Abstract
We provide new sufficient conditions for the existence of T-periodic solutions for the ϕ-laplacian pendulum equation (ϕ(x′))′ + k x′ + a sin x = e(t), where e ∈ C͠T. Our main tool is a continuation theorem due to Capietto, Mawhin and Zanolin and we improve or complement previous results in the literature obtained in the framework of the classical, the relativistic and the curvature pendulum equations.
1 Introduction
Let us consider the classical pendulum equation
where
k ≥ 0, a > 0, T > 0 and e ∈ C͠T the set of continuous T-periodic functions with mean value ē :=
The search for T-periodic solutions of (1.1) has been a fruitful subject over the last century, see [12, 13], and it is well-known the following general solvability result: there exist s− = s−(k, a, e) and s+ = s+(k, a, e) with
such that the equation
has a T-periodic solution if and only if s ∈ [s−, s+]. Moreover, whenever s ∈]s−, s+[ there exist at least two geometrically different solutions of (1.2). Surprisingly, it is still an open problem to know if the degeneracy condition s− = s+ can be attained or not.
Following the preceding notation, (1.1) has a T-periodic solution if and only if 0 ∈ [s−, s+]. This is always true in the conservative framework, i.e. when k = 0, as it was proven by Hamel in [9] and later improved in [16] by adding a second geometrically different T-periodic solution.
However, the solvability of (1.1) it is not longer ensured in the presence of a friction term: indeed, it has been proved in [19] that for each k, a, T > 0 there exists e ∈ C͠T such that (1.1) has not T-periodic solutions. Of course, sufficient conditions for the existence of T-periodic solutions for (1.1) are known, such as
or
where
Recently, Brézis and Mawhin, [3, Corollary 1], have proven that the relativistic conservative pendulum, that is
where c > 0 is the speed of the light in the vacuum, a > 0 and e ∈ C͠T, always has a T-periodic solution. Later, Bereanu and Torres, [2, Corollary 1.2], added the existence of a second T-periodic solution. So, in spite of the technical difficulties the solvability issue for (1.3) is analogous to the classical setting. To the contrary, Torres proved in [20] that the forced relativistic pendulum
always has a T-periodic solution for all k, a, T > 0 and e ∈ C͠T provided that
That condition has been improved later in [21, Corollary 3] (see also [7, Remark 1]) and the best bound until now for the right-hand side, up to our knowledge, was obtained in [1, Theorem 1], namely
However it is not know if there exist examples of non-continuation of periodic oscillations for (1.4) for bigger values of the period T.
Recently, it has been proven in [11, Theorem 2.1] the solvability of (1.4) under the following alternative condition to (1.5),
where the constant c* = c*(k, a, T, ∥e∥∞) is implicitly defined by the equation
Notice that 0 < c* < c but that neither (1.5) nor (1.6) are implied by each other, so they are independent.
Both equations, (1.1) and (1.4), fit into the so-called ϕ-Laplacian equations
where
ϕ :] −A, A[⟶] −B, B[ is an increasing and odd homeomorphism with 0 < A, B ≤ +∞.
Note that in the classical pendulum equation (1.1) we have ϕ(z) = z while in the relativistic pendulum (1.4) is
The present paper is organized as follows: in next section we present our main result, several consequences and we discuss their relevance with the literature. In Section 3 we collect the auxiliary results that we will need for the proof of our main result that is postponed until Section 4. Our main tool will be a Capietto-Mawhin-Zanolin continuation theorem given in [4]. Finally, in Section 5 we point out a striking difference on the dynamic behaviour between both the classical and relativistic pendulums and the “curvature” pendulum.
2 Main results
The following is the main result in this paper: a sufficient condition for the existence of multiple T-periodic solutions for equation (1.8). In the particular case of the relativistic pendulum equation (1.4) we improve simultaneously both conditions (1.5) and (1.6), see Corollary 2. It also provides an apparently new solvability condition even for the classical pendulum equation (1.1), see Theorem 6.
Theorem 1
Assume (H0), (H1) and moreover
and
Then there exist at least two geometrically different T-periodic solutions of (1.8), x1 and x2, such that:
We remark that condition (2.1) is implicitly assumed in (2.2) and it is trivially fulfilled for unbounded operators (that is, when B = +∞).
2.1 The relativistic pendulum
Taking
Theorem 2
Let us suppose (H0) and
where
Then equation (1.4) has at least two geometrically different T-periodic solutions satisfying (2.3) and (2.4).
Remark 3
As we have previously noticed condition (2.5) improves simultaneously both conditions (1.5) and (1.6) since 0 < ĉ < c* < c.
Example 4
For each ϵ > 0 let us consider the relativistic pendulum equation
where we have normalized c = 1.
Notice that
then Theorem 2 provides two T-periodic solutions of (2.7) for each 0 < ϵ < 0.0878689.
2.2 The p-laplacian pendulum
Taking now ϕ(z) = ∣z∣p−2z for p ≥ 2 in Theorem 1 we obtain the following existence result for the p-laplacian pendulum. This equation seems to be skipped in the literature and only few references explicitly deal with it, see [15].
2.3 The classical pendulum
If in Theorem 5 we take p = 2 we obtain an existence result for (1.1) that it is new to the best of our knowledge.
2.4 The curvature pendulum
Taking now
3 Auxiliary results
By means of the change of variables y = ϕ(x′) + kx, to find a T-periodic solution of (1.8) is equivalent to solve the following periodic boundary value problem for a first order system
This changes of variables was introduced in [14], inspired by the Liénard plane, and used also in [11]. Notice that in case ϕ is a bounded homeomorphism (that is, B < +∞), the right-hand side of system (3.1) is not longer defined on the whole plane ℝ2. With this idea in mind let us consider the periodic BVP
assuming that
where f : [0, T] × G × [0, 1] → ℝn, G ⊂ ℝn is an open set and f is a Carathéodory function such that for λ = 0 the map f is autonomous, that is, there exists a continuous function f0 : G → ℝn such that
for a.a. t ∈ [0, T] and all z ∈ G.
The following result is just a small modification of [4, Corollary 3] to deal with functions f(t, ⋅,λ) not defined on the whole ℝn.
Lemma 9
Let Ω be a bounded and open subset of ℝn such that Ω ⊂ G and suppose that the following conditions are satisfied:
(“Bound set” condition) For any λ ∈ [0, 1) and any z solution of
such that z(t) ∈ Ω for all t ∈ [0, T], it follows that z(t) ∈ Ω for all t ∈ [0, T];
dB(f0,Ω, 0) ≠ 0, where dB stands for the usual Brouwer degree in ℝn.
Then, problem (3.2) has at least one solution z(t) such that z(t) ∈ Ω for all t ∈ [0, T].
Proof
By the Tietze-Dugundji Theorem, see [22, Proposition 2.1], the function f : [0, T] × Ω × [0, 1] → ℝn admits a continuous extension f̃ : [0, T] × ℝn × [0, 1] → ℝn. Now, we can apply [4, Corollary 3] to obtain a solution z(t) of problem (3.2) with F(t, z) := f̃(t, z, 1) such that z(t) ∈ Ω. Then z(t) is also a solution of (3.2) with F(t, z) := f(t, z, 1).□
In order to apply Lemma 9 to problem (3.1) the following estimates about its possible solutions are essential.
Lemma 10
Let us assume (H0) and (H1). For any λ ∈ [0, 1], let (x, y) a solution of the problem
such that
Then, the following estimates hold:
There exists t0 ∈ [0, T] such that x(t0) = mπ,
∥y − kmπ∥∞ ≤
∥x − mπ∥∞ ≤
Proof
Define the functions x̃ = x − mπ and ỹ = y − kmπ which satisfy
Integrating the second equation over a period we have
where −π < −l ≤ x̃(t) ≤ l < π for all t ∈ [0, T]. Since a ≠ 0 it follows the existence of t0 ∈ [0, T] such that x̃(t0) = 0 which is equivalent to (i).
Now, integrating the first equation over a period we get
and by (H1) it follows the existence of t1 ∈ [0, T] such that ỹ(t1) = kx̃(t1) and then
Extending ỹ periodically, if needed, there exists t2 ∈ ℝ such that ỹ(t2) = ∥ỹ∥∞ with ∣t2 − t1∣ ≤
and thus (ii) is proven (take into account that since e ∈ C̃T then
Finally, from (i), (ii) and the first equation of (3.5) it follows (iii).□
4 Proof of Theorem 1
By (2.2) there exists l > 0 such that
and let us consider the open bounded set in ℝ2
Now, our strategy to find a T-periodic solution x1 satisfying (2.3) is to apply Lemma 9 to the homotopic system (3.4) and the set Ω1. Notice that by (2.1) we have that Ω1 ⊂ G := {(x, y) ∈ ℝ2 : ∣y − kx∣ < B}. So it is enough to verity the following two claims.
For any λ ∈ [0, 1] and any (x, y) solution of (3.4) such that (x(t), y(t)) ∈ Ω1 for all t ∈ [0, T], it follows that (x(t), y(t)) ∈ Ω1 for all t ∈ [0, T].
Since (x(t),y(t)) ∈ Ω1 for all t ∈ [0, T] then (x, y) satisfies the assumptions in Lemma 10 with m = 0. Then, from estimates (iii) and (ii) we have
and so (x(t), y(t)) ∈ Ω1 for all t ∈ [0, T].
dB(f0, Ω1, (0, 0)) ≠ 0 where f0(x, y) = (ϕ−1(y − kx), −a sin(x)).
Since 0 < l < π and a ≠ 0, the only zero of f0 in Ω1 is (0, 0) and then dB(f0, Ω1, (0, 0)) is well defined. Since Ω1 is symmetric with (0, 0) ∈ Ω1 and f0 is continuous in Ω1 and odd, then by Borsuk′s theorem it follows that dB(f0, Ω1, (0, 0)) is odd. Thus the claim follows.
Finally, the proof of the existence of a T-periodic solution x2 satisfying (2.4) is analogous by using the open bounded set
Indeed, the analogous to Claim 1 follows from the estimates provided in Lemma 10 with m = 1. On the other hand, for the analogous to Claim 2 consider the homeomorphism T : Ω1 → Ω2 defined by T(x, y) = (x + π, y + kπ). By the Product Formula (see [8, Theorem 5.1]) we have
since dB(T, Ω1, z) = 1 for any z ∈ Ω2. So we can apply again Borsuk′s theorem to the right hand side of (4.3) since (f0 ∘ T) (x, y) = (ϕ−1(y − kx), a sin(x)) is again a continuous and odd function.
Remark 11
Adding to (H1) the stronger regularity assumptions ϕ ∈ C1(−ϵ, ϵ), with ϵ > 0, and ϕ′(0) ≠ 0 we can compute exactly
5 A counterexample for the “curvature” pendulum
We have already stressed at Introduction that both equations (1.1), with k = 0, and (1.3) has a T periodic solution for any a ∈ ℝ and e ∈ C̃T. On the other hand, the “curvature” pendulum, also called the “sine-curvature” equation,
is a ϕ-laplacian type equation with a bounded operator, namely
Phase plane for (5.2) with the values a = 1 and H = 1.
Lemma 12
Let us consider a ∈ ℝ and H > 0. If x is any solution of the equation
defined on the interval [C, D] then
Proof
Note that (5.2) is an autonomous equation that admits the energy function
which is constant along the solutions of (5.2). Moreover, the potential V(u) := a cos(u) + (∣a∣ + H) u is an increasing homeomorphism from ℝ onto ℝ since V′(u) = −a sin(u) + ∣a∣ + H ≥ H > 0 for all u ∈ ℝ. If x is a solution of (5.2) from a phase plane analysis it follows that x vanishes at only one point, say t0, where x attains its global minimum. Then, if x is defined for any time t1 > t0 we have that x′(t) > 0 for all t ∈ (t0, t1] and since
by (5.3) we obtain
Thus
Let us define u(s) := E0 − V(s) and observe that u(s) ∈ (0, 1] for all s ∈ [x(t0),x(t1)] and moreover u′(s) = −V′(s) ≤ −H < 0. Then,
By an analogous reasoning in case t1 < t0 we get the desired result.□
Remark 13
Lemma 12 means that the maximal interval of definition for any solution of (5.2) is finite. Furthermore, the length of any of those maximal intervals is uniformly bounded by the explicit constant
On the other hand, from [6, Corollary 2] it follows that the Dirichlet problem
has a solution if and only if 0 < H < 2. So, the necessary condition given in Lemma 12 for the existence of a solution of (5.2) on an interval [C, D] is sharp.
Theorem 14
For any T > 0 and a ∈ ℝ, let us consider H >
Then equation (5.1) has not any solution defined on [0, T].
Proof
Clearly, (5.1) can not have a solution defined on [0, T] because in that case (5.2) would have a solution defined on [0, T/2], in contradiction with Lemma 12.□
Corollary 15
Given T > 0, a ∈ ℝ and e ∈ C̃T as in Theorem 14 the equation (5.1) has not T- periodic solutions.
Corollary 15 was obtained by other methods in [18], see also [17], but we stress that Theorem 14 can be applied not only for periodic, but also for any other kind of boundary conditions, like for instance Dirichlet or Neumann.
Acknowledgements
I am grateful to the anonymous reviewers of the manuscript for their comments and for pointing out an inconsistency in a former version of it.
Partially supported by Ministerio de Educación y Ciencia, Spain, and FEDER, Project MTM2017-85054-C2-1-P.
References
[1] C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded Φ–Laplacians, Journal of Dynamics and Differential Equations, 22 (2010), 463-471.10.1007/s10884-010-9172-3Search in Google Scholar
[2] C. Bereanu and P. J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Am. Math. Soc. 140 (2012), no. 8, 2713–2719.10.1090/S0002-9939-2011-11101-8Search in Google Scholar
[3] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations 23, (2010), 801-810.Search in Google Scholar
[4] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc. 329 (1992), no. 1, 41–72.10.1090/S0002-9947-1992-1042285-7Search in Google Scholar
[5] J. Chu, J. Lei and M. Zhang, The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator, J. Differential Equations, 247 (2009), 530-542.10.1016/j.jde.2008.11.013Search in Google Scholar
[6] J. A. Cid and P. J. Torres, Solvability for some boundary value problems with ϕ-Laplacian operators, Discrete Contin. Dyn. Syst., 23 (2009),727-732.10.3934/dcds.2009.23.727Search in Google Scholar
[7] J. A. Cid and P. J. Torres, On the existence and stability of periodic solutions for pendulum-like equations with friction and ϕ-Laplacian, Discrete Contin. Dyn. Syst., 33 (2013), 141–152.10.3934/dcds.2013.33.141Search in Google Scholar
[8] K. Deimling, Nonlinear Functional Analysis, Springer, 1985.10.1007/978-3-662-00547-7Search in Google Scholar
[9] G. Hamel, Ueber erzwungene Schingungen bei endlischen Amplituden, Math. Ann., 86 (1922), 1-13.10.1007/BF01458566Search in Google Scholar
[10] P. Habets and R. L. Pouso, Examples of the nonexistence of a solution in the presence of upper and lower solutions, ANZIAM J., 44 (2003), 591–594.10.1017/S1446181100012955Search in Google Scholar
[11] Q. Liu, L. Huang and G. Jiang, Periodic oscillations of the relativistic pendulum with friction, Electron. J. Differential Equations 2017, 40, 10 pp.Search in Google Scholar
[12] J. Mawhin, Seventy-Five Years of Global Analysis around the Forced Pendulum Equation, Proceedings of the EQUADIFF 9, 115–145, Brno, 1997.Search in Google Scholar
[13] J. Mawhin, Global results for the forced pendulum equation, in Handbook of Differential Equations, Vol. 1 (2004), Elsevier, 533-589.10.1016/S1874-5725(00)80008-5Search in Google Scholar
[14] J. Mawhin and G. Villari, Periodic solutions of some autonomous Liénard equations with relativistic acceleration, Nonlinear Anal. 160 (2017), 16–24.10.1016/j.na.2017.05.001Search in Google Scholar
[15] J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic, Le Matematiche, LXV (2010), 97-107.Search in Google Scholar
[16] J. Mawhin and M. Willem, Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Differential Equations, 52 (1984), 264-287.10.1016/0022-0396(84)90180-3Search in Google Scholar
[17] F. Obersnel and P. Omari, The periodic problem for curvature-like equations with asymmetric perturbations, J. Differential Equations 251 (2011), 1923–1971.10.1016/j.jde.2011.06.014Search in Google Scholar
[18] F. Obersnel and P. Omari, Multiple bounded variation solutions of a periodically perturbed sine-curvature equation, Commun. Contemp. Math. 13 (2011), 863–883.10.1142/S0219199711004488Search in Google Scholar
[19] R. Ortega, E. Serra and M. Tarallo, Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction, Proc. Amer. Math. Soc. 128, no. 9, (2000), 2659–2665.10.1090/S0002-9939-00-05389-2Search in Google Scholar
[20] P. J. Torres, Periodic oscillations of the relativistic pendulum with friction, Physics Letters A, 372 (2008), 6386-6387.10.1016/j.physleta.2008.08.060Search in Google Scholar
[21] P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with ϕ-Laplacian, Communications in Contemporary Mathematics, Vol. 13, No. 2 (2011) 283–292.10.1142/S0219199711004208Search in Google Scholar
[22] E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed-point theorems, Springer-Verlag, New York (1986).10.1007/978-1-4612-4838-5Search in Google Scholar
© 2021 J. Ángel Cid, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
