Time-maps and boundary value problems for ordinary differential equations. (English) Zbl 0987.34014

Zanolin, Fabio

Tricomi’s ideas and contemporary applied mathematics. Proceedings of the international conference held on the occasion of the 100th anniversary of the birth of Francesco G. Tricomi, Rome, Italy, November 28-29 and Turin, Italy, December 1-2, 1997. Rome: Accademia Nazionale dei Lincei, Atti Convegni Lincei. 147, 281-302 (1998).

This expository paper concerns the existence of periodic solutions to the equation (1) \(x''+g(x) =p(t)\), assuming \(g,p:\mathbb{R} \to\mathbb{R}\) are continuous and \(p\) is \(T\)-periodic with \(T>0\), with the help of the map \(\tau: (k_1,k_2)\to \mathbb{R}^+\) (the time map) associated with the homogeneous equation \[ \tau(k)= 2\int_{c_1(k)}^{c_2(k)}\frac {du}{ \sqrt {k^2-2G(u)}}, \tag{1} \] where \(G(u)=\int _0^ug(s) ds\), and \(k_i, c_i(k)\) depend on the particular form of \(g(x)\).
The author discusses the use of time-mapping approaches in the study of various boundary value problems (BVPs) for (1) and provides examples of their applications to periodic BVPs for (1). The time map technique together with the Poincaré-Birkhoff fixed-point theorem proves to be useful in the study of periodic BVPs, as is noted in the last part of the paper. The extensive list of references (92 items) makes the paper a valuable source of information on the subject.
For the entire collection see [Zbl 0948.00034].

Reviewer: Stanisław Sȩdziwy (MR 2001d:34043)

MSC:

34B15

Nonlinear boundary value problems for ordinary differential equations

Keywords:

boundary value problems; ordinary differential equations; time map technique; Poincaré-Birkhoff fixed-point theorem