Periodic solutions and asymptotic behavior in Liénard systems with \(p\)-Laplacian operators (Q637044)

The paper deals with the existence of \(T\)-periodic solutions to systems of the form \[ (\Phi _p(u'))'+\frac {\roman d}{{\roman d}t}\big (\nabla F(u)\big )+\nabla G(u)=e(t), \] where \(F\in C^2(\mathbb R^N,\mathbb R),\) \(G\in C^1(\mathbb R^N,\mathbb R)\) and \(e\in L^2(0,T).\) In particular, the authors show that the given system has a \(T\)-periodic solution if the following two conditions are satisfied: (i) there are \(r\geq 0\) and sets \(S,S'\) such that \(S\cup S'=\{1,2,\dots ,N\},\) \(S\cap S'=\emptyset ,\) \((\partial G(x)/\partial x_i-\bar {e}_i)x_i\geq 0\) for \(i\in S\) and \(| x_i| \geq r\) and \((\partial G(x)/\partial x_i-\bar {e}_i)x_i\leq 0\) for \(i\in S'\) and \(| x_i| \geq r\); (ii) there is \(m>0\) such that \(\langle F''(x)y,y\rangle\geq m| y| ^2\) for all \(x,y\in \mathbb {R}^N.\) The main tool is the continuation theorem due to \textit{R. Manásevich} and \textit{J. Mawhin} [J. Differential Equations 145, No.~2, 367--393 (1998; Zbl 0910.34051)]. Furthermore, the asymptotic behavior of all solutions is studied and sufficient conditions for their uniform ultimate boundedness are given.