Introduction

In this paper, we investigate the existence of periodic solutions for certain abstract wave equations. We are motivated by the papers of K. Ben-Naoum and J. Mawhin [1], and P.J. McKenna [8], where existence results of periodic solutions are proved for one-dimensional wave equations when the ratio between the space length and the period was irrational. Related equations are also studied by M. Yamaguchi in [10]. We proceeded in this direction in the paper [4]. We studied the equation

$$u_{tt}+Au=\varepsilon f(u,t)\, ,$$ (1)

where $A$ is a self-adjoint, unbounded linear operator with eigenvalues $\lambda _1\le \lambda _2\le \cdots \to \infty$, $\varepsilon$ is a small parameter and $f$ is $T$-periodic in $t\in {\bf R}$. By a $T$-periodic solution of (1) we mean a weak solution specified below. The following results are proved in [4] under additional assumptions on $A,\, f$.

Theorem 1   $($[4]$)$ Assume there exists a constant $c>0$ such that

$$\big \vert\alpha ^2- \frac {m^2}{\lambda _i}\big \vert\ge \frac {c}{\lambda _i}$$ (2)

$$\forall \, m\in {\bf N} ,\quad \forall \, \lambda _i>0\, ,$$

where $\alpha =\frac{T}{2\pi}$. Then $(\ref{eq:1.1})$ has a weak $T$-periodic solution for any $\varepsilon$ small.

Theorem 2   $($[4]$)$ Assume

$$\sum _{\lambda _i>0}\frac {1}{\sqrt {\lambda _i} }<\infty \, .$$

Then the Lebesque measure of the set of all positive $\alpha$ not satisfying $(\ref{eq:1.2})$ is zero.

We also studied the case when $0<\dim \ker A<\infty$. Finally we considered the example

$$u_{tt}-u_{xx}-n^2u=\varepsilon f(u,t)$$ (3)

$$u(t+T,\cdot )=u(t,\cdot )\quad \forall \, t\in S^T$$ (4)

$$u(t,0)=u(t,\pi )=0\quad \forall \, t\in S^T\, ,$$

where $f\: {\bf R} \times S^T\to {\bf R}$ is $C^1$-smooth and globally Lipschitz in $u$, $n\in {\bf N}$. Here $S^T= {\bf R} /[0,T]$ is the circle. The following result is proved in [4].

Theorem 3   $($[4]$)$ The equation $(\ref{eq:1.3})$ has a weak $T$-periodic solution, provided that it holds

$$\inf \limits_{i,m\in {\bf N} ,i>n}\vert i^2-n^2-\omega ^2m^2\vert>0\, ,$$ (5)

where $\omega =1/\alpha$, and there is a $z\in {\bf R}$ such that

$$\begin{eqnarray*} &&\int \limits_0^T\int \limits_0^\pi f(z.\sin nx,t)\sin nx\, d... ...\partial f}{\partial u}(z.\sin nx,t)\sin ^2nx\, dx\, dt\ne 0\, . \end{eqnarray*}$$

The purpose of this paper is two-fold. We firstly release the parameter $\varepsilon$ in (1), so we consider the equation

$$u_{tt}+Au=f(u,t)\, .$$ (6)

We have used in [4] the Banach fixed point theorem. To get our results in this paper, we apply the Leray-Schauder fixed point theorem. For this reason, we need more precision condition than (2), see (2.2) below. We also study the resonant case when $0<\dim \ker A<\infty$. We derive a Landesman-Lazer type result [7]. Finally, we present a forced beam equation as an example.

We secondly investigate more correctly and thoroughly the condition (5) than in [4]. By using some results of the number theory [2,3,6], we derive several conditions for $\omega$ when (5) is either satisfied or not.