Convergence of Subdifferentials versus Convergence of Slopes

Given a sequence of lower semicontinuous proper convex functions on a Banach space, the graph convergence of their subdifferentials as well as the epigraphical convergence of their slopes are investigated. Several (counter-)examples showing that equivalence between these convergences is uncommon in infinite dimension are provided. However, under suitable additional conditions it is possible to establish some relations of this kind. Herein, under a compactness type assumption it is shown that bounded mixed convergence of subdifferentials of the sequence of functions is equivalent to the convergence of subdifferentials in the sense of Painlevé-Kuratowski, which turns to be equivalent to the epigraphical convergence of slopes of the functions in separable Banach spaces.