Newton-Kantorovich theorem for a family of modified Halley's method under Hölder continuity conditions in Banach space

Abstract In this study, under p -Holder continuity conditions a family of new modified Halley’s method with (2 +  p )-order is proposed in Banach space which is used to solve the nonlinear operator equations. The Newton–Kantorovich convergence theorem for the family of new modified Halley’s method is established under Holder continuity conditions. The error estimate also is gotten. Finally, two examples are provided to show the application of our theorem.

[1]  Chong Li,et al.  Convergence of the family of the deformed Euler-Halley iterations under the Hölder condition of the second derivative , 2006 .

[2]  José Antonio Ezquerro,et al.  A modification of the super-Halley method under mild differentiability conditions , 2000 .

[3]  Miguel Ángel Hernández,et al.  Indices of convexity and concavity. Application to Halley method , 1999, Appl. Math. Comput..

[4]  José Antonio Ezquerro,et al.  On the R-order of the Halley method , 2005 .

[5]  Miguel Ángel Hernández,et al.  An acceleration of Newton's method: Super-Halley method , 2001, Appl. Math. Comput..

[6]  Wu Qing-biao,et al.  The convergence ball of the Secant method under Hölder continuous divided differences , 2006 .

[7]  Chong Li,et al.  Convergence of the variants of the Chebyshev-Halley iteration family under the Hölder condition of the first derivative , 2007 .

[8]  Ioannis K. Argyros,et al.  Concerning the “terra incognita” between convergence regions of two Newton methods , 2005 .

[9]  José M. Gutiérrez,et al.  Recurrence Relations for the Super-Halley Method , 1998 .

[10]  Miguel Ángel Hernández,et al.  Chebyshev method and convexity , 1998, Appl. Math. Comput..

[11]  Herbert H. H. Homeier.  A modified Newton method with cubic convergence: the multivariate case , 2004 .

[12]  Chong Li,et al.  Convergence of Newton's Method and Uniqueness of the Solution of Equations in Banach Spaces II , 2003 .

[13]  Ioannis K. Argyros,et al.  A note on the Halley method in Banach spaces , 1993 .

[14]  Qingbiao Wu,et al.  Mysovskii-type theorem for the Secant method under Hölder continuous Fréchet derivative , 2006 .

[15]  Sunethra Weerakoon,et al.  A variant of Newton's method with accelerated third-order convergence , 2000, Appl. Math. Lett..

[16]  Ioannis K. Argyros.  On the comparison of a weak variant of the Newton-Kantorovich and Miranda theorems , 2004 .

[17]  Yitian Li,et al.  On modified Newton methods with cubic convergence , 2006, Appl. Math. Comput..

[18]  José M. Gutiérrez,et al.  New recurrence relations for Chebyshev method , 1997 .

[19]  M. A. Salanova,et al.  Modification of the Kantorovich assumptions for semilocal convergence of the Chebyshev method , 2000 .

[20]  Ioannis K. Argyros.  The super-Halley method using divided differences , 1997 .

[21]  Ioannis K. Argyros,et al.  On the Newton-Kantorovich hypothesis for solving equations , 2004 .

[22]  Yitian Li,et al.  Modified Halley's method free from second derivative , 2006, Appl. Math. Comput..

[23]  Filomena Cianciaruso,et al.  Estimates of majorizing sequences in the Newton-Kantorovich method : A further improvement , 2006 .

[24]  José Antonio Ezquerro,et al.  Halley's method for operators with unbounded second derivative , 2007 .

[25]  Qingbiao Wu,et al.  Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space , 2006, Appl. Math. Comput..

[26]  M. Frontini,et al.  Some variant of Newton's method with third-order convergence , 2003, Appl. Math. Comput..

[27]  Xinghua Wang,et al.  Convergence of Newton's method and uniqueness of the solution of equations in Banach space , 2000 .

[28]  Ioannis K. Argyros.  On Newton's method under mild differentiability conditions and applications , 1999, Appl. Math. Comput..