Existence and uniqueness of solutions of higher-order antiperiodic dynamic equations (Q2388402)

The authors prove a theorem on existence and uniqueness of solutions of an \(n\) th order nonlinear dynamic equation \[ u^{\Delta^n} + \sum_{j=1}^{n-1} M_j u^{\Delta^j} = f(t,u) \] with antiperiodic boundary conditions \[ u^{\Delta^i} (a) = - u^{\Delta^i} (\sigma(b)), \quad 0 \leq i \leq n-1. \] The operators \(\Delta\) and \(\sigma\) refer to the general setting of time scales, which embraces the cases of ordinary differential equations, ordinary difference equations, and \(q\)-difference equations. The proof contains several steps: Solutions of the linear boundary value problem by Green's functions, the nonlinear problem is solved by using the notion of coupled lower and upper solutions in connection with the Arzela-Ascoli theorem. In a final section the theorem is applied to first and second order equations for the three classical cases mentioned above.