Search results for "Uniform norm"

Norm or numerical radius attaining polynomials on C(K)

2004

Abstract Let C(K, C ) be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on C(K, C ) attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to one. In the case of a compact metric space a characterization is obtained. As a consequence we show that, for a scattered compact Hausdorff space K, every continuous n-homogeneous complex polynomial on C(K, C ) can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the u…

Approximating the solutions of differential inclusions driven by measures

2019

The matter of approximating the solutions of a differential problem driven by a rough measure by solutions of similar problems driven by “smoother” measures is considered under very general assumptions on the multifunction on the right-hand side. The key tool in our investigation is the notion of uniformly bounded $$\varepsilon $$-variations, which mixes the supremum norm with the uniformly bounded variation condition. Several examples to motivate the generality of our outcomes are included.

A Structural Theorem for Metric Space Valued Mappings of Φ-bounded Variation

2009

In this paper we introduce the notion of $\Phi$-bounded variation for metric space valued mappings defined on a subset of the real line. Such a notion generalizes the one for real functions introduced by M. Schramm, and many previous generalized variations. We prove a structural theorem for mappings of $\Phi$-bounded variation. As an application we show that each mapping of $\Phi$-bounded variation defined on a subset of $\mathbb{R}$ possesses a $\Phi$-variation preserving extension to the whole real line.

C0-semigroups norm continuous at infinity

1996

Decay estimates in the supremum norm for the solutions to a nonlinear evolution equation

2014

We study the asymptotic behaviour, as t → ∞, of the solutions to the nonlinear evolution equationwhere ΔpNu = Δu + (p−2) (D2u(Du/∣Du∣)) · (Du/∣Du∣) is the normalized p-Laplace equation and p ≥ 2. We show that if u(x,t) is a viscosity solution to the above equation in a cylinder Ω × (0, ∞) with time-independent lateral boundary values, then it converges to the unique stationary solution h as t → ∞. Moreover, we provide an estimate for the decay rate of maxx∈Ω∣u(x,t) − h(x)∣.