On modification of Samoilenko's numerical-analytic method of solving boundary value problems for difference equations (Q686348)

The author studies some modification of Samoilenko's numerical analytic method in relation to the difference boundary value problem \(x(n) = f(x(n))\), \(n\in \{0,1,\dots,m-1\}\), \(Cx(0) + Dx(m) = d\), in any linear space equipped with a \(k\)-dimensional vector norm. He presents conditions which allow to state when the auxiliary equation \[ x(n) = x_ 0 + \sum^{n-1}_{i=0}f(x(i)) + \omega(n)[\varphi(x_ 0) + \sum^{m- 1}_{i=0}f(x(i))],\quad n\in\{0,1,\dots,m-1\} \] possesses a unique solution and this solution can be obtained by the method of successive approximations. Some generalizations are presented.