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Time-maps and boundary value problems for ordinary differential equations. (English) Zbl 0987.34014
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) ${x}^{\text{'}\text{'}}+g\left(x\right)=p\left(t\right)$, assuming $g,p:ℝ\to ℝ$ are continuous and $p$ is $T$-periodic with $T>0$, with the help of the map $\tau :\left({k}_{1},{k}_{2}\right)\to {ℝ}^{+}$ (the time map) associated with the homogeneous equation

$\tau \left(k\right)=2{\int }_{{c}_{1}\left(k\right)}^{{c}_{2}\left(k\right)}\frac{du}{\sqrt{{k}^{2}-2G\left(u\right)}},\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $G\left(u\right)={\int }_{0}^{u}g\left(s\right)ds$, and ${k}_{i},{c}_{i}\left(k\right)$ depend on the particular form of $g\left(x\right)$.

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.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE