Properties of generalized differentials
Let X be a real Banach space with unit ball B and with dual \(X^*\) and let \(\phi\) :X\(\to [-\infty,\infty]\) be any function with \(\phi(\overset\circ x)\in (-\infty,\infty)\). \(cl^{\downarrow}\phi\) denotes the lower semicontinuous envelope of \(\phi\). The author considers the generalized differential of \(\phi\) at \(\overset\circ x\) defined by the formula \(D^-\phi (\overset\circ x)=\cap_{\epsilon >0}cl\cup [\partial^-cl\phi (x);x\in B^{\epsilon}_{\phi}(\overset\circ x)\)] where \(\partial^-_{\epsilon}\phi (x)=\{x^*\in X^*;\lim \inf_{y\to x}[\phi(y)- \phi(x)- <x^*,y-x>/\| y-x\| \geq - \epsilon \}\), \(B^{\epsilon}_{\phi}(\overset\circ x)=\{x\in \overset\circ x+\epsilon B;| cl^{\downarrow}\phi(x)- \phi(\overset\circ x)| o\epsilon\}\) and cl denotes the closure. This definition refines the notion of generalized gradient of \textit{F. H. Clarke} [Trans. Am. math. Soc. 205, 247-262 (1975; Zbl 0307.26012)] and that of derived sets of \textit{J. Warga} [Optimal control of differential and functional equations (1972; Zbl 0253.49001)]. Several natural and useful properties of generalized differentials are established, the most notable being the inclusion \(D^-(\phi_ 1+\phi_ 2)\subset D^- \phi_ 1+D^-\phi_ 2\).
scientific article; zbMATH DE number 3960290
scientific article