Electronic Journal of Qualitative Theory of Differential Equations

We consider a relativistic particle under the action of a time-periodic central force field in the plane. Assuming an attractive type condition on some neighborhood of infinity there are many subharmonic and quasiperiodic motions. Moreover, the obtained information allows to give applications for many scalar problems involving the relativistic operator.


Introduction and main results
The motion of a particle subjected to the influence of an (autonomous) central force field in the plane may be mathematically modelled as a system of differential equations and it has had a great importance in Mechanics from the very beginning of this discipline in the seventeenth century.The central force field is determined by the function f and, from a physical point of view, we can have: 1. Attractive force fields: f is called an attractive force field whenever f (r) < 0 for all r > 0. For instance, letting f (r) = c/r 2 for some negative constant c < 0, the gravitational force created by a point mass fixed at the origin is attractive.This is the case of the well-known Kepler problem.
2. Repulsive force fields: f is called a repulsive force field whenever f (r) > 0 for all r > 0. For example considering f (r) = c/r 2 for some positive constant c > 0 gives the Coulombian force field created by an electrical charged particle fixed at the origin which works with another one with the same charge.
3. Mixed force fields: In this case f is considered positive at some levels and negative in others.The central force field can model the force field created by a charged particle which changes the sign depending on the position with respect to the origin.
These force fields are autonomous, i.e., they depend on the position but not directly on the time.However, Newton [18], in his study about Kepler's 2nd law had already considered the motion of a particle subjected to a periodic sequence of discrete time impulses.On the other hand, many problems involve particles which vary depending on the time they have been considered.
For example, the Gylden-Meshcherskii problem ẍ = M (t) x |x| 3  x ∈ R 2 \ {0}, even though it was originally proposed to explain the secular acceleration observed in the Moon's longitude, nowadays has many physical interpretations.Amongst others, it can be regarded as a Kepler problem with variable masses (M (t) = −G • m 1 (t) • m 2 (t), G is the gravitational constant and m 1 (t), m 2 (t) are the masses of the bodies) or a Coulombian problem with charged particles changing of sign (M (t) = κ•q 1 (t)•q 2 (t), κ is the Coulomb constant and q 1 (t), q 2 (t) are the charges of the particles).Mainly, the Gylden-Meshcherskii problem is used in the framework of the Kepler problem to describe a variety of phenomena including the evolution of binary stars, dynamics of particles around pulsating stars and many others (see [3,10,19,20] and the references there); but also it could be used as a Coulombian problem to study phenomenons such as: the stabilization of matter-wave breather in Bose-Einstein condensates, the propagation of guided waves in planar optical fibers, the electromagnetic trapping of a neutral atom near a charged wire (see [7] for the scalar versions).When dealing with particles moving at speed close to that of light it may be important to take into account the relativistic effects.Relativistic Dynamics is theoretically founded in the context of Special Relativity (see for instance [12,Chapter 33]), and the relativistic Kepler or Coulomb problem has been considered in previous works [1,6,17].However, it seems that for the most general non-autonomous center field force is still very unexplored, in this line we can cite the recent paper [22].When the mass of our particle at rest and the EJQTDE, 2013 No. 31, p. 2 speed of light are normalized to one, we are led to consider the following family of second-order systems in the plane: Here, f : R × (0, ∞) → R, f = f (t, r) is assumed to be Caratheodory and Tperiodic in the time variable t (i.e.f (t, •) : (0, ∞) → R is continuous for almost every t ∈ R, f (•, r) : R → R is T −periodic and medible for every r > 0 and for every fixed positive numbers a, b the function max r∈[a,b] |f (t, r)| is locally integrable on R).Notice however that it may be singular at r = 0. Solutions of (1) are understood in a classical sense, i.e., a C 1 function x : R → R 2 is a solution provided that and the equality (1) holds on almost everywhere.
In this paper we are interested in finding a certain class of functions which include the T −periodic ones but also some subharmonic and quasi-periodic solutions.To introduce this class of functions it will be convenient to use the polar coordinates and rewrite in complex notation each continuous function x : R → R 2 \{0} ≡ C\{0} as x(t) = r(t)e iθ(t) , where r(t) = |x(t)| and θ : R → R is some continuous determination of the argument along of the function x.We say that x is T −radially periodic whenever r(t) is T −periodic and there exists a real number ω such that θ(t) − ωt is T −periodic.This number was introduced in [22] as the rotational (or angular velocity) of x and denoted as rot(x).
For instance, the T -radially periodic function x : R → R 2 \ {0} is T -periodic if and only if rot x is an integer multiple of 2π/T .If rot x = (m/n) (2π/T ) for some relatively prime integers m = 0 = n then x will be subharmonic with minimal period nT .
Finally, if rot x 2π/T is irrational (and |x| is not constant) then x will not be periodic of any period and instead it will be quasi-periodic with two frequencies ω 1 = 2π T ; ω 2 = rot x.This is easy to check, since x(t) = r(t)e iθ(t) can be decomposed on the product of the T -periodic r(t)e i(θ(t)−rot(x)t) and the 2π/ rot x−periodic e i rot(x)t .
Some illustrative examples in order to understand better the above concepts may be seen in [11] and [22], especially in this last one there is a graphic representation with the meaning of the concept (see [22, Figure 1]).
It is well-known that if our force field is globally repulsive, i.e., has no T −radially periodic solutions.This fact is easily checkable multiplying in (1) by x and integrating on [0, T ].However, when our force field is attractive at some level r * > 0 and autonomous (it does not depend on t), i.e., f (r * ) < 0, EJQTDE, 2013 No. 31, p. 3 then an easy computation proves that there are T −radially periodic solutions of (1) with constant angular velocity equal to In addition to the above trivial results, recently we have proven that if we consider an attractive continuous (non-autonomous) force field at some level then there exists many infinitely T −radially periodic solutions of (1).
The above-mentioned results applied to the relativistic Gylden-Meshcherskii problem d dt imply that M must be negative in all R, which is too much restrictive for some type of physical problems above considered.The main objective in this paper is overcoming this restriction.For that we will prove the following statement.
Theorem 1 Assume the existence of r * > 0 such that Then either x is a T −rad.periodic solution of (1) = (0, ∞) or there exist T −rad.periodic solutions of (1) with angular velocity equal to 0. Now, according to Theorem 1 it can be proven that M := T 0 M (s)ds < 0 implies the existence of T −rad.periodic solutions of (2).In fact, M < 0 is a necessary and sufficient condition for the existence of non-constant T −rad.periodic solutions of (2).
Immediately the following questions arise from Theorem 1: what type of solutions are obtained in Theorem 1?, do T −periodic solutions of (1) exist?
The following theorem gives such answers.
In particular, taking ω = (2π)/(nT ) for some natural number large enough we can find the existence of sub-harmonic solutions having as minimal period a multiple of T .On the other hand, putting ω = (2π/T )s for some irrational number s we obtain the existence of infinitely many quasi-periodic orbits of (1).EJQTDE, 2013 No. 31, p. 4 At this moment, we would like to analyze the advantages and disadvantages of our results with respect to the known ones in the actual literature, not only in the relativistic case but also in the classical one.The only analytical result known for us assuming relativistic effects is [22,Theorem 1.1].Obviously, both results are independent and complementary.In both results, the singularity does not play any role, and the T −rad.periodic solutions are from the same nature, they rotate around the origin with very small angular velocity.Moreover, in both results an attractiveness condition is required on f .In the actual Theorems 1-2 we need something less than attractiveness on f , but, in contrast with the result proved in [22], such condition must be assumed not only at some fixed level but, at some neighborhood at infinity.However, the obtained information here is major and it will allow, since it will be seen, to get consequences even for scalar equations.With regard to the result in the classical case, we can find important differences.For example, on no account can be obtained analogous Theorems 1-2 for this case, because f must fulfil some sub-linear requirement in order to avoid the resonance phenomenon.
Finally, we point out that Theorems 1-2 also can become false if our particle is restricted to be on a line instead of on the plane.In this case problem ( 1) is reduced to Now T −rad.periodic solutions of ( 1) are reduced to T −periodic solutions of (3) (understanding by T −periodic solution, in the scalar case, a positive T −periodic function r in C 1 ); for that, obviously, if our force field is globally attractive, by contradiction, integrating in (3) on the period of some possible T −periodic solution r, it is proven that (3) cannot have any T −periodic solution.This implies, according to Theorem 1, that under the global attractiveness of f , for every positive number r there exists a T −rad.periodic solution of (1) x r such that min t∈R |x r (t)| = r.On the contrary, if one would know that there exists some level r 0 > 0 such that (1) has no any T −rad.periodic solutions then Theorem 1 provides the existence of T −periodic solutions of (3).On this idea is based our next theorem: Theorem 3 Under the assumption of Theorem 1.If there exists at level r 0 > 0 such that ) has at least one (positive) T −periodic solution.
Intuitively, in order to get some efficient conditions guaranteeing the existence of (positive) T −periodic solutions of (3), it is necessary that f is attractive in some neighborhood of infinity.Therefore, Theorem 3 can be applied to many equations, for example to where q 1 , q 2 are non-negative locally integrable and periodic functions, e is only locally integrable and periodic and γ, δ are positive constants.This type of equations can be important from the physical point of view for the studying process as trapless 3D Bose-Einstein condensate taking into account relativistic effects (see for the classical case [16,Section 5]).In addition, we do not know any analytical result from literature on them.As a particular case it can be considered the classic equation of Lazer and Solimini (case q 2 ≡ 0) with a weak type singularity (i.e.γ ∈ (0, 1)).In [4] it proved that, in this particular case, in addition of the necessary assumption T 0 e(s)ds < 0, more requirements must be assumed, but it was not possible to find them.Since it does not require any difficulty to apply Theorem 3 to study this type of equations, it will be convenient only to indicate that the above method works in order to avoid too many trivial arguments.
As a short and concrete application of Theorem 3, we will study the wellknown Mathieu-Duffing equations with relativistic effects.The obtained results can be compared with [5], where we got independent conditions using a variational approach; but the solutions could be non-positive (for more results about Mathieu-Duffing equations in the classical case see [21] and the references therein).
Actually, it will be important to recall some basic concepts on the equation (1).Let x = re iθ(t) be written in polar coordinates, it is a T −rad.periodic solution of (1) if and only if there exist a real number µ ∈ R (the relativistic angular moment) and (r, p) a T −periodic solution of ṙ = rp here p = ṙ/ 1 − ṙ2 − r 2 θ2 is called the relativistic linear momentum.Moreover, the rotational of any T −rad.periodic solution of (1) can be computed in terms of r, p and µ: The above process is reversible, i.e., if (r, p; µ) is a T −periodic solution of (HS) then x = re iθ , where θ is any primitive of µ/(r(t) µ 2 + r 2 (t) + r 2 (t)p 2 (t)), is a T −rad.periodic solution of (1).Taking into account this property we will equivalently use the system (HS) in order to study the equation (1) (see [22] for more details).
The paper is structured as follows: In section 2 some a priori bounds for the T −periodic solutions of (HS) are obtained.In Section 3, by using the continuation arguments of Leray-Schauder degree, our main Theorems 1-2 will be proven.In the last section, Section 4, we will apply Theorem 1 to study the existence of periodic solutions in the scalar case.An important example will illustrate our results.EJQTDE, 2013 No. 31, p. 6

A priori bounds
It is well-known that the fundamental principle on which is based the Relativistic Mechanic is: the particles cannot travel faster than light (which in our model is assumed to be 1).This basic principle implies the existence of bounds on the variation of T −rad.periodic solutions of (1), both in the angular and the radial components.More precisely: Lemma 1 Let (r, p; µ) a T −periodic solution of (HS).Then, Proof.The first part of the proof is obtained using that the oscillation of any T −periodic and continuously differentiable function r is bounded by ṙ ∞ T /2, i.e., it fulfils max t∈R r(t) − min t∈R r(t) ≤ ṙ ∞ T /2 (see [4,Lemma 6]).This elementary estimation will be frequently used in the presented paper.On the other hand, the definition (4) implies (b).
The main hypothesis of Theorems 1-2 is the attractiveness (in average) of the force field at someplace far from the origin, i.e., there exists r * > 0 such that Under this assumption one easily checks that any T −periodic solution (r, p; µ) with min t∈R r(t) ≥ r * has (relativistic) angular momentum µ = 0.
Proof.We use an argument by contradiction, we assume that there is (r, p; µ) a T −periodic solution of (HS) with min t∈R r(t) ≥ r * and µ = 0 (in view of (4) it is equivalent to assume that rot(r, p; µ) = 0).In particular, from (HS) we deduce that the T −periodic function r fulfils the equation (3).By integrating on the period of r and using (5) we get a contradiction.
The next goal will be to find some a priori estimates on the (relativistic) linear and angular momentum (p and µ) for every T −periodic solution (r, p; µ) of (HS) contained on some annular region.In order to proof the second part we integrate on the period of p the second equation of (HS) obtaining Assuming that |µ| ≥ 1 (on otherwise the proof is completed), the above identity allows to check This inequality will be used in the next section, so that it will be convenient to recall it.
Theorem 2bis.Under the assumption of Theorem 1bis., there exists ω * > 0 with the following property: for every ω ∈ (−ω * , ω * ) \ {0} there is some T −periodic solution (r, p; µ) of (HS) with rot(r, p; µ) = ω.This section will be destined to prove these results.For that, it will be useful to consider the following change of variable: Here r, p belong to the Banach spaces With regard to the new variables ( r, p and µ), (HS) can be rewritten as where are the Nemitskii operators defined by and Ω is the natural open subset of R × Y for which everything is well-defined, in this case Ω := {(λ, r, p; µ) ∈ R × Y : λ > 0, min t∈R r(t) > −1}.
We point out that whenever (r, p; µ) is a T −periodic solution of (HS) then, taking λ = r(0) and defining r, p by ( 7), ( r, p; µ) will be a T −periodic solution of ( HS); and vice versa, if there exist λ > 0 and ( r, p; µ) a T −periodic solution of ( HS) then (r, p; µ) defined by (7) will be a T −periodic solution of (HS) with r(0) = λ.
The key to prove Theorems 1bis-2bis will be the next result.
Proposition 1 Assume (5).Then there exists a connected subset C of the T −periodic solutions of ( HS) verifying and one of the following conditions: EJQTDE, 2013 No. 31, p. 9 The connection of C refers to the topology of R × Y .We postpone the proof of Proposition 1 to the end of the section; at this moment let us see how it can be used in order to obtain Theorems 1bis-2bis.
Let C be the connected set given by Proposition 1.Notice that the set C 1 := {(λ(1 + r), λ p; µ) : (λ, r, p; µ) ∈ C} is a subset of the T −periodic solutions of (HS) when λ > 0, because of the change of variable done in (7).Moreover, Proof of Theorem 1bis.The proof is concluded rewriting Proposition 1 using the change of variable (7) and the connected set C 1 .
It is clear from (HS) that whenever (r, p; µ) is a T −periodic solution of this system, (r, p; −µ) is another one; furthermore rot(r, p; µ) = − rot(r, p; −µ).This fact has the following consequence: for studying Theorem 2bis is sufficient to check that there exists ω * > 0 with the property that for any ω ∈ (0, ω * ) there is a T −periodic solution (r, p; µ) of (HS) with rot(r, p; µ) = ω.This will be our next goal.
At this moment it only remains to show Proposition 1.With this aim we rewrite ( HS) in an abstract way.The 1-dimensional subspace of C(R/T Z) composed by the constant functions will be identified with R; we use this identification in order to define the projections on this subspace Π, Q : C(R/T Z) → C(R/T Z): x(s)ds.
For any x ∈ Ker Q we denote by Kx to the primitive of x vanishing at t = 0, T , and the linear operator K : Ker Q → Ker Π defined in this way is compact.Taking into account the definitions of Nemitskii operator we can rewrite ( HS) as a fixed point problem (depending on a parameter) defined on suitable open set of Y y = F [λ; y], the (non-linear) operator F : Ω → Y is given by (we denote I to the identity operator of C(R/T Z)).We point out that F is completely continuous, i.e., it is continuous and maps bounded sets of R × Y whose closure is contained in Ω into relatively compact subsets of Y .
Before we prove Proposition 1, we compute the Leray-Schauder degree of Lemma 4 Assume the conditions of Proposition 1 and U λ an open set chosen before.Then F [λ, •] has no fixed point on ∂U λ and Proof.First, we check that F [λ; y] = y for any y ∈ ∂U λ .Indeed, this set can be divided in three parts non-disjoints: the set of the elements ( r, p; µ) ∈ Y such that r ∞ = T /(2λ), the set of elements ( r, p, µ) ∈ Y such that p ∞ = (P (λ − T /2) + 1)/λ or the set of elements ( r, p; µ) such that either µ = 0 or µ = M (λ−T /2).Recalling that whenever we define r, p as in (7), then F [λ; y] = y implies that (r, p; µ) is a T −periodic solution of (HS) with r(0) = λ ≥ λ 2 .
In order to prove the result we consider a homotopy to some fixed point problem whose associated operator has degree known and different from zero.Since the image of U λ by H[0; •] is contained on R 3 .By linking of ( 8) with Theorem 8.7 in the page 59 of [9] we see that where deg B denotes the Brouwer degree.Observe that for some suitable positive numbers a 1 , a 2 , and We are led to consider the functions ϕ : and their cartesian product The usual properties of the degree imply since ϕ and ψ change from negative to positive on their domain (see (6) with a = λ − T /2).It concludes the proof.
Proof of Proposition 1.We choose λ * a fixed number such that the bounded open set U λ * ⊂ Y is well define and is possible to apply Lemma 4, i.e., the degree is well defined and it fulfils deg Under these assumptions the classical Leray-Schauder continuation theorem ( [13], see also [2,8,14,15]) provides the existence of a connected set C composed by the elements (λ, r, p; µ) ∈ R × Y such that ( r, p; µ) is a T −periodic solution of ( HS) for that λ.Using the change of variable (7), Lemmas 1, 2 and 3 imply (i) and one of the following conditions: 1. {λ : (λ, r, p; µ) ∈ C} = (0, ∞).

inf{min
In order to finish the proof we will see that if (i a ) does not happen then (i b ) happens.Indeed, that (i a ) does not happen implies the existence of some constant k > 0 such that min t∈R r(t) ≥ k for any (r, p; µ) ∈ C 1 ; in particular EJQTDE, 2013 No. 31, p. 12 contradicting 1..Other consequence of the before one is . cannot be fulfilled.On the other hand, 3. do not hold.Indeed, r is clearly bounded; in addition, since k ≤ min t∈R r(t) < r * + T /2, with the same argument of Lemma 3 is proven that p = λ p and |µ| are uniformly bounded (they depend only on k and r * + T /2), so that 3. cannot be fulfilled.The only possibility is 4., but it is exactly the same that (i b ).
4 Applications to the existence of periodic solutions for the scalar case In this section, we will see how our Theorem 1 can be used in order to guarantee the existence of at least one T −periodic solution for the equation (3).
Since it has been mentioned in the introduction, our hypothesis ( 5) is not sufficient to guarantee the T −periodic solvability of (3).This is explained if one considers a global attractive function f , it fulfils hypothesis (5) but (3) has no periodic solutions.Therefore, in order to study the T −periodic solvability of (3) will be necessary, in some sense, that f is repulsive on somewhere, i.e., we will assume that there exists r 0 > 0 such that Now, under the hypothesis ( 5) and ( 9) we can guarantee the existence of at least one T −periodic solution for (3) (see Theorem 3).
Notice that if 2) happens, such solutions are T −periodic solutions of (HS) with the form (r, p; µ = 0), in particular r will be a T −periodic solution of (3).On the contrary, if (1) happens, we can choose a T −rad.periodic solution of ( 1 In order to prove the applicability of Theorem 3 we could consider many types of equations, even equations whose solvability is still unknown.In order to compare the results we will only concern the Mathieu-Duffing type equations, i.e., the equations like where q, p are locally integrable functions, with positive range for p, and both are T −periodic ones. It will be convenient to introduce a new notation which will be only used for this part.We denote P := T 0 p(s)ds, Q + := T 0 q + (s)ds, Q − = T 0 q − (s)ds, and Q := Q + − Q − ; here q + (s) := max{q(s), 0}, q − (s) = min{−q(s), 0}.With the previous notation we present the following results.
Let us define f (t, r) = q(t)r − p(t)r 3 .Because of P > 0 we can prove (5).On the other hand Notice that the function ξ(r) = rQ − (r + T 2 ) 3 P − T Q− 2 will attain its global maximum at r 0 , so that r 0 is the best election.Moreover, due to (11) it follows that ξ(r 0 ) > 0, i.e., (9) holds.Now, the results is proven applying Theorem 3. EJQTDE, 2013 No. 31, p. 14 In most of the cases, in literature is considered a particular case of (10), usually the considered equation is where b 1 , b 2 and c are real numbers; i.e., the particular case of (10) when q(t) = b 1 + b 2 cos t and p(t) = c.In this type of problems is usual to find 2π−periodic solutions (i.e.T = 2π).This will be done as example of applicableness of Corollary 1 (see [5,21]).

Lemma 3
For every a > 0 there exist P > 0 and M > 0 (depending only on a and f ) such that p ∞ ≤ P, |µ| < M, for any T −periodic solution (r, p; µ) of (HS) such that r(t) ∈ [a, a + T ].EJQTDE, 2013 No. 31, p. 7Proof.Let (r, p; µ) be any T −periodic solution of (HS) with r(t) ∈ [a, a + T ].We define t * ∈ [0, T ], t * ∈ [t * , t * + T ] the points where p attains its global maximum and minimum respectively.Therefore, by integrating on [t * , t * ] the second equation of (HS) we can prove the first part of the statement, i.e., a+T ] |f (t, r)|dt =: P.

Fixed any positive number
a, according to Lemma 3 we can define P (a) := T 0 max r∈[a,a+T ] |f (t, r)|dt and the function M (a) := (1+a) 3 + (a + T ) 2 1 + (a + T ) 2 + (a + T ) 2 P 2 (a) T T 0 max r∈[a,a+T ] |f (t, r)|dt in the way that p ∞ ≤ P (a) and |µ| < M (a) for any T −periodic solution (r, p; µ) of (HS) with r(t) ∈ [a, a + T ].Moreover, there exists λ 2 > 0 large enough such that the set of the continuous T −periodic functions on R) respectively.To simplify the notation, we will write Y := C 0 (R/T Z) × C(R/T Z) × R, the Banach space induced by the classical norm, i.e., if y = ( r, p; µ) ∈ Y we will write y = r ∞ + p ∞ + |µ|.

EJQTDE, 2013
No. 31, p. 10 Let us define λ 2 > r * in such a way that (6) holds and let consider the family of bounded open subset of Y U

Example 1 1 + b 2 ,
Assume b 1 > 0 and c > 0. If πb the equation (12) has at least one (positive) 2π−periodic solution.Obviously, by symmetry, it has another solution with different sign.