# Convex Sweeping Processes with Noncompact Perturbations and with Delay in Banach Spaces

Authors: A. G. Ibrahim; F. Aladsani
Journal: Abstract and Applied Analysis
 (2016)

---

## Abstract

We prove two results concerning the existence of solutions
for functional differential inclusions that are governed by sweeping
processes, with noncompact valued perturbations in Banach spaces. Indeed,
we have two goals. The first is to give a technique that allows considering
sweeping processes with noncompact valued perturbations and associated with
a multivalued function depending on time. The second is to give a technique
to overcome the arising problem from the nonlinearity of the normalized
mappings, when we deal with sweeping processes with noncompact valued
perturbations and associated with a multivalued function depending on time
and state.


---

## Body

## 1. Introduction

In his leading paper, Moreau [1] proposed and studied the following differential inclusion governed by sweeping process of first order:(1)
-
u
′
t
∈
N
G
t
u
t
a.e.  on
I
=
0
,
T
,
u
0
=
u
0
,where G is a multifunction from the interval I
=
[
0
,
T
] to the family of nonempty closed convex subsets of a Hilbert space H and N
G
(
t
)
(
u
(
t
)
) is the normal cone of the subset G
(
t
) at the point u
(
t
). Problem (1) corresponds to several important mechanical problems. For more details concerning the applications of (1), we refer to [1].Since then, important improvements have been developed by several authors. For some existence results of solutions for sweeping processes in Hilbert or Banach spaces, we refer to [2–10].LetE be a Banach space, r a positive real number, C
r
=
C
E
(
[
-
r
,
0
]
) the Banach space of continuous functions from [
-
r
,
0
] to E endowed with the uniform norm u
∞
=
S
u
p
{
u
(
t
)
:
t
∈
[
-
r
,
0
]
}, C a multifunction from I to the family of nonempty closed convex subsets of E, F a multifunction defined on I
×
C
r and with nonempty closed values in the topological dual space, E
∗, of E, and Γ a multifunction from I
×
E
∗ to the family of nonempty closed convex subsets of E. Let J
:
E
→
E
∗ and J
∗
:
E
∗
→
E be the normalized duality mappings and for each t
∈
I, τ
(
t
)
:
C
E
(
[
-
r
,
t
]
)
→
C
r and (
τ
(
t
)
g
)
(
s
)
=
g
(
s
+
t
), ∀
s
∈
[
-
r
,
0
]. Let Ψ
∈
C
r be given.In this paper, we prove two existence results. In the first result (Theorem15), we prove, in p-uniformly convex and q-uniformly smooth Banach space, the existence of solutions for the following sweeping process with noncompact valued perturbation and with delay:(2)
u
t
=
Ψ
t
,
for
t
∈
-
r
,
0
;
u
t
=
Ψ
0
+
∫
0
t
u
′
s
d
s
,
for
t
∈
I
,
u
t
∈
C
t
,
for
t
∈
I
,
u
∗
t
=
J
u
t
,
for
t
∈
I
,
u
∗
′
t
∈
-
N
C
t
u
t
+
F
t
,
τ
t
u
,
a.e.  for
t
∈
I
.In the second result (Theorem16), we prove, in p-uniformly convex and q-uniformly smooth Banach spaces, the existence of solutions for the following sweeping process with a noncompact valued perturbation and with delay:(3)
u
t
=
Ψ
t
,
for
t
∈
-
r
,
0
;
u
t
=
Ψ
0
+
∫
0
t
u
′
s
d
s
,
for
t
∈
I
,
u
∗
t
=
J
∗
u
t
,
for
t
∈
I
,
u
t
∈
Γ
t
,
u
∗
t
,
for
t
∈
I
,
u
∗
′
t
∈
-
N
Γ
t
,
u
∗
t
u
t
+
F
t
,
τ
t
u
,
a.e.  for
t
∈
I
.We prove the existence of solutions for (3) without imposing that the values of Γ are contained in a fixed convex compact subset; instead, we suppose that Γ satisfies a condition in terms of the Hausdorff measure of noncompactness.It is important to note that when the sweeping process is associated with a multifunction depending on time and state, the nonlinearity of the normalized duality mappings causes a difficulty when we deal with sweeping process with a perturbation; see, for example, the remark which has been given by Bounkhel and Castaing [6]. For this reason we define the multifunction Γ on I
×
E
∗
.In order to explain the mathematical motivation for this work, we mention some recent results in this domain.Aitalioubrahim [2] considered (2) when X is a Hilbert space and the value of C is nonempty compact and not necessarily convex.In Theorem (4.3) in [5], Bounkhel and Al-Yusof established the existence of solutions for (2), in a separable, p-uniformly convex (
p
>
1
) and q-uniformly smooth (
q
>
1
) Banach space without delay and when J
(
C
(
t
)
)
⊆
G, for any t
∈
I, for some convex compact set G
⊆
E
∗; F is upper semicontinuous with convex compact values and F
(
t
,
x
)
⊆
Z, ∀
(
t
,
x
)
∈
I
×
E, for some convex compact subset Z
⊂
E
∗.In Theorem (4.5) in [6], Bounkhel and Castaing considered (3) in a separable, p-uniformly convex (
p
≥
2
) and q-uniformly smooth Banach space, q
∈
(
1,2
], and when F
=
0 and Γ is defined from I
×
E to the family of nonempty closed convex subsets of E and satisfies a condition in terms of the Hausdorff measure of noncompactness.Castaing et al. [10] considered a second-order sweeping process without delay in a separable Hilbert space H in the case when K is a Lipschitz multifunction defined on I
×
H and taking closed ρ-proxy regular (ρ
>
0) values in H; and F is a convex weakly compact valued scalary upper semicontinuous defined on I
×
H
×
H.Ibrahim and AL-Adsani [7] considered a second order sweeping process with delay in a separable p-uniformly convex and q-uniformly smooth Banach space X and the values of perturbation F are nonempty convex weakly compact.Noel and Thibault [8] established the existence of solutions for nonconvex sweeping processes with a moving set depending on the state in Hilbert spaces.For other contributions on differential inclusions, see Gomaa [11].We note that ifE is a Hilbert space, then J is equal to the identity mapping, E is 2-uniformly convex and 2-uniformly smooth Banach space. Therefore, our technique allows discussing some sweeping process problems with noncompact perturbation in Hilbert spaces or in Banach spaces, whether the moving set depends on time or on time and state.The paper is organized as follows. Section2 is devoted to some definitions and notations needed later. In Section 3, we prove the existence of solutions for (2). In Section 4, we give existence of solutions for (3).


## 2. Preliminaries and Notations

LetP
c
(
E
)
=
A
⊆
E
:
A is nonempty closed, P
c
c
(
E
)
=
A
⊆
E
:
A is nonempty closed convex, and P
c
k
(
E
)
=
A
⊆
E
:
A is nonempty, convex and compact. Let B
=
x
∈
E
:
x
≤
1 and B
∗
=
{
z
∈
E
∗
:
z
≤
1
}.Definition 1 (see Def. 2.4.1 [12]).
The multivalued mappings(4)
J
:
E
⟶
2
E
∗
,
J
x
=
y
∈
E
∗
:
x
,
y
=
x
2
=
y
2
,
J
∗
:
E
∗
⟶
2
E
,
J
∗
y
=
x
∈
X
:
y
,
x
=
x
2
=
y
2are called the normalized duality mappings.In the following Lemma, we recall some properties ofJ and J
∗
.Lemma 2 (see [12], Prop. 2.4.5, 2.4.12, and 2.4.15).
We have the following:(1)
IfE is a Hilbert space, then J
(
x
)
=
{
x
}, for all x
∈
H.
(2)
For eachx
∈
E, J
(
x
) is nonempty closed convex and bounded subset of E
∗.
(3)
J
(
λ
x
)
=
λ
J
(
x
), for all x
∈
E and all λ
∈
R.
(4)
IfE
∗ is strictly convex, J is single-valued.
(5)
IfE is strictly convex, J is one to one; that is, x
≠
y
⇒
J
(
x
)
∩
J
(
y
)
=
ϕ.
(6)
IfE
∗is uniformly convex, then J is uniformly continuous on each bounded set in E; that is, ∀
ε
>
0 and α
>
0 there is a δ
=
δ
(
ε
,
α
)
>
0 such that for x and y with x
<
α and y
<
α we have(5)
x
-
y
<
δ
⟹
J
x
-
J
y
<
ε
(note that ifE
∗ is uniformly convex, then it is strictly convex and hence J is single-valued mapping).
(7)
IfE is reflexive, then J is a mapping from E onto E
∗, that is (6)
∪
J
x
:
x
∈
E
=
E
∗
;
(8)
IfE is reflexive strictly convex space with strictly convex conjugate space E
∗, then J and J
∗ are one-to-one, onto and single-valued mapping and (7)
J
-
1
=
J
∗
,
J
J
∗
=
I
X
∗
,
J
∗
J
=
I
X
,
whereI
X is the identity mapping on E and I
E
∗is the identity mapping on E
∗.For more properties of the duality mapping, we refer to [12–14].Definition 3 (see [12], Def. 2.8.1).
The Banach spaceE is said to be uniformly smooth if (8)
ρ
E
′
0
=
lim
t
→
0
⁡
ρ
E
t
t
=
0
,where ρ
E is the modulus of smoothness of E.Definition 4 (see [12–14]).
Letq
>
1 be a real number. A Banach space E is said to be q-uniformly smooth if there exists a constant c
>
0 such that ρ
E
(
t
)
≤
c
t
q, ∀
t
>
0.Clearly, everyq-uniformly smooth Banach space is uniformly smooth.Lemma 5 (see [12–14]).
The following properties are satisfied:(1)
E is uniformly smooth if and only if E
∗ is uniformly convex.
(2)
E is uniformly convex if and only if E
∗ is uniformly smooth.
(3)
IfE is uniformly smooth, then E is reflexive.Lemma 6 (see [12, 13]).
Letp
>
1:(1)
IfE is p-uniformly convex, then E
∗ is p
′-uniformly smooth where p
′
=
p
/
(
p
-
1
).
(2)
IfE is p-uniformly smooth, then E
∗ is p
′-uniformly convex where p
′
=
p
/
(
p
-
1
).Remark 7.
It is known that(1)
the Banach spaceL
p
(
I
,
R
) is m
i
n
{
2
,
p
} uniformly smooth;
(2)
if1
<
p
<
2, then L
p
(
I
,
R
) is 2-uniformly convex; if 2
≤
p, then L
p
(
I
,
R
) is p-uniformly convex.Now, let(9)
V
:
E
∗
×
E
⟶
R
,
V
∗
:
E
×
E
∗
⟶
Rbe two functions defined by(10)
V
φ
,
x
=
φ
2
-
2
φ
,
x
+
x
2
,
V
∗
x
,
φ
=
V
φ
,
x
.Observe thatV
(
J
(
x
)
,
x
)
=
0, ∀
x
∈
E, and if E is a Hilbert space, then V
(
φ
,
x
)
=
φ
-
x
2
.Definition 8 (see [12]).
LetZ be a nonempty subset of E and φ
∈
E
∗. If there exists a point z
∈
Z satisfying V
(
φ
,
z
)
=
d
Z
V
(
φ
), then z is called a generalized projection of φ onto Z, where d
Z
V
(
φ
)
=
i
n
f
x
∈
Z
V
(
φ
,
x
).The set of all such points is denoted byπ
Z
(
φ
); that is, (11)
π
Z
φ
=
z
∈
Z
:
V
φ
,
z
=
d
Z
V
φ
.The following lemma summarizes some important properties ofV and π
Z
(
φ
) (see [12]).Lemma 9.
LetX be a reflexive Banach space with dual space X
∗and let Z be a nonempty closed convex subset of X. The following properties hold:(1)
π
Z
(
φ
)
≠
ϕ, ∀
φ
∈
X
∗.
(2)
IfX is uniformly convex or uniformly smooth, then (12)
V
φ
,
x
=
0
⟺
φ
=
J
x
,
∀
x
∈
X
,
φ
∈
X
∗
.
(3)
X is strictly convex if and only if π
Z
(
φ
) is singleton for all φ
∈
X
∗.
(4)
IfX is also smooth, then, for any given φ
∈
X
∗, (13)
z
∈
π
Z
φ
⟺
φ
-
J
z
,
x
-
z
≤
0
,
∀
x
∈
X
.Definition 10 (see [4, 14]).
LetZ be a nonempty closed convex subset of E and z
∈
Z. The convex normal cone of Z at z is defined by (14)
N
Z
z
=
φ
∈
E
∗
:
φ
,
x
-
z
≤
0
,
∀
x
∈
E
.The following lemma gives a closedness property of the subdifferential of the distance function associated with a set-valued mapping with closed convex values.Lemma 11 (see [14]).
LetZ be a nonempty, closed, and convex subset of a Banach space X and z
∈
Z. Then,(1)
N
Z
(
z
)
∩
{
y
∈
X
∗
:
y
≤
1
}
=
∂
d
Z
(
z
), where ∂
d
Z
(
z
) is the subdifferential of the distance function;
(2)
ifX is reflexive and smooth, then (15)
z
∈
π
Z
φ
⟺
φ
-
J
z
∈
N
Z
z
,
∀
φ
∈
X
∗
,
∀
z
∈
Z
.Let us recall the following lemmas that will be used in the sequel.Lemma 12 (see [5], Lemma 4.1 and Prop. 4.2).
Letp, q
>
1, let X be a p-uniformly convex and q-uniformly smooth Banach space, and let Z be a nonempty bounded subset of X; then, there exist two constants a
>
0 and b
>
0 such that (16)
a
x
-
y
p
≤
V
J
x
,
y
≤
b
x
-
y
q
,
∀
x
,
y
∈
Z
,
d
Z
V
φ
≤
b
d
Z
J
∗
φ
q
,
∀
φ
∈
X
∗
.Moreover, if Z, in addition, is closed and φ
∈
X
∗, then (17)
d
Z
V
φ
=
0
⟺
J
∗
φ
∈
Z
.Lemma 13 (see [6], Prop. 4.3).
Letp
≥
2, and let X be a p-uniformly convex and q-uniformly smooth Banach space. The normalized duality mapping J is Lipschitz on bounded sets; that is, for any R
>
0, there is a positive constant dependent on R
,
C
(
R
), such that (18)
x
≤
R
,
y
≤
R
⟹
J
x
-
J
y
≤
C
R
x
-
y
.Lemma 14 (see [15], Lemma 2.3).
LetX be a separable Banach space, let G
:
[
a
,
b
]
→
2
X be a measurable multifunction, and let z
:
[
a
,
b
]
→
X be a measurable function. Then, for any positive measurable function r
:
[
a
,
b
]
→
R
+, there exists a measurable selection g of G such that for almost all t
∈
[
a
,
b
]
(19)
g
t
-
z
t
≤
d
z
t
,
G
t
+
r
t
.


## 3. Existence Results of Solutions for Problem (2)

Theorem 15.
Letp
>
1, q
∈
(
1,2
], E be a separable, p-uniformly convex and q-uniformly smooth Banach space, C
:
I
→
P
c
k
(
E
) and F
:
I
×
C
r
→
P
c
(
E
∗
). Suppose that the following conditions hold:(
H
1
)
There is a convex compact subsetK
⊆
E
∗, such that J
(
C
(
t
)
)
⊆
K, ∀
t
∈
I.
(
H
2
)
There are two constantsλ
1 and λ
2 such that for any t
1, t
2
∈
I and any x
1, x
2
∈
E we have (20)
d
C
t
1
x
1
q
/
q
/
-
d
C
t
2
x
2
q
/
q
/
≤
λ
1
t
1
-
t
2
+
λ
2
x
1
-
x
2
,
whereq
/
=
q
/
(
q
-
1
)
.
(
H
3
)
There is a continuous functionm
:
I
→
R
≥
0 such that, for any t
,
s
∈
I and ψ
1
,
ψ
2
∈
C
r, (21)
h
F
t
,
ψ
1
,
F
s
,
ψ
2
≤
m
t
-
m
s
+
ψ
1
-
ψ
2
.
(
H
4
)
There is a continuous functionP
:
I
→
R
≥
0 such that, for any φ
∈
C
r and any t
∈
I,(22)
sup
⁡
y
:
y
∈
F
t
,
φ
≤
P
t
1
+
φ
0
,
a
.
e
.Then, for any fixedΨ
∈
C
r with Ψ
(
0
)
∈
C
(
0
), there is a continuous function u
:
[
-
r
,
T
]
→
E such that u is Lipschitz on I and satisfies (2).Proof.
SinceE is uniformly convex, it is reflexive and strictly convex. Moreover, because E is uniformly smooth, its topological conjugate E
∗ is uniformly convex and hence E
∗ is strictly convex. Then, by property (8) in Lemma 2, the normalized duality maps J and J
′ are one-to-one, onto, single-valued maps and J
J
∗
=
I
E
∗,J
∗
J
=
I
E, where I
E
∗ and I
E are the identity maps on E
∗ and E, respectively. Moreover, since E is reflexive and strictly convex, Lemma 9 ensures that the generalized projection π
Z
(
φ
) is singleton for any closed convex subset Z of E and for any φ
∈
E
∗. Finally, from the reflexivity and smoothness of E, Lemma 11 tells us that(23)
y
∈
π
Z
φ
⟺
φ
-
J
y
∈
N
Z
y
.Now, for any fixed natural number n
≥
2, we consider the following partition for I: t
i
n
=
i
T
/
2
n, i
=
0,1
,
2
,
…
,
2
n
. We put I
0
n
=
[
t
0
n
,
t
1
n
], I
i
n
=
(
t
i
n
,
t
i
+
1
n
], i
=
0,1
,
2
,
…
,
2
n
-
1, and μ
n
=
T
/
2
n. Also, let θ
n
,
δ
n
:
I
→
I be defined as θ
n
(
0
)
=
0, θ
n
(
t
)
=
t
i
+
1
n, t
∈
(
t
i
n
,
t
i
+
1
n
], i
=
0,1
,
2
,
…
,
2
n
-
1, δ
n
(
t
)
=
t
i
n, t
∈
[
t
i
n
,
t
i
+
1
n
),  i
=
0,1
,
2
,
…
,
2
n
-
1, and δ
n
(
T
)
=
T. In order to make the proof easy for the reader, we divide the rest of the proof into steps.
Step 1. In this step, we show that if n
≥
2 is a fixed natural number and z
∈
L
1
(
I
,
E
∗
) is a fixed function, then there are u
n
∈
C
(
[
-
r
,
T
]
,
E
), u
n
∗
∈
C
(
[
-
r
,
T
]
,
E
∗
), and g
n
∈
L
1
(
I
,
E
∗
) such that u
n and u
n
∗ are absolutely continuous on I and the following properties hold:(24)
(i)
u
n
∗
t
=
J
Ψ
t
,
t
∈
-
r
,
0
;
(ii)
u
n
t
=
J
∗
u
n
∗
t
,
t
∈
-
r
,
T
;
(iii)
u
n
θ
n
t
∈
C
θ
n
t
,
t
∈
I
;
(iv)
g
n
t
∈
F
t
,
τ
δ
n
t
u
n
,
t
∈
I
;
(v)
g
n
t
-
z
t
≤
d
z
t
,
F
t
,
τ
δ
n
t
u
n
+
1
n
2
,
t
∈
I
;
(vi)
-
u
n
∗
′
t
+
g
n
t
∈
N
C
θ
n
t
u
n
θ
n
t
,
a.e.
t
∈
I
.
Indeed, in view of Lemma14, there is g
0
n
∈
L
1
(
[
t
0
n
,
t
1
n
]
,
E
∗
) such that g
0
n
(
t
)
∈
F
(
t
,
τ
(
0
)
Ψ
), t
∈
[
t
0
n
,
t
1
n
] and (25)
g
0
n
t
-
z
t
≤
d
z
t
,
F
t
,
τ
0
Ψ
+
1
n
2
,
for  a.e.
t
∈
t
0
n
,
t
1
n
.
Putx
0
n
=
ψ
(
0
) and(26)
x
1
n
=
Π
C
t
1
n
J
x
0
n
+
μ
n
g
0
n
t
0
n
.
Next, we defineu
n
∗
(
t
)
=
J
(
Ψ
(
t
)
) for t
∈
[
-
r
,
0
] and(27)
u
n
∗
t
=
t
1
n
-
t
μ
n
J
x
0
n
+
t
-
t
0
n
μ
n
J
x
1
n
,
t
∈
t
0
n
,
t
1
n
.
Observe that, according to Lemma9, this construction is well defined. Moreover, from the definition of the generalized projection, relation (26) gives us (28)
J
x
0
n
+
μ
n
g
0
n
t
0
n
-
J
x
1
n
μ
n
∈
N
C
t
1
n
x
1
nand this relation together with (27) implies (29)
-
u
n
∗
′
t
+
g
0
n
t
0
n
∈
N
C
t
1
n
x
1
n
,
a.e.  for
t
∈
t
0
n
,
t
1
n
.Next, by induction, we can define for i
=
1,2
,
…
,
2
n
-
1
(30)
u
n
∗
t
=
t
i
+
1
n
-
t
μ
n
J
x
i
n
+
t
-
t
i
n
μ
n
J
x
i
+
1
n
,
t
∈
t
i
n
,
t
i
+
1
n
,where(31)
x
i
+
1
n
=
Π
C
t
i
+
1
n
J
x
i
n
+
μ
n
g
i
n
t
i
n
,
g
i
n
∈
L
1
t
i
n
,
t
i
+
1
n
,
E
∗
,
g
i
n
t
∈
F
t
,
τ
δ
n
t
u
n
,
t
∈
t
i
n
,
t
i
+
1
n
,
u
n
t
=
J
∗
u
n
∗
t
,
t
∈
-
r
,
T
,
g
i
n
t
-
z
t
≤
d
z
t
,
F
t
,
τ
δ
n
t
u
n
+
1
n
2
,
a.e.
t
∈
t
i
n
,
t
i
+
1
n
.
Now, letg
n
:
I
→
E
∗ be such that g
n
(
t
)
=
g
i
n
(
t
), t
∈
[
t
i
n
,
t
i
+
1
n
), i
=
0,1
,
2
,
…
,
2
n
-
1, and g
n
(
T
)
=
g
2
n
n
(
T
). Note that g
i
n
(
t
i
n
)
=
g
i
n
(
δ
n
(
t
)
)
=
g
n
(
δ
n
(
t
)
), for t
∈
[
t
i
n
,
t
i
+
1
n
). Therefore, the functions u
n
∗, g
n, and u
n satisfy properties (i)–(vi) in (24).
Step 2. In this step, we show that there are three sequences (
u
n
)
n
≥
2
⊆
C
(
[
-
r
,
T
]
,
E
), (
u
n
∗
)
n
≥
2
⊆
C
(
[
-
r
,
T
]
,
E
∗
), and (
g
n
)
n
≥
2
⊆
L
1
(
I
,
E
∗
) such that u
n and u
n
∗ are absolutely continuous on [
0
,
T
] and for any n
≥
2 we have(32)
(i)
u
n
∗
t
=
J
Ψ
t
,
t
∈
-
r
,
0
;
(ii)
u
n
t
=
J
∗
u
n
∗
t
,
t
∈
-
r
,
T
;
(iii)
u
n
θ
n
t
∈
C
θ
n
t
,
t
∈
I
;
(iv)
g
n
t
∈
F
t
,
τ
δ
n
t
u
n
,
t
∈
I
;
(v)
g
n
+
1
t
-
g
n
t
≤
d
g
n
t
,
F
t
,
τ
δ
n
+
1
t
u
n
+
1
+
1
n
+
1
2
,
a.e.
t
∈
I
;
(vi)
-
u
n
∗
′
t
+
g
n
t
∈
N
C
θ
n
t
u
n
θ
n
t
,
a.e.
t
∈
I
.Indeed, let z be any fixed element in L
1
(
I
,
E
∗
); in view of Step 1, there are u
2
∈
C
(
[
-
r
,
T
]
,
E
), u
2
∗
∈
C
(
[
-
r
,
T
]
,
E
∗
), and g
2
∈
L
1
(
I
,
E
∗
) such that u
2
,
u
2
∗ are absolutely continuous on [
0
,
T
] and properties (i)–(vi) in (24) are satisfied for n
=
2. Now, since g
2
∈
L
1
(
I
,
E
∗
), then in view of Step 1, there are u
3
∈
C
(
[
-
r
,
T
]
,
E
), u
3
∗
∈
C
(
[
-
r
,
T
]
,
E
∗
), and g
3
∈
L
1
(
I
,
E
∗
) such that u
3
,
u
3
∗ are absolutely continuous on [
0
,
T
] and (32) is satisfied at n
=
3
. So, we can define inductively three sequences (
u
n
)
n
≥
2
⊆
C
(
[
-
r
,
T
]
,
E
)
,
(
u
n
∗
)
n
≥
2
⊆
C
(
[
-
r
,
T
]
,
E
∗
), and (
g
n
)
n
≥
2
⊆
L
1
(
I
,
E
∗
) such that properties (i)–(vi) in (32) are satisfied.
Step 3. In this step, we show that there is a positive number η such that for any n
≥
2
(33)
u
n
∗
′
t
≤
η
,
a.e.  for
t
∈
I
.
Letn
≥
2 be a fixed natural number. According to (H
1) and (H
4), for any t
∈
I,(34)
g
n
t
≤
P
t
1
+
τ
δ
n
t
u
n
0
=
P
t
1
+
u
n
δ
n
t
=
P
t
1
+
R
≤
max
t
∈
I
⁡
P
t
1
+
R
=
ρ
,where R
=
m
a
x
{
x
:
x
∈
K
}. Then, for every i
=
0,1
,
2
,
…
,
2
n
-
1,(35)
J
x
i
+
1
n
≤
R
,
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
R
+
T
ρ
.
LetB
=
{
x
∈
E
:
x
≤
R
+
T
ρ
} and Z
=
{
z
∈
E
∗
:
z
≤
R
+
T
ρ
}
. Note that E
∗ is q
/-uniformly convex (
q
/
=
q
/
(
1
-
q
)
) and p
/-uniformly smooth (p
/
=
p
/
(
p
-
1
)
)
. Therefore, in view of Lemma 12 and (35), there exists a positive constant a such that (36)
a
z
1
-
z
2
q
/
≤
V
∗
J
∗
z
1
,
z
2
,
∀
z
1
,
z
2
∈
Z
.Then, for every i
=
0,1
,
2
,
…
,
2
n
-
1,(37)
a
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
≤
V
∗
J
∗
J
x
i
+
1
n
,
J
x
i
n
+
μ
n
g
i
n
t
i
n
.
Further, sinceE is p-uniformly convex and q-uniformly smooth, then, again, by Lemma 12, there is a positive constant b such that (38)
d
B
V
φ
≤
b
d
B
J
∗
φ
q
,
∀
φ
∈
E
∗
.
So, for everyi
=
0,1
,
2
,
…
,
2
n
-
1, (39)
d
Γ
t
i
+
1
n
V
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
d
C
t
i
+
1
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
.
This relation together with (37) yields (40)
a
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
≤
V
∗
J
∗
J
x
i
+
1
n
,
J
x
i
n
+
μ
n
g
i
n
t
i
n
=
V
J
x
i
n
+
μ
n
g
i
n
t
i
n
,
x
i
+
1
n
=
d
C
t
i
+
1
n
V
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
d
C
t
i
+
1
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
.According to (
H
2
), the last inequality implies(41)
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
a
1
/
q
/
d
C
t
i
+
1
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
q
′
=
b
a
1
/
q
/
d
C
t
i
+
1
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
q
′
-
d
C
t
i
n
J
∗
J
x
i
n
q
/
q
′
≤
b
a
1
/
q
/
λ
1
μ
n
+
λ
2
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
-
J
∗
J
x
i
n
.
Moreover, from the assumptionq
∈
(
1,2
], we infer that q
/
≥
2
. Then, E
∗is q
/-uniformly convex (
q
/
≥
2
) and p
/-uniformly smooth. Hence, from Lemma 13, there is a constant β, depending on (
2
R
+
T
ρ
), such that, for all z
1,  z
1
∈
E
∗, z
1
≤
2
R
+
T
ζ, and z
1
≤
2
R
+
T
ρ, we have(42)
J
∗
z
1
-
J
∗
z
2
≤
β
z
1
-
z
2
.
Consequently,(43)
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
-
J
∗
J
x
i
n
≤
β
J
x
i
n
+
μ
n
g
i
n
t
i
n
-
J
x
i
n
=
β
μ
n
g
i
n
t
i
n
=
β
μ
n
ρ
.
This inequality and (41) yield (44)
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
a
1
/
q
/
λ
1
μ
n
+
β
λ
2
μ
n
ρ
.
Thus,(45)
J
x
i
+
1
n
-
J
x
i
n
≤
b
a
1
/
q
/
λ
1
μ
n
+
β
λ
2
μ
n
ρ
+
μ
n
ρ
=
μ
n
λ
1
¯
+
λ
2
¯
,where λ
1
¯
=
(
b
/
a
)
1
/
q
/
λ
1 and λ
2
¯
=
ρ
+
(
b
/
a
)
1
/
q
/
β
ρ
λ
2
.
Therefore, forn
≥
2,  i
=
0,1
,
2
,
…
,
2
n
-
1, and t
∈
(
t
i
n
,
t
i
+
1
n
), we get (46)
u
n
∗
′
t
≤
λ
1
¯
+
λ
2
¯
=
η
.
This proves that (33) is true.
Step 4. Our claim in this step is to show that the sequence (
u
n
∗
)
n
≥
2 has a convergent subsequence, still denoted by (
u
n
∗
)
n
≥
2, converging uniformly to a Lipschitz function u
∗
:
I
→
E
∗
,
(
u
n
∗
)
n
≥
2
′ converges weakly to u
∗
′, and (
u
n
) converges uniformly to u
=
J
∗
(
u
∗
).
Indeed, according to the definition ofu
n
∗ and (33), the sequence (
u
n
∗
)
n
≥
2 is equicontinuous. Moreover, let t
∈
I. Then, there are n
≥
2,  i
=
0,1
,
2
,
…
,
2
n
-
1, such that t
∈
[
t
i
n
,
t
i
+
1
n
]
. Hence, in view of (30) and (H
1), u
n
∗
(
t
)
∈
K
. This shows that the set {
u
n
∗
(
t
)
:
n
≥
2
} is relatively compact in E
∗. Therefore, Theorem 4. Ch.1. in [16] implies that there is a Lipschitz function u
∗
:
I
→
E
∗, such that the sequence (
u
n
∗
)
n
≥
2 has a subsequence, still denoted by (
u
n
∗
)
n
≥
2, which converges uniformly to u
∗, and (
u
n
∗
)
n
≥
2
′ converges weakly to u
∗
′
. We extend the definition of u
∗ on [
-
r
,
T
] by putting u
∗
(
t
)
=
J
(
Ψ
(
t
)
),  t
∈
[
-
r
,
0
]. Thus, (
u
n
∗
)
n
≥
2 converges uniformly to u
∗ on [
-
r
,
T
]. Since J
∗ is uniformly continuous on the compact set K, then the sequence (
u
n
)
n
≥
2 is uniformly convergent to u
:
[
-
r
,
T
]
→
E with u
=
J
∗
(
u
∗
). This proves our claim in this step.
Step 5. Let us show that u
(
t
)
∈
C
(
t
),  t
∈
I.
Lett
∈
I. According to (H
2), we have(47)
d
C
t
u
t
q
/
q
/
=
d
C
t
u
t
q
/
q
/
-
d
C
θ
n
t
u
n
θ
n
t
q
/
q
/
≤
λ
1
θ
n
t
-
t
+
λ
2
u
t
-
u
n
θ
n
t
.Since l
i
m
n
→
∞
λ
1
θ
n
t
-
t
+
λ
2
u
(
t
)
-
u
n
(
θ
n
(
t
)
)
=
0, therefore, u
(
t
)
∈
C
(
t
), ∀
t
∈
J
.
Step 6. In this step, we show that l
i
m
n
→
∞
τ
(
δ
n
(
t
)
)
u
n
-
τ
(
t
)
u
C
E
(
[
-
r
,
0
]
)
=
0, for every t
∈
I.
Lett
∈
I
. We have(48)
τ
δ
n
t
u
n
-
τ
t
u
C
E
-
r
,
0
≤
τ
δ
n
t
u
n
-
τ
t
u
n
C
E
-
r
,
0
+
τ
t
u
n
-
τ
t
u
C
E
-
r
,
0
≤
sup
-
r
≤
s
≤
0
⁡
u
n
δ
n
t
+
s
-
u
n
t
+
s
+
τ
t
u
n
-
τ
t
u
C
E
-
r
,
0
≤
sup
-
r
≤
s
1
≤
s
2
≤
1
s
1
-
s
2
≤
1
/
n
⁡
u
n
s
1
-
u
n
s
2
+
τ
t
u
n
-
τ
t
u
C
E
-
r
,
0
≤
sup
-
r
≤
s
1
≤
s
2
≤
0
s
1
-
s
2
≤
1
/
n
⁡
u
n
s
1
-
u
n
s
2
+
sup
-
r
≤
s
1
≤
0
≤
s
2
≤
1
s
1
-
s
2
≤
1
/
n
⁡
u
n
s
1
-
u
n
s
2
+
sup
0
≤
s
1
≤
s
2
≤
1
s
1
-
s
2
≤
1
/
n
⁡
u
n
s
1
-
u
n
s
2
+
τ
t
u
n
-
τ
t
u
C
E
-
r
,
0
≤
sup
-
r
≤
s
1
≤
s
2
≤
0
s
1
-
s
2
≤
1
/
n
⁡
Ψ
s
1
-
Ψ
s
2
+
sup
-
r
≤
s
1
≤
0
s
1
≤
1
/
n
⁡
u
n
s
1
-
u
n
0
+
sup
0
≤
s
2
≤
1
s
2
≤
1
/
n
⁡
u
n
0
-
u
n
s
2
+
sup
0
≤
s
1
≤
s
2
≤
1
s
1
-
s
2
≤
1
/
n
⁡
u
n
s
1
-
u
n
s
2
+
τ
t
u
n
-
τ
t
u
C
E
-
r
,
0
≤
2
sup
-
r
≤
s
1
≤
s
2
≤
0
s
1
-
s
2
≤
1
/
n
⁡
Ψ
s
1
-
Ψ
s
2
+
2
sup
0
≤
s
1
≤
s
2
≤
1
s
1
-
s
2
≤
1
/
n
⁡
u
n
s
1
-
u
n
s
2
+
τ
t
u
n
-
τ
t
u
.
By the continuity ofΨ, the uniform convergence of u
n towards u, and the preceding estimate, we get l
i
m
n
→
∞
τ
(
δ
n
(
t
)
)
u
n
-
τ
(
t
)
u
=
0.
Step 7. In this step, we show that the sequence (
f
n
)
n
≥
2, defined by f
n
(
t
)
=
g
n
(
δ
n
(
t
)
), converges almost everywhere to a function f
∈
L
1
(
I
,
E
∗
) and f
(
t
)
∈
F
(
t
,
τ
(
t
)
u
), a.e. t
∈
I.
To prove this, letn
≥
2 and let t
∈
I be a fixed point such that (32)(iv) and (v) are satisfied. In view of (H
4), we have (49)
f
n
+
1
t
-
f
n
t
=
g
n
+
1
δ
n
+
1
t
-
g
n
δ
n
t
=
d
g
n
δ
n
t
,
F
δ
n
+
1
t
,
τ
δ
n
+
1
δ
n
+
1
t
u
n
+
1
+
1
n
+
1
2
=
d
g
n
δ
n
t
,
F
δ
n
+
1
t
,
τ
δ
n
+
1
t
u
n
+
1
+
1
n
+
1
2
=
h
F
δ
n
t
,
τ
δ
n
δ
n
t
u
n
,
F
δ
n
+
1
t
,
τ
δ
n
+
1
δ
n
+
1
t
u
n
+
1
+
1
n
+
1
2
=
h
F
δ
n
t
,
τ
δ
n
t
u
n
,
F
δ
n
+
1
t
,
τ
δ
n
+
1
t
u
n
+
1
+
1
n
+
1
2
≤
m
δ
n
+
1
t
-
m
δ
n
t
+
τ
δ
n
+
1
t
u
n
+
1
-
τ
δ
n
t
u
n
+
1
n
+
1
2
≤
m
δ
n
+
1
t
-
m
δ
n
t
+
τ
δ
n
+
1
t
u
n
+
1
-
τ
δ
n
t
u
n
+
1
n
+
1
2
≤
m
δ
n
+
1
t
-
m
δ
n
t
+
τ
δ
n
+
1
t
u
n
+
1
-
τ
δ
n
t
u
n
+
1
n
+
1
2
≤
m
δ
n
+
1
t
-
m
δ
n
t
+
τ
δ
n
+
1
t
u
n
+
1
-
τ
δ
n
t
u
n
+
1
n
+
1
2
.Thus, for any two natural numbers n and q (2
<
n
<
q), we infer that (50)
f
n
t
-
f
q
t
=
f
n
t
-
f
n
+
1
t
+
f
n
+
1
t
-
f
n
+
2
t
+
⋯
+
f
q
-
1
t
-
f
q
t
≤
m
δ
n
+
1
t
-
m
δ
n
t
+
m
δ
n
+
2
t
-
m
δ
n
+
1
t
+
⋯
+
m
δ
q
-
1
t
-
m
δ
q
t
+
τ
δ
n
+
1
t
u
n
+
1
-
τ
δ
n
t
u
n
+
τ
δ
n
+
2
t
u
n
+
2
-
τ
δ
n
+
1
t
u
n
+
1
+
⋯
+
τ
δ
q
t
u
q
-
τ
δ
q
-
1
t
u
q
-
1
+
1
n
+
1
2
+
1
n
+
2
2
+
⋯
+
1
r
2
≤
m
δ
n
+
1
t
-
m
δ
n
t
+
m
δ
n
+
2
t
-
m
δ
n
+
1
t
+
⋯
+
m
δ
q
-
1
t
-
m
δ
q
t
+
τ
δ
n
+
1
t
u
n
+
1
-
τ
δ
n
t
u
n
+
τ
δ
n
+
2
t
u
n
+
2
-
τ
δ
n
+
1
t
u
n
+
1
+
⋯
+
τ
δ
q
t
u
q
-
τ
δ
q
-
1
t
u
q
-
1
+
r
-
n
n
+
1
2
.Since l
i
m
n
→
∞
δ
n
(
t
)
=
t, m is continuous, and l
i
m
n
→
∞
τ
(
δ
n
(
t
)
)
u
n
-
τ
(
t
)
u
C
E
(
[
-
r
,
0
]
)
=
0, then the right-hand side of the last inequality tends to zero when n
,
q
→
∞
. Hence, the sequence (
f
n
(
t
)
)
n
≥
2 is a Cauchy sequence in E
∗
. Thus, there is a function f
:
I
→
E
∗such that(51)
lim
n
→
∞
⁡
f
n
t
=
f
t
,
a.e.
t
∈
I
.Observe that, by (34), f
n
(
t
)
≤
ρ, ∀
t
∈
I
. So, f
∈
L
1
(
I
,
E
∗
). It remains to show that f
(
t
)
∈
F
(
t
,
τ
(
t
)
u
), a.e. t
∈
I. Indeed, for every n
≥
2 and almost everywhere t
∈
I, we have (52)
d
f
t
,
F
t
,
τ
t
u
≤
f
t
-
f
n
t
+
d
f
n
t
,
F
t
,
τ
t
u
≤
f
t
-
f
n
t
+
d
g
n
δ
n
t
,
F
t
,
τ
t
u
≤
f
t
-
f
n
t
+
h
F
δ
n
t
,
τ
δ
n
t
u
n
,
F
t
,
τ
t
u
.This relation and (H
4) imply (53)
d
f
t
,
F
t
,
τ
t
u
≤
f
t
-
f
n
t
+
m
δ
n
t
-
m
t
+
τ
δ
n
t
u
n
-
τ
t
u
.
Becausef
n converges to f almost everywhere, m is continuous, and l
i
m
n
→
∞
τ
(
δ
n
(
t
)
)
u
n
-
τ
(
t
)
u
C
E
(
[
-
r
,
0
]
)
=
0, for every t
∈
I, we conclude that f
(
t
)
∈
F
(
t
,
τ
(
t
)
u
), a.e. t
∈
I.
Step 8. We proceed to show that (54)
-
J
u
′
t
+
f
t
∈
N
C
t
u
t
,
a.e.
t
∈
I
.We follow the arguments used in [10]. In view of (32)(vi), (37), and Lemma 11, for every natural number n
≥
2 one obtains(55)
-
J
u
n
′
t
+
f
n
t
∈
λ
∂
d
C
θ
n
t
u
n
θ
n
t
,
a.e.
,where λ
=
η
+
ρ. (
H
2
). Observe that the uppersemicontinuity of the subdifferential of Lipschtz function ([17], Prop. 1.7) imply the multivalued function (
t
,
y
)
→
∂
d
C
(
t
)
(
y
) is scalary uppersemicontinuous with convex and w
e
a
k
∗compact values. That is, let t
n
→
t, in I, y
n
→
y, in E, y
n
∈
C
(
t
n
), and y
∈
C
(
t
)
. Then, for any v
∈
E, (56)
limsup
n
→
∞
⁡
σ
v
,
∂
d
C
t
n
y
n
≤
σ
v
,
∂
d
C
t
y
.
So, for anyv
∈
E,(57)
limsup
n
→
∞
⁡
σ
v
,
∂
d
C
θ
n
t
u
n
θ
n
t
≤
σ
v
,
∂
d
C
t
u
t
,where σ is the support function.
Now, let(
e
k
) be a sequence in E which separates the points. Hence, from the weak convergence of the sequence (
(
J
u
n
)
′
+
f
n
)
n
≥
2 to J
(
u
′
)
+
f in L
1
(
I
,
E
∗
), for any Lebesgue measurable subset A
⊆
J, we have (58)
∫
A
e
k
,
-
J
u
′
t
+
f
t
d
t
=
lim
n
→
∞
⁡
∫
A
e
k
,
-
J
u
n
′
t
+
f
n
t
d
t
.
This relation with (55) and (57) yields (59)
∫
A
e
k
,
-
J
u
′
t
+
f
t
d
t
≤
limsup
n
→
∞
⁡
∫
A
σ
e
k
,
λ
∂
d
C
θ
n
t
u
n
θ
n
t
d
t
≤
∫
A
limsup
n
→
∞
⁡
σ
e
k
,
λ
∂
d
C
θ
n
t
u
n
θ
n
t
d
t
≤
∫
A
σ
e
k
,
λ
∂
d
C
t
u
t
d
t
.
Therefore,(60)
e
k
,
-
J
u
′
t
+
f
t
≤
σ
e
k
,
λ
∂
d
C
t
u
t
,
∀
k
≥
1
.
Again, since the multivalued functiont
→
∂
d
C
(
u
(
t
)
) is measurable with convex and w
e
a
k
∗compact,  then by ([18], III.35) it follows that (61)
-
J
u
′
t
+
f
t
∈
λ
∂
d
C
t
u
t
,
a.e.
t
∈
I
.As u
(
t
)
∈
C
(
t
), ∀
t
∈
I, we get (62)
-
J
u
′
t
+
f
t
∈
N
C
t
u
′
t
,
a.e.This completes the proof.


## 4. Existence Result for (3)

In the following theorem, we present an existence theorem of solutions for (3). We do not assume that the values of Γ are contained in a convex compact fixed subset.Theorem 16.
Letp
≥
2, let q
∈
(
1,2
], let E be a separable, p-uniformly convex and q-uniformly smooth Banach space, let T
>
0, let Γ
:
I
×
E
∗
→
P
c
c
(
E
), and let F
:
I
×
C
r
→
P
c
c
(
E
∗
). Suppose that (
H
3
)
,
(
H
4
) and the following conditions hold:(
H
5
)
There is a positive constantR such that Γ
(
t
,
x
)
⊆
R
B
(
0,1
), ∀
(
t
,
x
)
∈
I
×
E
∗.
(
H
6
)
There are three positive constantsγ
1
,
γ
2
,
γ
3 with 0
<
γ
2
<
(
a
/
b
)
1
/
q
/ such that for any t
1
,
t
2
∈
I, any x
1
,
x
2
∈
E, and any y
1
,
y
2
∈
E
∗we have (63)
d
Γ
t
1
,
y
1
x
1
q
/
q
′
-
d
Γ
t
2
,
y
2
x
2
q
/
q
′
≤
γ
1
t
1
-
t
2
+
γ
2
y
1
-
y
2
+
γ
3
x
1
-
x
2
,
whereq
/
=
q
/
(
1
-
q
) and the coefficients a
,
b are given in Lemma 12.
(
H
7
)
For anyt
∈
I and any bounded subset A in E
∗ with χ
E
∗
(
A
)
>
0 and any r
>
0, one has (64)
χ
Γ
t
,
A
∩
r
B
0,1
<
χ
E
A
∗
,
whereA
∗
=
{
J
∗
(
x
)
:
x
∈
A
}
.Then, for any fixedΨ
∈
C
r satisfying Ψ
0
∈
Γ
(
0
,
J
(
Ψ
0
)
), there is a continuous function u
:
[
-
r
,
T
]
→
E such that u is absolutely continuous on [
0
,
T
] and (65)
u
t
=
Ψ
t
,
for
t
∈
-
r
,
0
;
u
t
=
Ψ
0
+
∫
0
t
u
′
s
d
s
,
for
t
∈
I
,
u
t
∈
Γ
t
,
J
u
t
,
for
t
∈
I
,
-
J
u
′
t
∈
N
Γ
t
,
J
u
t
u
t
+
F
t
,
τ
t
u
,
a.e.
for
t
∈
I
.Proof.
We divide the proof into the following steps.
Step 1. For any fixed natural number n
≥
2, let t
i
n
,
I
0
n
,
I
i
n,  i
=
0,1
,
2
,
…
,
2
n
-
1, μ
n,θ
n, and δ
n be as in the proof of Theorem 15.
By following the same lines in Steps 1 and 2 in the proof of Theorem15, with the following modifications: x
0
n
=
Ψ
(
0
),(66)
x
i
+
1
n
=
Π
Γ
t
i
+
1
n
,
J
x
i
n
J
x
i
n
+
μ
n
g
i
n
t
i
n
,
i
=
1,2
,
…
,
2
n
-
1
;
(67)
g
i
n
∈
L
1
t
i
n
,
t
i
+
1
n
,
E
∗
;
(68)
g
i
n
t
∈
F
t
,
τ
δ
n
t
u
n
,
t
∈
t
i
n
,
t
i
+
1
n
;
(69)
u
n
∗
t
=
t
i
+
1
n
-
t
μ
n
J
x
i
n
+
t
-
t
i
n
μ
n
J
x
i
+
1
n
,
t
∈
t
i
n
,
t
i
+
1
n
;
(70)
u
n
t
=
J
∗
u
n
∗
t
,
t
∈
-
r
,
T
,we can show that there are u
n
∈
C
(
[
-
r
,
T
]
,
E
), u
n
∗
∈
C
(
[
-
r
,
T
]
,
E
∗
), and g
n
∈
L
1
(
I
,
E
∗
) such that u
n and u
n
∗ are absolutely continuous on I and the following properties hold:(71)
(i)
u
n
t
=
Ψ
t
,
t
∈
-
r
,
0
;
ii
u
n
∗
t
=
J
u
n
t
,
t
∈
-
r
,
T
;
(iii)
u
n
θ
n
t
∈
Γ
θ
n
t
,
u
n
∗
δ
n
t
,
t
∈
I
;
(iv)
g
n
t
∈
F
t
,
τ
δ
n
t
u
n
,
t
∈
I
;
(v)
g
n
+
1
t
-
g
n
t
≤
d
g
n
t
,
F
t
,
τ
δ
n
+
1
t
u
n
+
1
+
1
n
+
1
2
,
a.e.
t
∈
I
;
(vi)
-
u
n
∗
′
t
+
g
n
t
∈
N
Γ
θ
n
t
,
u
n
∗
δ
n
t
u
n
θ
n
t
,
a.e.
t
∈
I
.
Step 2. In this step, we show that there is a positive number η such that, for any n
≥
2,(72)
u
n
∗
′
t
≤
η
,
a.e.  for
t
∈
I
.
Letn
≥
2 be a fixed natural number. According to (H
5) and (H
6), for any t
∈
I, we have(73)
g
n
t
≤
P
t
1
+
τ
δ
n
t
u
n
0
=
P
t
1
+
u
n
δ
n
t
=
P
t
1
+
R
≤
max
t
∈
I
⁡
P
t
1
+
R
=
ζ
.
Then, fori
∈
{
i
=
0,1
,
2
,
…
,
2
n
-
1
},(74)
J
x
i
+
1
n
≤
R
,
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
R
+
T
ζ
,
Note thatE
∗ is q
/-uniformly convex and p
/-uniformly smooth. Therefore, in view of (73), (74), and Lemma 12, there exists a positive constant a such that (75)
a
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
≤
V
∗
J
∗
J
x
i
+
1
n
,
J
x
i
n
+
μ
n
g
i
n
t
i
n
.
In addition, sinceE is p-uniformly convex and q-uniformly smooth, then, again, by Lemma 12, there is a positive constant b such that (76)
d
Γ
t
i
+
1
n
,
J
x
i
n
V
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
d
Γ
t
i
+
1
n
,
J
x
i
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
.
Then,(77)
a
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
≤
V
∗
J
∗
J
x
i
+
1
n
,
J
x
i
n
+
μ
n
g
i
n
t
i
n
=
V
J
x
i
n
+
μ
n
g
i
n
t
i
n
,
x
i
+
1
n
=
d
Γ
t
i
+
1
n
,
J
x
i
n
V
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
d
Γ
t
i
+
1
n
,
J
x
i
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
.This inequality with (66), (73), and (
H
6
), give us(78)
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
a
1
/
q
/
d
Γ
t
i
+
1
n
,
J
x
i
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
q
/
=
b
a
1
/
q
/
d
Γ
t
i
+
1
n
,
J
x
i
n
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
q
/
q
/
-
d
Γ
t
i
n
,
J
x
i
-
1
n
J
∗
J
x
i
n
q
/
q
/
≤
b
a
1
/
q
/
γ
1
μ
n
+
γ
2
J
x
i
n
-
J
x
i
-
1
n
+
γ
3
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
-
J
∗
J
x
i
n
.
Moreover, from the assumptionq
∈
1,2, we infer that q
/
≥
2
. Then, E
∗ is q
/-uniformly convex (
q
/
≥
2
) and p
/-uniformly smooth. Hence, from Lemma 13, there is a constant β, depending on (
2
R
+
T
ζ
), such that, for all z
1, z
1
∈
E
∗, z
1
≤
2
R
+
T
ζ, and z
2
≤
2
R
+
T
ζ, we have(79)
J
∗
z
1
-
J
∗
z
2
≤
β
z
1
-
z
2
.
Consequently,(80)
J
∗
J
x
i
n
+
μ
n
g
i
n
t
i
n
-
J
∗
J
x
i
n
≤
β
J
x
i
n
+
μ
n
g
i
n
t
i
n
-
J
x
i
n
=
β
μ
n
g
i
n
t
i
n
=
β
μ
n
ζ
.
This inequality and (78) yield (81)
J
x
i
+
1
n
-
J
x
i
n
+
μ
n
g
i
n
t
i
n
≤
b
a
1
/
q
/
γ
1
μ
n
+
γ
2
J
x
i
n
-
J
x
i
-
1
n
+
β
γ
3
μ
n
ζ
.
Thus,(82)
J
x
i
+
1
n
-
J
x
i
n
≤
b
a
1
/
q
/
γ
1
μ
n
+
γ
2
J
x
i
n
-
J
x
i
-
1
n
+
β
γ
3
μ
n
+
μ
n
ζ
=
μ
n
γ
1
¯
+
γ
3
¯
+
γ
2
¯
J
x
i
n
-
J
x
i
-
1
n
,where γ
1
¯
=
(
b
/
a
)
1
/
q
/
γ
1, γ
2
¯
=
(
b
/
a
)
1
/
q
/
γ
2, and γ
3
¯
=
ζ
+
(
b
/
a
)
1
/
q
/
β
ζ
γ
3. By repeating the same procedure as (82), (
i
-
1
)-times, we infer that(83)
J
x
i
+
1
n
-
J
x
i
n
≤
μ
n
γ
1
¯
+
γ
3
¯
1
+
γ
2
¯
+
γ
2
¯
2
+
γ
2
¯
3
+
⋯
+
γ
2
¯
i
-
1
+
γ
2
¯
i
J
x
1
n
-
J
x
0
n
.
Note thatΨ
0
∈
Γ
(
0
,
J
(
Ψ
0
)
) and x
0
n
=
Ψ
(
0
)
. Then, x
0
n
∈
Γ
(
0
,
J
(
x
0
n
)
). Hence, by arguing as in (78) and using (79), one obtains (84)
J
x
1
n
-
J
x
0
n
+
μ
n
g
0
n
t
0
n
q
/
≤
b
a
1
/
q
/
d
Γ
t
1
n
,
J
x
0
n
J
∗
J
x
0
n
+
μ
n
g
0
n
t
0
n
q
/
q
/
-
d
Γ
t
0
n
,
J
x
0
n
J
∗
J
x
0
n
q
/
q
/
=
b
a
1
/
q
/
γ
1
μ
n
+
γ
3
J
∗
J
x
0
n
+
μ
n
g
0
n
t
0
n
-
J
∗
J
x
0
n
≤
b
a
1
/
q
/
γ
1
μ
n
+
β
γ
3
μ
n
ζ
.Thus,(85)
J
x
1
n
-
J
x
0
n
≤
b
a
1
/
q
/
γ
1
μ
n
+
β
γ
3
μ
n
ζ
+
μ
n
ζ
=
μ
n
γ
1
¯
+
γ
3
¯
.Relations (83) and (85) and the fact that 0
<
γ
2
<
(
a
/
b
)
1
/
q
′yield (86)
J
x
i
+
1
n
-
J
x
i
n
≤
μ
n
γ
1
¯
+
γ
3
¯
1
+
γ
2
¯
+
γ
2
¯
2
+
γ
2
¯
3
+
⋯
+
γ
2
¯
i
-
1
+
γ
2
¯
i
≤
μ
n
γ
1
¯
+
γ
3
¯
1
1
-
γ
2
¯
.
Therefore, forn
≥
2, for i
=
0,1
,
2
,
…
,
2
n
-
1, and for t
∈
(
t
i
n
,
t
i
+
1
n
), we get (87)
u
n
∗
′
t
≤
γ
1
¯
+
γ
3
¯
1
1
-
γ
2
¯
=
η
.
This shows that (72) is true.
Step 3. We show that there is a Lipschitz function u
∗
:
I
→
E
∗, such that the sequence (
u
n
∗
)
n
≥
2 has a convergent uniform subsequence, still denoted by (
u
n
∗
)
n
≥
2, to u
∗ and (
u
n
∗
)
n
≥
2
′ converges weakly to u
∗
′
.
Indeed, by (72), for any s
1, s
2
∈
I
(
s
1
<
s
2
), we have(88)
u
n
∗
s
1
-
u
n
∗
s
2
≤
η
s
1
-
s
2
,
∀
n
≥
2
.
Then, the sequence(
u
n
∗
)
n
≥
2 is equicontinuous on I. We want to show that, for any t
∈
I, the subset {
u
n
∗
(
t
)
:
n
≥
2
} is relatively compact in E
∗.
As above, by Lemma12, there is a constant δ, depending on (
Ψ
(
0
)
+
T
η
), such that(89)
J
∗
u
n
∗
s
1
-
J
∗
u
n
∗
s
2
≤
δ
u
n
∗
s
1
-
u
n
∗
s
2
≤
δ
η
s
1
-
s
2
,
∀
n
≥
2
.So, for n
≥
2 and t
∈
I,(90)
u
n
θ
n
t
-
u
n
t
=
J
∗
u
n
∗
θ
n
t
-
J
∗
u
n
∗
t
≤
δ
η
θ
n
t
-
t
=
δ
η
μ
n
.But, by (66), (91)
u
n
θ
n
t
∈
Γ
θ
n
t
,
u
n
∗
δ
n
t
.Then, (90) yields(92)
u
n
t
∈
Γ
θ
n
t
,
u
n
∗
δ
n
t
+
u
n
θ
n
t
-
u
n
t
B
0,1
⊆
Γ
θ
n
t
,
u
n
∗
δ
n
t
+
δ
η
μ
n
B
0,1
,
∀
t
∈
I
.On the other hand, in view of (88) and (
H
7
), for any t
∈
I and any z
∈
Γ
(
θ
n
(
t
)
,
u
n
∗
(
δ
n
(
t
)
)
), we have (93)
d
Γ
t
,
u
n
∗
t
z
q
/
q
/
=
d
Γ
θ
n
t
,
u
n
∗
δ
n
t
z
q
/
q
/
-
d
Γ
t
,
u
n
∗
t
z
q
/
q
/
≤
γ
1
θ
n
t
-
t
+
γ
2
u
n
∗
δ
n
t
-
u
n
t
≤
γ
1
μ
n
+
γ
2
η
δ
n
t
-
t
≤
γ
1
μ
n
+
γ
2
η
μ
n
.This means that(94)
Γ
θ
n
t
,
u
n
∗
δ
n
t
⊆
Γ
t
,
u
n
∗
t
+
γ
1
μ
n
+
γ
2
η
μ
n
B
0,1
,
∀
t
∈
I
.In view of (92) and (94), one obtains(95)
u
n
t
⊆
Γ
t
,
u
n
∗
t
+
δ
η
μ
n
+
γ
1
μ
n
+
γ
2
η
μ
n
B
0,1
=
Γ
t
,
u
n
∗
t
+
μ
n
¯
B
0,1
,
∀
t
∈
I
,where μ
n
¯
=
μ
n
(
δ
η
μ
n
+
γ
1
μ
n
+
γ
2
η
).
Now, assume by contradiction that there ist
0
∈
I such that B
∗
(
t
0
)
=
{
u
n
∗
(
t
0
)
:
n
≥
2
} is not relatively compact in E
∗. Since J
∗ is continuous on bounded sets, the set B
(
t
0
)
=
{
u
n
(
t
0
)
:
n
≥
2
} is not relatively compact in E. Then, χ
(
B
(
t
0
)
)
>
0. Observe that, by (90), for n
≥
2,(96)
u
n
t
0
=
u
n
t
0
-
u
n
0
+
ψ
0
≤
δ
η
T
+
ψ
0
=
η
¯
.Therefore, in view of (
H
8
), (97)
χ
E
B
t
0
-
χ
E
Γ
t
0
,
B
∗
t
0
∩
η
¯
B
0,1
>
0
.Then, we can find ξ
>
0 such that(98)
χ
E
B
t
0
-
χ
E
Γ
t
0
,
B
∗
t
0
∩
η
¯
B
0,1
>
2
ξ
.Let n
0 be a natural number such that n
0
≥
2 and μ
n
¯
<
ξ
/
2, ∀
n
≥
n
0. So, by using (95) and (96),(99)
u
n
t
0
⊆
Γ
t
,
u
n
0
∗
t
∩
η
¯
B
+
ξ
2
B
0,1
.Then, from the properties of χ, (99), and (
H
8
), we infer that (100)
χ
B
t
0
=
χ
u
n
t
0
:
n
≥
n
0
≤
χ
Γ
t
0
,
B
∗
t
0
∩
η
¯
B
0,1
+
ξ
<
χ
B
t
0
-
2
ξ
+
ξ
=
χ
B
t
0
-
ξ
,which is a contradiction.
Therefore, Theorem 4. Ch.1. in [16] implies that there is a Lipschitz function u
∗
:
I
→
E
∗, such that the sequence (
u
n
∗
)
n
≥
2 has a convergent uniform subsequence, still denoted by (
u
n
∗
)
n
≥
2, to u
∗ and (
u
n
∗
)
n
≥
2
′ converges weakly to u
∗
′
. We extend the definition of u
∗ on [
-
r
,
T
] by putting u
∗
(
t
)
=
J
(
Ψ
(
t
)
), t
∈
[
-
r
,
0
]. Thus, (
u
n
∗
)
n
≥
2 converges uniformly to u
∗ on [
-
r
,
T
]. Since J
∗ is uniformly continuous on bounded sets, then the sequence (
u
n
)
n
≥
2
(
u
n
=
J
∗
u
n
∗
) is uniformly convergent to u
:
[
-
r
,
T
]
→
E with u
=
J
∗
(
u
∗
).
Step 4. Let us show that u
(
t
)
∈
Γ
(
t
,
u
∗
(
t
)
),  t
∈
I.
Lett
∈
I. According to (H
7), we have (101)
d
Γ
t
,
u
∗
t
u
t
q
/
q
/
=
d
Γ
t
,
u
∗
t
u
t
q
/
q
/
-
d
Γ
θ
n
t
,
u
∗
δ
n
t
u
n
θ
n
t
q
/
q
/
≤
γ
1
θ
n
t
-
t
+
γ
2
u
∗
t
-
u
n
∗
δ
n
t
+
γ
3
u
t
-
u
n
θ
n
t
.
Note thatl
i
m
n
→
∞
γ
1
|
θ
n
(
t
)
-
t
|
+
γ
2
u
∗
(
t
)
-
u
n
∗
(
δ
n
(
t
)
)
+
γ
3
u
(
t
)
-
u
n
(
θ
n
(
t
)
)
=
0
. Then, u
(
t
)
∈
Γ
(
t
,
u
∗
(
t
)
),  t
∈
I.
Step 5. Following the same lines in Steps 5 and 6 in the proof of Theorem 15, we can show that l
i
m
n
→
∞
τ
(
δ
n
(
t
)
)
u
n
-
τ
(
t
)
u
C
E
(
[
-
r
,
0
]
)
=
0, for every t
∈
I. Moreover, the sequence (
f
n
)
n
≥
2 defined by f
n
(
t
)
=
g
n
(
δ
n
(
t
)
), t
∈
I, converges almost everywhere to a function f
∈
L
1
(
1
,
E
∗
) and f
(
t
)
∈
F
(
t
,
τ
(
t
)
u
), a.e. t
∈
I.
Step 6. We proceed to show that (102)
-
J
u
′
t
+
f
t
∈
N
Γ
t
,
u
∗
t
u
t
,
a.e.
t
∈
I
.In view of (72) and (73) and Lemma 11, for every natural number n
≥
2, one obtains(103)
-
J
u
n
′
t
+
f
n
t
∈
η
+
ς
∂
d
Γ
θ
n
t
,
u
∗
δ
n
t
u
n
θ
n
t
,
a.e.
t
∈
I
.
By(
H
2
) and ([19], Prop. 1.7), the multivalued function (
t
,
y
)
→
∂
d
Γ
(
t
)
(
y
) is scalary uppersemicontinuous with convex and w
e
a
k
∗compact values. Then, for any v
∈
E,(104)
limsup
n
→
∞
⁡
σ
v
,
∂
d
Γ
θ
n
t
,
u
∗
δ
n
t
u
n
θ
n
t
≤
σ
v
,
∂
d
Γ
t
,
u
∗
t
u
t
,where σ is the support function.
Now, let(
e
k
) be a sequence in E which separates the points. Hence, from the weak convergence of the sequence (
(
J
u
n
)
′
+
f
n
)
n
≥
2 to J
(
u
′
)
+
f in L
1
(
I
,
E
∗
), for any Lebesgue measurable subset A
⊆
I, we have (105)
∫
A
e
k
,
-
J
u
′
t
+
f
t
d
t
=
lim
n
→
∞
⁡
∫
A
e
k
,
-
J
u
n
′
t
+
f
n
t
d
t
.
This relation with (103) and (104) yields (106)
∫
A
e
k
,
-
J
u
′
t
+
f
t
d
t
≤
limsup
n
→
∞
⁡
∫
A
σ
e
k
,
η
+
ς
∂
d
Γ
θ
n
t
,
u
∗
δ
n
t
u
n
θ
n
t
d
t
≤
∫
A
limsup
n
→
∞
⁡
σ
e
k
,
η
+
ς
∂
d
Γ
θ
n
t
,
u
∗
δ
n
t
u
n
θ
n
t
d
t
≤
∫
A
σ
v
,
∂
d
Γ
t
,
u
∗
t
u
t
d
t
.
So, for almostt
∈
I, (107)
e
k
,
-
J
u
′
t
+
f
t
≤
σ
v
,
∂
d
Γ
t
,
u
∗
t
u
t
,
∀
k
≥
1
.
Again, since the multivalued functiont
→
∂
d
Γ
(
t
)
(
u
(
t
)
) is measurable with convex and w
e
a
k
∗compact, then by ([18], III.35) it follows that (108)
-
J
u
′
t
+
f
t
∈
η
+
ς
∂
d
Γ
t
,
u
∗
t
u
t
,
a.e.As u
(
t
)
∈
Γ
(
t
,
u
∗
(
t
)
)
,
∀
t
∈
I, we get (109)
-
J
u
′
t
+
f
t
∈
N
Γ
t
,
u
∗
t
u
t
,
a.e.This completes the proof.



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*Source: 3853205-2016-02-23.xml*