scholarly journals Mysovskii-type theorem for the Secant method under Hölder continuous Fréchet derivative

2006 ◽  
Vol 320 (1) ◽  
pp. 415-424 ◽  
Author(s):  
Hongmin Ren ◽  
Qingbiao Wu
Keyword(s):  
Type Theorem ◽  
Fréchet Derivative ◽  
Secant Method ◽  
Hölder Continuous ◽  
Frechet Derivative

scholarly journals On Newton's method and nondiscrete mathematical induction

1988 ◽  
Vol 38 (1) ◽  
pp. 131-140 ◽  
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Lipschitz Condition ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Mathematical Induction ◽  
Fréchet Derivative ◽  
Hölder Continuous ◽  
Frechet Derivative ◽  
Nondiscrete Mathematical Induction ◽  
Sharp Error

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

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.

scholarly journals A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

1995 ◽  
Vol 36 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Fréchet Derivative ◽  
Necessary Condition ◽  
Control Problems ◽  
Partial Differential ◽  
Frechet Derivative ◽  
First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

On the non-reducibility of the (u,v)-derivative to the Fréchet derivative

1982 ◽  
Vol 37 (1) ◽  
pp. 153-154
Author(s):  
I A Bakhtin ◽  
V I Sobolev ◽  
V M Shcherbin
Keyword(s):  
Fréchet Derivative ◽  
Frechet Derivative

scholarly journals On the Solution of Equations by Extended Discretization

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 69
Author(s):  
Gus I. Argyros ◽  
Michael I. Argyros ◽  
Samundra Regmi ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equations ◽  
Error Bounds ◽  
Hölder Continuity ◽  
Fréchet Derivative ◽  
Holder Continuity ◽  
Frechet Derivative ◽  
Solution Of Equations ◽  
Discretization Technique

The method of discretization is used to solve nonlinear equations involving Banach space valued operators using Lipschitz or Hölder constants. But these constants cannot always be found. That is why we present results using ω− continuity conditions on the Fréchet derivative of the operator involved. This way, we extend the applicability of the discretization technique. It turns out that if we specialize ω− continuity our new results improve those in the literature too in the case of Lipschitz or Hölder continuity. Our analysis includes tighter upper error bounds on the distances involved.

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