In the study of nonexpansive mappings and fixed point theory the domain of the mapping is usually assumed to be bounded or, as in certain approximation results (see, e.g., [20]), fixed points are assumed to exist. However in [21] Penot used uniform asymptotic concepts which he had earlier introduced in [22] to extend the Browder-Göhde-Kirk theorem to unbounded sets. The term “asymptotic” is use...

Measures of risk appear in two categories: Risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures (safety measures). Pure risk measures (risk deviation measures) are natural generalizations of the standard deviation. While pur...

This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: sup-, sup⋆- and symmetric subdifferentials. They satisfy most fundamental properties convexity, weak∗-closedness and, if function is bounded, weak∗-compactness. Moreover, they are nonempty for Rainwater function, possess certain calculus rules, provide necessary sufficient optimality conditions not ne...

In this short note, we derive an upper estimate of Clarke's subdifferential marginal functions in Banach spaces. The structure the is very similar to other results already obtained literature. novelty lies on fact that our assertions general spaces, and avoid use Asplund assumption.

and Applied Analysis 3 Furthermore, if h is a convex function, then ∂ c h(?̂?) is in accordance with the subdifferential ∂h(?̂?) in convex analysis; that is,

We prove that the subdifferential of any semi-algebraic extended-real-valued function on Rn has n-dimensional graph. We discuss consequences for generic semi-algebraic optimization problems.