Periodic solutions of Liénard equations with singular forces of repulsive type. (English) Zbl 0863.34039

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, V. Bevc, J. L. Palmer, and C. Süsskind, J. British Inst. Radio Engineers 18, 696-708 (1958)]. In 1965, T. Ding proved an existence theorem of \(2\pi\)-periodic solution of (1) when \(a<1/16\), and in 1978, Y. Ye and X. Wang proved the same conclusion when \(a<2/(\pi^2+4) (\approx 0.1442)\).
Using the duality theorems developed recently by A. Capietto, J. Mawhin and 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 \(a\in(0,\chi^*)\).


34C25 Periodic solutions to ordinary differential equations
47H11 Degree theory for nonlinear operators
81V80 Quantum optics
78A60 Lasers, masers, optical bistability, nonlinear optics


Zbl 0718.34053
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