Limiting subdifferentials of perturbed distance functions in Banach spaces

We explore in arbitrary Banach spaces the Fréchet type εε-subdifferentials and the limiting subdifferentials for the perturbed distance function dSJ(⋅) determined by a closed subset SS and a lower semicontinuous function JJ defined on SS. In particular, upper and lower estimates for the Fréchet type εε-subdifferentials and for the limiting subdifferentials are provided in terms of the corresponding subdifferentials of the sum of the associated functions JJ and δSδS.