Ulam’s Type Stability and Generalized Norms

Generalized Proximity and Projection with Norms and Mixed-norms

We discuss generalized proximity operators (GPO) and their associated generalized projection problems. On inputs of size n, we show how to efficiently apply GPOs and generalized projections for separable norms and distance-like functions to accuracy in O(n log(1/ )) time. We also derive projection algorithms that run theoretically in O(n log n log(1/ )) time but can for suitable parameter range...

On the Integral Representations of Generalized Relative Type and Generalized Relative Weak Type of Entire Functions

In this paper we wish to establish the integral representations of generalized relative type and generalized relative weak type as introduced by Datta et al [9]. We also investigate their equivalence relation under some certain conditions.

The generalized spectral radius and extremal norms

The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generali...

Cauchy-Rassias Stability of linear Mappings in Banach Modules Associated with a Generalized Jensen Type Mapping

Logarithmic Norms and Nonlinear DAE Stability

Logarithmic norms are often used to estimate stability and perturbation bounds in linear ODEs. Extensions to other classes of problems such as nonlinear dynamics, DAEs and PDEs require careful modifications of the logarithmic norm. With a conceptual focus, we combine the extension to nonlinear ODEs [15] with that of matrix pencils [10] in order to treat nonlinear DAEs with a view to cover certa...