Convergence of the Newton–Kurchatov Method Under Weak Conditions

Improving Convergence Analysis of the Newton–Kurchatov Method under Weak Conditions

“…To get the radius of convergence we solve Equation ( 13) and similar one from [8]. Every such equation has two positive solutions.…”

“…Proof. The proof of Theorem 1 is carried out by mathematical induction and is similar to the one in [8] but there are some differences. By Equations (2) and ( 12), for n = 0 we have…”

“…Remark 1. The corresponding Equations ( 6) and (7) to conditions in [8] are given for each x, y, u, v ∈ D by…”

“…For some problems, the nonlinear function can be represented as the sum of the differentiable and nondifferentiable parts. In this case, methods with operator decomposition are often used [1,2,[6][7][8][9]. Numerical examples show that the convergence is faster than difference methods and the Newton-type method [10][11][12].…”

“…Here F is a differentiable operator, G is a continuous operator, D is an open convex set, X and Y are Banach spaces. We use Newton-Kurchatov method [8,[13][14][15][16] for solving Equation (1) numerically…”