Taylor's expansion for \(C^{1,1}\) functions in Asplund spaces
The author provides second-order Taylor's expansions for real-valued \(C^{1,1}\) functions in Asplund spaces using generalized second-order differentiability notions. By means of generalized subdifferentials for real-valued functions and associated coderivatives for set-valued mappings, second-order subdifferentials of a (first-order) subdifferentiable function \(\varphi: X\to\overline{\mathbb{R}}\) in a Banach space are introduced according to \[ \partial^2\varphi(\overline x,\overline y)(u):= (D^*\partial\varphi)(\overline x,\overline y)(u), \] where \(\overline y\in\partial\varphi(\overline x)\) and \(u\in X^{**}\). If \(\varphi\in C^{1,1}\) then of course \(\partial\varphi\) is single-valued and it can be shown that \[ \partial^2\varphi(\overline x)(u):= \partial^2\varphi(\overline x,\nabla\varphi(\overline x)(u)= \partial\langle u,\nabla\varphi\rangle(\overline x). \] On this base, the author presents the main theorem of the paper which points out that for a \(C^{1,1}\) function \(\varphi\) in an Asplund space \(X\) and for each interval \([a, b]\subset X\) there exists some \(c\in(a, b)\) such that \[ \varphi(b)\in \varphi(a)+ \langle\nabla\varphi(a), b-a\rangle+ \textstyle{{1\over 2}} \langle\partial^2\varphi(c)(\psi_{b-a}), b-a\rangle, \] where \(\psi_{b-a}\) is the natural embedding of \(b- a\in X\) in \(X^{**}\).
second-order subdifferential coderivative
scientific article; zbMATH DE number 5612995