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 $$a\in(0,\chi^*)$$.

MSC:

 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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