A generalized Kantorovich theorem for nonlinear equations based on function splitting (Q2269952)

The article deals with the solvability problem of the equation \[ F(u) = 0, \qquad F = f + g, \] where \(f\) and \(g\) are Fréchet differentiable operators between Banach spaces \(X\) and \(Y\); for the approximate solving this equation, the following generalized Newton-like scheme is proposed: \[ f'(u_m)u_{m+1} + g(u_{m+1}) = f'(u_m)u_m - f(u_m), \qquad m = 0,1,\dots. \] It is assumed that \(F'(u_0)^{-1}\) exists and \[ \|F'(u_0)^{-1}F(u_0)\| \leq a, \] \[ \|F'(u_0)^{-1}[f'(x) - f'(y)]\| \leq M\|x - y\|, \] \[ \|F'(u_0)^{-1}[g'(x) - g'(y)]\| \leq L\|x - y\|, \] \[ \|F'(u_0)^{-1}[F'(x) - F'(u_0)]\| \leq K_0\|x - u_0\|, \] \[ \|F'(u_0)^{-1}[g'(x) - g'(u_0)]\| \leq L_0\|x - u_0\|. \] Under these assumptions inequalities for \(a\), \(K\), \(L\), \(K_0\), \(L_0\) are presented that guarantee the solvability of the equation under consideration and the existence and convergence of the corresponding approximations. Moreover, the ball is described in which the solution exists, and the set, in which the solution is unique. The a priori and a posteriori error estimates are also presented. In the end of the article, an illustrative example (scalar equation) is given.