Local Convergence for a Frozen Family of Steffensen-Like Methods under Weak Conditions

Super Halley’s method is one of the most important iterative methods for solving nonlinear equations in Banach spaces. It’s local and semilocal convergence analysis is established using either majorizing sequences or recurrence relations under various continuity conditions such as Lipschitz or Holder using first/second order Frechet derivatives. In this paper, an attempt is made to establish it’s local convergence analysis under weaker continuity conditions on first order Frechet derivative. This work generalizes the earlier work in this direction and it is observed that it is applicable to cases whether they either fail to converge or give smaller balls of convergence.

We present a local convergence analysis for a family of super-Halley methods of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second Frechet-derivative of the operator involved. Earlier studies use hypotheses up to the third Frechet-derivative. Numerical examples are also provided in this study.

In the present paper, we consider a fifth-order method considered in Singh et al. (2016) to solve equations in Banach space under weaker assumptions. Using the idea of restricted convergence domains we extend the applicability of the method considered by

ABSTRACTLet X and Y be Banach spaces. Consider the following generalized equation problem:(1) where f is Fréchet differentiable on an open subset Ω of X and F is set-valued mapping with closed graph acting between Banach spaces. In the present paper, we introduce a variant of Newton-type method for solving generalized equation (1). Semi-local and local convergence analysis are provided under the weaker condition that of utilized by Jean-Alexis and Pietrus [A variant of Newton's method for generalized equations, Rev. Colombiana Mat. 39 (2005), pp. 97–112]. In particular, this result extends the corresponding ones Jean-Alexis and Pietrus (2005). Finally, we present a numerical example to validate the convergence result of this method.

In the present paper, we study the local convergence analysis of a fifth convergence order method considered by Sharma and Guha in [15] to solve equations in Banach space. Using our idea of restricted convergence domains we extend the applicability of this method. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.

Local convergence of a fifth convergence order method in Banach space

This paper is devoted to the study of the seventh-order Steffensen-type methods for solving nonlinear equations in Banach spaces. Using the idea of a restricted convergence domain, we extended the applicability of the seventh-order Steffensen-type methods. Our convergence conditions are weaker than the conditions used in the earlier studies. Numerical examples are also given in this study.

The semilocal and local convergence analyses of a two-step iterative method for nonlinear nondifferentiable operators are described in Banach spaces. The recurrence relations are derived under weaker conditions on the operator. For semilocal convergence, the domain of the parameters is obtained to ensure guaranteed convergence under suitable initial approximations. The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided. The region of accessibility and a way to enlarge the convergence domain are provided. Theorems are given for the existence-uniqueness balls enclosing the unique solution. Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.

In the present paper, we consider the generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable on a neighborhood $\Omega$ of a point $\bar{x}$ in $\mathcal X$, $g:\mathcal X\to \mathcal Y$ is differentiable at point $\bar{x}$ and linear as well as $\mathcal F$ is a set-valued mapping with closed graph acting between two Banach spaces $\mathcal X$ and $\mathcal Y$. We study the above generalized equation with the help of extended Newton-type method, introduced in [ M. Z. Khaton, M. H. Rashid, M. I. Hossain, Convergence Properties of extended Newton-type Iteration Method for Generalized Equations, Journal of Mathematics Research, 10 (4) (2018), 1--18, DOI:10.5539/jmr.v10n4p1, under the weaker conditions than that are used in Khaton et al. (2018). Indeed, semilocal and local convergence analysis are provided for this method under the conditions that the Frechet derivative of $f$ and the first-order divided difference of $g$ are Hölder continuous on $\Omega$. In particular, we show this method converges superlinearly and these results extend and improve the corresponding results in Argyros (2008) and Khaton $et$ $al.$ (2018).

Deformed super-Halley’s iteration for nonlinear equations is studied in Banach spaces with its local and semilocal convergence. The local convergence is established under Holder continuous first Frechet derivative. A theorem for the existence and uniqueness of solution is provided and the radii of convergence balls are obtained. For semilocal convergence, the second order Frechet derivative is Holder continuous. The Holder continuous first Frechet derivative is not used as it leads to lower R-order of convergence. Recurrence relations depending on two parameters are obtained. A theorem for the existence and uniqueness along with the estimation of bounds on errors is also established. The R-order convergence comes out to be $$(2+p), p \in (0,1]$$ . Nonlinear integral equations and a variety of numerical examples are solved to demonstrate our work.

We have developed a local convergence analysis for a general scheme of high-order convergence, aiming to solve equations in Banach spaces. A priori estimates are developed based on the error distances. This way, we know in advance the number of iterations required to reach a predetermined error tolerance. Moreover, a radius of convergence is determined, allowing for a selection of initial points assuring the convergence of the scheme. Furthermore, a neighborhood that contains only one solution to the equation is specified. Notably, we present the generalized convergence of these schemes under weak conditions. Our findings are based on generalized continuity requirements and contain a new semi-local convergence analysis (with a majorizing sequence) not seen in earlier studies based on Taylor series and derivatives which are not present in the scheme. We conclude with a good collection of numerical results derived from applied science problems.

We present a unified local convergence analysis for deformed Euler–Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study.

We present a local convergence analysis for deformed Chebyshev methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Chebyshev and other high-order methods under hypotheses up to the first Frechet derivative in contrast to earlier studies using hypotheses up to the second or third Frechet derivative. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study.

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems.

A new semi-local convergence analysis of the Gauss-Newton method for solving convex composite optimization problems is presented using restricted convergence domains. The results extend the applicability of the Gauss-Newton method under the same computational cost as in earlier studies. In particular, the advantages are: the error estimates on the distances involved are tighter and the convergence ball is at least as large. Moreover, the majorant function in contrast to earlier studies is not necessarily differentiable. Numerical examples are also provided in this study.