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 -plane.
Let the ends of the string move time-periodically on the -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 on
the -axis and the ends of the string are forced to move periodically
in in the vertical direction.

Let be a time-periodic noncylindrical domain in -plane defined by

(1.1) |

(1.2) |

where and , and , are periodic with period in , and is of order more than or equal to 2 with respect to . and satisfy some compatible boundary conditions (See (A4) later). As a typical example of , if identically vanish, then we give . and are small parameters and are supposed to satisfy ) continuous in . The above dependence of on is naturally imposed because we shall look for the small amplitude solutions and the external force working the whole string is of . We assume that and satisfy . This condition is natural in the sense that the boundaries oscillate with slower speed than the eigenspeed 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 as that of the given data.*

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

(1.3) |

where is an identity function, means the inverse function of and means the composition operation of functions

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

(1.4) |

(1.5) |

where 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 to the length of the interval 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

On the other hand, in our moving-boundary problem (1.1)-(1.2) the difficulty consists in the following. The length of the interval 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 . 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 -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 to the affine function, using the Herman-Yoccoz reduction theorem ([H], [Yoc]) (see Proposition 2.1) :

is the bijection of the noncylindrical domain to a cylindrical domain , maps the boundaries of , , onto the boundaries of , , (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 to BVP (1.1)-(1.2), we shall obtain BVP in the cylindrical domain :

(1.6) |

(1.7) |

where and , and , are -periodic in , and is of order more than or equal to with respect to . Then we shall show the existence of an -periodic solution of BVP (1.6)-(1.7) (Theorem 3.1). In case of , the problem (1.6)-(1.7) was considered by [BN-Ma] and [Mc]. Under some monotonicity conditions and the Lipshitz condition on and the Diophantine condition on the ratio of the length of the interval to the period of , 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

(1.8) |

(1.9) |

(1.10) |

(1.11) |

and nonlinear BVP

(1.12) |

(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, is the -periodic solution of BVP (1.6)-(1.7). Finally, by operating the inverse of the domain transformation to the above , we shall obtain the desired -periodic solution of BVP (1.1)-(1.2).

**Rotation Number.** Let
be one dimensional
periodic dynamical system. This means that is a continuous monotone
increasing function and is an -periodic function. We denote the set
of such functions by .
is the subgroup of
whose elements are of -class.
According to H. Poincaré, the rotation number of
is defined by

**Some Function Spaces.**
Let be a nonnegative integer. Let be an open set in .
Let , and be the usual Lebesgue
space and Sobolev spaces (resp.) with norms
and
. is defined as usual with norm
.
We omit in the norms if there is no confusion. We write
as
.

Let be denoted by . Let be a function space whose elements are defined in , of , -periodic in and have the supports contained in . We denote a set by . Let be the completion of with respect to norm . We define function spaces and in the same way, where is the noncylindrical domain defined by in section 1. In this paper, we write and as and (resp.). All the function spaces , , and are Hilbert spaces with the above norms.

We formulate our main result. Assume the following conditions. Let be an integer .

**(A1)**
, are of and -periodic,
and satisfy
and
for .

**(A2)**
, are of and -periodic.

**(A3)** The rotation number of satisfies
the following Diophantine condition : There exists a positive constant
such that
the Diophantine inequality

**(A4)** is of -class with respect to
and -periodic in . is
of -class with respect to
and -periodic in and satisfies

**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.** satisfying (A4) is written of the form

**Remark 3.** If and are constants, *e.g.*
and
, then we have
and
, whence
and
. This
means that 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.

*Proof.* Note that the rotation number of
is
equal to and the rotation number is conjugate-invariant.
For a given , we define by

Our main theorem is the following.