Periodic solutions of a forced Liénard equation
The author considers the forced Liénard equation \[ x''+ f(t,x,x') x'+ g(x)= e(t,x,x')\tag{1} \] under the following assumptions: \(e(t,x,y)\) is bounded; there is an \(\alpha> 0\) with \(f(t,x,y)\geq \alpha\); \(| g(x)|> | e(t,x,y)|\) for all large \(| x|\); \(g(x)/x\to 0\) as \(| x|\to \infty\); there is a \(T> 0\) with \(e(t+ T,x,y)= e(t,x,y)\) and \(f(t+ T,x,y)= f(t,x,y)\). He proves that under these assumptions (which do not imply \(g(x)\text{sgn }x\geq 0\) as \(| x|\to \infty\)) (1) has a \(T\)-periodic solution. The proof makes use of a fixed point theorem of \textit{H. Schaefer} [Math. Ann. 129, 415-416 (1955; Zbl 0064.357)].
scientific article; zbMATH DE number 711502
scientific article