Monotone method for a system of second order periodic boundary value problems
The paper considers the periodic boundary value problem \((*)\quad - u''\!_ i=f_ i(t,u,u'\!_ i),\quad u(0)=u(2\pi),\quad u'(0)=u'(2\pi),\) where \(f\in C([0,2\pi]\times R^ n\times R,R^ n)\) (1\(\leq i\leq n)\). Given two non-negative integers \(p_ i\) and \(q_ i\) such that \(p_ i+q_ i=n-1,\) the vector u is split up as \(u=(u_ i,U_{p_ i},U_{q_ i})\), picking out the \(u_ i\) component. Then the coupled periodic quasi-solutions u, v associated with (*) are defined as satisfying \(-u''\!_ i=f_ i(t,u_ i,U_{p_ i},V_{q_ i},u'\!_ i),\quad -v''\!_ i=f_ i(t,v_ i,V_{p_ i},U_{q_ i},v'\!_ i)\) as well as having period 2\(\pi\). Subject to various conditions on f, it is shown that there are monotonic sequences of solutions of the corresponding differential inequalities which converge to coupled minimal and maximal periodic quasi-solutions. The uniqueness of the solutions of (*) is also investigated.
scientific article; zbMATH DE number 3859527
scientific article