Error bounds of functions by proximal subdifferentials in Hilbert spaces

This paper is to study error bounds of a proper lower semicontinuous function defined on a Hilbert space and provide several dual conditions for the error bounds.By proximal subdifferentials of the given function, we give upper estimates on the error bounds moduli and thus obtain sufficient dual conditions for error bounds.Further, in terms of proximal subdifferentials of the concerned function at those points inside the solution set, we provide necessary dual conditions for the error bounds.These results are applied to the error bounds of the composite-convex function (i.e. a composition of a convex function with a continuously differentiable mapping between two Hilbert spaces) and some new dual conditions ensuring the error bounds are obtained.