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Delta-convex mappings between Banach spaces and applications. (English) Zbl 0685.46027
The concept of delta-convex functions (i.e. differences of two continuous convex functions) is generalized to mappings F: \(X\to Y\) between normed spaces. Such a mapping is called delta-convex if there exists a (continuous) convex function f: \(X\to {\mathbb{R}}\) such that \(f+y^*\circ F\) is a continuous convex function for all \(y^*\in Y^*\) with \(\| y^*\| =1\). It is shown that differentiability of f implies that of F. Thus for Asplund spaces one obtains generic differentiability for locally delta-convex mappings. A composition theorem for locally delta- convex mappings is reproved. And an inverse function theorem (assuming that \(F^{-1}\) is Lipschitz) is given. Furthermore it is shown that some mappings which naturally arise in the theory of integral and differential equations (e.g. the Nemyckii and Hammerstein operators) are often delta- convex. The paper ends with 10 open problems.
Reviewer: A.Kriegl

46G05 Derivatives of functions in infinite-dimensional spaces