An Estimate on Linear Functionals’ Kernels in Banach Spaces, and Regularity of Convex Functionals

Abstract

Motivated to obtain the second critical point of a nonlinear differential equation, which is expressed by derivatives of convex functional defined on a Banach space, an estimate with is given to see the relation between f -1 (0) and g -1 (0). And both the Fréchet differentiability and the continuity of Fréchet derivative of every convex functional defined on an open subset of a Banach space are shown.