Existence of nonconstant periodic solutions for a class of second-order systems with \(p(t)\)-Laplacian (Q1689752)

This article deals with the existence of periodic solutions for the \(p(t)\)-Laplacian system \[ (|u'(t)|^{p(t)-2}u'(t))' +\nabla F(t, u) =0, t\in [0, T], \] \[ u(0)-u(T) = u'(0)-u'(T) =0, \] where \(T>0\), \(u\in \mathbb{R}^N\), \(F: [0, T]\times \mathbb{R}^N \to \mathbb{R}\) and \(p(t)\) is a positive continuous \(T\)-periodic function. Under certain assumptions, it is shown that the system has at least one nonconstant \(T\)-periodic solution. Similar results can be found in the literature. It shows that the result obtained here covers some known results as particular cases, thus it extends the existing theories for the system.