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Local convergence theorems of Newton’s method for nonlinear equations using outer or generalized inverses

Ioannis K. Argyros (2000)

Czechoslovak Mathematical Journal

We provide local convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our convergence balls differ from earlier ones. In fact we show that with a simple numerical example that our convergence ball contains earlier ones. This way we have a wider choice of initial guesses than before. Our results can be used to solve undetermined systems, nonlinear...

Local entropy moduli and eigenvalues of operators in Banach spaces.

Bernd Carl, Thomas Kühn (1985)

Revista Matemática Iberoamericana

In the paper local entropy moduli of operators between Banach spaces are introduced. They constitue a generalization of entropy numbers and moduli, and localize these notions in an appropriate way. Many results regarding entropy numbers and moduli can be carried over to local entropy moduli. We investigate relations between local entropy moduli and s-numbers, spectral properties, eigenvalues, absolutely summing operators. As applications, local entropy moduli of identical and diagonal operators...

Local Exchange Potentials for Electronic Structure Calculations

Eric Cancès, Gabriel Stoltz, Gustavo E. Scuseria, Viktor N. Staroverov, Ernest R. Davidson (2009)

MathematicS In Action

The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock model as well as in some instances of the density functional theory. In a number of applications, it is convenient to approximate this integral operator by a multiplication operator, i.e. by a local potential. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator including the Slater potential, the optimized effective...

Local integrated C-semigroups

Miao Li, Fa-lun Huang, Quan Zheng (2001)

Studia Mathematica

We introduce the notion of a local n-times integrated C-semigroup, which unifies the classes of local C-semigroups, local integrated semigroups and local C-cosine functions. We then study its relations to the C-wellposedness of the (n + 1)-times integrated Cauchy problem and second order abstract Cauchy problem. Finally, a generation theorem for local n-times integrated C-semigroups is given.

Local Lipschitz continuity of the stop operator

Wolfgang Desch (1998)

Applications of Mathematics

On a closed convex set Z in ℝ N with sufficiently smooth ( 𝒲 2 , ∞ ) boundary, the stop operator is locally Lipschitz continuous from 𝐖 1 , 1 ( [ 0 , T ] , ℝ N ) × Z into 𝐖 1 , 1 ( [ 0 , T ] , ℝ N ) . The smoothness of the boundary is essential: A counterexample shows that 𝒞 1 -smoothness is not sufficient.

Local polynomials are polynomials

C. Fong, G. Lumer, E. Nordgren, H. Radjavi, P. Rosenthal (1995)

Studia Mathematica

We prove that a function f is a polynomial if G◦f is a polynomial for every bounded linear functional G. We also show that an operator-valued function is a polynomial if it is locally a polynomial.

Local properties of accessible injective operator ideals

F. Oertel (1998)

Czechoslovak Mathematical Journal

In addition to Pisier’s counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a “good behaviour” of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible...

Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

Joanna Janczewska (2004)

Open Mathematics

In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R k →Y is a C 2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R 1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R k and we describe the solution set of the studied equation in a small neighbourhood of this point.

Currently displaying 181 – 200 of 253