Montone enclosure for a class of discrete boundary value problems without monotone nonlinearities (Q1130429)

The paper is devoted to the study of the discrete boundary value problem \[ x_{k+1} =f_k(x_0, x_1, \dots, x_k),\quad 0\leq k\leq N-1,\quad Ax_0 +Bx_N=c,\tag{*} \] where \(f_k:R^{n (k +1)} \to R^n\) is a continuous vector function, \(A\) and \(B\) are known \(n\times n\) matrices, and \(c\in R^n\) is a known vector. The author introduces the concept of a pair of upper and lower solutions of problem (*) and establishes some comparison results. Then, the existence of at least one solution of (*) is studied. A sufficient condition assuring the monotone convergence of a new iterative scheme to the unique solution of problem (*) in some sector is given. This new approach is illustrated by an example.