Continuous selections for metric projection operators and for their generalizations

Existence, uniqueness, and stability of best and near-best approximations

“…Corollary 3.1 (see [64]). Let X be a Banach space and M ⊂ X be a stably monotone path-connected set (or P -compact monotone path-connected set) with Hausdorff continuous metric projection.…”

“…where E ⊂ X is an arbitrary bounded set and y 0 ∈ Y , tends to zero as δ → 0+. Theorem 4.3 (see [64]). Let (X, ∥ • |) be an asymmetric seminormed space whose asymmetric seminorm is equivalent to some seminorm on X with respect to which X is complete, let F : X → 2 X be a ∥ • |-stable mapping, and let, for all y ∈ X , the set M y = F (y) be closed and B-infinitely connected (that is, B-infinitely connected relative to the seminorm ∥ • |).…”

“…Theorem 4.4 (see [64]). Let X be a Banach space and M ⊂ X be an existence set such that the metric projection operator onto M is Hausdorff continuous and has B-infinitely connected values.…”

“…The ε-neighbourhood of a set E ⊂ X is defined by {x ∈ X | ρ(x, E) < ε}. Theorem 4.5 (see [64]). Let (X, ∥ • |) be an asymmetric seminormed space whose asymmetric seminorm ∥ • | is equivalent to some seminorm of X , and let a non-empty set A ⊂ X be B-infinitely connected.…”

Approximative and structural properties of sets in asymmetric spaces

“…We will next proceed as in the proof of Theorem 7 in [5]. The mapping f , which associates with each w from the set…”