Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

Shuang Wang; Dingbian Qian

doi:10.1515/ans-2021-2134

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Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

Shuang Wang und Dingbian Qian

Veröffentlicht/Copyright: 17. Juli 2021

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Aus der Zeitschrift Advanced Nonlinear Studies Band 21 Heft 3

Abstract

We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J⁢z′=∇⁡H⁢(t,z) from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂⁡H∂⁡x⁢(t,x,y), uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

Keywords: Subharmonic Solution; Indefinite Hamiltonian System; Spiral Function; Rotation Number; Poincaré–Birkhoff Theorem

MSC 2010: 34C25; 34B15; 34C15; 34D15

1 Introduction

We are interested in the multiplicity of subharmonic solutions for planar Hamiltonian systems

(1.1) J ⁢ z ′ = ∇ ⁡ H ⁢ ( t , z ) , z = ( x , y ) ∈ ℝ 2 ,

from a rotation number viewpoint, with a sign-changing term ∂⁡H∂⁡x⁢(t,x,y) (named “indefinite case”). We assume that H:ℝ×ℝ2→ℝ is C1 in the second variable, with ∇⁡H⁢(t,z) continuous and T-periodic in the first variable, and J denotes the standard symplectic matrix (0-110).

The results corresponding to the existence of periodic solutions for (1.1), based on the positively homogeneous Hamiltonian function or admissible spiral curves, are given in the papers [9, 10, 11, 12, 13]. Furthermore, the Poincaré–Birkhoff theorem is also an effective tool for studying the existence and multiplicity of periodic solutions for (1.1) (see [1, 4, 3]). It is well known that the global continuability of all solutions is generally a basic assumption for the applications of the Poincaré–Birkhoff theorem, even for the second-order scalar equation [2, 7, 6, 22, 24, 23].

For the second-order scalar equation

(1.2) x ′′ + f ⁢ ( t , x ) = 0 ,

influenced by works in the papers [14, 21, 8] and two classical papers [18, 19], Qian, Wang and Torres [25] recently developed a new approach to deal with the problems without both global continuability of the associated Cauchy problems and the sign assumption

sgn ⁡ ( x ) ⁢ f ⁢ ( t , x ) > 0 for any x≠0.

Precisely, they [25] used a rotation number approach and the phase-plane analysis of spiral properties to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems, at zero and infinity. We also refer [26, 27, 28] for the nice applications of the dynamical approach to rotation numbers. Particularly, [25] obtained multiplicity results with a piecewise linear setting, i.e.:

There exist functions a±∈L1⁢([0,T]) such that

a ± ⁢ ( t ) ≤ lim inf x → ± ∞ ⁡ f ⁢ ( t , x ) x uniformly a.e. in ⁢ t ∈ [ 0 , T ] .

There exist functions b±∈L1⁢([0,T]) such that

b ± ⁢ ( t ) ≥ lim sup x → 0 ± ⁡ f ⁢ ( t , x ) x uniformly a.e. in ⁢ t ∈ [ 0 , T ] .

The existence and multiplicity of the periodic solutions of planar systems (1.1) is relatively more complex. Recently, by using the Poincaré–Birkhoff theorem, Boscaggin and Garrione [4] investigated the problems with an asymptotically linear property at zero and infinity. Precisely, they [4] used the following assumptions:

∇ ⁡ H ⁢ ( t , 0 ) ≡ 0 and there exist V0∈𝒫,k0∈ℕ0 and γ0∈L1⁢(0,T), with

T τ V 0 < k 0 , 1 T ⁢ ∫ 0 T γ 0 ⁢ ( t ) ⁢ 𝑑 t = 1 ,

such that

lim sup | z | → 0 ⁡ 〈 ∇ ⁡ H ⁢ ( t , z ) , z 〉 2 ⁢ V 0 ⁢ ( z ) ≤ γ 0 ⁢ ( t ) uniformly for a.e. ⁢ t ∈ [ 0 , T ] .

There exist V∞∈𝒫,k∞∈ℕ0 and γ∞∈L1⁢(0,T), with

T τ V ∞ > k ∞ , 1 T ⁢ ∫ 0 T γ ∞ ⁢ ( t ) ⁢ 𝑑 t = 1 ,

such that

lim inf | z | → ∞ ⁡ 〈 ∇ ⁡ H ⁢ ( t , z ) , z 〉 2 ⁢ V ∞ ⁢ ( z ) ≥ γ ∞ ⁢ ( t ) uniformly for a.e. ⁢ t ∈ [ 0 , T ] ,

where 〈⋅,⋅〉 is the Euclidean scalar product in ℝ2, and 𝒫 is the class of the C2-functions V⁢(z):ℝ2→ℝ, which are positively homogeneous of degree 2 and strictly positive, i.e.,

0 < V ⁢ ( λ ⁢ z ) = λ 2 ⁢ V ⁢ ( z ) for all ⁢ λ > 0 , z ≠ 0 .

Notice that the mean value conditions

1 T ⁢ ∫ 0 T γ 0 ⁢ ( t ) ⁢ 𝑑 t = 1 and 1 T ⁢ ∫ 0 T γ ∞ ⁢ ( t ) ⁢ 𝑑 t = 1

give a partial answer to the problem of removing the sign assumption on ∇⁡H⁢(t,z). Indeed, by comparing with the linear systems in a positively homogeneous sense, the angular function of the solution satisfies θ′≤γ0⁢(t)⁢ϕ⁢(θ) for this case. This provides a way to estimate the increment of the angular function in time [0,T]. But this approach cannot deal with the case of ∫0Tγ0⁢(t)⁢𝑑t≤0, for instance, γ0⁢(t)=sin⁡t and T=2⁢π.

Besides assumptions (H${}_{0}$) and (H${}_{\infty}$), the uniqueness and global continuability of the solutions for the associated Cauchy problem are assumed in [4]. In addition, in [25], the uniqueness of solutions for the Cauchy problems associated with the equivalent system

(1.3) x ′ = - y , y ′ = f ⁢ ( t , x ) ,

of (1.2) is also required. Particularly, when y′=f⁢(t,x) is an indefinite potential at y=0, in order to describe the spiral property of the solution with a given initial value at y=0, the authors of [25] used a series of the solutions of analytic systems to approximate the unique solution of the original systems (1.3) with given initial value (see [25, Lemma 3.1]).

In this paper we provide an extension of the results presented in [25] for the second-order scalar equations, and of some results given in [4] to the indefinite planar Hamiltonian systems, without assuming uniqueness of solutions and other more restrictive assumptions.

In particular, we establish a unified approach to deal with the case of (1.1) in absence of the uniqueness, the global existence for the solutions of the associated Cauchy problems and the sign assumption on ∂⁡H∂⁡x⁢(t,x,y). To overcome the lack of uniqueness, we use a recent version of the Poincaré–Birkhoff theorem for Hamiltonian systems by Fonda and Ureña [15, 14]. We need some different method to improve the phase-plane analysis of the spiral properties when y′=f⁢(t,x) is an indefinite potential at y=0 in [25].

We will consider the “asymptotically semilinear” in the sense of rotation numbers, which is more general than that in [4] and [25] (see the discussion in Section 5 for details). That is, we introduce the following assumptions:

There exist positively homogeneous functions L0⁢(t,z)∈𝒬 and V0⁢(z)∈𝒫 such that, for each δ>0, there is rδ>0, with

〈 ∇ ⁡ H ⁢ ( t , z ) , z 〉 ≤ 〈 ∇ ⁡ L 0 ⁢ ( t , z ) , z 〉 + δ ⁢ V 0 ⁢ ( z )

for each 0<|z⁢(t)|<rδ and a.e. t∈[0,T].

There exist positively homogeneous functions L∞⁢(t,z)∈𝒬 and V∞⁢(z)∈𝒫 such that, for each δ>0, there is l⁢(t)∈L1⁢([0,T]), with

(1.4) 〈 ∇ ⁡ H ⁢ ( t , z ) , z 〉 ≥ 〈 ∇ ⁡ L ∞ ⁢ ( t , z ) , z 〉 - δ ⁢ V ∞ ⁢ ( z ) - l ⁢ ( t )

for all z∈ℝ2 and a.e. t∈[0,T].

ρ ⁢ ( L ∞ ) > ρ ⁢ ( L 0 ) .

In the above assumptions 𝒬 is the class of the functions L⁢(t,z):ℝ×ℝ2→ℝ, which are T-periodic in the first variable, differentiable and positively homogeneous of degree 2 in the second variable, i.e.,

L ⁢ ( t , λ ⁢ z ) = λ 2 ⁢ L ⁢ ( t , z ) for every λ > 0 and z ∈ ℝ 2 ;

ρ ⁢ ( L ∞ ) and ρ⁢(L0) are the rotation numbers of equations J⁢z′=∇⁡L∞⁢(t,z) and J⁢z′=∇⁡L0⁢(t,z), respectively, which are defined in Section 2.

With a sign-changing term ∂⁡H∂⁡x⁢(t,x,y), it is not easy to describe the rotation of the solution in the phase plane. So we introduce a definition of the spiral function of system (1.1). Using the spiral function, we can obtain two spiral curves, which make the solutions of the system spiral in the phase plane as they increase in norm.

Definition 1.1.

Suppose for sufficiently large r*, there are two strictly monotonically increasing functions ξ±⁢(r):[r*,+∞)→ℝ such that

ξ ± ⁢ ( r ) → + ∞ ⇔ r → + ∞ .

Moreover, for every solution z⁢(t) of system (1.1) satisfying r0=|z⁢(t0)|≥r*, either

ξ - ⁢ ( r ⁢ ( t 0 ) ) ≤ r ⁢ ( t ) ≤ ξ + ⁢ ( r ⁢ ( t 0 ) ) for ⁢ t ∈ [ t 0 , t 0 + T ] ,

or there exists t^∈(t0,t0+T) such that

θ ⁢ ( t ^ ) - θ ⁢ ( t 0 ) = - 2 ⁢ π

and

ξ - ⁢ ( r ⁢ ( t 0 ) ) ≤ r ⁢ ( t ) ≤ ξ + ⁢ ( r ⁢ ( t 0 ) ) for ⁢ t ∈ [ t 0 , t ^ ] .

Then we call ξ±⁢(⋅) the upper and lower spiral function of (1.1), respectively.

The following is the main result of this paper.

Theorem 1.1.

Suppose (1.1) satisfies (h${}_{0}$), (h${}_{\infty}$) and (h${}^{0}_{\infty}$). In addition, assume:

lim sup | z | → 0 ⁡ | ∇ ⁡ H ⁢ ( t , z ) | | z | < + ∞ .
There exist upper and lower spiral functions ξ ± ⁢ ( ⋅ ) of ( 1.1 ) for sufficiently large r * .
sgn ⁡ ( y ) ⁢ ∂ ⁡ H ∂ ⁡ y ⁢ ( t , 0 , y ) > 0 for y ≠ 0 and t ∈ [ 0 , T ] .

Then for any rational number jm∈(ρ⁢(L0),ρ⁢(L∞)), system (1.1) has at least two mT-periodic solutions. Furthermore, such mT-periodic solutions have exactly 2⁢j zeros in [0,m⁢T].

Remark 1.1.

Assumption (h1) implies that the solution with sufficiently small initial values always exist globally. This is the basic assumption of applying the Poincaré–Birkhoff theorem, see [18, 19] and [25]. Assumption (h3) is used to imply that the nonzero solutions of (1.1) make at most half counter-clockwise revolutions around the origin. The typical second-order equation x′=y,y′=-f⁢(t,x) always satisfies (h3). Assumption (h2) implies that, using the spiral function, we can obtain two spiral curves, which make the solutions of the system spiral in the phase plane as they increase in norm. We note that if all the solutions z⁢(t) of (1.1) are globally defined, we can obtain so called “elastic” property of the solutions, that is, for any fixed K>0 and any |z⁢(t0)|=r0 being sufficiently large, we have r±⁢(r0) such that

r - ⁢ ( r 0 ) ≤ | z ⁢ ( t ) | ≤ r + ⁢ ( r 0 ) for ⁢ t ∈ [ t 0 , t 0 + K ] ,

moreover,

r ± ⁢ ( r 0 ) → + ∞ ⇔ r 0 → + ∞ .

It is easy to see that we can chose two strictly monotonically increasing functions ξ±⁢(r):[r*,+∞)→ℝ as showed in Definition 1.1 such that

ξ - ⁢ ( r ) ≤ r - ⁢ ( r ) ≤ r + ⁢ ( r ) ≤ ξ + ⁢ ( r ) .

Hence, if all the solutions z⁢(t) of (1.1) are globally defined, then (h2) holds. Also, (H${}^{l}_{\infty}$)’ implies (h2) holds, see [25, Lemma 3.1].

The following example gives a specific application of Theorem 1.1, its proof will be given in Section 4.

Example 1.1.

Consider the asymmetric indefinite planar system

(1.5) { x ′ = y - x , y ′ = y - f ⁢ ( t , x ) ,

where f(t,x)=(α2+1)⁢(x+)31+(x+)2-|sint|⋅(x-)3, α>1, x+=max⁡{x,0} and x-=max⁡{-x,0}. For any rational number jm∈(0,2⁢α), system (1.5) has at least two m⁢π-periodic solutions. Furthermore, such m⁢π-periodic solutions have exactly 2⁢j zeros in [0,m⁢π].

The remaining sections of the paper are organized as follows. In Section 2, we introduce the definition and properties of rotation numbers for a planar positively homogeneous system. Section 3 gives the proof of Theorem 1.1. The spiral properties of large solutions are discussed in Section 4. In Section 5, we discuss the applications of Theorem 1.1 in the models studied in [4] and [25], then give the proof of Example 1.1. Finally, we give the proof of a technical lemma (Lemma 2.3) in Appendix A.

2 Definition and Properties of the Rotation Numbers

If z⁢(s)≠0 for every s in interval [0,t], we can define the t-rotation number of z⁢(t) as

Rot ⁢ ( z ⁢ ( t ) ; [ 0 , t ] ) = θ 0 - θ ⁢ ( t ) 2 ⁢ π for all ⁢ t ∈ [ 0 , T ] ,

where θ⁢(t) is the argument function of z⁢(t) associated with the polar coordinate

x ⁢ ( t ) = r ⁢ ( t ) ⁢ cos ⁡ θ ⁢ ( t ) , y ⁢ ( t ) = r ⁢ ( t ) ⁢ sin ⁡ θ ⁢ ( t ) ,

and (r0,θ0)=(r⁢(0),θ⁢(0)). Notice that

Rot ⁢ ( z ⁢ ( t ) ; [ 0 , t ] ) = 1 2 ⁢ π ⁢ ∫ 0 t y ⁢ ( t ) ⁢ x ′ ⁢ ( t ) - x ⁢ ( t ) ⁢ y ′ ⁢ ( t ) x 2 ⁢ ( t ) + y 2 ⁢ ( t ) ⁢ 𝑑 t = 1 2 ⁢ π ⁢ ∫ 0 t 〈 J ⁢ z ′ ⁢ ( t ) , z ⁢ ( t ) 〉 | z ⁢ ( t ) | 2 ⁢ 𝑑 t .

In the following, let z⁢(t)=(x⁢(t),y⁢(t)) be a solution of (1.1) with z⁢(0)=z0, and denote by

Rot H ⁢ ( t ; z ) := Rot ⁢ ( z ⁢ ( t ) ; [ 0 , t ] )

for short.

When (1.1) is a positively homogeneous Hamiltonian system

(2.1) J ⁢ z ′ = ∇ ⁡ L ⁢ ( t , z ) for ⁢ L ⁢ ( t , z ) ∈ 𝒬 ,

satisfied the uniqueness on the solution of the associated Cauchy problems, we have

(2.2) θ ′ = - 〈 J ⁢ z ′ , z 〉 | z | 2 = - 〈 ∇ ⁡ L ⁢ ( t , z ) , z 〉 | z | 2 = - 〈 ∇ L ( t , ( cos θ , sin θ ) ) , ( cos θ , sin θ ) 〉 = : Θ ( t , θ ; L ) .

Let θ⁢(t;0,θ0) be the unique solution of problem (2.2) satisfying the initial condition θ⁢(0;0,θ0)=θ0. As Θ⁢(t,θ;L) is 2⁢π-periodic in θ and T-periodic in t, we have

θ ⁢ ( t + m ⁢ T ; 0 , θ 0 ) = θ ⁢ ( t ; 0 , θ ⁢ ( m ⁢ T ; 0 , θ 0 ) )

and

(2.3) θ ⁢ ( t ; 0 , θ 0 + 2 ⁢ j ⁢ π ) = θ ⁢ ( t ; 0 , θ 0 ) + 2 ⁢ j ⁢ π

for all θ0∈ℝ and m,j∈ℤ. Thus, (2.2) is a differential equation on a torus. The rotation number of (2.2)

ρ ⁢ ( L ) := lim k → ∞ ⁡ θ 0 - θ ⁢ ( k ⁢ T ; 0 , θ 0 ) 2 ⁢ k ⁢ π

exists and is independent of θ0, see [17, Theorem 2.1, Chapter 2].

Since the solution of (2.1) is unique with respect to the initial value, we can prove that

z ⁢ ( t ; l ⁢ v ) = l ⁢ z ⁢ ( t ; v ) for ⁢ l ≠ 0 ,

where z⁢(t;l⁢v) and z⁢(t;v) are the solution of (2.1) such that z⁢(0;l⁢v)=l⁢v and z⁢(0;v)=v, respectively. Then the t-rotation number RotL⁢(t;z) of (2.1) is independent of l. For any t∈[0,T], we will write the t-rotation number RotL⁢(t;z) of (2.1) as RotL⁢(t;v), where v=z0|z0|.

Next, we will discuss the relations of ρ⁢(L) and RotL⁢(m⁢T;v) of system (2.1).

First, we recall the definition of the rotation number of an orientation-preserving homeomorphism. Let h:ℝ→ℝ be an orientation-preserving homeomorphism such that

(2.4) h ⁢ ( ϑ + 2 ⁢ j ⁢ π ) = h ⁢ ( ϑ ) + 2 ⁢ j ⁢ π for all ⁢ ϑ ∈ ℝ , j ∈ ℤ .

We can define the rotation number of h as

ρ ⁢ ( h ) = lim k → ∞ ⁡ ϑ - h k ⁢ ( ϑ ) 2 ⁢ k ⁢ π

(independent of the choice of ϑ), where hk is a map as the k times iterations of h:ℝ→ℝ.

The following lemma is analogous to [16, Propositions 2.1].

Lemma 2.1.

Let h be an orientation-preserving homeomorphism of R satisfying (2.4) and let jm be a rational number. Then:

ρ ⁢ ( h ) ≥ j m if and only if max ϑ ∈ ℝ ⁡ ( ϑ - h m ⁢ ( ϑ ) ) ≥ 2 ⁢ j ⁢ π ,
ρ ⁢ ( h ) ≤ j m if and only if min ϑ ∈ ℝ ⁡ ( ϑ - h m ⁢ ( ϑ ) ) ≤ 2 ⁢ j ⁢ π .

By means of Lemma 2.1, we can prove the following result.

Lemma 2.2.

Let jm be a rational number. Then:

ρ ⁢ ( L ) < j m if and only if Rot L ⁢ ( m ⁢ T ; v ) < j for all v ∈ S 1 ,
ρ ⁢ ( L ) > j m if and only if Rot L ⁢ ( m ⁢ T ; v ) > j for all v ∈ S 1 .

Proof.

Let h:ℝ→ℝ be the Poincaré map of (2.2), that is, h⁢(θ0)=θ⁢(T;0,θ0) for all θ0∈ℝ. Using the uniqueness of solutions of (2.2) with respect to the initial value, and since the equation is scalar, the solutions are ordered. It follows that h is an orientation-preserving homeomorphism. From (2.3), we know that

h ⁢ ( θ 0 + 2 ⁢ j ⁢ π ) = h ⁢ ( θ 0 ) + 2 ⁢ j ⁢ π for all ⁢ θ 0 ∈ ℝ , j ∈ ℤ .

Furthermore, by means of Lemma 2.1, we have

ρ ⁢ ( L ) < j m ⇔ max θ 0 ∈ ℝ ⁡ ( θ 0 - h m ⁢ ( θ 0 ) ) < 2 ⁢ j ⁢ π

⇔ max θ 0 ∈ ℝ ⁡ θ 0 - θ ⁢ ( m ⁢ T ; 0 , θ 0 ) 2 ⁢ π < j

⇔ Rot L ⁢ ( m ⁢ T ; v ) < j

for all v=(cos⁡θ0,sin⁡θ0). Thus we obtain (i). Result (ii) can be proved in a similar way. ∎

Furthermore, we give a comparison of the result (Lemma 2.3) associated with RotH⁢(t;z) of (1.1). The corresponding proof is inspired by the proof of [5, Lemma 3.4] and is postponed to the Appendix.

Lemma 2.3.

Let H⁢(t,z):R×R2→R be C1 in the second variable, with ∇⁡H⁢(t,z) continuous and T-periodic in the first variable. Then the following statements hold.

If H ⁢ ( t , z ) satisfies (h${}_{\infty}$) . Then, for each ε > 0 , there is R ε > 0 such that for each solution z ⁢ ( t ) of ( 1.1 ) with | z ⁢ ( t ) | ≥ R ε for all t ∈ [ 0 , T ] , it follows that

(2.5) Rot H ⁢ ( t ; z ) ≥ Rot L ⁢ ( t ; v ) - ε for all ⁢ t ∈ [ 0 , T ] , and ⁢ v = z 0 | z 0 | .
If H ⁢ ( t , z ) satisfies (h${}_{0}$) . Then, for each ε > 0 , there is r ε > 0 such that for each solution z ⁢ ( t ) of ( 1.1 ) with 0 < | z ⁢ ( t ) | ≤ r ε for all t ∈ [ 0 , T ] , it follows that

Rot H ⁢ ( t ; z ) ≤ Rot L ⁢ ( t ; v ) + ε for all ⁢ t ∈ [ 0 , T ] , and ⁢ v = z 0 | z 0 | .

Remark 2.1.

Similar to [5], Lemma 2.3 does not require the global continuablity of the solutions on [0,T]. The claims of this lemma have to be considered only in regard to those solutions z⁢(t) of (1.1) defined on [0,T] such that z⁢(t)≠0 for all t∈[0,T].

With these basic tools, the relations of RotH⁢(m⁢T;z) and the rotation number ρ⁢(L∞) (or ρ⁢(L0)) can be obtained.

Lemma 2.4.

Let H⁢(t,z):R×R2→R be C1 in the second variable, with ∇⁡H⁢(t,z) continuous and T-periodic in the first variable.

Assume that H ⁢ ( t , z ) satisfies (h${}_{\infty}$) and j m is a rational number. If ρ ⁢ ( L ∞ ) > j m , then there exists R ε > 0 such that every solution z ⁢ ( t ) of ( 1.1 ) with | z ⁢ ( t ) | ≥ R ε for all t ∈ [ 0 , m ⁢ T ] , satisfies Rot H ⁢ ( m ⁢ T ; z ) > j .
Assume that H ⁢ ( t , z ) satisfies (h${}_{0}$) and j m is a rational number. If ρ ⁢ ( L 0 ) < j m , then there exists r ε > 0 such that every solution z ⁢ ( t ) of ( 1.1 ) with 0 < | z ⁢ ( t ) | ≤ r ε for all t ∈ [ 0 , m ⁢ T ] , satisfies Rot H ⁢ ( m ⁢ T ; z ) < j .

Proof.

Since the proofs of the statements are identical, we will only prove (i). By Lemma 2.2, we have

Rot L ∞ ⁢ ( m ⁢ T ; v ) > j for ⁢ v = z 0 | z 0 | .

Hence, it is possible to choose a constant ε>0 such that

Rot L ∞ ⁢ ( m ⁢ T ; v ) - ε > j .

According to Lemma 2.3 and (h${}_{\infty}$), for such ε, there exists Rε>0 such that if the solution z⁢(t) of (1.1) satisfies |z⁢(t)|≥Rε for all t∈[0,T], then RotH⁢(m⁢T;z)≥RotL∞⁢(m⁢T;v)-ε>j. ∎

3 Modified Hamiltonian Systems and the Existence of Periodic Solutions

Notice that for (1.1), the global existence of the solutions of the associated Cauchy problem is not assumed. Thus the Poincaré map may not be well-defined. To overcome the lack of the global existence, we will consider a modified Hamiltonian system of (1.1).

From (h3) we can choose a C1 function H0⁢(x,y) such that

(3.1) H 0 ⁢ ( x , y ) ≥ max 0 ≤ t ≤ T ⁡ H ⁢ ( t , 0 , y ) + x 2 ,

(3.2) y ⁢ ∂ ⁡ H 0 ∂ ⁡ y ⁢ ( 0 , y ) > 0 for ⁢ y ≠ 0 ,

(3.3) H 0 ⁢ ( x , y ) → + ∞ ⇔ | x | + | y | → + ∞ .

Let

H λ ⁢ ( t , x , y ) = λ ⁢ ( x 2 ) ⁢ ( λ ⁢ ( y 2 ) ⁢ H ⁢ ( t , x , y ) + ( 1 - λ ⁢ ( y 2 ) ) ⁢ H 0 ⁢ ( x , y ) ) + ( 1 - λ ⁢ ( x 2 ) ) ⁢ H 0 ⁢ ( x , y ) ,

where λ⁢(s)∈C∞⁢(ℝ,ℝ) is a truncating function satisfying

λ ⁢ ( s ) = { 1 , | s | ≤ R * , smooth connection , R * < | s | < R * * , 0 , | s | ≥ R * * ,

where λ′⁢(s)≤0 and R*,R** are positive parameters, whose specific value will be given in the proof of Theorem 1.1.

Consider the Hamiltonian system

(3.4) J ⁢ z ′ = ∇ ⁡ H λ ⁢ ( t , z ) , z = ( x , y ) ∈ ℝ 2 .

The solution of (3.4) will also be simply denoted by z⁢(t)=(x⁢(t),y⁢(t)) with z⁢(0)=z0. Its polar coordinate will be denoted by (r⁢(t),θ⁢(t)) such that (r⁢(0),θ⁢(0))=(r0,θ0) for short.

We can obtain the following results.

Lemma 3.1.

Assume (h1). Then every solution z⁢(t) of (3.4) exists globally for t∈R. If a solution z⁢(t) of (3.4) satisfies z0≠(0,0), then z⁢(t)≠(0,0) for t∈R.

Proof.

Suppose that there exists a solution z*⁢(t) such that Iz*≠ℝ, where Iz* is the maximum existence interval of z*⁢(t). From the continuation theorem, there are tk∈Iz*, k=1,…, such that |tk|≤Mz*<+∞ and |z*⁢(tk)|→+∞ as k→∞. On the other hand, note that when |z|2≥2⁢R**, system (3.4) is equivalent to the autonomous Hamiltonian system

J ⁢ z ′ = ∇ ⁡ H 0 ⁢ ( z ) , z = ( x , y ) ∈ ℝ 2 .

Every solution of the above system lies on an integral curve, which implies that there exists c>0 such that H0⁢(z*⁢(tk))=c for sufficiently large k. Using (3.3), we have a constant b⁢(c) such that |z*⁢(tk)|≤b⁢(c). This leads to a contradiction.

When |z|2≤R*, system (3.4) is equivalent to the original system (1.1). By (h1), there exist two positive numbers M and rM such that

| ∇ ⁡ H ⁢ ( t , z ) | ≤ M ⁢ | z | for ⁢ 0 < | z | < r M .

It follows from (1.1) that

| r ′ ⁢ ( t ) | = | x ′ ⁢ ( t ) ⁢ x ⁢ ( t ) + y ′ ⁢ ( t ) ⁢ y ⁢ ( t ) | r ⁢ ( t )

≤ x 2 ⁢ ( t ) + y 2 ⁢ ( t ) + | ∂ ⁡ H ∂ ⁡ y ⁢ ( t , x , y ) | 2 + | ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) | 2 2 ⁢ r ⁢ ( t )

≤ 1 2 ⁢ ( 1 + M 2 ) ⁢ r ⁢ ( t ) .

For any given T~>0, we have

| z 0 | ⁢ e - ( 1 + M 2 ) ⁢ T ~ ≤ | z ⁢ ( t ) | ≤ | z 0 | ⁢ e ( 1 + M 2 ) ⁢ T ~ for ⁢ t ∈ [ - T ~ , T ~ ] ,

which implies that z⁢(t)≠(0,0) for t∈[-T~,T~] if z0≠(0,0). From the arbitrary choice of T~, we have proved the lemma. ∎

Lemma 3.2.

For any t2>t1, the nonzero solutions of (3.4) satisfy θ⁢(t2)-θ⁢(t1)<π.

Proof.

When x=0,

∂ ⁡ H λ ∂ ⁡ y ⁢ ( t , x , y ) = 2 ⁢ y ⁢ λ ′ ⁢ ( y 2 ) ⁢ ( H ⁢ ( t , 0 , y ) - H 0 ⁢ ( 0 , y ) ) + λ ⁢ ( y 2 ) ⁢ ∂ ⁡ H ∂ ⁡ y ⁢ ( t , 0 , y ) + ( 1 - λ ⁢ ( y 2 ) ) ⁢ ∂ ⁡ H 0 ∂ ⁡ y ⁢ ( 0 , y ) .

Then, by using (h3), (3.1), (3.2) and λ′⁢(y2)≤0, we have

x ′ ⁢ y = y ⁢ ∂ ⁡ H λ ∂ ⁡ y ⁢ ( t , x , y ) > 0 for ⁢ x = 0 , y ≠ 0 ⁢ and ⁢ t ∈ [ 0 , T ] .

Hence, a nonzero solution of (3.4) performs the clockwise rotations at y-axis, which implies that the increment of θ⁢(t) will be less than π. Thus, for any t2>t1, we have θ⁢(t2)-θ⁢(t1)<π. ∎

Moreover, we can use (h2) repeatedly to obtain the following spiral properties.

Lemma 3.3.

Assume (h2). Then for any fixed m,N0∈N and sufficiently large r*, there are two strictly monotonically increasing functions ξN0±⁢(r):[r*,+∞)→R such that

ξ N 0 ± ⁢ ( r ) → + ∞ ⇔ r → + ∞ .

If a solution z⁢(t) of system (1.1) satisfies r0≥r*, then either

ξ N 0 - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ N 0 + ⁢ ( r 0 ) for ⁢ t ∈ [ 0 , m ⁢ T ] ,

or there exists tN0^∈(0,m⁢T) such that

θ ⁢ ( t N 0 ^ ) - θ 0 = - 2 ⁢ N 0 ⁢ π

and

ξ N 0 - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ N 0 + ⁢ ( r 0 ) for ⁢ t ∈ [ 0 , t N 0 ^ ] .

To complete the proof of Theorem 1.1, we apply a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems (see [15, 14]). Precisely, let 0<R1<R2 and consider the annulus in ℝ2 defined as Ω=B¯R2\BR1, where BRi is an open ball of radius Ri, i=1,2. We write RotHλ⁢(m⁢T;z):=Rot⁢(z⁢(t);[0,m⁢T]), where z⁢(t) is a solution of system (3.4). As a consequence of [14, Theorem 5], we have:

Theorem 3.1.

Assume that every solution z⁢(t) of (3.4), departing from z0∈∂⁡Ω, is defined on [0,m⁢T] and satisfies

z ⁢ ( t ) ≠ ( 0 , 0 ) for ⁢ t ∈ [ 0 , m ⁢ T ] .

Assume moreover that there is a positive integer j such that

Rot H λ ⁢ ( m ⁢ T ; z ) < j if ⁢ | z 0 | = R 1 ,

Rot H λ ⁢ ( m ⁢ T ; z ) > j if ⁢ | z 0 | = R 2 .

Then the Hamiltonian system has at least two distinct mT-periodic solutions zi⁢(t) with zi⁢(0)∈Ω such that RotHλ⁢(m⁢T;zi⁢(t))=j, i=1,2.

Proof of Theorem 1.1.

We will divide the proof into the following steps.

Step 1. By ρ⁢(L0)<jm, (h${}_{0}$) and Lemma 2.4, there exists rε>0 such that every solution z⁢(t) of (1.1) with 0<|z⁢(t)|≤rε for all t∈[0,m⁢T] satisfies

(3.5) Rot H ⁢ ( m ⁢ T ; z ) < j .

Similar to the proof of Lemma 3.1, we have

| z 0 | ⁢ e - ( 1 + M 2 ) ⁢ m ⁢ T ≤ | z ⁢ ( t ) | ≤ | z 0 | ⁢ e ( 1 + M 2 ) ⁢ m ⁢ T for ⁢ t ∈ [ 0 , m ⁢ T ] .

We can find R1>0 is sufficiently small such that if |z0|=R1, then

0 < | z ⁢ ( t ) | ≤ r ε for all ⁢ t ∈ [ 0 , m ⁢ T ] .

Then, from (3.5), we have

(3.6) Rot H ⁢ ( m ⁢ T ; z ) < j if ⁢ | z 0 | = R 1 .

Let R*>rε. Then system (3.4) is equivalent to the original system (1.1). It follows from (3.6) that

(3.7) Rot H λ ⁢ ( m ⁢ T ; z ) < j if ⁢ | z 0 | = R 1 .

Step 2. By (h${}_{\infty}$), ρ⁢(L∞)>jm and Lemma 2.4, there exists Rε>rε such that, for each solution z⁢(t) of system (1.1) with |z⁢(t)|≥Rε for all t∈[0,m⁢T], it follows that

(3.8) Rot H ⁢ ( m ⁢ T ; z ) > j .

Choose

R 2 = ( ξ j + 1 - ) - 1 ⁢ ( R ε ) and R * ≥ R 2 ′ = ξ j + 1 + ⁢ ( R 2 ) ,

where ξj+1±⁢(⋅) are defined in Lemma 3.3. Thus, system (3.4) is equivalent to the original system (1.1) for |z⁢(t)|≤R2′. Now consider a solution of (3.4) with |z0|=R2. If Rε≤|z⁢(t)|≤R2′ for all t∈[0,m⁢T], it follows from (3.8) that

(3.9) Rot H λ ⁢ ( m ⁢ T ; z ) > j if ⁢ | z 0 | = R 2 .

If there exists t1∈(0,m⁢T) such that r⁢(t1)>R2′, then there exists t1′∈(0,t1) such that |z⁢(t1′)|=R2′ and |z⁢(t)|≤R2′ for t∈[0,t1′]. By Lemma 3.3, we know that

θ ⁢ ( t 1 ′ ) - θ 0 = - 2 ⁢ ( j + 1 ) ⁢ π .

Moreover, by using Lemma 3.2, we have

θ ⁢ ( m ⁢ T ) - θ 0 = ( θ ⁢ ( m ⁢ T ) - θ ⁢ ( t 1 ′ ) ) + ( θ ⁢ ( t 1 ′ ) - θ 0 ) < π - 2 ⁢ ( j + 1 ) ⁢ π < - 2 ⁢ j ⁢ π ,

i.e., (3.9) holds. Finally, if there exists t2∈(0,m⁢T) such that |z⁢(t2)|<Rε, the validity (3.9) is proven by the similar arguments as given above.

Step 3. Consider the annular Ω=B¯R2\BR1. By (3.7), (3.9) and Theorem 3.1, we can find at least two distinct mT-periodic solutions zm1⁢(t) and zm2⁢(t) such that zmi⁢(0)∈Ω and

(3.10) Rot H λ ⁢ ( m ⁢ T ; z m i ⁢ ( t ) ) = j for ⁢ i = 1 , 2 .

Step 4. We will show that zmi⁢(t) are in fact in r≤R*, i=1,2. Then zmi⁢(t) are two distinct mT-periodic solutions of (1.1), i=1,2. Note that zmi⁢(0)∈Ω. If there is t2∈(0,m⁢T) such that |zm1⁢(t2)|>R*, then we can find t2′∈(0,t2) satisfying |zm1⁢(t2′)|=R2′ and |zm1⁢(t)|≤R2′ for t∈[0,t2′]. We write the argument function of zm1⁢(t) as θm1⁢(t) for clarity. By Lemma 3.3, we have

θ m 1 ⁢ ( t 2 ′ ) - θ m 1 ⁢ ( 0 ) = - 2 ⁢ ( j + 1 ) ⁢ π .

Thus, using Lemma 3.2, we have

θ m 1 ⁢ ( m ⁢ T ) - θ m 1 ⁢ ( 0 ) = θ m 1 ⁢ ( m ⁢ T ) - θ m 1 ⁢ ( t 2 ′ ) + θ m 1 ⁢ ( t 2 ′ ) - θ m 1 ⁢ ( 0 ) < - 2 ⁢ j ⁢ π .

Then RotHλ⁢(m⁢T;zm1⁢(t))>j, which contradicts (3.10). Therefore, |zm1⁢(t)|≤R* for all t∈[0,m⁢T]. Namely, zm1⁢(t) is an mT-periodic solution of (1.1). Similarly, zm2⁢(t) is also an mT-periodic solution of (1.1). The proof is completed. ∎

4 The Spiral Properties of Large Solutions

In Remark 1.1 we show that the global existence of the solutions infers the spiral hypothesis (h2). In this section we discuss the sufficient conditions of hypothesis (h2) when (1.1) has no global existence of the solutions.

Inspired by [14, 25], we introduce the following conditions:

Let φ∈C1⁢(ℝ,ℝ) be a monotone increasing function such that φ⁢(0)=0, and let 𝒟i, i=1,2,3,4, be regions in ℝ2 as

𝒟 1 = { ( x , y ) : x ≥ 0 , y > φ ⁢ ( x ) } , 𝒟 2 = { ( x , y ) : x > 0 , y ≤ φ ⁢ ( x ) } ,

𝒟 3 = { ( x , y ) : x ≤ 0 , y < φ ⁢ ( x ) } , 𝒟 4 = { ( x , y ) : x < 0 , y ≥ φ ⁢ ( x ) } .

Let V⁢(x,y):ℝ×ℝ→ℝ+ be a piecewise continuously differentiable function such that V⁢(x,y)=Vi⁢(x,y) for (x,y)∈𝒟i,

V i ⁢ ( x , y ) → + ∞ ⇔ x 2 + y 2 → + ∞ ,

and Vi⁢(x,φ⁢(x)) are monotone functions with respect to x, i=1,2,3,4. For each solution (x⁢(t),y⁢(t)) of (1.1) with (x⁢(t),y⁢(t))∈𝒟i for t∈I, there exists ci⁢(t)∈L1⁢(ℝ/T⁢ℤ,ℝ+) such that

(4.1) d ⁢ V i ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) d ⁢ t ≤ c i ⁢ ( t ) ⁢ V i ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) for a.e. ⁢ t ∈ I .

Moreover, let U⁢(x,y)=Ui⁢(x,y) for (x,y)∈𝒟i have the same properties as Vi⁢(x,y) except inequality (4.1) is replaced by the following inequality:

(4.2) d ⁢ U i ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) d ⁢ t ≥ - b i ⁢ ( t ) ⁢ U i ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) for a.e. ⁢ t ∈ I , i = 1 , 2 , 3 , 4 .
∂ ⁡ H ∂ ⁡ y ⁢ ( t , x , y ) = 0 for y=φ⁢(x) and t∈[0,T], sgn⁡(y-φ⁢(x))⁢∂⁡H∂⁡y⁢(t,x,y)>0 for y≠φ⁢(x) and t∈[0,T].

Under (h4) and (h5), we will prove that (h2) holds, that is, the large solutions of system (1.1) have spiral properties. Notice that now for (1.1), the uniqueness of the Cauchy problem is not assumed. So, in the part of dealing with the spiral property of the solution from y=0, we can not follow the approximation method used in [25], which is based on the uniqueness of the solution with given initial point. We will make some improvements of phase-plane analysis of the spiral properties.

In particular, we will replace the analysis of the solution starting from a small neighborhood of y=φ⁢(x) instead of the analysis of the solution starting from y=0. Precisely, we will find a priori estimates of a compact set Eδ, which can be used to show that large solutions of (1.1) from y=φ⁢(x) only meet {y=φ(x)±δ}∩Eδ finite times. Then, by the properties of Eδ, V⁢(x,y) and U⁢(x,y), we can discuss the spiral properties of the solution starting from |y-φ⁢(x)|<δ recursively (see the following proof of Lemma 4.1, Case 3).

Lemma 4.1.

Assume (h4) and (h5). Then (h2) holds.

Proof.

Without loss of generality, let t0=0 and z0=(0,r0) with r0 is sufficiently large. We divide our proof into the following two steps.

Step 1. We first prove that there exist ξ2±⁢(r0), with

ξ 2 ± ⁢ ( r 0 ) → + ∞ ⇔ r 0 → + ∞ ,

such that either

z ⁢ ( t ) ∈ 𝒟 1 ∪ 𝒟 2 , ξ 2 - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ 2 + ⁢ ( r 0 )

for t∈[0,T], or there exists t2∈(0,T) such that above inequality holds for t∈[0,t2) and z⁢(t) meet x=0 entering into 𝒟3 at t=t2 (see Figure 1). We will discuss the estimates of z⁢(t) in the following cases.

$Figure 1 The trajectories in 𝒟1{\mathcal{D}_{1}} and 𝒟2{\mathcal{D}_{2}}.$

Figure 1

The trajectories in 𝒟1 and 𝒟2.

Case 1. Let z⁢(t)∈𝒟1 for t∈[0,t1), where t1≤T. By (4.1)–(4.2) and the Gronwall inequality, we have

(4.3) V 1 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≤ V 1 ⁢ ( x ⁢ ( 0 ) , y ⁢ ( 0 ) ) ⁢ e ∫ 0 t c 1 ⁢ ( s ) ⁢ 𝑑 s ≤ V 1 ⁢ ( 0 , r 0 ) ⁢ e ∫ 0 T c 1 ⁢ ( s ) ⁢ 𝑑 s

and

(4.4) U 1 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≥ U 1 ⁢ ( x ⁢ ( 0 ) , y ⁢ ( 0 ) ) ⁢ e - ∫ 0 t b 1 ⁢ ( s ) ⁢ 𝑑 s ≥ U 1 ⁢ ( 0 , r 0 ) ⁢ e - ∫ 0 T b 1 ⁢ ( s ) ⁢ 𝑑 s

for t∈[0,t1). Then

(4.5) ξ 1 - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ 1 + ⁢ ( r 0 ) for ⁢ t ∈ [ 0 , t 1 ) ,

where

ξ 1 - ⁢ ( r 0 ) = min ⁡ { x 2 + y 2 : U 1 ⁢ ( x , y ) = U 1 ⁢ ( 0 , r 0 ) ⁢ e - B 1 } ,

ξ 1 + ⁢ ( r 0 ) = max ⁡ { x 2 + y 2 : V 1 ⁢ ( x , y ) = V 1 ⁢ ( 0 , r 0 ) ⁢ e C 1 }

and B1=∫0Tb1⁢(s)⁢𝑑s, C1=∫0Tc1⁢(s)⁢𝑑s. Note that

V 1 ⁢ ( x , y ) = V 1 ⁢ ( 0 , r 0 ) ⁢ e C 1 → + ∞ ⇔ r 0 → + ∞ ,

which implies x2+y2→+∞⇔r0→+∞. Thus,

(4.6) ξ 1 + ⁢ ( r 0 ) → + ∞ ⇔ r 0 → + ∞ .

Similarly, we have ξ1-⁢(r0)→+∞⇔r0→+∞.

If t1=T, the proof is completed. If t1<T, then z⁢(t) will meet y=φ⁢(x) entering into 𝒟2 for the first time at t=t1. Note that ∂⁡H∂⁡x⁢(t,x,φ⁢(x)) could change sign for sufficiently large |x|. Thus, z⁢(t) could eventually return back to 𝒟1 for sufficiently large |x|.

Let t2′∈[t1,t2) be the time such that y⁢(t2′)=φ⁢(x⁢(t2′)), and let t2* be any given time. Next, we will estimate z⁢(t) in the following two cases (see Figure 1):

z ⁢ ( t ) ∈ 𝒟 2 for t∈[t2′,t2*), t2*∈(t2′,t2],
z ⁢ ( t ) ∈ 𝒟 1 ∪ 𝒟 2 for t∈[t1,t2*), t2*∈(t1,t2].

Case 2. For case (a), by (4.1)–(4.2) and the Gronwall inequality, we have

(4.7) V 2 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≤ V 2 ⁢ ( x ⁢ ( t 2 ′ ) , y ⁢ ( t 2 ′ ) ) ⁢ e ∫ t 2 ′ t c 2 ⁢ ( s ) ⁢ 𝑑 s

and

(4.8) U 2 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≥ U 2 ⁢ ( x ⁢ ( t 2 ′ ) , y ⁢ ( t 2 ′ ) ) ⁢ e - ∫ t 2 ′ t b 2 ⁢ ( s ) ⁢ 𝑑 s

for t∈[t2′,t2*). Then

(4.9) ξ t 2 ′ - ≤ r ⁢ ( t ) ≤ ξ t 2 ′ + for ⁢ t ∈ [ t 2 ′ , t 2 * ) ,

where

ξ t 2 ′ - = min ⁡ { x 2 + y 2 : U 2 ⁢ ( x , y ) = U 2 ⁢ ( x ⁢ ( t 2 ′ ) , y ⁢ ( t 2 ′ ) ) ⁢ e - B 2 } ,

ξ t 2 ′ + = max ⁡ { x 2 + y 2 : V 2 ⁢ ( x , y ) = V 2 ⁢ ( x ⁢ ( t 2 ′ ) , y ⁢ ( t 2 ′ ) ) ⁢ e C 2 }

and B2=∫0Tb2⁢(s)⁢𝑑s, C2=∫0Tc2⁢(s)⁢𝑑s. Since ξt2′- and ξt2′+ are only dependent on r⁢(t2′), so is r⁢(t) for t∈[t2′,t2*).

Case 3. Now we discuss the estimates of case (b). Using (h4), without loss of generality, we assume Vi⁢(x,φ⁢(x)) and Ui⁢(x,φ⁢(x)) are monotonically increasing with respect to x along the solution of system (1.1) in 𝒟i, i=1,2. We write

w x 1 - = min ⁡ { V 1 ⁢ ( x 1 - , φ ⁢ ( x 1 - ) ) , U 2 ⁢ ( x 1 - , φ ⁢ ( x 1 - ) ) } , w x 1 + = max ⁡ { V 1 ⁢ ( x 1 + , φ ⁢ ( x 1 + ) ) , U 2 ⁢ ( x 1 + , φ ⁢ ( x 1 + ) ) } ,

where

x 1 - := min ⁡ { | x | : x 2 + φ 2 ⁢ ( x ) = ξ 1 - ⁢ ( r 0 ) } , x 1 + := max ⁡ { | x | : x 2 + φ 2 ⁢ ( x ) = ξ 1 + ⁢ ( r 0 ) } .

Denote by E1 the compact set

{ ( x , y ) : x t 1 - ≤ x ≤ x t 1 + , | y - φ ⁢ ( x ) | ≤ 1 } ,

where

x t 1 - = min ⁡ { | x | : V 1 ⁢ ( x , φ ⁢ ( x ) ) ≥ w x 1 - ⁢ e - 2 ⁢ K b ⁢ c ⁢ and ⁢ U 2 ⁢ ( x , φ ⁢ ( x ) ) ≥ w x 1 - ⁢ e - 2 ⁢ K b ⁢ c } - T ,

x t 1 + = max ⁡ { | x | : V 1 ⁢ ( x , φ ⁢ ( x ) ) ≤ w x 1 + ⁢ e 2 ⁢ K b ⁢ c ⁢ and ⁢ U 2 ⁢ ( x , φ ⁢ ( x ) ) ≤ w x 1 + ⁢ e 2 ⁢ K b ⁢ c } + T ,

and Kb⁢c=max⁡{B1,B2,C1,C2}. Recalling ξ1±⁢(r0)→+∞⇔r0→+∞, we have

(4.10) x t 1 ± → + ∞ ⇔ r 0 → + ∞ .

Let

(4.11) K 1 = max ( x , y ) ∈ E 1 ⁡ { | ∂ ⁡ V 1 ∂ ⁡ x ⁢ ( x , y ) | , | ∂ ⁡ V 1 ∂ ⁡ y ⁢ ( x , y ) | , | ∂ ⁡ U 2 ∂ ⁡ x ⁢ ( x , y ) | , | ∂ ⁡ U 2 ∂ ⁡ y ⁢ ( x , y ) | } ,

(4.12) K 2 = K b ⁢ c T min { V 1 ( x t 1 - , φ ( x t 1 - ) , U 2 ( x t 1 - , φ ( x t 1 - ) } , K φ = max x ∈ [ x t 1 - , x t 1 + ] | φ ′ ( x ) |

and denote λ=K1⁢(1+Kφ)K2. Then there exists ε>0 such that

(4.13) 1 - ε ⁢ λ ⁢ Δ ⁢ t ≥ e - Δ ⁢ t and 1 + ε ⁢ λ ⁢ Δ ⁢ t ≤ e Δ ⁢ t for ⁢ Δ ⁢ t > 0 .

Note that ∂⁡H∂⁡y⁢(t,x,y)=0 for y=φ⁢(x) and t∈[0,T]. Then, for such ε, there exists sufficiently small δ∈(0,1) such that

| x ′ ⁢ ( t ) | = | ∂ ⁡ H ∂ ⁡ y ⁢ ( t , x , y ) | ≤ ε for ⁢ | y - φ ⁢ ( x ) | < δ , x t 1 - ≤ x ≤ x t 1 + , t ∈ [ 0 , T ] .

Next, we estimate the solution z⁢(t) for t∈[t1,t2*) in the following cases.

Case 3 (i). If |y⁢(t)-φ⁢(x⁢(t))|<δ for all t∈[t1,t2*), then |x′⁢(t)|≤ε for all t∈[t1,t2*). It follows that

| x ⁢ ( t ) - x ⁢ ( t 1 ) | ≤ ε ⁢ | t - t 1 | ≤ ε ⁢ T for ⁢ t ∈ [ t 1 , t 2 * ) ,

which implies that x⁢(t)∈[xt1-,xt1+] for t∈[t1,t2*). Thus, z⁢(t) is in Eδ for t∈[t1,t2*), where

E δ = { ( x , y ) : x t 1 - ≤ x ≤ x t 1 + , | y - φ ⁢ ( x ) | < δ } .

Then we can find a uniform bound of r⁢(t) for t∈[t1,t2*).

Case 3 (ii). Let t*1∈(t1,t2*) be the first time such that

y ⁢ ( t * 1 ) = φ ⁢ ( x ⁢ ( t * 1 ) ) - δ or y ⁢ ( t * 1 ) = φ ⁢ ( x ⁢ ( t * 1 ) ) + δ .

That is, |y⁢(t)-φ⁢(x⁢(t))|<δ for all t∈[t1,t*1) (see Figure 2 or Figure 3). We can use a similar argument in Case 3 (i) to obtain that

z ⁢ ( t ) ∈ E δ ⁢ and ⁢ | x ⁢ ( t ) - x ⁢ ( t 1 ) | ≤ ε ⁢ ( t - t 1 ) for ⁢ t ∈ [ t 1 , t * 1 ) .

From (4.11)–(4.13), we have

V 1 ⁢ ( x ⁢ ( t ) , φ ⁢ ( x ⁢ ( t ) ) ) ≥ V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) - K 1 ⁢ ( 1 + K φ ) ⁢ ε ⁢ ( t - t 1 )

= V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ ( 1 - ε ⁢ K 1 ⁢ ( 1 + K φ ) V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ ( t - t 1 ) )

≥ V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ ( 1 - ε ⁢ λ ⁢ ( t - t 1 ) ⁢ K b ⁢ c T )

(4.14) ≥ V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ e - K b ⁢ c T ⁢ ( t - t 1 )

for t∈[t1,t*1). On the other hand, from (4.11)–(4.13), we have

V 1 ⁢ ( x ⁢ ( t ) , φ ⁢ ( x ⁢ ( t ) ) ) ≤ V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) + K 1 ⁢ ( 1 + K φ ) ⁢ ε ⁢ ( t - t 1 )

= V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ ( 1 + ε ⁢ K 1 ⁢ ( 1 + K φ ) V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ ( t - t 1 ) )

≤ V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ ( 1 + ε ⁢ λ ⁢ ( t - t 1 ) ⁢ K b ⁢ c T )

(4.15) ≤ V 1 ⁢ ( x ⁢ ( t 1 ) , φ ⁢ ( x ⁢ ( t 1 ) ) ) ⁢ e K b ⁢ c T ⁢ ( t - t 1 )

for t∈[t1,t*1). Inequalities (4.14) and (4.15) imply that

(4.16) V 1 t 1 ⁢ e - K b ⁢ c T ⁢ ( t - t 1 ) ≤ V 1 ⁢ ( x ⁢ ( t ) , φ ⁢ ( x ⁢ ( t ) ) ) ≤ V 1 t 1 ⁢ e K b ⁢ c T ⁢ ( t - t 1 ) for ⁢ t ∈ [ t 1 , t * 1 ) ,

where V1t1=V1⁢(x⁢(t1),φ⁢(x⁢(t1))). Similar to (4.16), we have

(4.17) U 2 t 1 ⁢ e - K b ⁢ c T ⁢ ( t - t 1 ) ≤ U 2 ⁢ ( x ⁢ ( t ) , φ ⁢ ( x ⁢ ( t ) ) ) ≤ U 2 t 1 ⁢ e K b ⁢ c T ⁢ ( t - t 1 ) for ⁢ t ∈ [ t 1 , t * 1 ) ,

where U2t1=U2⁢(x⁢(t1),φ⁢(x⁢(t1))).

If y⁢(t*1)=φ⁢(x⁢(t*1))-δ and y⁢(t)<φ⁢(x⁢(t)) for all t∈[t*1,t2*), then z⁢(t)∈𝒟2 for t∈[t*1,t2*). Thus, we can use an argument similar to Case 2, to obtain a uniform bound of r⁢(t) for t∈[t*1,t2*).

If not, there would be t1′′ such that y⁢(t1′′)=φ⁢(x⁢(t1′′)). Therefore, we can find τ1′∈[t1,t*1) such that y⁢(τ1′)=φ⁢(x⁢(τ1′)) and y⁢(t)<φ⁢(x⁢(t)) for t∈(τ1′,t1′′) (see Figure 2). Similar to (4.7) and (4.8), we have

(4.18) V 2 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≤ V 2 ⁢ ( x ⁢ ( τ 1 ′ ) , y ⁢ ( τ 1 ′ ) ) ⁢ e ∫ τ 1 ′ t c 2 ⁢ ( s ) ⁢ 𝑑 s = V 2 ⁢ ( x ⁢ ( τ 1 ′ ) , φ ⁢ ( x ⁢ ( τ 1 ′ ) ) ) ⁢ e ∫ τ 1 ′ t c 2 ⁢ ( s ) ⁢ 𝑑 s ,

(4.19) U 2 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≥ U 2 ⁢ ( x ⁢ ( τ 1 ′ ) , y ⁢ ( τ 1 ′ ) ) ⁢ e - ∫ τ 1 ′ t b 2 ⁢ ( s ) ⁢ 𝑑 s = U 2 ⁢ ( x ⁢ ( τ 1 ′ ) , φ ⁢ ( x ⁢ ( τ 1 ′ ) ) ) ⁢ e - ∫ τ 1 ′ t b 2 ⁢ ( s ) ⁢ 𝑑 s

for t∈[τ1′,t1′′], respectively.

If y⁢(t*1)=φ⁢(x⁢(t*1))+δ and y⁢(t)>φ⁢(x⁢(t)) for all t∈[t*1,t2*), then z⁢(t)∈𝒟1 for t∈[t*1,t2*). Thus, we can use an argument similar to Case 1, to obtain a uniform bound of r⁢(t) for t∈[t*1,t2*).

If not, there would be t1′′ such that y⁢(t1′′)=φ⁢(x⁢(t1′′)). Furthermore, there exists τ1′∈[t1,t*1) such that y⁢(τ1′)=φ⁢(x⁢(τ1′)) and y⁢(t)>φ⁢(x⁢(t)) for t∈(τ1′,t1′′) (see Figure 3).

Similar to (4.3) and (4.4), we have

(4.20) V 1 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≤ V 1 ⁢ ( x ⁢ ( τ 1 ′ ) , y ⁢ ( τ 1 ′ ) ) ⁢ e ∫ τ 1 ′ t c 1 ⁢ ( s ) ⁢ 𝑑 s = V 1 ⁢ ( x ⁢ ( τ 1 ′ ) , φ ⁢ ( x ⁢ ( τ 1 ′ ) ) ) ⁢ e ∫ τ 1 ′ t c 1 ⁢ ( s ) ⁢ 𝑑 s ,

(4.21) U 1 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≥ U 1 ⁢ ( x ⁢ ( τ 1 ′ ) , y ⁢ ( τ 1 ′ ) ) ⁢ e - ∫ τ 1 ′ t b 1 ⁢ ( s ) ⁢ 𝑑 s = U 1 ⁢ ( x ⁢ ( τ 1 ′ ) , φ ⁢ ( x ⁢ ( τ 1 ′ ) ) ) ⁢ e - ∫ τ 1 ′ t b 1 ⁢ ( s ) ⁢ 𝑑 s

for t∈[τ1′,t1′′], respectively.

Notice that (4.16) and (4.17) imply that z⁢(t)∈Eδ for t∈[t1,τ1′]. Moreover, using (4.18)–(4.21), letting

w δ 1 + = max ⁡ { x 2 + y 2 : ( x , y ) ∈ E δ } ,

w δ 1 - = min ⁡ { x 2 + y 2 : ( x , y ) ∈ E δ } ,

w V 2 + = max ⁡ { x 2 + y 2 : V 2 ⁢ ( x , y ) ≤ V 2 ⁢ ( x t 1 + , φ ⁢ ( x t 1 + ) ) ⁢ e K b ⁢ c } ,

w V 1 + = max ⁡ { x 2 + y 2 : V 1 ⁢ ( x , y ) ≤ V 1 ⁢ ( x t 1 + , φ ⁢ ( x t 1 + ) ) ⁢ e K b ⁢ c } ,

w U 2 - = min ⁡ { x 2 + y 2 : U 2 ⁢ ( x , y ) ≥ U 2 ⁢ ( x t 1 - , φ ⁢ ( x t 1 - ) ) ⁢ e - K b ⁢ c } ,

w U 1 - = min ⁡ { x 2 + y 2 : U 1 ⁢ ( x , y ) ≥ U 1 ⁢ ( x t 1 - , φ ⁢ ( x t 1 - ) ) ⁢ e - K b ⁢ c }

and

ξ t 1 + = max ⁡ { w δ 1 + , w V 1 + , w V 2 + } , ξ t 1 - = min ⁡ { w δ 1 - , w U 1 - , w U 2 - } ,

we have

(4.22) ξ t 1 - ≤ r ⁢ ( t ) ≤ ξ t 1 + for ⁢ t ∈ [ t 1 , t 1 ′′ ] .

Similar to (4.6), it follows from (4.10) that ξt1±→+∞⟺r0→+∞.

$Figure 2 Trajectory intersects Γ and Γ-δ{\Gamma_{-\delta}}.$

Figure 2

Trajectory intersects Γ and Γ-δ.

$Figure 3 Trajectory intersects Γ and Γδ{\Gamma_{\delta}}.$

Figure 3

Trajectory intersects Γ and Γδ.

Furthermore, by the monotony increasing property of V1⁢(x⁢(t),φ⁢(x⁢(t))) along the solution of (1.1) in 𝒟1 and the monotony decreasing property of U2⁢(x⁢(t),φ⁢(x⁢(t))) along the solution of (1.1) in 𝒟2, we have

V 1 ⁢ ( x ⁢ ( t 1 ′′ ) , φ ⁢ ( x ⁢ ( t 1 ′′ ) ) ) ≥ V 1 ⁢ ( x ⁢ ( τ 1 ′ ) , φ ⁢ ( x ⁢ ( τ 1 ′ ) ) )

and

U 2 ⁢ ( x ⁢ ( t 1 ′′ ) , φ ⁢ ( x ⁢ ( t 1 ′′ ) ) ) ≤ U 2 ⁢ ( x ⁢ ( τ 1 ′ ) , φ ⁢ ( x ⁢ ( τ 1 ′ ) ) ) ,

respectively. Combining (4.16) with (4.20), we have

V 1 t 1 ⁢ e - ( K b c T ⁢ ( t 1 ′′ - t 1 ) + ∫ t 1 t 1 ′′ c 1 ⁢ ( s ) ⁢ 𝑑 s ) ≤ V 1 ⁢ ( x ⁢ ( t 1 ′′ ) , φ ⁢ ( x ⁢ ( t 1 ′′ ) ) ) ≤ V 1 t 1 ⁢ e K b c T ⁢ ( t 1 ′′ - t 1 ) + ∫ t 1 t 1 ′′ c 1 ⁢ ( s ) ⁢ 𝑑 s .

Similarly, combining (4.17) with (4.19), we have

U 2 t 1 ⁢ e - ( K b c T ⁢ ( t 1 ′′ - t 1 ) + ∫ t 1 t 1 ′′ b 2 ⁢ ( s ) ⁢ 𝑑 s ) ≤ U 2 ⁢ ( x ⁢ ( t 1 ′′ ) , φ ⁢ ( x ⁢ ( t 1 ′′ ) ) ) ≤ U 2 t 1 ⁢ e K b c T ⁢ ( t 1 ′′ - t 1 ) + ∫ t 1 t 1 ′′ b 2 ⁢ ( s ) ⁢ 𝑑 s .

Next, we will estimate the solution z⁢(t) for t∈[t1′′,t2*). If |y⁢(t)-φ⁢(x⁢(t))|<δ for all t∈[t1′′,t2*), we can use an argument similar to Case 3 (i), for obtaining a uniform bound of r⁢(t) for t∈[t1′′,t2*). If not, the solution z⁢(t) would be similar to Case 3 (ii). We can prove that there are finite times

t 1 ′ ≤ τ 1 ′ < t * 1 < t 1 ′′ ≤ τ 1 ′′ < t * 2 < t 1 3 ⁢ ⋯ < t 1 k - 1 ≤ τ 1 k - 1 < t * k - 1 < t 1 k

in [t1,t2*), with the similar properties as t*1, τ1′ and t1′′. That is, for j=2,…,k, either

y ⁢ ( t * j - 1 ) = φ ⁢ ( x ⁢ ( t * j - 1 ) ) - δ , y ⁢ ( τ 1 j - 1 ) = φ ⁢ ( x ⁢ ( τ 1 j - 1 ) ) , y ⁢ ( t 1 j ) = φ ⁢ ( x ⁢ ( t 1 j ) ) , | y ⁢ ( t ) - φ ⁢ ( x ⁢ ( t ) ) | < δ for ⁢ t ∈ [ t 1 j - 1 , τ 1 j - 1 ) and y ⁢ ( t ) < φ ⁢ ( x ⁢ ( t ) ) for ⁢ t ∈ ( τ 1 j - 1 , t 1 j )

y ⁢ ( t * j - 1 ) = φ ⁢ ( x ⁢ ( t * j - 1 ) ) + δ , y ⁢ ( τ 1 j - 1 ) = φ ⁢ ( x ⁢ ( τ 1 j - 1 ) ) , y ⁢ ( t 1 j ) = φ ⁢ ( x ⁢ ( t 1 j ) ) , | y ⁢ ( t ) - φ ⁢ ( x ⁢ ( t ) ) | < δ for ⁢ t ∈ [ t 1 j - 1 , τ 1 j - 1 ) and y ⁢ ( t ) > φ ⁢ ( x ⁢ ( t ) ) for ⁢ t ∈ ( τ 1 j - 1 , t 1 j ) .

Indeed, since z⁢(t1) and z⁢(t*1) are on two different curves Γ and Γδ (or Γ-δ), and z⁢(t)∈Eδ for all t∈[t1,t*1) (see Figure 2 or Figure 3), we have

δ ≤ d ⁢ ( z ⁢ ( t * 1 ) , z ⁢ ( t 1 ) ) ≤ | x ⁢ ( t * 1 ) - x ⁢ ( t 1 ) | + | y ⁢ ( t * 1 ) - y ⁢ ( t 1 ) | ≤ ( K 3 + K 4 ) ⁢ ( t * 1 - t 1 ) ,

where

K 3 = max ( x , y ) ∈ E δ , t ∈ [ 0 , T ] ⁡ { | ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) | } and K 4 = max ( x , y ) ∈ E δ , t ∈ [ 0 , T ] ⁡ { | ∂ ⁡ H ∂ ⁡ y ⁢ ( t , x , y ) | } .

It follows that

(4.23) t 1 ′′ - t 1 ≥ t * 1 - t 1 ≥ δ K 3 + K 4 .

Similarly, for j=3,…,k, we have

(4.24) t 1 j - t 1 j - 1 ≥ t * j - 1 - t 1 j - 1 ≥ δ K 3 + K 4 .

Inequalities (4.23) and (4.24) imply that k≤2⁢(KF1+KF2)⁢πδ+1.

Similar to (4.18), (4.19), (4.20) and (4.21), for j=3,…,k, recursively, we have either

V 2 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≤ V 2 ⁢ ( x ⁢ ( τ 1 j - 1 ) , φ ⁢ ( x ⁢ ( τ 1 j - 1 ) ) ) ⁢ e ∫ τ 1 j - 1 t c 2 ⁢ ( s ) ⁢ 𝑑 s ,

U 2 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≥ U 2 ⁢ ( x ⁢ ( τ 1 j - 1 ) , φ ⁢ ( x ⁢ ( τ 1 j - 1 ) ) ) ⁢ e - ∫ τ 1 j - 1 t b 2 ⁢ ( s ) ⁢ 𝑑 s

for t∈[τ1j-1,t1j], or

V 1 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≤ V 1 ⁢ ( x ⁢ ( τ 1 j - 1 ) , φ ⁢ ( x ⁢ ( τ 1 j - 1 ) ) ) ⁢ e ∫ τ 1 j - 1 t c 1 ⁢ ( s ) ⁢ 𝑑 s ,

U 1 ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) ≥ U 1 ⁢ ( x ⁢ ( τ 1 j - 1 ) , y ⁢ ( τ 1 j - 1 ) ) ⁢ e - ∫ τ 1 j - 1 t b 1 ⁢ ( s ) ⁢ 𝑑 s

for t∈[τ1j-1,t1j]. Moreover, it follows that either

(4.25) V 1 t 1 ⁢ e - ( K b c T ⁢ ( t 1 j - t 1 ) + ∫ t 1 t 1 j c 1 ⁢ ( s ) ⁢ 𝑑 s ) ≤ V 1 ⁢ ( x ⁢ ( t 1 j ) , φ ⁢ ( x ⁢ ( t 1 j ) ) ) ≤ V 1 t 1 ⁢ e K b c T ⁢ ( t 1 j - t 1 ) + ∫ t 1 t 1 j c 1 ⁢ ( s ) ⁢ 𝑑 s

(4.26) U 2 t 1 ⁢ e - ( K b c T ⁢ ( t 1 j - t 1 ) + ∫ t 1 t 1 j b 2 ⁢ ( s ) ⁢ 𝑑 s ) ≤ U 2 ⁢ ( x ⁢ ( t 1 j ) , φ ⁢ ( x ⁢ ( t 1 j ) ) ) ≤ U 2 t 1 ⁢ e K b c T ⁢ ( t 1 j - t 1 ) + ∫ t 1 t 1 j b 2 ⁢ ( s ) ⁢ 𝑑 s

for j=3,…,k. Using (4.25) or (4.26), similar to (4.16) and (4.17), it follows that z⁢(t)∈Eδ for t∈[t1j-1,τ1j-1], j=3,…,k. Then we can obtain the same estimation as (4.22), that is,

(4.27) ξ t 1 - ≤ r ⁢ ( t ) ≤ ξ t 1 + for ⁢ t ∈ [ t 1 j - 1 , t 1 j ] , j = 3 , … , k .

For t>t1k, we have one of the following possibilities:

z ⁢ ( t ) in 𝒟1 for all t∈(t1k,T],
z ⁢ ( t ) in 𝒟2 for all t∈(t1k,T] or there exists t2∈(0,T) such that z⁢(t) meet x=0 entering into 𝒟3 at t=t2,
| y ⁢ ( t ) - φ ⁢ ( x ⁢ ( t ) ) | < δ for all t∈(t1k,T].

Therefore, by (4.5), (4.9), (4.22), (4.27) and the discussion in Case 3 (i), there exist ξ2±⁢(r0), with

ξ 2 ± ⁢ ( r 0 ) → + ∞ ⇔ r 0 → + ∞ ,

such that either

z ⁢ ( t ) ∈ 𝒟 1 ∪ 𝒟 2 , ξ 2 - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ 2 + ⁢ ( r 0 )

for t∈[0,T], or the above inequality holds for t∈[0,t2).

Step 2. Similarly, we can discuss the cases of z⁢(t)∈𝒟3 and 𝒟4 and find ξ4±⁢(r0) such that

ξ 4 - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ 4 + ⁢ ( r 0 ) for ⁢ t ∈ ( t 2 , T ] .

Taking

ξ + ⁢ ( r 0 ) = max ⁡ { ξ i + ⁢ ( r 0 ) : i = 2 , 4 } and ξ - ⁢ ( r 0 ) = min ⁡ { ξ i - ⁢ ( r 0 ) : i = 2 , 4 } ,

we have either

ξ - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ + ⁢ ( r 0 ) for ⁢ t ∈ [ 0 , T ] ,

or there exists t1^∈[0,T) such that z⁢(t) intersects x=0 at t=t1^ and z⁢(t) completes one clockwise turn around the origin when t∈[0,t1^]. Moreover,

ξ - ⁢ ( r 0 ) ≤ r ⁢ ( t ) ≤ ξ + ⁢ ( r 0 ) for ⁢ t ∈ [ 0 , t 1 ^ ] .

Finally, it is clear that ξ±⁢(r0) can be chosen as strictly increasing functions. ∎

5 Applications

In this section we discuss the applications of Theorem 1.1 in the models studied in [4] and [25]. Then we give the proof of Example 1.1.

(1) At first, we note that, taking L0⁢(t,z)=γ0⁢(t)⁢V0⁢(z) and L∞⁢(t,z)=γ∞⁢(t)⁢V∞⁢(z), (H0) and (H∞) implies (h${}_{0}$) and (h${}_{\infty}$) are satisfied. Then Theorem 5.1 in [4] is a special case of Theorem 1.1.

(2) Now we consider the main result in [25]. The second-order equation (1.2) is equivalent to

x ′ = y , y ′ = - f ⁢ ( t , x ) .

In this case, we have

〈 ∇ ⁡ H ⁢ ( t , z ) , z 〉 = f ⁢ ( t , x ) ⁢ x + y 2 ,

where H⁢(t,z)=y22+∫0xf⁢(t,s)⁢𝑑s. From (H${}^{r}_{0}$), for each δ>0, there exists rδ>0 for each |x|<rδ, it follows that

f ⁢ ( t , x ) ⁢ x ≤ ( b + ⁢ ( t ) + δ ) ⁢ ( x + ) 2 + ( b - ⁢ ( t ) + δ ) ⁢ ( x - ) 2

for a.e. t∈[0,T], where x+=max⁡{x,0}, x-=max⁡{-x,0}. That is, for each δ>0, there exists rδ>0, for each 0<|z|<rδ, it follows that

〈 ∇ ⁡ H ⁢ ( t , z ) , z 〉 ≤ ( b + ⁢ ( t ) ⁢ x + - b - ⁢ ( t ) ⁢ x - ) ⁢ x + y 2 + δ ⁢ ( x 2 + y 2 )

for a.e. t∈[0,T]. If we take

L 0 ⁢ ( t , z ) = 1 2 ⁢ ( b + ⁢ ( t ) ⁢ ( x + ) 2 + b - ⁢ ( t ) ⁢ ( x - ) 2 + y 2 ) and V 0 ⁢ ( z ) = x 2 + y 2 ,

then (h${}_{0}$) holds.

According to (H${}^{l}_{\infty}$)’ and f⁢(t,x)∈C⁢(ℝ×ℝ,ℝ), for each δ>0, there exists l⁢(t)∈L1⁢([0,T]) such that

f ⁢ ( t , x ) ⁢ x ≥ ( a + ⁢ ( t ) - δ ) ⁢ ( x + ) 2 + ( a - ⁢ ( t ) - δ ) ⁢ ( x - ) 2 - l ⁢ ( t )

for all x∈ℝ and a.e. t∈[0,T]. Then

〈 ∇ ⁡ H ⁢ ( t , z ) , z 〉 ≥ ( a + ⁢ ( t ) ⁢ x + - a - ⁢ ( t ) ⁢ x - ) ⁢ x + y 2 - δ ⁢ ( x 2 + y 2 ) - l ⁢ ( t )

for all z∈ℝ2 and a.e. t∈[0,T]. So that (h${}_{\infty}$) holds with

L ∞ ⁢ ( t , z ) = 1 2 ⁢ ( y 2 + a + ⁢ ( t ) ⁢ ( x + ) 2 + a - ⁢ ( t ) ⁢ ( x - ) 2 ) and V ∞ ⁢ ( z ) = x 2 + y 2 .

It is easy to see that ρ⁢(L∞)=ρ∞ and ρ⁢(L0)=ρ0, where ρ∞ and ρ0 are the rotation numbers of the piecewise linear equations x′′+a+⁢(t)⁢x+-a-⁢(t)⁢x-=0 and x′′+b+⁢(t)⁢x+-b-⁢(t)⁢x-=0, respectively.

Therefore, Theorem 1.1 is a generalization of the main result in [25] to indefinite planar Hamiltonian systems.

Next, we will prove Example 1.1.

Proof of Example 1.1.

We will verify the hypotheses in Theorem 1.1 by the following steps.

Step 1. We will prove that system (1.5) is an asymptotically semilinear planar system in the sense of rotation numbers. Namely, system (1.5) satisfies (h${}_{0}$), (h${}_{\infty}$) and (h${}^{0}_{\infty}$). Set Iεk={t∈[0,2π]:|sint|≤εk}, where εk satisfies εk→0 as k→+∞. For each k∈ℕ, there exists dk>0 such that

f ( t , x ) x = { | sin t | x 4 ≥ 0 for ⁢ x < - d k ⁢ and ⁢ t ∈ I ε k , | sin t | x 4 > ε k x 4 ≥ k 2 ⁢ x 2 for ⁢ x < - d k ⁢ and ⁢ t ∈ [ 0 , 2 ⁢ π ] \ I ε k , ( α 2 + 1 ) ⁢ x 4 1 + x 2 ≥ ( α 2 + 1 - 1 k ) ⁢ x 2 for ⁢ x ≥ d k ⁢ and ⁢ t ∈ [ 0 , 2 ⁢ π ] .

Consider system (1.5), we have 〈∇⁡H⁢(t,z),z〉=f⁢(t,x)⁢x-2⁢x⁢y+y2, where H⁢(t,z)=y22+∫0xf⁢(t,s)⁢𝑑s-x⁢y. Let

(5.1) L ∞ k ( t , z ) = { y 2 2 - x ⁢ y for ⁢ x < 0 ⁢ and ⁢ t ∈ I ε k , y 2 2 + k 2 ⁢ x 2 2 - x ⁢ y for ⁢ x < 0 ⁢ and ⁢ t ∈ [ 0 , 2 ⁢ π ] \ I ε k , y 2 2 + ( α 2 + 1 - 1 k ) ⁢ x 2 2 - x ⁢ y for ⁢ x ≥ 0 ⁢ and ⁢ t ∈ [ 0 , 2 ⁢ π ] ,

and V∞⁢(z)=x2+y2. Then (h${}_{\infty}$) holds. On the other hand, it is obvious that

lim sup x → 0 ⁡ f ⁢ ( t , x ) x ≤ β for ⁢ t ∈ [ 0 , 2 ⁢ π ] ,

where β>1. For each δ>0, there exists rδ with |x|<rδ, it follows that

f ⁢ ( t , x ) ⁢ x ≤ ( β + δ ) ⁢ x 2 for ⁢ t ∈ [ 0 , 2 ⁢ π ] .

So (h${}_{0}$) is valid with

L 0 β ⁢ ( t , z ) = y 2 2 + β ⁢ x 2 2 - x ⁢ y and V 0 ⁢ ( z ) = x 2 + y 2 .

Next, we estimate ρ⁢(L∞k) and ρ⁢(L0β). Let z⁢(t) be the solution of J⁢z′=∇⁡L∞k⁢(t,z) such that |z0|=1. Then z⁢(t)≠0 for t∈[0,2⁢π]. We can easily prove that z⁢(t) moves in turn on the right and left phase planes around the origin. Let △k⁢t=△k+⁢t+△k-⁢t be the time when z⁢(t) completes one clockwise turn, where △k+⁢t is the time of z⁢(t) going through the right half plane and △k-⁢t is the time of z⁢(t) going through the left half plane. From (5.1), we have

- θ ′ = sin 2 ⁡ θ + ( α 2 + 1 - 1 k ) ⁢ cos 2 ⁡ θ - 2 ⁢ sin ⁡ θ ⁢ cos ⁡ θ for ⁢ x ≥ 0 .

It follows that

∫ - π 2 π 2 d ⁢ θ sin 2 ⁡ θ + ( α 2 + 1 - 1 k ) ⁢ cos 2 ⁡ θ - 2 ⁢ sin ⁡ θ ⁢ cos ⁡ θ = π α 2 - 1 k ,

which implies △k+⁢t→πα as k→+∞. Similarly, we have

| θ ′ | = | 2 ⁢ sin ⁡ θ ⁢ cos ⁡ θ - sin 2 ⁡ θ | ≤ 2 for ⁢ x < 0 ⁢ and ⁢ t ∈ I ε k .

Moreover,

(5.2) - θ ′ = sin 2 ⁡ θ + k 2 ⁢ cos 2 ⁡ θ - 2 ⁢ sin ⁡ θ ⁢ cos ⁡ θ for ⁢ x < 0 ⁢ and ⁢ t ∈ [ 0 , 2 ⁢ π ] \ I ε k .

Note that

(5.3) ∫ π 2 3 ⁢ π 2 d ⁢ θ sin 2 ⁡ θ + k 2 ⁢ cos 2 ⁡ θ - 2 ⁢ sin ⁡ θ ⁢ cos ⁡ θ = π k 2 - 1 .

If △k-⁢t≥2⁢πk2-1+mes⁢(Iεk), then there is an interval [t1,t2]⊂[0,2⁢π]\Iεk such that z⁢(t) is always in the left half plane {(x,y):x<0} for all t∈[t1,t2] and t2-t1>πk2-1. From (5.2)–(5.3), it implies that θ⁢(t1)-θ⁢(t2)>π, which is a contradiction. Hence, we can deduce that

△ k - ⁢ t < 2 ⁢ π k 2 - 1 + mes ⁢ ( I ε k ) → 0 as ⁢ k → ∞ .

To conclude, one has

ρ ⁢ ( L ∞ k ) = lim k → ∞ ⁡ - θ ⁢ ( k ⁢ T ) 2 ⁢ k ⁢ π → 2 ⁢ α as ⁢ k → ∞ .

Using the same arguments as given above, one has ρ⁢(L0β)→0 as β→1+.

Step 2. We will prove that system (1.5) satisfies conditions (h1), (h2) and (h3). Note that we have δ>0 such that

| f ⁢ ( t , x ) | ≤ ( β + 1 ) ⁢ | x | for ⁢ | x | < δ , t ∈ [ 0 , 2 ⁢ π ] .

Then, for sufficiently small |z|, we have

| ∇ ⁡ H ⁢ ( t , z ) | = ( f ⁢ ( t , x ) - y ) 2 + ( y - x ) 2 ≤ [ ( β + 1 ) ⁢ | x | + | y | ] 2 + ( | x | + | y | ) 2 ≤ 4 ⁢ ( β + 1 ) ⁢ x 2 + y 2 ,

which implies that

lim sup | z | → 0 ⁡ | ∇ ⁡ H ⁢ ( t , z ) | | z | ≤ 4 ⁢ ( β + 1 ) < + ∞ .

Then condition (h1) holds; (h3) is obvious satisfied. Next, we will prove that (h2) holds.

It is clear that system (1.5) satisfies (h5) with φ⁢(x)=x. Moreover, by using the straight line Γ:y=x and y-axis, we divide ℝ2 into four regions 𝒟i,i=1,2,3,4 as

𝒟 1 = { ( x , y ) : x ≥ 0 , y > x } , 𝒟 2 = { ( x , y ) : x > 0 , y ≤ x } ,

𝒟 3 = { ( x , y ) : x ≤ 0 , y < x } , 𝒟 4 = { ( x , y ) : x < 0 , y ≥ x } .

Consider a function

I ⁢ ( x , y ) = y 2 2 + F + ⁢ ( x ) - x ⁢ y ,

where F+⁢(x)=∫0xf+⁢(s)⁢𝑑s, f+⁢(x)=sgn⁡(x)⁢max⁡{|x|,max0≤t≤2⁢π⁡|f⁢(t,x)|}. Then

I ⁢ ( x , y ) → + ∞ ⇔ x 2 + y 2 → + ∞ .

When z⁢(t)∈𝒟1, we have

I ′ ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) = y ⁢ y ′ + f + ⁢ ( x ) ⁢ x ′ - y ⁢ x ′ - x ⁢ y ′

= y ⁢ ( y - f ⁢ ( t , x ) ) + f + ⁢ ( x ) ⁢ ( y - x ) - y ⁢ ( y - x ) - x ⁢ ( y - f ⁢ ( t , x ) )

= ( y - x ) ⁢ ( f + ⁢ ( x ) - f ⁢ ( t , x ) ) ≥ 0 .

On the other hand, we consider a function

R ⁢ ( x , y ) = 1 2 ⁢ ( x 2 + ( x - y ) 2 ) .

Then R⁢(x,y)→+∞⇔x2+y2→+∞. Note that f⁢(t,x)=(α2+1)⁢x31+x2≥0 for x≥0. When z⁢(t)∈𝒟1, we have

R ′ ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) = x ⁢ x ′ + ( x - y ) ⁢ ( x ′ - y ′ )

= x ⁢ ( y - x ) + ( x - y ) ⁢ ( f ⁢ ( t , x ) - x )

= 2 ⁢ x ⁢ ( y - x ) - ( y - x ) ⁢ f ⁢ ( t , x )

≤ 2 ⁢ R ⁢ ( x ⁢ ( t ) , y ⁢ ( t ) ) .

Similarly, we can estimate I′⁢(x⁢(t),y⁢(t)) and R′⁢(x⁢(t),y⁢(t)) for z⁢(t)∈𝒟i,i=2,3,4. Thus, we can find that

V ⁢ ( x , y ) = { R ⁢ ( x , y ) for ⁢ z ⁢ ( t ) ∈ 𝒟 1 ∪ 𝒟 3 , I ⁢ ( x , y ) for ⁢ z ⁢ ( t ) ∈ 𝒟 2 ∪ 𝒟 4 ,

and

U ⁢ ( x , y ) = { I ⁢ ( x , y ) for ⁢ z ⁢ ( t ) ∈ 𝒟 1 ∪ 𝒟 3 , R ⁢ ( x , y ) for ⁢ z ⁢ ( t ) ∈ 𝒟 2 ∪ 𝒟 4 ,

and we can check that (h4) holds for system (1.5). Combing (h5) with φ⁢(x)=x and using Lemma 4.1, we have proved that (h2) holds for system (1.5).

Therefore, Example 1.1 is thus proved by Theorem 1.1. ∎

Communicated by Fabio Zanolin

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11901507

Award Identifier / Grant number: 12071327

Award Identifier / Grant number: 12071410

Funding source: Natural Science Foundation of Jiangsu Province

Award Identifier / Grant number: BK20181058

Funding statement: This work is supported by the National Natural Science Foundation of China (No. 11901507, No. 12071327 and No. 12071410), the Natural Science Foundation of Jiangsu Province (No. BK20181058), and the Qing Lan Project of the Jiangsu Higher Education Institutions of China.

A Appendix

Here we give the proof of Lemma 2.3.

At first, we claim that the t-rotation number RotL⁢(t;v) of a solution of system (2.1) depends upon the function L⁢(t,z) described by the following lemma.

Lemma A.1.

Suppose that Li⁢(t,z)∈Q, i=1,2, and

(A.1) 〈 ∇ ⁡ L 2 ⁢ ( t , z ) - ∇ ⁡ L 1 ⁢ ( t , z ) , z 〉 ≥ 0

holds for all z∈R2 and a.e. in t∈[0,T]. Then

Rot L 2 ⁢ ( t ; v ) ≥ Rot L 1 ⁢ ( t ; v ) for all t ∈ [ 0 , T ] and all ⁢ v ∈ S 1 .

Proof.

For each θ0∈ℝ, let θ⁢(t;θ0,L) be the unique solution of equation (2.2) satisfying θ⁢(0;θ0,L)=θ0. Denote by ϑ(t)=:θ1(t)-θ2(t), where θi⁢(t)=θ⁢(t;θ0,Li), for i=1,2. Note that ϑ⁢(0)=0. By (2.2), we have

d ⁢ ϑ ⁢ ( t ) d ⁢ t = Θ ⁢ ( t , θ 1 ; L 1 ) - Θ ⁢ ( t , θ 2 ; L 2 )

= ( Θ ( t , θ 1 ; L 2 ) - Θ ( t , θ 2 ; L 2 ) ) + ( Θ ( t , θ 1 ; L 1 ) - Θ ( t , θ 1 ; L 2 ) ) = : a ( t ) ϑ ( t ) + b ( t ) ,

where

a ⁢ ( t ) = ∂ ⁡ Θ ∂ ⁡ θ ⁢ ( t , ξ ; L 2 ) , ξ = ξ ⁢ ( t ) ∈ [ θ 1 ⁢ ( t ) , θ 2 ⁢ ( t ) ] ,

b ⁢ ( t ) = 〈 ∇ ⁡ L 2 ⁢ ( t , ( cos ⁡ θ 1 ⁢ ( t ) , sin ⁡ θ 1 ⁢ ( t ) ) ) - ∇ ⁡ L 1 ⁢ ( t , ( cos ⁡ θ 1 ⁢ ( t ) , sin ⁡ θ 1 ⁢ ( t ) ) ) , ( cos ⁡ θ 1 ⁢ ( t ) , sin ⁡ θ 1 ⁢ ( t ) ) 〉 .

Therefore,

ϑ ⁢ ( t ) = ∫ 0 t b ⁢ ( s ) ⁢ exp ⁡ ( ∫ s t a ⁢ ( τ ) ⁢ 𝑑 τ ) ⁢ 𝑑 s .

It follows from (A.1) that b⁢(t)≥0 for a.e. t∈[0,T]. Thus ϑ⁢(t)≥0 for all t∈[0,T], that is, θ1⁢(t)≥θ2⁢(t) for all t∈[0,T]. Hence,

Rot L 2 ⁢ ( t ; v ) ≥ Rot L 1 ⁢ ( t ; v ) for all t ∈ [ 0 , T ] and all ⁢ v ∈ S 1 . ∎

Lemma A.2.

Suppose that Li⁢(t,z)∈Q, i=1,2, and V⁢(z)∈P. Then, for each ε>0, there exists δ>0 such that, if

(A.2) 〈 ∇ ⁡ L 1 ⁢ ( t , z ) - ∇ ⁡ L 2 ⁢ ( t , z ) , z 〉 ≤ δ ⁢ V ⁢ ( z )

holds for all z∈R2 and a.e. t∈[0,T], then

(A.3) Rot L 1 ⁢ ( t ; v ) - Rot L 2 ⁢ ( t ; v ) ≤ ε for all t ∈ [ 0 , T ] and all ⁢ v ∈ S 1 .

Proof.

Let ε>0 be fixed. For δ>0 is small enough, suppose that (A.2) holds for all z∈ℝ2 and a.e. t∈[0,T]. According to Lemma A.1, we know that

Rot L 1 - δ ⁢ ( t ; v ) ≤ Rot L 2 ⁢ ( t ; v ) for all t ∈ [ 0 , T ] and all ⁢ v ∈ S 1 ,

where RotL1-δ⁢(t;v) is the t-rotation number of the solution z⁢(t) for system

J ⁢ z ′ = ∇ ⁡ L 1 ⁢ ( t , z ) - 1 2 ⁢ δ ⁢ ∇ ⁡ V ⁢ ( z ) .

Hence, to prove the result, it will be sufficient to show that there exists a constant δ>0, and it follows that

Rot L 1 - δ ⁢ ( t ; v ) ≥ Rot L 1 ⁢ ( t ; v ) - ε for all t ∈ [ 0 , T ] and all ⁢ v ∈ S 1 .

If not, for each n∈ℕ, there would be tn∈[0,T] and vn∈S1 such that

(A.4) Rot L 1 - 1 n ⁢ ( t n ; v n ) < Rot L 1 ⁢ ( t n ; v n ) - ε .

Without loss of generality we can assume that tn→τ and vn→ω∈S1. Since RotL1⁢(⋅;⋅) is continuous on [0,T]×S1, by passing to the upper limit on both sides of (A.4), we have

(A.5) lim sup n → ∞ ⁡ Rot L 1 - 1 n ⁢ ( t n ; v n ) ≤ Rot L 1 ⁢ ( τ ; ω ) - ε .

Let zn⁢(t) and z⁢(t) be the solutions of

J ⁢ z ′ = ∇ ⁡ L 1 ⁢ ( t , z ) - 1 2 ⁢ n ⁢ ∇ ⁡ V ⁢ ( z ) and J ⁢ z ′ = ∇ ⁡ L 1 ⁢ ( t , z ) ,

respectively, with zn⁢(0)=vn and z⁢(0)=ω. According to the continuous dependence theorem of solutions with respect to initial values and parameters [17], it follows that zn⁢(t)→z⁢(t) (n→∞), uniformly on [0,T]. Note that

Rot L 1 - 1 n ⁢ ( t n ; v n ) = 1 2 ⁢ π ⁢ ∫ 0 t n 〈 J ⁢ z n ′ , z n 〉 x n 2 + y n 2 ⁢ 𝑑 t = 1 2 ⁢ π ⁢ ∫ 0 t n 〈 ∇ ⁡ L 1 ⁢ ( t , z n ) , z n 〉 - 1 n ⁢ V ⁢ ( z n ) x n 2 + y n 2 ⁢ 𝑑 t .

Letting n→∞, we have

Rot L 1 - 1 n ⁢ ( t n ; v n ) → 1 2 ⁢ π ⁢ ∫ 0 τ 〈 ∇ ⁡ L 1 ⁢ ( t , z ) , z 〉 x 2 + y 2 ⁢ 𝑑 t = Rot L 1 ⁢ ( τ ; ω ) ,

which is in contradiction with (A.5). Therefore, (A.3) is proved and this completes the proof of Lemma A.2. ∎

Proof of Lemma 2.3.

Let ε>0 be fixed. By Lemma A.2, there is a small enough δ>0 such that

(A.6) Rot L ⁢ ( t ; v ) - Rot L - δ ⁢ ( t ; v ) ≤ ε 2 for all t ∈ [ 0 , T ] and all ⁢ v ∈ S 1 ,

where RotL-δ⁢(t;v) is the t-rotation number of the solution z⁢(t) for system

J ⁢ z ′ = ∇ ⁡ L ⁢ ( t , z ) - 1 2 ⁢ δ ⁢ ∇ ⁡ V ⁢ ( z ) .

By assumption (h${}_{\infty}$), for such δ, there exists l⁢(t)∈L1⁢([0,T]) such that (1.4) holds.

Next, we assume, by contradiction, that the statement of (2.5) is not true. This implies that, for each n∈ℕ, there is a solution zn⁢(t) of (1.1) defined on [0,T] with |zn⁢(t)|≥n for all t∈[0,T] and such that, for some tn∈[0,T],

(A.7) Rot H ⁢ ( t n ; z n ) < Rot L ⁢ ( t n ; v n ) - ε ,

where vn=zn⁢(0)|zn⁢(0)|=(cos⁡αn,sin⁡αn). Without loss of generality, we can assume that if n→∞, then tn→τ and vn→ω=(cos⁡α,sin⁡α)∈S1 with αn→α. Since RotL⁢(⋅;⋅) is continuous on [0,T]×S1, by passing to the upper limit on both sides of (A.7), we have

(A.8) lim sup n → ∞ ⁡ Rot H ⁢ ( t n ; z n ) ≤ Rot L ⁢ ( τ ; ω ) - ε .

Then, from (1.1) and (1.4) in the polar coordinates xn⁢(t)=rn⁢(t)⁢cos⁡θn⁢(t), yn⁢(t)=rn⁢(t)⁢sin⁡θn⁢(t), we have

θ n ′ ⁢ ( t ) = - 〈 ∇ ⁡ H ⁢ ( t , z n ) , z n 〉 | z n | 2 ≤ δ ⁢ V ⁢ ( e i ⁢ θ n ⁢ ( t ) ) + l ⁢ ( t ) n 2 - 〈 ∇ ⁡ L ⁢ ( t , e i ⁢ θ n ⁢ ( t ) ) , e i ⁢ θ n ⁢ ( t ) 〉

for a.e. t∈[0,T], where ei⁢θn⁢(t)=(cos⁡θn⁢(t),sin⁡θn⁢(t)). By a result on differential inequality [20], we have

(A.9) Rot H ⁢ ( t n ; z n ) = θ n ⁢ ( 0 ) - θ n ⁢ ( t n ) 2 ⁢ π ≥ α n - ϑ n ⁢ ( t n ) 2 ⁢ π ,

where ϑn⁢(tn) is the solution of

{ θ ′ ⁢ ( t ) = δ ⁢ V ⁢ ( e i ⁢ θ ⁢ ( t ) ) + l ⁢ ( t ) n 2 - 〈 ∇ ⁡ L ⁢ ( t , e i ⁢ θ ⁢ ( t ) ) , e i ⁢ θ ⁢ ( t ) 〉 , θ 0 = α n ,

and ei⁢θ⁢(t)=(cos⁡θ⁢(t),sin⁡θ⁢(t)). By the continuous dependence of the solutions to above equation, we obtain that, as n→∞, ϑn⁢(t)→θ¯⁢(t), uniformly on t∈[0,T], where θ¯⁢(t) is the solution of

{ θ ′ ⁢ ( t ) = δ ⁢ V ⁢ ( e i ⁢ θ ⁢ ( t ) ) - 〈 ∇ ⁡ L ⁢ ( t , e i ⁢ θ ⁢ ( t ) ) , e i ⁢ θ ⁢ ( t ) 〉 , θ 0 = α .

In particular, we know from the uniform convergence that ϑn⁢(tn)→θ¯⁢(τ) for n→∞. Observe that

Rot L - δ ⁢ ( τ ; ω ) = 1 2 ⁢ π ⁢ ∫ 0 τ 〈 ∇ ⁡ L ⁢ ( t , z ) - 1 2 ⁢ δ ⁢ ∇ ⁡ V ⁢ ( z ) , z 〉 x 2 + y 2 ⁢ 𝑑 t = - 1 2 ⁢ π ⁢ ∫ 0 τ ( θ ¯ ⁢ ( t ) ) ′ ⁢ 𝑑 t = θ ¯ ⁢ ( 0 ) - θ ¯ ⁢ ( τ ) 2 ⁢ π .

Then it follows from (A.9) that

(A.10) lim inf n → ∞ ⁡ Rot H ⁢ ( t n ; z n ) ≥ α - θ ¯ ⁢ ( τ ) 2 ⁢ π = Rot L - δ ⁢ ( τ ; ω ) .

Combining (A.10) and (A.8), we have

Rot L - δ ⁢ ( τ ; ω ) ≤ Rot L ⁢ ( τ ; ω ) - ε ,

which is in contradiction with (A.6).

Hence, the first claim in Lemma 2.3 is proved. The proof of the other statement is quite similar to that given earlier for the first claim, and hence it is omitted. ∎

Acknowledgements

The authors thank the referee for the careful reading of the paper and the useful suggestions.

References

[1] A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: A rotation number approach, Adv. Nonlinear Stud. 11 (2011), no. 1, 77–103. 10.1515/ans-2011-0104Suche in Google Scholar

[2] A. Boscaggin, One-signed harmonic solutions and sign-changing subharmonic solutions to scalar second order differential equations, Adv. Nonlinear Stud. 12 (2012), no. 3, 445–463. 10.1515/ans-2012-0302Suche in Google Scholar

[3] A. Boscaggin, Periodic solutions to superlinear planar Hamiltonian systems, Port. Math. 69 (2012), no. 2, 127–140. 10.4171/PM/1909Suche in Google Scholar

[4] A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem, Nonlinear Anal. 74 (2011), no. 12, 4166–4185. 10.1016/j.na.2011.03.051Suche in Google Scholar

[5] F. Dalbono and F. Zanolin, Multiplicity results for asymptotically linear equations, using the rotation number approach, Mediterr. J. Math. 4 (2007), no. 2, 127–149. 10.1007/s00009-007-0108-zSuche in Google Scholar

[6] M. A. del Pino and R. F. Manásevich, Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity, J. Differential Equations 103 (1993), no. 2, 260–277. 10.1006/jdeq.1993.1050Suche in Google Scholar

[7] T. R. Ding and F. Zanolin, Periodic solutions of Duffing’s equations with superquadratic potential, J. Differential Equations 97 (1992), no. 2, 328–378. 10.1016/0022-0396(92)90076-YSuche in Google Scholar

[8] T. R. Ding and F. Zanolin, Subharmonic solutions of second order nonlinear equations: A time-map approach, Nonlinear Anal. 20 (1993), no. 5, 509–532. 10.1016/0362-546X(93)90036-RSuche in Google Scholar

[9] C. Fabry and A. Fonda, Periodic solutions of perturbed isochronous Hamiltonian systems at resonance, J. Differential Equations 214 (2005), no. 2, 299–325. 10.1016/j.jde.2005.02.003Suche in Google Scholar

[10] A. Fonda, Positively homogeneous Hamiltonian systems in the plane, J. Differential Equations 200 (2004), no. 1, 162–184. 10.1016/j.jde.2004.02.001Suche in Google Scholar

[11] A. Fonda and M. Garrione, Double resonance with Landesman–Lazer conditions for planar systems of ordinary differential equations, J. Differential Equations 250 (2011), no. 2, 1052–1082. 10.1016/j.jde.2010.08.006Suche in Google Scholar

[12] A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential systems in the plane, J. Differential Equations 252 (2012), no. 2, 1369–1391. 10.1016/j.jde.2011.08.005Suche in Google Scholar

[13] A. Fonda and A. Sfecci, Periodic solutions of a system of coupled oscillators with one-sided superlinear retraction forces, Differential Integral Equations 25 (2012), no. 11–12, 993–1010. 10.57262/die/1356012248Suche in Google Scholar

[14] A. Fonda and A. Sfecci, Periodic solutions of weakly coupled superlinear systems, J. Differential Equations 260 (2016), no. 3, 2150–2162. 10.1016/j.jde.2015.09.056Suche in Google Scholar

[15] A. Fonda and A. J. Ureña, A higher dimensional Poincaré–Birkhoff theorem for Hamiltonian flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 3, 679–698. 10.1016/j.anihpc.2016.04.002Suche in Google Scholar

[16] S. Gan and M. Zhang, Resonance pockets of Hill’s equations with two-step potentials, SIAM J. Math. Anal. 32 (2000), no. 3, 651–664. 10.1137/S0036141099356842Suche in Google Scholar

[17] J. K. Hale, Ordinary Differential Equations, Robert E. Krieger, Huntington, 1980. Suche in Google Scholar

[18] P. Hartman, On boundary value problems for superlinear second order differential equations, J. Differential Equations 26 (1977), no. 1, 37–53. 10.1016/0022-0396(77)90097-3Suche in Google Scholar

[19] H. Jacobowitz, Periodic solutions of x′′+f⁢(x,t)=0 via the Poincaré–Birkhoff theorem, J. Differential Equations 20 (1976), no. 1, 37–52. 10.1016/0022-0396(76)90094-2Suche in Google Scholar

[20] V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications. Vol. I: Ordinary Differential Equations, Academic Press, New York, 1969. Suche in Google Scholar

[21] A. Margheri, C. Rebelo and P. J. Torres, On the use of Morse index and rotation numbers for multiplicity results of resonant BVPs, J. Math. Anal. Appl. 413 (2014), no. 2, 660–667. 10.1016/j.jmaa.2013.12.005Suche in Google Scholar

[22] D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer–Leach–Dancer condition, J. Differential Equations 171 (2001), no. 2, 233–250. 10.1006/jdeq.2000.3847Suche in Google Scholar

[23] D. Qian, L. Chen and X. Sun, Periodic solutions of superlinear impulsive differential equations: A geometric approach, J. Differential Equations 258 (2015), no. 9, 3088–3106. 10.1016/j.jde.2015.01.003Suche in Google Scholar

[24] D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal. 36 (2005), no. 6, 1707–1725. 10.1137/S003614100343771XSuche in Google Scholar

[25] D. Qian, P. J. Torres and P. Wang, Periodic solutions of second order equations via rotation numbers, J. Differential Equations 266 (2019), no. 8, 4746–4768. 10.1016/j.jde.2018.10.010Suche in Google Scholar

[26] S. Wang and D. Qian, Periodic solutions of p-Laplacian equations via rotation numbers, Comm. Pure Appl. Anal. 20 (2021), no. 5, 2117–2138. 10.3934/cpaa.2021060Suche in Google Scholar

[27] C. Zanini, Rotation numbers, eigenvalues, and the Poincaré–Birkhoff theorem, J. Math. Anal. Appl. 279 (2003), no. 1, 290–307. 10.1016/S0022-247X(03)00012-XSuche in Google Scholar

[28] M. Zhang, The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentials, J. Lond. Math. Soc. (2) 64 (2001), no. 1, 125–143. 10.1017/S0024610701002277Suche in Google Scholar

Received: 2020-11-26

Revised: 2021-04-05

Accepted: 2021-04-09

Published Online: 2021-07-17

Published in Print: 2021-08-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/ans-2021-2134

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Subharmonic Solution; Indefinite Hamiltonian System; Spiral Function; Rotation Number; Poincaré–Birkhoff Theorem

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