On the existence of periodic oscillations for pendulum-type equations

Abstract: We provide new su cient 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 ∈ C̃T . Ourmain 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.


Introduction
Let us consider the classical pendulum equation 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 −a ≤ s− ≤ s + ≤ a, such that the equation . This is always true in the conservative framework, i.e. when k = , as it was proven by Hamel in [9] and later improved in [16] by adding a second geometrically di erent 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 > there exists e ∈ C T such that (1.1) has not T-periodic solutions. Of course, su cient conditions for the existence of T-periodic solutions for (  Notice that < 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), t into the so-called ϕ-Laplacian equations 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 di erence on the dynamic behaviour between both the classical and relativistic pendulums and the "curvature" pendulum.

Main results
The following is the main result in this paper: a su cient 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. and Then there exist at least two geometrically di erent T-periodic solutions of (1.8), x and x , such that: We remark that condition (2.1) is implicitly assumed in (2.2) and it is trivially ful lled for unbounded operators (that is, when B = +∞).

. The relativistic pendulum
Taking in Theorem 1 we obtain the following existence result for equation (1.4).

Example 4.
For each ϵ > let us consider the relativistic pendulum equation where we have normalized c = .

. The p-laplacian pendulum
Taking now ϕ(z) = |z| p− z for p ≥ 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].

. The classical pendulum
If in Theorem 5 we take p = we obtain an existence result for (1.1) that it is new to the best of our knowledge. Clearly, if e L < π T then (2.10) is satis ed for small enough k, a > . This observation is applied in the following result.

Corollary 7.
Let us x T > and e ∈ C T such that e L < π T . (2.11) Then for any k, a > such that

. The curvature pendulum
Taking now ϕ(z) = z √ + z in Theorem 1 we obtain the following existence result for the "curvature" pendulum. Di erent su cient conditions for the solvability of curvature pendulum equations were given in [1,17,18].

Auxiliary results
By means of the change of variables y = ϕ(x ) + kx, to nd a T-periodic solution of (1.8) is equivalent to solve the following periodic boundary value problem for a rst 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 de ned on the whole plane R . With this idea in mind let us consider the periodic BVP In order to apply Lemma 9 to problem (3.1) the following estimates about its possible solutions are essential. such that x − mπ ∞ ≤ l, for some < l < π and m ∈ Z.
Then, the following estimates hold: Proof. De ne the functionsx = x − mπ andỹ = y − kmπ which satisfy Extendingỹ periodically, if needed, there exists t ∈ R such thatỹ(t ) = ỹ ∞ with |t − t | ≤ T . Without loss of generality let us suppose that t ≤ t . Using the second equation of (3.5) we obtain Finally, from (i), (ii) and the rst equation of (3.5) it follows (iii).

Proof of Theorem 1
By (2.2) there exists l > such that Now, our strategy to nd a T-periodic solution x satisfying (2.3) is to apply Lemma 9 to the homotopic system (3.4) and the set Ω . Notice that by (2.1) we have that Ω ⊂ G := {(x, y) ∈ R : |y − kx| < B}. So it is enough to verity the following two claims.

Claim 1. For any λ ∈ [ , ] and any (x, y) solution of (3.4) such that (x(t), y(t)) ∈ Ω for all t ∈ [ , T], it follows that (x(t), y(t)) ∈ Ω for all t ∈ [ , T].
Since (x(t), y(t)) ∈ Ω for all t ∈ [ , T] then (x, y) satis es the assumptions in Lemma 10 with m = . Then, from estimates (iii) and (ii) we have and so (x(t), y(t)) ∈ Ω for all t ∈ [ , T]. Ω , ( , )) ≠ where f (x, y) = (ϕ − (y − kx), −a sin(x)). Since < l < π and a ≠ , the only zero of f in Ω is ( , ) and then d B (f , Ω , ( , )) is well de ned. Since Ω is symmetric with ( , ) ∈ Ω and f is continuous in Ω and odd, then by Borsuk's theorem it follows that d B (f , Ω , ( , )) is odd. Thus the claim follows.   of the solution tends to in nity 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.

Lemma 12.
Let us consider a ∈ R and H > . If x is any solution of the equation 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 > 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 t , where x attains its global minimum. Then, if x is de ned for any time t > t we have that x (t) > for all t ∈ (t , t ] and since by (5.3) we obtain Thus Let By an analogous reasoning in case t < t we get the desired result. 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.