Subdifferentials of convex functions on time scales
The first part of this paper is devoted to convex functions. The authors consider real-valued functions defined on a time scale interval, give the definition of convexity, and present several necessary and sufficient conditions for convexity. They investigate the properties of convex and concave functions such as the relation to monotonicity of \(f^{\Delta}\) and \(f^{\nabla}\), and to the sign of \(f^{\Delta\Delta}\) and \(f^{\nabla\nabla}\). In the second part, they introduce the notion of a subdifferential. Consider a function \(f:I\to\mathbb R\), where \(I\) is a time scale interval. The subdifferential of \(f\) at \(t_0\in I\) is denoted by \(\partial f(t_0)\), and it is the set of all real numbers \(s\) such that \[ f(t)\geq f(t_0)+(t-t_0)s\;\;\;\text{ for every }t\in I. \] The authors prove several theorems which relate subdifferentiability, convexity and the existence of \(\Delta\) and \(\nabla\)-derivatives, including the following statements: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}]A function is subdifferentiable if and only if it is convex. \item[{\(\bullet\)}] If \(f\) is convex, \(t\in I\) and both \(f^\Delta(t)\) and \(f^\nabla(t)\) exist, then \(\partial f(t)=\{\alpha f^\Delta(t)+(1-\alpha)f^\nabla(t),\;\alpha\in[0,1]\}\). \item[{\(\bullet\)}] If \(f\) is convex and both \(\Delta\) and \(\nabla\)-differentiable, then \(\int_a^b\varphi(t)\Delta t\leq f(b)-f(a)\) and \(\int_a^b\varphi(t)\nabla t\geq f(b)-f(a)\) for every \(\varphi\in \partial f\). \item[{\(\bullet\)}] If \(f\) is subdifferentiable and attains a minimum at \(t\), then \(0\in\partial f(t)\). \end{itemize}}
scientific article; zbMATH DE number 5866058