On the semilocal convergence of Newton–Kantorovich method under center-Lipschitz conditions

Local Convergence for an Efficient Eighth Order Iterative Method with a Parameter for Solving Equations Under Weak Conditions

“…which shows (18) for n = 0 and x 1 ∈ U (x * , r ). By simply replacing x 0 , y 0 , z 0 , x 1 by x k , y k , z k , x k+1 in the preceding estimates we arrive at estimates (13) and (4).…”

Unified convergence domains of Newton-like methods for solving operator equations

“…This problem is very important, since many problems can be formulated to find such a zero x * using Mathematical Modeling [8,[10][11][12]14,18,20] . It is well known that the zero x * can be found in explicit form only in special cases.…”

“…is undoubtedly the most popular iterative method for generating a sequence { x n } approximating x * [1,2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . The most famous semilocal convergence results for Newton's method are the celebrated Newton -Kantorovich theorem and its refinement [8,14,20] .…”

Expanding the Applicability of Secant Method With Applications

“…It has been used in order to find weaker convergence criteria for Newton's method by Argyros in [8]. Gutiérrez et al in [13] give sufficient conditions for Newton's method using both Lipschitz and center-Lipschitz conditions, Magreñán in [18] for the damped Newton's methods and Amat et al in [2,4] or García-Olivo [12] for other methods.…”