Multiple existence of periodic solutions for Liénard system
The authors consider the existence of multiple solutions, to the Liénard system (E) \(x''(t)- {d\over dt} G(x(t))+ f(t, x(t))= e\), (B) \(x(0)= x(2\pi)\); \(x'(0)= x'(2\pi)\), where \(e\in \mathbb{R}^ N\), \(G: \mathbb{R}^ N\to \mathbb{R}^ N\), \(f: \mathbb{R}\times \mathbb{R}^ N\to \mathbb{R}^ N\) are all continuous functions. The authors show that for a sufficiently large constant vector \(R_ 0\in \mathbb{R}^ N\) the equation (E) will have at least \(2^ N\) solutions if \(e> R_ 0\), meaning that each component of \(e\) is greater than the corresponding component of \(R_ 0\). The technique of proof uses the invariance of degree under homotopy argument introduced by J. Mawhin in the late 1970-s. The reviewer comments that the assumption of \(e\) being ``large'' permits one to offer fairly elementary conclusions based on non-standard analysis arguments, if some assumptions are made about \(G'\) and \(f\) for large values of \(t\), or for all values of \(x\).
scientific article; zbMATH DE number 775648