Infinite Subharmonic Solutions of the Forced Relativistic Oscillators ()
1. Introduction
We are concerned with the multiplicity of subharmonic solutions of the nonlinear differential equation of the forced relativistic oscillators
(1.1)
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 [1] - [11] and the references therein). In [1] , using variational methods, Brezis and Mawhin proved the existence of a T-periodic solution of the forced relativistic Pendulum
(1.2)
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 [2] 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 [3] 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 [4] that Equation (1.1) has at least one
-periodic solution provided that g satisfies
(g1)
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 [12] , 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
(2.1)
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
Set
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
,
(2.2)
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
Hence,
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
(2.3)
and
(2.4)
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
(2.5)
provided that
is large enough. From the expression of
we know
Furthermore, there exists
such that, for
,
(2.6)
Therefore, if
is large enough and
, then we have
and
which, together with (2.6), implies that
(2.7)
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
,
Proof. Since
we have that, for any sufficiently small
, there exists
such that
Set
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,
(2.8)
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
(2.9)
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
(2.10)
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
and
Set
and
Then we have
(2.11)
and
(2.12)
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
Set
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
we get
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 [4] 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
,
(3.1)
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 [13] . 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
(3.2)
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
(3.3)
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
(3.4)
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
(3.5)
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 [12] .
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
or
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
Set
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
,
(4.2)
It follows from Lemma 3.3 that there is a sufficiently large constant
such that
(4.3)
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
(4.4)
with
,
. It follows that
are the
-periodic solutions of (3.1). Using standard methods as in [14] 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
(4.5)
Firstly, we prove
(4.6)
Otherwise, there are
(
) such that
Write
,
,
. Set
Obviously,
are the
-periodic solutions of Equation (3.1) satisfying
(4.7)
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
(4.8)
Otherwise, there are a subsequence
and a constant
such that, for
,
Set
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
(4.9)
Let us denote by
the polar coordinates of
. Then we get from (4.4) and (4.9) that
which implies
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
and
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
(4.10)
Since
where
is defined in section 2, we have
Consequently, we get from (4.10) that
Furthermore,
The proof is complete.
NOTES
*Research supported by National Natural Science Foundation of China, No.11501381.