• The Pure and Applied Mathematics
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• v.13 no.4 s.34
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• pp.261-267
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• 2006

A local convergence analysis of Newton's method for perturbed generalized equations is provided in a Banach space setting. Using center Lipschitzian conditions which are actually needed instead of Lipschitzian hypotheses on the $Fr\'{e}chet$-derivative of the operator involved and more precise estimates under less computational cost we provide a finer convergence analysis of Newton's method than before [5]-[7].

• East Asian mathematical journal
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• v.24 no.2
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• pp.161-168
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• 2008

We revisit the study of finding solutions of equations containing a differentiable and a continuous term on a Banach space setting [1]-[5], [9]-[11]. Using more precise majorizing sequences than before [9]-[11], we provide a semilocal convergence analysis for the generalized Newton's method as well the generalized modified Newton's method. It turns out that under the same or even weaker hypotheses: finer error estimates on the distances involved, and an at least as precise information on the location of the solution can be obtained. The above benefits are obtained under the same computational cost.

• Journal of applied mathematics & informatics
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• v.6 no.3
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• pp.685-696
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• 1999

We present local and semilocal convergence results for New-ton's method in a Banach space setting. In particular using Lipschitz-type assumptions on the second Frechet-derivative we find results con-cerning the radius of convergence of Newton's method. Such results are useful in the context of predictor-corrector continuation procedures. Finally we provide numerical examples to show that our results can ap-ply where earlier ones using Lipschitz assumption on the first Frechet-derivative fail.

• East Asian mathematical journal
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• v.24 no.3
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• pp.295-304
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• 2008

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.

• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.111-120
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• 2008

A semi local convergence analysis is provided for Newton's method in a Banach space setting. The operators involved are only locally Holderian. We make use of a point-based approximation and center-Holderian hypotheses. This approach can be used to approximate solutions of equations involving nonsmooth operators.

• East Asian mathematical journal
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• v.25 no.4
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• pp.425-431
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• 2009

We provide a local convergence analysis for Newton-like methods for the solution of generalized equations in a Banach space setting. Using some ideas of ours introduced in [2] for nonlinear equations we show that under weaker hypotheses and computational cost than in [7] a larger convergence radius and finer error bounds on the distances involved can be obtained.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.1051-1067
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• 2023

In this paper, an algorithm for approximating zeros of sum of three monotone operators is introduced and its convergence properties are studied in the setting of 2-uniformly convex and uniformly smooth Banach spaces. Unlike the existing algorithms whose step-sizes usually depend on the knowledge of the operator norm or Lipschitz constant, a nice feature of the proposed algorithm is the fact that it requires only a diminishing and non-summable step-size to obtain strong convergence of the iterates to a solution of the problem. Finally, the proposed algorithm is implemented in the setting of a classical Banach space to support the theory established.

• East Asian mathematical journal
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• v.26 no.3
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• pp.365-370
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• 2010

A local convergence result is provided for the continuous analog of Newton's method in a Banach space setting. The radius of convergence is larger, the error bounds tighter, and under the same or weaker hypotheses than before [12].

• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.23-41
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• 2015

We present a semilocal convergence analysis of a third order method for approximating a locally unique solution of an equation in a Banach space setting. Recently, this method was studied by Chun, Stanica and Neta. These authors extended earlier results by Kou, Li and others. Our convergence analysis extends the applicability of these methods under less computational cost and weaker convergence criteria. Numerical examples are also presented to show that the earlier results cannot apply to solve these equations.

• The Pure and Applied Mathematics
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• v.16 no.4
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• pp.405-416
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• 2009

In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.