Mean value theorem for functions possessing first order convex approximations. Applications in optimization theory (Q1062326)

The main result of the paper is the following: Let X be an l.t.s., \(A\subset X\), [a,b]\(\subset int A\) and \(f: A\to R\) satisfying the conditions a) f is hemicontinuous on [a,b], b) for every \(x\in]a,b[\) there exists \(\phi_ x: X\to R\) a first order convex approximation for f at x and c) \(x\to \phi_ x(a-b)\), \(x\to \phi_ x(b-a)\) are upper semicontinuous. Then there exists \(c\in]a,b[\) and \(x^*\in \delta \phi_ c(0)\) such that \(f(b)-f(a)=<x^*,b-a>\). An application to optimization problems is given.