Semilocal convergence and R-order for modified Chebyshev-Halley methods

In this paper, we study the semilocal convergence and R-order for a class of modified Chebyshev-Halley methods for solving non-linear equations in Banach spaces. To solve the problem that the third-order derivative of an operator is neither Lipschitz continuous nor Hölder continuous, the condition of Lipschitz continuity of third-order Fréchet derivative considered in Wang et al. (Numer Algor 56:497–516, 2011) is replaced by its general continuity condition, and the latter is weaker than the former. Furthermore, the R-order of these methods is also improved under the same condition. By using the recurrence relations, a convergence theorem is proved to show the existence-uniqueness of the solution and give a priori error bounds. We also analyze the R-order of these methods with the third-order Fréchet derivative of an operator under different continuity conditions. Especially, when the third-order Fréchet derivative is Lipschitz continuous, the R-order of the methods is at least six, which is higher than the one of the method considered in Wang et al. (Numer Algor 56:497–516, 2011) under the same condition.     

Semilocal convergence on a family of root-finding multi-point methods in Banach spaces under relaxed continuity condition

In this paper, we consider the semilocal convergence on a family of root-finding multi-point methods. Compared with the results in reference (Hernández, M.A., Salanova, M.A., J. Comput. Appl. Math. 126, 131–143 3), these multi-point methods do not require the second derivative, Hölder continuity condition is relaxed, and the R-order is also enhanced. We prove an existence-uniqueness theorem of the solution. The R-order for these multi-point methods is at least 6 + q with relaxed continuous second derivative, where q∈[0,1].     

Semilocal convergence for Halley’s method under weak Lipschitz condition

The semilocal convergence properties of Halley’s method for nonlinear operator equations are studied under the hypothesis that the second derivative satisfies some weak Lipschitz condition. The method employed in the present paper is based on a family of recurrence relations which will be satisfied by the involved operator. An application to a nonlinear Hammerstein integral equation of the second kind is provided.     

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

We introduce the new idea of recurrent functions to provide a new semilocal convergence analysis for Newton-type methods, under mild differentiability conditions. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in some interesting cases (Chen, Ann Inst Stat Math 42:387–401, 1990; Chen, Numer Funct Anal Optim 10:37–48, 1989; Cianciaruso, Numer Funct Anal Optim 24:713–723, 2003; Cianciaruso, Nonlinear Funct Anal Appl 2009; Dennis 1971; Deuflhard 2004; Deuflhard, SIAM J Numer Anal 16:1–10, 1979; Gutiérrez, J Comput Appl Math 79:131–145, 1997; Hernández, J Optim Theory Appl 109:631–648, 2001; Hernández, J Comput Appl Math 115:245–254, 2000; Huang, J Comput Appl Math 47:211–217, 1993; Kantorovich 1982; Miel, Numer Math 33:391–396, 1979; Miel, Math Comput 34:185–202, 1980; Moret, Computing 33:65–73, 1984; Potra, Libertas Mathematica 5:71–84, 1985; Rheinboldt, SIAM J Numer Anal 5:42–63, 1968; Yamamoto, Numer Math 51: 545–557, 1987; Zabrejko, Numer Funct Anal Optim 9:671–684, 1987; Zinc̆ko 1963). Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, and a differential equation are also provided in this study.     

Convergence of iterations of Euler family under weak condition

The iteration maps of Euler family for finding zeros of an operatorf in Banach spaces is defined as the partial sum of Taylor expansion of the local inversef _z ^-1 off atz. The unified convergence theorem is established for the iterations of Euler family under the assumption that , while the strong condition thatf is analytic in Smale’s criterion α is replaced by the weak condition thatf is of finite order derivative.     

Remark on the convergence of the midpoint method under mild differentiability conditions

We establish a convergence theorem for the Midpoint method using a new system of recurrence relations. The purpose of this note is to relax its convergence conditions. We also given an example where our convergence theorem can be applied but other ones cannot.     

On the R-order of convergence of Newton's method under mild differentiability conditions

A new technique is used instead of the classical majorant principle to analyze the R-order of convergence of the Newton process when more general conditions than the Kantorovich ones are considered.     

Semilocal convergence of a modified multi-point Jarratt method in Banach spaces under general continuity condition

The semilocal convergence for a modified multi-point Jarratt method for solving non-linear equations in Banach spaces is established with the third-order Fréchet derivative of the operator under a general continuity condition. The recurrence relations are derived for the method, and from this, we prove an existence-uniqueness theorem, and give a priori error bounds. The R-order of the method is also analyzed with the third-order Fréchet derivative of the operator under different continuity conditions. Numerical application on non-linear integral equation of the mixed type is given to show our approach.     

Semilocal convergence of a family of iterative methods in Banach spaces

In this work, we prove a third and fourth convergence order result for a family of iterative methods for solving nonlinear systems in Banach spaces. We analyze the semilocal convergence by using recurrence relations, giving the existence and uniqueness theorem that establishes the R-order of the method and the priori error bounds. Finally, we apply the methods to two examples in order to illustrate the presented theory.     

Semilocal convergence of multi-point improved super-Halley-type methods without the second derivative under generalized weak condition

In this paper, we consider the semilocal convergence of multi-point improved super-Halley-type methods in Banach space. Different from the results of super-Halley method studied in reference Gutiérrez, J.M. and Hernández, M.A. (Comput. Math. Appl. 36,1–8, 1998) these methods do not require second derivative of an operator, the R-order is improved and the convergence condition is also relaxed. We prove a convergence theorem to show existence and uniqueness of the solution.     

Convergence analysis for a family of improved super-Halley methods under general convergence condition

In this paper, we focus on the semilocal convergence for a family of improved super-Halley methods for solving non-linear equations in Banach spaces. Different from the results in Wang et al. (J Optim Theory Appl 153:779–793, 2012), the condition of Hölder continuity of third-order Fréchet derivative is replaced by its general continuity condition, and the latter is weaker than former. Moreover, the R-order of the methods is also improved. By using the recurrence relations, we prove a convergence theorem to show the existence-uniqueness of the solution. The R-order of these methods is analyzed with the third-order Fréchet derivative of the operator satisfies general continuity condition and Hölder continuity condition.     

Strong convergence of modified Mann iterations for a countable family of nonexpansive mappings

In this paper we propose a new modified Mann iteration for computing common fixed points of nonexpansive mappings in a Banach space. We give certain different control conditions for the modified Mann iteration. Then, we prove strong convergence theorems for a countable family of nonexpansive mappings in uniformly smooth Banach spaces. These results improve and extend results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], Yao, et al. [Y. Yao, R. Chen and J. Yao, Strong convergence and certain control conditions for modified Mann iteration, Nonlinear Anal. 68 (2008) 1687–1693], Qin and Su [X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007) 415–424], and many others.     

Semilocal convergence of a family of third-order methods in Banach spaces under Hölder continuous second derivative

In this paper, the semilocal convergence of a family of multipoint third-order methods used for solving F(x)=0$F\left(x\right)=0$ in Banach spaces is established. It is done by using recurrence relations under the assumption that the second Fréchet derivative of F$F$ satisfies Hölder continuity condition. Based on two parameters depending upon F$F$, a new family of recurrence relations is defined. Using these recurrence relations, an existence–uniqueness theorem is established to prove that the R$R$-order convergence of the method is (2+p)$(2+p)$. A priori error bounds for the method are also derived. Two numerical examples are worked out to demonstrate the efficacy of our approach.     

Local convergence analysis of inexact Gauss-Newton like methods under majorant condition

In this paper, we present a local convergence analysis of inexact Gauss-Newton like methods for solving nonlinear least squares problems. Under the hypothesis that the derivative of the function associated with the least squares problem satisfies a majorant condition, we obtain that the method is well-defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the least squares problem. It also allows us to obtain an estimate of convergence ball for inexact Gauss-Newton like methods and some important, special cases.     

The Ulm method under mild differentiability conditions

The Ulm method is considered to approximate a solution of a nonlinear operator equation F(x) = 0. We study the convergence of this method when F′ is ω-conditioned and prove that the R-order of convergence is at least 1 + p if ω is quasi-homogeneous of type ω(tz)≤ t ^p ω(z), for z > 0, tϵ[0,1] and pϵ[0,1]. Preparation of this paper was partly supported by the Ministry of Education and Science (MTM 2005-03091).     

Directional differentiability of optimal solutions under Slater's condition

For convex parametric optimization problems it is shown that the optimal solution is directionally differentiable provided that a strong second-order sufficient optimality condition and Slater's condition are satisfied for the unperturbed problem. This directional derivative is equal to the optimal solution of a certain quadratic programming problem. For the construction of this quadratic problem, a preliminary choice of a suitable KKT-multiplier is necessary, which under additional assumptions may be taken as a vertex of the set of KKT-multipliers of the unperturbed problem. In the last part of this paper, the contingent derivative of the optimal solution is investigated.     

On the local convergence of a family of Euler-Halley type iterations with a parameter

This paper aims to study the local convergence of a family of Euler-Halley type methods with a parameter α for solving nonlinear operator equations under the second-order generalized Lipschitz assumption. The radius r _α of the optimal convergence ball and the error estimation of the method corresponding to α are estimated for each α ∈ ( − ∞ , + ∞ ). For each α > 0, we get r _α  ≥ r _− α and the upper bound of the error estimation of the method with α > 0 is not larger than the one with α < 0. For each α ≤ 0, we get the precise value of r _α , which is closely linked to the dynamical property of the method applied to a real or a complex function, and the optimal error estimation, which decreases when α→0⁻. Results show that the method corresponding to α is better than the one corresponding to − α for each α > 0 and the Chebyshev-Euler method is the best among all methods in the family with α ∈ ( − ∞ , 0] from the view of both safe choice of the initial point and error estimation.     

Semilocal convergence theorems for a certain class of iterative procedures

We provide semilocal convergence theorems for Newton-like methods in Banach space using outer and generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least square problems and ill-posed nonlinear operator equations.