Porosity of ill-posed problems

Abstract

Abstract. We prove that in several classes of optimization problems, including the setting of smooth variational principles, the complement of the set of well-posed problems is σ-porous.

Generic well‐posedness in minimization problems

“…The latter is suited to give a unified approach to get results for problems with functional constraints, which will be described in the next section. Finally, an extension of the Deville-Godefroy-Zizler principle, involving σ-porosity rather than Baire category, has been proved by Deville-Revalski in [7].…”

An Extension of the Banach–stone Theorem

“…We give below some examples. Let us mention here that the axioms (A 1 ) and (A 3 ) are related to the variational principle of Deville et al [14] and Deville and Revalski [16] (see Theorem 2.7 below) and the axiom (A β 4 ) was introduced and studied in [6] and is a part of the hypothesis of [6,Theorem 2.8]. The theorems [6,Theorem 2.8] and Theorem 2.7 will be used in a crucial way in the proof of our main result (Theorem 1.2).…”

“…The proof of Theorem 1.2 will be given in Section 3. It is based on the differentiability of some convex functions generalizing the norm • ∞ and a duality result introduced in [6] together with the Deville-Godefroy-Zizler variational principle (see [14,16]). Note that the original proof of the Banach-Stone theorem in the compact metric case, given by Banach in [9], is based on the Gâteaux differentiability of the norm…”

On σ‐porous sets in abstract spaces

“…On the other hand, in many recent papers (see, e.g., [23,25,26,105]) concerning Banach (and other abstract) spaces σ-porosity means σ-lower porosity. In fact, in these papers σ-porosity is defined in a formally different but equivalent way: by the condition (ii) of the following well-known proposition.…”