scholarly journals A CONVERGENCE ANALYSIS FOR NEWTON-LIKE METHODS IN BANACH SPACE UNDER WEAK HYPOTHESES AND APPLICATIONS

1999 ◽  
Vol 30 (4) ◽  
pp. 253-261
Author(s):  
IOANNIS K. ARGYROS
Keyword(s):  
Banach Space ◽  
Lipschitz Continuity ◽  
Real Life ◽  
Approximate Solutions ◽  
Continuity Condition ◽  
Point Boundary ◽  
Hölder Continuous ◽  
Life Problems ◽  
Convergence Results ◽  
Bending Of Beams

In this study we use Newton-like methods to approximate solutions of nonlinear equations in a Banach space setting. Most convergence results for Newton-like methods involve some type of a Lipschitz continuity condition on the Frechet-derivative of the operator involved However there are many interesting real life problems already in the literature where the operator can only satisfy a Holder continuity condition. That is why here we chose the Frcchet-derivativc of the operator involved to be only Holder continuous, which allows us to consider a wider range of problems than before. Special choices of our parameters reduce our results to earlier ones. An error analysis is also provided for our method. At the end of our study, we provide applications to show that our results apply where earlier results do not. In paricular we solve a two point boundary value problem appearing in physics in connection with the problem of bending of beams.

scholarly journals A New Higher-Order Iterative Scheme for the Solutions of Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 271 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros
Keyword(s):  
Mathematical Modeling ◽  
Banach Space ◽  
Nonlinear Equations ◽  
Numerical Experiments ◽  
Iterative Scheme ◽  
Real Life ◽  
System Of Nonlinear Equations ◽  
Life Problems ◽  
Lipschitz Constants ◽  
Better Than

Many real-life problems can be reduced to scalar and vectorial nonlinear equations by using mathematical modeling. In this paper, we introduce a new iterative family of the sixth-order for a system of nonlinear equations. In addition, we present analyses of their convergences, as well as the computable radii for the guaranteed convergence of them for Banach space valued operators and error bounds based on the Lipschitz constants. Moreover, we show the applicability of them to some real-life problems, such as kinematic syntheses, Bratu’s, Fisher’s, boundary value, and Hammerstein integral problems. We finally wind up on the ground of achieved numerical experiments, where they perform better than other competing schemes.

scholarly journals A Third Order Newton-Like Method and Its Applications

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 31 ◽  
Author(s):  
D. Sahu ◽  
Ravi Agarwal ◽  
Vipin Singh
Keyword(s):  
Nonlinear Operator ◽  
Lipschitz Continuity ◽  
Approximate Solutions ◽  
Continuity Condition ◽  
Convergence Theory ◽  
Fredholm Integral Equations ◽  
Third Order ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
First Derivative

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.

scholarly journals A Novel Picture Fuzzy n-Banach Space with Some New Contractive Conditions and Their Fixed Point Results

2020 ◽  
Vol 2020 ◽  
pp. 1-12 ◽  
Author(s):  
Awais Asif ◽  
Hassen Aydi ◽  
Muhammad Arshad ◽  
Zeeshan Ali
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Linear Space ◽  
Fuzzy Set ◽  
Fixed Point Theorems ◽  
Normed Linear Space ◽  
Real Life ◽  
Intuitionistic Fuzzy ◽  
Life Problems ◽  
Picture Fuzzy Set

A picture fuzzy n-normed linear space (NPF), a mixture of a picture fuzzy set and an n-normed linear space, is a proficient concept to cope with uncertain and unpredictable real-life problems. The purpose of this manuscript is to present some novel contractive conditions based on NPF. By using these contractive conditions, we explore some fixed point theorems in a picture fuzzy n-Banach space (BPF). The discussed modified results are more general than those in the existing literature which are based on an intuitionistic fuzzy n-Banach space (BIF) and a fuzzy n-Banach space. To express the reliability and effectiveness of the main results, we present several examples to support our main theorems.

scholarly journals AN INEXACT NEWTON METHOD WITH INNER PRECONDITIONED CG FOR NON-UNIFORMLY MONOTONE ELLIPTIC PROBLEMS

2021 ◽  
Vol 26 (3) ◽  
pp. 383-394
Author(s):  
Benjámin Borsos
Keyword(s):  
Banach Space ◽  
Newton Method ◽  
Lower Bounds ◽  
Numerical Experiments ◽  
Real Life ◽  
Elliptic Problems ◽  
Upper And Lower Bounds ◽  
Preconditioned Conjugate Gradient ◽  
Inexact Newton Method ◽  
Convergence Results

The present paper introduces an inexact Newton method, coupled with a preconditioned conjugate gradient method in inner iterations, for elliptic operators with non-uniformly monotone upper and lower bounds. Convergence is proved in Banach space level. The results cover real-life classes of elliptic problems. Numerical experiments reinforce the convergence results.

scholarly journals Local Convergence of a Family of Weighted-Newton Methods

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 103
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros ◽  
J.A. Machado ◽  
Ali Alshomrani
Keyword(s):  
Banach Space ◽  
Fourth Order ◽  
Local Convergence ◽  
Real Life ◽  
Newton Methods ◽  
Convergence Radius ◽  
Life Problems ◽  
Scope Of Application ◽  
Error Estimations ◽  
Derivatives Of

This article considers the fourth-order family of weighted-Newton methods. It provides the range of initial guesses that ensure the convergence. The analysis is given for Banach space-valued mappings, and the hypotheses involve the derivative of order one. The convergence radius, error estimations, and results on uniqueness also depend on this derivative. The scope of application of the method is extended, since no derivatives of higher order are required as in previous works. Finally, we demonstrate the applicability of the proposed method in real-life problems and discuss a case where previous studies cannot be adopted.

scholarly journals Numerical solvability of generalized Bagley–Torvik fractional models under Caputo–Fabrizio derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shatha Hasan ◽  
Nadir Djeddi ◽  
Mohammed Al-Smadi ◽  
Shrideh Al-Omari ◽  
Shaher Momani ◽  
...  
Keyword(s):  
Reproducing Kernel ◽  
Initial Conditions ◽  
Reproducing Kernel Hilbert Space ◽  
Real Life ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Point Of View ◽  
The Novel ◽  
Life Problems ◽  
Fractional Models

AbstractThis paper deals with the generalized Bagley–Torvik equation based on the concept of the Caputo–Fabrizio fractional derivative using a modified reproducing kernel Hilbert space treatment. The generalized Bagley–Torvik equation is studied along with initial and boundary conditions to investigate numerical solution in the Caputo–Fabrizio sense. Regarding the generalized Bagley–Torvik equation with initial conditions, in order to have a better approach and lower cost, we reformulate the issue as a system of fractional differential equations while preserving the second type of these equations. Reproducing kernel functions are established to construct an orthogonal system used to formulate the analytical and approximate solutions of both equations in the appropriate Hilbert spaces. The feasibility of the proposed method and the effect of the novel derivative with the nonsingular kernel were verified by listing and treating several numerical examples with the required accuracy and speed. From a numerical point of view, the results obtained indicate the accuracy, efficiency, and reliability of the proposed method in solving various real life problems.

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

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Dharmendra Kumar Gupta ◽  
Eulalia Martínez ◽  
Sukhjit Singh ◽  
Jose Luis Hueso ◽  
Shwetabh Srivastava ◽  
...  
Keyword(s):  
Recurrence Relations ◽  
Lipschitz Continuity ◽  
Hölder Continuity ◽  
Continuity Condition ◽  
Fréchet Derivative ◽  
Second Order ◽  
Holder Continuity ◽  
Hölder Continuous ◽  
Dynamical Study ◽  
Frechet 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.

The Psychologist in the Medical School: An Example of Solving Real-Life Problems

1970 ◽  
Author(s):  
Matisyohu Weisenberg ◽  
Carl Eisdorfer ◽  
C. Richard Fletcher ◽  
Murray Wexler
Keyword(s):  
Medical School ◽  
Real Life ◽  
Life Problems

scholarly journals AnANN Model Trained on Regional Data in the Prediction of Particular Weather Conditions

2021 ◽  
Vol 11 (11) ◽  
pp. 4757
Author(s):  
Aleksandra Bączkiewicz ◽  
Jarosław Wątróbski ◽  
Wojciech Sałabun ◽  
Joanna Kołodziejczyk
Keyword(s):  
Atmospheric Pressure ◽  
Real Life ◽  
Real Data ◽  
Weather Conditions ◽  
Maximum Temperature ◽  
Air Masses ◽  
Weather Forecasts ◽  
Life Problems ◽  
The World ◽  
The City

Artificial Neural Networks (ANNs) have proven to be a powerful tool for solving a wide variety of real-life problems. The possibility of using them for forecasting phenomena occurring in nature, especially weather indicators, has been widely discussed. However, the various areas of the world differ in terms of their difficulty and ability in preparing accurate weather forecasts. Poland lies in a zone with a moderate transition climate, which is characterized by seasonality and the inflow of many types of air masses from different directions, which, combined with the compound terrain, causes climate variability and makes it difficult to accurately predict the weather. For this reason, it is necessary to adapt the model to the prediction of weather conditions and verify its effectiveness on real data. The principal aim of this study is to present the use of a regressive model based on a unidirectional multilayer neural network, also called a Multilayer Perceptron (MLP), to predict selected weather indicators for the city of Szczecin in Poland. The forecast of the model we implemented was effective in determining the daily parameters at 96% compliance with the actual measurements for the prediction of the minimum and maximum temperature for the next day and 83.27% for the prediction of atmospheric pressure.

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