We are concerned with the multiplicity of subharmonic solutions of the nonlinear differential equation of the forced relativistic oscillators
where is locally Lipschitz continuous, is continuous and periodic, whose least period is .
The dynamical properties of relativistic oscillators are being studied with an increasing interest because of its extensive applications in different branches of theoretical physics such as quantum mechanics, statistical mechanics, superconductivity theory, nuclear physics and so on (see  -  and the references therein). In  , using variational methods, Brezis and Mawhin proved the existence of a T-periodic solution of the forced relativistic Pendulum
where a is a positive constant and h is a continuous and T-periodic function with mean value . Under same conditions, using Szulkins critical point theory, Bereanu and Torres  proved the existence of a second T-periodic solution of Equation (1.2) which is not different from the previous by a multiple of . When the mean value , using degree arguments, Bereanu and Mawhin  proved that Equation (1.2) has at least two solutions not differing by a multiple of if
For the existence of periodic solutions of Equation (1.1) when g is not periodic, it was proved in  that Equation (1.1) has at least one -periodic solution provided that g satisfies
A natural question is whether Equation (1.1) has multiple periodic solutions when (g1) holds. In the present paper, we shall study this problem. We assume that g still satisfies at most linear condition, i.e. there are two constants such that
(g2) , for every .
By using the generalized Poincaré-Birkhoff fixed point theorem  , we prove the following theorem.
Theorem 1.1. Assume that conditions (g1) and (g2) hold. Then there is an integer such that, for any integer , Equation (1.1) has at least two subharmonic solutions ( ) of order n and these subharmonic solutions extend to the infinity; that is
Throughout this paper, we always use , to denote the real number set and the natural number set, respectively. For the continuous -periodic function , we set .
The rest of the paper is organized as follows. Section 2 presents several preliminary lemmas for the equivalent system of Equation (1.1). Section 3 gives some estimates on the angle variable of the transformed system. Section 4 proves the main conclusion (Theorem 1.1).
2. Basic Lemmas
Firstly, we consider the equivalent planar system of Equation (1.1). Let us set
Then we have
Obviously, is continuously differentiable. Thus Equation (1.1) is equivalent to the system
For any , we denote by the solution of Equation (2.1) satisfying the initial value
Next, we shall perform some phase plane analysis for Equation (2.1). Set
Lemma 2.1. Assume that (g1) holds. Then every solution of Equation (2.1) exists uniquely on the whole t-axis.
Proof. We define a function ,
Obviously, we have
Then we have
Since , for any and is continuous, we know that
which implies that, for any positive constant ,
Therefore, there is no blow-up for the solution on any finite interval . Consequently, exists on the whole interval . Similarly, we can prove that exists on the whole interval . The uniqueness of follows directly from the local Lipschitzian condition of g and the differentiability of .
We now take the transformation
to Equation (2.1) and get the equations for and ,
whenever . Let be the solution of Equation (2.2) through the initial point .
Lemma 2.2. Assume that (g1) and (g2) hold. Then, for any fixed constant , there exist positive constants and such that, for and ,
(1) ; (2) .
Proof. (1) Since and p are bounded, it follows from (g2) that there is a constant such that
It follows that
Obviously, we have that
uniformly with respect to . Consequently, there are constants and such that, for and ,
(2) From (g1) we know that there exist and such that
Therefore, if , then we infer from (2.3) that
Since for any , we have that
Consequently, if , then
On the other hand, if , then we know from the conclusion in (1) that, for large enough, . It follows from (2.4) that
provided that is large enough. From the expression of we know
Furthermore, there exists such that, for ,
Therefore, if is large enough and , then we have
which, together with (2.6), implies that
It follows from (2.4) and (2.7) that, for large enough and ,
The proof of Lemma 2.2 is complete.
Remark 2.3. From the proof of Lemma 2.2 we know that there exists a constant such that, if , , then , , where I is an interval.
Lemma 2.4. Assume that (g1) and (g2) hold. Then, for any , there exists such that, for ,
we have that, for any sufficiently small , there exists such that
Next, we shall estimate the time needed for the solution to pass through each region of , respectively. If , and , , then we get from Lemma 2.2 and (g2) that, for and large enough,
Consequently, we have
Owing to and , for , we obtain
provided that is small enough and is large enough. Similarly, we can prove that the time needed for the solution to pass through each region of is greater than provided that is large enough. Therefore, The conclusion of Lemma 2.4 holds.
Lemma 2.5. Assume that (g1) and (g2) hold. Then for any and , there exists an such that
provided that is large enough.
Proof. Assume by contradicition that there is an integer such that
for any sufficiently large and . We will proceed in two cases.
(1) For , , where is defined in Remark 2.3. In this case, and then is decreasing on the interval . From (2.9) we know that
Therefore, the orbit has a asymptotical ray . If , then as . It follows that, for t large enough, and then
which implies that as . This is a contradicition. Hence, , . Without loss of generality, we assume the asymptotical ray is , , where . If , then we have as . But, it follows from that there is a sufficiently large constant such that for large enough, which implies as . This is a contradicition. If , then as . But, since , we have that for t large enough. Consequently, is bounded from above. Thus we get a contradicition.
(2) There is an such that and , . We next show that there is a large such that, for ,
To this end, we shall construct a continuous counter-clockwise rotating spiral curve , which is injective and makes infinite rotations around the origin . Moreover, the curve satisfies
and every time when the solution of Equation (2.1) intersects with the curve only from the inner part to the outer part. Let us take a positive constant with . We define
Then we have
We now take a large constant such that the curve is a simply closed curve and the circle lies inside this closed curve. Consider the curve
which intersects with the x-axis at exactly two points and with . Then we have
Let us consider the curve
Assume that intersects with the positive x-axis at the point . Then we have
We shall prove that . In fact, since
which implies that for c large enough because is increasing for large enough. Set
Next we consider the curve
Applying the same method as above we can define the curve . Successively, we can construct the curves and ( ). Let us set
We now take a starting point and define the parametrization of in polar coordinates
where denotes the Euclidian norm of a point on , whose argument is s. From the construction of we know that its parametrization is continuous and satisfies (2.10) and makes infinite rotations around the origin as . Moreover, it follows from (2.11) and (2.12) that all solutions cross the curve only from the inner part to the outer part.
For the fixed integer above. Let us take a sufficiently large constant such that the spiral curve ( ) lies inside the circle . If and there is a sufficiently large such that
then the orbit will move clock-wise during the period . Since can cross spiral only from the inner part to the outer part, it will make at least l rotations when it finally reaches the circle . Consequently, we get
which contradicts with (2.9).
3. Estimates on the Angle Variable
When the condition (g1) holds, it was proved in  that Equation (2.1) has at least one -periodic solution.
Let be an -periodic solution of Equation (2.1). We now take a transformation
to Equation (2.1) and get the equations for and ,
Let be the solution of Equation (3.1) satisfying the initial value . From Lemma 2.1 we know that exists on the whole t-axis uniquely. Thus we can define the Poincaré map P of Equation (3.1),
It is well-known that P is an area-preserving homeomorphism.
Obviously, Equation (3.1) has a trivial solution ( ), which corresponds to the -periodic solution . Let be the solution of Equation (3.1) satisfying the initial condition . It follows that for all . Hence, it can be represented by polar coordinates
where and are continuous for all .
Using a similar method as in proving Lemma 2.2, we can prove the following lemma.
Lemma 3.1. Assume that (g1) and (g2) hold. Then there is an such that, if , , then
where I is an interval.
Lemma 3.2. If and m are two positive constants such that
then, for any ,
Proof. The proof follows an argument in  . Since and g is locally Lipschitz continuous, the solutions of Cauchy problems of Equation (3.1) are unique. Therefore, the solution can not go through the origin. Obviously, is increasing. Since , the orbit moves in the clockwise direction when it intersects with the v-axis. Therefore, if the orbit intersects the positive (or the negative) v-axis at the time and intersects subsequently the negative (or the positive) v-axis at the time , then we have
On the other hand, if the orbit stays in the right half-plane (or in the left half-plane) during the time interval , the increase of the angle satisfies
It follows from (3.2) and (3.3) that
Lemma 3.3. Assume that (g1) and (g2) hold. Then, for any , there exists such that, for ,
Proof. We denote by the orbit of the -periodic solution of Equation (2.1) in the -plane. Let be the orbit of the solution of Equation (2.1) satisfying the initial value in the -plane. Consider the moving points
Let be the triangle with the vertices . Obviously, the vector has the argument and the vector has the argument . It follows from Lemma 2.2 that, if is large enough, then is also large enough for and then we have . Furthermore,
Therefore, we have
From Lemma 2.4, Lemma 3.1 and (3.4) we get
Lemma 3.4. Assume that (g1), (g2) hold. Then for any and , there is an such that for and any sufficiently large ,
Proof. We still use some notations in the proof of Lemma 3.3. From the proof of Lemma 2.5 we know that we can enlarge such that and
where is a constant given in Lemma 2.5. Then we have
From Lemma 2.5 and (3.5) we get that, for (or ) large enough,
According to Lemma 3.2, we get that, for any ,
4. Proof of Main Theorem
We first recall a generalized version of the Poincaré-Birkhoff fixed point theorem by Rebelo  .
A generalized form of the Poincaré-Birkhoff fixed point theorem Let be an annular region bounded by two strictly star-shaped curves around the origin, and , , where denotes the interior domain bounded by . Suppose that is an area-preserving homeomorphism and admits a lifting, with the standard covering projection , of the form
where w and h are continuous functions of period in the second variable. Correspondingly, for and , assume the twist condition
Then, F has two fixed points , in the interior of , such that
Proof of Theorem 1.1. According to Lemma 3.4, we can take a prime and a sufficiently large constant ( is defined in Lemma 3.1) such that, every solution of Equation (3.1) with and , satisfies the following property:
(P) There is a constant such that
Since is compact and the solution is continuous dependence on the initial value , we can take a suitable for every solution such that S is bounded from above. Write
Choosing , we infer from Lemma 3.2 that, for any ,
It follows from Lemma 3.3 that there is a sufficiently large constant such that
From (4.2) and (4.3) we know that the n-iteration of the Poincaré map P is twisting on the annulus:
Obviously, is an area-preserving homeomorphism. According to the generalized Poincaré-Birkhoff fixed point theorem, has at least two fixed points ( ), whose polar coordinates are , satisfying
with , . It follows that
are the -periodic solutions of (3.1). Using standard methods as in  and (4.4), we can further prove that is the minimal period. Therefore, are subharmonic solutions of order n of Equation (2.1).
In what follows, we shall prove
Firstly, we prove
Otherwise, there are ( ) such that
Write , , . Set
Obviously, are the -periodic solutions of Equation (3.1) satisfying
Let be the polar coordinates expression of . From the definition of and (4.1), (4.7) we know that, for ,
which contradicits with (4.4) because the orbits of the solutions and are the same.
Secondly, we prove
Otherwise, there are a subsequence and a constant such that, for ,
Since D is compact, it follows from Lemma 3.1 that there is a constant such that, if a solution ( , I is an interval) of Equation (3.1) lies in D, then
Let us denote by the polar coordinates of . Then we get from (4.4) and (4.9) that
This is impossible since as .
Finally, we prove (4.5). Assume by contradicition that (4.5) does not hold. Then we know from (4.8) that there exist a subsequence and a constant such that
Since are -periodic, there are and such that
Using the similar method as in proving Lemma 3.4, we can prove that, for l large enough,
From Lemma 3.1 we know that, for l large enough,
which contradicts with (4.4).
According to (4.5), we know that, for any integer , Equation (2.1) has at least two subharmonic solutions ( ) of order n satisfying
where is defined in section 2, we have
Consequently, we get from (4.10) that
The proof is complete.
*Research supported by National Natural Science Foundation of China, No.11501381.
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