On the convergence of Newton-type methods under mild differentiability conditions

Argyros, Ioannis K.; Hilout, Saïd

doi:10.1007/s11075-009-9308-x

Cited by 10 publications

(7 citation statements)

References 20 publications

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“…The Newton method and the Newton-like method are attractive because it converges rapidly from any sufficient initial guess. A number of researchers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] have generalized and established local as well as semilocal convergence analysis of the Newton method Equation (2) under the following conditions:…”

Section: Introductionmentioning

confidence: 99%

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A Third Order Newton-Like Method and Its Applications

2018

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In this paper, we design a new third order Newton-like method and establish its convergence theory for finding the approximate solutions of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis of our third order Newton-like method under the ω -continuity condition. Then we apply our approach to solve nonlinear fixed point problems and Fredholm integral equations, where the first derivative of an involved operator does not necessarily satisfy the Hölder and Lipschitz continuity conditions. Several numerical examples are given, which compare the applicability of our convergence theory with the ones in the literature.

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Section: Introductionmentioning

confidence: 99%

“…for each n ≥ 0. More precisely, Parhi and Gupta [21] have studied the semilocal convergence analysis of Equation (8) for computing a solution of the operator Equation 5, where G : D ⊂ X → X is a nonlinear Fréchet differentiable operator defined on an open convex subset D under the condition:…”

Section: Introductionmentioning

confidence: 99%

A Third Order Newton-Like Method and Its Applications

2018

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“…for each n ≥ 0, where F ′ x denotes the Fréchet derivative of F at point x ∈ D. The Newton method and the Newton-like methods are attractive because it converges rapidly from any sufficient initial guess. A number of researchers [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] have generalized and established local as well as semilocal convergence analysis of the Newton method (1.2) under the following conditions:…”

Section: Introductionmentioning

confidence: 99%

A Third Order Newton-Like Method and Its Applications

Sahu¹,

Agarwal²,

Cho³

et al. 2018

Preprint

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In this paper, we study the third order semilocal convergence of the Newton-like method for finding the approximate solution of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis under ω-continuity condition, which is weaker than the Lipschitz and Hölder continuity conditions. Second, we apply our approach to solve Fredholm integral equations, where the first derivative of involved operator not necessarily satisfy the Hölder and Lipschitz continuity conditions. Finally, we also prove that the R-order of the method is 2p + 1 for any p $\in$ (0,1].

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“…In order to avoid the use of the inverse of the derivative of the operator F involved in S, some authors (see, for example, [6,4,3,8,18,26]) studied the Newton-like method given by…”

Section: Introductionmentioning

confidence: 99%

“…4) where ω is a nondecreasing and non-negative function on R + . They considered a function h : [0, 1] → R + such that ω(st) ≤ h(s)ω(t)…”

Section: Introductionmentioning

confidence: 99%

Accessibility of solutions of operator equations by Newton-like methods

Sahu

Cho

Agarwal

et al. 2015

Journal of Complexity

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a b s t r a c tThe concept of a majorizing sequence introduced and applied by Rheinboldt in 1968 is taken up to develop a convergence theory of the Picard iteration x n+1 = G(x n ) for each n ≥ 0 for fixed points of an iteration mapping G : D 0 ⊂ X → X in a complete metric space X satisfying iterated contraction-like condition:Here J 3 is a suitable set of (R + ) 3 to be defined in Section 2. We study the region of accessibility of fixed points of G by the Picard iteration u n+1 = G(u n ), where the starting point u 0 ∈ D 0 is not necessarily x 0 . Our convergence theory is applied to the Newtonlike iterations in Banach spaces under the center Lipschitz condi-Our results extend and improve the previous ones in the sense of the center Lipschitz condition and the region of accessibility of solutions. We apply our results to solve the nonlinear Fredholm operator equations of second kind.

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On the convergence of Newton-type methods under mild differentiability conditions

Cited by 10 publications

References 20 publications

A Third Order Newton-Like Method and Its Applications

A Third Order Newton-Like Method and Its Applications

A Third Order Newton-Like Method and Its Applications

Accessibility of solutions of operator equations by Newton-like methods

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