Generalized Projections on Closed Nonconvex Sets in Uniformly Convex and Uniformly Smooth Banach Spaces

Abstract

The present paper is devoted to the study of the generalized projection : * → , where is a uniformly convex and uniformly smooth Banach space and is a nonempty closed (not necessarily convex) set in . Our main result is the density of the points * ∈ * having unique generalized projection over nonempty close sets in . Some minimisation principles are also established. An application to variational problems with nonconvex sets is presented.( ) ≥ (resp., ( ) ≤ ) .Obviously from the definition of -uniform convexity anduniform smoothness the constants and satisfy ∈ (1, 2] and ≥ 2. It is known (see, e.g., [8,9]) that uniformly convex

Existence Results for Second Order Nonconvex Sweeping Processes in q-Uniformly Convex and 2-Uniformly Smooth Separable Banach Spaces

“…We point out that this set may be empty when the space is not reflexive even if S is closed and convex (see Example 1.4. in [5]). In addition, we notice that for nonconvex sets S, the set π S (x * ) may be empty for some points x * ∈ X * (see Example 4.1 in [6]).…”

Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

“…To do that, we need the following new version of Borwein-Preiss variational principle in terms of the functional . For its proof we refer the reader to [13]. has a unique minimum at = .…”

“…We close this section by proving the well known Bishop-Phelps density theorem in terms of -proximal normal cone in smooth Banach spaces (see, e.g., [7]). To do that we need the following recent result on the generalised projection on nonconvex closed sets [13]. It is an analogue result to Lau's theorem for metric projections in reflexive Banach spaces ( [14]).…”

V -Prox-Regular Functions in Smooth Banach Spaces

“…Throughout this work, X will denote a reflexive smooth Banach space unless otherwise specified. We recall from [1] the concept of V-proximal subdifferential. Definition 1.…”

“…It has been proved in [2] that in general, we have the inclusion ∂ π f ðxÞ ⊂ ∂ π G f ðxÞ. The generalized projection on a closed nonempty set S is defined as follows: x ∈ π S ðx * Þ if and only if Vðx * , xÞ = inf x∈S Vðx * , xÞ (see [3] for convex sets and see [1] for nonconvex sets). We also recall (see for instance [2]) the definition of the Fréchet subdifferential and Fréchet normal cone as follows: x * ∈ ∂ F f ð xÞ if and only if for all ε > 0, there exists δ > 0 such that…”