Introduction

In the study of the motions of nonlinear vibrating string with periodically oscillating ends, it seems to be interesting to investigate under which conditions periodic motions exist.

In this paper, we shall consider an oscillating string of finite length in the $(x,u)$-plane. Let the ends of the string move time-periodically on the $(x,u)$-plane and a nonlinear time-periodic vertical external force work on the string. We shall be concerned with the existence of the time-periodic motions of the vibrating string under small vertical external forces. This problem is mathematically formulated as the existence problem of periodic solutions of the Dirichlet boundary value problem for one-dimensional wave equation with a time-periodic nonlinear forcing term, where the boundaries oscillate periodically in $t$ on the $x$-axis and the ends of the string are forced to move periodically in $t$ in the vertical direction.

Let $\Omega$ be a time-periodic noncylindrical domain in $(x,t)$-plane defined by

$$a_1(t) < x < a_2(t), \quad t \in R^1.$$

Here $a_1(t)$ and $a_2(t)$ are periodic functions. The period is normalized to $1$, for simplicity. Consider BVP (the boundary value problem) for a nonlinear one-dimensional wave equation :

$$\partial_t^2 \,u - \partial_x^2 \,u = \mu p(x,t) + f(x,t,u), \,\,\, (x,t) \in \Omega,$$ (1.1)

$$u(a_1(t),t) = \nu b_1(t), \quad u(a_2(t),t) = \nu b_2(t), \quad t \in R^1,$$ (1.2)

where $p(x,t)$ and $f(x,t,u)$, and $b_i(t),\, i = 1,2$, are periodic with period $1$ in $t$, and $f(x,t,u)$ is of order more than or equal to 2 with respect to $u$. $p(x,t)$ and $f(x,t,u)$ satisfy some compatible boundary conditions (See (A4) later). As a typical example of $f$, if $b_i(t),\, i = 1,2,$ identically vanish, then we give $f(x,t,u) = \pm u^m\,(m \ge 2)$. $\mu$ and $\nu$ are small parameters and are supposed to satisfy $\nu = \nu(\mu) = O(\mu) \,(\mu \to 0$) continuous in $\mu$. The above dependence of $\nu$ on $\mu$ is naturally imposed because we shall look for the small amplitude solutions and the external force working the whole string is of $O(\mu)\,(\mu \to 0)$. We assume that $a_1(t)$ and $a_2(t)$ satisfy $\vert\,a_i'(t)\,\vert < 1$ $(i = 1,2)$. This condition is natural in the sense that the boundaries oscillate with slower speed than the eigenspeed $1$ of waves by (1.1). Otherwise, the shock waves come out.

The aim of this paper is to show the existence of time-periodic solutions with small amplitude of BVP (1.1)-(1.2) with the same period $1$ as that of the given data.

We define the following composed function $A$ that is a fundamental tool in this research. Let $A$ be a composed function defined by

$$A = A_1^{-1} \circ A_2, \quad A_i = (I + a_i) \circ(I - a_i)^{-1}, \quad i = 1,2,$$ (1.3)

where $I$ is an identity function, $f^{-1}$ means the inverse function of $f$ and $\circ$ means the composition operation of functions i.e. $f \circ g(x) = f(g(x))$. Geometrically $A$ is a map naturally defined by the reflected characteristics in the $(x,t)$-plane. $A$ is one dimensional periodic dynamical system. It is known in a series of works ([Ya1]-[Ya4], [Ya6]) that $A$ and its rotation number $\rho(A)$ play an essential role in studying the qualitative behavior of solutions of IBVP and BVP in domain with periodically oscillating boundaries. For the definition of the rotation number, see Notation and Definitions in this section.

For the case where the ends of the string are fixed, BVP is of the form

$$\partial_t^2 \,u - \partial_x^2 \,u = F(x,t,u), \,\,\, (x,t) \in (0,a) \times R^1,$$ (1.4)

$$u(0,t) = u(a,t) = 0, \quad t \in R^1,$$ (1.5)

where $a$ is a positive constant. In this case there are very many works on the existence of time-periodic solutions of BVP (1.4)-(1.5) (see [R1][R2][B-C-N][W] etc. and see the references therein). It should be noted that the ratio of the period of the forcing term $F(x,t,u)$ to the length $a$ of the interval $[\,0,a \,]$ plays an important role in the study of the behavior of the solution. That is, the behaviors depend on the rationality or irrationality of the ratio. As is shown in [Ya8], even in the linear case i.e., $F(x,t,u) = F(x,t)$ in (1.4) it happens that there are no bounded solutions, as a matter of course, no periodic solutions of (1.1)-(1.2) if the Diophantine order of the irrational ratio is large and the differentiability of $F(x,t)$ is small. It is known that if the Diophantine order of a real number is large, the number is well-approximated by the rational numbers.

On the other hand, in our moving-boundary problem (1.1)-(1.2) the difficulty consists in the following. The length of the interval $[\,a_1(t),a_2(t)\,]$ varies continuously as time varies continuously. Hence the ratio takes both rational and irrational values as time proceeds. However, this difficulty is essentially overcome by introducing the rotation number of $A$. In a series of papers ([Ya4], [Ya6] and [Ya-Yo]) we clarified the interesting fact that the rotation number plays the same role as the length of the interval as the ends are fixed.

We shall show that under the Diophantine condition on the rotation number (See the assumption (A3) in this section) there exists a small $1$-periodic solution of BVP (1.1)-(1.2) (Theorem 1.1). It is well-known in number theory ([Kh]) that all real numbers with periodic continued fraction expansions satisfy the above Diophantine condition. Especially the set of all algebraic numbers of degree 2 is equal to the above set.

Our steps to show the results on the existence of periodic solutions are as follows. First we shall reduce the function $A$ to the affine function, using the Herman-Yoccoz reduction theorem ([H], [Yoc]) (see Proposition 2.1) :

$$H\circ A\circ H^{-1}(x) = x + \omega.$$

Here $\omega$ is the rotation number of $A$ and $H$ is a conjugate function that is one-dimensional periodic dynamical system of $C^\infty$. Then, using the conjugate function $H$, we shall construct a domain transformation $\Phi : R^2 \to R^2$ in section 2 :

$$\xi = \left(H\circ A_1^{-1}(x + t) - H(-x + t)\right)/2,$$

$$\tau = \left(H\circ A_1^{-1}(x + t) + H(-x + t)\right)/2.$$

$\Phi$ is the bijection of the noncylindrical domain $\Omega$ to a cylindrical domain $D = (0,\omega/2) \times R^1$, maps the boundaries of $\Omega$, $x = a_1(t)$, $x = a_2(t)$ onto the boundaries of $D$, $\xi = 0$, $\xi = \omega/2$ (resp.) and preserves the d'Alembertian form (Proposition 2.2). The last statement means that the transformed differential operator contains only d'Alembertian but has no lower order differential operators. Such transformations were developped in [Ya4], [Ya6] and [Ya-Yo]. It should be noted that the above d'Alembertian preserving property has good advantage to study the qualitative behavior of the solutions. Second, applying the domain transformation $\Phi$ to BVP (1.1)-(1.2), we shall obtain BVP in the cylindrical domain $D$ :

$$\partial_\tau^2 \, v - \partial_\xi^2 \, v = \mu q(\xi,\tau) + g(\xi,\tau,v), \,\,\, (\xi,\tau) \in D,$$ (1.6)

$$v(0,\tau) = \nu c_1(\tau), \quad v(\omega/2,\tau) = \nu c_2(\tau), \quad \tau \in R^1,$$ (1.7)

where $q(\xi,\tau)$ and $g(\xi,\tau,v)$, and $c_i(\tau),\,i = 1,2$, are $1$-periodic in $\tau$, and $g(\xi,\tau,v)$ is of order more than or equal to $2$ with respect to $v$. Then we shall show the existence of an $1$-periodic solution of BVP (1.6)-(1.7) (Theorem 3.1). In case of $c_i(t) \equiv 0,\,i = 1,2$, the problem (1.6)-(1.7) was considered by [BN-Ma] and [Mc]. Under some monotonicity conditions and the Lipshitz condition on $g$ and the Diophantine condition on the ratio of the length of the interval to the period of $g$, they showed the existence of periodic weak solution.

To show our results, first we shall decompose BVP (1.6)-(1.7) into two linear homogeneous BVPs

$$\partial_\tau^2 \,v_1 - \partial_\xi^2 \,v_1 = 0, \,\,\, (\xi,\tau) \in D,$$ (1.8)

$$v_1(0,\tau) = c_1(\tau), \quad v_1(\omega/2,\tau) = 0, \quad \tau \in R^1,$$ (1.9)

$$\partial_\tau^2 \,v_2 - \partial_\xi^2 \,v_2 = 0, \,\,\, (\xi,\tau) \in D,$$ (1.10)

$$v_2(0,\tau) = 0, \quad v_2(\omega/2,\tau) = c_2(\tau), \quad \tau \in R^1,$$ (1.11)

and nonlinear BVP

$$\partial_\tau^2 \,w - \partial_\xi^2 \,w = \mu q(\xi,\tau) + g(\xi,\tau,\nu(v_1 + v_2) + w), \,\,\, (\xi,\tau) \in D,$$ (1.12)

$$w(0,\tau) = 0, \quad w(\omega/2,\tau) = 0, \quad \tau \in R^1.$$ (1.13)

Then we shall show the existence of periodic solutions of BVP (1.8)-(1.9) and (1.10)-(1.11) (Proposition 3.1), using the method of [Ya3]. In order to show the existence of a periodic solution of BVP (1.12)-(1.13), we shall apply the standard contracting mapping principle in suitable function space to our BVP (1.6)-(1.7). This is similar to the existence theorem ([Ya5], pp.519-521) of periodic solutions of nonlinear evolution equations of second order. Then by the principle of superposition, $v = \nu(v_1 + v_2) + w$ is the $1$-periodic $C^2$ solution of BVP (1.6)-(1.7). Finally, by operating the inverse $\Phi^{-1}$ of the domain transformation $\Phi$ to the above $v$, we shall obtain the desired $1$-periodic solution of BVP (1.1)-(1.2).

Notation and Definitions.

Rotation Number. Let $F(x) = x + G(x)$ be one dimensional periodic dynamical system. This means that $F(x)$ is a continuous monotone increasing function and $G(x)$ is an $1$-periodic function. We denote the set of such functions $F$ by $D(T^1)$. $D^\infty(T^1)$ is the subgroup of $D(T^1)$ whose elements are of $C^\infty$-class. According to H. Poincaré, the rotation number $\rho(F)$ of $F \in D(T^1)$ is defined by

$$\rho(F) = \lim_{n \to \pm \infty} \frac{F^n(x) - x}{n},$$

where $F^n(x)$ is the $n$-th iterate of $F(x)$. It is well-known ([H]) that $\rho(F)$ is independent of $x$ and the convergence is uniform with respect to $x$. As we regard $\rho(F)$ as a functional of $F$, $\rho(F)$ is continuous with respect to $\sup_{0 \le x \le 1}\vert F(x)\vert$. Note that the rotation number has the conjugate-invariant property. Namely, one has the following identity

$$\rho(F) = \rho(\phi \circ F \circ \phi^{-1})$$

for any $\phi \in D(T^1)$. Since clearly the rotation number of $R_\alpha(x) = x + \alpha$ ($\alpha$ : constant) is equal to $\alpha$, it follows that $\rho(\phi \circ R_\alpha \circ \phi^{-1}) = \alpha$ for any $\phi \in D(T^1)$. For more details of the rotation numbers, see [H].

Some Function Spaces. Let $s$ be a nonnegative integer. Let $G$ be an open set in $R^n$. Let $L^2(G)$, $H^s(G)$ and $H_0^s(G)$ be the usual Lebesgue space and Sobolev spaces (resp.) with norms $\vert\cdot\vert _{L^2(G)}$ and $\vert\cdot\vert _{H^s(G)}$. $C^s(G)$ is defined as usual with norm $\vert\cdot\vert _{C^s(G)}$. We omit $G$ in the norms if there is no confusion. We write $\vert\cdot\vert _{C^0}$ as $\vert\cdot\vert _{C}$.

Let $(0,\omega/2) \times R^1$ be denoted by $D$. Let $D_0^\infty(D)$ be a function space whose elements $f(x,t)$ are defined in $D$, of $C^\infty(D)$, $1$-periodic in $t$ and have the supports contained in $D$. We denote a set $(0,\omega/2) \times (0,1)$ by $D_0$. Let $K^s_0(D)$ be the completion of $D_0^\infty(D)$ with respect to norm $\vert\cdot\vert _{H^s(D_0)}$. We define function spaces $D_0^s(\Omega)$ and $K_0^s(\Omega)$ in the same way, where $\Omega$ is the noncylindrical domain defined by in section 1. In this paper, we write $H^s(D_0)$ and $L^2(D_0)$ as $H^s(D)$ and $L^2(D)$ (resp.). All the function spaces $K^s_0(D)$, $H^s(D)$, $K_0^s(\Omega)$ and $H^s(\Omega)$ are Hilbert spaces with the above norms.

Main Theorem

We formulate our main result. Assume the following conditions. Let $s$ be an integer $\ge 4$.

(A1) $a_i(t),\,i = 1,2$, are of $C^\infty$ and $1$-periodic, and satisfy $a_1(t) < a_2(t)$ and $\vert\,a_i'(t)\vert < 1$ for $t \in R^1$.

(A2) $b_i(t),\, i = 1,2$, are of $C^\infty$ and $1$-periodic.

(A3) The rotation number $\omega$ of $A$ satisfies the following Diophantine condition : There exists a positive constant $C$ such that the Diophantine inequality

$$\vert n - m/\omega\vert \ge C n^{-1}$$

holds for all $n,\,m \in N$.

(A4) $p(x,t)$ is of $C^s$-class with respect to $(x,t) \in \bar{\Omega}$ and $1$-periodic in $t$. $f(x,t,u)$ is of $C^{s+2}$-class with respect to $(x,t,u) \in \bar{\Omega} \times R^1$ and $1$-periodic in $t$ and satisfies

$$f(x,t,0)= \partial_u f(x,t,0) = 0.$$

$p(x,t)$ and $f(x,t,u)$ satisfy compatible boundary conditions :

$$p(a_i(t),t) = 0,\quad i = 1,2,$$

holds for all $t \in R^1$, and there exists a positive constant $\nu_0$ such that for any $\nu$ with $\vert\nu\vert \le \nu_0$,

$$f(a_i(t),t,\nu b_i(t)) = 0,\quad i = 1,2,$$

holds for all $t \in R^1$.

Remark 1. It is well-known in number theory ([Kh]) that all numbers with periodic continued fraction expansion satisfy (A3). Note that the set of all algebraic numbers of degree 2 coinsides with the above set.

Remark 2. $f(x,t,u)$ satisfying (A4) is written of the form

$$f(x,t,u) = u^2 r(x,t,u),$$

where $r(x,t,u)$ is of $C^s$-class with respect to $(x,t,u) \in \bar{\Omega} \times R^1$. As an example of $f(x,t,u)$ that satisfies the compatible boundary condition in (A4), we can take $r(x,t,u)$ with $r(a_i(t),t,u) = 0,\, i = 1,2$, for all $(t,u) \in R^1_t \times R^1_u$. $f$ possibly depends on the parameter $\nu$. As such an example we give $r(x,t,u) = R(x,t,(u-\nu b_1(t))(u-\nu b_2(t)))$, where $R(x,t,U)$ satisfies $R(a_i(t),t,0) = 0$ for all $t \in R^1$.

Remark 3. If $a_1(t)$ and $a_2(t)$ are constants, e.g. $a_1(t) \equiv a$ and $a_2(t) \equiv b$, then we have $A_1(t) = t + 2a$ and $A_2(t) = t + 2b$, whence $A(t) = t + 2b - 2a$ and $\rho(A) = 2b - 2a$. This means that $\rho(A)/2$ is equal to the length of the interval.

The existence of the boundary functions that satisfy both of an analytical condition (A1) and a number-theoretic condition (A3) is assured by the following proposition.

Proposition 1.1   Let $\omega$ be any real number. Then there exists $1$-periodic $C^\infty$ functions $a_i(t),\,i = 1,2$, such that $\rho(A) = \omega$. Here $A$ is defined by (1.3).

Proof. Note that the rotation number of $R_\alpha(x) = x + \alpha$ is equal to $\alpha$ and the rotation number is conjugate-invariant. For a given $\omega$, we define $A_\phi$ by

$$A_\phi(x) = \phi \circ R_\omega \circ \phi^{-1}(x)$$

for $\phi \in D^\infty(T^1)$. Then clearly $\rho(A_\phi) = \omega$. For simplicity, we set $a_1(t) \equiv 0$. Then $A_1 = I$ and $A_\phi = A_2$. We set $a_2(t) = a(t)$, for simplicity. Then by (1.3) we have an equality

$$(I + a) \circ (I - a)^{-1}(t) = A_\phi(t) \equiv (I + B_\phi)(t),$$

where $B_\phi(t)$ is an 1-periodic $C^\infty$ function and satisfy $\vert B_\phi'(t)\vert < 1$. Then setting $y = (I - a)^{-1}(t)$, we have

$$a(y) = (1/2) B_\phi \circ (I - a)(y).$$

We consider a function $G(y,a) = a - (1/2) B_\phi(y - a)$ of $C^\infty$-class with respect to $(y,a)$ and apply the implicit function theorem to a functional equation $G(y,a) = 0$. Since $B_\phi$ satisfies $\vert B_\phi'(t)\vert < 1$ for any $t \in R^1$ and hence we have

$$\vert G_a(y,a)\vert \ge 1 - (1/2) \vert B_\phi'(y - a)\vert > 1/2 > 0,$$

this functional equation has a $C^\infty$ solution $a(y)$.         Q.E.D.

Our main theorem is the following.

Theorem 1.1   Assume (A1), (A2), (A3) and (A4). Then there exists a positive constant $\mu_0$ such that for any $\mu$ satisfying $\vert\mu\vert < \mu_0$ BVP (1.1)-(1.2) has an $1$-periodic solution of $C^2$-class with respect to $(x,t) \in \bar{\Omega}$.