On Fréchet subdifferential of supremum for arbitrary family of continuous functions

The paper focuses on the Fréchet subdifferential of the pointwise supremum of the family of functions taken over arbitrary index sets. The all functions in the corresponding family are defined on a Fréchet smooth space; this class of Banach spaces includes all reflexive spaces and all separable Asplund spaces. The new upper estimates, including non-convex ones, establish for the Fréchet subdifferentials of the suprema of continuous functions and lower semicontinuous functions. In these estimates, an additional requirement is imposed on every $\varepsilon$-active index that corresponds continuous function: the $\varepsilon$-closeness of the considered point of the graph of the supremum to the graph of this continuous function. The key two-sided inequalities with respect to the graph of continuous function, corresponding to $\varepsilon$-active index, are based on the two-sided unidirectional mean value inequality. The method of proving upper estimates combines the approaches of works of J. S. Treiman, Y. S. Ledyaev, B. S. Mordukhovich, T. Nghia, and P. Pérez–Aros.