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equj_HTML.gif
      Define an operator J : C T 2 k C T 2 k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq60_HTML.gif by
      ( J x ) ( t ) = 0 t x ( s ) d s T 2 π i = 0 b 2 i + 1 2 i + 1 = T 2 π i = 0 [ a 2 i + 1 2 i + 1 sin 2 π ( 2 i + 1 ) t T b 2 i + 1 2 i + 1 cos 2 π ( 2 i + 1 ) t T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equk_HTML.gif
      Clearly,
      d J x ( t ) d t = x ( t ) , ( J x ) ( 0 ) = T 2 π i = 0 b 2 i + 1 2 i + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equl_HTML.gif
      which implies that
      d 2 ( J 2 x ( t ) ) d t 2 = x ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ9_HTML.gif
      (2.1)
      Furthermore, we obtain
      | J x ( t ) | 0 T | x ( s ) | d s + T 2 π i = 0 | b 2 i + 1 | 2 i + 1 T x C k + T 2 π ( i = 0 b 2 i + 1 2 ) 1 2 ( i = 0 1 ( 2 i + 1 ) 2 ) 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equm_HTML.gif
      Note that
      ( i = 0 1 ( 2 i + 1 ) 2 ) 1 2 = π 2 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equn_HTML.gif
      using the Parseval equality 0 T | x ( s ) | 2 d s = T 2 i = 0 [ a 2 i + 1 2 + b 2 i + 1 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq61_HTML.gif, we get
      | J x ( t ) | T x C k + T 4 2 i = 0 [ a 2 i + 1 2 + b 2 i + 1 2 ] T x C k + T 4 2 2 T 0 T | x ( s ) | 2 d s 5 T 4 x C k , t [ 0 , T ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equo_HTML.gif

      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 : Ω ¯ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq62_HTML.gif is completely continuous and p ( I K ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq63_HTML.gif. Then the equation ( I K ) ( x ) = p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq64_HTML.gif has a solution in Ω if deg ( I K , Ω , p ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq65_HTML.gif.

      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 θ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq66_HTML.gif. Assume that K : Ω ¯ R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq67_HTML.gif is completely continuous and odd with θ ( I K ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq68_HTML.gif. Then deg ( I K , Ω , θ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq69_HTML.gif is odd.

      3 Proof of Theorem 1.1

      Proof of Theorem 1.1 Consider the following homotopy problem:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ10_HTML.gif
      (3.1)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ11_HTML.gif
      (3.2)

      where ( λ , λ ) [ p 1 , p 2 ] × [ q 1 , q 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq70_HTML.gif, μ [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq71_HTML.gif.

      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 { μ n } [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq72_HTML.gif and corresponding solutions { x n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq73_HTML.gif of (3.1)-(3.2) such that
      x n C 1 + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ12_HTML.gif
      (3.3)
      Set z n = x n x n C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq74_HTML.gif. Obviously, z n C 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq75_HTML.gif and z n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq76_HTML.gif satisfies
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ13_HTML.gif
      (3.4)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ14_HTML.gif
      (3.5)
      By (1.4), (3.3) and the fact that f is continuous, there exist n 0 Z + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq77_HTML.gif, C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq78_HTML.gif such that
      | f ( t , x n ) | x n C 1 = | f ( t , x n ) x n | | x n | x n C 1 C 1 for  n n 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equp_HTML.gif
      In view of μ n [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq79_HTML.gif, together with the choice of ( λ , λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq80_HTML.gif, it follows that there exists M 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq81_HTML.gif such that, for all n n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq82_HTML.gif,
      | z n ( t ) | M 1 , t [ 0 , T ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equq_HTML.gif
      It is easily seen that { z n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq83_HTML.gif and { z n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq84_HTML.gif are uniformly bounded and equicontinuous on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq85_HTML.gif. Then, using the Arzela-Ascoli theorem, there exist uniformly convergent subsequences on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq85_HTML.gif for { z n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq83_HTML.gif and { z n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq84_HTML.gif respectively, which are still denoted as { z n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq83_HTML.gif and { z n ( t ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq84_HTML.gif, such that
      lim n z n ( t ) = z ( t ) , lim n z n ( t ) = z ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ15_HTML.gif
      (3.6)
      Clearly, z C 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq86_HTML.gif. Since x n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq87_HTML.gif is a solution of (3.1)-(3.2), for each n, we get
      0 T x n ( t ) d t = 0 T 2 x n ( t ) d t + 0 T 2 x n ( t + T 2 ) d t = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equr_HTML.gif
      which implies that there exists t n [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq88_HTML.gif such that x n ( t n ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq89_HTML.gif. Then
      lim n z n ( t n ) = lim n x n ( t n ) x n C 1 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ16_HTML.gif
      (3.7)
      Owing to that the sequences { t n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq90_HTML.gif and { μ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq91_HTML.gif are uniformly bounded, there exist t 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq92_HTML.gif and μ 0 [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq93_HTML.gif such that, passing to subsequences if possible,
      lim n t n = t 0 , lim n μ n = μ 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ17_HTML.gif
      (3.8)
      Multiplying both sides of (3.4) by z n ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq94_HTML.gif and integrating from t n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq95_HTML.gif to t, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equs_HTML.gif
      Taking a superior limit as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq96_HTML.gif, by (3.3) and (3.6)-(3.8), we obtain
      [ z ( t 0 ) ] 2 [ z ( t ) ] 2 = μ 0 lim sup n 2 F ( t , x n ( t ) ) x n 2 ( t ) z 2 ( t ) + ( 1 μ 0 ) λ z 2 ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equt_HTML.gif
      By the assumption (ii) and the choice of λ, if z ( t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq97_HTML.gif, we have
      [ z ( t 0 ) ] 2 [ z ( t ) ] 2 p 2 z 2 ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equu_HTML.gif
      Similarly, we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equv_HTML.gif
      Note that z ( t ) C 1 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq98_HTML.gif, the above inequalities can be rewritten as the following equivalent forms:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ18_HTML.gif
      (3.9)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ19_HTML.gif
      (3.10)

      It is easy to see that z ( t 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq99_HTML.gif. In fact, if not, in view of (3.7)-(3.10), we get z ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq100_HTML.gif, z ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq101_HTML.gif, t [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq102_HTML.gif, which is contrary to z C 1 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq86_HTML.gif.

      We claim that z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq103_HTML.gif has only finite zero points on [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq104_HTML.gif. In fact, if not, we may assume that there are infinitely many zero points { ζ i } [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq105_HTML.gif of z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq103_HTML.gif. Without loss of generality, we assume that there exists ζ 0 [ 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq106_HTML.gif such that lim i ζ i = ζ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq107_HTML.gif. Letting t = ζ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq108_HTML.gif in (3.9)-(3.10) and taking i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq109_HTML.gif, we can obtain that z ( ζ 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq110_HTML.gif. Without loss of generality, we assume that z ( ζ 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq111_HTML.gif. Since z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq112_HTML.gif is continuous, there exist η , δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq113_HTML.gif such that z ( t ) η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq114_HTML.gif, t [ t 0 δ , t 0 + δ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq115_HTML.gif. Then there exists n 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq116_HTML.gif such that, if n > n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq117_HTML.gif, we have
      z n ( t ) η , t [ t 0 δ , t 0 + δ ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ20_HTML.gif
      (3.11)
      Clearly, z n ( t ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq118_HTML.gif, t [ t 0 δ , t 0 + δ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq115_HTML.gif. Take ζ , ζ [ t 0 δ , t 0 + δ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq119_HTML.gif with ζ < ζ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq120_HTML.gif such that z ( ζ ) = z ( ζ ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq121_HTML.gif. Integrating (3.4) from ζ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq122_HTML.gif to ζ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq123_HTML.gif,
      z n ( ζ ) z n ( ζ ) = μ n 1 x n C 1 ζ ζ f ( t , x n ( s ) ) d s + ( 1 μ n ) ζ ζ λ z n ( s ) d s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ21_HTML.gif
      (3.12)
      By (3.3), (3.11), we obtain
      x n ( t ) = z n ( t ) x n C 1 η x n C 1 + as  n + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equw_HTML.gif
      holds uniformly for t [ ζ , ζ ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq124_HTML.gif. Thus, using (1.4), we get
      f ( t , x n ( t ) ) x n ( t ) ρ , t [ ζ , ζ ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equx_HTML.gif
      which implies that
      f ( t , x n ( t ) ) x n C 1 = f ( t , x n ( t ) ) x n ( t ) z n ( t ) ρ η > 0 , t [ ζ , ζ ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equy_HTML.gif
      Then, together with (3.6), (3.8) and (3.12), we obtain
      0 μ 0 ρ η ( ζ ζ ) + ( 1 μ 0 ) λ η ( ζ ζ ) > 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equz_HTML.gif

      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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq125_HTML.gif. Without loss of generality, we assume t 0 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq126_HTML.gif. Firstly, we consider the case that z ¯ ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq127_HTML.gif. Assume that z 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq128_HTML.gif, z 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq129_HTML.gif satisfy the following equations respectively:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ22_HTML.gif
      (3.13)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ23_HTML.gif
      (3.14)
      with
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ24_HTML.gif
      (3.15)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ25_HTML.gif
      (3.16)
      Take t 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq130_HTML.gif as the first zero point of z ¯ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq125_HTML.gif on ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq131_HTML.gif. Then by (3.13)-(3.16) it follows that
      z 1 ( t ) z ¯ ( t ) z 2 ( t ) , t [ 0 , t 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ26_HTML.gif
      (3.17)
      In fact, by (3.15)-(3.16) and the fact that z ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq132_HTML.gif, z 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq128_HTML.gif, z 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq129_HTML.gif are continuous differential, it is easy to see that there exists sufficiently small ϵ ( 0 , t 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq133_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equaa_HTML.gif

      If there is t ¯ ( ϵ , t 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq134_HTML.gif such that z ¯ ( t ¯ ) = z 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq135_HTML.gif, then comparing (3.9) with (3.13), we can obtain that z ¯ ( t ¯ ) z 1 ( t ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq136_HTML.gif, which implies that if t > t ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq137_HTML.gif, we have z ¯ ( t ¯ ) z 1 ( t ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq138_HTML.gif. Then z ¯ ( t ) z 1 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq139_HTML.gif for t ( 0 , t 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq140_HTML.gif. Similarly, we have z ¯ ( t ) z 2 ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq141_HTML.gif, t [ 0 , t 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq142_HTML.gif. Hence, (3.17) holds.

      Similarly, if z 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq128_HTML.gif, z 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq129_HTML.gif satisfy
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ27_HTML.gif
      (3.18)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ28_HTML.gif
      (3.19)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equab_HTML.gif
      then we obtain
      z 1 ( t ) z ¯ ( t ) z 2 ( t ) , t [ t 1 , t 2 ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equac_HTML.gif

      where t 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq143_HTML.gif is the first zero point on ( t 1 , T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq144_HTML.gif.

      Since z ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq103_HTML.gif has finite zero points, (3.13), (3.14), (3.18), (3.19) can be transformed into the following equations respectively:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ29_HTML.gif
      (3.20)
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ30_HTML.gif
      (3.21)
      Then there exist A , B , C , D > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq145_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equad_HTML.gif
      It is easy to get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equae_HTML.gif
      Since z ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq132_HTML.gif is anti-periodic and z ¯ ( 0 ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq127_HTML.gif, there exists m Z + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq42_HTML.gif such that
      ( m + 1 ) π q 2 + m π p 2 t m = T 2 ( m + 1 ) π q 1 + m π p 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equaf_HTML.gif
      which implies that there exists a real number pair ( p , q ) [ p 1 , p 2 ] × [ q 1 , q 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq146_HTML.gif such that
      ( m + 1 ) π q + m π p = T 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ31_HTML.gif
      (3.22)
      On the other hand, in view of the assumption (ii), by the definition of Σ and ( p , q ) [ p 1 , p 2 ] × [ q 1 , q 2 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq146_HTML.gif, it follows that
      ( m + 1 ) π p + m π q T 2 , m Z + , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equag_HTML.gif

      which is contrary to (3.22).

      If z ¯ ( 0 ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq147_HTML.gif, then by the assumption (ii), we can obtain a contradiction using similar arguments.

      In a word, we can see that there exists C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq148_HTML.gif independent of μ such that
      x C 1 C . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equ32_HTML.gif
      (3.23)
      Set
      Ω = { x C T 2 1 : x C 1 < C + 1 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equah_HTML.gif
      Clearly, Ω is a bounded open set in C T 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq149_HTML.gif. Note that, for x C T 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq150_HTML.gif, using the assumption on f, we obtain
      http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equai_HTML.gif

      which implies that φ C T 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq151_HTML.gif.

      Define G μ : Ω ¯ C T 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq152_HTML.gif by
      G μ ( x ( t ) ) = J 2 φ ( μ , t , x ( t ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equaj_HTML.gif
      Clearly, G μ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq153_HTML.gif is completely continuous, and by (2.1) and (3.1) it follows that the fixed point of G 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq154_HTML.gif in Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq155_HTML.gif is the anti-periodic solution of problem (1.1). Define the homotopy H : Ω ¯ × [ 0 , 1 ] C T 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq156_HTML.gif as follows:
      H ( x , μ ) = x G μ ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equak_HTML.gif
      In view of (3.23), it follows that
      H ( x , μ ) 0 , ( x , μ ) Ω × [ 0 , 1 ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equal_HTML.gif
      Hence,
      deg ( I G 1 , Ω , 0 ) = deg ( I G 0 , Ω , 0 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equam_HTML.gif
      Note that the operator G 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq157_HTML.gif is odd. By Lemma 2.2 it follows that deg ( I G 0 , Ω , 0 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq158_HTML.gif. Thus,
      deg ( I G 1 , Ω , 0 ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_Equan_HTML.gif

      Now, using Lemma 2.1, we can see that (1.2) has a solution and hence (1.1) has a T 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-149/MediaObjects/13661_2012_Article_249_IEq16_HTML.gif-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

      (1)
      College of Information Technology, Jilin Agricultural University
      (2)
      College of Mathematics, Jilin University

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