In this paper we address some of the most fundamental questions regarding the differentiability structure of locally Lipschitz functions defined on separable Banach spaces. For example, we examine the relationship between integrability, D-representability, and strict differentiability. In addition to this, we show that on any separable Banach space there is a significant family of locally Lipsc...

Peano differentiability generalizes ordinary differentiability to higher order. There are two ways to define Peano differentiability for functions defined on non-open sets. For both concepts, we investigate the question under which conditions a function defined on a closed set can be extended to a Peano differentiable function on the ambient space if the sets and functions are definable in an o...

In this article, issues of both stability and dissipativity for a type bidirectional associative memory (BAM) neural systems with time delays are investigated. By using generalized Halanay inequalities constructing appropriate Lyapunov functionals, some novelty criteria obtained the asymptotic BAM delays. Also, without assuming boundedness differentiability activation functions, new sufficient ...

• Banach-Alaoglu: compactness of polars • Variant Banach-Steinhaus/uniform boundedness • Second polars • Weak boundedness implies boundedness • Weak-to-strong differentiability The comparison of weak and strong differentiability is due to Grothendieck, although the original sources are not widely available.

This article proposes a general gH-gradient efficient-direction method and $${\mathcal{W}}$$ -gH-gradient efficient for the optimization problems with interval-valued functions. The convergence analysis step-wise algorithms of both methods are presented. It is observed that converges linearly strongly convex objective function. To develop proposed to study their convergence, ideas strong convex...

Recent work in the areas of nonparametric regression and spatial smoothing has focused on modelling functions of inhomogeneous smoothness. In the regression literature, important progress has been made in fitting free-knot spline models in a Bayesian context, with knots automatically being placed more densely in regions of the covariate space in which the function varies more quickly. In the sp...

The Hukuhara-Levelt-Turrittin decomposition theorem gives a classification of differential modules over the field C((z)) of formal Laurent series resembling the decomposition of a finite-dimensional vector space equipped with a linear endomorphism into generalized eigenspaces. It implies that after adjoining a suitable root of z, one can express any differential module as a successive extension...

in this paper, the (m+1)-step adams-bashforth, adams-moulton, and predictor-correctormethods are used to solve rst-order linear fuzzy ordinary dierential equations. the conceptsof fuzzy interpolation and generalised strongly dierentiability are used, to obtaingeneral algorithms. each of these algorithms has advantages over current methods. moreover,for each algorithm a convergence formula can b...