V-Proximal Trustworthy Banach Spaces

Abstract

In a recent work (2016), the first author proved the fuzzy sum rule for the V-proximal subdifferential under some natural assumptions on an equivalent norm of the Banach spaces. In the present paper, we are going to prove that the class of Banach spaces satisfying the fuzzy sum rule is very large and contains all Lp spaces 1<p<∞ as well as the sequence spaces lp1<p<∞, the Sobolev spaces Wp,n1<p<∞, and the Schatten trace ideals Cp1<p<∞.

Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

“…In [5], we introduced and studied the class of V-proximal trustworthy spaces as follows: Definition 2 ([5]). We will say that a Banach space X is V-proximal trustworthy provided that for any > 0, any two functions f 1 , f 2 : X → R ∪ {∞}, and any u ∈ X such that f 1 is lower semicontinuous and f 2 is Lipschitz around u, the following fuzzy sum rule holds:…”

V -Prox-Regular Functions in Smooth Banach Spaces

“…We recall from [2] that X is V -proximal trustworthy provided that for any ε > 0, any two functions f 1 , f 2 : X ⟶ ℝ ∪ f∞g and any u ∈ X such that f 1 is lower semicontinuous and f 2 is Lipschitz around u the following fuzzy sum rule holds:…”

“…Here, U f i ðu, εÞ ≔ fx ∈ u + εB such that | f i ðxÞ − f i ðuÞ|< εg, and B * denotes the closed unit ball in X * . The proof of the following proposition can be found in [2]. Proposition 5.…”

“…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).…”

“…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…”