Strongly unique minimization of functionals in Banach spaces with applications to theory of approximation and fixed points (Q1076327)

The paper proves a theorem on the existence of a strong minimum point for a real valued functional defined on a closed convex set and fulfilling some growth conditions. A strong minimum for a functional means that every minimizing sequence converges to the minimum point (in other papers the property is called well-posedness, see e.g. \textit{T. Zolezzi} [Appl. Math. Optimization 4, 209-223 (1978; Zbl 0381.90105)]. Moreover, there are some estimates for the best approximation problem in Hilbert and \(L^ p\)-spaces and an existence theorem for (strongly unique) centers in a Banach space. The last result gets as a corollary an existence theorem of fixed points for a uniformly Lipschitz mapping on a Hilbert space.