Applications of variational methods to an anti-periodic boundary value problem of a second-order differential system (Q2409617)

The authors prove the existence of multiple solutions of the following anti-periodic problem: \[ x''(t)+M\, x(t)+\nabla\,F(t,x(t)) =0, \; \text{a.e. \(t \in [0,T]\)}, \quad x(0)=-x(T), \; x'(0)=-x'(T), \] where \(x=(x_1,x_2, \ldots,x_n)^T \in R^n\), \(M\) is a \(n \times n\) real symmetric matrix, and \(F:[0,T] \times R^n \to R\) is measurable in \(t\) for any \(x \in R^n\) and continuously differentiable in \(x\) for a.e. \(t \in [0,T]\). In particular, the existence of at least two solutions is proved when the eigenvalues of the matrix \(M\) are less than \(\pi^2/T^2\). Moreover, the existence of \(2T\)-periodic solutions is also deduced. The results follow from variational methods and critical point theory.