文摘
Based on a primal regularity criterion we provide lower bounds for the regularity modulus of a nonlinear single-valued mapping F from a Banach space X into another Banach space Y . We focus on the case when F is defined on a proper (closed convex) subset of X only rather than on the whole of X. Three possible ways of approximating F around the reference point are considered. First, we use a tangential approximation by set-valued mappings associated with the Bouligand’s tangent cone to the graph of F. Then we move on to approximations by positively homogeneous set-valued mappings whose graphs contain the graph of F, for example, by the strict prederivative. Finally, we use an approximation by bunches of continuous linear operators. In the first two cases finding approximating objects is relatively easy while in the third case the approximating object is very convenient to work with. On examples, we illustrate that these approaches are different and neither of them implies the other, unless the spaces in question are finite dimensional. Keywords Metric regularity linear openness tangential approximation contingent tangent cone strict prederivative measure of noncompactness Clarke’s inverse function theorem cofinal family renorming nonsmooth analysis