scholarly journals On Global Convergence of Third-Order Chebyshev-Type Method under General Continuity Conditions

2022 ◽  
Vol 6 (1) ◽  
pp. 46
Author(s):  
Fouad Othman Mallawi ◽  
Ramandeep Behl ◽  
Prashanth Maroju
Keyword(s):  
Integral Equation ◽  
Global Convergence ◽  
Iterative Method ◽  
Boundary Value Problem ◽  
Banach Spaces ◽  
Continuity Condition ◽  
Third Order ◽  
Type Method ◽  
First Order ◽  
Theoretical Results

There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method is proposed under the assumptions that Fréchet derivative of first order satisfies continuity condition of the Hölder. Finally, we consider some integral equation and boundary value problem (BVP) in order to illustrate the suitability of theoretical results.

Controllability of Abstract Systems of Fractional Order

2015 ◽  
Vol 18 (6) ◽  
Author(s):  
Therese Mur ◽  
Hernán R. Henríquez
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Integral Equation ◽  
Control Systems ◽  
Banach Spaces ◽  
Fractional Differential Equation ◽  
Fractional Integral ◽  
First Order ◽  
Fractional Integral Equation ◽  
Fractional Differential

AbstractIn this paper we are concerned with the controllability of control systems governed by a fractional differential equation in Banach spaces. Using the properties of the Mittag-Leffler function we generalize to these systems a result of Korobov and Rabakh, which was established for first order systems. We apply our results to study the controllability of a system modeled by a fractional integral equation in a Hilbert space.

scholarly journals Solving a nonlinear biharmonic boundary value problem

2018 ◽  
Vol 33 (4) ◽  
pp. 308-324
Author(s):  
Dang Quang A ◽  
Truong Ha Hai ◽  
Nguyen Thanh Huong ◽  
Ngo Thi Kim Quy
Keyword(s):  
Iterative Method ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Nonlinear Operator ◽  
Nonlinear Term ◽  
Boundary Value ◽  
New Approach ◽  
Nonlinear Elastic ◽  
Uniqueness Of A Solution ◽  
Theoretical Results

In this paper we study a boundary value problem for a nonlinear biharmonic equation, which models a bending plate on nonlinear elastic foundation. We propose a new approach to existence and uniqueness  and numerical solution of the problem. It is based on the reduction of the problem to finding fixed point of a nonlinear operator for the nonlinear term. The result is that under some easily verified conditions we have established the existence and uniqueness of a solution and the convergence of an iterative method for the solution. The positivity of the solution and the monotony of iterations are also considered. Some examples demonstrate the applicability of the obtained theoretical results and the efficiency of the iterative method.

scholarly journals Iteration for a Third-Order Three-Point BVP with Sign-Changing Green's Function

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Ya-Hong Zhao ◽  
Xing-Long Li
Keyword(s):  
Iterative Method ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Positive Solution ◽  
Green's Function ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Monotone Positive Solution

We are concerned with the following third-order three-point boundary value problem:u‴(t)=f(t,u(t)),t∈[0,1],u′(0)=u(1)=0,u″(η)+αu(0)=0, whereα∈[0,2)andη∈[2/3,1). Although corresponding Green's function is sign-changing, we still obtain the existence of monotone positive solution under some suitable conditions onfby applying iterative method. An example is also included to illustrate the main results obtained.

scholarly journals Positive solution for singular third-order BVPs on the half line with first-order derivative dependence

2021 ◽  
Vol 13 (1) ◽  
pp. 105-126
Author(s):  
Abdelhamid Benmezaï ◽  
El-Djouher Sedkaoui
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Positive Constant ◽  
Space Variable ◽  
Boundary Value ◽  
Order Derivative ◽  
Half Line ◽  
Third Order ◽  
First Order ◽  
The Third

Abstract In this paper, we investigate the existence of a positive solution to the third-order boundary value problem { - u ‴ ( t ) + k 2 u ′ ( t ) = φ ( t ) f ( t , u ( t ) , u ′ ( t ) ) ,       t > 0 u ( 0 ) = u ′ ( 0 ) = u ′ ( + ∞ ) = 0 , \left\{ \matrix{- u'''\left( t \right) + {k^2}u'\left( t \right) = \phi \left( t \right)f\left( {t,u\left( t \right),u'\left( t \right)} \right),\,\,\,t > 0 \hfill \cr u\left( 0 \right) = u'\left( 0 \right) = u'\left( { + \infty } \right) = 0, \hfill \cr} \right. where k is a positive constant, ϕ ∈ L1 (0;+ ∞) is nonnegative and does vanish identically on (0;+ ∞) and the function f : ℝ+ × (0;+ ∞) × (0;+ ∞) → ℝ+ is continuous and may be singular at the space variable and at its derivative.

ON THE R-ORDER CONVERGENCE OF A THIRD ORDER METHOD IN BANACH SPACES UNDER MILD DIFFERENTIABILITY CONDITIONS

2009 ◽  
Vol 06 (02) ◽  
pp. 291-306 ◽  
Author(s):  
P. K. PARIDA ◽  
D. K. GUPTA
Keyword(s):  
Banach Spaces ◽  
Recurrence Relations ◽  
Hölder Continuity ◽  
A Priori ◽  
Continuity Condition ◽  
Order Method ◽  
Holder Continuity ◽  
Third Order ◽  
A Priori Error Bounds ◽  
Two Parameters

The aim of this paper is to discuss the convergence of a third order method for solving nonlinear equations F(x)=0 in Banach spaces by using recurrence relations. The convergence of the method is established under the assumption that the second Fréchet derivative of F satisfies a condition that is milder than Lipschitz/Hölder continuity condition. A family of recurrence relations based on two parameters depending on F is also derived. An existence-uniqueness theorem is also given that establish convergence of the method and a priori error bounds. A numerical example is worked out to show that the method is successful even in cases where Lipschitz/Hölder continuity condition fails.

scholarly journals Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Abhimanyu Kumar ◽  
Dharmendra K. Gupta ◽  
Eulalia Martínez ◽  
Sukhjit Singh
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Local Convergence ◽  
Recurrence Relations ◽  
Semilocal Convergence ◽  
Convergence Domain ◽  
Differentiability Condition ◽  
Type Integral ◽  
Weaker Conditions ◽  
Theoretical Results

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.

scholarly journals Solvability of a Third-Order Singular Generalized Left Focal Problem in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-18
Author(s):  
Youwei Zhang
Keyword(s):  
Fixed Point ◽  
Boundary Value Problem ◽  
Banach Spaces ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Multiple Positive Solutions ◽  
Third Order ◽  
The Third ◽  
Priori Estimates

We consider the existence of positive solution for a third-order singular generalized left focal boundary value problem with full derivatives in Banach spaces. Green’s function and its properties, explicit a priori, estimates will be presented. By means of the theories of the fixed point in cones, we establish some new and general results on the existence of single and multiple positive solutions to the third-order singular generalized left focal boundary value problem. Our results are generalizations and extensions of the results of the focal boundary value problem. An example is included to illustrate the results obtained.

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