Convergences of subgradients of sequences of convex functions (Q388473)

Let \(E\) be a real Banach space with dual \(E^*,\) and \(f,f_n:E\to\mathbb R\cup\{\infty\},\, n\in\mathbb N,\, \) lsc convex functions. The author considers conditions under which for every \(x^*\in\partial f(x)\) there exist sequences \(\{x_n\}_{n=1}^\infty\subset E\) and \(x^*_n\in \partial f(x_n),\, n\in \mathbb N,\,\) such that \(x_n\to x,\, x_n^*\to x^* \) (convergence in norms) and \(f_n(x_n)\to f(x).\) Such a condition is given in Theorem 3.1, namely, the existence of a sequence \(u_n\to x\) with \(f_n(u_n)\to f(x)\) and of \(r>0\) such that NEWLINE\[NEWLINE \liminf_{n\to\infty}\inf\{f_n(u)-f_n(u_n)-\langle x^*,u-u_n\rangle : u\in B[x,r]\}\geq 0.NEWLINE\]NEWLINENEWLINENEWLINEHe shows that this condition is fulfilled when \(\{f_n\}\) is a sequence of lsc convex functions which is epigraphical slice convergent to the lsc convex function \(f\), as well as when \(\{f_n\}\) is an increasing sequence of lsc convex functions whose limit \(f\) is Fréchet derivable at \(x.\)NEWLINENEWLINEThe case of \(w^*\)-convergence of \(\{x^*_n\}\) to \(x^*\) was treated by the author in several previous papers, see, e.g., [\textit{D. Zagrodny}, J. Convex Anal. 12, No. 1, 213--219 (2005; Zbl 1071.49016)].NEWLINENEWLINEAs application, one gives a new proof to a result of \textit{L. P. Vlasov}, [Mat. Zametki 8, 545--550 (1970; Zbl 0203.12101)], asserting that in a Banach space \(E\) with strictly convex dual, the closure of the union of any increasing sequence of balls with radii tending to infinity is a half space or the whole space.