Available at: http://publications.ictp.it IC/2009/070

United Nations Educational, Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FIRST AND SECOND ORDER CONVEX SWEEPING PROCESSES

IN REFLEXIVE SMOOTH BANACH SPACES

M. Bounkhel1

King Saud University, College of Science, Department of Mathematics,P.O. Box 2455, Riyadh 11451, Riyadh, Saudi-Arabia

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

R. Al-Yusof2

King Saud University, College of Science, Department of Mathematics,P.O. Box 2455, Riyadh 11451, Riyadh, Saudi-Arabia.

Abstract

In this paper we establish new characterizations of the normal cone of closed convex sets in

reflexive smooth Banach spaces and then we use those results to prove the existence of solutions

for first order convex sweeping processes and their variants in reflexive smooth Banach spaces.

The case of second order convex sweeping processes is also studied.

MIRAMARE – TRIESTE

December 2009

1Corresponding author: bounkhel@ksu.edu.sa2ralyusof@hotmail.com

1. Introduction

In [15], Moreau introduced and studied the following differential inclusion

(1.1) −x′(t) ∈ N(C(t);x(t)) a. e. on [0, T ], x(0) = x0 ∈ C(0),

where C : [0, T ]H is a set-valued mapping defined from I = [0, T ] to a Hilbert space H

and takes closed convex values. N(C(t);x(t)) denotes the convex normal cone, in the sense of

convex analysis, to the set C(t) at x(t). The differential inclusion (1.1) is known as the sweeping

process problem. This evolution differential inclusion corresponds to several important mechanical

problems. For more details concerning the applications of (1.1), we refer the reader to [15]. Since

the work [15], several extensions of (1.1) in different ways have been given. We state some of

them. In [7] the author introduced some new techniques from which many results can be derived,

essentially the existence of a solution of (1.1) for C(t) = S + ν(t), where S is a fixed non convex

closed set and ν is a mapping with finite variation. Recently, in [5], the authors proved the

existence of solutions of the perturbed sweeping process

(1.2) −x′(t) ∈ N(C(t);x(t)) + F (t, x(t)) a. e. on [0, T ], x(0) = x0 ∈ C(0),

where C has not necessarily convex values and F is an u.s.c. set-valued mapping with convex

compact values. The class of differential inclusions (1.2) appears in particular in mathematical

economies. Some new variants of (1.1) has been studied recently in [4, 14] in the convex and non

convex settings, we are interested here by the following:

(1.3) −x(t) ∈ N(C(t);x′(t)) a. e. on [0, T ].

(For the motivations of (1.3) and its applications we refer to [14]). Many other extension of (1.1)have been studied and we cannot list them here. Accordingly to the best of our knowledge, theextension of (1.1) from the Hilbert space setting to the Banach space setting does not appear inany previous work till now even in the convex case. In the present paper we try to extend (1.2)and (1.3) to the reflexive smooth Banach space setting when C is a convex set-valued mapping.A second order version of (1.2) has been the subject of various papers (see for instance [19, 20, 21]).The last and recent extension of (SDI), (accordingly to our best knowledge), has been given in[19], in which the author proved the existence of solution for the following general extension of(SDI)

x′′(t) ∈ −N(K(x(t));x′(t)) + F (t, x(t), x′(t)) +G(t, x(t), x′(t)) a.e. t ∈ I, (SSPMP )

where K is a non convex set-valued mapping with compact values, H is Hilbert separable, F is

scalarly u.s.c. convex set-valued mapping and G is a non convex continuous set-valued mapping.

(It is called Second Order Sweeping Process with Mixed Perturbations in short (SSPMP )). He

also studied some topological properties of (SSPMP ). We have to mention that (SSPMP ) is

the more general form of second order sweeping process with perturbation studied until now, in

the Hilbert space setting. Our objective in the second part of the present paper is to extend

(SSPMP ) to the Banach space setting (when K has convex values).

The paper is organized as follows. Section 2 is devoted to some definitions and notations

needed in the paper. In Section 3, we establish new characterizations, in terms of projections,2

of the convex normal cone (Propositions 3.1 and 3.3) that will be the main tool in the proofs in

the next sections. In Section 4, we assume that X is p-uniformly convex and q-uniformly smooth

and we propose and study the existence of solutions for appropriate extensions of (1.2) and (1.3)

in this case. The last section is reserved to the second order convex sweeping processes.

2. Preliminaries

In the sequel, X is a Banach space with topological dual space X∗. We denote by dS the usual

distance function to S, i.e., dS(x) := infu∈S ‖x − u‖. We need first to recall some notation and

definitions needed in the paper. Let S be a nonempty closed convex set of X and x be a point in

S. The convex normal cone of S at x is defined by (see for instance [8])

(2.1) N(S; x) = ϕ ∈ X∗ : 〈ϕ, x− x〉 ≤ 0 for all x ∈ S.

Assume that S is not necessarily convex and let u ∈ X. If there exists a point x in S whose

distance to u is minimal, then x is called a projection of u onto S. The set of all such points, if

any, is denoted by PS(u). It is well known (see for example [9]) that the proximal normal cone

of S at x ∈ S can be defined, when X is Hilbert, in terms of the projection PS as follows

(2.2) NP (S; x) = ξ ∈ X∗ : ∃α > 0 so that x ∈ PS(x+ αξ),

and also that NP (S; x) = N(S; x) for closed convex sets.

Let f : X → R ∪ +∞ be a lower semicontinuous convex extended real valued function,

x ∈ dom f := x ∈ X : f(x) <∞. The convex subdifferential of f at x is given by

(2.3) ∂f(x) = ϕ ∈ X∗ : 〈ϕ, x− x〉 ≤ f(x) − f(x), for all x ∈ X.

An equivalent definition is given (see for instance [8]) by

(2.4) ∂f(x) = ϕ ∈ X∗ : (ϕ,−1) ∈ N(epi f ; (x, f(x))).

It is well known that the distance function dS is always Lipschitz continuous with ratio 1 and

it is convex if and only if S is a closed and convex set. The relation between normal cones and

subdifferentials is given in the following proposition. (For the proof, we can refer the reader, for

instance, to [8]).

Proposition 2.1. Let S be a nonempty, closed and convex subset of a Banach space X and x be

a point in S. Then

N(S; x) ∩ B∗ = ∂dS(x).

Here B∗ denotes the closed unit ball centered at the origin of X∗.

The following proposition states an important result that will be used in our proofs, which is

the closedness of the subdifferential of the distance function to images of set-valued mappings

whose images are closed and convex.3

Proposition 2.2. Let I be an interval of R, X be a Banach space and C : IX be a continuous

set-valued mapping with nonempty, closed and convex values. The following closedness property

of the subdifferential of the distance function holds: “ for any t ∈ I, x ∈ C(t), xn → x, tn → t

(xn not necessarily in C(tn)), with tn ∈ I, and ϕn →w ϕ, one has ϕ ∈ ∂dC(t)(x) ”. Here →w

means the weak convergence in X∗.

Proof. The proof follows directly from the definition.

A Banach space (X, ‖ · ‖) is said to be smooth provided the limit

limt→0

‖x+ ty‖ − ‖x‖

t

exists for each x, y ∈ X satisfying ‖x‖ = ‖y‖ = 1. In this case, the norm of X is said to be

Gateaux differentiable. The normalized duality mapping J : XX∗ is defined by

J(x) = j(x) ∈ X∗ : 〈j(x), x〉 = ‖j(x)‖‖x‖ = ‖x‖2 = ‖j(x)‖2.

Many properties of the normalized duality mapping J have been studied. For the details, one

may see Takahashi’s book [17] or Vainberg’s book [18]. We list some properties of J needed in

our proofs:

(J1) For any x ∈ X, J(x) is nonempty;

(J2) For any x ∈ X and any real number α, J(αx) = αJ(x);

(J3) If X is reflexive, then J is a mapping of X onto X∗ ;

(J4) If X∗ is strictly convex, then J is a single valued mapping;

(J5) J is a continuous operator in smooth Banach spaces;

(J6) If X is strictly convex, then J is one-to-one ;

(J7) If X∗ is uniformly convex, then J is uniformly continuous on bounded subsets of X ;

(J8) If X is reflexive strictly convex space with strictly convex conjugate space X∗ and if

J∗ : X∗X is a normalized duality mapping in X∗, then J−1 = J∗, JJ∗ = IX∗ , J∗J = IX ,

where IX∗ (resp. IX) is the identity mapping on X∗ (resp. X) ;

(J9) J is the identity operator in Hilbert spaces.

It is known (see [10, 11]) that a reflexive Banach space X is smooth if and only if X∗ is strictly

convex. Hence by (J3) and (J4), if X is reflexive smooth Banach space, then J is a single valued

mapping from X onto X∗. Furthermore, by (J8), if X is reflexive smooth strictly convex Banach

space, then J−1 = J∗, JJ∗ = IX∗ , and J∗J = IX .

Let V : X∗ ×X → R be defined by

V (ϕ, x) = ‖ϕ‖2 − 2〈ϕ, x〉 + ‖x‖2, for any ϕ ∈ X∗ and x ∈ X.

We list now some important properties of V needed in our proofs, when X is a reflexive smooth

Banach space.

i) V (ϕ, x) ≥ 0, ∀x ∈ X, ∀ϕ ∈ X∗;

ii) (‖ϕ‖ − ‖x‖)2 ≤ V (ϕ, x) ≤ (‖ϕ‖ + ‖x‖)2, ∀x ∈ X, ∀ϕ ∈ X∗;4

iii) V (J(x), x) = 0, ∀x ∈ X;

iv) (ϕ, x) 7→ V (ϕ, x) is continuous and V is convex with respect to x when ϕ is fixed and

convex with respect to ϕ when x is fixed;

v) V (ϕ, x) is differentiable with respect to x when ϕ is fixed;

vi) gradxV (ϕ, x) = 2(J(x) − ϕ).

vii) V (ϕ, x) = 0 if and only if ϕ = J(x). This property is true in uniformly convex and

uniformly smooth Banach spaces.

Based on the functional V , a set πS(ϕ) of generalized projections of ϕ ∈ X∗ onto S is defined as

follows (see [1, 2]).

Definition 2.1. Let S be a nonempty subset of X and ϕ ∈ X∗. If there exists a point x ∈ S

satisfying

V (ϕ, x) = infx∈S

V (ϕ, x),

then x is called a generalized projection of ϕ onto S. The set of all such points is denoted by

πS(ϕ).

The following example (given in Example 1.4. in [13]) shows that if the Banach space is not

reflexive, πS(ϕ) may be empty for some elements ϕ ∈ X∗ even when S is closed and convex.

Example 2.3. For any positive integer n, let en ∈ l1, such that its nth entry is n+1n

and all other

entries are 0. Let S = coe1, e2, . . . , en, . . . , that is the closed convex hull. Then πS(θ) = ∅,

where θ = (0, 0, . . . ) is the origin of both l1 and l∞.

The following theorem summarizes some important properties of the set πS(ϕ). For the proof

of these results we refer the reader to [1, 2].

Theorem 2.4. Let X be a reflexive Banach space with dual space X∗ and S be a nonempty,

closed and convex subset of X. The following properties hold:

(π1) πS(ϕ) 6= ∅, for any ϕ ∈ X∗;

(π2) If X is also smooth, then for any given ϕ ∈ X∗, x ∈ πS(ϕ) if and only if 〈ϕ−J(x), x−x〉 ≤

0 for all x ∈ S;

(π3) πS(ϕ) is singleton for all ϕ ∈ X∗ if and only if X is strictly convex.

We have seen that if X is a reflexive smooth Banach space, J is a single valued mapping from

X onto X∗. Then Theorem 3.1 in [16] ensures the following result.

Theorem 2.5. For a nonempty closed convex subset S of a reflexive smooth Banach space X

and u ∈ S, the following assertions are equivalent:

i) x ∈ S is a projection of u onto S, that is x ∈ PS(u);

ii) 〈J(u − x), x− x〉 ≤ 0 for all x ∈ S;

iii) J(u− x) ∈ N(S; x).

5

3. New Characterization of the convex normal cone in Banach spaces

The main aim of this section is to add more characterizations of the convex normal cone given

in (2.1) in terms of various concepts of projection.

We start by introducing a new definition of the proximal normal cone in reflexive smooth

Banach spaces strongly based on the projection concept.

Definition 3.1. Let X be a reflexive smooth Banach space, S be a nonempty closed subset of

X, and x ∈ S. The proximal normal cone NP (S; x) to S at x is defined by

NP (S; x) = J(x) : ∃α > 0 so that x ∈ PS(x+ αx).

It is clear that NP (S; x) is in fact a cone.

The following proposition shows that the proximal normal cone NP (S; x) and the well known

convex normal cone N(S; x) given in (2.1) coincide when S is closed convex.

Proposition 3.1. Let X be a reflexive smooth Banach space, S be a nonempty closed and convex

subset of X, and x ∈ S. Then

NP (S; x) = N(S; x).

Proof. Since J is an onto single-valued mapping, for any ϕ ∈ X∗, there exists xϕ ∈ X so that

J(xϕ) = ϕ. So, from the definitions of NP (S; x) and N(S; x) and Theorem 2.5, we have

ϕ ∈ NP (S; x) ⇔ J(xϕ) ∈ NP (S; x), where ϕ = J(xϕ)

⇔ x ∈ PS(x+ αxϕ), for some α > 0

⇔ 〈J(x+ αxϕ − x), x− x〉 ≤ 0, for some α > 0 and for all x ∈ S

⇔ α〈J(xϕ), x− x〉 ≤ 0, for some α > 0 and for all x ∈ S

⇔ 〈J(xϕ), x− x〉 ≤ 0, for all x ∈ S

⇔ ϕ = J(xϕ) ∈ N(S; x),

which proves that NP (S; x) = N(S; x).

If X is also strictly convex, then NP (S; x) has another characterization which is given in the

following proposition.

Proposition 3.2. Let S be a nonempty closed convex subset of a reflexive smooth strictly convex

Banach space X and x be a point in S. Then

NP (S; x) = ϕ ∈ X∗ : ∃α > 0 so that x ∈ PS(x+ αJ∗(ϕ)).

Proof. The proof follows immediately from property (J6) of the normalized duality mapping.

Using the generalized projection πS given in Definition 2.1, we introduce another definition of

the proximal normal cone for a closed subset in reflexive smooth Banach spaces.6

Definition 3.2. Let X be a reflexive smooth Banach space, S be a nonempty closed subset of

X and x ∈ S. The generalized proximal normal cone Nπ(S; x) to S at x is defined by

Nπ(S; x) = ϕ ∈ X∗ : ∃α > 0 so that x ∈ πS(J(x) + αϕ).

It is clear that Nπ(S; x) is a cone.

The following proposition gives a new characterization of the convex normal cone in terms of

the generalized projection πS .

Proposition 3.3. Let S be a nonempty closed and convex subset of a reflexive smooth Banach

space X and x be a point in S. Then

Nπ(S; x) = N(S; x).

Proof. We have from the definitions of Nπ(S; x) and N(S; x), and the property (π2) of the

generalized projection πS(·) that

ϕ ∈ Nπ(S; x) ⇔ x ∈ πS(J(x) + αϕ), for some α > 0

⇔ 〈J(x) + αϕ− J(x), x− x〉 ≤ 0, for some α > 0 and for all x ∈ S

⇔ α〈ϕ, x − x〉 ≤ 0, for some α > 0 and for all x ∈ S

⇔ 〈ϕ, x− x〉 ≤ 0, for all x ∈ S

⇔ ϕ ∈ N(S; x).

Hence Nπ(S; x) = N(S; x).

We end this list of characterization by the following one studied in Borwein and Strojwas [6]

(see also Ioffe [12]).

Proposition 3.4. Let X be a reflexive smooth strictly convex Banach space, S be a nonempty

closed and convex subset of X, x ∈ S and 0 6= ϕ ∈ X∗. Then ϕ ∈ N(S; x) if and only if there

exists u 6∈ S such that

(3.1) dS(u) = ‖u− x‖ and 〈ϕ, u − x〉 = ‖ϕ‖‖u − x‖.

Proof. By Propositions 3.1 and 3.3 we have that if ϕ ∈ N(S; x) = NP (S; x), there exists

α > 0 so that x ∈ PS(x + αJ∗(ϕ)). Put u = x + αJ∗(ϕ). Then dS(u) = ‖u − x‖ and dS(u) =

α‖J∗(ϕ)‖ = α‖ϕ‖ 6= 0 and hence u 6∈ S. Also we have

〈ϕ, u − x〉 = α〈ϕ, J∗(ϕ)〉 = α‖ϕ‖‖J∗(ϕ)‖ = ‖ϕ‖‖u − x‖.

Conversely, assume that (3.1) is satisfied and put α = ‖u−x‖‖ϕ‖ , then 〈αϕ, u− x〉 = α‖ϕ‖‖u− x‖, so

by the definition of the normalized duality mapping αϕ ∈ J(u − x) and hence by Property (J6)

of the duality mapping we obtain u = x+αJ∗(ϕ). This implies that ϕ ∈ NP (S; x) = N(S; x).

7

4. Variants of the sweeping process in p-uniformly convex and q-uniformly

smooth Banach spaces

The main objective of this section is to prove the existence of solutions of the two following

variants of the sweeping process problem : Find y : I → X∗ such that

(SPP )

−y′(t) ∈ N(C(t);J∗(y(t))) + F (t, J∗(y(t))) a.e. on I

y(0) = J(x0) ∈ J(C(0)).

(V SPP )

−y(t) ∈ N(C(t);J∗(y′(t))) + F (t, J∗(y(t))) a.e. on I

y(0) = J(x0) ∈ J(C(0)), y′(0) ∈ J(C(0)).

Note that (SPP) is an appropriate extension of the sweeping process (see [15]) from Hilbert

spaces to reflexive smooth Banach spaces. While (V SPP ) is an appropriate extension of one of

the variants of the sweeping process studied in [4] and [14] from Hilbert spaces to reflexive smooth

Banach spaces.

In order to prove the existence of solutions of these two differential inclusions, we have to prove

some results concerning the p-uniformly convex and q-uniformly smooth Banach space needed in

the proofs of our main theorems in this section. Let X be a Banach space. We recall that (see

[10, 11]) the modulus of convexity and smoothness of a Banach space X are defined respectively

by

δX(ǫ) = inf1 − ‖1

2(x+ y)‖ : ‖x‖ = ‖y‖ = 1 and ‖x− y‖ = ǫ, 0 ≤ ǫ ≤ 2,

and

ρX(t) = sup1

2(‖(x+ y)‖ + ‖(x− y)‖) − 1 : ‖x‖ = 1, ‖y‖ = t, t > 0.

The space X is said to be uniformly convex if δX(ǫ) > 0 for all 0 < ǫ ≤ 2 and uniformly smooth

if limt↓0

ρX(t)

t= 0. Let p, q > 1 be real numbers. Then X is said to be p-uniformly convex (resp.

q-uniformly smooth) if there is a constant c > 0 such that δX(ǫ) ≥ cǫp (resp. ρX(t) ≤ ctp).

It is known ( see for instance [10, 11]) that either uniformly convex or uniformly smooth Banach

space are reflexive and also the uniformly convex Banach space is strictly convex. If X is a p-

uniformly convex Banach space, then X∗ is a p′-uniformly smooth Banach space with p′ = pp−1 is

the conjugate number of p. If X is a q-uniformly smooth Banach space, then X∗ is a q′-uniformly

convex Banach space with q′ = qq−1 .

An important result is given in the following lemma.

Lemma 4.1. Let p, q > 1, X be a p-uniformly convex and q-uniformly smooth Banach space,

and let S be a bounded set. Then there exist two constants α > 0 and β > 0 so that

α‖x− y‖p ≤ V (J(x), y) ≤ β‖x− y‖q, for all x, y ∈ S.

8

Proof. Assume that S ⊂ RB. From Remark 7.7 in [2] there exist two constants α1 > 0, β1 > 0

depending on R so that

α1δX

(

‖x− y‖

4R

)

≤ V (J(x), y) ≤ β1ρX

(

4‖x− y‖

R

)

,

for all x, y in S, where δX(·) and ρX(·) are the modulo of convexity and smoothness of X respec-

tively. Since X is p-uniformly convex and q-uniformly smooth, there exist two constants c > 0

and d > 0 so that

c

(4R)p‖x− y‖p ≤ δX

(

‖x− y‖

4R

)

and ρX

(

4‖x− y‖

R

)

≤ d

(

4

R

)q

‖x− y‖q,

which gives the required constants.

Now, we introduce the function dVS that will play the role of the usual distance function dS

in Hilbert spaces. Let S be a nonempty closed subset of a Banach space X. The function

dVS : X∗ → [0,∞[, is given by

dVS (ϕ) = inf

x∈SV (ϕ, x).

The following result is needed in the proof of Theorem 4.3 and it has its own interest.

Proposition 4.2. Let X be a p-uniformly convex and q-uniformly smooth Banach space, S be a

nonempty closed subset of X, and ϕ ∈ X∗. Then dVS (ϕ) = 0 if and only if J∗(ϕ) ∈ S.

Proof. If J∗(ϕ) ∈ S, then V (ϕ, J∗(ϕ)) = V (J(J∗(ϕ)), J∗(ϕ)) = 0, and hence by the definition

of dVS we obtain dV

S (ϕ) = 0. Conversely, suppose dVS (ϕ) = 0. From the definition of the infimum,

for every n ∈ N there exists xn ∈ S so that

(4.1) 0 ≤ V (ϕ, xn) <1

n2.

Since J is an onto mapping, there exists x ∈ X so that ϕ = J(x). So, we have from (5.2),

0 ≤ V (J(x), xn) <1

n2,

which implies by property (ii) of the functional V given in section 2

0 ≤ (‖x‖ − ‖xn‖)2 ≤ V (J(x), xn) <

1

n2,

and hence ‖xn‖ < ‖x‖ + 1n≤ ‖x‖ + 1 := R. Now, from Lemma 4.1 there exists a constant α > 0

so that α‖xn − x‖p ≤ V (J(x), xn) < 1n, which implies xn → x and hence x ∈ S, because S is

closed. Finally, J∗(ϕ) = J∗(J(x)) = x ∈ S and so the proof is complete.

Now, we are ready to prove the first main result of this section which is an existence result for

(SPP ).

Theorem 4.3. Let p, q > 1, X be a separable p-uniformly convex and q-uniformly smooth Banach

space, T > 0, I = [0, T ] and let C : IX be a set-valued mapping with nonempty closed convex

values satisfying for any ϕ,ψ ∈ X∗ and t, t′ ∈ I

(4.2) |(dVC(t))

1

q′ (ψ) − (dVC(t′))

1

q′ (ϕ)| ≤ λ|t′ − t| + γ‖ϕ − ψ‖,9

where q′ = qq−1 , and λ, γ > 0. Assume that J(C(t)) ⊂ K for any t ∈ I, for some convex compact

set K ⊂ X∗. Let F : I × XX∗ be an upper semicontinuous set-valued mapping with convex

compact values in X∗ such that F (t, x) ⊂ L for all (t, x) ∈ I ×X, for some convex compact set

L ⊂ X∗. Then, for any x0 ∈ C(0), there exists a Lipschitz continuous mapping y : I → X∗ such

that

(SPP )

−y(t)′ ∈ N(C(t);J∗(y(t))) + F (t, J∗(y(t))) a.e. on Iy(0) = J(x0) ∈ J(C(0)).

Proof. Consider for every n ∈ N , the following partition of I: tn,i := iT/2n (0 ≤ i ≤ 2n) and

In,i+1 :=]tn,i, tn,i+1] if 0 ≤ i ≤ 2n − 1 and In,0 := 0.

Put µn := T/2n. Fix n0 ∈ N so that µn0< 1. For every n ≥ n0, we define by induction :

(4.3)

un,0 := x0, z∗n,0 ∈ F (tn,0, un,0),

z∗n,i ∈ F (tn,i, un,i),

un,i+1 := πC(tn,i+1)

(

J(un,i) − µnz∗n,i

)

.

By Theorem 2.4, the last equality is well defined.

For every n ≥ n0, these points (un,i)(0≤i≤2n) and (z∗n,i)(0≤i≤2n) are used to construct two mappings

z∗n and u∗n from I to X∗ by defining their restrictions to each interval In,i as follows:

for t = 0, set z∗n(t) := z∗n,0 and u∗n(t) := J(un,0) = J(x0);

for all t ∈ In,i(0 ≤ i ≤ 2n), set z∗n(t) := z∗n,i, and

(4.4) u∗n(t) := J(un,i) +t− tn,i

µn

(

J(un,i+1) − J(un,i) + µnz∗n,i

)

− (t− tn,i)z∗n,i.

For every n ≥ n0, the mappings un from I to X is given by

(4.5) un(t) := J∗(u∗n(t)), for all t ∈ I.

For every t, t′ in In,i (0 ≤ i ≤ 2n) one has

u∗n(t′) − u∗n(t) =t′ − t

µn

(

J(un,i+1) − J(un,i) + µnz∗n,i

)

− (t′ − t)z∗n,i.

Thus

(4.6) ‖u∗n(t′) − u∗n(t′)‖ ≤ |t′ − t|

(

‖J(un,i+1) − J(un,i) + µnz∗n,i‖

µn+ ‖z∗n,i‖

)

Note that X∗ is p′-uniformly smooth and q′-uniformly convex where p′ and q′ are the conjugate

numbers of p and q respectively. Note also that J(un,i+1), J(un,i) − µnz∗n,i ∈ RB∗, where R :=

k + T l, k, l are the positive constants satisfying K ⊂ kB∗, L ⊂ lB∗ respectively.

Then, Lemma 4.1 ensures the existence of a constant α > 0 so that

α‖J(un,i+1) − J(un,i) + µnz∗n,i‖

q′ ≤ V∗(J∗(J(un,i+1)), J(un,i) − µnz

∗n,i)

= ‖J∗(J(un,i+1))‖2 − 2〈J∗(J(un,i+1)), J(un,i) − µnz

∗n,i〉 + ‖J(un,i) − µnz

∗n,i‖

2

= ‖un,i+1‖2 − 2〈un,i+1, J(un,i) − µnz

∗n,i〉 + ‖J(un,i) − µnz

∗n,i‖

2

= V (J(un,i) − µnz∗n,i, un,i+1)

10

= dVC(tn,i+1)(J(un,i) − µnz

∗n,i),

and so

α1

q′ ‖J(un,i+1) − J(un,i) + µnz∗n,i‖ ≤ (dV

C(tn,i+1))1

q′ (J(un,i) − µnz∗n,i)

− (dVC(tn,i)

)1

q′ (J(un,i))

≤ λ(tn,i+1 − tn,i) + γ‖µnz∗n,i‖

≤ (λ+ γl)µn,

where V∗ : X∗∗ ×X∗ → [0,∞[ defined by

V (ξ, ϕ) = ‖ξ‖2 − 2〈ξ, ϕ〉 + ‖ϕ‖2,

for any ξ ∈ X∗∗ and ϕ ∈ X∗. Then, one obtains for every 0 ≤ i ≤ 2n

(4.7) ‖J(un,i+1) − J(un,i)) + µnz∗n,i‖ ≤

[

λ+ γl

α1

q′

]

µn.

Thus, in view of (4.6), if t, t′ ∈ In,i (0 ≤ i ≤ 2n), one obtains

(4.8) ‖u∗n(t′) − u∗n(t)‖ ≤

[

λ+ (γ + α1

q′ )l

α1

q′

]

|t′ − t|,

and, by addition this also holds for all t, t′ ∈ I. This inequality entails that u∗n is Lipschitz

continuous and hence un is continuous because J∗ is continuous.

Coming back to the definition of u∗n in (4.4), one observes that for 0 ≤ i ≤ 2n

(4.9) (u∗n)′(t) =1

µn(J(un,i+1) − J(un,i) + µnz

∗n,i) − z∗n,i, for a.e. t ∈ In,i.

Then one obtains, in view of (4.7), for a.e. t ∈ I

(4.10) ‖(u∗n)′(t) + z∗n(t)‖ ≤

[

λ+ γl

α1

q′

]

:= δ.

Now, let θn, ρn be defined from I to I by θn(0) = 0,ρn(0) = 0, and

(4.11) θn(t) = tn,i+1, ρn(t) = tn,i if t ∈ In,i(0 ≤ i ≤ 2n).

Then, by (4.3), the construction of u∗n and z∗n and the new characterization of the normal cone

given in Proposition 3.3, we have for a.e. t ∈ I

(4.12) z∗n(t) ∈ F (ρn(t), un(ρn(t))) and (u∗n)′(t) + z∗n(t) ∈ −N(C(θn(t);un(θn(t)))).

This last inclusion, relation (4.10), and Proposition 2.1 entail for a.e. t ∈ I

(4.13) (u∗n)′(t) + z∗n(t) ∈ −δ∂dC(θn(t)(un(θn(t))).

Observe that

(4.14) ‖(u∗n)′(t)‖ ≤ δ + l,

and

u∗n(t) =

(

1 −(t− tn,i)

µn

)

J(un,i) +(t− tn,i)

µnJ(un,i+1) ∈ K.

11

Thus, for every t ∈ I, the set u∗n(t) : n ≥ n0 is relatively compact in X∗. Therefore, the estimate

(4.14) and Theorem 1.4.4 in [3] ensure that there exists a Lipschitz mapping y : I → X∗ such

that u∗n → y uniformly on I and (u∗n)′(t) in L1(I,X∗). Hence J∗u∗n = un converges uniformly

to J∗y, because J∗ is uniformly continuous on the compact set K and u∗n(t) ∈ K for all t ∈ I.

Moreover, for a.e. t ∈ I, by definition (4.11) of θn(t) one has |θn(t) − t| ≤ µn, and by (4.8) we

have

‖u∗n(θn(t)) − y(t)‖ ≤ ‖u∗n(t) − y(t)‖ + (δ + l)µn.

So

(4.15) limn→∞

θn(t) = t, limn→∞

u∗n(θn(t)) = y(t),

and

(4.16) limn→∞

un(θn(t)) = limn→∞

J∗(u∗n(θn(t))) = J∗(y(t)).

As un(θn(t)) ∈ C(θn(t)) it follows from Proposition 5.8 and (4.2)

(dVC(t))

1

q′ (y(t)) = (dVC(t))

1

q′ (y(t)) − (dVC(θn(t)))

1

q′ (J(un(θn(t))))

≤ λ|t− θn(t)| + γ‖u∗n(θn(t)) − y(t)‖ → 0 as n→ +∞,

and hence, by Proposition 5.8 once again, one obtains J∗(y(t)) ∈ C(t), because C(t) is closed.

Now, let us define Z∗n(t) :=

∫ t

0 z∗n(s)ds. Observe that for all t ∈ I the set Z∗

n(t) : n ≥ n0 is

contained in the compact set TL and so it is relatively compact in X∗. Therefore, as ‖z∗n(t)‖ ≤ l,

Theorem 1.4.4 in [3] ensures the existence of an absolutely continuous mapping Z∗ : I → X∗ such

that Z∗n → Z∗ uniformly on I and (Z∗

n)′ = z∗n converges weakly to (Z∗)′ = z∗ in L1(I,X∗).

We proceed now to prove that

y′(t) + z∗(t) ∈ −N(C(t);J∗(y(t))) for a.e. t ∈ I.

The weak convergence in L1(I,X∗) of (u∗n)′n and z∗nn to y′ and z∗ respectively entail for

almost all t ∈ I (by Mazur’s lemma)

y′(t) + z∗(t) ∈⋂

n

co(u∗j )′(t) + z∗j (t) : j ≥ n.

Fix any such t ∈ I and consider any ξ ∈ X. The last relation above yields

〈y′(t) + z∗(t), ξ〉 ≤ infn

supj≥n

〈(u∗j )′(t) + z∗j (t), ξ〉,

and hence according to (4.13)

〈y′(t) + z∗(t), ξ〉 ≤ lim supn

σ(−δ∂dC(θn(t))(un(θn(t)), ξ) ≤ σ(−δ∂dC(t)(J∗(y(t))), ξ),

where the second inequality follows from the upper hemicontinuity property in Proposition 2.2.

As the set ∂dC(t)(J∗(y(t))) is closed and convex and J∗(y(t)) ∈ C(t), we obtain

y′(t) + z∗(t) ∈ −δ∂dC(t)(J∗(y(t))) ⊂ −N(C(t);J∗(y(t))).

12

By the upper semicontinuity of F and the convexity of its values and with the same techniques

used above we can prove that z∗(t) ∈ F (t, u(t)) and so we get

−y′(t) ∈ N(C(t);J∗(y(t))) + F (t, J∗(y(t))),

which completes the proof.

The second main theorem is an existence result of a variant of sweeping process with pertur-

bation given in the following theorem.

Theorem 4.4. Let p, q > 1, X be a separable p-uniformly convex and q-uniformly smooth Banach

space, T > 0, I = [0, T ] and C : IX be a set-valued mapping with nonempty closed and convex

values satisfying for any ϕ,ψ ∈ X∗ and any t, t′ ∈ I

(4.17) |(dVC(t′))

1

q′ (ψ) − (dVC(t))

1

q′ (ϕ)| ≤ λ|t′ − t| + γ‖ψ − ϕ‖,

where q′ = qq−1 and λ, γ > 0 are two constants. Assume that J(C(t)) ⊂ K for any t ∈ I, for

some convex compact set K ⊂ X∗. Let F : I ×X → X∗ be an upper semicontinuous set-valued

mapping with convex compact values in X∗ such that F (t, x) ⊂ L for all (t, x) ∈ I × X, for

some convex compact set L in X∗. Then, for any x0 ∈ C(0), there exists a Lipschitz continuous

mapping y : I → X∗ such that

(V SPP )

−y(t) ∈ N(C(t);J∗(y′(t))) + F (t, J∗(y(t))) a.e. on Iy(0) = J(x0) ∈ J(C(0)), y′(0) ∈ J(C(0)).

Proof. Consider for every n ∈ N , the following partition of I:

tn,i := iT/2n (0 ≤ i ≤ 2n) and In,i :=]tn,i, tn,i+1] if 0 ≤ i ≤ 2n − 1,

and In,0 := tn,0. Put µn := T/2n. Fix n0 ∈ N so that µn0< 1. For every n ≥ n0, we choose by

induction :un,0 := x0 ∈ C(0), z∗n,0 ∈ F (tn,0, un,0)

z∗n,i ∈ F (tn,i, un,i)

un,i+1 := πC(tn,i+1)

(

J(un,i) − µnz∗n,i

)

.

This induction is well defined by using the assumption that C(t) is nonempty closed and convex

for all t ∈ I and Theorem 2.4.

We use the sequences (un,i)(0≤i≤2n) and (z∗n,i)(0≤i≤2n) to construct two mappings z∗n and u∗n from

I to X∗ by defining their restrictions to each interval In,i as follows:

for t = 0, set z∗n(t) := z∗n,0, and u∗n(t) := J(un,0) = J(x0),

for all t ∈ In,i+1(0 ≤ i ≤ 2n − 1), set z∗n(t) := z∗n,i and

(4.18) u∗n(t) := J(un,i+1)(t− tn,i+1) +J(un,i+1) − J(un,i)

µn.

For every n ∈ N , the mapping un from I to X is defined by

(4.19) un(t) := J∗(u∗n(t)), for all t ∈ I.13

Clearly the mappings (u∗n)n are Lipschitz continuous on In,i. Indeed, for any t, t′ in In,i+1 we

have by construction

‖u∗n(t′) − u∗n(t)‖ = ‖J(un,i+1)‖|t′ − t| ≤ k|t′ − t|,

where k is the positive constant satisfying K ⊂ kB∗, and so (u∗n)n are differentiable a.e. on I

with

(4.20) (u∗n)′(t) = J(un,i) for a.e. t ∈ In,i.

Now by the new characterization of the normal cone given in Proposition 3.3, we get for every

0 ≤ i ≤ 2n

(4.21) −J(un,i+1) + J(un,i) − µ+ nz∗n,i ∈ N(C(tn,i+1);un,i).

Let us define the step functions θn, ρn from I to I by θn(0) = 0,ρn(0) = 0, and

θn(t) = tn,i+1, ρn(t) = tn,i if t ∈ In,i+1(0 ≤ i ≤ 2n).

Then we get by (4.18), (4.20), and (4.21) for a.e. t ∈ I that

(4.22) u∗n(θn(t)) + z∗n(t) ∈ −N(C(θn(t);J∗((u∗n)′(t)))).

On the other hand, we have

‖u∗n(θn(t)) + z∗n(t)‖ =

∥

∥

∥

∥

J(un,i+1) − J(un,i)

µn+ z∗n,i

∥

∥

∥

∥

=

∥

∥

∥

∥

J(un,i+1) − J(un,i) + µnz∗n,i

µn

∥

∥

∥

∥

.(4.23)

Since X∗ is p′-uniformly smooth and q′-uniformly convex where p′ and q′ are the conjugate

numbers of p and q respectively, and since J(un,i+1), J(un,i)− µnz∗n,i ∈ RB∗, where R := k+ T l,

and l is the positive constant satisfying L ⊂ lB∗.

Then, by Lemma 4.1 there exists a constant α > 0 so that

α‖J(un,i+1 − J(un,i) + µnz∗n,i‖

q′ ≤ V (J∗(J(un,i+1)), J(un,i) − µnz∗n,i)

= V (J(un,i) − µnz∗n,i, un,i+1)

= dVC(tn,i+1)(J(un,i) − µnz

∗n,i),

and so by (4.17) we obtain

α1

q′ ‖J(un,i+1 − J(un,i) + µnz∗n,i‖ ≤ (dV

C(tn,i+1))1

q′ (J(un,i) − µnz∗n,i)

− (dVC(tn,i)

)1

q′ (J(un,i))

≤ λ|tn,i+1 − tn,i| + γ‖µnz∗n,i‖

≤ (λ+ γl)µn.

So, we get

(4.24)‖J(un,i+1) − J(un,i) + µnz

∗n,i‖

µn≤

[

λ+ γl

α1

q′

]

.

14

Then one obtains by (4.23)

(4.25) ‖u∗n(θn(t)) + z∗n(t)‖ ≤

[

λ+ γl

α1

q′

]

:= δ.

Inclusion (4.22), relation (4.25), and Proposition 2.1 entail for a.e. t ∈ I

(4.26) u∗n(θn(t)) + z∗n(t) ∈ −δ∂dC(θn(t))(J∗((u∗n)′(t))).

We define now the piecewise affine mapping v∗n for t ∈ In,i+1

(4.27) v∗n(t) = J(un,i+1) +(t− tn,i+1)

µn

(

J(un,i+1) − J(un,i) + µnz∗

n,i

)

− (t− tn,i+1)z∗

n,i.

Observe that

v∗n(θn(t)) = J(un,i+1) ∈ J(C(θn(t))) ⊂ K ⊂ kB∗.

Now, we show that the mappings v∗n are equi-Lipschitz with ratio δ+l. Indeed, for any t, t′ ∈ In,i+1

we have by (4.4) and (4.27)

‖v∗n(t′) − v∗n(t)‖ ≤‖J(un,i+1) − J(un,i) + µnz

∗n,i‖

µn|t′ − t| + ‖z∗n,i‖|t

′ − t| ≤ (δ + l)|t′ − t|.

It is also clear by the construction of v∗n and u∗n that

‖v∗n(t) − (u∗n)′(t)‖ ≤‖J(un,i+1) − J(un,i) + µnz

∗n,i‖

µn|t− tn,i+1| + ‖z∗n,i‖|t− tn,i|,

(4.28) ≤ (δ + l)|t− tn,i| ≤ (δ + l)µn,

and hence ‖v∗n − (u∗n)′‖∞ → 0 as n→ ∞.

Uniform convergence of v∗n. Observe that

(4.29) ‖(v∗n)′(t) ≤‖J(un,i+1) − J(un,i) + µnz

∗n,i‖

µn+ ‖z∗n,i‖ ≤ (δ + l),

and

v∗n(t) =(tn,i+1 − t)

µnJ(un,i) +

(

1 −(tn,i+1 − t)

µn

)

J(un,i+1) ∈ K.

Thus, for every t ∈ I, the set v∗n(t) : n ≥ n0 is relatively compact in X∗. Therefore, the

estimate (4.29) and Theorem 1.4.4 in [3] ensure the existence of a Lipschitz continuous mapping

v∗ : I → X∗ with ratio (δ + l) such that v∗n → v∗ uniformly on I. This with (4.28) prove the

uniform convergence of (u∗n)′ to v∗ on I.

Now, we define the Lipschitz continuous mapping y : I → X∗ as follows

y(t) = J(x0) +

∫ t

0v∗(s)ds, for all t ∈ I.

Then y′(t) = v∗(t) a.e. on I. By the definition of u∗n and y we obtain for all t ∈ I

‖u∗n(t) − y(t)‖ =

∥

∥

∥

∥

∫ t

0

(

(u∗n)′(s) − v∗(s))

ds

∥

∥

∥

∥

≤ T‖(u∗n)′ − v∗‖∞

and so by (4.28) we get

‖u∗n − y‖∞ ≤ T‖(u∗n)′ − v∗‖∞ ≤ T‖(u∗n)′ − v∗n‖∞ + T‖v∗n − v∗‖∞ → 0, as n→ ∞.15

This proves the uniform convergence of u∗n to y on I. Since |θn(t) − t| ≤ µn on I, then θn(t) → t

uniformly on I and so u∗n(θn(·)) converges uniformly to y on I. Now, as J∗((u∗n)′(t)) ∈ C(θn(t))

a.e. on I, and by Proposition 5.8 and (4.17) we get for a.e. t ∈ I

(dVC(t))

1

q′ (y′(t)) = (dVC(t))

1

q′ (y′(t)) − (dVC(θn(t)))

1

q′ ((u∗n)′(t))

≤ λ|t− θn(t)| + γ‖y′(t) − (u∗n)′(t)‖ → 0, as n→ ∞,

and since C(t) is closed, Proposition 5.8 ensures, once again, that

(4.30) J∗(y′(t)) ∈ C(t), a.e. on I.

Now, let us define Z∗n(t) :=

∫ t

0 z∗n(s)ds. Observe that for all t ∈ I the set Z∗

n(t) : n ≥ n0

is contained in the compact set TL and so it is relatively compact in X∗. Therefore, by the

inequality ‖z∗n(t)‖ ≤ l, and Theorem 1.4.4 in [3] we have the existence of an absolutely continuous

mapping Z : I → X∗ such that Z∗n → Z∗ uniformly on I and (Z∗

n)′ = z∗n converges weakly to

(Z∗)′ = z∗ in L1(I,X∗).

The weak convergence in L1(I,X∗) of u∗n(θn(·))n and z∗nn to y and z∗ respectively entail

for almost all t ∈ I (by Mazur’s lemma)

y(t) + z∗(t) ∈⋂

n

cou∗j(θj(t)) + z∗j (t) : j ≥ n.

Fix any such t ∈ I and consider any ξ ∈ X. The last relation above yields

〈y(t) + z∗(t), ξ〉 ≤ infn

supj≥n

〈u∗j(θj(t)) + z∗j (t), ξ〉,

and hence by (4.26) we obtain

〈y(t) + z∗(t), ξ〉 ≤ lim supn

σ(−δ∂dC(θn(t))(J∗((u∗n)′(t))), ξ) ≤ σ(−δ∂dC(t)(J

∗(y′(t))), ξ),

where the second inequality follows from the upper hemicontinuity property in Proposition 2.2.

Finally, we have ∂dC(t)(J∗(y′(t))) is a closed convex set in X∗ and so the last inequality entails

for a.e. t ∈ I

y(t) + z∗(t) ∈ −δ∂dC(t)(J∗(y′(t))) ⊂ −N(C(t);J∗(y′(t)).

By the upper semicontinuity of F and the convexity of its values and with the same techniques

used above we can prove that z∗(t) ∈ F (t, J∗(y(t))) and so we get

−y(t) ∈ N(C(t);J∗(y′(t))) + F (t, J∗(y(t))) a.e. on I,

which completes the proof.

Remark 4.1. It is very important to mention that the Lipschitz continuity of the function

(dVC(t′))

1

q′ assumed in (4.2) and (4.17) coincides, in Hilbert spaces, with the Lipschitz continu-

ity of the distance function to the images of C (i.e., (t, x) 7→ dC(t)(x)) and so the existence results

proved in our papers extend the results in from Hilbert spaces to uniformly convex and uniformly

smooth Banach spaces.

16

5. An existence result for a second order convex sweeping process with

perturbations

In the present section, letX be a separable, p-uniformly convex and q-uniformly smooth Banach

space (where p, q > 1), x0 ∈ X, ν∗0 be an open neighborhood of J(x0) in X∗, ν0 := J∗(ν∗0),K :

cl(ν0)X be a set-valued mapping taking nonempty closed values in X and satisfying: for any

x, x′ ∈ cl(ν0) and any ϕ,ϕ′ ∈ X∗,

(5.1) |(dVK(x′))

q−1

q (ϕ′) − (dVK(x))

q−1

q (ϕ)| ≤ λ‖J(x′) − J(x)‖ + γ‖ϕ′ − ϕ‖,

where λ, γ > 0 be two constants and u0 ∈ K(x0). It is important to mention that, in the

Hilbert spaces setting, relation (5.1) coincides with the Lipschitz continuity of the usual distance

function to the images of K. Our aim in this section is to prove the local existence of the following

appropriate extension of (SSPMP) on cl(ν0), that is, we prove the existence of T > 0 and Lipschitz

continuous mappings y : [0, T ] → J(cl(ν0)) and z : [0, T ] → X∗ such that

(GSSPMP )

z(0) = J(u0), J∗(z(t)) ∈ K(J∗(y(t))), for all t ∈ [0, T ];

y(t) = J(x0) +

∫ t

0z(s)ds, for all t ∈ [0, T ];

z′(t) ∈ −N(K(J∗(y(t)));J∗(z(t))) + F (t, y(t), z(t))+

G(t, y(t), z(t)) a.e. on [0, T ].

We begin by recalling the following lemma proved in [25].

Lemma 5.1. Let (X, dX ) and (Y, dY ) be two metric spaces and let h : X → Y be a uniformly

continuous mapping. Then for every sequence (ǫn)n≥1 of positive numbers there exists a strictly

decreasing sequence of positive numbers (en)n≥1 converging to 0 such that

• for any n ≥ 2, 1en−1

and en−1

enare integers ≥ 2;

• for any n ≥ 1, and any x, x′ ∈ X, one has

dX(x, x′) ≤ en =⇒ dY (h(x), h(x′)) ≤ ǫn.

The following proposition proves a closedness property of the subdifferential of the distance

function associated with a set-valued mapping with closed convex values. Its proof follows directly

from the definition of the normal cone.

Proposition 5.2. Let Ω be an open subset in a normed vector space N, X be a Banach space

and C : ΩX be a Hausdorff-continuous set-valued mapping with compact convex values. The

following closedness property of the subdifferential of the distance function holds: “ for any z ∈

Ω, x ∈ C(z), xn → x, zn → z (xn not necessarily in C(zn)), with zn ∈ Ω, and ϕn →w ϕ, one has

ϕ ∈ ∂dC(z)(x).”

The following result is needed in the proof of Theorem 5.4 and it has its own interest.17

Proposition 5.3. Let X be a p-uniformly convex and q-uniformly smooth Banach space, S be a

nonempty closed subset of X, and ϕ ∈ X∗. Then dVS (ϕ) = 0 if and only if J∗(ϕ) ∈ S.

Proof. If J∗(ϕ) ∈ S, then V (ϕ, J∗(ϕ)) = V (J(J∗(ϕ)), J∗(ϕ)) = 0, and hence by the definition

of dVS we obtain dV

S (ϕ) = 0. Conversely, suppose dVS (ϕ) = 0. From the definition of the infimum,

for every n ∈ N there exists xn ∈ S so that

(5.2) 0 ≤ V (ϕ, xn) <1

n2.

Since J is an onto mapping, there exists x ∈ X so that ϕ = J(x). So, we have from (5.2),

0 ≤ V (J(x), xn) <1

n2,

which implies by property (ii) of the functional V given in Section 2

0 ≤ (‖x‖ − ‖xn‖)2 ≤ V (J(x), xn) <

1

n2,

and hence ‖xn‖ < ‖x‖ + 1n≤ ‖x‖ + 1 := R. Now, from Lemma 4.1 there exists a constant α > 0

so that α‖xn − x‖p ≤ V (J(x), xn) < 1n, which implies xn → x and hence x ∈ S, because S is

closed. Finally, J∗(ϕ) = J∗(J(x)) = x ∈ S and so the proof is complete.

Now we prove our first main theorem in this section.

Theorem 5.4. Let F,G : [0,∞[×X∗ ×X∗X∗ be two set-valued mappings and let µ > 0 such

that J(x0) + µB∗ ⊂ ν∗0 . Assume that the following assumptions are satisfied:

(i) For all x ∈ cl(ν0), J(K(x)) ⊂ L ⊂ lB∗ for some convex compact set L in X∗ and some

l > 0;

(ii) F is scalarly u.s.c. on [0, µl] × J(gphK) with nonempty convex weakly compact values;

(iii) G is uniformly continuous on [0, µl] × αB∗ × lB∗ into nonempty compact subsets of X∗,

for α := ‖x0‖ + µ;

(iv) F and G satisfy the following condition:

F (t, x∗, u∗) ⊂ δ1(1 + ‖x∗‖ + ‖u∗‖)B∗

and

G(t, x∗, u∗) ⊂ δ2(1 + ‖x∗‖ + ‖u∗‖)B∗,

for all (t, x∗, u∗) ∈ [0, µl] × J(gphK) for some δ1, δ2 ≥ 0.

Let β be the constant given in Lemma 4.1 satisfying

β‖φ− ψ‖q′ ≤ V∗(J∗(φ), ψ), for all φ,ψ ∈ RB∗,

where R := l + (δ1 + δ2)(1 + α+ l), q′ = qq−1 , and V∗ : X∗∗ ×X∗ → [0,∞[ is defined by

V (ξ, ϕ) = ‖ξ‖2 − 2〈ξ, ϕ〉 + ‖ϕ‖2,18

for any ξ ∈ X∗∗ and ϕ ∈ X∗. Then for every T ∈]0, µl] there exist Lipschitz continuous mappings

y : [0, T ] → J(cl(ν0)) and z : [0, T ] → X∗ such that

z(0) = J(u0), J∗(z(t)) ∈ K(J∗(y(t))), y(t) = J(x0) +

∫ t

0z(s)ds, for all t ∈ [0, T ];

z′(t) ∈ −N(K(J∗(y(t)));J∗(z(t))) + F (t, y(t), z(t)) +G(t, y(t), z(t)) a.e. [0, T ],

with ‖y′(t)‖ ≤ l, and

‖z′(t)‖ ≤

[

λl + γ(δ1 + δ2)(1 + α+ l)

β1

q′

]

+ (δ1 + δ2)(1 + α+ l) a.e. [0, T ].

In other words, there is a Lipschitz continuous solution y : [0, T ] → J(cl(ν0)) to the following

second order differential inclusion:

−y′′(t) ∈ N(K(J∗(y(t)));J∗(y′(t))) + F (t, y(t), y′(t)) +G(t, y(t), y′(t)) a.e. [0, T ]

y(0) = J(x0), y′(0) = J(u0), J∗(y′(t)) ∈ K(J∗(y(t))), for all t ∈ [0, T ],

with ‖y′(t)‖ ≤ l, and

‖y′′(t)‖ ≤

[

λl + γ(δ1 + δ2)(1 + α+ l)

β1

q′

]

+ (δ1 + δ2)(1 + α+ l) a.e. on [0, T ].

Proof. We give the proof in four steps.

Step 1 Construction of approximants. Let T ∈]0, µl[ and put I := [0, T ] and K := I × αB∗ × lB∗.

Then by the assumption (v) we have

(5.3) ‖F (t, x∗, u∗)‖ ≤ δ1(1 + ‖x∗‖ + ‖u∗‖) ≤ δ1(1 + α+ l) =: ξ1,

(5.4) ‖G(t, x∗, u∗)‖ ≤ δ2(1 + ‖x∗‖ + ‖u∗‖) ≤ δ2(1 + α+ l) =: ξ2,

for all (t, x∗, u∗) ∈ K ∩ (I × J(gph K)).

Note that K ∩ (I × J(gph K)) 6= ∅ because (J(x0), J(y0)) ∈ (αB∗ × lB∗) ∩ J(gph K).

Let ǫn = 12n (n = 1, 2, . . . ). Then by the uniform continuity of G on the set K and Lemma 5.1,

there is a strictly decreasing of positive numbers (en) converging to 0 such that en ≤ 1 and T2n−1

and en−1en

are integers ≥ 2 and the following implication holds:

(5.5) ‖(t1, x∗1, u

∗1) − (t2, x

∗2, u

∗2)‖ ≤ µen =⇒ H (G(t1, x

∗1, u

∗1), G(t2, x

∗2, u

∗2)) ≤ ǫn,

for every (t1, x∗1, u

∗1), (t2, x

∗2, u

∗2) ∈ K where ‖(t, x∗, u∗)‖ = |t| + ‖x∗‖ + ‖u∗‖ and µ = 1 + l +

[

λl+γ(ξ1+ξ2)

β1

q′

]

+ (ξ1 + ξ2).

For each n ∈ N , we consider the partition of I given by

(5.6) Pn = tn,i = ien : i = 0, 1, . . . , µn =T

en.

We recall (see [25]) some important properties of the sequence of the partitions (Pn)n needed in

the sequel:

(Pr1) Pn ⊂ Pn+1, for all n ∈ N.19

(Pr2) For every n ∈ N and for every tn,i ∈ Pn \P1 there exists a unique couple (m, j) of positive

integers depending on tn,i, such that m < n, tn,i /∈ Ps for every s ≤ m, tn,i ∈ Ps for every

s > m, 0 ≤ j < µn and tm,j < tn,i < tm,j+1.

Put In,i := [tn,i, tn,i+1[, for all i = 0, . . . , µn − 1 and In,µn := T.

For every n ∈ N we define the following approximating mappings on each interval In,i as follows

(5.7)

u∗n(t) := J(un,i),

x∗n(t) = J(x0) +∫ t

0 u∗n(s)ds,

fn(t) := fn,i ∈ F (tn,i, x∗n(tn,i), u

∗n(t)), and

gn(t) := gn,i ∈ G(tn,i, x∗n(tn,i), u

∗n(t)),

where un,0 = u0 and for all i = 0, . . . , µn − 1 the point un,i+1 is given by

(5.8) un,i+1 = πK(xn(tn,i+1))(J(un,i) + en(fn,i + gn,i)).

As

x∗n(tn,i) = J(x0) +

∫ tn,i

0u∗n(s)ds ∈ J(x0) + ltn,iB∗ ⊂ J(x0) + µB∗ ⊂ ν∗0 ,

so

xn(tn,i) = J∗(x∗n(tn,i)) ⊂ J∗(ν∗0 ) = ν0,

and as K has nonempty closed convex values and defined on cl(ν0), by Theorem 2.4 one can

choose a point un,1. Similarly, we can define, by induction, the points (un,i)i (fn,i)i, and (gn,i)i.

Let us define θn(t) := tn,i, if t ∈ In,i. Then, the definition of x∗n(·) and u∗n(·) and the assumption

(i) yield for all t ∈ I,

(5.9) J∗(u∗n(t)) ∈ K(J∗(x∗n(θn(t)))) ⊂ lB.

So, the mappings x∗n(·) are Lipschitz with ratio l and they are also equibounded, with ‖x∗n‖∞ ≤

‖x0‖ + lT .

Observe also that for all n ∈ N and t ∈ I one has

(5.10) J∗(x∗n(t)) ∈ αB ∩ ν0

Indeed, the definition of xn(·) and un(·) ensure that, for all t ∈ I,

x∗n(t) = J(x0) +

∫ t

0u∗n(s)ds ⊂ J(x0) + ltB∗ ⊂ J(x0) + µB∗ ⊂ αB∗ ∩ ν

∗0 ,

and so

J∗(x∗n(t)) ∈ J∗(αB∗ ∩ ν∗0) ⊂ αB ∩ ν0,

and hence K(J∗(x∗n(t))) is well defined for all t ∈ I.

Now we define the piecewise affine approximants from I to X∗ as follows

(5.11) v∗n(t) := J(un,i) + e−1n (t− tn,i)(J(un,i+1) − J(un,i)), if t ∈ In,i.

20

Observe that v∗n(θn(t)) = J(un,i) for all i = 0, . . . , µn and so by (5.8),(5.10), and the assumption

(i), one has

J∗(v∗n(θn(t))) ∈ K(J∗(x∗n(tn,i))) = K(J∗(x∗n(θn(t)))) ⊂ lB.

Then by (5.3),(5.4),(5.7),(5.10), and the last relation we obtain for all t ∈ I and all n ∈ N

(5.12)

fn(t) ∈ F (θn(t), x∗n(θn(t)), v∗n(θn(t))) ∩ ξ1B∗, and

gn(t) ∈ G(θn(t), x∗n(θn(t)), v∗n(θn(t))) ∩ ξ2B∗.

Now, we check that the mappings v∗n are equi-Lipschitz. Indeed, by (5.8) and the assumption (ii)one has

V1

q′ (J(un,i) + en(fn,i + gn,i), un,i+1) = (dVK(J∗(x∗

n(tn,i+1))))

1

q′ (J(un,i) + en(fn,i + gn,i))

− (dVK(J∗(x∗

n(tn,i))))

1

q′ (J(un,i))

≤ λ‖x∗n(tn,i+1) − x∗n(tn,i)‖ + γ‖en(fn,i + gn,i))‖

≤ [λl + γ(ξ1 + ξ2)]en,

Since J(un,i+1), J(un,i) + en(fn,i + gn,i) ∈ RB∗, we have

β‖J(un,i+1) − J(un,i) − en(fn,i + gn,i)‖q′ ≤ V∗(J

∗(J(un,i+1)), J(un,i) + en(fn,i + gn,i))

= V (J(un,i) + en(fn,i + gn,i), un,i+1)

≤ [λl + γ(ξ1 + ξ2)]eq′

n ,

and hence

‖J(un,i+1) − J(un,i) − en(fn,i + gn,i)‖ ≤ [λl + γ(ξ1 + ξ2)

β1

q′

]en.

Then we have

‖J(un,i+1) − J(un,i)‖ ≤ ‖J(un,i+1) − J(un,i) − en(fn,i + gn,i)‖ + en‖fn,i + gn,i‖

≤

(

λl + γ(ξ1 + ξ2)

β1

q′

+ (ξ1 + ξ2)

)

en.(5.13)

So, for any t, t′ ∈ In,i one has

‖v∗n(t′) − v∗n(t)‖ = e−1n |t′ − t|‖J(un,i+1) − J(un,i)‖ ≤ |t′ − t|

(

λl + (γ + β1

q′ )(ξ1 + ξ2)

β1

q′

)

.

This shows that the mappings v∗n are equi-Lipschitz. By the definition of u∗n(·) and v∗n(·) we have

‖v∗n(t) − u∗n(t)‖ ≤ e−1n |t− tn,i|‖J(un,i+1) − J(un,i)‖

≤

(

λl + (γ + β1

q′ )(ξ1 + ξ2)

β1

q′

)

en,

and hence

(5.14) ‖v∗n − u∗n‖∞ → 0.21

Let us define ρn(t) := tn,i+1, if t ∈ In,i and i = 0, . . . , µn − 1. The definition of v∗n(·) and the

relation (5.8) yield

(5.15) J∗(v∗n(ρn(t))) ∈ K(J∗(x∗n(ρn(t)))), for all t ∈ In,i, (i = 0, . . . , µn − 1),

and by the definition of v∗n(·), one has for a.e. t ∈ In,i

(5.16) (v∗n)′(t) = e−1n (J(un,i+1) − J(un,i))).

So, by the new characterization of the normal cone given in Proposition 3.3, we get for a.e. t ∈ I

(5.17) (v∗n)′(t) − (fn(t) + gn(t)) ∈ −N(K(J∗(x∗n(ρn(t))));J∗(v∗n(ρn(t)))).

In fact,

un,i+1 = πK(J∗(x∗

n(tn,i+1)))(J(un,i) + en(fn,i + gn,i))

= πK(J∗(x∗

n(tn,i+1)))(J(un,i+1) − [J(un,i+1) − J(un,i)] + en(fn,i + gn,i))

⇔ J(un,i+1) − J(un,i) − en(fn,i + gn,i) ∈ −N(K(J∗(x∗n(tn,i+1)));un,i+1)

⇔ e−1n (J(un,i+1) − J(un,i)) − fn,i − gn,i ∈ −N(K(J∗(x∗n(tn,i+1)));un,i+1).

On the other hand, by (5.13) and (5.16), it is clear that

(5.18) ‖(v∗n)′(t)‖ ≤

(

λl + (γ + β1

q′ )(ξ1 + ξ2)

β1

q′

)

.

Put δ :=

(

λl+(γ+2β1

q′ )(ξ1+ξ2)

β1

q′

)

. Therefore, the relations (5.12), (5.17), and (5.18), and Proposition

2.1 entail for a.e. t ∈ I

(5.19) (v∗n)′(t) − (fn(t) + gn(t)) ∈ −δ∂d(K(J∗(x∗

n(ρn(t))))(J∗(v∗n(ρn(t)))).

Step 2. Uniform convergence of the sequences x∗n(·) and v∗n(·). :

Since e−1n (t − tn,i) ≤ 1, for all t ∈ In,i and J(un,i), J(un,i) ∈ L, and L is a convex set in X∗ one

gets for all t ∈ I,

v∗n(t) = J(un,i) + e−1n (t− tn,i)[J(un,i+1) − J(un,i)]

=

(

1 −t− tn,i

en

)

J(un,i) +t− tn,i

enJ(un,i+1) ∈ L.

Thus for every t ∈ I, the set v∗n(·) : n ∈ N is relatively compact in X∗. Therefore, the estimate

(5.18) and Theorem 0.4.4 in [3] ensure that there exists a Lipschitz mapping u∗ : I → X∗ such

that:

• v∗n converges uniformly to u∗ on I;

• ((v∗n)′) weakly converges to (u∗)′ in L1(I,X∗).22

Now, we define the Lipschitz mapping x∗ : I → X∗ as

(5.20) x∗(t) = J(x0) +

∫ t

0u∗(s)ds, for all t ∈ I.

Then by the definition of x∗n one obtains for all t ∈ I,

‖x∗n(t) − x∗(t)‖ = ‖

∫ t

0(u∗n(s) − u∗(s))ds‖ ≤ T‖u∗n − u∗‖∞,

and by (5.14) we get

(5.21) ‖x∗n − x∗‖∞ ≤ T‖u∗n − v∗n‖∞ + T‖v∗n − u∗‖∞ → 0 as n→ ∞.

Hence x∗n converges uniformly to x∗ on I, which completes the proof of the second step.

Step 3. Relative compactness of (gn). :

By the relation of θn(·) we have for all t ∈ I and all n ≥ n0, |θn(t) − t| ≤ en. Then (x∗n θn)

and (v∗n θn) converge uniformly on I to x∗ and u∗ respectively. Now, by (5.12) and the uniform

continuity of G on I × αB∗ × lB∗ one has

dG(t,x∗(t),u∗(t))(gn(t)) ≤ H(G(θn(t), x∗n(θn(t)), v∗n(θn(t))), G(t, x∗(t), u∗(t))) → 0 as n→ ∞.

This implies the relative compactness of the set gn(t) : n ∈ N in X∗ for all t ∈ I because

G(t, x∗(t), u∗(t)) is a compact set in X∗. Now, we have to show that the sequence is an equi-

oscillating family of bounded functions. Recall that a family F of bounded mappings x : I → X is

equi-oscillating if for every ǫ > 0, there exists a finite partition of I into intervals Ij (j = 0, . . . ,m)

such that for all x ∈ F and all j = 0, . . . ,m one has wIj≤ ǫ, where wIj

denotes the oscillation of

x in Ij defined by

(5.22) wIj:= sup‖x(t) − x(t′)‖ : t, t′ ∈ Ij.

Fix any ǫ > 0 and let m0 ∈ N such that 4ǫm0≤ ǫ. Consider the finite partition Ij :=

[tm0,j, tm0,j+1[, (j = 0, . . . , µm0− 1) of I. We shall prove that

(5.23) wIj(gn) ≤ ǫ, for all n ∈ N and all j = 0, . . . , µm0

− 1.

For that purpose, we have to choose gn,i in (5.7) in such way that the following condition holds

for every n ∈ N and i = 0, . . . , µm0− 1:

‖gn(tn,i) − gn(tn,i−1)‖ ≤ ǫn, if tn,i ∈ P1,

‖gn(tn,i) − gn(tn,p)‖ ≤ ǫn, if tn,i 6∈ P1,(5.24)

where (m, p) is the unique pair of integers assigned to tn,i such that m < n, tn,i 6∈ Pj for j ≤ m,

tn,i ∈ Pj for j > m and tm,p < tn,i < tm,p+1. For i = 0 we take gn,0 ∈ G(0, J(x0), J(u0)). By

induction we assume that gn,j ∈ G(tn,j , x∗n(tn,j), J(un,j)) have been defined for all j ∈ 0, . . . , i−

1.

If tn,i ∈ P1, it suffices to take gn,j ∈ G(tn,i, x∗n(tn,i), J(un,i)) such that

‖gn,i − gn,i−1‖ ≤ H(G(tn,i, x∗n(tn,i), J(un,i)), G(tn,i−1, x

∗n(tn,i−1), J(un,i−1))).

23

Indeed, by virtue of (5.9) and (5.13) we have

‖(tn,i, x∗n(tn,i), J(un,i)) − (tn,i−1, x

∗n(tn,i−1), J(un,i−1))‖ ≤

(

1 + l +

(

λl + (γ + β1

q′ )(ξ1 + ξ2)

β1

q′

))

en ≤ µen,

which, in combining with (5.5), gives

‖gn(tn,i) − gn(tn,i−1)‖ = ‖gn,i − gn,i−1‖ ≤ ǫn.

If tn,i ∈ P1, then tm,p ∈ Pn (because m < n) and so there is a unique integer q < i such that

tm,p = tn,q. Hence tn,i − tn,q = tn,i − tm,p < tm,p+1 − tm,p ≤ em. This with (5.9) and (5.18)

‖(tn,i, x∗n(tn,i), J(un,i)) − (tn,q, x

∗n(tn,q), J(un,q))‖ ≤

(

1 + l +

(

λl + (γ + β1

q′ )(ξ1 + ξ2)

β1

q′

))

em ≤ µem,

which together with (5.5) yield

H(G(tn,i, x∗n(tn,i), J(un,i)), G(tn,q, x

∗n(tn,q), J(un,q))) ≤ ǫm.

Since gn(tm,p) = gn(tn,q) = gn,q ∈ G(tn,q, x∗n(tn,q), J(un,q)), we may choose gn,i ∈ G(tn,i, x

∗n(tn,i), J(un,i))

such that

‖gn(tn,i) − gn(tm,p)‖ = ‖gn,i − gn,q‖ ≤ ǫm,

which is the second inequality in (5.24).

Next, we prove that (5.23) holds.

If n ≤ m0, then en

em0

is an integer and every Ij is contained in some interval [tn,k, tn,k+1[ in which

gn is constant. Thus (5.20) is trivial in this case:

wIj(gn) = 0, for all j = 0, . . . , µm0

and all n ≤ m0.

Let n > m0. Asem0

enis an integer, then 2en ≤ em0

. By Property (Pr1), it follows that

tm0,j, tm0,j+1 ∈ Pn. Thus, there exist c, d such that 0 ≤ c ≤ d, tm0,j = tn,c and tm0,j+1 = tn,d. The

values of the mapping gn on Ij = [tm0,j, tm0,j+1[= [tn,c, tn,d[ are gn(tn,s) = gn,s, with c < s < d.

So, we shall prove that, for all c < s < d,

(5.25) ‖gn(tn,s) − gn(tm0,j)‖ ≤ 2ǫm0,

and so ‖gn(t)− gn(tm0,j)‖ ≤ 2ǫm0, for all t ∈ Ij and all n > m. Then it will follows that, for all t

and t′ in Ij,

‖gn(t′) − gn(t)‖ ≤ ‖gn(t′) − gn(tm0,j)‖ + ‖gn(tm0,j) − gn(t)‖ ≤ 4ǫm0≤ ǫ.

Hence wIj(gn) ≤ ǫ, and (5.23) holds.

Let tn,s ∈ Pn such that c < s < d. Then tn,s 6∈ Pm0and consequently tn,s 6∈ Pn. Now by

Property (Pr2), there exists a unique couple (m1, p1) such that m1 < n, tn,s ∈ Pm1+1 \ Pm1and

24

tm1,p1< tn,s < tm1,p1+1 with p1 < µm1

. By virtue of the second inequality in (5.24), we obtain

that

(5.26) ‖gn(tm2,p2) − gn(tm1,p1

)‖ ≤ ǫm2,

because tm1,p1∈ Pn (m1 < n implies Pm1

⊂ Pn). As mentioned above for the couple (m1, p1), it

is not hard to check that tm0,j ≤ tm2,p2. If tm0,j = tm2,p2

, then (5.21) follows by summing (5.22)

and (5.23), since ǫm1+ ǫm2

≤ ǫm0(because m1,m2 > m0). The case if tm0,j < tm2,p2

is treated

as above.

The inductive procedure is now clear: There exists a finite sequence (mi, pi), i = 0, . . . , k such

that

m0 ≤ mk ≤ mk−1 < · · · < m1 < n, tmk,pk= tm0,j, tmi,pi

∈ Pmi⊂ Pn, for all i and

‖gn(tmi,pi) − gn(tmi+1,pi+1

)‖ ≤ ǫmi+1, i = 0, . . . , k − 1.

Consequently, by applying these inequalities, (5.26), and the triangle inequality, we obtain

‖gn(tn,s) − gn(tm1,p1)‖ ≤ ǫm1

+ ǫm2+ · · · + ǫmk

≤ 2ǫm0.

Thus completing the proof of (5.25) and so we get the relative compactness for the uniform

convergence in the space of bounded mappings of the sequence gn(·). Therefore, there exists a

bounded mapping g(·) : I → X∗ such that ‖gn − g‖∞ → 0 for some subsequence of (gn) which

can be denoted without loss of generality by (gn).

Step 4. Existence of a solution. :

Since (x∗n θn) and (v∗n θn) converge uniformly on I to x∗ and u∗ respectively, then by the

uniform continuity of G on I×αB∗× lB∗, the closedness of the set G(t, x∗(t), u∗(t)), and the fact

that gn(t) ∈ G(θn(t), x∗n(θn(t)), v∗n(θn(t))) a.e. on I (by (5.12)), we obtain g(t) ∈ G(t, x∗(t), u∗(t),

a.e. on I.

Recall that J∗(v∗n(θn(t))) ∈ K(J∗(xn(θn(t)))), for all t ∈ I and n ∈ N. It follows then by

assumption (ii) that

(dVK(J∗(x∗(t))))

1

q′ (u∗(t)) = (dVK(x(t)))

1

q′ (u∗(t)) − (dVK(xn(θn(t))))

1

q′ (v∗n(θn(t)))

≤ λ‖x∗(t) − x∗n(θn(t))‖ + γ‖u∗(t) − v∗n(θn(t))‖ → 0.

Hence dVK(J∗(x∗(t)))(u

∗(t)) = 0, and so by the closedness of K and Proposition 5.8, one gets

J∗(u∗(t)) ∈ K(J∗(x∗(t))), for all t ∈ I.

By (5.12) we can assume without loss of generality that the sequence fn converges in the

weak star topology in L∞(I,X∗) to some mapping f . Therefore, from (5.12) once again, we can

classically (see Theorem V-14 in [24]) conclude that f(t) ∈ F (t, x∗(t), u∗(t)) a.e. on I, because

by hypothesis F is scalarly u.s.c. with convex weakly compact values. We apply now Castaing

techniques (see for example [22]). The weak star convergence of (v∗n)′− (fn +gn) to (u∗)′− (f +g)25

in L∞(I,X∗) (by what precedes and Step 2) entails by (Mazur’s lemma) that for a.e. t ∈ I

(u∗)′(t) − (f(t) + g(t)) ∈⋂

n

co(v∗j )′(t) − (fj(t) + gj(t)), j ≥ n.

Fix such t in I and any ξ ∈ X. Then the last relation gives

〈(u∗)′(t) − (f(t) + g(t)), ξ〉 ≤ infn

supj≥n

〈(v∗j )′(t) − (fj(t) + gj(t)), ξ〉.

Hence by 5.19, one obtains

〈(u∗)′(t) − (f(t) + g(t)), ξ〉 ≤ lim supn

σ(−δ∂dK(J∗(x∗

n(ρn(t))))(J∗(v∗n(ρn(t)))), ξ).

Since |ρn(t)− t| ≤ en on [0, T [, then ρn(t) → t uniformly on [0, T [. It follows then by Proposition

5.2 that for a.e. t ∈ I and any ξ ∈ X,

〈(u∗)′(t) − (f(t) + g(t)), ξ〉 ≤ σ(−δ∂dK(J∗(x∗(t)))(J∗(u∗(t))), ξ).

Since ∂dK(J∗(x∗(t)))(J∗(u∗(t))) is a convex and closed set in X∗ and so that the last inequality

entails

(u∗)′(t) − (f(t) + g(t)) ∈ −δ∂dK(J∗(x∗(t)))(J∗(u∗(t))) ⊂ −N(K(J∗(x∗(t)));J∗(u∗(t))),

because J∗(u∗(t)) ∈ K(J∗(x∗(t))). Thus

(u∗)′(t) ∈ −N(K(J∗(x∗(t)));J∗(u∗(t))) + F (t, x∗(t), u∗(t)) +G(t, x∗(t), u∗(t)),

and so the proof of the theorem is complete by taking y ≡ x∗ and z ≡ u∗.

In the previous theorem we have proved an existence result for the problem (GSSPMP) when

the perturbation F is assumed to be globally upper semicontinuous. Our aim in the next theorem

is to prove the existence of solutions of the problem (GSSPCP) (the Generalized Second order

Sweeping Process with a Convex Perturbation F , i.e., the case when G = 0), when the global

scalarly upper semicontinuity of F on [0, µl] × J(gph K) is replaced by the following weaker

assumptions:

(A1) For any t ∈ [0, µl], the set-valued mapping F (t, ·, ·) is scalarly u.s.c. on J(gph K);

(A2) F is scalarly measurable with respect to the σ-field of [0, µl]× J(gph K) generated by the

Lebesgue sets in [0, µl] and the Borel sets in the space X∗.

Our proof here is based on an approximation method. The idea is to approximate a set-valued

mapping F that satisfies (A1) and (A2) and study the convergence of the solutions xn of

(GSSPCP)n associated with each Fn (the existence of such solutions is ensured by our result

in Theorem 5.4). We will use a special approximation Fn of F defined by

Fn(t, x∗, u∗) :=1

µn

∫

It,µn

F (s, x∗, u∗)ds

for all (t, x∗, u∗) ∈ I × X∗ × X∗, where I is some compact interval, µn is a sequence of strictly

positive numbers converging to zero and It,µn := I ∩ [t, t + µn]. For more details concerning

this approximation we refer the reader to [26, 23] and the references therein. We need the two

following lemmas. For their proofs we refer to [26, 23].26

Recall that a Polish space is a separable complete metrizable topological space and a Suslin space

is a continuous image of a Polish space.

Lemma 5.5. Let T > 0, S be a Suslin metrizable space, X be a separable Banach space and

F : [0, T ]×SX be a set-valued mapping with nonempty convex weakly compact values. Assume

that F satisfies the following assumptions:

(a) For any t ∈ [0, T ], F (t, ·) is scalarly u.s.c. on S;

(b) F is scalarly measurable w.r.t. the σ-field of [0, T ] × S generated by the Lebesgue sets in

[0, T ] and the Borel sets in the topological space S;

(c) F (t, y) ⊂ ρ(1 + ‖y‖)B, for all (t, y) ∈ [0, T ] × S and for some ρ > 0.

Then F is a globally scalarly u.s.c. set-valued mapping on [0, T ]×S with nonempty convex compact

values satisfying:

F (t, y) ⊂ Tρ(1 + ‖y‖)B,

for all (t, y) ∈ [0, T ] × S.

Lemma 5.6. Let T > 0, S be a Suslin metrizable space, X be a separable Banach space and

F : [0, T ]×SX be a set-valued mapping with nonempty convex weakly compact values. Assume

that F is bounded on [0, T ] × S and that satisfies the hypothesis (a) and (b) in Lemma 5.5.

Then for any sequence (yn)n of Lebesgue measurable mappings from [0, T ] to S which converges

pointwisely to a Lebesgue measurable mapping to y, any sequence (zn)n in L1([0, T ],X) weakly

converging to z in L1([0, T ],X) and satisfying zn(t) ∈ F (t, yn(t)) a.e. on [0, T ] and one has

z(t) ∈ F (t, y(t)), a.e. on [0, T ].

Now we are able to prove our result in this section.

Theorem 5.7. Let F : [0,∞[×X∗ × X∗X∗ be a set-valued mapping and µ > 0 such that

J(x0) + µB∗ ⊂ ν∗0 . Assume that the hypothesis (i), (iv) in Theorem 5.4 are satisfied and assume

that F satisfies (A1) and (A2). Let β be the constant given in Lemma 4.1 satisfying

β‖φ− ψ‖q′ ≤ V∗(J(φ), ψ), for all φ,ψ ∈ RB∗,

where R := l+Tδ1(1+α+ l) and α := ‖x0‖+µ. Then for every T ∈]0, µl] there exists a Lipschitz

continuous solution y : [0, T ] → J(cl(ν0)) of (GSSPCP) satisfying ‖y′(t)‖ ≤ l and

‖y′′(t)‖ ≤

[

λl + (γ + β1

q′ )(Tδ1)(1 + α+ l)

β1

q′

]

, a.e. on [0, T ].

Proof. Let T ∈]0, µl] and put I := [0, T ] and S := αB∗ × lB∗ where α := ‖x0‖ + µ. Clearly S

is a Suslin metrizable space. Let µn be a sequence of strictly positive numbers that converges to

zero. For each n ≥ 1 we put

Fn(t, x∗, u∗) :=1

µn

∫

It,µn

F (s, x∗, u∗)ds

27

for all (t, x∗, u∗) ∈ I ×X∗ ×X∗. By Lemma 5.5 the set-valued mapping Fn is scalarly u.s.c. on

I × S with nonempty convex compact values and satisfies

Fn(t, x∗, u∗) ⊂ Tδ1(1 + ‖x∗‖ + ‖u∗‖)B∗ ⊂ Tδ1(1 + α+ l)B∗ := Tξ1B∗,

for any (t, x∗, u∗) ∈ I ×S and all n ≥ 1. So that we can apply the result of Theorem 5.4 For each

n ≥ 1, there exists a Lipschitz continuous mapping yn : I → J(cl(ν0)) satisfying

y′′n(t) ∈ −N(K(J∗(yn(t)));J∗(y′n(t))) + Fn(t, yn(t), y′n(t)) a.e. on I;

y(0) = J(x0), y′n(0) = J(u0), y′n(t) ∈ J(K(J∗(yn(t)))), for all t ∈ I,

with ‖y′n(t)‖ ≤ l, and

‖y′′n(t)‖ ≤

[

λl + (γ + β1

q′ )(Tξ1)

β1

q′

]

a.e. on I and for all n ≥ 1.

Since y′n(t) ∈ J(K(J∗(yn(t)))) ⊂ L for all n ≥ 1 and all t ∈ I, then we set the relative compactness

of the set y′n(t) : n ≥ 1 in X∗ for all t ∈ I. Therefore, by Arzela Ascoli’s theorem we may extract

from y′n a subsequence that converges uniformly to some Lipschitz mapping y′. By integrating

we get the uniform convergence of the sequence yn to y because they have the same initial value

yn(0) = J(x0), for all n ≥ 1. Now, by (GSSPCP)n there is for any n ≥ 1 a Lebesgue measurable

mapping fn : I → X∗ such that

(5.27) fn(t) ∈ Fn(t, yn(t), y′n(t)) ⊂ Tδ1(1 + ‖xn(t)‖ + ‖y′n(t)‖)B∗ ⊂ Tξ1B∗

and

(5.28) fn(t) − y′′n(t) ∈ N(K(J∗(yn(t)));J∗(y′n(t))) = δ∂dK(J∗(yn(t)))(J∗(y′n(t))),

for a.e. t ∈ I, where δ :=

[

λl + γ(Tξ1)

β1

q′

]

and ξ1 := δ1(1 + α + l). Observe by (5.27) and

(GSSPCP )n that (fn)n and (y′′n)n are equibounded in L∞(I,X∗) and so subsequences may be

extracted that converge in the weak star topology of L∞(I,X∗). Without loss of generality, we

may suppose that these sequences are denoted (fn)n and (y′′n)n respectively. Denote by f and w

their weak star limits respectively. Then for each t ∈ I

J(u0) +

∫ t

0y′′(s)ds = y′(t) = lim

n→∞y′n(t)

= J(u0) + limn→∞

∫ t

0y′′n(s)ds

= J(u0) + limn→∞

∫ t

0w(s)ds,

which gives the equality y′′(t) = w(t) for almost all t ∈ I, that is, y′′n converges in the weak star

topology in L1(I,X∗) to y′′.

It follows then from (GSSPCP )n and the hypothesis (ii) in the theorem that

(dVK(J∗(y(t))))

1

q′ (y′(t)) = (dVK(J∗(y(t))))

1

q′ (y′(t)) − (dVK(J∗(yn(t))))

1

q′ (y′n(t))

≤ λ‖yn(t) − y(t)‖ + γ‖y′n(t) − y′(t)‖ → 0,28

and so dVK(J∗(y(t))))(y

′(t)) = 0, for all t ∈ I. Hence Proposition 5.8 yields J∗(y′(t)) ∈ K(J∗(y(t)))

because the set K(J∗(y(t))) is closed.

The weak star convergence in L∞(I,X∗) of y′′n and fn to y′′ and f respectively entail for almost

all t ∈ I (by Mazur’s lemma)

f(t) − y′′(t) ∈⋂

n

cofj(t) − y′′j (t) : j ≥ n.

Fix any such t ∈ I and consider any ζ ∈ X. The last relation ensures

〈f(t) − y′′(t), ζ〉 ≤ infn

supj≥n

〈fj(t) − y′′j (t), ζ〉,

and hence according to (5.28) and Proposition 5.2 we get

〈f(t) − y′′(t), ζ〉 ≤ lim supn

σ(

δ∂dK(J∗(yn(t)))(J∗(y′n(t))), ζ

)

≤ σ(

δ∂dK(J∗(y(t)))(J∗(y′(t))), ζ

)

.

As the set ∂dK(J∗(y(t)))(J∗(y′(t))) is closed and convex, we obtain

f(t) − y′′(t) ∈ δ∂dK(J∗(y(t)))(J∗(y′(t)))

⊂ N(K(J∗(y(t)));J∗(y′(t))),(5.29)

because J∗(y′(t)) ∈ K(J∗(y(t))). Now we check that f(t) ∈ F (t, y(t), y′(t)) a.e. on I. Since f is

bounded on I × S, fn converges in the weak star topology in L∞(I,X∗) to f , and (yn, y′n) is a

sequence of Lebesgue measurable mappings from I to S (because J∗(y′n(t)) ∈ K(J∗(yn(t))) ⊂ lB∗

and ‖yn(t)‖ ≤ α for all t ∈ I) converging uniformly to (y, y′), it follows then from Lemma 5.6

that f(t) ∈ F (t, y(t), y′(t)) a.e. on I. Consequently, we obtain by (5.29)

y′′(t) ∈ −N(K(J∗(y(t)));J∗(y′(t))) + F (t, y(t), y′(t)).

Thus completing the proof of the theorem.

Throughout the rest of the paper, let X be a separable p-uniformly convex q-uniformly smooth

Banach space, Ω∗ be an open subset in X∗, Ω := J∗(Ω∗), F : [0,∞[×X∗ × X∗X∗ be a set-

valued mapping, andK : cl(Ω)X be a set-valued mapping satisfying assumption (ii) in Theorem

5.4 and taking nonempty closed convex values in X. In the sequel, we are interested by some

topological properties of the solution set of the problem (GSSPCP ). Let x0 ∈ Ω, u0 ∈ K(x0),

and T > 0 such that J(x0) + T lB∗. We denote by SF (J(x0), J(u0)) the set of all Lipschitz

continuous mappings (y, z) : [0, T ] → cl(Ω∗) ×X∗ such that

z(0) = J(u0);

y(t) = J(x0) +∫ t

0 z(s)ds, for all t ∈ [0, T ];

J∗(z(t)) ∈ K(J∗(y(t))), for all t ∈ [0, T ];

z′(t) ∈ −N(K(J∗(y(t)));J∗(z(t))) + F (t, y(t), z(t)) a.e. [0, T ].

(GSSPCP )

29

Proposition 5.8. Assume that the hypothesis of Theorem 5.4 are satisfied and that gph K is

compact in cl(Ω)× lB. Then the set SF (J(x0), J(u0)) is relatively compact in C([0, T ],X∗ ×X∗).

Proof. By Theorem 5.4 the set of solution (y, z) of (GSSPCP ) is equi-continuous and for

any t ∈ [0, T ] one has the set (y(t), z(t)) : (y, z) ∈ SF (J(x0), J(u0)) is relatively compact in

X∗ ×X∗ because it is contained in the compact set K := (v,w) ∈ X∗ ×X∗ : (J∗(v), J∗(w)) ∈

gph K. Then Arzela-Ascoli’s theorem gives the relative compactness of the set SF (J(x0), J(u0))

in C([0, T ],X∗ ×X∗).

Remark 5.1. Assume that Ω = X and let T be any strictly positive number. Put

SF (K) :=⋃

(x0,u0)∈ gph K

SF (J(x0), J(u0)).

With the same arguments, as in the proof of Proposition 5.8, we can show that under the same

hypothesis in Proposition 5.8 the set SF (K) is relatively compact in C([0, T ],X∗ ×X∗).

Now we wish to prove the closedness of the set-valued mapping SF .

Proposition 5.9. Assume that the hypothesis of Theorem 5.4 are satisfied. Then the set-valued

mapping SF has a closed graph in Ω∗ × J(K(Ω)) × C([0, T ],X∗ ×X∗).

Proof. Let (xn0 , u

n0 )n ∈ Ω×K(Ω) and (yn, zn)n ∈ C([0, T ],X∗×X∗) with (yn, zn) ∈ SF (J(xn

0 ), J(un0 ))

such that (xn0 , u

n0 ) → (x0, u0) ∈ Ω ×K(Ω) uniformly, and (yn, zn) → (y, z) ∈ C([0, T ],X∗ ×X∗)

uniformly. We have to show that (y, z) ∈ SF (J(x0), J(u0)). First observe that for n sufficiently

large xn0 ∈ x0 + lTB. Now, it is not difficult to check that the closedness of gph K in X ×X and

so K in X∗ ×X∗. Indeed, if (xn, un) ∈ gphK = (x, u) : u ∈ K(x) such that (xn, un) → (x, u).

Then, by the relation (5.1), we have

(dVK(x))

1

q′ (J(u)) = (dVK(x))

1

q′ (J(u)) − (dVK(xn))

1

q′ (J(un))

≤ λ‖J(x) − J(xn)‖ + γ‖J(u) − J(un)‖ −→ 0,

which implies, by Proposition 5.8 and the closedness of K(x), that u = J∗(J(u)) ∈ K(x) and

hence gph K is closed in X ×X. The closedness of K follows from the closdness of gph K and

the bicontinuity of J . The uniform convergence of both sequences (xn0 , u

n0 )n and (yn, zn)n and

the continuity of J imply that (y(0), z(0)) = (J(x0), J(u0)) and that J∗(z(t)) ∈ K(J∗(y(t))) for

all t ∈ [0, T ]. On the other hand one has for all t ∈ [0, T ]

y(t) = limnyn(t) = J(x0) + lim

n

∫ t

0zn(s)ds = J(x0) +

∫ t

0z(s)ds.

It remains then to show that

z′(t) ∈ −N(K(J∗(y(t));J∗(z(t)))) + F (t, y(t), z(t)), a.e. [0, T ].

By the assumptions we have

(zn)′(t) ∈ −N(K(J∗(yn(t));J∗(zn(t)))) + F (t, yn(t), zn(t)), a.e. [0, T ].30

Then for every n there exists a measurable selection fn such that

(5.30) fn(t) ∈ F (t, yn(t), zn(t))

and

(5.31) −(zn)′(t) + fn(t) ∈ N(K(J∗(yn(t)));J∗(zn(t))),

for a.e. t ∈ [0, T ]. By Theorem 5.4 one has for n sufficiently large

‖(zn)′(t)‖ ≤

[

λl + (γ + β1

q′ )δ1(1 + ‖xn0‖ + T l + l)

β1

q′

]

≤

[

λl + (γ + β1

q′ )δ1(1 + ‖x0‖ + T l + l)

β1

q′

]

.(5.32)

By (v) in Theorem 5.4 and the fact that J∗(zn(t)) ∈ K(J∗(yn(t))) one gets

(5.33) ‖fn(t)‖ ≤ δ1(1 + ‖x0‖ + T l+ l).

Therefore, we may suppose without loss of generality that (zn)′ → z′ and fn → f in the weak star

topology in L∞([0, T ],X∗). Since F (t, ·, ·) is scalarly upper semicontinuous with convex compact

values, then we get easily that f(t) ∈ F (t, y(t), z(t)), a.e. t ∈ [0, T ]. Now by (5.30),(5.32),(5.33),

and Proposition 2.1 we have for δ :=

[

λla+(γ+β1

q′ )δ1(1+‖x0‖+T l+l)

β1

q′

]

, the following inclusion

−(zn)′(t) + fn(t) ∈ δ∂dK(J∗(yn(t)))(J∗(zn(t))), for a.e. t ∈ [0, T ].

Then by using Mazur’s lemma and Proposition 5.2, it is easy to conclude that for a.e. t ∈ [0, T ]

f(t) − z′(t) ∈ δ∂dK(J∗(y(t)))(J∗(z(t))) ⊂ N(K(J∗(y(t)));J∗(z(t))), for a.e. t ∈ [0, T ],

because J∗(z(t)) ∈ K(J∗(y(t))). Thus we get for a.e. t ∈ [0, T ]

z′(t) ∈ −N(K(J∗(y(t)));J∗(z(t))) + F (t, y(t), z(t)),

which completes the proof of the proposition.

Remark 5.2. The proof of Proposition 5.8 shows that the solution set SF (J(x0), J(u0)) associated

to the problem (GSSPMP ) is relatively compact in C([0, T ],X∗×X∗) whenever the graph gph K

is compact in X. Contrarily, our proof in Proposition 5.9 cannot provide the closedness of the

graph of the set-valued mapping SF associated to the problem (GSSPMP ). The difficulty that

prevents to conclude is the absence of the convexity of G.

Acknowledgements. We thank the referees for some comments that allowed us to improve

the presentation of the paper. The first author would like to thank the Abdus Salam International

Centre for Theoretical Physics (ICTP) for providing excellent facilities for finalizing the paper

during his short visit to ICTP in July 2009.31

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