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BY 4.0 license Open Access Published by De Gruyter May 27, 2020

On the existence of periodic oscillations for pendulum-type equations

  • J. Ángel Cid EMAIL logo


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 eT. 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.

MSC 2010: 34C25

1 Introduction

Let us consider the classical pendulum equation

x+kx+asinx=e(t), (1.1)


  1. k ≥ 0, a > 0, T > 0 and eT the set of continuous T-periodic functions with mean value ē := 1T0Te(t)dt=0.

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

x+kx+asinx=e(t)+s, (1.2)

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 eT 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

Te2<kπ3,([16, Remark 1]),
e<a,([16, Remark 2]),


T(2T+kπ+δE)<4,([6, Remark 4]),

where δE=maxt[0,T]0te(s)dsmint[0,T]0te(s)ds.

Recently, Brézis and Mawhin, [3, Corollary 1], have proven that the relativistic conservative pendulum, that is

x1x2c2+asinx=e(t), (1.3)

where c > 0 is the speed of the light in the vacuum, a > 0 and eT, 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

x1x2c2+kx+asinx=e(t), (1.4)

always has a T-periodic solution for all k, a, T > 0 and eT 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

2cT<23π. (1.5)

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),

2cT<2π, (1.6)

where the constant c* = c*(k, a, T, ∥e) is implicitly defined by the equation

c=ckTc+3kπ+2(a+e)Tc2+kTc+3kπ+2(a+e)T2. (1.7)

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

ϕ(x)+kx+asinx=e(t), (1.8)


  1. 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 ϕ(z)=z1z2c2. Notice also that other important homeomorphisms, like the p-laplacian ϕ (z) = zzp−2, with p ≥ 2, and the mean curvature operator ϕ(z)=z1+z2 also satisfy (H1).

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

2kπ+aT2+eL12<B, (2.1)


T2ϕ1(2kπ+aT2+eL12)<π. (2.2)

Then there exist at least two geometrically different T-periodic solutions of (1.8), x1 and x2, such that:

π<x1(t)<πforalltRandx1(t1)=0forsomet1[0,T], (2.3)
0<x2(t)<2πforalltRandx2(t2)=πforsomet2[0,T]. (2.4)

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 ϕ(z)=z1z2c2 in Theorem 1 we obtain the following existence result for equation (1.4).

Theorem 2

Let us suppose (H0) and

2Tc^<4π, (2.5)


c^=c(2kπ+aT2+eL12)c2+(2kπ+aT2+eL12)2. (2.6)

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

(x1x2)+ϵ2x+ϵ3sinx=ϵ3sinϵt, (2.7)

where we have normalized c = 1.

Notice that T=2πϵ and limϵ0+T=+, so condition (1.5) is not satisfied for small enough ϵ > 0. On the other hand, since


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) = ∣zp−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].

Theorem 5

Suppose (H0), p ≥ 2 and

T2(2kπ+aT2+eL12)1p1<π. (2.8)

Then equation

(|x|p2x)+kx+asinx=e(t), (2.9)

has at least two geometrically different T-periodic solutions satisfying (2.3) and (2.4).

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.

Theorem 6

Suppose (H0) and

T2(2kπ+aT2+eL12)<π. (2.10)

Then equation (1.1) has at least two geometrically different T-periodic solutions satisfying (2.3) and (2.4).

Clearly, if eL12<2πT then (2.10) is satisfied for small enough k, a > 0. This observation is applied in the following result.

Corollary 7

Let us fix T > 0 and eT such that

eL12<2πT. (2.11)

Then for any k, a > 0 such that


the equation (1.1) has at least two geometrically different T-periodic solutions satisfying (2.3) and (2.4).

2.4 The curvature pendulum

Taking now ϕ(z)=z1+z2 in Theorem 1 we obtain the following existence result for the “curvature” pendulum. Different sufficient conditions for the solvability of curvature pendulum equations were given in [1, 17, 18].

Theorem 8

Suppose (H0) and moreover

2kπ+aT2+eL12<1 (2.12)


T2(2kπ+aT2+eL12)1(2kπ+aT2+eL12)2<π. (2.13)

Then equation

(x1+x2)+kx+asinx=e(t), (2.14)

has at least two geometrically different T-periodic solutions satisfying (2.3) and (2.4).

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

x=ϕ1(ykx),x(0)=x(T),y=asin(x)+e(t),y(0)=y(T). (3.1)

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

z=F(t,z),z(0)=z(T), (3.2)

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 zG.

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 ofn such that ΩG and suppose that the following conditions are satisfied:

  1. (“Bound set” condition) For any λ ∈ [0, 1) and any z solution of

    z=f(t,z;λ),z(0)=z(T), (3.3)

    such that z(t) ∈ Ω for all t ∈ [0, T], it follows that z(t) ∈ Ω for all t ∈ [0, T];

  2. dB(f0,Ω, 0) ≠ 0, where dB stands for the usual Brouwer degree inn.

Then, problem (3.2) has at least one solution z(t) such that z(t) ∈ Ω for all t ∈ [0, T].


By the Tietze-Dugundji Theorem, see [22, Proposition 2.1], the function f : [0, T] × Ω × [0, 1] → ℝn admits a continuous extension : [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) := (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

x=ϕ1(ykx),x(0)=x(T),y=asin(x)+λe(t),y(0)=y(T). (3.4)

such that


Then, the following estimates hold:

  1. There exists t0 ∈ [0, T] such that x(t0) = ,

  2. ykmπ kl+aT2+eL12,

  3. x maxt[0,T]x(t)mint[0,T]x(t)T2ϕ1(2kl+aT2+eL12).


Define the functions = x and = yk which satisfy

x~~=ϕ1(y~kx~),x~(0)=x~(T),y~=asin(x~+mπ)+λe(t),y~(0)=y~(T). (3.5)

Integrating the second equation over a period we have


where −π < −l(t) ≤ l < π for all t ∈ [0, T]. Since a ≠ 0 it follows the existence of t0 ∈ [0, T] such that (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 (t1) = kx̃(t1) and then


Extending periodically, if needed, there exists t2 ∈ ℝ such that (t2) = ∥ with ∣t2t1∣ ≤ T2. Without loss of generality let us suppose that t1t2 . Using the second equation of (3.5) we obtain


and thus (ii) is proven (take into account that since eT then 0Te+(s)ds=0Te(s)ds and ∥eL1 = 20Te+(s)ds).

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

Ω1={(x,y)R2:|x|<l,|y|<kπ+aT2+eL12}. (4.1)

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 Ω1G := {(x, y) ∈ ℝ2 : ∣ykx∣ < B}. So it is enough to verity the following two claims.

  1. 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].

  2. dB(f0, Ω1, (0, 0)) ≠ 0 where f0(x, y) = (ϕ−1(ykx), −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

    Ω2={(x,y)R2:|xπ|<l,|ykπ|<kπ+aT2+eL12}. (4.2)

    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 : Ω1Ω2 defined by T(x, y) = (x + π, y + ). By the Product Formula (see [8, Theorem 5.1]) we have

    dB(f0,Ω2,(0,0))=dB(f0T,Ω1,(0,0))dB(T,Ω1,z)=dB(f0T,Ω1,(0,0)), (4.3)

    since dB(T, Ω1, z) = 1 for any zΩ2. So we can apply again Borsuk′s theorem to the right hand side of (4.3) since (f0T) (x, y) = (ϕ−1(ykx), a sin(x)) is again a continuous and odd function.

Remark 11

Adding to (H1) the stronger regularity assumptions ϕC1(−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 eT. On the other hand, the “curvature” pendulum, also called the “sine-curvature” equation,

(x1+x2)+asinx=e(t), (5.1)

is a ϕ-laplacian type equation with a bounded operator, namely ϕ(z)=z1z2(1,1). This fact leads to the following dramatic difference with respect to the classical/relativistic pendulum: for each T > 0 and a ∈ ℝ there exists eT such that equation (5.1) has not any solution defined on the whole interval [0, T]. In particular, for such eT the equation (5.1) has not T periodic solutions in clear contrast with the classical and the relativistic settings. The key difference is that the kinetic energy of the “curvature” pendulum, that is Ekin(x):=11+x2, is a priori bounded independently of the solution (quite surprinsingly, even if the velocity of the solution tends to infinity its kinetic energy remains bounded!). The approach in this section is inspired on some previous ideas developed in [10] to obtain counterexamples to the existence of solutions between well-ordered lower and upper solutions without the so-called Nagumo condition.

Fig. 1 
Phase plane for (5.2) with the values a = 1 and H = 1.
Fig. 1

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

(x1+x2)+asinx=|a|+H, (5.2)

defined on the interval [C, D] then DC<2H.


Note that (5.2) is an autonomous equation that admits the energy function

E(u,v):=11+v2+acos(u)+(|a|+H)u, (5.3)

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∣ + HH > 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

E(x(t),x(t))=E(x(t0,x(t0))):=E0for all t[t0,t1],

by (5.3) we obtain

x(t)=1(E0V(x(t)))2E0V(x(t))for all t[t0,t1].



Let us define u(s) := E0V(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 2H. To the contrary, note that for −2∣a∣ ≤ H ≤ 0 the equation (5.2) admit constant solutions, thus defined on the whole real line.

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 > 4T and eT such that


Then equation (5.1) has not any solution defined on [0, T].


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 eT 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.


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.


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Received: 2019-07-22
Accepted: 2020-03-06
Published Online: 2020-05-27

© 2021 J. Ángel Cid, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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