Fixed points of increasing operators in ordered space with applications. (Q698938)

This article deals with some modifications of well-known results about minimal-maximal fixed points of increasing operators in ordered spaces. As applications, the following boundary value problem: \[ \ddot x+f(t,x)=0,\quad a<t<b, \] \[ ax(a)-\beta\dot x(a) =0,\;\gamma x(b)+\delta\dot x(b)=0,\;(\alpha,\beta,\gamma, \delta\geq 0, (b-a)\alpha\gamma+ \alpha\delta+ \beta\gamma>0) \] is considered. The author formulates some natural conditions for the existence of minimal-maximal solutions to this problem, assuming that the function \(f(t,\cdot)\) is increasing without any assumptions on continuity, compactness, convexity or concavity.