Recurrence relations for a Newton-like method in Banach spaces

Abstract

The convergence of iterative methods for solving nonlinear operator equations in Banach spaces is established from the convergence of majorizing sequences. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton's method [L.B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, 1979] and the third order methods such as Halley's, Chebyshev's and super Halley's [V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing 44 (1990) 169–184; V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Halley method, Computing 45 (1990) 355–367; J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000) 227–236; J.M. Gutiérrez, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997) 171–183; J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the Super–Halley method, Comput. Math. Appl. 7(36) (1998) 1–8; M.A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–445 [10]]. In this paper, an attempt is made to use recurrence relations to establish the convergence of a third order Newton-like method used for solving a nonlinear operator equation F ( x ) = 0 , where F : Ω ⊆ X → Y be a nonlinear operator on an open convex subset Ω of a Banach space X with values in a Banach space Y . Here, first we derive the recurrence relations based on two constants which depend on the operator F. Then, based on this recurrence relations a priori error bounds are obtained for the said iterative method. Finally, some numerical examples are worked out for demonstrating our approach.