A Steffensen's type method in Banach spaces with applications on boundary-value problems

In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is analyzed.     

Majorizing sequences for Newton’s method from initial value problems

The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton’s method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovich’s technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems.     

Solving nonlinear integral equations of Fredholm type with high order iterative methods

The application of high order iterative methods for solving nonlinear integral equations is not usual in mathematics. But, in this paper, we show that high order iterative methods can be used to solve a special case of nonlinear integral equations of Fredholm type and second kind. In particular, those that have the property of the second derivative of the corresponding operator have associated with them a vector of diagonal matrices once a process of discretization has been done.     

On the local convergence of inexact Newton-type methods under residual control-type conditions

A local convergence analysis of inexact Newton-type methods using a new type of residual control was recently presented by C. Li and W. Shen. Here, we introduce the center-Hölder condition on the operator involved, and use it in combination with the Hölder condition to provide a new local convergence analysis with the following advantages: larger radius of convergence, and tighter error bounds on the distances involved. These results are obtained under the same hypotheses and computational cost. Numerical examples further validating the theoretical results are also provided in this study.     

Applying a fixed point theorem of Krasnosel’skii type to the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation

Using a fixed point theorem of Krasnosel’skii type, the paper proves the existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation.     

Note on the cubic decreasing region of the Chebyshev method

In this paper we reduce the two-dimensional cubic decreasing region considered in Hernandez and Salanova (2000) [1], [2] into one-dimensional region or interval for the Chebyshev method. It means that we find a simple sufficient condition for the semilocal convergence of the method.     

A unified approach for constructing fast two-step Newton-like methods

We introduce some new very general ways of constructing fast two-step Newton-like methods to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. We provide existence-uniqueness theorems as well as an error analysis for the iterations involved using Newton-Kantorovich-type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function defined on a closed ball centered at a certain point and of a fixed radius and with values into the positive real axis. Special choices of this function lead to favorable comparisons with results already in the literature. Some applications to the solution of nonlinear integral equations appearing in radiative transfer as well as to the solution of integral equations of Uryson-type are also provided.     

Extending the Newton-Kantorovich hypothesis for solving equations

The famous Newton-Kantorovich hypothesis (Kantorovich and Akilov, 1982 [3], Argyros, 2007 [2], Argyros and Hilout, 2009 [7]) has been used for a long time as a sufficient condition for the convergence of Newton’s method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here, using Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we show that the Newton-Kantorovich hypothesis can be weakened, under the same information. Moreover, the error bounds are tighter than the corresponding ones given by the dominating Newton-Kantorovich theorem (Argyros, 1998 [1]; [2] and [7]; Ezquerro and Hernández, 2002 [11]; [3]; Proinov 2009, 2010 [16] and [17]).Numerical examples including a nonlinear integral equation of Chandrasekhar-type (Chandrasekhar, 1960 [9]), as well as a two boundary value problem with a Green’s kernel (Argyros, 2007 [2]) are also provided in this study.     

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.     

On the convergence of Newton's method for a class of nonsmooth operators

We provide an analog of the Newton–Kantorovich theorem for a certain class of nonsmooth operators. This class includes smooth operators as well as nonsmooth reformulations of variational inequalities. It turns out that under weaker hypotheses we can provide under the same computational cost over earlier works [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] a semilocal convergence analysis with the following advantages: finer error bounds on the distances involved and a more precise information on the location of the solution. In the local case not examined in [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] we can show how to enlarge the radius of convergence and also obtain finer error estimates. Numerical examples are also provided to show that in the semilocal case our results can apply where others [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305] fail, whereas in the local case we can obtain a larger radius of convergence than before [S.M. Robinson, Newton's method for a class of nonsmooth functions, Set-Valued Anal. 2 (1994) 291–305].     

Improved local convergence of Newton’s method under weak majorant condition

We provide a local convergence analysis for Newton’s method under a weak majorant condition in a Banach space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before [14]. Special cases and numerical examples are also provided in this study.     

Kantorovich-type semilocal convergence analysis for inexact Newton methods

We provide a new semilocal convergence analysis for generating an inexact Newton method converging to a solution of a nonlinear equation in a Banach space setting. Our analysis is based on our idea of recurrent functions. Our results are compared favorably to earlier ones by others and us (Argyros (2007, 2009) [5] and [6], Argyros and Hilout (2009) [7], Guo (2007) [15], Shen and Li (2008) [18], Li and Shen (2008) [19], Shen and Li (2009) [20]). Numerical examples are provided to show that our results apply, but not earlier ones [15], [18], [19] and [20].     

On the convergence of modified Newton methods for solving equations containing a non-differentiable term

We provide a semilocal convergence analysis for certain modified Newton methods for solving equations containing a non-differentiable term. The sufficient convergence conditions of the corresponding Newton methods are often taken as the sufficient conditions for the modified Newton methods. That is why the latter methods are not usually treated separately from the former. However, here we show that weaker conditions, as well as a finer error analysis than before can be obtained for the convergence of modified Newton methods. Numerical examples are also provided.     

Convergence of the variants of the Chebyshev–Halley iteration family under the Hölder condition of the first derivative

The present paper is concerned with the convergence problem of the variants of the Chebyshev–Halley iteration family with parameters for solving nonlinear operator equations in Banach spaces. Under the assumption that the first derivative of the operator satisfies the Hölder condition of order p$p$, a convergence criterion of order 1+p$1+p$ for the iteration family is established. An application to a nonlinear Hammerstein integral equation of the second kind is provided.     

Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings

Let X$X$ be a reflexive and smooth real Banach space which has a weakly sequentially continuous duality mapping. In this paper, we consider the following viscosity approximation scheme _xn+1=_λn+1f(_xn)+(1−_λn+1)_Tn+1xn$x_{n+1}={\lambda }_{n+1}f\left(x_n\right)+(1-{\lambda }_{n+1})T_{n+1}{x}_n$ (where f$f$ is a generalized contraction mapping) for infinitely many nonexpansive self-mappings _T1,_T2,_T3,…$T_1,T_2,T_3,\dots$ in X$X$. We establish a strong convergence result which generalizes some results in the literature.     

Majorizing functions and two-point Newton-type methods

The semi-local convergence of a Newton-type method used to solve nonlinear equations in a Banach space is studied. We also give, as two important applications, convergence analyses of two classes of two-point Newton-type methods including a method mentioned in [5] and the midpoint method studied in [1], [2] and [12]. Recently, interest has been shown in such methods [3] and [4].     

The existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general Banach space

Motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n-dimensional real space, we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on R₊ with values in a general Banach space, via a fixed point theorem of Krasnosel’skii type. In order to illustrate the result obtained here, an example is given.