On the solution of nonlinear equations containing a non-differentiable term
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We approximate a locally unique solution of a nonlinear operator equation containing a non-differentiable operator in a Banach space setting using Newton's method. Sufficient conditions for the semilocal convergence of Newton's method in this case have been given by several authors using mainly increasing sequences [1]-[6]. Here, we use center as well as Lipschitz conditions and decreasing majorizing sequences to obtain new sufficient convergence conditions weaker than before in many interesting cases. Numerical examples where our results apply to solve equations but earlier ones cannot [2], [5], [6] are also provided in this study.
Abstract
We approximate a locally unique solution of a nonlinear operator equation containing a non-differentiable operator in a Banach space setting using Newton's method. Sufficient conditions for the semilocal convergence of Newton's method in this case have been given by several authors using mainly increasing sequences [1]-[6]. Here, we use center as well as Lipschitz conditions and decreasing majorizing sequences to obtain new sufficient convergence conditions weaker than before in many interesting cases. Numerical examples where our results apply to solve equations but earlier ones cannot [2], [5], [6] are also provided in this study.
참고문헌 (6)
- J. Appel, "On the two-step Newton method for the solution of nonlinear operator equations", Math. Nachr, 172, pp.5~14 (1995)
- I.K. Argyros, "On the solution of equations with non dierentiable operators and the Ptak error estimates", BIT, 30, pp.725~754 (1990)
- I.K. Argyros, "A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space", J. Math. Anal. and Appl, 298, pp.374~397 (2004)
- I.K. Argyros, "Computational theory of iterative methods" (2007), Elsevier
- P.P. Zabrejko, "Generalization of the Newton-Kantorovich method for an equation with non-dierentiable operator (Russian)", Ukr. Mat. Zhurn, 34, pp.365~369 (1985)
- P.P. Zabrejko, "The majorant method in the theory of Newton-Kantorovich approximations, and the Ptak error estimates", Numer. Funct. Anal. Optimiz, 9, pp.671~684 (1987)