Periodic solutions of semilinear Duffing equations with impulsive effects

J. Math. Anal. Appl. 467 (2018) 349–370

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Periodic solutions of semilinear Duffing equations with impulsive effects Yanmin Niu, Xiong Li ∗,1 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China

a r t i c l e

i n f o

Article history: Received 7 January 2018 Available online 5 July 2018 Submitted by Y. Huang Keywords: Impulsive differential equations Poincaré–Birkhoff twist theorem Periodic solutions

a b s t r a c t In this paper we are concerned with the existence of periodic solutions for semilinear Duffing equations with impulsive effects. Firstly for its autonomous equation, any motion of the solution is same as the motion of the corresponding equation without impulses until it meets the first impulse time. Under the influence of impulses, these two motions are likely to be quite different. We introduce some reasonable assumptions on the impulsive functions to control these differences such that the information valid for the equation without impulses can always be used for the impulsive one. Basing on Poincaré–Birkhoff twist theorem, we prove the existence of infinitely many periodic solutions. Secondly, as for the nonautonomous equation where the autonomous case is taken as an auxiliary one, we find the relation between the solutions of these two equations and then obtain the existence of infinitely many periodic solutions also by Poincaré–Birkhoff twist theorem. Lastly, an example with special impulses satisfying the above assumptions is given. © 2018 Elsevier Inc. All rights reserved.

1. Introduction We are concerned in this paper with the existence of periodic solutions for the second order impulsive differential equation ⎧  t = tj ; ⎪ ⎨ x + g(x) = p(t),  (1.1) x(tj +) = I(x(tj −), x (tj −)), ⎪ ⎩   x (tj +) = J(x(tj −), x (tj −)), j ∈ Z, where 0 ≤ t1 < 2π, g(x), p(t) ∈ C(R, R) and p(t) is 2π-periodic, I, J : R × R → R are continuous maps, and the impulsive time is 2π-periodic, that is, tj+1 = tj + 2π for j ∈ Z. We also use x(tj ), x (tj ) to denote x (tj −), x(tj −) for simplicity. * Corresponding author. 1

E-mail addresses: [email protected] (Y. Niu), [email protected] (X. Li). Partially supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities.

https://doi.org/10.1016/j.jmaa.2018.07.008 0022-247X/© 2018 Elsevier Inc. All rights reserved.

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This problem comes from Duffing equation x + g(x) = p(t)

(1.2)

and there is a wide literature dealing with the existence of periodic solutions for the above Duffing equation, not only because of its physical significance, but also the application of various mathematical techniques on it, such as Poincaré–Birkhoff twist theorem in [9], [11], the variational method in [1], [16] and topological degree or index theories in [3], [4]. Under different assumptions on the function g, for example being superlinear, sublinear, semilinear and so on, there are many interesting results on the existence and multiplicity of periodic solutions of (1.2), see [8], [20], [23], [24] and the references therein. Among these, the existence problem of periodic solutions for semilinear Duffing equations challenges more attentions for its special resonance phenomenon. At resonance, equation (1.2) may have no bounded solutions, therefore the crucial point of solving this problem is to exclude the resonance, and there are also many studies on it, see [5], [6] and [14]. Recently, as impulsive equations widely arise in applied mathematics, they attract a lot of attentions and many authors study the basic theory in [2], [13], along with the existence of periodic solutions of impulsive differential equations via fixed point theory in [17], topological degree theory in [10], [22], and the variational method in [18], [25]. As we all know, the existence of impulses, even the simplest impulsive function, may cause complicated dynamic phenomena and bring great difficulties to study. The changes between the behaviors of solutions with and without impulsive effects may be great. Here we take the simplest linear equation as an example and consider x + x = 0

(1.3)

with the impulsive conditions x(tj +) = 2x(tj −),

x (tj +) =

1  x (tj −), 2

(1.4)

where tj = jπ for j ∈ Z. It is easy to see that without impulses, all solutions of (1.3) are 2π-periodic and satisfy x2 (t) + x 2 (t) = C,

(1.5)

where C is a constant related to the initial values. However, under the influence of impulsive functions in (1.4), all solutions except for the trivial one are unbounded. In fact, with the initial point (x(0), x (0)) = (x0 , 0) (x0 = 0), the solution at each tj ± is located on the x-axis and the radius of the trajectory (1.5) at tj + becomes two times larger than the previous one at tj −, that is, (x(π+), x (π+)) = (−2x0 , 0), (x(2π+), x (2π+)) = (4x0 , 0), · · · , which implies the solution tends to infinity as t → +∞. From the above example, we find that the existence of periodic solutions for impulsive semilinear Duffing equations deserves exploring. What impulsive functions do guarantee the existence of periodic solutions is the key point. Different from the extensive study for second order differential equations without impulsive terms, there are only a few results on the existence and multiplicity of periodic solutions for impulsive ones. In [21], Qian et al. considered the superlinear impulsive differential equation ⎧   t = tj ; ⎪ ⎨ x + g(x) = p(t, x, x ),  Δx|t=tj = Ij (x (tj −) , x (tj −)), ⎪ ⎩ Δx |t=tj = Jj (x (tj −) , x (tj −)), j ∈ Z,

(1.6)

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where 0 ≤ t1 < · · · < tk < 2π, Ij , Jj : R × R → R are continuous maps, tj+k = tj + 2π for j ∈ Z, and g is a continuous function with the superlinear growth condition lim

|x|→+∞

g(x) = +∞. x

The authors proved via Poincaré–Birkhoff twist theorem the existence of infinitely many periodic solutions of (1.6) with p = p(t), and also the existence of periodic solutions for non-conservative case with degenerate impulsive terms by developing a new twist fixed point theorem. In [19], the authors of the present paper discussed the existence of periodic solutions for the sublinear impulsive differential equation ⎧ ⎪ x + g(x) = p(t, x, x ), ⎪ ⎨ Δx|t=tj = ax (tj −) , ⎪ ⎪ ⎩ Δx |t=tj = ax (tj −) ,

t = tj ; (1.7) j ∈ Z,

where 0 ≤ t1 < · · · < tk < 2π, a > 0 is a constant, tj+k = tj + 2π for j ∈ Z, and g is a continuous function with the sublinear growth condition lim

|x|→+∞

g(x) = 0. x

Basing on the Poincaré–Bohl fixed point theorem and the twist fixed point theorem established by Qian et al. in [21], they obtained the existence of harmonic solutions and subharmonic solutions, respectively. The impulsive functions are especially chosen to keep the arguments of trajectories unchanged in the polar coordinates, such that the difficulties causing by the impulses can be reduced. In this article, we discuss the semilinear impulsive Duffing equation (1.1), which is different from the superlinear or sublinear case and there is few papers on it up to now. Ding in [5] obtained an infinite class of periodic solutions for (1.2) without impulses under three conditions (H1 )–(H3 ) (mentioned in Section 2). On the basis of Ding’s work, Jiang et al. [12] revised the condition (H3 ) in [5] and proved the same result for the following impulsive equation ⎧ ⎪ x + g(x) = p(t), ⎪ ⎨ x(tk +) = ak x(tk −), ⎪ ⎪ ⎩  x (tk +) = bk x(tk −),

t = tk ; (1.8) k = 1, 2, · · · ,

where g(x), p(t) ∈ C(R, R) and p(t) is 2π-periodic, 0  t0 < t1 < · · · < tk < tk+1 < · · · ↑ ∞, ak > 0, ak bk = 1, there exists a positive integer q such that a1 · · · ak = 1, b1 · · · bk = 1, ak+q = ak , bk+q = bk , k = 1, 2, · · · , tk+q = tk + 2π, k = 0, 1, 2, · · · . If the solutions of general autonomous semilinear Duffing equations are a family of closed curves in the phase plane, but due to the existence of impulses, the closed curves may not be preserved and even be broken. This can be seen from Eq. (1.3) with the impulse (1.4). Denote the closed curve of the autonomous equation of (1.8) by Γ0 with the initial value (x(0), x (0)), while through each impulsive time tj , the curve Γ0 becomes Γj with the point (x(tj +), x (tj +)), j = 1, 2, · · · . Generally, under the impulses in (1.8), Γ1 is different from Γ0 . Even though through q impulsive effects, the motion Γq comes back to Γ0 , the time in which the argument decreases 2π is no longer the least positive period of Γ0 . Therefore the assumption on the original curve Γ0 is invalid for the jumped solutions on Γj , j = 1, · · · , q − 1. In this paper, in order to avoid the uncertainties brought by impulses, we will look for the optimal assumption about impulsive functions. Under the condition (H3 ), which only provides some information

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about the least positive period of the closed curve of the autonomous equation without impulses, we have to control the solutions of (1.1) at impulsive times such that the change between the jumped closed curve Γ1 and the original one Γ0 is tiny. Only by doing this can we make the analysis of the motions on the subsequent curves Γj available. Furthermore, in order to use Poincaré–Birkhoff twist theorem, we should verify the twist condition on the boundaries of an annulus. But the impulse can bring great changes for the arguments of solutions. All these difficulties due to the existence of impulses are considered in this article and we overcome these by choosing a class of impulses given in Sections 3 and 4. The rest is organized as follows. A generalized form of Poincaré–Birkhoff twist theorem due to Ding, together with some basic lemmas and symbols, are given in Section 2. Section 3 provides an existence theorem of infinitely many 2π-periodic solutions for the autonomous form of Eq. (1.1) via the twist theorem. The conditions in [5] are not enough for the impulses one, hence we put forward some new reasonable assumptions. Subsequently in Section 4, we establish the relation between arguments of the Poincaré mapping for the autonomous and nonautonomous equations under the polar coordinates. Again it follows from the twist theorem that there are infinitely many 2π-periodic solutions of (1.1). As an application of the results achieved in above sections, we present a special example and give more details in the last section. 2. Preliminaries For convenience, we introduce some basic lemmas and symbols which are used in the sequel. Firstly consider Duffing equation x + g(x) = 0

(2.1)

and its equivalent system x = y,

y  = −g(x).

(2.2)

This is a planar autonomous system whose orbits are curves determined by the following equation V (x, y) =

1 2 y + G(x) = c, 2

(2.3)

x where G(x) = 0 g(u)du and c is a parameter. The description of V −1 (c) is given in the following lemma, the proof of which we omit and one can refer to [5]. Lemma 2.1. ([5]) If the condition (H2 ) below holds, then V −1 (c) is a closed curve for c ≥ c0 which is star-shaped about the origin, where c0 is a positive constant. Denote the curve V −1 (c) by Γc . It follows from Lemma 2.1 that each curve Γc (c ≥ c0 ) intersects the x-axis at two points: (h(c), 0) and (−h1 (c), 0), where h(c), h1 (c) > 0 are uniquely determined by the formula G(h(c)) = G(−h1 (c)) = c.

(2.4)

Let (x(t), y(t)) be any solutions of (2.2) whose orbits are Γc (c ≥ c0 ). Obviously, this solution is periodic. Denote its least positive period by τ (c). It can be induced from equation (2.3) that h(c) √  du  τ (c) = 2 . c − G(u) −h1 (c)

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In [5], Ding proved the existence of periodic solutions under the following hypotheses: (H1 ) Let g(x) ∈ C 1 (R, R), and K be a positive constant, such that |g  (x)| ≤ K,

x ∈ R;

(H2 ) There exist two constants M0 > 0 and A0 > 0, such that x−1 g(x) ≥ A0 ,

|x| ≥ M0 ;

(H3 ) There exist a constant α > 0, an integer m > 0, and two sequences ak and bk , such that ak → ∞ and bk → ∞ as k → ∞, and moreover τ (ak ) <

2π − α, m

τ (bk ) >

2π + α. m

Ding also in [5] constructed a concrete example

1 x2 2 2 2 cos log(1 + x ) − sin log(1 + x ) g(x) = x m + 10 10(1 + x2 )

(2.5)

satisfying (H1 )–(H3 ) very skillfully, where m ∈ Z+ . In the sequel, we preserve hypothesis (H3 ), which is denoted by (A2 ) newly, and combine (H1 ) and (H2 ) into a little stronger one (A1 ) g(x) ∈ C 1 (R, R), g(0) = 0, and there exist two positive constants M, K such that M ≤ g  (x) ≤ K,

x ∈ R.

It is easy to verify that the function g in (2.5) also satisfies (A1 ) and under (A1 ) Lemma 2.1 holds. Now we introduce the Poincaré–Birkhoff twist theorem. Generally speaking, it is a powerful tool to obtain the existence of periodic solutions. To use this theorem more conveniently, there are many generalizations and here we briefly restate a version due to Ding in [7]. Let D denote an annular region in the (x, y)-plane. The boundary of D consists of two simple closed curves: the inner boundary curve C1 and the outer boundary curve C2 . Let Di denote the simple connected open set bounded by Ci , i = 1, 2. Consider an area-preserving mapping T : R2 → R2 . Suppose that T (D) ⊂ R2 − 0, where 0 is the origin and T |D admits a continuous lifting, with the standard covering projection Π : (r, θ) → (r cos θ, r sin θ), of the form r∗ = f (r, θ),

θ∗ = θ + g(r, θ),

(2.6)

where f and g are continuous in (r, θ) and 2π-periodic in θ. Lemma 2.2. ([7], [15]) Besides the above mentioned assumptions, we assume that 1. C1 is star-shaped about the origin; 2. there exists an area-preserving homeomorphism T0 : D2 → R2 , which satisfies T0 |D = T and 0 ∈ T0 (D1 ); 3. T has a lifting T defined by (2.6) such that g(r, θ) > 0 (< 0) on Π−1 (C1 ) and g(r, θ) < 0 (> 0) on Π−1 (C2 ). Then T has at least two fixed points such that their images under Π are two different fixed points of T in D.

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3. Autonomous Duffing impulsive equations We start with the autonomous equation, which is a special case of (1.1) with p(t) = 0 and is equivalent to ⎧   ⎪ ⎨ z = w, w = −g(z), z(tj +) = I(z(tj −), w(tj −)), ⎪ ⎩ w(tj +) = J(z(tj −), w(tj −)),

t = tj ; (3.1) j ∈ Z,

where g, I, J, tj are same as in (1.1). Moreover, let the jumping map Ψ : (z, w) → (I(z, w), J(z, w))

(3.2)

satisfy the following assumption (A3 ) Ψ is an area-preserving homeomorphism. Let (z(t, z, w), w(t, z, w)) be the solution of (3.1) with the initial point (z(0), w(0)) = (z, w). It is not hard to show that every such solution exists on the whole t-axis under the condition (A1 ) (see [2]). Then the Poincaré mapping P1 : R2 → R2 is well defined by (z, w) → (z(2π, z, w), w(2π, z, w)) . For each solution (z(t, z, w), w(t, z, w)), let c = V (z, w) =

1 2 w + G(z), 2

and Vc (t) = V (z(t, z, w), w(t, z, w)) =

1 2 w (t, z, w) + G(z(t, z, w)). 2

Since the motion of (z(t, z, w), w(t, z, w)) is always on the curve Γc without impulses, then for all t ∈ [0, t1 ], Vc (t) = Vc (0) = c. Therefore Vc (t1 −) = c and Vc (t1 +) =

1 2 1 w (t1 +) + G(z(t1 +)) = J 2 (w(t1 ), z(t1 )) + G(I(w(t1 ), z(t1 ))), 2 2

here and hereafter (w(t1 ), z(t1 )) = (z(t1 −, z, w), w(t1 −, z, w)). Since the solutions (z(t, z, w), w(t, z, w)) of (3.1) are a family of closed curves Γc with c > 0 in the phase plane, but due to the existence of impulses, the closed curves Γc may not be preserved and even be broken. Under the condition (A2 ), which only provides some information about the least positive period of the closed curve of the autonomous equation without impulses, we have to control the solutions of (3.1) at impulsive times such that the change between the jumped closed curve and the original one is tiny. Only by doing this can we make the analysis of the motions on the subsequent curves available. For this purpose, we assume that (A4 ) there exists some positive constant ε1 such that Vc (t1 +) = Vc (t1 −) + O(c−ε1 ),

c → ∞.

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Next, we take the transform z = ρ cos ϕ, w = ρ sin ϕ for system (3.1). Then the resulting equations for ρ and ϕ are ⎧ ⎨ ρ = ρ cos ϕ · sin ϕ − g(ρ cos ϕ) sin ϕ, 1 ⎩ ϕ = − sin2 ϕ − g(ρ cos ϕ) cos ϕ, t = tj . ρ

(3.3)

Meanwhile, from this transformation we have ρ=



z 2 + w2

(3.4)

and ϕ = arctan

w + kπ, z

k ∈ Z.

Obviously, such definition of ϕ is not clear for ϕ(tj +) and ϕ(tj −) since the difference of these two arguments is up to arbitrary integer multiple of 2π. We need an exact expression to make sure that the jumping map Ψ has a continuous lifting in polar coordinates (ρ, ϕ). Then for (tj +)  (z(tj +), w(tj +)) = (I(z(tj ), w(tj )), J(z(tj ), w(tj ))), we define ϕ(tj +) = arg (tj +) + 2kπ,

(3.5)

where k is chosen to satisfy |ϕ(tj +) − ϕ(tj −)| ≤ π and arg denotes the argument of with arg ∈ [0, 2π). Particularly, if for some j ∈ Z there exists k such that |ϕ(tj +) − ϕ(tj −)| = π, we only choose the suitable k satisfying ϕ(tj +) − ϕ(tj −) = π or ϕ(tj +) − ϕ(tj −) = −π for all those j. By this definition, ϕ(tj +) can be decided by ϕ(tj −) with the only k. Moreover, if ϕ(ti −) = ϕ(tj −) + 2π, then ϕ(ti +) = ϕ(tj +) + 2π. This implies that for the jumping map about ϕ: ϕ → ϕ + Δϕ(ρ, ϕ) in the polar coordinates, Δϕ(ρ, ϕ) is continuous and 2π periodic in ϕ. Then at the impulsive times, ρ(tj +) =

z(tj +)2 + w(tj +)2 =

I(z(tj ), w(tj ))2 + J(z(tj ), w(tj ))2

and ϕ(tj +) is defined in (3.5) for each j ∈ Z. Let (ρ(t, ρ, ϕ), ϕ(t, ρ, ϕ)) be the solution of (3.3) with the initial point (ρ, ϕ), where ρ cos ϕ = z and ρ sin ϕ = w. To guarantee the twist condition on the boundaries of an annulus, the change of ϕ(t) at tj + and tj −, which is denoted by Δϕ(tj )  Δϕ(ρ(tj −), ϕ(tj −)) = ϕ(tj +) − ϕ(tj −) should not be too large. That is, we assume that

(3.6)

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356



2

(A5 ) C Δϕ(t )

< mαC1 , where C1 = min{1, M }, C2 = max{1, K}, and M , K, α, m are given in (A1 ) 1 C1 and (A2 ). The estimation about the variation of ϕ(t, ρ, ϕ) during the time interval [0, 2π] will be given in the following lemma. Lemma 3.1. Assume that (A1 )–(A5 ) hold. Let Φ(ρ, ϕ) = ϕ(2π, ρ, ϕ) − ϕ. Then for k sufficiently large, there exist two positive constants β1 , β2 , such that 

Φ(ρ, ϕ) ≤ −2mπ − β1 ,

(ρ cos ϕ, ρ sin ϕ) ∈ Γak ;

Φ(ρ, ϕ) ≥ −2mπ + β2 ,

(ρ cos ϕ, ρ sin ϕ) ∈ Γbk ,

(3.7)

where ak , bk are given in (A2 ). Proof. Firstly, suppose (z, w) = (ρ cos ϕ, ρ sin ϕ) ∈ Γak . Consider the solution (ρ(t, ρ, ϕ), ϕ(t, ρ, ϕ)) of (3.3) and assume (ρ∗ (t, ρ, ϕ), ϕ∗ (t, ρ, ϕ)) is the solution with the same initial value (ρ, ϕ) of (3.3) without impulses. From (A1 ) and Lemma 2.1, there exist constants c0 > 0, C1 = min{1, M } and C2 = max{1, K} such that if c ≥ c0 , then C1 (z 2 + w2 ) ≤ w2 + zg(z) ≤ C2 (z 2 + w2 ),

(z, w) ∈ Γc ,

(3.8)

where c = V (0). By (3.8) and the second equation of (3.3), for k large enough satisfying ak ≥ c0 , −C2 ≤ ϕ (t, ρ, ϕ) ≤ −C1 ,

t ∈ [0, 2π] − t1 .

(3.9)

It also holds for ϕ∗ (t, ρ, ϕ), i.e. −C2 ≤ (ϕ∗ ) (t, ρ, ϕ) ≤ −C1 ,

t ∈ [0, 2π].

(3.10)

By the definition in (3.6), we have Φ(ρ, ϕ) = ϕ(2π, ρ, ϕ) − ϕ = ϕ(2π, ρ, ϕ) − ϕ(t1 +, ρ, ϕ) + ϕ(t1 +, ρ, ϕ) − ϕ(t1 −, ρ, ϕ) + ϕ(t1 −, ρ, ϕ) − ϕ

(3.11)

= ϕ(2π, ρ, ϕ) − ϕ(t1 +, ρ, ϕ) + Δϕ(t1 ) + ϕ∗ (t1 , ρ, ϕ) − ϕ, the last equality of which is due to that without impulse effect, ϕ(t1 −, ρ, ϕ) = ϕ∗ (t1 , ρ, ϕ). The next discussions will be divided into two parts according to the sign of Δϕ(t1 ). Case i: Δϕ(t1 ) > 0. Assume that the solution (ρ(t, ρ, ϕ), ϕ(t, ρ, ϕ)) jumps from Γak to another curve Γak at the repulsive time t = t1 . By (A4 ), for k large enough with ak = c c0 , Γak is very close to Γak such that there exist εak > 0 (εak → 0 as k → ∞) and Δtak > 0, such that ϕ(t1 +, ρ, ϕ) = ϕ∗ (t1 − Δtak , ρ, ϕ), |ρ(t1 +, ρ, ϕ) − ρ∗ (t1 − Δtak , ρ, ϕ)| ≤ εak , and for each t ∈ (t1 , 2π],

(3.12)

Y. Niu, X. Li / J. Math. Anal. Appl. 467 (2018) 349–370

|ϕ (t, ρ, ϕ) − (ϕ∗ ) (t − Δtak , ρ, ϕ)| ≤ εak .

357

(3.13)

Such εak can be found for the inequality (3.13) due to the continuous dependence on initial values for (ρ∗ (t − Δtak , ρ, ϕ), ϕ∗ (t1 − Δtak , ρ, ϕ)) and (ρ(t, ρ, ϕ), ϕ(t, ρ, ϕ)) with ‘initial values’ at t1 +. In other words, Δϕ(t1 ) = ϕ(t1 +, ρ, ϕ) − ϕ(t1 −, ρ, ϕ) = ϕ∗ (t1 − Δtak , ρ, ϕ) − ϕ∗ (t1 , ρ, ϕ)

(3.14)

and 2π ϕ(2π, ρ, ϕ) − ϕ(t1 +, ρ, ϕ) =

ϕ (s, ρ, ϕ)ds

t1 +



2π−Δtak

(ϕ∗ ) (s, ρ, ϕ)ds + (2π − t1 )εak

≤

(3.15)

t1 −Δtak

= ϕ∗ (2π − Δtak , ρ, ϕ) − ϕ∗ (t1 − Δtak , ρ, ϕ) + εak , where we denote (2π − t1 )εak by εak . By (3.14) and (3.15), the further estimate for (3.11) is Φ(ρ, ϕ) ≤ ϕ∗ (2π − Δtak , ρ, ϕ) − ϕ∗ (t1 − Δtak , ρ, ϕ) + ϕ∗ (t1 , ρ, ϕ) − ϕ + Δϕ(t1 ) + εak = ϕ∗ (2π, ρ, ϕ) − ϕ − [ϕ∗ (2π, ρ, ϕ) − ϕ∗ (2π − Δtak , ρ, ϕ)] − Δϕ(t1 ) + Δϕ(t1 ) + εak

2π = ϕ∗ (2π, ρ, ϕ) − ϕ − ϕ∗ 2π−Δt + εak , ak

(3.16)

where

2π ϕ∗ 2π−Δt

ak

 ϕ∗ (2π, ρ, ϕ) − ϕ∗ (2π − Δtak , ρ, ϕ).

2π Now we first estimate ϕ∗ 2π−Δt . From (3.10) and (3.14), we have ak

t1 −C2 Δtak ≤ −Δϕ(t1 ) =

(ϕ∗ ) (s, ρ, ϕ)ds ≤ −C1 Δtak ,

t1 −Δtak

which implies that Δϕ(t1 ) Δϕ(t1 ) ≤ Δtak ≤ . C2 C1

(3.17)

Therefore,

2π C2 − Δϕ(t1 ) ≤ ϕ∗ 2π−Δt = ak C1

2π

(ϕ∗ ) (s, ρ, ϕ)ds ≤ −

C1 Δϕ(t1 ). C2

(3.18)

2π−Δtak

The remaining term we need to calculate in the last line of (3.16) is ϕ∗ (2π, ρ, ϕ) − ϕ, where we recall that ϕ∗ (t, ρ, ϕ) is the solution of (3.3) without impulses. We now show that ϕ∗ (2π, ρ, ϕ) − ϕ ≤ −2mπ − mαC1 .

(3.19)

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In fact, since (ρ∗ (t, ρ, ϕ), ϕ∗ (t, ρ, ϕ)) lying in Γak has the least positive period τ (ak ), we know that the time in which ϕ∗ (t) has a decrement 2π is just τ (ak ). Denote ϕ∗ (2π, ρ, ϕ) − ϕ = −2lπ − σ, where l ≥ 0 is an integer, and 0 < σ ≤ 2π. Let tσ denote the time in which ϕ∗ (t) decreases from ϕ − 2lπ to ϕ − 2lπ − σ. Then l · τ (ak ) + tσ = 2π. Since 0 < tσ ≤ τ (ak ), by (A2 ) we have  2π = l · τ (ak ) + tσ ≤ (l + 1)τ (ak ) < (l + 1)

 2π −α . m

It follows that l ≥ m. If l ≥ m + 1, then ϕ∗ (2π, ρ, ϕ) − ϕ < −2lπ ≤ −2(m + 1)π.

(3.20)

Now assume l = m. Then we have  tσ = 2π − m · τ (ak ) ≥ 2π − m

2π −α m

 = mα.

(3.21)

By (3.10) and (3.21), we obtain l·τ (a k )+tσ

(ϕ∗ ) (s, ρ, ϕ)ds ≤ −C1 tσ ≤ mαC1 .

−σ = l·τ (ak )

Thus ϕ∗ (2π, ρ, ϕ) − ϕ = −2lπ − σ ≤ −2mπ − mαC1 .

(3.22)

Combining (3.20) and (3.22) yields the result since mα < 2π by (A2 ) and C1 ≤ 1. (3.16), together with (3.18) and (3.19) shows that 2 Φ(ρ, ϕ) ≤ −2mπ − mαC1 + C ak C1 Δϕ(t1 ) + ε   2 = −2mπ − mαC1 − C Δϕ(t ) − ε  . 1 a k C1

By the assumptions (A4 ) and (A5 ), we choose k large enough such that ak = c c0 and εak <  C2 C1 Δϕ(t1 ) . With β1 

1 2

 mαC1 −

1 2



mαC1 −

 C2 Δϕ(t1 ) , C1

the first inequality of (3.7) is proved in this case. Case ii: Δϕ(t1 ) < 0. The discussion about this case is similar with case i, and we just focus on some differences. Again assume that the solution (ρ(t), ϕ(t)) jumps from Γak to another curve Γak at the impulsive time t = t1 , which is very close to Γak for k large enough, then there exist εak > 0 small enough and Δtak > 0, such that ϕ(t1 +, ρ, ϕ) = ϕ∗ (t1 + Δtak , ρ, ϕ), |ρ(t1 +, ρ, ϕ) − ρ∗ (t1 + Δtak , ρ, ϕ)| ≤ εak ,

(3.23)

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359

and for each t ∈ (t1 , 2π], |ϕ (t, ρ, ϕ) − (ϕ∗ ) (t + Δtak , ρ, ϕ)| ≤ εak .

(3.24)

Δϕ(t1 ) = ϕ(t1 +, ρ, ϕ) − ϕ(t1 −, ρ, ϕ) = ϕ∗ (t1 + Δtak , ρ, ϕ) − ϕ∗ (t1 , ρ, ϕ)

(3.25)

Similarly, we obtain

and 2π ϕ(2π, ρ, ϕ) − ϕ(t1 +, ρ, ϕ) =

ϕ (s, ρ, ϕ)ds

t1 + 2π+Δta



(3.26)

k

(ϕ∗ ) (s, ρ, ϕ)ds + (2π − t1 )εak

≤ t1 +Δta

k

= ϕ∗ (2π + Δtak , ρ, ϕ) − ϕ∗ (t1 + Δtak , ρ, ϕ) + ε ak , where ε ak  (2π − t1 )εak . From (3.10) and (3.25), it holds that t1 +Δta

−C2 Δtak



k

(ϕ∗ ) (s, ρ, ϕ)ds ≤ −C1 Δtak ,

≤ Δϕ(t1 ) = t1

which implies

2π+Δta C2 k = Δϕ(t1 ) ≤ ϕ∗ 2π C1

2π+Δta



k

(ϕ∗ ) (s, ρ, ϕ)ds ≤

C1 Δϕ(t1 ). C2

(3.27)

2π

Then by (3.11), (3.15) and (3.27), Φ(ρ, ϕ) ≤ ϕ∗ (2π + Δtak , ρ, ϕ) − ϕ∗ (t1 + Δtak , ρ, ϕ) + ϕ∗ (t1 , ρ, ϕ) − ϕ + Δϕ(t1 ) + ε ak   = ϕ∗ (2π, ρ, ϕ) − ϕ + ϕ∗ (2π + Δtak , ρ, ϕ) − ϕ∗ (2π, ρ, ϕ) − Δϕ(t1 ) + Δϕ(t1 ) + ε ak 1 ≤ ϕ∗ (2π, ρ, ϕ) − ϕ + C  ak C2 Δϕ(t1 ) + ε    1 ≤ −2mπ − mαC1 − C Δϕ(t ) − ε  1 ak , C2

(3.28)

where the estimate about ϕ∗ (2π, ρ, ϕ) − ϕ is same as in (3.19). By letting β1  mαC1 , 1 for sufficiently large k with ε ak < − C C2 Δϕ(t1 ), the first inequality of (3.7) in this case is then proved. We are now in a position to prove the second inequality of (3.7). Suppose (z, w) = (ρ cos ϕ, ρ sin ϕ) ∈ Γbk , where k is sufficiently large such that bk ≥ c0 , and the solution (ρ(t), ϕ(t)) jumps from Γbk to another curve

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Γbk at the impulsive time t = t1 , which is very close to Γbk with bk = c c0 . The discussions here are similar with those on Γak , so that we just need to estimate ϕ∗ (2π, ρ, ϕ) − ϕ. We assert that ϕ∗ (2π, ρ, ϕ) − ϕ ≥ −2mπ + mαC1 .

(3.29)

In fact, since (ρ∗ (t, ρ, ϕ), ϕ∗ (t, ρ, ϕ)) lying in Γbk has the least period τ (bk ), we know that the time in which ϕ∗ (t) has a decrement 2π is just τ (bk ). Denote ϕ∗ (2π, ρ, ϕ) − ϕ = −2kπ + η where k ≥ 1 is an integer, and 0 < η ≤ 2π. Let tη denote the time in which ϕ∗ (t) decreases from ϕ − 2kπ + η to ϕ − 2kπ. Then k · τ (bk ) − tη = 2π. Since 0 < tη ≤ τ (bk ), by (A2 ) we have  2π = k · τ (bk ) − tη ≥ (k − 1)τ (bk ) ≥ (k − 1)

 2π +α . m

It follows that k ≤ m. If k ≥ m − 1, then ϕ∗ (2π, ρ, ϕ) − ϕ ≥ −2kπ ≥ −2(m − 1)π.

(3.30)

Now assume k = m. Then we have  tη = k · τ (bk ) − 2π ≥ −2π + m

2π +α m

 = mα.

(3.31)

By (3.10) and (3.31), we obtain k·τ (b k )−tη

(ϕ∗ ) (s, ρ, ϕ)ds ≥ C1 tη ≥ mαC1 .

η= k·τ (bk )

Thus ϕ∗ (2π, ρ, ϕ) − ϕ = −2kπ + η ≥ −2mπ + mαC1 ,

(3.32)

which results the conclusion by (3.30) and (3.32). Therefore in case i, (3.11), (3.13) and (3.18) deduce

2π Φ(ρ, ϕ) ≥ ϕ∗ (2π, ρ, ϕ) − ϕ − ϕ∗ 2π−Δt

ak

≥ −2mπ + mαC1 + For sufficiently large k with εak <

C1 C2 Δϕ(t1 ),

C1 C2 Δϕ(t1 )

− εak ,

− εak .

(3.33)

we obtain the second inequality of (3.7) by letting β2  mαC1 .

In case ii, by (3.11), (3.24) and (3.27), we estimate

2π+Δta k − ε Φ(ρ, ϕ) ≥ ϕ∗ (2π, ρ, ϕ) − ϕ + ϕ∗ 2π  ak ≥ −2mπ + mαC1 +

C2 C1 Δϕ(t1 )

− ε ak .

(3.34)

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Since the assumption (A5 ) holds, we choose k large enough such that bk = c c0 and ε ak <  C2 C1 Δϕ(t1 ) . With β2 

1 2

361

1 2

 mαC1 +

  C2 mαC1 + Δϕ(t1 ) , C1

the second inequality of (3.7) is proved. The proof of Lemma 3.1 is then completed. 2 Under the polar coordinates, the Poincaré mapping P1 can be written in the form ρ∗ = ρ(2π, ρ, ϕ),

ϕ∗ = ϕ(2π, ρ, ϕ) + 2kπ,

(3.35)

where k is an arbitrary integer. It is easy to see that if ρ(t, ρ, ϕ) > 0,

t ∈ [0, 2π],

ϕ(2π, ρ, ϕ) is well defined and is continuous in (ρ, ϕ). Moreover, according to the definition of ϕ(t1 +) in (3.5), we have ϕ(2π, ρ, ϕ + 2π) = ϕ(2π, ρ, ϕ) + 2π.

(3.36)

Now, let Γak and Γbk be the curves given by Lemma 2.1, where the specified parameters ak , bk ≥ c0 are given by (A2 ), for k ≥ n0 . We can rearrange ak and bk such that ak < bk < ak+1 for k ≥ n0 . Then Γak and Γbk bound an annular region Ak , and Γbk and Γak+1 bound another annular region Bk , for k ≥ n0 . It is well known that each fixed point of P1 corresponds to a 2π-periodic solution of (3.1). In the following theorem, we will apply Lemma 2.2 to show that P1 has at least two fixed points in each Ak and Bk for sufficiently large k. As a consequence, Eq. (3.1) has an infinite class of 2π-periodic solutions. Theorem 3.2. Under the assumptions (A1 )–(A5 ), Eq. (3.1) has infinitely many 2π-periodic solutions. Proof. Assume that ak , bk ≥ c0 for k ≥ n0 , where n0 is large enough and c0 is given in Lemma 2.1. Denote by Ak the region bounded by Γak and Γbk . Thus the restriction P1 |Ak can be written in (3.35). Let the integer k = m, then (3.35) can be rewritten in the form of ρ∗ = ρ(2π, ρ, ϕ),

ϕ∗ = ϕ + Φ1 (ρ, ϕ),

with Φ1 (ρ, ϕ) = ϕ(2π, ρ, ϕ) − ϕ + 2mπ = Φ(ρ, ϕ) + 2mπ. By Lemma 3.1, we obtain 

Φ1 (ρ, ϕ) ≤ −β1 < 0,

(ρ cos ϕ, ρ sin ϕ) ∈ Γak ;

Φ1 (ρ, ϕ) ≥ β2 > 0,

(ρ cos ϕ, ρ sin ϕ) ∈ Γbk ,

for k ≥ n0 . This proves the validity of condition 3 of Lemma 2.2 for the restriction P1 |Ak (k ≥ n0 ). With Lemma 2.1 and ak , bk c0 for large k, conditions 1 and 2 of Lemma 2.2 are obvious. In addition, we need to clarify P1 is an area-preserving map. Indeed, the Poincaré mapping P1 : (z, w) → (z(2π, z, w), w(2π, z, w)) can be expressed by P1 = P 1 ◦ Ψ ◦ P 0 , where

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P 0 : (z, w) → (z(t1 −, z, w), w(t1 −, z, w)) , P 1 : (z(t1 +, z, w), w(t1 +, z, w)) → (z(2π, z, w), w(2π, z, w)) and Ψ is the impulsive map defined in (3.2). Since P j , j = 0, 1 are symplectic by equation z  + g(z) = 0 being conservative and Ψ is area-preserving via (A3 ), the Poincaré mapping P1 is area-preserving. Therefore, we obtain the existence of at least two fixed points of P1 in Ak (k ≥ n0 ) by Lemma 2.2. This means that (3.1) has at least two 2π-periodic solutions in Ak . Similarly, there are at least two 2π-periodic solutions in Bk . Since each periodic solution of (3.1) is bounded by Γak and Γbk , then the above specified 2π-periodic solutions of (3.1) constitute an infinite class. The proof of Theorem 3.2 is thus completed. 2 4. Nonautonomous Duffing impulsive equations We rewrite (1.1) as the equivalent system of the form ⎧  x = y, y  = −g(x) + p(t), ⎪ ⎪ ⎨ x(tj +) = I(x(tj −), y(tj −)), ⎪ ⎪ ⎩ y(tj +) = J(x(tj −), y(tj −)),

t = tj ; (4.1) j ∈ Z.

Let (x(t, x, y), y(t, x, y)) be the solution of (4.1) with the initial point (x, y). Also it follows from the condition (A1 ) that every such solution exists on the whole t-axis (see [2]). Then the Poincaré mapping P2 : R2 → R2 is well defined by (x, y) → (x(2π, x, y), y(2π, x, y)) . Applying the polar transform x = γ cos θ, y = γ sin θ to (4.1), we get the equations for γ and θ, ⎧  ⎪ ⎨ γ = γ cos θ · sin θ − g(γ cos θ) sin θ + p(t) sin θ, 1 ⎪ ⎩ θ = − sin2 θ − [g(γ cos θ) cos θ − p(t) cos θ] , γ

t = tj .

(4.2)

At the impulsive times, γ(tj +) =

x(tj +)2 + y(tj +)2 =

I(x(tj ), y(tj ))2 + J(x(tj ), y(tj ))2 ,

θ(tj +) is defined similarly with ϕ(tj +) in (3.5) and hereafter we always denote Δθ(tj )  Δθ(θ(tj ), γ(tj )) = θ(tj +) − θ(tj −).     Let γ(t, γ, θ), θ(t, γ, θ) be the solution of (4.2) through the initial point γ(0), θ(0) = (γ, θ). Then P2 is rewritten in the form of γ ∗ = γ(2π, γ, θ),

θ∗ = θ(2π, γ, θ) + 2lπ,

(4.3)

where l is an arbitrary integer. Similar with the discussion for autonomous equations, the existence, continuity about the initial values and (3.36) are also true for γ(t, γ, θ) and θ(t, γ, θ) on [0, 2π]. In order to find the twist condition for θ(t), we first will study the intimate relation between equations (3.3) and (4.2), especially their arguments under the polar coordinates. We remark that the values of θ(t1 +) and ϕ(t1 +) are effected not only by θ(t1 −) and ϕ(t1 −) but also by γ(t1 −) and ρ(t1 −). The differences between the first equations in (3.3) and (4.2) imply that γ(t1 −) and ρ(t1 −) are quite different, therefore the differences of θ(t1 +) and ϕ(t1 +) may be large even though θ(t1 −) and ϕ(t1 −) are of little difference.

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To ensure the tiny change between these two arguments, we need to assume that the argument ϕ(t1 +) after the impulsive time depends only on the argument ϕ(t1 −) before the impulsive time, that is, (A6 ) there are ε2 > 0 and a function h ∈ C(R), such that ϕ(t1 +) = h(ϕ(t1 −)) + O(c−ε2 ), c → ∞. Since the impulsive functions in Eq. (3.1) and Eq. (4.1) are same, assumption (A6 ) is also satisfied for θ(t1 +). Let Θ(γ, θ) = θ(2π, γ, θ) − θ. The following lemma will estimate the difference between Θ(γ, θ) and Φ(γ, θ) with the same initial point. Lemma 4.1. Assume that (A1 ) and (A6 ) hold. For any ε > 0, there exists γ ∗ > 0 such that, for γ ≥ γ ∗ , |Θ(γ, θ) − Φ(γ, θ)| = |θ(2π, γ, θ) − ϕ(2π, γ, θ)| < ε. Proof. For 0 < ε < π, let (z(t, x, y), w(t, x, y)) be the solution of (3.1) with the initial point (z(0), w(0)) = (x, y). Denote 

u(t) = u(t, x, y) = x(t, x, y) − z(t, x, y), v(t) = v(t, x, y) = y(t, x, y) − w(t, x, y).

Then we have dv = p(t) − g  (ϑ(t))u(t), dt

du = v, dt

t = tj , j ∈ Z,

where ϑ(t) = x(t) + λ(t) (z(t) − x(t)), 0 ≤ λ(t) ≤ 1. Since the impulsive functions in (3.1) and (4.1) are same, then for each tj , j ∈ Z, u(tj +) = x(tj +) − z(tj +) = I(x(tj ), y(tj )) − I(z(tj ), w(tj )), v(tj +) = y(tj +) − w(tj +) = J(x(tj ), y(tj )) − J(z(tj ), w(tj )).

(4.4)

 1 Let η(t) = u2 (t) + v 2 (t) 2 . Then for t = tj , we have η

dη = uv + p(t)v − g  (ϑ(t))uv. dt

It follows from (A1 ) that

dη 1



≤ (1 + K)η + B, dt 2

(4.5)

where B = sup |p(t)|. t∈[0,2π]

First we consider θ(t, γ, θ) − ϕ(t, γ, θ) on [0, t1 ]. The differential inequality (4.5) together with η(0) = 0 yields η(t) ≤ for t ∈ [0, t1 ]. Denote

 2B  t1 (1+K) − 1  H0 e2 1+K

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ψ(t) = ψ(t, γ, θ) = θ(t, γ, θ) − ϕ(t, γ, θ), where (γ, θ) is the polar coordinate of (x, y), that is, (γ cos θ, γ sin θ) = (x, y). It is clear that if |ψ(t)| < π, then ψ(t) is just the angle between the vectors (x(t), y(t)) and (z(t), w(t)). By the law of cosines, we have cos ψ(t) =

H02 γ 2 (t) + ρ2 (t) − η 2 (t) ≥1− , 2γ(t)ρ(t) 2γ(t)ρ(t)

t ∈ [0, t1 ].

On the other hand, γ(t) ≥ ρ(t) − η(t) ≥ ρ(t) − H0 . Therefore, under the assumptions that |ψ(t)| < π and ρ(t) − H0 > 0, we have cos ψ(t) ≥ 1 −

H02 , 2ρ(t) (ρ(t) − H0 )

t ∈ [0, t1 ].

(4.6)

Notice that ρ(t) becomes arbitrarily large for all t ∈ [0, 2π] if the initial value γ is sufficiently large. Then there is a constant γ0 > 0 such that, for γ ≥ γ0 and t ∈ [0, t1 ], ρ(t) − H0 > 0,

H02 < 1 − cos δ, 2ρ(t) (ρ(t) − H0 )

(4.7)

where δ = min{ π2 , 2ε }. From (4.6) and (4.7), we conclude that if |ψ(t)| < π on [0, t1 ], then the inequality cos ψ(t) > cos δ holds, which implies |ψ(t)| < δ,

t ∈ [0, t1 ].

|ψ(t)| < π,

t ∈ [0, t1 ].

Now we just need to verify (4.8)

Since ψ(0) = 0 and ψ(t) is continuous on [0, t1 ], there exists a > 0 such that |ψ(t)| < π,

t ∈ [0, a).

(4.9)

To prove (4.8), we need to prove a > t1 . If it is not true, then the a given above belongs to (0, t1 ] such that (4.9) holds and |ψ(a)| = π. Noticing that inequality (4.6), (4.7) and (4.9) hold on t ∈ [0, a), again we can find a γ  such that for γ ≥ γ  |ψ(t)| < δ, Thus we obtain |ψ(a)| ≤ δ ≤ for t ∈ [0, t1 ],

π 2

t ∈ [0, a).

< π, which contradicts with |ψ(a)| = π. Hence (4.8) holds and subsequently

|ψ(t)| < δ ≤

ε < ε. 2

Next we turn to consider ψ(t) on (t1 , 2π]. It follows (4.4) that  1 η(t1 +) = u2 (t1 +) + v 2 (t1 +) 2  η1 .

(4.10)

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Integrating the inequality (4.5) on (t1 , 2π], we have η(t) ≤



2B1 B e(1+K)π −  H1 , 1+K B1

t ∈ (t1 , 2π],

where B1 = 1+K 2 η1 + B. In the same way as the discussion on [0, t1 ], we can prove that there is a constant γ1 > 0, such that for γ ≥ γ1 and t ∈ (t1 , 2π], cos ψ(t) ≥ 1 −

H12 , 2ρ(t) (ρ(t) − H1 )

(4.11)

and ρ(t) − H1 > 0,

H12 < 1 − cos δ, 2ρ(t) (ρ(t) − H1 )

(4.12)

provided |ψ(t)| < π,

t ∈ (t1 , 2π].

(4.13)

That is to say if |ψ(t)| < π on (t1 , 2π], then the inequality |ψ(t)| < δ

(4.14)

holds for t ∈ (t1 , 2π]. Similarly, we just need to prove (4.13). By (4.10), ψ(t1 ) = ψ(t1 −) < ε. Since the impulsive functions in (4.2) and (3.3) are same and assumption (A6 ) holds, ψ(t1 +) is continuous in initial value ψ(t1 ) and thereby ψ(t1 +) can be small enough for large γ. Thus there exists a > 0 such that |ψ(t)| < π,

t ∈ (t1 , a).

(4.15)

We just need to verify a > 2π. The proof of this is similar with the proof of (4.8) therefore we omit here. In particular, |ψ(2π)| < δ ≤ 2ε < ε by (4.14). Consequently, choosing γ ∗ = max{γ0 , γ1 }, for γ ≥ γ ∗ , we have |Θ(γ, θ) − Φ(γ, θ)| = |θ(2π, γ, θ) − ϕ(2π, γ, θ)| = |ψ(2π)| < ε. The proof of Lemma 4.1 is then completed. 2 By Lemma 3.1, together with the small difference between θ(2π) and ϕ(2π), we can obtain the existence of 2π-periodic solutions by the twist theorem easily. The main result of Eq. (4.1) is stated as following. Theorem 4.2. Assume that (A1 )–(A6 ) hold, then Eq. (4.1) has infinitely many 2π-periodic solutions. Proof. Let c1 ≥ c0 be sufficiently large, such that for c ≥ c1 , (γ, θ) ∈ Γc implies γ ≥ γ ∗ , where γ ∗ is specified in Lemma 4.1. There is no loss of generality to assume that ak , bk ≥ c1 for k ≥ n0 , where n0 is large enough. It follows from Lemma 4.1 that 

γ(t, γ, θ) ≥ ρ(t) − H0 > 0, t ∈ [0, t1 ], γ(t, γ, θ) ≥ ρ(t) − H1 > 0, t ∈ (t1 , 2π],

(4.16)

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provided that (γ, θ) ∈ Ak for k ≥ n0 . Thus the restriction P2 |Ak can be written in (4.3), where we put the integer l = m. Now (4.3) is in the following form γ ∗ = γ(2π, γ, θ),

θ∗ = θ + Θ1 (γ, θ)

with Θ1 (γ, θ) = Θ(γ, θ) + 2mπ. By Lemma 4.1, choosing 0 < ε < min{β1 , β2 }, we obtain |Θ1 (γ, θ) − Φ1 (γ, θ) − 2mπ| < ε, which together with (3.7) yields 

Θ1 (γ, θ) < 0,

(ρ cos ϕ, ρ sin ϕ) ∈ Γak ;

Θ1 (γ, θ) > 0,

(ρ cos ϕ, ρ sin ϕ) ∈ Γbk .

This proves the validity of condition 3 of Lemma 2.2 for the restriction P2 |Ak (k ≥ n0 ). By Lemma 2.1 and (4.16), conditions 1 and 2 of Lemma 2.2 also hold. Similar to the proof of Theorem 3.2, area-preserving property of P2 is obtained. Therefore, applying Lemma 2.2 on P2 , we ensure the existence of at least two fixed points, which corresponds to the 2π-periodic solutions of (1.1) in Ak (k ≥ n0 ). Similarly, the analysis on Bk concludes the existence of 2π-periodic solutions of (1.1) in Bk (k ≥ n0 ). Since each period solutions of (1.1) is bounded by Γak , Γbk , then the above specified 2π-periodic solutions of (1.1) constitute an infinite class of solutions. The proof is thus completed. 2 Remark 4.3. One can easily obtain the same results if (1.1) has finitely many impulsive times in [0, 2π). 5. An example In this section, we first give a class of Duffing equations with g being odd and with special impulses, which satisfy the assumptions (A3 ), (A4 ), (A6 ), and then make more precise analysis about the existence of periodic solutions. Consider the second order impulsive differential equation ⎧  x + g(x) = p(t), ⎪ ⎪ ⎨ Δx|t=tj = −2x (tj −) , ⎪ ⎪ ⎩ Δx |t=tj = −2x (tj −) ,

t = tj ; (5.1) j ∈ Z,

where 0 ≤ t1 < 2π, Δx|t=tj = x (tj +) − x (tj −), Δx |t=tj = x (tj +) − x (tj −), g(x), p(t) ∈ C(R, R) and p(t) is 2π-periodic. In addition, we assume that the impulsive time is 2π-periodic, that is, tj+1 = tj + 2π for j ∈ Z. It is equivalent to the system of the form ⎧  x = y, y  = −g(x) + p(t), t = tj ; ⎪ ⎪ ⎨ x(tj +) = −x(tj −), ⎪ ⎪ ⎩ j ∈ Z, y(tj +) = −y(tj −), the autonomous equation of which is

(5.2)

Y. Niu, X. Li / J. Math. Anal. Appl. 467 (2018) 349–370

⎧   ⎪ ⎪ z = w, w = −g(z), ⎨ z(tj +) = −z(tj −), ⎪ ⎪ ⎩ w(tj +) = −w(tj −),

367

t = tj ; (5.3) j ∈ Z.

For consistence, we also use the marks in the previous sections. Let (z(t, z, w), w(t, z, w)) be the solution of (5.3) with (z(0), w(0)) = (z, w) and (x(t, x, y), y(t, x, y)) be the solution of (5.2) with (x(0), y(0)) = (x, y), respectively. We always assume that (A1 ) and (A2 ) hold, and furthermore assume g(x) is an odd function. Under these hypotheses, we find the trajectories of (5.3) in (z, w) plane, V (z, w) = 12 w2 + G(z) = c, are closed curves which are symmetric with respect to the z-axis and w-axis. Under the influence of these special impulses, the solution at impulsive times jumps from (z(tj −), w(tj −)) to the origin-symmetric position (−z(tj− ), −w(tj− )), and hence the motion of (z(t), w(t)) cannot escape the closed curve for t ∈ R. Thus the jumping map is area-preserving and Vc (t1 +) = Vc (t1 −), that is, the assumption (A4 ) holds with O(c−ε1 ) = 0. In the polar coordinates, by choosing Δϕ(tj ) = ϕ(tj +) − ϕ(tj −) = −π, therefore the assumption (A6 ) is valid. Moreover, from the proof of Theorem 3.1, in (3.27) and (3.34) of case ii,

2π+Δta k = −π, ϕ∗ 2π which results that 

Φ(ρ, ϕ) ≤ −2mπ − mαC1 − π, (ρ cos ϕ, ρ sin ϕ) ∈ Γak ; Φ(ρ, ϕ) ≥ −2mπ + mαC1 − π, (ρ cos ϕ, ρ sin ϕ) ∈ Γbk .

(5.4)

Then we have the existence of periodic solutions for Eq. (5.1) in the next theorem. Theorem 5.1. Assume that (A1 ) and (A2 ) hold, then Eq. (5.1) has finitely many 2π-periodic solutions. Furthermore, if π < mαC1 , it has infinitely many 2π-periodic solutions. Proof. If π < mαC1 , by letting β1 = mαC1 + π > 0 and β2 = mαC1 − π > 0 in (5.4), Lemma 3.1 holds and all assumptions of Theorem 4.2 are satisfied. Therefore, applying Theorem 4.2 we obtain infinitely many 2π-periodic solutions for Eq. (5.1). If the condition π < mαC1 is no longer valid, from (5.4) the twist condition cannot be guaranteed. Therefore we need more precise estimation for the second inequality in (5.4). In the polar coordinates, we have ⎧  ρ = ρ cos ϕ · sin ϕ − g(ρ cos ϕ) sin ϕ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ ϕ = − sin2 ϕ − g(ρ cos ϕ) cos ϕ, ρ ⎪ ⎪ ⎪ ρ(t1 +) = ρ(t1 −), ⎪ ⎪ ⎪ ⎪ ⎩ ϕ(t1 +) = ϕ(t1 −) − π.

t = t1 .

(5.5)

Suppose (z, w) = (ρ cos ϕ, ρ sin ϕ) ∈ Γbk , where k is sufficiently large such that bk ≥ c0 . We see that the time in which ϕ(t) has a decrement 2π without the effect of impulses is just τ (bk ). Denote

Y. Niu, X. Li / J. Math. Anal. Appl. 467 (2018) 349–370

368

Φ(ρ, ϕ) = ϕ(2π, ρ, ϕ) − ϕ = −2qπ + ξ, where q ≥ 1 is an integer, and 0 < ξ ≤ 2π. Let tξ denote the time in which ϕ(t) decreases from ϕ − 2qπ + ξ to ϕ − 2qπ. In the following proof, we always assume that during the time interval from 2π to 2π + tξ , there is under no influence of impulses. In the interval [0, 2π + tξ ], ϕ(t) moves q loops, and only one loop is effected by the impulse. Then there exists τ ∧ ∈ (0, τ (bk )) such that (q − 1)τ (bk ) + τ ∧ − tξ = 2π. In fact τ ∧ = 12 τ (bk ) due to the symmetry of Γbk . By the condition (A2 ) and 0 < tξ ≤ τ (bk ), it holds that 2π = (q − 1)τ (bk ) + τ ∧ − tξ ≥



3 2

q−

 τ (bk ) >

   3 2π +α , q− 2 m

(5.6)

provided q > 1. Notice that if q = 1, Φ(ρ, ϕ) = −2π + ξ ≥ −2mπ + ξ.

(5.7)

On one hand for m > 1, Φ(ρ, ϕ) ≥ −2mπ + ξ > −2(m − 1)π = −2mπ + 2π. On the other hand, for m = 1, then q = m = 1 and this case has the same result with (5.11). Next let just consider the case q > 1. It follows (5.6) that q ≤ m. Notice that if m given in (A2 ) equals to 1, then q = 1 which we have discussed. Assume that m > 1. For q ≤ m − 1, we have Φ(ρ, ϕ) = −2qπ + ξ ≥ −2qπ ≥ −2(m − 1)π.

(5.8)

Now if q = m, by (5.6), tξ = −2π + (m − 1)τ (bk ) + τ ∧ = −2π + (m − 12 )τ (bk )   ≥ −2π + (m − 12 ) 2π m +α   = mα − 12 2π m +α . Since ξ > 0, tξ > 12 tξ > 0, then there exists εξ > 0, such that 

1 tξ ≥ max εξ , mα − 2



2π +α m

  Mξ > 0.

(5.9)

By (3.9) and (5.9), we have 2π ξ =

ϕ (s, ρ, ϕ)ds = −

2π+tξ

2π+t  ξ

ϕ (s, ρ, ϕ)ds

(5.10)

2π

≥ −(−C1 tξ ) ≥ Mξ C1 . Thus Φ(ρ, ϕ) = −2qπ + ξ ≥ −2mπ + Mξ C1 . (5.7), along with (5.8) and (5.11) yields

(5.11)

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369

Φ(ρ, ϕ) ≥ −2mπ + β2

(5.12)

where β2 = min{2π, Mξ C1 } > 0. We notice that β2 is related to k since Mξ defined in (5.9) and ξ all depend on Γbk . Then the result in this theorem is different with Theorem 4.2 while Theorem 3.2 and Lemma 4.1 are valid for (5.1). The following proof is similar with the proof of Theorem 4.2 such that we focus on the difference. Instead of all k ≥ n0 , we choose arbitrary finitely many integers k1 , k2 , · · · , kn with ki ≥ n0 , i = 1, 2, · · · n. Correspondingly, for each ki , denote the curve by Γaki , Γbki and the annulus by Aki , Bki . To be precise, in Lemma 3.1, we denote β2 = β2i by (5.12) for each ki and β1 = mαC1 + π by (5.4). Let ε0 = min{β1 , β21 , β22 , · · · , β2n }. Since β1 > 0 and β2i > 0, i = 1, 2, · · · n, then ε0 > 0. By Lemma 4.1, choosing ε = ε0 , we obtain |Θ1 (γ, θ) − Φ1 (γ, θ) − 2mπ| < ε0 , which together with (3.7) yields 

Θ1 (γ, θ) < 0,

(ρ cos ϕ, ρ sin ϕ) ∈ Γaki ;

Θ1 (γ, θ) > 0,

(ρ cos ϕ, ρ sin ϕ) ∈ Γbki .

Lastly, apply Lemma 2.2 on each Aki and Bki (ki ≥ n0 , i = 1, 2, · · · n), we obtain finitely many 2π-periodic solutions. 2 From the proof of Theorem 5.1, we find the finiteness of periodic solutions due to that β2 in (5.12) is not a definite constant. From this point, we can improve Theorem 5.1 with a weaker assumption. Theorem 5.2. Assume that (A1 ) and (A2 ) hold. If mα > solutions. Proof. In fact, if mα >

1 2

 2π m

1 2

 2π m

 + α , Eq. (5.1) has infinitely many 2π-periodic

 + α , by (5.9) and (5.10) in the proof of Theorem 5.1, we have 1 tξ ≥ mα − 2



2π +α m

 >0

and therefore Φ(ρ, ϕ) ≥ −2mπ + β2 ,    where β2 = min{2π, mα − 12 2π C1 } > 0 and β2 is a constant having no relationship with Γbk . Then m +α together with the first inequality in (5.4), we can apply the twist theorem on infinitely many annular regions Ak and Bk (k ≥ n0 ) instead of finite Aki and Bki . The proof of the existence of infinitely many 2π-periodic solutions is same with the proof of Theorem 4.2. 2 Remark 5.3. The condition π < mαC1 is stronger than the condition mα > symmetry of Γbk , t1 + 12 τbk



−π = Δϕ(t1 ) =

1 1 (ϕ) (s, ρ, ϕ)ds ≤ − τbk C1 ≤ − 2 2



1 2

 2π m

 + α . Indeed, by the

 2π + α C1 m

t1

which implies

1 2

 2π m

 +α <

π C1 .

Therefore, by π < mαC1 , we deduce mα >

1 2

 2π m

 +α .

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1 : mα > 2



 2π +α , m

such that Theorem 4.2 also can be proved by the similar proof with Theorem 5.1. Corollary 5.4. Assume that (A1 )–(A6 ) hold with (A5 ) replaced by (A∗5 ), moreover g is an odd function. Then Eq. (4.1) has infinitely many 2π-periodic solutions. References [1] A. Bahri, H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Comm. Pure Appl. Math. 37 (1984) 403–442. [2] D. Bainov, P. Simenov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, Essex, England, 1993. [3] A. Capietto, J. Mawhin, F. Zanolin, A continuation approach for superlinear periodic value problems, J. Differential Equations 88 (1990) 347–395. [4] A. Capietto, J. Mawhin, F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992) 41–72. [5] T. Ding, An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance, Proc. Amer. Math. Soc. 86 (1982) 47–54. [6] T. Ding, Nonlinear oscillations at a point of resonance, Sci. China Ser. A 1 (1982) 1–13. [7] W. Ding, A generalization of the Poincaré–Birkhoff theorem, Proc. Amer. Math. Soc. 88 (1983) 341–346. [8] T. Ding, F. Zanolin, On periodic solutions of sublinear Duffing equations, J. Math. Anal. Appl. 158 (1991) 316–332. [9] T. Ding, F. Zanolin, Periodic solutions of Duffing’s equations with superquadratic potential, J. Differential Equations 97 (1992) 328–378. [10] Y. Dong, Sublinear impulse effects and solvability of boundary value problems for differential equations with impulses, J. Math. Anal. Appl. 264 (2001) 32–48. [11] H. Jacobowitz, Periodic solutions of x + f (x, t) = 0 via the Poincaré–Birkhoff theorem, J. Differential Equations 20 (1976) 37–52. [12] F. Jiang, J. Shen, Y. Zeng, Applications of the Poincaré–Birkhoff theorem to impulsive Duffing equations at resonance, Nonlinear Anal. Real World Appl. 13 (2012) 1292–1305. [13] V. Lakshmikantham, D. Bainov, P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [14] A.C. Lazer, D.E. Leach, Bounded perturbations of forced harmonic oscillations at resonance, Ann. Math. Pures Appl. 82 (1969) 49–68. [15] P. Le Calvez, J. Wang, Some remarks on the Poincaré–Birkhoff theorem, Proc. Amer. Math. Soc. 138 (2010) 703–715. [16] Y. Long, Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. Amer. Math. Soc. 311 (1989) 749–780. [17] J.J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett. 15 (2002) 480–493. [18] J.J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl. 10 (2009) 680–690. [19] Y. Niu, X. Li, Periodic solutions of sublinear impulsive differential equations, Taiwanese J. Math. (2017), online publication. [20] D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer–Leach–Dancer condition, J. Differential Equations 171 (2001) 233–250. [21] D. Qian, L. Chen, X. Sun, Periodic solutions of superlinear impulsive differential equations: a geometric approach, J. Differential Equations 258 (2015) 3088–3106. [22] D. Qian, X. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl. 303 (2005) 288–303. [23] D. Qian, P.J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal. 36 (2005) 1707–1725. [24] C. Rebelo, F. Zanolin, Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities, Trans. Amer. Math. Soc. 348 (1996) 2349–2389. [25] J. Sun, H. Chen, J.J. Nieto, Infinitely many solutions for second-order Hamiltonian system with impulsive effects, Math. Comput. Modelling 54 (2011) 544–555.