Existence of solutions for time second order nonmonotone differential equations involving convex potentials with periodic and anti-periodic boundary conditions
The paper is devoted to the second order ordinary differential equation \[ \left\{\begin{array}{l} u\in W^{2,2}(0,T;\mathbb{R}^N),\\ -u''+u'+A(u)=f(t),\text{ for a.e. }t\in (0,T), \end{array}\right.\tag{1} \] subject to the periodic boundary conditions \[ u(0)=u(T),\quad u'(0)=u'(T),\tag{2} \] or the anti-periodic boundary conditions \[ u(0)=-u(T),\quad u'(0)=-u'(T).\tag{3} \] Here \(A(u)=\nabla_{g(t)}E(u)\) or \(A(u)=Q(t,u,u')^{-1}\nabla \widetilde E(u)\), where \(\nabla_{g(t)}E\) denotes the gradient of a potential \(E\) on \(\mathbb{R}^N\) which is taken with respect to time-dependent inner products \(\langle\cdot,\cdot\rangle_{g(t)}\) on \(\mathbb{R}^N\), and \(\nabla \widetilde E\) denotes the euclidian gradient operator on \(\mathbb{R}^N\), \(Q:[0,T]\times \mathbb{R}^N\times \mathbb{R}^N\to \mathcal{L}(\mathbb{R}^N)\) and \(f:[0,T]\times \mathbb{R}^N\to \mathbb{R}^N\) are given functions. The authors prove the existence of solutions for problems \((1),(2)\) and \((1),(3)\) by using the continuity method, a global inverse functions theorem and the Schaefer fixed point theorem. Based on the Galerkin method and the monotonicity method, they also study the existence of solutions for nonmonotone second order differential equations in infinite dimensional spaces with periodic and anti-periodic boundary conditions. Some applications for partial differential equations involving Dirichlet and Neumann \(p\)-Laplace operators are finally presented.
scientific article; zbMATH DE number 7632572
scientific article