# Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

Authors: Messaoud Bounkhel
Journal: Journal of Function Spaces
 (2016)

---

## Abstract

In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, inp-uniformly convex and q-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.


---

## Body

## 1. Introduction

In [1], the authors introduced the concept of geometric proximal subdifferential ∂
G
π
f
(
x
) and the concept of analytic proximal subdifferential ∂
A
π
f
(
x
) in reflexive Banach spaces for x
∈
dom
⁡
f. They also introduced the concept of generalised proximal normal cone N
π
(
S
;
x
) for x
∈
S. The main tool in their approach is the generalised projection operator π
S and the functional V (see [1, 2]). They proved various important properties for their concepts.In order to avoid any confusion with other existing proximal subdifferential and proximal normal cones in Banach spaces (see, e.g., [3–6]) we will call ∂
A
π
f, ∂
G
π
f, and N
π
(
S
;
·
) analytic V-proximal subdifferential, geometric V-proximal subdifferential, and V-proximal normal cone, respectively. These names are more natural since these concepts are strongly based on V and we use these names in all the paper.The main aim of this work is to prove the fuzzy sum rule and the chain rule for both proximal subdifferentials. Our results extend the existing results on the usual proximal subdifferential from Hilbert spaces setting to our Banach spaces setting. We start by proving that∂
A
π
f
=
∂
G
π
f whenever the space admits an equivalent norm which is C
2 away from the origin. Sections 3 and 4 are devoted to proving the fuzzy sum rule and the chain rule. In Section 5, we define the limiting constructions for the V-proximal concepts (the analytic and geometric V-proximal subdifferential and the V-proximal normal cone). In this section we prove that these limiting constructions coincide with the limiting constructions in [7] whenever the space admits an equivalent norm with a power of that norm being C
2 away from the origin. Section 5 is ended by proving the well known Bishop-Phelps theorem for the V-proximal normal cone and its limiting construction. The paper is closed by an application, in which, we derive a necessary optimality condition for nonconvex minimisation problems and nonconvex variational inequalities in terms of generalised projections.


## 2. Preliminaries

LetX be a reflexive smooth Banach space with topological dual space X
∗. We will denote by B and B
∗ the closed unit ball centred at the origin in X and X
∗, respectively. Let f
:
X
→
R
∪
{
+
∞
} be a function and x
∈
X where f is finite. We recall from [1] the concept ofanalytic
V
-proximal subdifferential
∂
A
π
f
(
x
) (called in [1]analytic proximal subdifferential). An element x
∗
∈
X
∗ belongs to ∂
A
π
f
(
x
) provided that there exists σ
>
0 so that(1)
x
∗
,
x
′
-
x
≤
f
x
′
-
f
x
+
σ
V
J
x
,
x
′
,for x
′ very close to x, where J
:
X
→
X
∗ is the normalized duality mapping and V
:
X
∗
×
X
→
R is a functional defined by(2)
V
x
∗
,
x
=
x
∗
2
-
2
x
∗
,
x
+
x
2
,
for  any
x
∗
∈
X
∗
,
x
∈
X
.For a closed nonempty setS in X and x
¯
∈
S, the authors in [1] defined the concept of V
-proximal normal cone
N
π
(
S
;
x
¯
) (called in [1]proximal normal cone) by N
π
(
S
;
x
¯
)
=
∂
A
π
ψ
S
(
x
¯
). Using this V-proximal normal cone, the authors in [1] defined another concept of V-proximal subdifferential, calledgeometric proximal subdifferential, as follows: An element x
∗
∈
X
∗ belongs to ∂
G
π
f
(
x
) provided (
x
∗
,
-
1
)
∈
N
π
(
e
p
i
f
;
x
¯
,
f
x
¯
), where e
p
i
f is the epigraph of f defined by e
p
i
f
=
{
(
x
,
r
)
∈
X
×
R
:
f
(
x
)
≤
r
}. In this paper ∂
G
π
f will be calledgeometric
V
-proximal subdifferential. We recall, respectively, the well known concepts of proximal subdifferential and Fréchet subdifferential (see, e.g., [5]):(i)
x
∗
∈
∂
P
f
(
x
) if and only if there exist δ
,
σ
>
0 such that(3)
x
∗
,
x
′
-
x
≤
f
x
′
-
f
x
+
σ
x
′
-
x
2
,
for  any
x
′
∈
x
+
δ
B
.
(ii)
x
∗
∈
∂
F
f
(
x
) if and only if for any ϵ
>
0 there exists δ
>
0 such that(4)
x
∗
,
x
′
-
x
≤
f
x
′
-
f
x
+
ϵ
x
′
-
x
,
for  any
x
′
∈
x
+
δ
B
.The Fréchet and proximal normal cone are defined as N
F
(
S
;
x
¯
)
=
∂
F
ψ
S
(
x
¯
) and N
P
(
S
;
x
¯
)
=
∂
P
ψ
S
(
x
¯
)
. Notice that (see [7, 8]) Fréchet and proximal subdifferential can be defined geometrically by the formulas ∂
F
f
(
x
¯
)
=
{
x
∗
∈
X
∗
:
(
x
∗
,
-
1
)
∈
N
F
(
S
;
x
¯
)
} and ∂
P
f
(
x
¯
)
=
{
x
∗
∈
X
∗
:
(
x
∗
,
-
1
)
∈
N
P
(
S
;
x
¯
)
}.We recall (see, e.g., [1, 9]) the definition of p-uniformly convex and q-uniformly smooth Banach spaces. The space X is said to be p-uniformly convex (resp., q-uniformly smooth) if there is a constant c
>
0 such that (5)
δ
X
ϵ
≥
c
ϵ
p
resp
.
ρ
X
t
≤
c
t
q
,where δ
X and ρ
X are defined, respectively, by (6)
δ
X
ϵ
=
i
n
f
1
-
x
+
y
2
:
x
=
y
=
1
,
x
-
y
=
ϵ
,
0
≤
ϵ
≤
2
,
ρ
X
t
=
sup
⁡
1
2
x
+
y
+
x
-
y
-
1
:
x
=
1
,
y
=
t
,
t
>
0
.Notice that the constants p and q in the previous definition always satisfy p
≥
2 and q
∈
(
1,2
].The following inclusions have been proved in [1]:(7)
N
π
S
;
x
¯
⊂
N
F
S
;
x
¯
,
∂
A
π
f
x
⊂
∂
G
π
f
x
⊂
∂
F
f
x
.Consider the following two assumptions on the norm on X at a given point x
¯
∈
X: (A1)
V
(
J
(
x
¯
)
,
x
)
≤
α
x
-
x
¯
q, for x close to x
¯ and for some α
>
0 and q
∈
(
1,2
].
(A2)
α
x
-
x
¯
p
≤
V
(
J
(
x
¯
)
,
x
), for x close to x
¯ and for some α
>
0 and p
≥
2.Using Lemma  4.1 in [1], we get that (A1) is satisfied for any q-uniformly smooth Banach spaces and (A2) is satisfied for any p-uniformly convex Banach spaces. Easily we can check that if (A1) is satisfied with q
=
2, then we have N
π
(
S
;
x
¯
)
⊂
N
P
(
S
;
x
¯
) and ∂
G
π
f
(
x
)
⊂
∂
P
f
(
x
) and so ∂
A
π
f
(
x
)
⊂
∂
G
π
f
(
x
)
⊂
∂
P
f
(
x
)
⊂
∂
F
f
(
x
) and N
π
(
S
;
x
¯
)
⊂
N
P
(
S
;
x
¯
)
⊂
N
F
(
S
;
x
¯
). This is the case, for instance, of L
p spaces with p
≥
2. The reverse inclusions N
P
(
S
;
x
¯
)
⊂
N
π
(
S
;
x
¯
) and ∂
P
f
(
x
)
⊂
∂
A
π
f
(
x
) hold whenever (A2) is satisfied with q
=
2 and so N
P
(
S
;
x
¯
)
⊂
N
π
(
S
;
x
¯
)
⊂
N
F
(
S
;
x
¯
) and ∂
P
f
(
x
)
⊂
∂
A
π
f
(
x
)
⊂
∂
G
π
f
(
x
)
⊂
∂
F
f
(
x
) which is the case of L
p spaces with p
∈
(
1,2
]. In the general case we cannot provide a relationship between ∂
P
f
(
x
) and the two V-proximal subdifferentials ∂
G
π
f
(
x
) and ∂
A
π
f
(
x
).We recall from [1] the following important characterisation of V-proximal normal cone in terms of the generalised projection.Proposition 1.
For any closed setS and any point x
¯
∈
S we have (8)
N
π
S
;
x
¯
=
x
∗
∈
X
∗
:
∃
α
>
0
s
u
c
h
t
h
a
t
x
¯
∈
π
S
J
x
¯
+
α
x
∗
.Hereπ
S
:
X
∗
→
S is the generalised projection operator defined as follows (see [10, 11]): (9)
π
S
x
∗
=
x
∈
S
:
V
x
∗
,
x
=
inf
s
∈
S
⁡
V
x
∗
,
s
.We summarise in the following proposition some needed results of the analytic and geometricV-proximal subdifferentials proved in [1].Proposition 2.
LetX be a p-uniformly convex and q-uniformly smooth Banach space and let f
:
X
→
R
∪
{
+
∞
} be a lower semicontinuous function and x
¯
∈
dom
⁡
f:(i)
Ifg is C
2 on a neighbourhood of x
¯, then (10)
∂
A
π
f
+
g
x
¯
=
∂
A
π
f
x
¯
+
g
′
x
¯
.
(ii)
Lety
∗
,
x
∗
∈
X
∗, and α
>
0. Then (11)
∂
A
π
f
+
y
∗
,
·
+
α
V
x
∗
,
·
x
¯
=
y
∗
+
2
α
J
x
¯
-
x
∗
+
∂
A
π
f
x
¯
.It has been proved in [1] that the equality form ∂
A
π
f
(
x
)
=
∂
G
π
f
(
x
) holds for convex l.s.c. functions and for indicator functions associated with closed nonempty sets in any reflexive smooth Banach space. So, it is a natural question to ask whether this equality is valid for general forms of functions. In the following proposition we give a positive answer with additional assumptions on the space for any l.s.c. function.Proposition 3.
LetX be p-uniformly convex and q-uniformly smooth Banach space. Assume that X admits an equivalent norm
⁡
· which is C
2 on X
∖
{
0
}. Let V be the functional associated with ·. Let f
:
X
→
R
∪
{
+
∞
} be a lower semicontinuous function at x
∈
dom
⁡
f. Then ∂
A
π
f
(
x
)
=
∂
G
π
f
(
x
).Proof.
The direct inclusion∂
A
π
f
(
x
)
⊂
∂
G
π
f
(
x
) has been proved in [1]. Let x
∗
∈
∂
G
π
f
(
x
); that is, (
x
∗
,
-
1
)
∈
N
π
(
e
p
i
f
;
(
x
,
f
(
x
)
)
) and hence there exists σ
>
0 such that (12)
x
∗
,
-
1
,
x
′
,
r
-
x
,
f
x
≤
σ
V
J
x
,
f
x
;
x
′
,
r
,
∀
x
′
,
r
∈
epi
⁡
f
.It follows that (13)
x
∗
,
x
′
-
x
≤
f
x
′
-
f
x
+
σ
V
J
x
,
x
′
+
f
x
′
-
f
x
2
∀
x
′
∈
dom
⁡
f
.By lower semicontinuity of f we may choose some η
>
0 such that f
(
x
′
)
>
f
(
x
)
-
1
/
2
σ, for all x
′
∈
x
+
η
B. Also, we can take η too small so that (14)
V
J
x
,
x
′
+
-
σ
-
1
x
∗
;
x
′
-
x
<
1
2
σ
2
.Therefore, for any x
′
∈
x
+
η
B we have (15)
0
<
1
σ
x
∗
,
x
′
-
x
-
V
J
x
,
x
′
+
1
2
σ
2
≤
f
x
′
-
f
x
+
1
2
σ
2which yields(16)
f
x
-
1
2
σ
+
1
σ
x
∗
,
x
′
-
x
-
V
J
x
,
x
′
+
1
2
σ
2
1
/
2
≤
f
x
′
,
∀
x
′
∈
x
+
η
B
.Set h
x
′
:
=
f
x
′
-
g
x
′
-
f
x
+
1
/
2
σ and g
x
′
:
=
〈
1
/
σ
x
∗
,
x
′
-
x
〉
-
V
(
J
(
x
)
,
x
′
)
+
1
/
[
2
σ
]
2
1
/
2. Then h
x
′
≥
h
x
=
0, for all x
′
∈
x
+
η
B; that is, x is a local minimum of h and so 0
∈
∂
A
π
h
x
=
∂
A
π
f
-
g
x. Now, we can easily check that g is C
1 around x with g
′
x
′
=
(
2
σ
)
-
1
x
∗
+
J
x
-
J
x
′
/
g
x
′ for x
′ very close to x. Indeed, since g
x
=
1
/
2
σ
>
0 we can find δ
>
0 smaller than η for which we have g
x
′
>
1
/
4
σ
>
0, for any x
′
∈
x
+
δ
B
⊂
x
+
η
B. Thus by the differentiability of the equivalent norm we have for any x
′
∈
x
+
δ
B
(17)
g
′
x
′
,
y
=
l
i
m
t
→
0
+
t
-
1
g
x
′
+
t
y
-
g
x
′
=
l
i
m
t
→
0
+
1
g
x
′
+
t
y
+
g
x
′
g
2
x
′
+
t
y
-
g
2
x
′
t
=
l
i
m
t
→
0
+
1
/
σ
x
∗
,
x
′
+
t
y
-
x
-
V
J
x
,
x
′
+
t
y
+
1
/
2
σ
2
-
1
/
σ
x
∗
,
x
′
-
x
-
V
J
x
,
x
′
+
1
/
2
σ
2
2
g
x
′
t
=
1
2
g
x
′
1
σ
x
∗
,
y
-
l
i
m
t
→
0
+
V
J
x
,
x
′
+
t
y
-
V
J
x
,
x
′
t
=
1
2
g
x
′
1
σ
x
∗
,
y
-
l
i
m
t
→
0
+
x
′
+
t
y
2
-
x
′
2
-
2
J
x
,
x
′
+
t
y
-
x
′
t
=
1
2
g
x
′
1
σ
x
∗
+
2
J
x
,
y
-
l
i
m
t
→
0
+
x
′
+
t
y
2
-
x
′
2
t
=
1
2
g
x
′
1
σ
x
∗
+
2
J
x
-
2
J
x
′
,
y
=
1
g
x
′
1
2
σ
x
∗
+
J
x
-
J
x
′
,
y
;that is, the function g is differentiable at any x
′
∈
x
+
δ
B and g
′
(
x
′
)
=
(
2
σ
)
-
1
x
∗
+
J
x
-
J
x
′
/
g
(
x
′
). Now, we use the C
2 property of the norm around x, that is, the C
1 property of the duality mapping J around x, and we obtain the C
1 property of g around x with g
′
(
x
)
=
x
∗. Now, we return to 0
∈
∂
A
π
(
f
-
g
)
(
x
) and we use the previous proposition to get 0
∈
∂
A
π
f
(
x
)
-
g
′
(
x
); that is, x
∗
∈
∂
A
π
f
(
x
) and hence the proof is complete.Remark 4.
The assumption on the norm used in the previous proposition is true for any Hilbert space and for many other Banach spaces (see, e.g., [9, 12]). Using Theorem 1.1 in Section  5.1 in [9] we get that all L
p spaces for p
≥
2 satisfy the C
2 property with their canonical norm and the same result is also true for the Sobolev spaces W
p
,
m (p
≥
2). We refer the reader for more examples and discussions to [9, 12] and the references therein.


## 3. Fuzzy Sum Rules

In this section, we are going to prove the fuzzy sum rule for the analyticV-proximal subdifferential. To do that, we need the following new version of Borwein-Preiss variational principle in terms of the functional V. For its proof we refer the reader to [13].Theorem 5.
LetX be p-uniformly convex and q-uniformly smooth Banach space and let f
:
X
→
R be bounded below lower semicontinuous function, and ϵ
>
0. Suppose that x
¯ is a point satisfying f
(
x
¯
)
<
inf
x
∈
X
⁡
f
(
x
)
+
ϵ. Then for any λ
>
0 there exist points y and z with(18)
J
z
-
J
x
¯
<
λ
,
V
J
z
,
y
<
λ
,
f
y
≤
f
x
¯and having the property that the function (19)
x
⟼
f
x
+
ϵ
λ
V
J
z
;
xhas a unique minimum at x
=
y.Let us prove the following lemma needed in the proof of the main result in this section. Its proof follows the same lines of the proof of Proposition8.2 in [8].Lemma 6.
LetX be p-uniformly convex and q-uniformly smooth Banach space. Let f
i
:
X
→
R
∪
∞, i
=
1,2, be l.s.c. functions at x
¯
∈
dom
⁡
f
1
∩
dom
⁡
f
2; one of them is Lipschitz continuous around x
¯. Let C be a closed convex subset of X with C
∩
dom
⁡
f
1
∩
dom
⁡
f
2
≠
∅. Let s
>
0 and let · be any equivalent norm on X. Then for any r
n
>
0 with r
n
→
∞ we have (20)
lim
n
→
∞
⁡
inf
x
,
y
∈
C
⁡
f
1
x
+
f
2
y
+
r
n
x
-
y
s
=
inf
x
∈
C
⁡
f
1
x
+
f
2
x
.Proof.
Obviously we always have(21)
inf
x
,
y
∈
C
⁡
f
1
x
+
f
2
y
+
r
n
x
-
y
s
≤
inf
x
∈
C
⁡
f
1
x
+
f
2
x
,
∀
n
≥
1and so (22)
lim
n
→
∞
⁡
inf
x
,
y
∈
C
⁡
f
1
x
+
f
2
y
+
r
n
x
-
y
s
≤
inf
x
∈
C
⁡
f
1
x
+
f
2
x
.Conversely, let (
x
n
,
y
n
)
∈
C
×
C such that (23)
inf
x
,
y
∈
C
⁡
f
1
x
+
f
2
y
+
r
n
x
-
y
s
+
1
n
≥
f
1
x
n
+
f
2
y
n
+
r
n
x
n
-
y
n
s
.Since f
1 and f
2 are bounded below on C we get x
0
∈
int
⁡
C
∩
dom
⁡
f
1
∩
dom
⁡
f
2
(24)
c
1
+
c
2
+
r
n
x
n
-
y
n
s
≤
f
1
x
n
+
f
2
y
n
+
r
n
x
n
-
y
n
s
≤
f
1
x
0
+
f
2
x
0
+
1
n
.This ensures that (25)
r
n
x
n
-
y
n
s
≤
m
,for some m
>
0 which yields that x
n
-
y
n
→
0 since r
n
→
∞.
We use now the Lipschitz continuity off
1 over C for some K
>
0 to get(26)
inf
x
,
y
∈
C
⁡
f
1
x
+
f
2
y
+
r
n
x
-
y
s
≥
f
1
x
n
+
f
2
y
n
+
r
n
x
n
-
y
n
s
-
1
n
≥
f
1
y
n
-
K
x
n
-
y
n
+
f
2
y
n
-
1
n
≥
inf
x
∈
C
⁡
f
1
x
+
f
2
x
-
K
x
n
-
y
n
-
1
n
.Taking the limit as n
→
∞ yields the desired inequality and hence the proof is finished.Now, we assume that the spaceX satisfies an additional assumption. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
}. We refer the reader to [9, 12] for many examples of Banach space satisfying this assumption. So, we are ready to prove the main result.Theorem 7.
LetX be q-uniformly smooth and p-uniformly convex Banach space. Let f
i
:
X
→
R
∪
∞,  i
=
1,2, be l.s.c. functions at x
¯
∈
dom
⁡
f
1
∩
dom
⁡
f
2; one of them is Lipschitz continuous around x
¯. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and let V be the functional associated with that norm
⁡
·. Let x
∗
∈
∂
A
π
(
f
1
+
f
2
)
(
x
¯
). Then for any ϵ
>
0, there exist x
1
,
x
2
∈
x
¯
+
ϵ
B with f
i
x
i
-
f
i
x
¯
<
ϵ
,
i
=
1,2, such that (27)
x
∗
∈
∂
A
π
f
1
x
1
+
∂
A
π
f
2
x
2
+
ϵ
B
∗
.Proof.
Letx
∗
∈
∂
A
π
(
f
1
+
f
2
)
(
x
¯
). There exist η
,
σ
>
0 such that (28)
x
⟼
f
1
x
+
f
2
x
+
σ
V
J
x
¯
,
x
-
x
∗
;
x
-
x
¯attains a minimum over C
:
=
x
¯
+
η
B at x
=
x
¯. We take also η small enough so that f
1 and f
2 are bounded below on C and f
1 is Lipschitz on C which is satisfied for any x
,
y
∈
x
¯
+
η
B. Let (29)
φ
n
x
,
y
≔
f
1
x
+
f
2
y
+
σ
V
J
x
¯
,
y
-
x
∗
;
x
-
x
¯
+
r
n
x
-
y
s
+
I
C
x
+
I
C
y
,where r
n
↑
∞. Now consider the problem of minimising φ
n
(
·
,
·
) over X
×
X. Applying Lemma 6 with the functions f
1 and f
2
+
σ
V
(
J
(
x
¯
)
,
·
)
-
〈
x
∗
;
x
-
x
¯
〉 yields the conclusion that the nonnegative quantity q
n
:
=
f
1
(
x
¯
)
+
f
2
(
x
¯
)
-
inf
x
∈
X
⁡
φ
n
(
x
,
x
)
→
0 as n
→
∞; that is, for n large enough the couple (
x
¯
,
x
¯
) satisfies (30)
φ
n
x
¯
,
x
¯
=
f
1
x
¯
+
f
2
x
¯
≤
inf
x
∈
X
⁡
φ
n
x
,
x
+
1
n
.We apply now the variational principle stated in Theorem 5 with f
:
=
φ
n, ϵ
:
=
q
n
+
1
/
n, and λ
:
=
ϵ. Then there exist two points (
x
1
n
,
x
2
n
) and (
z
1
n
,
z
2
n
)
∈
X
×
X such that(31)
z
1
n
,
z
2
n
-
x
¯
,
x
¯
<
q
n
+
1
n
,
V
X
×
X
J
X
×
X
x
¯
,
x
¯
,
x
1
n
,
x
2
n
≤
q
n
+
1
n
,and φ
n
(
x
1
n
,
x
2
n
)
≤
φ
n
(
x
¯
,
x
¯
) such that the function (32)
x
,
y
⟼
φ
n
x
,
y
+
V
X
×
X
J
X
×
X
z
1
n
,
z
2
n
;
x
,
yhas a unique minimum at (
x
,
y
)
=
(
x
1
n
,
x
2
n
) over the space X
×
X. Thus for n sufficiently large (since q
n
→
0) both x
1
n and x
2
n are in int
⁡
(
C
)
:
=
x
¯
+
η
(
int
⁡
B
), so that the fact of the minimisation implies the separate necessary conditions:(33)
0
∈
∂
A
π
f
1
·
-
x
∗
;
·
-
x
¯
+
r
n
·
-
x
2
n
s
+
V
J
z
1
n
;
·
x
1
n
,
0
∈
∂
A
π
f
2
·
+
σ
V
J
x
¯
;
·
+
r
n
x
1
n
-
·
s
+
V
J
z
2
n
;
·
x
2
n
.Thus using the fact that the functions x
1
n
-
·
s and ·
-
x
2
n
s are C
2 on int
⁡
C with (34)
∇
x
1
n
-
·
s
u
=
-
s
x
1
n
-
u
s
-
2
J
x
1
n
-
u
,
∇
·
-
x
2
n
s
u
=
s
u
-
x
2
n
s
-
2
J
u
-
x
2
n
,and by using Proposition 2, we can write (35)
0
∈
∂
A
π
f
1
x
1
n
-
x
∗
+
r
n
s
x
1
n
-
x
2
n
s
-
2
J
x
1
n
-
x
2
n
+
2
J
x
1
n
-
2
J
z
1
n
,
0
∈
∂
A
π
f
2
x
2
n
+
2
σ
J
x
2
n
-
J
x
¯
-
r
n
s
x
1
n
-
x
2
n
s
-
2
J
x
1
n
-
x
2
n
+
2
J
x
2
n
-
2
J
z
2
n
.Adding these two inclusions yields (36)
x
∗
∈
∂
A
π
f
1
x
1
n
+
∂
A
π
f
2
x
2
n
+
2
σ
J
x
2
n
-
J
x
¯
-
J
X
×
X
z
1
n
,
z
2
n
+
J
X
×
X
x
1
n
,
x
2
n
.Let γ
n
:
=
J
X
×
X
(
z
1
n
,
z
2
n
)
-
J
X
×
X
(
x
1
n
,
x
2
n
)
+
σ
J
(
x
2
n
)
-
J
(
x
¯
). For n large enough we can ensure that γ
n
<
ϵ and η
<
ϵ and so (37)
x
∗
∈
∂
A
π
f
1
x
1
n
+
∂
A
π
f
2
x
2
n
+
ϵ
B
∗
with
x
1
n
,
x
2
n
∈
x
¯
+
ϵ
B
.Thus, it remains to show that |
f
i
(
x
i
n
)
-
f
(
x
¯
)
|
<
ϵ, i
=
1,2, for n large enough. Since φ
n
(
x
1
n
,
x
2
n
)
≤
φ
n
(
x
¯
,
x
¯
), we have(38)
f
1
x
¯
+
f
2
x
¯
≥
f
1
x
1
n
+
f
2
x
2
n
+
σ
V
J
x
¯
,
x
2
n
-
x
∗
;
x
1
n
-
x
¯
+
r
n
x
1
n
-
x
2
n
s
+
I
C
x
1
n
+
I
C
x
2
n
≥
f
1
x
1
n
+
f
2
x
2
n
-
x
∗
;
x
1
n
-
x
¯and so (39)
f
1
x
1
n
+
f
2
x
2
n
≤
f
1
x
¯
+
f
2
x
¯
+
x
∗
x
1
n
-
x
¯
,for n large enough. From this inequality we derive (40)
limsup
n
→
∞
⁡
f
1
x
1
n
+
f
2
x
2
n
≤
f
1
x
¯
+
f
2
x
¯
.Using now the l.s.c. of both functions f
1 and f
2 and x
1
n
→
x
¯ and x
2
n
→
x
¯ we obtain(41)
f
1
x
¯
+
f
2
x
¯
≤
liminf
n
→
∞
⁡
f
1
x
1
n
+
liminf
n
→
∞
⁡
f
2
x
2
n
≤
liminf
n
→
∞
⁡
f
1
x
1
n
+
f
2
x
2
nand so we deduce from the previous inequalities that (42)
lim
n
→
∞
⁡
f
1
x
1
n
+
f
2
x
2
n
=
f
1
x
¯
+
f
2
x
¯
.Using this equality and the l.s.c. of f
2 we get(43)
f
1
x
¯
=
f
1
x
¯
+
f
2
x
¯
-
f
2
x
¯
≥
lim
n
→
∞
⁡
f
1
x
1
n
+
f
2
x
2
n
+
limsup
n
→
∞
⁡
-
f
2
x
2
n
≥
limsup
n
→
∞
⁡
f
1
x
1
n
.This inequality with the l.s.c. of f
1 ensures that (44)
lim
n
→
∞
⁡
f
1
x
1
n
=
f
1
x
¯
.Similarly for f
2
(
x
2
n
) we obtain l
i
m
n
→
∞
f
2
(
x
2
n
)
=
f
2
(
x
¯
)
. These two equalities ensure |
f
i
(
x
i
n
)
-
f
i
(
x
¯
)
|
<
ϵ, i
=
1,2, for n large enough.We notice that the fuzzy sum rule for the geometricV-proximal subdifferential cannot be deduced directly from the previous one. However, it will be used for that purpose. First we need to prove the following proposition.Proposition 8.
LetX be q-uniformly smooth and p-uniformly convex Banach space. Let f
:
X
→
R
∪
{
∞
} be a l.s.c. function at x
¯
∈
dom
⁡
f. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and let V be the functional associated with that norm
⁡
·. For any x
∗
∈
∂
F
f
(
x
¯
) and any ϵ
>
0, there exist x
1
∈
x
¯
+
ϵ
B with |
f
(
x
1
)
-
f
(
x
¯
)
|
<
ϵ such that(45)
x
∗
∈
∂
A
π
f
x
1
+
ϵ
B
∗
.Proof.
Letx
∗
∈
∂
F
f
(
x
¯
). Fix any ϵ
>
0. Then by definition of the Fréchet subdifferential there exists δ
>
0 such that (46)
x
⟼
h
x
≔
f
x
+
ϵ
2
x
-
x
¯
-
x
∗
;
x
-
x
¯attains a minimum over C
:
=
x
¯
+
δ
B at x
=
x
¯. Hence (47)
0
∈
∂
A
π
h
x
¯
=
∂
A
π
f
+
ϵ
2
·
-
x
¯
-
x
∗
;
·
-
x
¯
x
¯
.Using the fuzzy sum rule for the analytic V-proximal subdifferential in Theorem 7 we can find x
1
,
x
2
∈
x
¯
+
ϵ
B with |
f
(
x
1
)
-
f
(
x
¯
)
|
≤
ϵ such that(48)
0
∈
∂
A
π
f
x
1
+
∂
A
π
ϵ
2
·
-
x
¯
x
2
-
x
∗
+
ϵ
2
B
∗
⊂
∂
A
π
f
x
1
+
ϵ
2
B
∗
-
x
∗
+
ϵ
2
B
∗
.Thus,(49)
x
∗
∈
∂
A
π
f
x
1
+
ϵ
B
∗
,and so the proof of the proposition is complete.Now, we are ready to prove the fuzzy sum rule for the geometricV-proximal subdifferential.Theorem 9.
LetX be q-uniformly smooth and p-uniformly convex Banach space. Let f
i
:
X
→
R
∪
∞, i
=
1,2, be l.s.c. functions at x
¯
∈
dom
⁡
f
1
∩
dom
⁡
f
2; one of them is Lipschitz continuous around x
¯. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and let V be the functional associated with that norm
⁡
·. If x
∗
∈
∂
G
π
(
f
1
+
f
2
)
(
x
¯
), then for any ϵ
>
0, there exist x
1
,
x
2
∈
x
¯
+
ϵ
B with f
i
x
i
-
f
i
x
¯
<
ϵ, i
=
1,2, such that (50)
x
∗
∈
∂
G
π
f
1
x
1
+
∂
G
π
f
2
x
2
+
ϵ
B
∗
.Proof.
Fix anyϵ
>
0 and let K be the Lipschitz constant of f
2 around x
¯. Let ϵ
0
=
ϵ
/
4
m
a
x
{
K
,
1
}. Assume that x
∗
∈
∂
G
π
(
f
1
+
f
2
)
(
x
¯
). By Proposition 2 we get x
∗
∈
∂
F
(
f
1
+
f
2
)
(
x
¯
). Then, by Proposition 8 there exists x
0
∈
x
¯
+
ϵ
0
B with |
(
f
1
+
f
2
)
(
x
0
)
-
(
f
1
+
f
2
)
(
x
¯
)
|
<
ϵ
0, such that (51)
x
∗
∈
∂
A
π
f
1
+
f
2
x
0
+
ϵ
0
B
∗
.We use now the fuzzy sum rule for the analytic V-proximal subdifferential to obtain some points x
1
,
x
2
∈
x
0
+
ϵ
0
B with |
f
i
(
x
i
)
-
f
i
(
x
0
)
|
<
ϵ
0, i
=
1,2, such that (52)
x
∗
∈
∂
A
π
f
1
x
1
+
∂
A
π
f
2
x
2
+
ϵ
0
B
∗
+
ϵ
0
B
∗
⊂
∂
G
π
f
1
x
1
+
∂
G
π
f
2
x
2
+
ϵ
B
∗
.Obviously we have x
1
,
x
2
∈
x
0
+
ϵ
0
B
⊂
x
¯
+
ϵ
0
B
+
ϵ
0
B
⊂
x
¯
+
ϵ
B. Also we have (53)
f
2
x
2
-
f
2
x
¯
≤
f
2
x
2
-
f
2
x
0
+
f
2
x
0
-
f
2
x
¯
≤
ϵ
0
+
K
x
0
-
x
¯
≤
ϵ
0
+
K
ϵ
0
≤
ϵ
,
f
1
x
1
-
f
1
x
¯
≤
f
1
x
1
-
f
1
x
0
+
f
1
x
0
-
f
1
x
¯
≤
ϵ
0
+
f
1
+
f
2
x
0
-
f
1
+
f
2
x
¯
+
f
2
x
0
-
f
2
x
¯
≤
ϵ
0
+
ϵ
0
+
K
x
0
-
x
¯
≤
2
ϵ
0
+
K
ϵ
0
≤
ϵ
.This completes the proof.The following corollary is a direct consequence of Theorem7. It is also valid for the geometric V-proximal subdifferential.Corollary 10.
LetX be q-uniformly smooth and p-uniformly convex Banach space. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and let V be the functional associated with that norm
⁡
·. Let f
:
X
→
R
∪
{
∞
} be locally Lipschitz function around x
¯
∈
S. If x
¯ minimises f over S, then for any ϵ
>
0, there exist x
1
∈
x
¯
+
ϵ
B with |
f
(
x
1
)
-
f
(
x
¯
)
|
<
ϵ and x
2
∈
[
x
¯
+
ϵ
B
]
∩
S such that (54)
0
∈
∂
A
π
f
x
1
+
N
π
S
;
x
2
+
ϵ
B
∗
.


## 4. Chain Rule

Using the fuzzy sum rule established in the previous section for the analyticV-proximal subdifferential, we prove the chain rule for both analytic and geometric V-proximal subdifferentials. Let Y be another p
′-uniformly convex and q
′-uniformly smooth Banach space and let g
:
Y
→
R
∪
{
∞
} and F
:
X
→
Y. In this section, ·
E will denote the norm on E and the closed unit ball in E will be denoted by B
E. We will denote by V
E functional (2) associated with the norm
⁡
·
E. We will say that F is a locally V-Lipschitz mapping around x
¯
∈
X provided that there exist k
>
0 and η
>
0 such that(55)
V
Y
J
F
x
,
F
y
≤
k
V
X
J
x
,
y
,
F
x
-
F
y
Y
≤
k
x
-
y
X
,
∀
x
,
y
∈
x
¯
+
η
B
X
.Theorem 11.
Letx
¯
∈
X, F be a locally V-Lipschitz mapping around x
¯, and let g be locally Lipschitz around F
(
x
¯
). Assume that X (resp., Y) admit an equivalent norm
⁡
·
X (resp., ·
Y) such that ·
X
s (resp., ·
Y
r) (for some s
,
r
≥
2) is C
2-differentiable on X
∖
{
0
} (resp., Y
∖
{
0
}) and let V
X (resp., V
Y) be the functional associated with ·
X (resp., ·
Y). Let x
∗
∈
∂
A
π
(
g
∘
F
)
(
x
¯
). Then for all ϵ
>
0 there exist x
~
∈
x
¯
+
ϵ
B
X, y
~
∈
F
(
x
¯
)
+
ϵ
B
Y, and y
∗
∈
∂
A
π
g
(
y
~
) such that F
(
x
~
)
-
F
(
x
¯
)
Y
≤
ϵ and (56)
x
∗
∈
∂
A
π
y
∗
,
F
·
x
~
+
ϵ
B
X
∗
.Proof.
Fixϵ
>
0. By Lipschitz property of F around x
¯ and by definition of the analytic V-proximal subdifferential there exist a positive number k (we may assume that k
≥
1), σ
>
0, and η
∈
(
0
,
ϵ
/
2
) such that(57)
F
x
-
F
y
Y
≤
k
x
-
y
X
,
∀
x
,
y
∈
x
¯
+
η
B
X
,
x
∗
,
x
-
x
¯
≤
g
F
x
-
g
F
x
¯
+
σ
V
X
J
x
¯
,
x
,
∀
x
∈
x
¯
+
η
B
X
.Set C
:
=
g
p
h
F, the graph of F in X
×
Y. The previous inequalities ensure that (58)
g
F
x
¯
-
x
∗
,
x
¯
≤
g
y
-
x
∗
,
x
+
σ
V
X
J
x
¯
,
x
+
ψ
C
x
,
y
,for any (
x
,
y
)
∈
(
x
¯
,
F
(
x
¯
)
)
+
η
B
X
×
Y
. This means that the function (
x
,
y
)
↦
g
(
y
)
-
〈
x
∗
,
x
〉
+
σ
V
X
(
J
(
x
¯
)
,
x
)
+
ψ
C
(
x
,
y
) attains a local minimum at (
x
¯
,
F
(
x
¯
)
) and hence (59)
0,0
∈
∂
A
π
g
y
-
x
∗
,
x
+
σ
V
X
J
x
¯
,
x
+
ψ
C
x
,
y
x
¯
,
F
x
¯
.Applying our fuzzy sum rule in Theorem 7 with δ
∈
(
0
,
m
i
n
{
η
/
2
k
,
η
/
k
+
4
}
) there exist (
x
1
,
y
1
) and (
x
2
,
y
2
) in (
x
¯
,
F
(
x
¯
)
)
+
δ
B
X
×
Y such that for some (60)
x
1
∗
,
y
1
∗
∈
∂
A
π
g
y
-
x
∗
,
x
+
σ
V
X
J
x
¯
,
x
x
1
,
y
1we have (61)
-
x
1
∗
,
-
y
1
∗
∈
∂
A
π
ψ
C
x
2
,
y
2
+
δ
B
X
∗
×
Y
∗
=
N
π
C
;
x
2
,
y
2
+
δ
B
X
∗
×
Y
∗
.This ensures that (
x
2
,
y
2
)
∈
C; that is, y
2
=
F
(
x
2
). Using Proposition 2 we obtain for some y
∗
∈
∂
A
π
g
(
y
1
)
(62)
x
1
∗
,
y
1
∗
=
-
x
∗
+
2
J
x
1
-
2
J
x
¯
,
y
∗
.Thus (63)
x
∗
-
2
J
x
1
+
2
J
x
¯
,
-
y
∗
∈
N
π
C
;
x
2
,
y
2
+
δ
B
X
∗
×
Y
∗
.Hence there exist u
2
∗
∈
δ
B
X
∗, v
2
∗
∈
δ
B
Y
∗, σ
0
>
0, and δ
0
∈
(
0
,
δ
/
2
) such that for any (
x
,
y
)
∈
g
p
h
F
∩
[
(
x
2
,
F
(
x
2
)
+
δ
0
B
X
∗
×
Y
∗
] we have(64)
x
∗
-
2
J
x
1
+
2
J
x
¯
-
u
2
∗
,
-
y
∗
-
v
2
∗
,
x
,
y
-
x
2
,
F
x
2
≤
σ
0
V
X
×
Y
J
x
2
,
F
x
2
;
x
,
y
≤
σ
0
V
X
J
x
2
,
x
+
σ
0
V
Y
J
F
x
2
,
y
.The last inequality follows from Lemma  3.5 in [1] and hence for any x
∈
x
2
+
δ
0
/
k
B
X
⊂
x
2
+
δ
0
B
X we have F
(
x
)
∈
F
(
x
2
)
+
δ
0
B
X and hence we get(65)
x
∗
-
2
J
x
1
+
2
J
x
-
-
u
2
∗
,
x
-
x
2
≤
σ
0
V
X
J
x
2
,
x
+
σ
0
V
Y
J
F
x
2
,
F
x
+
y
∗
+
v
2
∗
,
F
x
-
F
x
2
≤
σ
0
1
+
k
V
X
J
x
2
,
x
+
G
x
-
G
x
2
,where G
(
x
)
=
〈
y
∗
+
v
2
∗
,
F
(
·
)
〉
(
x
)
. Thus (66)
x
∗
-
2
J
x
1
+
2
J
x
¯
-
u
2
∗
∈
∂
A
π
y
∗
,
F
·
+
v
2
∗
,
F
·
x
2
.We use the fuzzy sum rule once again to obtain two points x
3 and x
4 in x
2
+
δ
0
B
X such that(67)
x
∗
∈
∂
A
π
y
∗
,
F
·
x
3
+
∂
A
π
v
2
∗
,
F
·
x
4
+
u
2
∗
+
2
J
x
1
-
J
x
¯
+
δ
0
B
X
∗
∈
∂
A
π
y
∗
,
F
·
x
3
+
∂
A
π
v
2
∗
,
F
·
x
4
+
2
J
x
1
-
J
x
¯
+
2
δ
B
X
∗
.The Lipschitz property of F with ratio k
>
0 around x
¯ ensures that 〈
v
2
∗
,
F
(
·
)
〉 is Lipschitz with ratio δ
k around x
¯ and hence by Proposition 2 we obtain (68)
∂
A
π
v
2
∗
,
F
·
x
4
⊂
δ
k
B
X
∗
.Also the uniform continuity of J ensures by shrinking δ
0 that J
(
x
1
)
-
J
(
x
¯
)
∈
δ
B
X
∗ and hence (69)
x
∗
∈
∂
A
π
y
∗
,
F
·
x
3
+
k
+
4
δ
B
X
∗
.Therefore, by taking x
~
:
=
x
3 and y
~
:
=
y
1, we get x
~
∈
x
¯
+
ϵ
B
X and y
~
,
F
(
x
~
)
∈
F
(
x
¯
)
+
ϵ
B
Y with x
∗
∈
∂
A
π
{
〈
y
∗
,
F
(
·
)
〉
}
(
x
~
)
+
ϵ
B
X
∗ and y
∗
∈
∂
A
π
g
(
y
~
), thus completing the proof.Remark 12.
As for the fuzzy sum rule for geometricV-proximal subdifferential, we combine the chain rule for the analytic V-proximal subdifferential and Proposition 8 to prove easily the chain rule for the geometric V-proximal subdifferential.


## 5. LimitingV-Proximal Subdifferential and Limiting V-Proximal Normal Cone

After proving the density theorem for∂
A
π
f in [1] and consequently for ∂
G
π
f, it is a natural question to consider the limiting object of both V-proximal subdifferential concepts as follows:(70)
∂
A
L
π
f
x
=
limsup
x
′
→
x
⁡
∂
A
π
f
x
′
≔
w
-
lim
n
⁡
x
n
∗
:
x
n
∗
∈
∂
A
π
f
x
n
with
x
n
⟶
f
x
,
∂
G
L
π
f
x
=
limsup
x
′
→
x
⁡
∂
G
π
f
x
′
≔
w
-
lim
n
⁡
x
n
∗
:
x
n
∗
∈
∂
G
π
f
x
n
with
x
n
⟶
f
x
,where x
n
→
f
x means x
n
→
x with f
(
x
n
)
→
f
(
x
). These two subdifferentials will be called, respectively,the limting analytic
V-proximal subdifferential andthe limiting geometric
V-proximal subdifferential. Our main aim in this section is to prove, under the assumptions of Theorem 7, that these two constructions coincide with the limiting Fréchet subdifferential (also called Mordukhovich subdifferential) (see [7]).Theorem 13.
LetX be q-uniformly smooth and p-uniformly convex Banach space. Let f
:
X
→
R
∪
{
∞
} be a l.s.c. function at x
¯
∈
dom
⁡
f. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and let V be the functional associated with that norm
⁡
·. Then(71)
∂
A
L
π
f
x
¯
=
∂
G
L
π
f
x
¯
=
∂
F
L
f
x
¯
=
∂
P
L
f
x
¯
.Here ∂
F
L
f
(
x
¯
) and ∂
P
L
f
(
x
¯
) are limiting Fréchet subdifferential and limiting proximal subdifferential, respectively; that is, ∂
F
L
f
(
x
¯
)
=
limsup
x
′
→
x
⁡
∂
F
f
(
x
′
) and ∂
P
L
f
(
x
¯
)
=
limsup
x
′
→
x
⁡
∂
P
f
(
x
′
)
.Proof.
The last equality∂
F
L
f
(
x
¯
)
=
∂
P
L
f
(
x
¯
) has been proved in [6] which is our setting. So, it remains to prove the other two equalities. It will be enough to prove the inclusion ∂
F
L
f
(
x
¯
)
⊂
∂
A
L
π
f
(
x
¯
). Fix x
∗
∈
∂
F
L
f
(
x
¯
). There exists x
n
∗
→
w
x
∗ with x
n
∗
∈
∂
F
f
(
x
n
) such that x
n
-
x
¯
≤
1
/
n, |
f
(
x
n
)
-
f
(
x
¯
)
|
≤
1
/
n. Using Proposition 8 with ϵ
=
1
/
n we get a sequence x
n
′
∈
x
n
+
1
/
n
B with |
f
(
x
n
′
)
-
f
(
x
n
)
|
≤
1
/
n such that x
n
∗
∈
∂
A
π
f
(
x
n
)
+
1
/
n
B
∗. So, there exists b
n
∈
B
∗ such that y
n
∗
:
=
x
n
∗
+
1
/
n
b
n
∈
∂
A
π
f
(
x
n
). Clearly x
n
′
→
f
x
¯ and y
n
∗
→
w
x
∗. Thus, x
∗
∈
∂
A
L
π
f
(
x
¯
) and hence the proof is complete.As a first consequence of this result we have the following theorem, which is a new result in Banach spaces.Theorem 14.
LetX be q-uniformly smooth and p-uniformly convex Banach space. Let S be any closed nonempty set of X. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and let V be the functional associated with that norm
⁡
·. Then (72)
N
F
L
S
;
x
¯
=
N
P
L
S
;
x
¯
=
N
L
π
S
;
x
¯
=
x
∗
∈
X
∗
:
∃
x
n
⟶
S
x
¯
,
x
n
∗
⟶
w
x
∗
,
α
n
>
0
s
.
t
.
x
n
∈
π
S
J
x
n
+
α
n
x
n
∗
.We establish in the following theorem a new characterisation in terms of analytic and geometricV-proximal subdifferentials of the singular subdifferential ∂
∞
f
(
x
¯
) defined (see [7]) by (73)
∂
∞
f
x
¯
≔
x
∗
∈
X
∗
:
x
∗
,
0
∈
N
F
L
epi
⁡
f
;
x
¯
,
f
x
¯
.Theorem 15.
LetX be q-uniformly smooth and p-uniformly convex Banach space. Let f
:
X
→
R
∪
{
∞
} be a l.s.c. function at x
¯
∈
dom
⁡
f. Assume that X admits an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and let V be the functional associated with that norm
⁡
·. Then (74)
∂
∞
f
x
¯
=
limsup
x
→
f
x
¯
,
λ
→
0
+
⁡
λ
∂
F
f
x
=
limsup
x
→
f
x
¯
,
λ
→
0
+
⁡
λ
∂
A
π
f
x
=
limsup
x
→
f
x
¯
,
λ
→
0
+
⁡
λ
∂
G
π
f
x
.Proof.
The first equality is proved in [7]. To conclude, it is enough to prove the inclusion (75)
∂
∞
f
x
¯
⊂
limsup
x
→
f
x
¯
,
λ
→
0
+
⁡
λ
∂
A
π
f
x
. Fix any x
∗
∈
∂
∞
f
(
x
¯
); that is, (
x
∗
,
0
)
∈
N
F
L
(
epi
⁡
f
;
(
x
¯
,
f
(
x
¯
)
)
). Then there exist (
x
n
,
r
n
)
→
e
p
i
f
(
x
¯
,
f
(
x
¯
)
) and (
x
∗
,
-
ϵ
n
)
→
w
(
x
∗
,
0
) such that (76)
x
n
∗
,
-
ϵ
n
∈
N
F
epi
⁡
f
;
x
n
,
r
n
.Clearly ϵ
n
≥
0, for all n
≥
1. We distinguish two cases.
Case  1. ϵ
n
>
0 for n sufficiently large. For this case we have (
x
n
∗
/
ϵ
n
,
-
1
)
∈
N
F
(
e
p
i
f
;
(
x
n
,
r
n
)
); that is, x
n
∗
/
ϵ
n
∈
∂
F
f
(
x
n
). Using Proposition 8 we get a sequence y
n
∈
x
n
+
1
/
n
B with |
f
(
y
n
)
-
f
(
x
n
)
|
≤
1
/
n such that x
n
∗
/
ϵ
n
∈
∂
A
π
f
(
y
n
)
+
1
/
n
B
∗; that is, x
n
∗
/
ϵ
n
+
1
/
n
b
n
∈
∂
A
π
f
(
y
n
) for some b
n
∈
B
∗. Let y
n
∗
:
=
x
n
∗
/
ϵ
n
+
1
/
n
b
n and λ
n
:
=
ϵ
n. Then y
n
∗
∈
∂
A
π
f
(
y
n
) with y
n
→
f
x
¯ and λ
n
y
n
∗
→
w
x
∗.
Case  2. ϵ
n
=
0 for some subsequence of {
ϵ
n
}
n. In this case we get subsequences of {
x
n
}
n, {
x
n
∗
}
n, {
r
n
}
n (still denoted by the same notation) for which we have (
x
n
∗
,
0
)
∈
N
F
(
epi
⁡
f
;
(
x
n
,
r
n
)
). Then we use Theorem  4 in [6] to get sequences {
z
n
}
n, {
z
n
∗
}
n, {
λ
n
}
n such that z
n
→
f
x
¯, λ
n
→
0
+, and λ
n
z
n
∗
-
x
n
∗
→
0 such that (
z
n
∗
,
-
1
)
∈
N
F
(
epi
⁡
f
;
(
z
n
,
f
(
z
n
)
)
). Hence z
n
∗
∈
∂
F
f
(
z
n
) with z
n
→
f
x
¯ and λ
n
z
n
∗
→
w
x
∗. Similarly to the first case, by using Proposition 8 we get a sequence {
y
n
}
n such that z
n
∗
∈
∂
A
π
f
(
y
n
)
+
1
/
n
B
∗ with y
n
∈
z
n
+
1
/
n
B and |
f
(
y
n
)
-
f
(
z
n
)
|
≤
1
/
n. Then for some b
n
∈
B
∗ we have y
n
∗
:
=
z
n
∗
+
1
/
n
b
n
∈
∂
A
π
f
(
y
n
) with y
n
→
f
x
¯ and λ
n
y
n
∗
=
λ
n
[
z
n
∗
+
1
/
n
b
n
]
→
w
x
∗.
Thus, from both cases we getx
∗
∈
limsup
x
→
f
x
¯
,
λ
→
0
+
⁡
λ
∂
A
π
f
(
x
) and so the proof is complete.We close this section by proving the well known Bishop-Phelps density theorem in terms ofV-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]).Theorem 16.
LetX be p-uniformly convex and q-uniformly smooth Banach space, and let S be any closed nonempty set of X. Then there is a dense set of points in X
∗ admitting unique generalised projection on S; that is, for any x
∗
∈
X
∗, there exists x
n
∗
→
x
∗ with π
S
(
x
n
∗
)
≠
∅, ∀
n.Now we are ready to prove the following result.Theorem 17.
LetX be p-uniformly convex and q-uniformly smooth Banach space and let S be a nonempty closed subset of X. Then the set of points x
∈
S with N
π
(
S
;
x
)
≠
{
0
} is dense in the boundary of S.Proof.
Fix anyϵ
>
0 and x
¯
∈
b
d
S. There exist x
ϵ
∉
S such that x
ϵ
-
x
¯
≤
ϵ
/
3. Using Theorem 16 we can find z
ϵ
∗
∈
X
∗
∖
J
(
S
) with z
ϵ
∗
-
J
x
ϵ
≤
ϵ
/
3 and π
S
(
z
ϵ
∗
)
≠
∅. Let z
ϵ
∈
π
S
(
z
ϵ
∗
); that is, z
ϵ
∈
S and V
(
z
ϵ
∗
,
z
ϵ
)
≤
V
(
z
ϵ
∗
,
x
¯
)
. Then by definition of the V-proximal normal cone we get z
ϵ
∗
-
J
z
ϵ
∈
N
π
(
S
;
z
ϵ
). Obviously z
ϵ
∗
-
J
z
ϵ
≠
0 since J
z
ϵ
∈
J
(
S
) and z
ϵ
∗
∉
J
(
S
), which ensures N
π
(
S
;
z
ϵ
)
≠
{
0
}
. Let n
≥
1 and take ϵ
=
1
/
n. From what precedes we can find for any n
≥
1 a sequence x
n
∉
S, z
n
∈
S, z
n
∗
∉
J
(
S
) with x
n
→
x
¯ and z
n
∗
→
J
x
¯. We are going to prove that z
n
→
x
¯. By the q-uniform smoothness of the space X, there exists by Lemma  4.1 in [1] two positive numbers δ
>
0 and α
>
0 such that(77)
V
J
x
,
y
≤
α
x
-
y
q
,
∀
x
,
y
∈
x
¯
+
δ
B
.Then for n large enough we can write (78)
V
z
n
∗
,
x
¯
=
V
J
J
∗
z
n
∗
,
x
¯
≤
α
J
∗
z
n
∗
-
x
¯
q
.Here J
∗ is the normalized duality mapping on X
∗. Therefore, since z
n
∈
π
S
(
z
n
∗
) we have (79)
V
z
n
∗
,
z
n
=
inf
s
∈
S
⁡
V
z
n
∗
,
s
≤
V
z
n
∗
,
x
¯
≤
α
J
∗
z
n
∗
-
x
¯
q
.This ensures that V
(
z
n
∗
,
z
n
)
→
0 as n
→
∞. Hence the sequence z
n is bounded by some M
>
0 (since we have z
n
≤
z
n
∗
+
V
(
z
n
∗
,
z
n
)) and so(80)
V
J
x
¯
,
z
n
≤
V
z
n
∗
,
z
n
+
2
z
n
+
x
¯
+
z
n
∗
z
n
∗
-
J
x
¯
⟶
0
,
as
n
⟶
∞
.We use now the p-uniform convexity of X to obtain a constant β
>
0 (depending on X, x
¯, and M) such that (81)
x
-
y
p
≤
β
V
J
x
,
y
,
∀
x
,
y
∈
x
¯
+
M
B
.Thus, (82)
z
n
-
x
¯
p
≤
β
V
J
x
¯
,
z
n
⟶
0
.This ensures that z
n
→
S
x
¯, thus completing the proof.In the next theorem we establish an analogue result to the previous theorem with the limitingV-proximal normal cone. First, we need to define the following concept of normal compactness with respect to the V-proximal normal cone.Definition 18.
A closed setS is said to be normally compact around x
¯
∈
S if there exist positive numbers γ, δ and a compact subset K of X such that (83)
N
π
S
;
x
⊂
x
∗
∈
X
∗
:
γ
x
∗
≤
m
a
x
v
∈
K
x
∗
,
v
,
∀
x
∈
S
∩
x
¯
+
δ
B
.Clearly, this normal compactness with respect toV-proximal normal cone is ensured by the normal compactness with respect to the Fréchet normal cone introduced in [15] and used by many authors (see, e.g., [7]). We point out that using Theorem 13 the next result is covered by Corollary  3.7 in [7] provided that the space X satisfies the assumptions in Theorem 13 and the normal compactness is satisfied with the Fréchet normal cone. However, our result is proved for any p-uniformly convex and q-uniformly smooth Banach space and under the normal compactness relatively to the V-proximal normal cone.Theorem 19.
LetX be p-uniformly convex and q-uniformly smooth Banach space and let S be a nonempty closed subset of X. Assume that S isnormally compact around x
¯ with respect to the V-proximal normal cone. If x
¯
∈
bd
⁡
S, then N
L
π
(
S
;
x
¯
)
≠
{
0
}.Proof.
Letx
¯
∈
b
d
S. Then by the previous proposition there exists a sequence x
n
→
S
x
¯ with N
π
(
S
;
x
n
)
≠
{
0
}. Let x
n
∗
∈
N
π
(
S
;
x
n
) with x
n
∗
≠
0. Set y
n
∗
=
x
n
∗
/
x
n
∗. By extracting a subsequence we may assume that the sequence y
n
∗ weakly converges to some limit y
∗
∈
X
∗. Obviously y
∗
∈
N
L
π
(
S
;
x
¯
). It remains to prove that y
∗
≠
0. Assume by contradiction that y
∗
=
0; that is, the sequence y
n
∗
→
w
0 and so 〈
y
n
∗
,
v
〉
→
0, ∀
v
∈
X. Using the normal compactness of S we get (84)
γ
y
n
∗
≤
max
v
∈
K
⁡
y
n
∗
,
v
⟶
0and so y
n
∗
→
0 which contradicts the fact that y
n
∗
=
1. Thus y
∗
≠
0 and so N
L
π
(
S
;
x
¯
)
≠
{
0
}. This completes the proof.We use this theorem to prove a useful result for the nonemptiness of the limiting geometricV-proximal subdifferential for locally Lipschitz functions and consequently the nonemptiness of the limiting geometric V-proximal subdifferential.Theorem 20.
LetX be p-uniformly convex and q-uniformly smooth Banach space and let S be a nonempty closed subset of X. Let f
:
X
→
R be a locally Lipschitz around x
¯
∈
X. Then ∂
G
L
π
f
(
x
¯
)
≠
∅
.Proof.
LetS
:
=
e
p
i
f
⊂
X
×
R, which is obviously closed and epi-Lipschitz around (
x
¯
,
f
(
x
¯
)
) due to the Lipschitz continuity of f around x
¯ and so S is normally compact with respect to the V-proximal normal cone (see Proposition  2.7 in [15]). Also, obviously (
x
¯
,
f
(
x
¯
)
) is a boundary point of S. Using the previous proposition we get N
L
π
(
epi
⁡
f
;
(
x
¯
,
f
(
x
¯
)
)
)
≠
{
(
0,0
)
}. Then there exist (
x
n
∗
,
-
α
n
)
→
w
(
x
∗
,
-
α
)
≠
(
0,0
) and (
x
n
,
r
n
)
→
epi
⁡
f
(
x
¯
,
f
(
x
¯
)
) with (
x
n
∗
,
-
α
n
)
∈
N
π
(
epi
⁡
f
;
(
x
n
,
r
n
)
). Notice first that obviously we have r
n
=
f
(
x
n
) and α
n
≥
0. In order to prove that α
n
>
0, we assume by contradiction that α
n
=
0. Then α
=
0 and so (85)
x
∗
,
0
∈
N
L
π
epi
⁡
f
;
x
¯
,
f
x
¯
⊂
N
F
L
epi
⁡
f
;
x
¯
,
f
x
¯
,which means that x
∗
∈
∂
∞
f
(
x
¯
) and so Corollary  3.9 in [7] ensures by the Lipschitz continuity of f that x
∗
=
0 and this contradicts the fact that (
x
∗
,
-
α
)
≠
(
0,0
). So this case is not possible. Consequently, we have α
n
>
0. Then (
x
n
∗
/
α
n
,
-
1
)
∈
N
π
(
S
;
(
x
n
,
r
n
)
); that is, x
n
∗
/
α
n
∈
∂
G
π
f
(
x
n
) and so x
∗
/
α
∈
∂
G
L
π
f
(
x
¯
) and so ∂
G
L
π
f
(
x
¯
) is nonempty and the proof is complete.Remark 21.
The nonemptiness of the limiting analyticV-proximal subdifferential for locally Lipschitz functions in p-uniformly convex and q-uniformly smooth Banach space cannot be deduced from the previous result in Theorem 20. However, the nonemptiness of ∂
A
L
π
f
(
x
¯
) is ensured under the assumptions of Theorem 7.


## 6. Applications

Throughout this section the spaceX is p-uniformly convex and q-uniformly smooth Banach space admitting an equivalent norm
⁡
· such that ·
s (for some s
≥
2) is C
2-differentiable on X
∖
{
0
} and the functional V is associated with that norm
⁡
·. Our objective in this section is to present two applications of the results proved in the previous sections. Our results in this section are new in the Banach spaces setting. We establish in terms of the generalised projection π
S some necessary optimality conditions to nonconvex minimisation problems and nonconvex variational inequalities.


### 6.1. Necessary Optimality Conditions to Nonconvex Minimisation Problems

Consider the following constrained minimisation problem:P
inf
x
∈
S
⁡
f
x
,where f
:
X
→
R is a l.s.c. function and S is a closed nonempty set in X.Proposition 22.
Assume thatf is locally Lipschitz at x
¯. If x
¯ is a solution of P, then there exist x
∗
∈
∂
F
L
f
(
x
¯
), x
n
→
S
x
¯, x
n
∗
→
w
x
∗, and α
n
>
0 such that x
n
∈
π
S
(
J
(
x
n
)
-
α
n
x
n
∗
).Proof.
Assume thatx
¯ is a solution of P. Then x
¯ is a global minimum of the function f
+
ψ
S over X and so 0
∈
∂
F
L
(
f
+
ψ
S
)
(
x
¯
). Using the exact sum rule for the limiting Fréchet subdifferential (see, e.g., [7]) we get 0
∈
∂
F
L
f
(
x
¯
)
+
∂
F
L
ψ
S
(
x
¯
)
=
∂
F
L
f
(
x
¯
)
+
N
F
L
(
S
;
x
¯
). Then there exists x
∗
∈
∂
F
L
f
(
x
¯
) such that -
x
∗
∈
N
F
L
(
S
;
x
¯
). Using Theorem 14 we obtain x
n
→
S
x
¯, x
n
∗
→
w
x
∗, and α
n
>
0 such that x
n
∈
π
S
(
J
(
x
n
)
-
α
n
x
n
∗
) and hence the proof is complete.A direct consequence of this proposition is given in the following corollary.Corollary 23.
Assume thatx
¯ is a V-proximal normally regular point of S; that is, N
L
π
(
S
;
x
¯
)
=
N
π
(
S
;
x
¯
) and assume that f is locally Lipschitz at x
¯. If x
¯ is a solution of P, then x
¯
∈
π
S
(
J
(
x
¯
)
-
α
x
∗
), for some α
>
0 and for some x
∗
∈
∂
F
L
f
(
x
¯
).Proof.
Following the same lines of the proof of Proposition22, we obtain some x
∗
∈
∂
F
L
f
(
x
¯
) such that -
x
∗
∈
N
F
L
(
S
;
x
¯
). Since x
¯ is a V-proximal normally regular point of S, that is, N
L
π
(
S
;
x
¯
)
=
N
π
(
S
;
x
¯
) we have by Theorem 13 the equality N
F
L
(
S
;
x
¯
)
=
N
π
(
S
;
x
¯
) and so -
x
∗
∈
N
π
(
S
;
x
¯
). Consequently, by the characterisation of the V-proximal normal cone given in Proposition 1 there exists α
>
0 such that x
¯
∈
π
S
(
J
(
x
¯
)
-
α
x
∗
) and hence the proof is complete.


### 6.2. Necessary Optimality Conditions to Nonconvex Variational Inequalities

Letf
:
X
→
X
∗ be an arbitrary mapping, let S
⊂
X be a nonempty closed set not necessarily convex, and let F
:
X
×
X
→
R be a bifunction satisfying F
(
x
,
x
)
=
0 on S. Consider the following nonconvex variational inequality:(86)
Find
x
¯
∈
S
,
such  that
f
x
¯
;
y
-
x
¯
+
F
x
¯
,
y
≥
0
,
∀
y
∈
S
.Our aim is to derive necessary conditions for a solution to (86) in terms of the generalised projection on S. First, we observe that this variational inequality coincides with the one studied in [16] when S is convex and F is taken to satisfy F
(
x
,
y
)
=
0 on S.Proposition 24.
Assume thatx
¯ is a solution of (86) and that F
(
x
¯
,
·
) is locally Lipschitz at x
¯. Then there exists some x
∗
∈
∂
F
L
F
(
x
¯
,
·
)
(
x
¯
) and sequences {
x
n
}, {
x
n
∗
}, and α
n such that x
n
→
S
x
¯, x
n
∗
→
w
x
∗, and α
n
>
0 and satisfying x
n is a fixed point of the set-valued mapping x
⇉
π
S
(
J
(
x
)
-
α
n
[
x
n
∗
+
f
(
x
¯
)
]
), for any n
≥
1.Proof.
Assume thatx
¯
∈
S is a solution of (86). Then x
¯ is a global minimum of the function h
(
x
)
:
=
〈
f
(
x
¯
)
;
x
-
x
¯
〉
+
F
(
x
¯
,
x
)
+
ψ
S
(
x
). Hence (87)
0
∈
∂
F
L
h
x
¯
=
∂
F
L
f
x
¯
;
·
-
x
¯
+
F
x
¯
,
·
+
ψ
S
x
¯
.Using the exact sum rule for the Fréchet limiting subdifferential (see, e.g., [7]) and the fact that F
(
x
¯
,
·
) is locally Lipschitz at x
¯, we obtain(88)
0
∈
f
x
¯
+
∂
F
L
F
x
¯
,
·
x
¯
+
∂
F
L
ψ
S
x
¯
,
that  is
,
-
f
x
¯
∈
∂
F
L
F
x
¯
,
·
x
¯
+
N
F
L
S
;
x
¯
.Then there exists some x
∗
∈
∂
F
L
F
(
x
¯
,
·
)
(
x
¯
)
≠
∅ such that (89)
-
f
x
¯
-
x
∗
∈
N
F
L
S
;
x
¯
.By the characterisation given in Theorem 14, there exist x
n
→
S
x
¯, α
n
>
0, and y
n
∗
→
w
x
∗
+
f
(
x
¯
) such that x
n
∈
π
S
(
J
(
x
n
)
-
α
n
y
n
∗
). Let x
n
∗
:
=
y
n
∗
-
f
(
x
¯
). Then x
n
∗
→
w
x
∗ and x
n
∈
π
S
(
J
(
x
n
)
-
α
n
[
x
n
∗
+
f
(
x
¯
)
]
) and hence x
n is a fixed point of the set-valued mapping x
⇉
π
S
(
J
(
x
)
-
α
n
[
x
n
∗
+
f
(
x
¯
)
]
), for any n
≥
1 and hence the proof is complete.The following corollary is a direct consequence of the previous proposition atV-proximal normally regular points.Corollary 25.
Assume thatx
¯ is a V-proximal normally regular point of S and that x
¯ is a solution of (86). Assume that F
(
x
¯
,
·
) is locally Lipschitz at x
¯. Then there exist x
∗
∈
∂
F
L
F
(
x
¯
,
·
)
(
x
¯
) and α
>
0 such that x
¯ is a fixed point of the set-valued mapping x
⇉
π
S
(
J
(
x
)
-
α
[
x
∗
+
f
(
x
¯
)
]
).


## 6.1. Necessary Optimality Conditions to Nonconvex Minimisation Problems

Consider the following constrained minimisation problem:P
inf
x
∈
S
⁡
f
x
,where f
:
X
→
R is a l.s.c. function and S is a closed nonempty set in X.Proposition 22.
Assume thatf is locally Lipschitz at x
¯. If x
¯ is a solution of P, then there exist x
∗
∈
∂
F
L
f
(
x
¯
), x
n
→
S
x
¯, x
n
∗
→
w
x
∗, and α
n
>
0 such that x
n
∈
π
S
(
J
(
x
n
)
-
α
n
x
n
∗
).Proof.
Assume thatx
¯ is a solution of P. Then x
¯ is a global minimum of the function f
+
ψ
S over X and so 0
∈
∂
F
L
(
f
+
ψ
S
)
(
x
¯
). Using the exact sum rule for the limiting Fréchet subdifferential (see, e.g., [7]) we get 0
∈
∂
F
L
f
(
x
¯
)
+
∂
F
L
ψ
S
(
x
¯
)
=
∂
F
L
f
(
x
¯
)
+
N
F
L
(
S
;
x
¯
). Then there exists x
∗
∈
∂
F
L
f
(
x
¯
) such that -
x
∗
∈
N
F
L
(
S
;
x
¯
). Using Theorem 14 we obtain x
n
→
S
x
¯, x
n
∗
→
w
x
∗, and α
n
>
0 such that x
n
∈
π
S
(
J
(
x
n
)
-
α
n
x
n
∗
) and hence the proof is complete.A direct consequence of this proposition is given in the following corollary.Corollary 23.
Assume thatx
¯ is a V-proximal normally regular point of S; that is, N
L
π
(
S
;
x
¯
)
=
N
π
(
S
;
x
¯
) and assume that f is locally Lipschitz at x
¯. If x
¯ is a solution of P, then x
¯
∈
π
S
(
J
(
x
¯
)
-
α
x
∗
), for some α
>
0 and for some x
∗
∈
∂
F
L
f
(
x
¯
).Proof.
Following the same lines of the proof of Proposition22, we obtain some x
∗
∈
∂
F
L
f
(
x
¯
) such that -
x
∗
∈
N
F
L
(
S
;
x
¯
). Since x
¯ is a V-proximal normally regular point of S, that is, N
L
π
(
S
;
x
¯
)
=
N
π
(
S
;
x
¯
) we have by Theorem 13 the equality N
F
L
(
S
;
x
¯
)
=
N
π
(
S
;
x
¯
) and so -
x
∗
∈
N
π
(
S
;
x
¯
). Consequently, by the characterisation of the V-proximal normal cone given in Proposition 1 there exists α
>
0 such that x
¯
∈
π
S
(
J
(
x
¯
)
-
α
x
∗
) and hence the proof is complete.


## 6.2. Necessary Optimality Conditions to Nonconvex Variational Inequalities

Letf
:
X
→
X
∗ be an arbitrary mapping, let S
⊂
X be a nonempty closed set not necessarily convex, and let F
:
X
×
X
→
R be a bifunction satisfying F
(
x
,
x
)
=
0 on S. Consider the following nonconvex variational inequality:(86)
Find
x
¯
∈
S
,
such  that
f
x
¯
;
y
-
x
¯
+
F
x
¯
,
y
≥
0
,
∀
y
∈
S
.Our aim is to derive necessary conditions for a solution to (86) in terms of the generalised projection on S. First, we observe that this variational inequality coincides with the one studied in [16] when S is convex and F is taken to satisfy F
(
x
,
y
)
=
0 on S.Proposition 24.
Assume thatx
¯ is a solution of (86) and that F
(
x
¯
,
·
) is locally Lipschitz at x
¯. Then there exists some x
∗
∈
∂
F
L
F
(
x
¯
,
·
)
(
x
¯
) and sequences {
x
n
}, {
x
n
∗
}, and α
n such that x
n
→
S
x
¯, x
n
∗
→
w
x
∗, and α
n
>
0 and satisfying x
n is a fixed point of the set-valued mapping x
⇉
π
S
(
J
(
x
)
-
α
n
[
x
n
∗
+
f
(
x
¯
)
]
), for any n
≥
1.Proof.
Assume thatx
¯
∈
S is a solution of (86). Then x
¯ is a global minimum of the function h
(
x
)
:
=
〈
f
(
x
¯
)
;
x
-
x
¯
〉
+
F
(
x
¯
,
x
)
+
ψ
S
(
x
). Hence (87)
0
∈
∂
F
L
h
x
¯
=
∂
F
L
f
x
¯
;
·
-
x
¯
+
F
x
¯
,
·
+
ψ
S
x
¯
.Using the exact sum rule for the Fréchet limiting subdifferential (see, e.g., [7]) and the fact that F
(
x
¯
,
·
) is locally Lipschitz at x
¯, we obtain(88)
0
∈
f
x
¯
+
∂
F
L
F
x
¯
,
·
x
¯
+
∂
F
L
ψ
S
x
¯
,
that  is
,
-
f
x
¯
∈
∂
F
L
F
x
¯
,
·
x
¯
+
N
F
L
S
;
x
¯
.Then there exists some x
∗
∈
∂
F
L
F
(
x
¯
,
·
)
(
x
¯
)
≠
∅ such that (89)
-
f
x
¯
-
x
∗
∈
N
F
L
S
;
x
¯
.By the characterisation given in Theorem 14, there exist x
n
→
S
x
¯, α
n
>
0, and y
n
∗
→
w
x
∗
+
f
(
x
¯
) such that x
n
∈
π
S
(
J
(
x
n
)
-
α
n
y
n
∗
). Let x
n
∗
:
=
y
n
∗
-
f
(
x
¯
). Then x
n
∗
→
w
x
∗ and x
n
∈
π
S
(
J
(
x
n
)
-
α
n
[
x
n
∗
+
f
(
x
¯
)
]
) and hence x
n is a fixed point of the set-valued mapping x
⇉
π
S
(
J
(
x
)
-
α
n
[
x
n
∗
+
f
(
x
¯
)
]
), for any n
≥
1 and hence the proof is complete.The following corollary is a direct consequence of the previous proposition atV-proximal normally regular points.Corollary 25.
Assume thatx
¯ is a V-proximal normally regular point of S and that x
¯ is a solution of (86). Assume that F
(
x
¯
,
·
) is locally Lipschitz at x
¯. Then there exist x
∗
∈
∂
F
L
F
(
x
¯
,
·
)
(
x
¯
) and α
>
0 such that x
¯ is a fixed point of the set-valued mapping x
⇉
π
S
(
J
(
x
)
-
α
[
x
∗
+
f
(
x
¯
)
]
).



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*Source: 1917387-2016-06-13.xml*