K e y w o r d s G a u s s N e w t o n ' s method, Lipschitz conditions with L average, Convergence ball, Uniqueness bali. 1. I N T R O D U C T I O N Finding the solution of a nonlinear operator equation f(x) = 0 (1.1) in Banach space X is a very general subject which is widely used in both theoretical and applied areas of mathematics, where f is a nonlinear operator from some domain D in a real...

The Kantorovich Theorem is a fundamental tool in nonlinear analysis which has been extensively used in classical numerical analysis. In this paper we show that it can also be used in analyzing interior point methods. We obtain optimal bounds for Newton’s method when relied upon in a path following algorithm for linear complementarity problems. Given a point z that approximates a point z(τ ) on ...

In this paper, we show that the Kleinman algorithm can be used well to solve the algebraic Riccati equation (ARE) of singularly perturbed systems, where the quadratic term of the ARE may be inde1nite. The quadratic convergence property of the Kleinman algorithm is proved by using the Newton–Kantorovich theorem when the initial condition is chosen appropriately. In addition, the numerical method...

On the background of a careful analysis of linear DAEs, linearizations of nonlinear index2 systems are considered. Finding appropriate function spaces and their topologies allows to apply the standard Implicit Function Theorem again. Both, solvability statements as well as the local convergence of the Newton-Kantorovich method (quasilinearization) result immediately. In particular, this applies...

The recent developments of our method to design two freeform optical surfaces (either refractive or reflective) using an optimized ray mapping computation are presented. The procedure is based on the Monge-Kantorovich theory of optimal mass transport, and the related conditions for an optimal mapping. The procedure is illustrated by designing an efficient street lighting lens, achieving an opti...

Uko and Argyros provided in [18] a Kantorovich–type theorem on the existence and uniqueness of the solution of a generalized equation of the form f(u)+g(u) ∋ 0, where f is a Fréchet–differentiable function, and g is a maximal monotone operator defined on a Hilbert space. The sufficient convergence conditions are weaker than the corresponding ones given in the literature for the Kantorovich theo...

We propose a model to describe the optimal distributions of residents and services in a prescribed urban area. The cost functional takes into account the transportation costs (according to a Monge–Kantorovich-type criterion) and two additional terms which penalize concentration of residents and dispersion of services. The tools we use are the Monge–Kantorovich mass transportation theory and the...

Received: January 6, 2011 Accepted: January 24, 2011 doi:10.5539/jmr.v3n2p66 The research is supported by Zhejiang Provincial Education Department research projects (Y201016421) Abstract Monge-Kantorovich transportation problem in a bounded region of Euclidean space is transferred into a partial differential equations group. And then the explicit formula of the optimal coupling is achieved. The...