We present simple proofs that spaces of homogeneous polynomials on Lp[0, 1] and ℓp provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976)., Seán Dineen, Jorge Mujica., and Obsahuje seznam literatury
We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.
We extend the applicability of Newton's method for approximating a solution of a nonlinear operator equation in a Banach space setting using nondiscrete mathematical induction concept introduced by Potra and Pták. We obtain new sufficient convergence conditions for Newton's method using Lipschitz and center-Lipschitz conditions instead of only the Lipschitz condition used in F. A. Potra, V. Pták, Sharp error bounds for Newton's process, Numer. Math., 34 (1980), 63–72, and F. A. Potra, V. Pták, Nondiscrete Induction and Iterative Processes, Research Notes in Mathematics, 103. Pitman Advanced Publishing Program, Boston, 1984. Under the same computational cost as before, we provide: weaker sufficient convergence conditions; tighter error estimates on the distances involved and more precise information on the location of the solution. Numerical examples are also provided in this study.
Let $\mathcal B_c$ denote the real-valued functions continuous on the extended real line and vanishing at $-\infty $. Let $\mathcal B_r$ denote the functions that are left continuous, have a right limit at each point and vanish at $-\infty $. Define $\mathcal A^n_c$ to be the space of tempered distributions that are the $n$th distributional derivative of a unique function in $\mathcal B_c$. Similarly with $\mathcal A^n_r$ from $\mathcal B_r$. A type of integral is defined on distributions in $\mathcal A^n_c$ and $\mathcal A^n_r$. The multipliers are iterated integrals of functions of bounded variation. For each $n\in \mathbb N$, the spaces $\mathcal A^n_c$ and $\mathcal A^n_r$ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to $\mathcal B_c$ and $\mathcal B_r$, respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space $\mathcal A_c^1$ is the completion of the $L^1$ functions in the Alexiewicz norm. The space $\mathcal A_r^1$ contains all finite signed Borel measures. Many of the usual properties of integrals hold: Hölder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.
Starting from Lagrange interpolation of the exponential function ${\rm e}^z$ in the complex plane, and using an integral representation formula for holomorphic functions on Banach spaces, we obtain Lagrange interpolating polynomials for representable functions defined on a Banach space $E$. Given such a representable entire funtion $f\colon E \to \mathbb C$, in order to study the approximation problem and the uniform convergence of these polynomials to $f$ on bounded sets of $E$, we present a sufficient growth condition on the interpolating sequence.
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 least squares problems and ill-posed nonlinear operator equations.
We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “Lipschitz-type” and center-“Lipschitz-type” instead of just “Lipschitz-type” conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
We consider a nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in a Banach space. The existence of at least one solution is proved by using the set-valued analog of Mönch fixed point theorem associated with the technique of measures of noncompactness.
Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated $\sigma $-ideals are studied. These $\sigma $-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.
Let $B(X)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B(X)$ satisfying the relation $\sigma _{T_1}(x) = \sigma _{T_2}(x)$ for any vector $x\in X$, where $\sigma _T(x)$ denotes the local spectrum of $T\in B(X)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1 - T_2$ is a quasinilpotent operator, that is $\Vert (T_1 - T_2)^n\Vert ^{1/n} \rightarrow 0$ as $n \rightarrow \infty $. Without these conditions, we describe the operators with the same local spectra only in some particular cases.