The equivalence of several basic theorems for subdifferentials
In the paper it is shown that special basic tools for \(\beta\)-subdifferentials introduced by different authors are equivalent. Let \(X\) be a Banach space and \(\beta\) be a bornology on \(X\). Then the lower-semicontinuous function \(g:X\to \mathbb{R}\) is called to be \(\beta\)-differentiable at \(x\) with \(\beta\)-derivative \(\nabla g(x)\) if the limit \[ \lim_{x\to 0} {g(x+ tu)- g(x)- t\cdot\langle g(x),u\rangle\over t}= 0 \] is uniform in \(u\in V\) for every set \(V\in\beta\). It is \(\beta\)-smooth at \(x\) if the mapping \(\nabla g:X\to X^*_\beta\) is continuous around \(x\) (where \(X^*_\beta\) is the dual space of \(X\) provided with the topology of uniform convergence on \(\beta\)-sets). Finally, the function \(f: X\to\mathbb{R}\) is called to be \(\beta\)-subdifferentiable and \(x^*\) is a \(\beta\)-subgradient at \(x\) if there exists a \(\beta\)-smooth function \(g\) such that \(f\)-\(g\) admits a minimum at \(x\). Using special bornologies for \(\beta\) (e.g., the family of singleton sets, bounded sets,\dots) then of course we get special differentiability notions. In detail, regarding this general differentiability concept, the author compares the following sum rules for \(N\) functions which are assumed to be lower-semicontinuous: -- the nonlocal fuzzy sum rule of Borwein, Treiman and Zhu; -- the local fuzzy sum rule of Ioffe; -- the Lipschitz local fuzzy sum rule (which is the same, but all functions are locally Lipschitz \phantom{-- }at \(x\)); -- the multidirectional mean-value theorem of Clarke, Ledyaev, Stern and Wolenski; -- the extremal principle of Mordukhovich. In the main theorem of the paper, the author proves the equivalence of these conditions -- without additional assumptions. Beside this, further interesting consequences of this result are given, e.g., a sufficient condition for these tools in form of the existence of a \(\beta\)-smooth Lipschitz bump-function and a new characterization of Asplund spaces by means of the mentioned sum rules in the special case of Fréchet subdifferentiability.
scientific article; zbMATH DE number 1230453