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 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.

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.

scholarly journals On the Convergence of Newton-like Method for Variational Inclusions under Pseudo-Lipschitz Mapping

2020 ◽  
Vol 40 (1) ◽  
pp. 43-53
Author(s):  
Mst Zamilla Khaton ◽  
MH Rashid ◽  
MI Hossain
Keyword(s):  
Differentiable Function ◽  
Divided Difference ◽  
Admissible Function ◽  
Fréchet Derivative ◽  
Variational Inclusion ◽  
Lipschitz Mapping ◽  
Lipschitz Continuous ◽  
Frechet Derivative ◽  
Set Valued Mapping ◽  
Fréchet Differentiable

In the present paper, we study a Newton-like method for solving the variational inclusion defined by the sums of a Frechet differentiable function, divided difference admissible function and a set-valued mapping with closed graph. Under some suitable assumptions on the Frechet derivative of the differentiable function and divided difference admissible function, we establish the existence of any sequence generated by the Newton-like method and prove that the sequence generated by this method converges linearly and superlinearly to a solution of the variational inclusion. Specifically, when the Frechet derivative of the differentiable function is continuous, Lipschitz continuous, divided difference admissible function admits first order divided di_erence and the setvalued mapping is pseudo-Lipschitz continuous, we show the linear and superlinear convergence of the method. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 43-53

scholarly journals On the Semilocal Convergence of the Multi–Point Variant of Jarratt Method: Unbounded Third Derivative Case

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 540 ◽  
Author(s):  
Zhang Yong ◽  
Neha Gupta ◽  
J. P. Jaiswal ◽  
Kalyanasundaram Madhu
Keyword(s):  
Nonlinear Operator ◽  
A Priori ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Fréchet Derivative ◽  
Third Order ◽  
Frechet Derivative ◽  
Initial Iterate ◽  
Jarratt Method ◽  
Third Derivative

In this paper, we study the semilocal convergence of the multi-point variant of Jarratt method under two different mild situations. The first one is the assumption that just a second-order Fréchet derivative is bounded instead of third-order. In addition, in the next one, the bound of the norm of the third order Fréchet derivative is assumed at initial iterate rather than supposing it on the domain of the nonlinear operator and it also satisfies the local ω -continuity condition in order to prove the convergence, existence-uniqueness followed by a priori error bound. During the study, it is noted that some norms and functions have to recalculate and its significance can be also seen in the numerical section.

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