Submonotone mappings in Banach spaces and applications

The notions 'submonotone' and 'strictly submonotone' mapping, introduced by J. Spingarn in R-n, are extended in a natural way to arbitrary Banach spaces. Several results about monotone operators are proved for submonotone and strictly submonotone ones: Rockafellar's result about local boundedness of monotone operators; Kenderov's result about single-valuedness and upper-semicontinuity almost everywhere of monotone operators in Asplund spaces; minimality (as w(*)-cusco mappings) of maximal strictly submonotone mappings, etc. It is shown that subdifferentials of various classes of nonconvex functions defined as pointwise suprema of quasi-differentiable functions possess submonotone properties. Results about generic differentiability of such functions are obtained (among them are new generalizations of an Ekeland and Lebourg's theorem). Applications are given to the properties of the distance function in a Banach space with a uniformly Gateaux differentiable norm.