On a continuation lemma for the study of a certain planar system with applications to Liénard and Rayleigh equations (Q1112994)

A continuation lemma is presented for the study of the existence of T- periodic solutions \((T>0)\) to the planar system \[ (1)\quad x'=\Phi (y)- H(x)+E_ 1(t),\quad y'=-\psi (x)+E_ 2(t), \] with \(\Phi\),H,\(\psi\) : \({\mathbb{R}}\to {\mathbb{R}}\) continuous and \(E_ 1,E_ 2: {\mathbb{R}}\to {\mathbb{R}}\) continuous and T-periodic. Then some applications are given to the Liénard equation (2) \(x''+f(x)x'+g(x)=e(t)\) and to the Rayleigh equation (3) \(x''+f(x')+g(x)=e(t),\) where f,g:\({\mathbb{R}}\to {\mathbb{R}}\) are continuous and e: \({\mathbb{R}}\to {\mathbb{R}}\) is continuous and T-periodic. In particular the following theorems are produced. Theorem 1. Assume \((g_ 1)\) \(\lim_{| x| \to +\infty}g(x)sign(x)=+\infty,\) and \[ (g_ 2)\quad (\limsup_{x\to -\infty}g(x)/x)^{-1/2}+(\limsup_{x\to - \infty}g(x)/x)^{-1/2}>T/\pi. \] Then equation (2) has at least one T- periodic solution, for every e(.). Theorem 2. Assume \((g_ 1)\), \((G_ 1)\) \(\limsup_{x\to +\infty}2G(x)/x^ 2<(1/T)^ 2,\) where \(G(x)=\int^{x}_{0}g(u)du,\) and \((f_ 1)\) \(\liminf_{x\to - \infty}f(x)/x\geq 0\) (or \((f_ 2)\) \(\limsup_{x\to -\infty}f(x)/x\leq 0).\) Then equation (3) has at least one T-periodic solution, for every e(.).