Nonlinear Equations In Banach Spaces Research Articles

On the Kantorovich Theory for Nonsingular and Singular Equations

We develop a new Kantorovich-like convergence analysis of Newton-type methods to solve nonsingular and singular nonlinear equations in Banach spaces. The outer or generalized inverses are exchanged by a finite sum of linear operators making the implementation of these methods easier than in earlier studies. The analysis uses relaxed generalized continuity of the derivatives of operators involved required to control the derivative and on real majorizing sequences. The same approach can also be implemented on other iterative methods with inverses. The examples complement the theory by verifying the convergence conditions and demonstrating the performance of the methods.

Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces

In this article, we propose a new methodology to construct and study generalized three-step numerical methods for solving nonlinear equations in Banach spaces. These methods are very general and include other methods already in the literature as special cases. The convergence analysis of the specialized methods is been given by assuming the existence of high-order derivatives which are not shown in these methods. Therefore, these constraints limit the applicability of the methods to equations involving operators that are sufficiently many times differentiable although the methods may converge. Moreover, the convergence is shown under a different set of conditions. Motivated by the optimization considerations and the above concerns, we present a unified convergence analysis for the generalized numerical methods relying on conditions involving only the operators appearing in the method. This is the novelty of the article. Special cases and examples are presented to conclude this article.

Local Convergence of a Seventh Order Derivative-Free Method for Solving Nonlinear Equations in Banach Spaces

Local Convergence of a Seventh Order Derivative-Free Method for Solving Nonlinear Equations in Banach Spaces

Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers

In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented.

Recurrence relations for a family of iterations assuming Hölder continuous second order Fréchet derivative

Abstract The semilocal convergence using recurrence relations of a family of iterations for solving nonlinear equations in Banach spaces is established. It is done under the assumption that the second order Fréchet derivative satisfies the Hölder continuity condition. This condition is more general than the usual Lipschitz continuity condition used for this purpose. Examples can be given for which the Lipschitz continuity condition fails but the Hölder continuity condition works on the second order Fréchet derivative. Recurrence relations based on three parameters are derived. A theorem for existence and uniqueness along with the error bounds for the solution is provided. The R-order of convergence is shown to be equal to 3 + q when θ = ±1; otherwise it is 2 + q, where q ∈ (0, 1]. Numerical examples involving nonlinear integral equations and boundary value problems are solved and improved convergence balls are found for them. Finally, the dynamical study of the family of iterations is also carried out.

On the Local Convergence of Two-Step Newton Type Method in Banach Spaces under Generalized Lipschitz Conditions

The motive of this paper is to discuss the local convergence of a two-step Newton-type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e., L-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L-average particularly it is assumed that L is positive integrable function but not necessarily non-decreasing. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.

A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable.

Generalizing the local convergence analysis of a class of $k$-step iterative algorithms with H

We present the local convergence analysis of a class of $k$-step iterative algorithms using the Hölder continuity condition for approximating solutions of nonlinear equations in Banach spaces. Our analysis is based on the Hölder continuity of the first-or

Fixed point methods and accretivity for perturbed nonlinear equations in Banach spaces

In this paper we use fixed point theorems to guarantee the existence of solutions for inclusions of the form Au+λu+Fu∋g, where A is a quasi-m-accretive operator defined in a Banach space, λ>0, and the nonlinear perturbation F satisfies some suitable conditions. We apply the obtained results, among other things, to guarantee the existence of solutions of boundary value problems of the type −Δρ(u(x))+λu(x)+Fu(x)=g(x), x∈Ω, and ρ(u)=0 on ∂Ω, where the Laplace operator Δ should be understood in the sense of distributions over Ω and to study the existence and uniqueness of solution for a nonlinear integro-differential equation posed in L1(Ω).

Local convergence of fourth and fifth order parametric family of iterative methods in Banach spaces

This paper deal with the study of local convergence of fourth and fifth order iterative method for solving nonlinear equations in Banach spaces. Only the premise that the first order Frechet derivative fulfills the Lipschitz continuity condition is needed. Under these conditions, a convergence theorem is established to study the existence and uniqueness regions for the solution for each method. The efficacy of our convergence study is shown solving various numerical examples as a nonlinear integral equation and calculating the radius of the convergence balls. We compare the radii of convergence balls and observe that by our approach, we get much larger balls as existing ones. In addition, we also include the real and complex dynamic study of one of the methods applied to a generic polynomial of order two.

On the local convergence of modified Weerakoon’s method in Banach spaces

Based on Lipschitz continuity condition, we study the local convergence analysis for the fifth-order convergent modified Weerakoon’s method for solving nonlinear equations in Banach spaces. Lipschitz continuity condition on the first derivative is assumed to extend the applicability of the scheme. This analysis enables us to solve such problems for which previous studies based on higher-order derivatives unable to find the solution. A theorem showing the existence and uniqueness of the solution along with computable error bounds is established. Standard numerical examples like nonlinear integral equation and system of nonlinear equations are solved to demonstrate the productiveness of our theoretical outcomes.

Convergence of a Stirling-like method for fixed points in Banach spaces

We have very few literature on fixed point iterative methods for solving nonlinear equations. We consider the Stirling method given by Rall (1969). Based on Stirling method, in this paper, we propose a third order Stirling-like method for finding fixed point for nonlinear equations in Banach spaces. We study the local and semilocal convergence of this method for finding the fixed points of nonlinear equations in Banach spaces. The convergence is established under the assumption that the first order Fréchet derivative satisfies the Lipschitz continuity condition. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We consider the numerical examples for local and semilocal convergence case and calculate the existence and uniqueness region of convergence balls even if we fail to apply the results in Rall (1975), Argyros (1995) and Parhi and Gupta (2010)[11,12] due to the F is not contraction on Ω .

On the Local Convergence of Modified Homeier-Like Method in Banach Spaces

The aim of this article is to investigate the local convergence analysis of the multi-step Homeier-like approach in order to approximate the solution of nonlinear equations in Banach spaces, which fulfilled the Lipschitz as well as Holder continuity condition. The Holder condition is more relax than Lipschitz condition. Also, the existence and uniqueness theorem has been derived and found their error bounds. Numerical examples are available to appear the importance of theoretical discussions.

Local convergence for a Chebyshev-type method in Banach space free of derivatives

This paper is devoted to the study of a Chebyshev-type method free of derivatives for solving nonlinear equations in Banach spaces. Using the idea of restricted convergence domain, we extended the applicability of the Chebyshev-type methods. Our convergence conditions are weaker than the conditions used in earlier studies. Therefore the applicability of the method is extended. 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 multi-step high order method with divided differences under hypotheses on the first derivative

Abstract This paper is devoted to the study of a multi-step method with divided differences for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the sixth order of the operator under consideration is used to prove the convergence of the method. That restricts the applicability of the method. In this paper we extended the applicability of the sixth-order multi-step method by using only hypotheses on the first derivative of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study.

Semilocal convergence of a computationally efficient iterative method in Banach spaces under weak condition

In this article, we discuss the semilocal convergence of well established efficient eighth-order method for solving nonlinear equations in Banach spaces under relaxed condition. The semilocal convergence of this scheme is established by using recurrence relations. We derive a system of recurrence relations for the method and then prove the existence and uniqueness result that shows the R-order of the method. Finally, numerical example is presented to validate the theoretical discussions.

Local convergence of Super Halley’s method under weaker conditions on Fréchet derivative in Banach spaces

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.

Analysis of semilocal convergence in banach spaces under relaxed condition and computational efficiency

The current paper is concerned with the study of semilocal convergence of a fifth-order method for solving nonlinear equations in Banach spaces under mild condition. The existence and uniqueness theorem has been proved followed by the error estimates. The computational efficiency of the considered scheme over the identical order methods is also examined, which endorses the nobility of the present scheme from computational point of view. Lastly, application of theoretical development has been made in nonlinear integral equation.

On the local convergence of a Newton–Kurchatov-type method for non-differentiable operators

By means of a nice idea, a Newton–Kurchatov type iterative process is constructed for solving nonlinear equations in Banach spaces. We analyze the local convergence of this iterative process. This study have an important and novel feature, since it is applicable to non-differentiable operators. So far, most of the local convergence results considered by other authors may apply only to differentiable operators due to the conditions that are required on the solution of the nonlinear equation.

On the existence of solutions of nonlinear equations

Nonlinear equations in Banach spaces are considered. Solvability conditions are obtained for them. These results are a generalization of the Hadamard diffeomorphism theorem. Additionally, conditions for the existence of coincidence points of two mappings acting on metric spaces are obtained.