Periodic solutions of Liénard equations with singular forces of repulsive type
This paper is motivated by the work of seeking positive \(2\pi\)-periodic solutions of the Brillouin electron beam system \[ x''+a(1+\cos t)x={\textstyle {b\over x'}} \qquad \bigl({}'= {\textstyle{d\over dt}}\bigr), \tag{1} \] where \(a\) and \(b\) are positive parameters. The electricians think that (1) has at least one positive \(2\pi\)-periodic solution when \(a<1/4\) [see, \textit{V. Bevc}, \textit{J. L. Palmer}, and \textit{C. Süsskind}, J. British Inst. Radio Engineers 18, 696-708 (1958)]. In 1965, \textit{T. Ding} proved an existence theorem of \(2\pi\)-periodic solution of (1) when \(a<1/16\), and in 1978, \textit{Y. Ye} and \textit{X. Wang} proved the same conclusion when \(a<2/(\pi^2+4) (\approx 0.1442)\). Using the duality theorems developed recently by \textit{A. Capietto}, \textit{J. Mawhin} and \textit{F. Zanolin} [J. Differ. Equations 88, No. 2, 347-395 (1990; Zbl 0718.34053)] the present author studies the existence of positive periodic solutions of a general singular periodic equation in the form \(x''+f(x)x'+ g(t,x)=0\). As an example, he proves the existence of positive \(2\pi\)-periodic solutions of (1) when \(a<\chi_2 (\approx 0.1532)\). It is thus an interesting problem to find the best bound \(\chi^*\), such that (1) has at least one \(2\pi\)-periodic solution whenever \(a\in(0,\chi^*)\).
scientific article; zbMATH DE number 944386