People’s Democratic Republic of Algeria
Ministry of Higher Education and Scientific Research
University of Mohammed Seddik Ben Yahya-Jijel
Faculty of Exact Sciences and Informatics
Departement of Mathematics
Dissertation
Presented to obtain the diploma of
Master
Specialty Applied Mathematics Option PDE and applications
Theme
A class of evolution variational inequalities
Presented by : Sarra Gaouir.
Board of Examiners :
Dr. Ilyas Kecis University Mohamed Seddik Ben Yahia. President Pr. Tahar Haddad University Mohamed Seddik Ben Yahia. Supervisor Dr. Wahiba khellaf University Mohamed Seddik Ben Yahia. Examiner
Acknowledgments
First of all and foremost, I thank ALLAH who helped me and gave me the strength, patience and courage during my years of study.
I warmly thank supervisor Pr. Haddad T ahar for his incredibly carfule and efficient guidance and I am greatly indebted for his understanding, patience, help, all advice and notes during my preparation of dissertation. I would like to say
”thank you for everything”.
I would like also to thank the members of jury for kindly devoting their precious time to read and evaluate this research.
Last but not least, big thanks to all my teachers of our departement, all my colleagues and all my beloved family.
Contents
Abstract 3
General Introduction 4
1 Preliminaries and auxiliary results 5
1.1 Prox-regular sets . . . 5
1.2 Stability . . . 7
1.3 Other results and theorems . . . 9
2 Integral Perturbation of Moreau’s Sweeping Processes 15 2.1 Technical Assumptions . . . 16
2.2 Integral Perturbation of Moreau’s Sweeping Processes . . . 16
2.2.1 Existence result . . . 17
2.2.2 Uniqueness and continuous dependence . . . 22
2.2.3 Stability result . . . 26
General Conclusion 31
Glossary of Notation
Glossary of Notation
Operations and Symbols
:= equal by definition
≡ identically equal
h·, ·i Scalair product in H
k · k or | · | norms
dH (C(t), C(s)) Pompieu-Hausdorff distance between sets
end of proof
Spaces
R := ]−∞, +∞[ real line
Rd d-dimensional Euclidean space
H real Hilbert space
AC the space of absolutely continuous functions
L1([0, T ], H) H-valued Lebesgue integrable functions over [0, T ]
σ(E, E0) weak topology
Sets
B closed unit ball
P rojC(x) projection of x to C
Np(C,x) proximal normal cone to C at x
Nconv(C, x) the normal cone to C at x in the sens of convex analysis
N (C, x) or NC(x) Clarke normal cone to C at x
Functions
d(·, C) or dC(·) distance function
∂dC(x) basic/limiting subdifferential of distance at x
Mappings
f : X −→ Y single-valued mappings from X to Y F : X ⇒ Y set-valued mappings from X to Y
Abstract
We establish, in the setting of finite dimensional, a result concerning the well posedness and the nature of Carath´eodory solutions for a sweeping process driven by a prox-regular moving set with an integral forcing term, where the integrand is Lipschitz with respect to the state variable.
R´esum´e
Nous ´etablissons, dans le cadre de la dimension finie, un r´esultat concernant la bonne position et la nature des solutions de Carath´eodory pour un processus de rafle conduit par un ensemble mobile prox-r´egulier avec un terme de for¸cage int´egral, o`u l’integrande est Lipschitz par rapport `a la variable d’´etat.
General Introduction
This dissertation deals with the Cauchy problem for the time dependent evolution inclusion ˙x(t) ∈ −NC(t)(x(t)) + Z t 0 f (s, x(s))ds a.e. in [0, T ] x(0) = x0 ∈ C(0),
where the state variable x belongs to H for each t ∈ [0, T ], and x(0) = x0 is the initial condition,
C(t) ⊂ H is a (mildly non-convex) moving set and f is Lipschitz with respect to x.
This dissertation is inspired first from the article of Colombo-Kozaily [6], which is only intersted the existence and uniqueness in the infinite dimensional Hilbert space of a solution. It is a detail of its results but using a different method with a generalization of some conditions in the finite dimensional. Finally, we will discuss the stability of the solution of the problem without the integration of the second part as follows :
˙x(t) ∈ −NC(t)(x(t)) − f (t, x(t)) a.e. in [0, T ]
x(0) = x0 ∈ C(0),
which is taken from the article of Wadippuli-Gudoshnikov-Makarenkov [19].
All the results remain correct in the infinite dimensional, which is noted in our research, only in the proof of existence the adopted method is based on the finite dimensional. For this we will take in all the paper H = Rd.
The dissertation is organized as follows. In chapter 2 : in section 1, for sake of readability we collect the hypotheses used along the dissertation. In section 2, we establish the existence, the uniqueness and the nature of a solution whenever the sets C(t) are r-prox-regular (r > 0) and varie in an absolutely continuous way and f is Lipschitz continuous with respect to the second variable on any bounded subset of H and which satisfies the natural growth condition
Chapter
1
Preliminaries and auxiliary results
This introductory section is to study prox-regular sets with constant thickness r and to recall some fundamental results related to the proximal normal cone. The stability of the solution has been well presented in this section. We also give some notions and theorems used in the other sections. Throughout the paper H is a real Hilbert space endowed with the inner product h·, ·i and k · k is the associated norm.
The keywords of this chapter are : Prox-regular set-Proximal normal cone- Stability-Integro-differential.
1.1
Prox-regular sets
Given a set C ⊂ H, associate with it the distance function d(x, C) = dC(x) := inf
z∈C
kx − zk, x ∈ H, and define the projector of x ∈ H to C by
P rojC(x) := {y ∈ C : dC(x) := kx − yk}.
The proximal normal cone enjoys a geometrical characterization (see, e.g., [5]) given by the equality
Np(C, x) = {v ∈ H : x ∈ P rojC(x + αv) for some α > 0}.
Definition 1.1.1. Let C ⊂ H be a closed set and r > 0. We say that C is r- prox-regular if for all x ∈ C and all v ∈Np(C, x) with kvk < 1, x is the unique nearest point of C to x + rv,
i.e.,
1.1. Prox-regular sets Here P rojC denotes the unique nearest point of C to x.
Remark 1.1.1.
(1) We notice that, for r-prox-regular sets, the proximal normal cone coincides (see Edmond-Thibault [9], Rockafellar-Wets [16]) with both the limiting normal cone ( also known as Mordukhovich normal cone, see [9]) and Clarke normal cone.
(2) The closed r-prox-regular set C is convex if and only if r = +∞, and in this case all normal cones coincide with Nconv(C, x) ( see Colombo-Thibault [7]) as shown in the
following :
Nconv(C, x) = {v ∈ H : hv, y − xi ≤ 0, ∀y ∈ C}.
(3) For x ∈ C (r-prox-regular sets) the proximal normal cone is also related to the distance function to C via the equality (see, e.g., [5, 2])
∂dC(x) = N (C, x) ∩ B.
Remark 1.1.2.
(1) The Clarke subdifferential of the distance function d(·, C) has values weak closed compact convex and upper semicontinuous, which proved (by using [15] the following proposition): Proposition 1.1.1. Let f : W → R be a locally lipschitz function on an open subset W and x ∈ W. Then
(a) The set-valeud mapping ∂f (·) is nonempty compact closed convex and upper semi-continuous on W for the weak topology on H.
(b) for all r ∈ R one has ∂(rf )(x) = r∂f (x).
Since the function dC(·) is Lipschitz continuous with constant l = 1 on Rd. So the results
stated above holde are realized. Indeed,
recall further that a mapping f : Rd→ Rp is Lipschitz continuous with constant l ≥ 0 on
some set C ⊂ Rd if we have the estimate
1.2. Stability Note that the Lipschitz continuity of f in (1.1).
For any z ∈ C, we have
dC(x) := inf z∈C
kx − zk which implies in turn that
∀ε > 0 ∃ z ∈ C; kx − zk ≤ dC(x) + ε
that implies
dC(y) ≤ ky − zk ∀y ∈ W
≤ ky − xk + kx − zk ≤ ky − xk + dC(x) + ε.
Similarly, we get dC(x) ≤ kx − yk + dC(y) + ε and thus |dC(x) − dC(y)| ≤ kx − yk + ε.
Letting ε → 0, we have
|dC(x) − dC(y)| ≤ kx − yk,
which justifies by (1.1) the Lipschitz continuity of dC(·) on Rd with constant l = 1.
(2) Since the set C is r-prox-regular, so all the subdiferential of distance function coincides. Definition 1.1.2. Let C ⊂ H be a closed set and r > 0 be given. We say that C is r-prox-regular provided the inequality
hζ, y − xi ≤ ky − xk
2
2r (1.2)
holds for all x, y ∈ C and all unit vectors ζ ∈ NC(x).
1.2
Stability
Stability theory plays a central role in systems theory and engineering. There are different kinds of stability problems that arise in the study of dynamical systems. This section is con-cerned mainly with stability of solution. We shall see the relationship between the solution of linear system and inclusion.
Definition 1.2.1. Let F : I ⇒ X. A function f : I → X will be said to be a selection of F if f (t) ∈ F (t) for every t.
1.2. Stability Consider a differential inclusion
˙x(t) ∈ F (t, x(t)),
where F : R × Rd → Rd is a set-valued map. The problem that we consider in this section is
to find a selection f (·, x) ∈ F (·, x), one has the differential equation ˙x(t) = f (t, x(t)).
Definition 1.2.2. Consider the Cauchy problem
˙x = f (t, x) (1.3)
with initial condition x(t0) = z0. It is assumed that the solution of this problem exists on
[t0, +∞[.
Definition 1.2.3. Let x be the maximal solution of (1.3) such as x(t0, z) = z.
• the solution x(t, z0) is stable if there exist a closed ball B(z0, r) and constant C ≥ 0 such
as
(i) for all z ∈ B(z0, r), t 7→ x(t, z)is well defined on [t0, +∞[;
(ii) for all z ∈ B(z0, r) and t ≥ t0 one has
kx(t, z) − x(t, z0)k ≤ Ckz − z0k.
• the solution x(t, z0) is asymptotically stable if it is stable and the following condition
is satisfied
(ii)0 there is a closed ball B(z0, r) and continuous function γ : [t0, +∞[→ R+ with
lim
t→+∞γ(t) = 0 such as for all z ∈ B(z0, r) and t ≥ t0 we have
kx(t, z) − x(t, z0)k ≤ γ(t)kz − z0k.
• the solution x(t, z0) is exponentially stable if it is asymptotically stable and there exists
positive constants α, β and δ such that if kx(t, z) − x(t, z0)k ≤ δ, then
kx(t, z) − x(t, z0)k ≤ αkz − z0ke−βt.
1.3. Other results and theorems Lemma 1.2.1. Let x be the solution of (1.3). Then the following implications are satisfied :
x is exponentially stable ⇓
x is asymptotically stable ⇓
x is stable. Proof. Just we use the Definition 1.2.3.
1.3
Other results and theorems
We will discuss the existence and uniqueness of the integro-differential equation of the forme ˙x(t) = g(t, x(t)) + Z t 0 k(t, s)f (s, x(s))ds a.e. in [0, T ] x(0) = x0. (1.4)
Theorem 1.3.1 ([1], Theorem 2.1.3). Let k : [0, T ] × [0, T ] → R and f, g : [0, T ] × Rd → Rd.
Suppose the following conditions hold :
t 7→ g(t, x) is mesurable for all x ∈ Rd (1.5)
there exists µ ∈ L1[0, T] with kg(t, x)k ≤ µ(t)
for a.e t ∈ [0, T] and all x ∈ Rd
(1.6)
t 7→ f (t, x) is mesurable for all x ∈ Rd (1.7)
there exists τ ∈ L1[0, T] with kf(t, x)k ≤ τ (t)
for a.e t ∈ [0, T] and all x ∈ Rd
(1.8) k : [0, T ] × [0, T ] → R is continuous and k(t) = sups∈[0,t]|k(t, s)| is bounded on [0, T ] (1.9)
1.3. Other results and theorems
for every x ∈ Rd there exists a neighborhood U x
of x ∈ Rd and a Lx ∈ L1[0, T] here Lx : [0, T] → [0, ∞[
with kg(t, x1) − g(t, x2)k ≤ Lx(t)kx1− x2k for every t ∈ [0, T] and x1, x2 ∈ Ux
(1.10) and
for every x ∈ Rd there exists a neighborhood V x
of x ∈ Rd and a Nx ∈ L1[0, T] here Nx: [0, T] → [0, ∞[
with kf (t, x1) − f (t, x2)k ≤ Nx(t)kx1− x2k for every t ∈ [0, T] and x1, x2 ∈ Vx.
(1.11)
Then there exists a unique solution x ∈ AC to (1.4).
Theorem 1.3.2. [4](Existence of measurable selection). Let X be a separable metric space, (I, Σ) a measurable space. F : I ⇒ X a multifunction with nonempty closed valued. If F is measurable, then it admits a measurable selection.
Right now Kakutani Fixed Point Theorem.
Theorem 1.3.3. ( [17], Theorem 6.4). If K is a nonvoid compact, convex subset of Rd and
Φ : K ⇒ K is a multifunction with compact, convex values and closed graph, then Φ has a fixed point.
Remark 1.3.1. Denot by Gr(Φ) the graph of Φ, which defines as Gr(Φ) := {(x, y) ∈ X ⇒ Y / y ∈ Φ(x)}.
Theorem 1.3.4. ( [4], Theorem VI-4) Let U be a topological space and let φ be a multi-function from [0, T ] ×U with non empty convex compact values in a Hausdorff locally convex space E such that for every t ∈ [0, T ], φ(t, ·) is upper semicontinuous. Let xn and x belongs to
U[0,T ] and y
n, y be scalarly ds-integrable mapping from [0, T ] to E. We assume the following
hypotheses :
(a) There exists a sequense (e0n) in E0 which separates the points of E (b) lim
1.3. Other results and theorems (c) For every fixed x0 ∈ E0, the sequence h x0, yn(·) i converges to h x
0
, y(·) i with respect to the weak topology σ L1[0, T ], L∞[0, T ].
(d) yn(t) ∈ φ t, xn.
Then y(t) ∈ φ t, x.
Remark 1.3.2. The arguments in the following results give us some measurable selections results which will be used in the proof of the existence of solutions of a multivalued differential equation.
Definition 1.3.1. (Equicontinuity) Consider a subset A ⊂ C(X). We call A equicontinuous if, ∀ > 0, ∃δ > 0 such that for x, y ∈ X :
d(x, y) < δ ⇒ |f (x) − f (y)| < ∀f ∈ A where d is the metric on X.
Theorem 1.3.5. ( [18])(The Arzela-Ascoli Theorem) If a sequence {xn} in C(X) is bounded
and equicontinuous then it has a uniformly convergent subsequence. Proposition 1.3.6. ( [14]) Let E is normed space we have :
(1) E0 separable ⇔ B is metrizable in σ(E, E0). (2) B is weakly compact.
Theorem 1.3.7. ( [8], Theorem 3 )(Dominated Convergence Theorem (DCT)). Let (Ω, Σ, µ) be a finite measure space and (fn) be a sequence of Bochner integrable X valued functions on Ω.
If lim
n fn= f in µ-measurable and if there exists a real-valued Lebesgue integrable function g on
Ω with kfnk ≤ g µ-almost every where, then f is Bochner integrable and lim n Z E fndµ = Z E f dµ for each E ∈ Σ.
Lemma 1.3.8. (Gronwall-Bellman)( [13], Lemma A.1 ) Let an absolutely continuous function a : [τ, T ] → R satisfies
˙a(t) ≤ λa(t) + b(t), for a.e. t ∈ [τ, T],
where τ ≤ T and λ ∈ R are constants, and b : [τ, T ] → R is an integrable function. Then a(t) ≤ eλta(τ ) +
Z t
τ
1.3. Other results and theorems Lemma 1.3.9. Let r : [0, T ] → R be a nonnegative absolutely continuous function and let K1, K2 > 0. Suppose that r(0) ≤ η and, for some ε > 0,
˙r(t) ≤ ε + K1r(t) + K2 p r(t) Z t 0 p r(s) ds ∀t ∈ [0, T ]. (1.12) Then r(t) ≤ 2(η + ε)e(max{K1,K2}+2)t+ e (max{K1,K2}+2)2t − 12 2(K + 1)2 ε ∀t ∈ [0, T ].
Proof. Assumption (1.12) implies
˙r(t) ≤ ε + K1(r(t) + ε) + K2 p r(t) + ε Z t 0 p r(s) + ε ds ∀t ∈ [0, T ]. (1.13) Setting zε(t) :=pr(t) + ε ⇒ ˙zε(t) = ˙r(t) 2pr(t) + ε = ˙r(t) 2zε(t) ⇒ ˙r(t) = 2zε(t) ˙zε(t) and z2
ε(t) := r(t) + ε, one obtains from (1.13)
2zε(t) ˙zε(t) ≤ ε + K1zε(t)2+ K2zε(t) Z t 0 zε(s) ds, implies ˙zε(t) ≤ ε 2zε(t) +1 2 K1zε(t) + K2 Z t 0 zε(s) ds, set again vε(t) := Z t 0 zε(s) ds ⇒ ˙vε(t) := zε(t) and .. vε(t) := ˙zε(t). Then .. vε(t) ≤ ε 2 ˙vε(t) +1 2 K1˙vε(t) + K2vε(t)
like that r is nonnegative,√ε ≤pr(t) + ε := zε(t) := ˙vε(t) ⇒
1 ˙vε(t) ≤ √1 ε implies .. vε(t) ≤ √ ε 2 + 1 2 K1˙vε(t) + K2vε(t) ≤ √ ε 2 + K( ˙vε+ vε), where K = max{K1 2 , K2
2 }. Set at last wε(t) := ˙vε(t) + vε(t). It follows from the above inequality
that, for all t ∈ [0, T ], ˙ wε(t) := .. vε(t) + ˙vε(t) ≤ √ ε 2 + (K + 1) ˙vε(t) + Kvε(t) ≤ √ ε 2 + (K + 1)wε(t). Applying Gronwall’s Lemma to wε(t)(with a ≡ wε , λ ≡ K + 1 and b ≡
√ ε
2 ) and recalling that wε(0) ≤ √ η + ε because wε(0) = ˙vε(0) + vε(0) = zε(0) =pr(0) + ε ≤ √ η + ε, one obtains wε(t) ≤ e(K+1)twε(0) + √ ε 2 Z t 0 e(K+1)(t−s)ds,
1.3. Other results and theorems implies wε(t) ≤ √ η + ε e(K+1)t+e (K+1)t− 1 K + 1 √ ε 2 .
By observing that both vε and ˙vε are nonnegative, so
p r(t) + ε := vε(t) ≤ ˙vε(t) + vε(t) := wε(t) ≤ √ η + ε e(K+1)t+ e (K+1)t− 1 K + 1 √ ε 2 ⇒ r(t)+ε ≤√η + ε e(K+1)t+e (K+1)t− 1 K + 1 √ ε 2 2 (a+b)2≤2(a2+b2) ≤ 2(η+ε)e2(K+1)t+(e (K+1)t− 1)2 2(K + 1)2 ε. Then r(t) ≤ 2(η + ε)e(max{K1,K2}+2)t+ e (max{K1,K2}+2)2t − 12 2(K + 1)2 ε ∀t ∈ [0, T ].
Lemma 1.3.10. Let x(·) ∈ AC, then d dtkx(t)k 2 = 2 ˙x(t), x(t) , a.e. t ∈ [0, T ]. Proof. d dtkx(t)k 2 = lim h→0 kx(t + h)k2− kx(t)k2 h = lim h→0 hx(t + h), x(t + h)i − kx(t)k2 h = lim h→0 x(t + h) − x(t) + x(t), x(t + h) − x(t) + x(t) − kx(t)k2 h = lim h→0 kx(t + h) − x(t)k2+ 2x(t + h) − x(t), x(t) + kx(t)k2− kx(t)k2 h = 2 ˙x(t), x(t) .
Definition 1.3.2. ( Cauchy-Schwarz inequality (C-S)). Let X be a inner product space. Then : ∀(x, y) ∈ X2, |hx, yi| ≤ kxk · kyk·
Definition 1.3.3. Let f : R × Rd→ R
(i) f is monotone on Rd, in the sense that for all x
1, x2 ∈ Rd,
1.3. Other results and theorems (ii) f is strongly monotone on Rd, in the sense that for all x
1, x2 ∈ Rd, x1 6= x2,
hf (t, x1) − f (t, x2), x1− x2i ≥ ηkx1− x2k2 for all t ∈ R, for some fixed η > 0. (1.15)
Lemma 1.3.11. ( [3], Lemma 1.4) (A Gronwall-like inequality.) Let α, β, r : [0, T ] → [0, ∞[ be three non-negative Lebesgue integrable functions such that for almost all t ∈ [0, T ]
r(t) ≤ α(t) + β(t) Z t 0 r(s)ds. Then r(t) ≤ α(t) + β(t)α(s) Z t 0 a(s)eRstβ(τ )dτds for all t ∈ [0, T ].
Chapter
2
Integral Perturbation of Moreau’s Sweeping
Processes
In this chapter, we are going to prove a well posedness result and we study the nature of Carath´eodory solutions for a sweeping process driven by a prox-regular moving set with an integral forcing term, where the integrand is Lipschitz with respect to the state variable.
The first part of this chapter is devoted to the existence of perturbed sweeping processes. That is, we consider the Cauchy problem for the time dependent evolution inclusion
˙x(t) ∈ −NC(t)(x(t)) + Z t 0 f (s, x(s))ds a.e. in [0, T ] x(0) = x0 ∈ C(0), (2.1)
where the state variable x belongs to H, C(t) ⊂ H is a (mildly non-convex) moving set and f is Lipschitz with respect to x. The dynamics can be seen as an integral perturbation of the so called sweeping process, namely the differential inclusion
˙x(t) ∈ −NC(t)(x(t)) a.e. in [0, T ] x(0) = x0 ∈ C(0), (2.2)
that was introduced by J.J.Moreau in the Seventies.
The second part of the chapter we will prove uniqueness and continuous dependence. Finally, we will study the stability of solution to integral perturbed sweeping processes and we analyze the conditions that made stability or not.
2.1. Technical Assumptions
2.1
Technical Assumptions
For the sake of readability, in this section we collect the hypotheses used along the chapter. Through this chapter H = Rd, Ω ⊂ H be open and such that C(t) ⊂ Ω for all t ∈ [0, T ].
Hypotheses on the set-valued map C : [0, T ] ⇒ H: C is a set-valued map with nonempty and closed values. Moreover, we will consider the following conditions :
(C1) C(t) varies in an absolutely continuous way, i.e. there exists an absolutely continuous
function ζ : [0, T ] → R, which is monotone increasing and
dH(C(t), C(s)) ≤ |ζ(t) − ζ(s)| for all s, t ∈ [0, T];
(C2) There exists r > 0 such that C(t) is r-prox-regular for each t ∈ [0, T ].
Remark 2.1.1. By the definition of the Hausdorff distance dH(C(t), C(s)) := sup
x∈H
|dC(t)(x) − dC(s)(x)|,
we can relate (C1) like that :
|dC(t)(x) − dC(s)(x)| ≤ |ζ(t) − ζ(s)| for all x ∈ H and all s, t ∈ [0, T].
Hypotheses on the valued mappings f : [0, T ] × Ω → H : f is a the single-valued mappings. Moreover, we will consider the following conditions :
(f0) f (·, x) is measurable for all x ∈ Ω;
(f1) there exists non-negative function β(·) ∈ L2([0, T ], R) such that for a.e.t ∈ [0, T ]and all
x ∈ Ω
kf (t, x)k ≤ β(t);
(f2) there exists non-negative function γ(·) ∈ L2([0, T ], R) such that for a.e.t ∈ [0, T ]and all
x, y ∈ Ω
kf (t, y) − f (t, x)k ≤ γ(t)ky − xk.
2.2
Integral Perturbation of Moreau’s Sweeping Processes
In this section, we will prove a well posedness result for Carath´eodory solutions for a sweeping process (2.1) and its nature.
2.2. Integral Perturbation of Moreau’s Sweeping Processes Theorem 2.2.1. Let the assumptions stated above hold. Then the Cauchy problem (2.1) ad-mits one and one Carath´eodory solution, that is defined on [0, T ], is Lipschitz and depends continuously (actuallyLipschitz) on the initial datum x0.
Proof. We will demonstrate this Theorem in the form of the following subsection :
2.2.1
Existence result
In this subsection we will prove the existence of solutions of (2.1). The basic idea is to look for a problem that is equivalent (2.1) and this problem is a special case of another one who has the existence. As shown in the following steps :
step 1 :
Theorem 2.2.2. Under assumption (C1), (C2),(f0),(f1) and (f2). The original problem
(2.1) is equivalent to the following problem : ˙x(t) ∈ − | ˙ζ(t)| + kz(t)k∂dC(t)(x(t)) + z(t) a.e. in [0, T ] x(0) = x0 ∈ C(0), (2.3) where z(t) := Z t 0 f (s, x(s))ds.
Proof. According to the problem (2.2) which is equivalent to the following problem ˙x(t) ∈ −| ˙ζ(t)|∂dC(t)(x(t)) a.e. in [0, T ] x(0) = x0 ∈ C(0). (2.4)
See Haddad-Kecis-Thibault [10], just take L = 0 and C(t, x) ≡ C(t). Then assume t 7→ x(t) as solution of (2.1), and we put z(t) =
Z t 0 f (s, x(s))ds then (2.1) becomes ˙x(t) − z(t) ∈ −NC(t)(x(t)) a.e. in [0, T ] x(0) = x0 ∈ C(0),
we put ˙y(t) = ˙x(t) − z(t) ⇒ y(t) = x(t) − Z t
0
z(s)ds, with y(0) = x(0) ∈ C(0) i.e, ˙
y(t) ∈ −NC(t)(y(t) +
Z t
0
z(s)ds ) such as
2.2. Integral Perturbation of Moreau’s Sweeping Processes Indeed, let ζ ∈ NC+a(x + a) ⇔ hζ, y − x − ai ≤ ky − x − ak2 2r ∀x ∈ C, ∀y ∈ C + a, we put y = c + a we get hζ, c + a − x − ai ≤ kc + a − x − ak 2 2r ∀c, x ∈ C, then hζ, c − xi ≤ kc − xk 2 2r ∀c, x ∈ C. Equivalently, ζ ∈ NC(x).
For each t ∈ [0, T ] let us set ψ(t) :=
Z t
0
z(s)ds and D(t) := C(t) − ψ(t).
Obviously, the set-valued map D(·) satisfies (C2). Now, let x ∈ H and t, s ∈ [0, T ],
one has d(x, D(t)) − d(x, D(s)) = d(x, C(t) − ψ(t)) − d(x, C(s) − ψ(s)) = d(x + ψ(t), C(t)) − d(x + ψ(s), C(s))
≤ kψ(t) − ψ(s)k + |ζ(t) − ζ(s)| ( because dC(·) is Lipschitz and C1)
≤ |V (t) − V (s)|, where V (t) := Z t 0 | ˙ζ(s)| + kz(s)kds.
Hense D(·) satisfies also (C1) with the absolutely continuous function V (·).
As x0 ∈ C(0) = D(0), we know that the following sweeping process
˙
y(t) ∈ −ND(t)(y(t)) a.e.t ∈ [0, T ]
y(0) = x0.
Then according to (2.4) y(·) satisfies also the inclusion ˙
y(t) ∈ −| ˙V (t)|∂dD(t)(y(t)) a.e. t ∈ [0, T ]
⇔ ˙x(t) ∈ −| ˙V (t)|∂dC(t)(x(t)) +
Z t
0
2.2. Integral Perturbation of Moreau’s Sweeping Processes Thus,
k ˙y(t)k ≤ | ˙V (t)| = | ˙ζ(t)| + kz(t)k a.e. t ∈ [0, T ] (2.5) by (2.5), we obtain the estimation
k ˙x(t) − z(t)k ≤ kz(t)k + | ˙ζ(t)| a.e. t ∈ [0, T ], one has
k ˙x(t)k ≤ k ˙x(t) − z(t)k + kz(t)k ≤ 2kz(t)k + | ˙ζ(t)| a.e. t ∈ [0, T ]. (2.6) step 2 :
As the convex and closed compact valued multifunction x 7→ −| ˙V (t)|∂dC(t)(x(·)) is upper
semicontinuous on H and mesurable with respect to t ( follow respectively, from Remark 1.1.2 or [11] and [12] ). We notice that (2.3) is a special case of the following general problem : ˙x(t) ∈ F (t, x(t)) + Z t 0 f (s, x(s))ds a.e. in [0, T ] x(0) = x0. (2.7)
With : F : [0, T ] × H ⇒ H be L([0, T ]) ⊗ B(H)-measurable set-valued mapping with nonempty convex and closed compact valued and such that for each t ∈ [0, T ] the set-valued maping F (t, ·) is upper semicontinuos. Assume that F is integrably bounded, that is, there exists σ ·) ∈ L2([0, T ], [0, +∞[ such that
F (t, x) ⊂ T (t) := σ(t)B, ∀(t, x) ∈ [0, T ] × H.
Theorem 2.2.3. The convex differential inclusion with integral perturbed (2.7) has a solution x(·) ∈ AC, for which there is a lebesgue mesurable selection ϕ of F (·, x(·)) such that kϕ(t)k ≤ σ(t) and ˙x(t) = ϕ(t) +
Z t
0
f (s, x(s))ds for a.e. t ∈ [0, T ].
Proof. Recall that a AC function x : [0, T ] → H is solution of the problem (2.7) as F is nonempty closed measurable valued mapping then it admits a mesurable selection by virtue of ( Theorem 1.3.2). Let us denote by ST2 the set of all L2([0, T ], H)-selection of T
S2
T := {g ∈ L2([0, T ], H) : g(t) ∈ T (t) a.e. t ∈ [0, T ]}.
By virtue of ( Theorem 1.3.1), (just take k = 1 and g(t, x(t)) ≡ g(t)) for each g ∈ ST2, we find
2.2. Integral Perturbation of Moreau’s Sweeping Processes (i) k(t, s) ≡ 1 is a constant f unction, then it is continuous and bounded on [0, T ], so (1.9)
is checked
(ii) g(t, x(t)) ≡ g(t), which mesurable and L2([0, T ], H) integrable, so (1.5), (1.6) and (1.10) are checked
(iii) according to (f0) , (f1) and (f2), so (1.7), (1.8) and (1.11) are checked.
Then, there is a unique AC solution xg of
˙xg(t) = g(t) + Z t 0 f (s, xg(s))ds a.e. in [0, T ] xg(0) = x0. (2.8) For each g ∈ S2
T, let us define the multifunction
Ψ(g) := {h ∈ L2([0, T ], H) : h(t) ∈ F (t, xg(t)) a.e. t ∈ [0, T ]}.
As F is a convex compact then it is clear that Ψ(g) is a nonempty convex weakly compact subset of S2
T, here the nonemtiness follows from ([4], Theorem VI-6). From the above
consideration, we need to prove that the convex weakly compact valued mapping
Ψ(g) : ST2 ⇒ ST2 admits a fixed point. As L2([0, T ], H) is separable, ST2 is compact metrizable with respect to the weak topology of L2([0, T ], H). By virtue of Kakutani
Theorem, it is enough to prove that the graphe Gr(Ψ) is sequentially weakly closed in S2
T × ST2. Let (hn, gn) ∈ Gr(Ψ) weakly converging to (h, g) ∈ ST1 × ST1. We have to verify
that h ∈ Ψ(g). The proof’s idea is given by virtue of ( Theorem 1.3.4). From the definition xgn is the AC solution of
˙xgn(t) = gn(t) + Z t 0 f (s, xgn(s))ds a.e. in [0, T ] xgn(0) = x0, (2.9)
with gn ∈ ST1 and hn ∈ F (t, xgn(t)). Then
kgnkL2([0,T ],H) ≤ kσkL2([0,T ],H); ∀n. (2.10)
Now, we show ˙xgn is bounded in L
2([0, T ], H), i.e; k ˙x gnkL2([0,T ],H)≤ constante; ∀n by (f1), (2.9) and C-S we have k ˙xgn(t)k ≤ kgn(t)k + Z T 0 β(s)ds ≤ kgn(t)k + √ T kβkL2([0,T ],H)
2.2. Integral Perturbation of Moreau’s Sweeping Processes and so k ˙xgn(t)k 2 ≤ kg n(t)k2+ T kβk2L2([0,T ],H)+ 2 √ T kgn(t)kkβkL2([0,T ],H) so Z T 0 k ˙xgn(t)k 2dt ≤ kg n(t)k2L2([0,T ],H)+ T2kβk2L2([0,T ],H)+ 2 √ T kβkL2([0,T ],H) Z T 0 kgn(t)kdt by (2.10) we get k ˙xgn(t)k 2 L2([0,T ],H) ≤ M ; ∀n where M := kσk2L2([0,T ],H)+ T2kβk2L2([0,T ],H)+ 2T |βkL2([0,T ],H)kσkL2([0,T ],H) is a constant. Therefore k ˙xgn(t)kL2([0,T ],H)≤ √ M ; ∀n. (2.11)
This implies that
kxgn(t)k ≤ kx0k + Z T 0 k ˙xgn(t)kdt ≤ kx0k + √ T k ˙xgnkL2([0,T ],H) ≤ kx0k + √ T√M setting R := kx0k + √ T√M < +∞, hence kxgn(t)k ≤ R; ∀n,
we conclude that, for all t ∈ [0, T ] xgn(t); n ≥ 1 is relatively compact in H. On the other
hand, the sequence(xgn)n is equicontinuous because : for all s ∈ [0, T ]; (s ≤ t), we have
kxgn(t) − xgn(s)k = k Z t s ˙xgn(z)dzk ≤ Z t s k ˙xg(z)kdz ≤ k ˙xgnkL2([0,T ],H) √ t − s ≤√M√t − s; ∀n.
Applying the Ascoli-Arz´ela Theorem, we conclude that (xgn) converge informley and
2.2. Integral Perturbation of Moreau’s Sweeping Processes Also by (2.11) we have ˙xgn converges weakly to a function v in L
2([0, T ], H) so x(t) := lim n xgn(t) = limn xgn(0) + Z t 0 ˙xgn(s))ds = lim n x0+ Z t 0 ˙xgn(s))ds = x0+ Z t 0 v(s)ds; ∀t ∈ [0, T ] i.e; x is AC and ˙x = v a.e.t ∈ [0, T ].
Observe that, for every s ∈ [0, T ], lim
n f (s, xgn(s)) = f (s, x(s)).
By continuity of f (s, ·) ( because f is lipschitz compared to x by (f2) ) and xgn converge
to x. In addition by (f0), (f1) we can aplly (DCT) to obtain
lim n Z t 0 f (s, xgn(s))ds = Z t 0 f (s, x(s))ds. ∀t ∈ [0, T ].
So, we obtain the convergence in L2([0, T ], H) of
Z t 0 f (s, xgn(s))ds = ˙xgn(t) − gn(t) to Z t 0 f (s, x(s))ds, and then ˙x(t) − g(t) = Z t 0 f (s, x(s))ds a.e. in [0, T ] x(0) = x0,
by uniqueness we get x := xg. As hn ∈ F (t, xgn(t)) a.e. And invoking the ( Theorem
1.3.4) we get h ∈ F (t, xg(t)), and hense (h, g) ∈ Gr(Ψ) a.e. The proof is therefore
complete.
So we conclude the problem (2.3) has at least solution.
Remark 2.2.1. In order to prove the uniqueness of the problem (2.1) we take γ(t) as constant (we will put γ(t) ≡ γ) as following :
2.2.2
Uniqueness and continuous dependence
Let x1, x2 be solutions of
˙x(t) ∈ −NC(t)(x(t)) +
Z t
0
2.2. Integral Perturbation of Moreau’s Sweeping Processes with x1(0) = u1, x2(0) = u2. We have, for a.e.
− ˙xi(t) + Z t 0 f (s, xi(s))ds ∈ NC(t)(xi(t)) ∀i = 1, 2. We put ζi := − ˙xi(t) + Z t 0
f (s, xi(s))ds ∀i = 1, 2. And invoking the (Definition 1.1.2) that
hζ1, x2− x1i ≤ kx2− x1k2 2r (with y = x2 and x = x1) (2.12) and hζ2, x1− x2i ≤ kx2− x1k2 2r (with y = x1 and x = x2) ⇔ h−ζ2, x2− x1i ≤ kx2− x1k2 2r (2.13) (2.12) +(2.13) gives hζ1− ζ2, x2− x1i ≤ kx2− x1k2 r ∀ ζi ∈ NC(t)(xi(t)) ∀i = 1, 2, i.e; − ˙x1(t) + Z t 0 f (s, x1(s))ds + ˙x2(t) − Z t 0 f (s, x2(s))ds, x2(t) − x1(t) ≤ kx2(t) − x1(t)k 2 r from which we obtain
˙x2(t)− ˙x1(t), x2(t)−x1(t) ≤ Z t 0 f (s, x2(s))ds− Z t 0 f (s, x1(s))ds, x2(t)−x1(t) +kx2(t) − x1(t)k 2 r C−S ≤ Z t 0 f (s, x2(s)) − f (s, x1(s)) ds x2(t) − x1(t) + kx2(t) − x1(t)k2 r ≤ Z t 0 f (s, x2(s)) − f (s, x1(s)) ds x2(t) − x1(t) + kx2(t) − x1(t)k2 r .
By the condition (f2) we obtain
˙x2(t)− ˙x1(t), x2(t)−x1(t) ≤ γ Z t 0 x2(s)−x1(s) ds x2(t)−x1(t) +kx2(t) − x1(t)k 2 r .
Applying Lemma(1.3.10), one obtains d dt 1 2kx2(t) − x1(t)k 2 ≤ γ x2(t) − x1(t) Z t 0 x2(s) − x1(s) ds + kx2(t) − x1(t)k2 r (2.14) setting r(t) := 1 2kx2(t) − x1(t)k
2, one obtaines from (2.14)
˙r(t) ≤ γp2r(t) Z t 0 p 2r(s)ds + 2 rr(t) = K1r(t) + K2 p r(t) Z t 0 p r(s)ds,
2.2. Integral Perturbation of Moreau’s Sweeping Processes
where K1 =
2
r and K2 = 2γ. Invoking Lemma(1.3.9) with η = r(0) and ε arbitrary, we obtain r(t) ≤ 2(r(0) + ε)e2(K+1)t+ e (K+1)t− 12 2(K + 1)2 ε ∀t ∈ [0, T ], (2.15) where K = max{K1 2 , K2
2 }, it remains to observe that x(t) is the only global solution.
Finally, by (2.6) and (2.15), we obtain the solution of (2.1) is Lipschitz and depends contin-uously on the initial datum x0 x(t) = x0+
Z t 0
˙x(s)ds and all the data. Which completes the
proof of Theorem 2.2.1.
Remark 2.2.2. The above Theorem is a generalization of the Theorem 3.1 [6] ; just we take a special cases of the previous conditions as follow :
(1) we keep (f0), (C2) as they are
(2) in the conditions (f1), (f2) we take the functions β(·), γ(·) as constants
(3) in the condition (C1) we take ζ(t) = αt.
Remark 2.2.3. In the next Lemma, we need it to prove a new theorem of the existence of the solutions for (2.1) with a change in the condition (f1).
Lemma 2.2.4. Assume that (C1), (C2), (f0) and (f2) are satisfied and let f satisfies the next
condition : (f1)
0
there exists two non-negative functions p(·), q(·) ∈ L2([0, T ], R) such that for a.e.t ∈ [0, T ] and all x ∈ Ω
kf (t, x)k ≤ p(t) + q(t)kxk,
if x(·) is an absolutely continuous solution to the problem (2.1). Then for almost all t ∈ [0, T ] k ˙x(t)k ≤ a(t) + b(t), where a(t) = | ˙ζ(t)| + 2T (p(t) + q(t)kx0k) and b(t) = 2T q(t) Z t 0 a(s)e2TRstq(τ )dτds.
Proof. Suppose that x(·) is an absolutely continuous solution of (2.1). By (f1)
0
one has f is Lebesgue-integrable on [0, T ] because p, q are integrable and x is bounded on [0, T ]. In addition by the assumption stated above hold we can apply the (2.6) in the Theorem 2.2.1 to obtain
2.2. Integral Perturbation of Moreau’s Sweeping Processes using the condition (f1)
0 we get k ˙x(t)k ≤ ˙ζ(t)| + 2T p(t) + 2T q(t)kx(t)k = | ˙ζ(t)| + 2T p(t) + 2T q(t) x0+ Z t 0 ˙x(s)ds ≤ | ˙ζ(t)| + 2T p(t) + 2T q(t)kx0k + 2T q(t) Z t 0 k ˙x(s)kds, by setting a(t) := | ˙ζ(t)| + 2T p(t) + 2T q(t)kx0k we have
k ˙x(t)k ≤ a(t) + 2T q(t) Z t
0
k ˙x(s)kds.
We proceed now to applying a Gronwall-like inequality we obtain the estimation k ˙x(t)k ≤ a(t) + 2T q(t) Z t 0 a(s)e2TRstq(τ )dτds | {z } b(t) , hence k ˙x(t)k ≤ a(t) + b(t).
Remark 2.2.4. The conditions of the next theorem are the same conditions for the Theorem 2.2.1, just we replace (f1) by (f1)
0
.
Theorem 2.2.5. Let the assumptions stated above hold. Then, for each x0 ∈ C(0), there is an
absolutely continuous solution x : [0, T ] → H for the problem (2.1) and for any solution x(·) one has
k ˙x(t)k ≤ a(t) + b(t),
for almost all t ∈ [0, T ], where a(t) and b(t) are given in Lemma 2.2.4. Proof. Suppose f satisfies (f0), (f2) and (f1)
0
. For a(t) and b(t) given by Lemma 2.2.4, put γ(t) := kx0k +
Z t 0
(a(s) + b(s))ds. Let us consider the mapping Π : [0, T ] × H → H with
Π(t, x) = x if kxk ≤ γ(t) γ(t) x kxk if kxk > γ(t),
and put f0(t, ·) = f (t, Π(t, x)). Then f0(t, x) inherits the conditions (f0), (f2) from f (·, x) and
we have
2.2. Integral Perturbation of Moreau’s Sweeping Processes using the condition (f1)
0 we get kf0(t, x)k ≤ p(t) + q(t)kΠ(t, x)k ≤ p(t) + q(t)γ(t). By setting β(t) := p(t) + q(t)γ(t) we have kf0(t, x)k ≤ β(t) ∈ L2([0, T ]) ∀x ∈ H,
hence f0 satisfies all the conditions of the Theorem 2.2.1. Then x(·) is the solution of the
problem
˙x(t) ∈ −NC(t)(x(t)) +
Z t
0
f0(s, x(s))ds a.e, x(0) = x0. (2.16)
And we must justify x(·) is a solution to (2.16) if and only if x(·) is a solution to (2.1). Firstly, we justify that x(·) is a solution to (2.16), then x(·) is a solution to (2.1).
Let x(·) the solution of (2.16), i.e;
˙x(t) ∈ −NC(t)(x(t)) + Z t 0 f (s, Π(t, x(s)))ds specially if kxk ≤ γ(t) we find : ˙x(t) ∈ −NC(t)(x(t)) + Z t 0 f (s, x(s))ds.
Second, we justify that x(·) is a solution to (2.1), then x(·) is a solution to (2.16) .
Let x(·) the solution of (2.1) and according to the Lemma 2.2.4 we have k ˙x(t)k ≤ a(t) + b(t) so always in this case kx(t)k ≤ kx0k +
Z t
0
(a(s) + b(s))ds := γ(t) then f (t, x) = f0(t, x) i.e x(·) is
the solution of (2.16).
2.2.3
Stability result
In this subsection, we will study the nature of solution of the problem (2.1); we analyze the conditions that made stability or not.
Remind : In [19], it was shown that f satisfies the strong monotonicity assumption(1.15) with
η > α + β r ,
then the corresponding perturbed Moreau sweeping process is given as :
2.2. Integral Perturbation of Moreau’s Sweeping Processes has at least one solution 7→ x(t) is globally exponentially stable.
And the problem (2.1) given as :
˙x ∈ −N (C(t), x) − − Z t
0
f (s, x(s))ds. Remark 2.2.5.
(1) We want to prove the stability of the solution of the problem (2.1), it is better to use the Definition 1.2.3 but in the previous definition, the conditions to find the stability are neccessary. So this is why in the proof of uniqueness (2.15) we can’t judge or conclude because there are two random solutions are lower in norm with an exponentiality which is increasing. Then this is what makes us think about the second point.
(2) In [19], they have shown the solution of (2.17) is exponentially stable if we assume f is strongly monotone. In addition, they indicated that the less of the degree of monotony lead to the less of the degree of stability. For this we assume that in the absence of a monotone condition may cause the falling into the solution unstable and we ask if this condition only neccessary or more sufficient?
The basic idea is : to understand the efficiency of the monotonicity condition. We Consider the following examples :
Example 2.2.1. In (2.1) let f (t, x(t)) := x(t) and C(t) := B(0, 1). Then the normal cone given by NB(0,1)(x) = R+x if kx(t)k = 1 {0} if kx(t)k < 1, we have two cases :
(1) if kx(t)k < 1,
one has (2.1) equivalence
˙x(t) = Z t
0
x(s)ds, the characteristic equation is :
r2− 1 = 0, then
2.2. Integral Perturbation of Moreau’s Sweeping Processes with x(0) = x0 and ˙x(0) = 0, then x(t) =x20(e−t+ et) it is clear that
kx(t) − y(t)k = (et+e−t
2 )kx0 − y0k and limt→+∞( et+e−t
2 ) = +∞, hence the solution x(·) is
unstable. (2) If kx(t)k = 1,
one has (2.1) equivalence the scalar integro-differential equation : ˙x(t) = −λx(t) +
Z t
0
x(s)ds, λ > 0, the characteristic equation is :
r2+ λr − 1 = 0, we have the discriminant is positive, ∆ = λ2+ 4 > 0.
Then x(t) = c1er1t+ c2er2t with x(0) = x0 and ˙x(0) = −λx0, we get
x(t) = x0 r2− r1 (r2+ λ)er1t− (r1+ λ)er2t, where r1 = −λ− √ ∆ 2 < 0 and r2 = −λ+√∆ 2 > 0,
then it is clear lim
t→+∞x(t) = +∞, hence the solution x(·) is unstable.
Example 2.2.2. If the same gives a first example; the difference is f (t, x(t)) := -x(t) (the ”-” sign).
Also, we have two cases : (1) If kx(t)k < 1,
one has (2.1) equivalence
˙x(t) = − Z t
0
x(s)ds, the characteristic equation is :
r2+ 1 = 0
then x(t) = c1cos(t) + c2sin(t) with x(0) = x0 and ˙x(0) = 0, then x(t) =x0cos(t) it is
2.2. Integral Perturbation of Moreau’s Sweeping Processes (2) If kx(t)k = 1,
one has (2.1) equivalece the scalar integro-differential equation : ˙x(t) = −λx(t) −
Z t
0
x(s)ds, λ > 0, the characteristic equation is :
r2+ λr + 1 = 0, the discriminant ∆ = λ2 − 4 = (λ − 2)(λ + 2). So,
(i) if λ < 2 ⇒ ∆ < 0,
we have r = ±i√4 − λ2 := ±iβ, then x(t) = c
1cos(βt) + c2sin(βt),
with x(0) = x0 and ˙x(0) = −λx0,then x(t) = x0cos(βt) − βλx0sin(βt) it is clear that
kx(t)k ≤ Cx0, hence the solution x(·) is stable.
(ii) If λ > 2 ⇒ ∆ > 0, we have x(t) = c1er1t+ c2er2t, then x(t) = x0 r1− r2 (r1+ λ)er2t− (r2+ λ)er1t, where r1 = −λ −√∆ 2 and r2 = −λ +√∆ 2 ,
it is clear that r1 and r2 are strictly negative, then kx(t)k ≤ Cx0er2t ∀t ∈ [0, T ],
hence the solution x(·) is exponentially stable. (iii) If λ = 2 ⇒ ∆ = 0,
we have
x(t) = (c1t + c2)ert,
where r = −λ
2 = −1 and x(0) = x0, ˙x(0) = −λx0, then
x(t) = x0((1 − λ)t + 1)e−t = x0(1 − t)e−t it is clear that kx(t)k ≤ x0e−t ∀t ∈ [0, T ],
hence the solution x(·) is exponentially stable.
We summarize the solution of the first example is unstable, but the second is at least stable. The question is : what is the reason?.
The answer is : we notice that the difference related to the function ”f (·, x(·))”; we are inspired by Wadippul-Gudoshnikov-Makarenkov [19] the possibility problem of monotony. We will take care of the first case kx(t)k < 1 and the same for the second case.
2.2. Integral Perturbation of Moreau’s Sweeping Processes So we notice that in the first example
− Z t 0 f (s, x(s))ds = x0 2 (e −t− et)
is monotone for all t ∈ [0, T ].
And the second example we find the solution stable but the −
Z t
0
f (s, x(s))ds = x0sin(t)
is not monotone every t ∈ [0, T ].
So it is now possible to answer our note in the Remark 2.2.5; there is an unstable solution (Example 2.2.1) of the problem (2.1) and the monotone condition is necessary for obtaining the stability, but it is not sufficient ( Example 2.2.2 is the counter example ) i.e, the monotone condition guarantees stability, but the stability does not mean that monotone condition exist.
Conclution.
We can generalized the result of Wadippul-Gudoshnikov-Makarenkov for founding stability of solution with the problem (2.1). And the nature of the solution of (2.1) we can only say that it is bounded and its stability is not fixed.
General Conclusion
In this dissertation we proved the existence of at least one solution to a nonconvexe sweeping process with lipschitz integral perturbations, using the Kakutani Fixed Point Theorem in the finite dimensional. The uniqueness of the solution follows when the vector field of the sweeping process is uniformly bounded and its stability is not fixed.
In the future, we will study the existence of the solution of our problem, using any type of fixed point theorems in the infinite dimensional. Also we will discuss the sufficient conditions to assure always the stability and we see its applications · · ·
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