Implicit multifunction theorems (Q1970321)

In this interesting paper the aim of the authors is to solve the implicit multifunction equation \(0\in F(x,p),\) where \(F:U\rightarrow 2^{Y}\) with \( U\subset X\times P,\) \(X,Y\) being Banach spaces and \(P\) a metric space. In order to solve this equation they prove first an implicit multifunction theorem for \(f(x,p)\leq 0\) for \(f:U\rightarrow R.\) For proving this result they use the smooth variational principle, a fuzzy sum rule for Fréchet subdifferentials, the multidirectional mean value inequality and a theorem on the subdifferential of marginal functions. In order to obtain Robinson-Ursescu's theorem the authors also refine the multidirectional mean value inequality.