A unified exhaustive study of monotone iterative method for initial value problems. (Q2783615)

Consider the initial value problem NEWLINE\[NEWLINE\frac{du}{dt}=F(t,u),\quad 0\leq t\leq T,\quad u(0)=u_0\in\mathbb R.\tag{*}NEWLINE\]NEWLINE Under the assumption that \(F\) is a sum of two monotone functions, the authors use the techniques of upper and lower solutions and coupled upper and lower solutions to (*) to construct iteration schemas yielding monotone or alternating sequences which converge to extremal solutions or coupled extremal solutions to (*). They derive also conditions implying that the coupled extremal solutions converge to the unique solution of (*).