Kantorovich Type Research Articles

Chebyshev method and convexity

From convexity we obtain new convergence theorems for the Chebyshev method applied to a nonlinear scalar equation f( x) = 0. Moreover, Kantorovich type conditions for the convergence of the Chebyshev method in the complex plane are given. Finally, a new type of assumptions are considered to prove the convergence of this method in Banach spaces.

Concerning the convergence of inexact Newton methods

In this study, we use inexact Newton methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semi-local results of Kantorovich type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately, we can find sharper error bounds on the distances involved than before. Furthermore, we show that special cases of our results reduce to the corresponding ones already in the literature. Finally, our results are used to solve integral equations that cannot be solved with existing methods.

Standard kantorovich theorem of the chebyshev method on complex plane

We redefine the Chebyshev method in the equivalent form. Then we use analysis techniques which we have already developed in a series of our research works [1, 2, 3, 4, 5, 6] to establish a Kantorovich type convergence theorem. Also, a lower and an upper bound is clearly given under the standard Kantorovich assumptions for Newton's method.

Error bounds for Newton-like methods under Kantorovich type assumptions, II

Error bounds for Newton-like methods under Kantorovich type assumptions, II

Error bounds for Newton-like methods under Kantorovich type assumptions

This paper provides a method for deriving newa posteriori error bounds for Newton-like methods in a Banach space under Kantorovich type assumptions. The bounds found are sharper than those of Miel [10] and include those recently obtained by Moret [12]. The applicability of our method is studied for other types of iterations including Newton’s method.

On the behaviour of approximate Newton methods

Newton's method for solving non linear operator equations requests at each step the solution of a linear equation. When these equations are solved only approximately we have a so called Approximate Newton Method (A.N.M.). In this paper we examine the convergence and the order of convergence of A.N.M.'s under Kantorovich type hypotheses, giving criteria for controlling the behaviour of the iterations. Moreover a posteriori error estimates are proposed. The application of the general results to the case of Newton-Iterative methods is illustrated.

Saturation of Kantorovich type operators

It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szasz—Mirakjan and Baskakov operators inL1 metric in the optimal order {n−1} if and only ifϕ2f′ is of bounded variation where\(\varphi (x) = \sqrt x \) and\(\varphi (x) = \sqrt {x(1 + x)} \), respectively.

Some properties of a new kind of modulus of smoothness

The modulus of smoothness \omega (f, \delta)_{\varphi, p} = \mathrm {sup}_{0 < h ≤ \delta} \| \Delta^2_{h, \varphi} \|_{L^p} has arisen during the investigation of positive operators of the Kantorovich type. Here we show that \omega_{\varphi, p} resembles the ordinary case \varphi = 1 and we give the characterization of those functions f for which \omega (f, \delta)_{\varphi, p} = O (\delta^2) . The results obtained have applications to positive operators.

Necking of a bar

The necking process of an axisymmetric tension specimen made of elastic-plastic, strain-hardening material is analyzed by a generalized J2-flow theory for large deformations. The governing equations are solved in a Kantorovich type approach based on a variational principle. The effects of both geometric nonlinearities resulting from large deformation and the physical nonlinearities arising from plastic material behavior are considered. Numerical results have been found for the stress and deformation histories in the specimen up to 50 per cent reduction of the neck radius. The shape of the neck, for the first time, is rationally calculated.