ﻢﻴﻠﻌﺘﻟا ةرازو
ﱄﺎﻌﻟا
ﻲﻤﻠﻌﻟا ﺚﺤﺒﻟاو
Ministère de l’enseignement supérieur et de la recherche scientifique
Université Djillali Liabes
Sidi-Bel-Abbes
Djillali Liabes University
Sidi-Bel-Abbes
ﺔﻌﻣﺎﺟ
سبﺎيل يلﻼيجلا
يديس
ب
سﺎبﻌل
Année 2018
Faculté des Sciences Exactes Département de Mathématiques
BP 89 SBA 22000 ALGERIE TEL/FAX : 048-54-43-44
THÈSE
Présentée en vue de l’obtention du diplôme de Doctorat en sciences
Option :
Equations différentielles ordinaires
Présentée par :
HALIMI Berrezoug
Thèse soutenue le 24 / 06 / 2018
Directeur De Thèse Mr OUAHAB Abdelghani Professeur U.D.L Sidi Bel Abbes
Devant le jury composé de :
Président : Mr BENCHOHRA Mouffak Professeur U.D.L Sidi Bel Abbes
Examinateur : Mr ABBAS Said Professeur U.T.M. Saida
Examinateur : Mr HEDIA Benouda M.C.A Univ. Tiaret
Examinateur : Mr LAZREG Jamel Eddine M.C.A U.D.L Sidi Bel Abbes
Examinateur : Mr SLIMANI Boualem Attou M.C.A Univ. A.B.B Tlemcen
Stabilité des équations différentielles impulsives
et inclusion différentielles
Contents
1 Preliminaries 5
1.1 Multivalued Analysis . . . 5
1.2 C0-Semigroups . . . 7
1.3 Analytic semigroups . . . 9
1.4 Fractional Powers of Closed Operators . . . 10
1.5 Fixed point theorems . . . 11
2 Vector metric spaces 13 2.1 Generalized metric space . . . 13
2.2 Matrix convergent . . . 18
2.3 Fixed point results in generalized metric spaces . . . 20
3 Impulsive differential equations with delay 23 3.1 Stability via Banach fixed point . . . 23
3.2 Stability via Krasnoselskii fixed point theorem . . . 27
3.3 Perturbated problem . . . 31
4 Impulsive differential equations on the half-line 35 4.1 Uniqueness and continuous dependence on initial data . . . 36
4.2 Existence and compactness of solution sets . . . 41
5 Differential Inclusions 49 5.1 Filippov’s Theorem . . . 49
5.2 Relaxation Theorem . . . 61
6 Impulsive Semilinear Differential Inclusions 63 6.1 Mild Solutions . . . 64
6.2 Existences result . . . 64
6.3 An example . . . 72
Bibliography 77
Remerciements
Je tiens avant tout `a remercier Allah pour la force et la volont´e qu’il m’a donn´ees pour pouvoir achever ce travail.
Je dois beaucoup `a mon directeur de th`ese M.Ouahab Abdelghani qui a su me
faire profiter de sa science. Il m’a offert son temps et sa patience. Ses conseils, remar-ques et critiremar-ques ont toujours ´et´e une aide pr´ecieuse pour moi. J’ai beaucoup appris `a son contact et eu grand plaisir `a travailler avec lui. Il m’est tr´es agr´eable de lui adresser
mes vives remerciements et de lui t´emoigner ma sinc`ere reconnaissance. Merci aussi
pour toutes les fois o`u j’ai fait appel `a lui pour une aide, scientifique ou autre, car il n’a jamais ni h´esit´e, ni m´enag´e sa peine pour r´epondre `a mes sollicitations.
Je tiens ´egalement `a remercier M.Benchohra Mouffak qui m’a fait l’honneur de
pr´esider le jury ainsi que M.Abbas Said, M.Slimani Boualem Attou, M.Hedia
Benouda et M.Lazreg Djamael Eddine pour avoir accept´e de faire partie du jury
et d’y avoir consacr´e une partie de leurs temps.
Je voudrais aussi remercier l’organisme qui m’a permis de poursuivre mes recherches
sans me faire de soucis: le D´epartement de Math´ematiques de l’universit´e Djillali
Li-abes de Sidi Bel Abbes - Alg´erie, les professeurs, les ´etudiants et le personnel de
soutien.
Je tiens `a remercier aussi M. Ait Dads El Hadi qui m’a accueilli et pour toutes
les facilit´es offertes dans son laboratoire des Math´ematiques Appliqu´ees et Dynamique
des Populations de l’universit´e Cadi Ayyad, Facult´e des Sciences Semlalia,
Mar-rakech - Maroc.
Je remercie ma femme qui m’a toujours soutenu. Elle a du subir mon ´eloignement
et se retrouver seule `a s’occuper de nos enfants. Je sais que cela n’a pas ´et´e facile.
Enfin, merci `a mes parents. Sans leurs sacrifices je ne serais pas devenu ce que je suis aujourd’hui.
Dedicace
Je d´edie ce travail `a mes chers parents,
maman et mon p`ere
`
a mes enfants Mohamed salah Eddine, Khaled Ibn El Oualid et Aness Ibn Malek `a ma
femme `
a mes fr`eres et soeurs `
a tous mes enseignants.
Publications
• H. Berrezoug, J. Henderson, A. Ouahab , Existence and uniqueness of solu-tions for a system of impulsive differential equasolu-tions on the half-line. J. Nonl. Functional Anal., Vol.2017(2017), Article ID 38, pp.1-16.
• H. Berrezoug, J. J. Nieto, A. Ouahab, Stability by fixed point theory of impul-sive differential equations with delay,submitted.
Introduction
Impulsive diiferential equations, that is, differential equations involving impulse ef-fect, appear as a natural description of observed evolution phenomena of several real world problems. Many processes studied in applied sciences are represented by differ-ential equations. However, the situation is quite different in many physical phenomena that have a sudden change in their states such as mechanical systems with impact, biological systems such as heart beats, blood flows, population dynamics, theoretical physics, radiophysics, pharmacokinetics, mathematical economy, chemical technology, electric technology, metallurgy, ecology, industrial robotics, biotechnology processes, chemistry, engineering, control theory, medicine and so on. Adequate mathematical models of such processes are systems of differential equations with impulses.
The theory of impulsive differential equations is a new and important branch of differ-ential equations. The first paper in this theory is related to
Milman and Myshkis [44] and then followed by a period of active research which culminated with the monograph by Halanay and Wexler [33].
A comprehensive introduction to the basic theory is well developed in the mono-graphs by Benchohra et al. [11], Graef et al. [31], Laskshmikantham et al. [42], and Samoilenko and Perestyuk [61].
This thesis is devoted to the existence and stability via fixed point theorem of so-lutions for different classes of initial values problems for impulsive differential equation and inclusions with fixed and variable moments. This thesis is arranged as follows:
• In Chapters 1, 2, we introduce definitions, lemmas, notions of semigroup and fixed point theorem which are used throughout this thesis.
• In Chapter 3, we consider the following impulsive delay equations. x0(t) = −a(t)x(t − r), t ∈ J := [0, ∞), t 6= tk, k = 1, . . . , ∆xt=tk = Ik(x(t − k)), k = 1, . . . , x(t) = ψ(t), t ∈ [−r, 0] (0.0.1)
where a : [0, ∞) → R be bounded and continuous, r be a positive constant, 0 = t0 < t1 < . . . < tk< . . . , lim k→∞tk = ∞, limh→0x(tk+ h) = x(t + k), limh→0x(tk− h) = x(t−k) and ∆xt=tk = x(t + k) − x(t − k). 1
For any function x defined on [−r, +∞) and any t ∈ J , we denote by xt the
element of C([−r, 0], R) defined by.
xt(θ) = x(t + θ), θ ∈ [−r, 0]
Here xt(.) represents the history of the state from time t-r, up to the present time
t.
By Krasnoselskii fixed point we studied the asymptotic stability as zero solution of problem 0.0.1 is provided in the first section of chapter 3. In the least section we investigate the stability of zero solution for some class of impulsive perturbation problem with delay.
• In Chapter 4, we study the existence, uniqueness, continuous dependance on ini-tial condition and boundedness of solution for a system of impulsive differenini-tial equation using the fixed point approach in vector Banach space. Also the com-pactness and u.s.c of operator solution are investigated, we consider the following system x0(t) = f (t, x, y), t ∈ J := [0, ∞), t 6= tk, k = 1, . . . , y0(t) = g(t, x, y), t ∈ J, t 6= tk, k = 1, . . . , x(t+k) − x(t−k) = Ik(x(tk), y(tk)), k = 1, . . . ,
y(t+k) − y(t−k) = Ik(x(tk), y(tk)), k = 1, . . . ,
x(0) = x0,
y(0) = y0,
(0.0.2)
where x0, y0 ∈ R, f, g : J ×R×R → R are a given functions, Ik, Ik∈ C(R×R, R).
The notations x(t+k) = lim
h→0+x(tk+ h) and x(t
−
k) = lim
h→0+x(tk− h) stand for the right and the left limits of the functions y at t = tk, respectively.
• In Chapter 5, we establish the measurable Filippov’s theorem and relaxation problem for the following system of differential inclusions with impulse effects.
x0(t) ∈ F1(t, x(t), y(t)), y
0
(t) ∈ F2(t, x(t), y(t)), a.e.t ∈ [0, b] (0.0.3)
x(0) = x0, y(0) = y0. (0.0.4)
where Fi : [0, b] × Rn× Rn→ P(Rn) are multifunctions and x0, y0 ∈ Rn.
• In Chapter 6 we prove existence result based on a nonlinear alternative of Leray-Schauder type theorem in generalized Banach spaces.
x0(t) ∈ Ax(t) + F1(t, x(t), y(t)), t ∈ [0, b], t 6= t k, y0(t) ∈ Ay(t) + F2(t, x(t), y(t)), t ∈ [0, b], t 6= t k, ∆x(t) ∈ Ik(x(tk)), t = tk k = 1, 2, . . . , m ∆y(t) ∈ Ik(y(tk)), x(0) = x0 ∈ E, y(0) = y0 ∈ E, (0.0.5)
3
where J := [0, b], E is a real separable Banach space with inner product h·, ·i
induced by norm k · k, A : D(A) ⊂ E −→ E is the infinitesimal generator of a
strongly continuous semigroup of bounded linear operators (S(t))t≥0 in X and
F1, F2 : [0, b] × E × E −→ P(E) are given set-valued functions, where P(E)
denotes the family of nonempty subsets of X, Ik: E → P(E), (k = 1, 2, . . . , m).
Key words and phrases: Impulsive differential equation, multifunction, fixed point theorems, differential inclusion, generalized metric space, vector Banach space, stabil-ity, Filippov theorem, relaxation, compactness, delay equation.
AMS (MOS) Subject Classification. 34A12, 34A34, 34A37, 34A60, 34B15, 34B27, 34B37.
Chapter 1
Preliminaries
In this chapter, we introduce notations, definitions, and preliminary facts from multi-valued analysis, which are used throughout this thesis. We denote by
P(X) = {Y ⊂ P : Y 6= ∅}; Pcl(X) = {Y ∈ P(X) : Y closed}; Pb(X) = {Y ∈ P(X) : Y bounded}; Pcv(X) = {Y ∈ P(X) : Y convex}; Pcp(X) = {Y ∈ P(X) : Y compact}; Pcv,cp(X) = Pcv(X) ∩ Pcp(X).
1.1
Multivalued Analysis
Let (X, d) and (Y, ρ) be two metric spaces and F : X → P(Y ) be a multi-valued
mapping. The map F is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X
the set F (x0) is a nonempty, closed subset of X, and if for each open set N of Y
containing F (x0), there exists an open neighborhood M of x0 such that F (M ) ⊆ Y.
That is, if the set F−1(V ) is closed for any closed set V in Y . Equivalently, F is u.s.c. if the set F+1(V ) is open for any open set V in Y .
The mapping F is said to be completely continuous if it is u.s.c. and, for every bounded subset A ⊆ X, F (A) is relatively compact, i.e., there exists a relatively compact set K = K(A) ⊂ X such that
F (A) = [{F (x) : x ∈ A} ⊂ K.
Also, F is compact if F (X) is relatively compact, and it is called locally compact if for each x ∈ X, there exists an open set U containing x such that F (U ) is relatively compact.
A multivalued map F : X → P(X) has convex (closed) values if F (x) is convex (closed) for all x ∈ X. We say that F is bounded on bounded sets if F (B) is bounded
in X for each bounded set B of X, that is, supx∈B{sup{k y k: y ∈ F (x)}} < ∞. Finally, we say that F has a fixed point if there exists x ∈ X such that x ∈ F (x).
A multivalued map F : J → Pcl(X) is said to be measurable if for each x ∈ E, the
function Y : J → X defined by
Y (t) = dist(x, F (t)) = inf{k x − z k: z ∈ F (t)} (1.1.1)
is Lebesgue measurable
Theorem 1.1.1. [24] Let F : X → Pcp(Y ) be a closed locally compact multifunction.
Then F is u.s.c.
In what follows, by E we will denote a Banach space over the field of real numbers R and by J a closed interval in R. We let
C(J, E) = {Y : J → E y is continuous}.
C([0, b], Rn) is the Banach space of all continuous functions from [0, b] into Rn with the
norm
kyk∞= sup{ky(t)k : 0 ≤ t ≤ b}.
L1([0, b], Rn) denotes the Banach space of measurable functions y : [0, b] → Rn which
are Lebesgue integrable and normed by kykL1 =
Z b
0
| y(t) | dt for all y ∈ L1
([0, b], Rn).
Definition 1.1.1. The map f : [0, b] × B → Rn is said to be L1-Carath´eodory if
(i) t 7→ f (t, x) is measurable for each x ∈ B;
(ii) x 7→ f (t, x) is continuous for almost all t ∈ [0, b];
(iii) For each q > 0, there exists hq ∈ L1([0, b], R+) such that
kf (t, x)k ≤ hq(t) for all kxkB ≤ q and for almost all t ∈ [0, b].
Theorem 1.1.2. (Kuratowki, Ryll and Nardzewski) [39] Let E be a separable Banach
space and let F : J → Pcl(E) be a measurable map, then there exists a measurable map
7 Let A be a subset of J × B, A is L ⊗ B measurable if A belongs to the σ− algebra generated by all sets of the form N × D, where N is Lebesgue measurable in J and D is
Borel measurable in B. A subset A of L1(J, E) is decomposable if for all u, v ∈ A and
N ⊂ J measurable, the function uχN+vχJ −N ∈ A, where χ stands for the characteristic
function.
Let X be a nonempty closed subset of E and G : X → P(E) a multivalued operator with nonempty closed values. G is lower semicontinuous (l.s.c.) if the set {x ∈ X : G(X) ∩ B 6= ∅} is open for any open set B in E. The following two results are easily deduced from the limit properties.
Lemma 1.1.1. (See e.g. [8], Theorem 1.4.13) If G : X −→ Pcp(Y ) is u.s.c., then for
any x0 ∈ X,
lim sup
x→x0
G(x) = G(x0).
Lemma 1.1.2. (See e.g. [8], Lemma 1.1.9) Let (Kn)n∈N ⊂ K ⊂ X be a sequence of
subsets where K is compact in the separable Banach space X. Then co (lim sup n→∞ Kn) = \ N >0 co ([ n≥N Kn),
where co A refers to the closure of the convex hull of A.
Lemma 1.1.3. [41]. Given a Banach space X, let F : [a, b] × X −→ Pcp,cv(X) be an
L1-Carath´eodory multi-valued map such that for each y ∈ C([a, b], X), S
F,y 6= ∅ and let
Γ be a linear continuous mapping from L1([a, b], X) into C([a, b], X). Then the operator
Γ ◦ SF : C([a, b], X) −→ Pcp,cv(C([a, b], X)),
y 7−→ (Γ ◦ SF)(y) := Γ(SF,y)
has a closed graph in C([a, b], X) × C([a, b], X).
1.2
C
0-Semigroups
Let E be a Banach space and B(E) be the Banach space of linear bounded operators.
Definition 1.2.1. A semigroup of class C0 is a one parameter family {S(t) | t ≥ 0} ⊂
B(E) satisfying the conditions:
(i) S(t) ◦ S(s) = S(t + s), for t, s ≥ 0, (ii) S(0) = I,
(iii) the map t → S(t)(x) is strongly continuous, for each x ∈ E, i.e˙, lim
A semigroup of bounded linear operators S(t), is uniformly continuous if lim
t→0kS(t) − Ik = 0.
Here I denotes the identity operator in E.
We note that if a semigroup S(t) is class (C0) then satisfies the growth condition.
Proposition 1.2.1. Let {S(t)}t≥0 be a semigroup of bounded linear operator. Then
there exists some constant M ≥ 0 and ω ∈ R such that
kS(t)kB(E) ≤ M eωt, for t ≥ 0.
If, in particular M = 1 and β = 0, i.e˙, kS(t)kB(E) ≤ 1, for t ≥ 0, then the semigroup
S(t) is called a contraction semigroup (C0).
Definition 1.2.2. Let S(t) be a semigroup of class (C0) defined on E. The infinitesimal
generator A of S(t) is the linear operator defined by
A(x) = lim
h→0
S((h)(x) − x)
h , for x ∈ D(A),
where D(A) = {x ∈ E | limh→0
S(h)(x)−x
h exists in E}.
Let us recall the following property:
Proposition 1.2.2. The infinitesimal generator A is closed linear and densely defined
operator in E. If x ∈ D(A), then S(t)(x) is a C1-map and
d
dtS(t)(x) = A(S(t)(x)) = S(t)(A(x)) on [0, ∞).
Theorem 1.2.1. (Hille and Yosida) [54]. Let A be a densely defined linear operator with domain and range in a Banach space E. Then A is the infinitesimal generator of
uniquely determined semigroup S(t) of class (C0) satisfying
kS(t)kB(E)≤ M exp(ωt), t ≥ 0,
where M > 0 and ω ∈ R if and only if (λI − A)−1 ∈ B(E) and k(λI − A)−nk ≤
M/(λ − ω)n, n = 1, 2, . . ., for all λ ∈ R.
For more details on strongly operators, we refer the reader to the books of Gold-stein [32], Pazy [54].
9
1.3
Analytic semigroups
Definition 1.3.1. Let ∆ = {z : ϕ1 < argz << ϕ2, ϕ1 < 0 << ϕ2} and for z ∈ ∆ let
S(z) be a bounded linear operator. The family S(z), z ∈ ∆ is an analytic semigroup in ∆ if
(i) z → S(z) is analytic in ∆. (ii) S(0) = I and lim
z−→0S(z)x = x for every x ∈ E.
(iii) S(z1+ z2) = S(z1)S(z2) for z1, z2 ∈ ∆.
A semigroup S(t) will be called analytic if it is analytic in some sector ∆ containing the nonnegative real axis.
Clearly, the restriction of an analytic semigroup to the real axis is a C0 semigroup.
We will be interested below in the possibility of extending a given C0 semigroup to an
analytic semigroup in some sector ∆ around the nonnegative real axis.
Theorem 1.3.1. [54] Let S(t) be a uniformly bounded C0 semigroup. Let A be the
infinitesimal generator of S(t) and assume 0 ∈ ρ(A). The following statements are equivalent:
(a) S(t) can be extended to an analytic semigroup in a sector ∆δ = {z : |argz| < δ}
and kS(z)k is uniformly bounded in every closed subsector ∆δ0, δ
0
< δ, of ∆δ.
(b) There exists a constant C such that for every σ > 0, τ 6= 0
kR(σ + it : A)k ≤ C
τ. (c) There exist 0 < δ < π/2 and M > 0 such that
ρ(A) ⊃ Σ = {λ : |argλ| < π
2 + δ} ∪ {0}
and
kR(λ : A)k ≤ M
|λ| for λ ∈ Σ, λ 6= 0.
(d) S(t) is differentiable for t > 0 and there is a constant C such that
kAS(t)k ≤ C
1.4
Fractional Powers of Closed Operators
For our definition we will make the following assumption.
(DDC) Let A be a densely defined closed linear operator for which ρ(A) ⊃ Σ+ = {λ : 0 < ω < |argλ| ≤ π} ∪ V where V is a neighborhood of zero, and
kR(λ : A)k ≤ M
1 + |λ| for λ ∈ Σ
+
.
If M = 1 and w = π2 then −A is the infinitesimal generator of a C0 semigroup.
If w < π
2 then, by Theorem 1.3.1, −A is the infinitesimal generator of an analytic
semigroup. The assumption that 0 ∈ ρ(A) and therefore a whole neighborhood V of zero is in ρ(A) was made mainly for convenience. Most of the results on fractional powers that we will obtain in this section remain true even if 0 ∈ ρ(A).
Definition 1.4.1. Let A satisfy Assumption (DDC) with w < π2. For every α > 0 we
define
Aα = (A−α)−1.
For α = 0, Aα = I.
Theorem 1.4.1. [54] Let Aα be defined by Definition 1.4.1 then,
(a) Aα is a dosed operator with domain D(Aα) = R(A−α) = the range of A−α.
(b) α ≥ β > 0 implies D(Aα) ⊂ D(Aβ) .
(c) D(Aα) = E for every α ≥ 0.
(d) If α, β are real then
Aα+βx = Aα· Aβx
for every x ∈ D(Aγ) where γ = max(α, β, α + β).
Theorem 1.4.2. [54] Let −A be the infinitesimal generator of an analytic semigroup S(t). if 0 ∈ ρ(A) then,
(a) S(t) : E → D(Aα) for every t > 0 and α ≥ 0.
(b) For every x ∈ D(Aα) we have S(t)Aαx = AαS(t)x.
(c) For every t > 0 the operator AαS(t) is bounded and
kAαS(t)k ≤ M
αt−αe−δt.
(d) Let 0 < α ≤ 1 and x ∈ D(Aα) then
11
1.5
Fixed point theorems
In this section we present some classical fixed point theorems.
Theorem 1.5.1. (Schaefer’s fixed point theorem) (see also [62], page 29). Let X be a Banach space and let N : X → X be a completely continuous map. If the set
Φ = {x ∈ X : λx = N (x) for some λ > 1} is bounded, then N has a fixed point.
Theorem 1.5.2. (nonlinear alternative [30]). let X be a a Banach space with C ⊂ X closed and convex. Assume U is a relatively open subset of C with 0 ∈ U and G : U → C is a compact map. Then either,
(i) G has a fixed point in U ; or
(ii) there is a point u ∈ ∂U and λ ∈ (0, 1) with u = λG(u).
Before stating our next fixed point theorem, we need some preliminaries.
Let (X, d) be a metric space induced from the normed space(X, | . |). Consider Hd :
P(X) × P(X) → R+∪ {∞}, given by Hd(A, B) = max{sup a∈A d(a, B), sup b∈B d(A, b)}, (1.5.1)
where d(A, b) = infa∈Ad(a, b), d(a, B) = infb∈Bd(a, b). Then (Pb,cl(X), Hd) is a metric
space and (Pcl(X), Hd) is a generalized (complete) metric space (see [39]).
Definition 1.5.1. A multivalued operator G : X → Pcl(X) is called
(a) γ− Lipschitz if there exists γ > 0 such that
Hd(G(x), G(y)) ≤ γd(x, y), for each x, y ∈ X; (1.5.2)
(b) a contraction if it is γ− Lipschitz with γ < 1. The following is due to Schauder.
Theorem 1.5.3. [50] Let B is a closed, convex and nonempty subset of a Banach space E. Let N : B → B be a continuous mapping such that N (B) is a relatively compact subset of E. Then N has at least one fixed point in B. That is, there exists y ∈ B such that N y = y.
Theorem 1.5.4. (Krasnoselskii) [43] Let X be a Banach space. Suppose that A and B map X into X such that
(ii) B is a contraction with constant 0 < α < 1. (iii) the set
M = {x ∈ X : x = λBx
λ
+ λA(x), λ ∈ (0, 1)} is bounded. Then there exists x ∈ X with Ax + Bx = x.
For further readings and details on multi-valued analysis and fixed point theory, we refer the reader to the books by Aubin and Celina [6], Aubin and Frankowska [8],
Deimling [24], Djebali et al [25], Dugundji and Granas, [27], G´orniewicz [29], Hu and
Chapter 2
Vector metric spaces
The classical Banach contraction principle was extended for contractive maps on spaces endowed with vector-valued metric space by Perov in 1964 [57] and Perov and Kibenko [56]. Till now, there have been a number of attempts to generalize the Perov fixed point theorem in several directions and also have been a number of applications in various fields of nonlinear analysis, system of ordinary differential and semilinear differential equations.
2.1
Generalized metric space
In this section we define generalized metric space (or vector metric spaces) a prove some properties. If, x, y ∈ Rn, x = (x
1, . . . , xn), y = (y1, . . . , yn), by x ≤ y we mean xi ≤ yi
for all i = 1, . . . , n. Also |x| = (|x1|, . . . , |xn|) and max(x, y) = max(max(x1, y1), . . . , max(xn, yn)).
If c ∈ R, then x ≤ c means xi ≤ c for each i = 1, . . . , n. For x ∈ Rn, (x)i = xi, i =
1, . . . , n.
Definition 2.1.1. Let X be a nonempty set. By a generalized metric on X (or
vector-valued metric) we mean a map d : X × X → Rn with he following properties:
(i) d(u, v) ≥ 0 for all u, v ∈ X; if d(u, v) = 0 then u = v (ii) d(u, v) = d(v, u) for all u, v ∈ X
(iii) d(u, v) ≤ d(u, w) + d(w, v) for all u, v, w ∈ X.
Note that for any i ∈ {1, . . . , n} (d(u, v))i = di(u, v) is a metric space in X.
We call the pair (X, d) generalized metric space. For r = (r1, r2, . . . , rn) ∈ Rn+,we
will denote by
B(x0, r) = {x ∈ X : d(x0, x) < r}
the open ball centrad in x0 with radius r and
B(x0, r) = {x ∈ X : d(x0, x) ≤ r}
the closed ball centered in x0 with radius r = (r1, . . . , rn) > 0, ri > 0, i = 1; . . . , n.
Definition 2.1.2. Let (X, d) be a generalized metric space. A subset A ⊆ X is called open if, for any x0 ∈ A, there exists r ∈ Rn+ with r > 0 such that
B(x0, r) ⊆ A.
Any open ball is an open set and the collection of all open balls of X generates the generalized metric topology on X.
Definition 2.1.3. Let (X, d) be a generalized metric space
(a) A sequence (xp) in X converge (or Rn+-converges) to some x ∈ X, if for every
∈ Rn
+, > 0 there exists p0() ∈ N such that for each
d(xp, x) ≤ for all p ≥ p0().
(b) A sequence (xp) is called Cauchy sequence if for every ∈ Rn+, > 0 there exists
p0() ∈ N such that for each
d(xp, xq) ≤ for all p, q ≥ p0().
(c) A generalized metric space X is called complete if each Cauchy sequence in X converges to a limit inX.
(d) A subset Y of a generalized metric space X is said to be closed whenever (xp) ⊆ Y
and xp → x, as p → ∞ imply x ∈ Y .
Using the above definitions, we have the following properties: If xp → x as p → ∞,
then
(i) The limit x is unique.
(ii) Every subsequence of (xp) converges to x.
(iii) If also xp → x as p → ∞, then
d(xp, yp) → d(x, y) as p → ∞.
Theorem 2.1.1. For the generalized metric space (X, d) the followings hold: (a) Every convergent sequence is an Cauchy sequence,
15
(c) If an Cauchy sequence (xp) has a subsequence (xpk) such that
xpk → x as pk→ ∞,
then
xp → x as p → ∞.
Proof. (a) Let (xp)p∈N be a convergent sequence in X. The for every ∈ Rn+ there
exists p0() ∈ N such that
d(xp, x) ≤
2 for all p ≥ p0().
Then for every p, q ≥ p0() we have
d(xp, xq) ≤ d(xp, x) + d(xq, x) ⇒ d(xp, xq) ≤ .
Hence (xp)p∈N is an Cauchy sequence in X.
(b) Let (xp)p∈N be an Cauchy sequence. Fixe ∈ Rn+ there exists p0() ∈ N such
that
d(xp, xq) ≤ , for all p, q ≥ p0().
Hence for each p ∈ N, we get
xp ∈ B(xp0(), + r), r = max
1≤i,j≤p0()−1
d(xi, xj),
this implies that (xp)p∈N bounded in X.
(c) Let (xp)p∈N be an Cauchy sequence and let (xpk)pk∈N be a subsequence of (xp)p∈N such that limpk→∞xpk = x. The for every ∈ R
n
+ there exist p∗(), q∗() ∈ N such that
d(xp, xq) ≤ 2 for allp, q ≥ p∗() and d(xpk, x) ≤ 2 for allpk≥ q∗() Then
d(xp, x) ≤ d(xp, xpk) + d(xpk, x) ≤ for all p ≥ max(q∗(), p∗()). Hence
xp → x as p → ∞.
(a) A function f : X → Y is said to be continuous (or topologically continuous) at x if for every ∈ Rn+, > 0 there exists some δ() ∈ Rn+, δ() > 0 such that
ρ(f (x), f (y)) < whenever x, y ∈ X and d(x, y) < δ().
The function f is said to be topologically continuous if it is topologically continuous at each point of X.
Definition 2.1.5. Let (X, d) be an generalized metric space. We say that a subset
Y ⊂ X is closed if, (xp) ⊂ Y and xp → x as p → ∞ imply x ∈ Y.
Definition 2.1.6. Let (X, d) be a generalized metric space. A subset C of X is called compact if, every open cover of C has a finite subcover. A subset C of X is sequentially compact if, every sequence in C contains a convergent subsequence with limit in C.
Definition 2.1.7. A subset C of X is totally bounded if, for each Rn
+ with > 0,
there exists a finite number of elements x1, x2, . . . , xp ∈ X such that
C ⊆ ∪pi=1B(xi, ).
The set x1, . . . , xpis called a finite -net.
Theorem 2.1.2. If C is a subset of X, then the following affirmations hold:
i) C is compact if and only if, C is sequentially compact if and only if , C is closed and totally bounded;
ii) C relatively compact, if and only if, C sequentially relatively compact, if and only if, C totally bounded.
Definition 2.1.8. Let (X, d) be an generalized metric space. If A ⊂ X is a nonempty set, then the function
δ(A) = sup{d(x, y) : x; y ∈ A}
is called the diameter of A. If δ(A) < ∞, then A is called an bounded set.
Let (X, d) be a generalized metric space we define the following metric spaces: Let Xi = X, i = 1, . . . , n. Consider Qn i=1Xi with ¯d: ¯ d((x1, . . . , xn), (y1, . . . , yn)) = n X i=1 di(xi, yi).
The diagonal space of Qn
i=1Xi defined by e X = {(x, . . . , x) ∈ n Y i=1 Xi : x ∈ X, i = 1, . . . , n}.
17 Thus is a metric space with the following distance
d∗((x, . . . , x), (y, . . . , y)) = n
X
i=1
di(x, y), for each x, y ∈ X.
It is clear that eX is closed set in Qn
i=1Xi.
Intuitively, X and eX. This is showed in the following result.
Lemma 2.1.1. Let (X, d) be a generalized metric space. Then there exists h : X → eX
an homeomorphism map.
Proof. Consider h : X → eX defined by
h(x) = (x, . . . , x) for all x ∈ X. Obviously h is bijective.
• To prove that h is a continuous map. Let x, y ∈ X. Thus d∗(h(x), h(y)) ≤ n X i=1 di(x, y).
For > 0 we take δ = (n, . . . ,n), let fixed x0 ∈ X and B(x0, δ) = {x ∈ X : d(x0, x) <
δ}, then for every x ∈ B(x0, δ) we have
d∗(h(x0), h(x)) ≤ .
• Now, h−1 : eX → X is a continuous map defined by
h−1(x, . . . , x) = x, (x, . . . , x) ∈ eX. To show that h−1 is continuous. Let (x, . . . , x), (y, . . . , y) ∈ eX, then
d(h−1(x, . . . , x), h−1(y, . . . , y)) = d(x, y). Let = (1, . . . , n) > 0 we take δ =
min
1≤i≤ni
n and we fix (x0, . . . , x0) ∈ eX. Set
B((x0, . . . , x0), δ) = {(x, . . . , x) ∈ eX : d∗((x0, . . . , x0), (x, . . . , x)) < δ}. For (x, . . . , x) ∈ B((x0, . . . , x0), δ) we have d∗((x0, . . . , x0), (x, . . . , x)) < δ ⇒ n X i=1 di(x0, x) < min 1≤i≤ni n . Then di(x0, x) < min 1≤i≤ni n , i = 1, . . . , n ⇒ d(x0, x) < . Hence h−1 is continuous.
Theorem 2.1.3. Every generalized metric space is paracompact.
Proof. Let X be a generalized metric space, by Lemma 2.1.1 there exists eX metric space
homeomorphic to X. Since every metric space is paracompact hence X is paracompact.
2.2
Matrix convergent
Definition 2.2.1. A square matrix M ∈ Mn×n(R) of real numbers is said to be
con-vergent to zero if
Mk → 0, as k → ∞.
Lemma 2.2.1. [60] Let M be a square matrix of nonnegative numbers. The following assertions are equivalent:
(i) M is convergent towards zero;
(ii) the matrix I − M is non-singular and
(I − M )−1 = I + M + M2+ . . . + Mk+ . . . ; (iii) |λ| < 1 for every λ ∈ C with det(M − λI) = 0
(iv) (I − M ) is non-singular and (I − M )−1 has nonnegative elements;
Proof. Assume that M is converge to zero. We show that I − M is non-singular it suffices to prove that the linear system
(I − M )x = 0 (2.2.1)
has only the null solution. Let x ∈ C be a solution of the system (2.2.1), then x = M x = M2x = . . . Mkx = . . .
and letting k → ∞ we deduce x = 0. Hence I − M is non-singular. Furthermore, we have
I − (I − M )(I + M + M2+ . . . Mk) = Mk+1 → 0 as k → ∞.
This implies that
(I − M )−1 = I + M + M2+ . . . Mk. . . .
Lemma 2.2.2. A square matrix M ∈ Mn×n(R) of real numbers is said to be convergent
to zero if and only if its spectral radius ρ(M ) is strictly less than 1. In other words ,this means that all the eigenvalues of M are in the open unit disc.
19
Lemma 2.2.3. Let M ∈ Mn×n(R+) is convergent towards zero, then
z ≤ (I − M )−1z for every z ∈ Rn+.
Proof. Since M ∈ Mn×n(R+) is convergent towards zero, then from lemma 2.2.1,
(I − M )−1 ∈ Mn×n(R+) and
(I − M )−1 = I + M + M2+ . . .
Thus for every z ∈ Rn
+ we have (I − M )−1z = ∞ X i=0 Miz ⇒ z ≤ (I − M )−1z.
Lemma 2.2.4. Let M ∈ Mn×n(R+) is convergent towards zero, then
PM = {z ∈ Rn+ : (I − M )z > 0},
is nonempty and coincides with the set
{(I − M )−1z0 : z0 ∈ Rn, z0 > 0}.
Proof. Clear that I − M ∈ Mn×n(R+) and singular matrix, then for every z ∈ Rn+, z =
(z1, . . . , zn) with zi > 0, i = 1, . . . , n, we get (I − M )z > 0. This implies that Pm 6= ∅.
Now we show that
PM = {(I − M )−1z0 : z0 ∈ Rn, z0 > 0}.
Indeed, if z0 ∈ Rn and z0 > 0, then
z := (I − M )−1z0 ≥ z0 ⇒ z > 0.
Hence (I − M )z > 0 and so z ∈ PM. Conversely, if z ∈ PM, then z0 := (I − M )z > 0
and z = (I − M )−1z.
Definition 2.2.2. We say that a non-singular matrix A = (aij)1≤i,j≤n ∈ Mn×n(R) has
the absolute value property if
A−1|A| ≤ I, where
|A| = (|aij|)1≤i,j≤n ∈ Mn×n(R+).
Some examples of matrices convergent to zero are the following:
1) A = a 0
0 b
2) A = a −c 0 b , where a, b, c ∈ R+ and a + b < 1, c < 1 3) A = a −a b −b , where a, b, c ∈ R+ and |a − b| < 1, a > 1, b > 0.
Lemma 2.2.5. Let M = (aij)1≤i,j≤n ∈ Mn×n(R+) be a triangular matrix with
max{|aii|i = 1, . . . , n} <
1 2.
Then the matrix A = (I − M )−1M is convergent to zero.
Proof. Suppose M := a11 . . . a1n .. . ... · · · 0 . . . ann
∈ Mn×n(R+). Then the eigenvalues of M
are λi = 1−aaiiii, for all i = 1, . . . , n. Since all of the eigenvalues of M are in the open
unit disc, the conclusion follows from Theorem 2.2.1.
Example 2.2.1. Some examples of matrices convergent to zero are:
1. M = a a b b , where a, b ∈ R+ and a + b < 1; 2. M = a b a b , where a, b ∈ R+ and a + b < 1; 3. M = a b 0 c
, where a, b, c ∈ R+ and max{a, c} < 1.
2.3
Fixed point results in generalized metric spaces
Now, we recall how to define the contraction and other known helpful results for the proof of Krasnoselskii’s theorem for single valued operators in generalized Banach spaces.
Definition 2.3.1. Let (X, d) be a generalized metric space and let f : X → X be a single valued operator. Then, f is called a single valued M -contraction if and only if,
M ∈ Mm×m(R+) is a matrix convergent to zero and
21 Theorem 2.3.1. [57]Let (X, d) be a complete generalized metric space with d : X ×
X −→ Rn and let N : X −→ X be such that
d(N (x), N (y)) ≤ M d(x, y)
for all x, y ∈ X and some square matrix M of nonnegative numbers. If the matrix M
is convergent to zero, that is Mk −→ 0 as k −→ ∞, then N has a unique fixed point
x∗ ∈ X
d(Nk(x0), x∗) ≤ Mk(I − M )−1d(N (x0), x0)
for every x0 ∈ X and k ≥ 1.
Proof. Let x ∈ X and define the sequence xn= Nn(x), where Nn = N ◦ . . . ◦ N. Using
the fact N is M −contraction, we get
d(xk+1, xk) ≤ Mkd(N (x), x)
and, as a consequence,
d(xk, xk+m) ≤ (Mk+ Mk+1+ . . . + Mk+m−1)d(N (x), x)
From lemma 2.2.1 we deduce that
d(xk, xk+m) ≤ Mk(I − M )−1d(N (x), x).
Hence (xk) is a Cauchy sequence with respect to d and thus converges to some limit
x∗ ∈ X. The continuity of N guarantees that
x∗ = N (x∗).
For uniqueness let y1, y2 be two fixed points of N , then
d(y1, y2) = d(Nk(y1), Nk(y2)) ≤ Mkd(N (y1), N (y2)).
Since Mk → 0 as k → ∞, this implies d(y
1, y2) = 0, so yl= y2.
In [65], the following version of the Krasnoselskii’s fixed point theorem in generalized Banach space was obtained.
Theorem 2.3.2. (Krasnoselskii type) [65] Let X be a generalized Banach space. Sup-pose that A and B map X into X such that
(i) A is a completely continuous operator. (ii) B is a contraction with constant α < 1.
Then there exists x ∈ M with A(x) + B(x) = x.
Theorem 2.3.3. Let X be a generalized Banach space, C be a nonempty compact
convex subset of X, G : C → Pcp,cv(C) be an u.s.c. multivalued map, then the operator
inclusion G has at leat one fixed point, that is there exists x ∈ C such that x ∈ G(x). As a consequence of the above result we present the multivalued version of Schae-fer’s fixed point theorem and nonlinear alternative Leray-Schauder type theorem in generalized Banach spaces.
Theorem 2.3.4. Let (X, k · k) be a generalized Banach space and F : X → Pcp,cv(X)
be a completely continuous multivalued mapping and u.s.c. Moreover assume that the set
A = {x ∈ X : x ∈ λF (x) for some λ ∈ (0, 1)}
Chapter 3
Impulsive differential equations
with delay
The fixed point theory has been proven to be a powerful tool for dealing with the stability of functional differential equations was studied in [13–20].
In this chapter we consider the following impulsive delay equations x0(t) = −a(t)x(t − r), t ∈ J := [0, ∞), t 6= tk, k = 1, . . . , ∆xt=tk = Ik(x(t − k)), k = 1, . . . , x(t) = ψ(t), t ∈ [−r, 0] (3.0.1)
where a : [0, ∞) → R be bounded and continuous, r be a positive constant, 0 = t0 < t1 < . . . < tk < . . . , lim k→∞tk = ∞. limh→0x(tk+ h) = x(t + k), limh→0x(tk − h) = x(t − k), ∆xt=tk = x(t + k) − x(t − k) and Ik ∈ C(R, R)
For any function x defined on [−r, +∞) and any t ∈ J , we denote by xt the element
of C([−r, 0], R) defined by.
xt(θ) = x(t + θ), θ ∈ [−r, 0]
Here xt(.) represents the history of the state from time t-r, up to the present time t.
3.1
Stability via Banach fixed point
Consider the Banach space
P Cb = {y ∈ P C([−r, ∞), R) : y is bounded},
where
P C([−r, ∞), R) = {y : [−r, ∞) → R, xk ∈ C((tk, tk+1], R), x(t−k) and x(t +
k) exist and satisfy
x(tk) = x(t−k) for k = 1, . . .}
and xk:= y|(tk,tk+1]. Endowed with the norm
kxkb = sup{|x(t)| : t ∈ [−r, ∞)},
P Cb is a Banach space. Next we define what we mean by a solution to problem (3.0.1).
Lemma 3.1.1. ( [10]) Let C ⊂ P Cb. Then C is relatively compact if it satisfies the
following conditions:
(a) C is uniformly bounded in P Cb(R+, R).
(b) The functions belonging to C are almost equicontinuous on R+, i.e. equicontinuous
on every compact interval of R+.
(c) The functions from C are equiconvergent, that is, given ε > 0, there corresponds T (ε) > 0 such that |x(τ1) − x(τ2)| < ε for any τ1, τ2 ≥ T (ε) and x ∈ C.
Definition 3.1.1. A function x ∈ P C(J, R) is said to be a solution of (3.0.1) if x0(t) = −a(t)x(t − r), t ∈ R+ t 6= tk, k = 1, . . . , ∆xt=tk = Ik(x(t
−
k)), k = 1, . . . ,
and x(t) = ψ(t), t ∈ [−r, 0].
Lemma 3.1.2. The solution of above problem can be expressed by the formula. x(t) = x(0)e−R0ta(s+r)ds+
Z t t−r
a(u + r)x(u)du − e−R0ta(u+r)du
Z 0 −r a(u + r)x(u)du − Z t 0 a(s + r)e− Rt sa(u+r)du Z s s−r a(u + r)x(u)duds + X 0<tk<t Ik(x(tk))e −Rt tka(s+r)ds. and x(t) = ψ(t), t ∈ [−r, 0].
Proof. Let x ∈ P C be solution of problem (3.0.1), then for t ∈ [0, t1] we have
Z t 0 x(s)eR0s+ra(u)du 0 ds = Z t 0 eR0s+ra(u)du d ds Z s s−r a(u + r)x(u)du ds, then Z t 0 x(s)eR0s+ra(u)du 0 ds = Z t 0 eR0s+ra(u)du( d ds Z s s−r a(u + r)x(u)du ds x(t)eR0t+ra(u)du− x(0)e
Rr 0 a(u)du = e Rt+r 0 a(u)du Z t t−r a(u + r)x(u)du −eR0ra(u)du Z 0 −r a(u + r)x(u)du − Z t 0 d ds(e Rs+r 0 a(u)du) Z s s−r a(u + r)x(u)du ds.
25 Thus x(t) = x(0)e− Rt 0a(s + r)ds + Z t t−r a(u + r)x(u)du −e−R0ta(u+r)du Z 0 −r a(u + r)x(u)du − Z t 0
a(s + r)e−Rsta(u+r)du)
Z s s−r a(u + r)x(u)duds. For t ∈ (t1, t2], we get Z t t1 x(s)e Rs+r t1 a(u)du 0 ds = Z t t1 e Rs+r t1 a(u)du( d ds Z s s−r a(u + r)x(u)du ds x(t)e Rt+r t1 a(u)du− x(t+ 1)e Rt1+r t1 a(u)du = e Rt+r t1 a(u)du Z t t−r a(u + r)x(u)du −eRt1t1+ra(u)du Z t1 t1−r a(u + r)x(u)du − Z t t1 a(s + r)e Rs+r t1 a(u)du) Z s s−r a(u + r)x(u)duds x(t) = x(t+1)e− Rt t1a(s+r)ds+ Z t t−r a(u + r)x(u)du −e− Rt t1a(u+r)du Z t1 t1−r a(u + r)x(u)du − Z t t1 a(s + r)e− Rt sa(u+r)du Z s s−r a(u + r)x(u)duds. Then x(t) = x(0)e−R0ta(s+r)ds+ Z t t−r a(u + r)x(u)du −e−R0ta(u+r)du Z 0 −r a(u + r)x(u)du − Z t 0
a(s + r)e−Rsta(u+r)du
Z s s−r a(u + r)x(u)duds +I1(x(t1))e −Rt t1a(s+r)ds.
We continue this process we obtain for t ∈ [0, b], we concluded
x(t) = x(0)e−R0ta(s+r)ds+ Z t
t−r
a(u + r)x(u)du − e−R0ta(u+r)du
Z 0 −r a(u + r)x(u)du − Z t 0
a(s + r)e−Rsta(u+r)du
Z s s−r a(u + r)x(u)duds + X 0<tk<t Ik(x(tk))e −Rt tka(s+r)ds.
We will use the Banach fixed point theorem to prove that under the Lipschitz conditions of the jumps functions Ik, k = 1, . . . , and for each small initial condition ψ
a solution of problem (3.0.1) bounded and tends to zero as t → ∞. Theorem 3.1.1. Assume that:
(H1) Rt t−r|a(u + r)|du + Rt 0 |a(s + r)|e −Rt sa(u+r)duRs s−r|a(u + r)|duds ≤ α (H2) Rt 0 a(s + r)ds → ∞ as t → ∞
(H3) There exist ck ≥ 0, k = 1, . . . such that
Ik(0) = 0, |Ik(x) − Ik(y)| ≤ ck|x − y|, for all x, y ∈ R.
hold. If α +
∞
X
k=1
ck < 1. Then the problem (3.0.1) has unique bounded solution and
tends to zero as t → ∞ Proof. Consider N : P Cb → P Cb by (N x)(t) = ψ(0)e−R0ta(s+r)ds+Rt t−ra(u + r)x(u)du −e−Rt 0a(u+r)duR0 −ra(u + r)x(u)du −Rt 0 a(s + r)e −Rt sa(u+r)duRs s−ra(u + r)x(u)duds +P 0<tk<tIk(x(tk))e −Rt tka(s+r)ds if t ∈ [0, ∞), ψ(t) if t ∈ [−r, 0].
From (H1)-(H3), we can easily prove that N (P Cb) ⊂ P Cb and N is contraction
opera-tor, then by Banach fixed point there exists unique x ∈ P Cb such that x = N (x) which
27 t → ∞. Let t ∈ [0, ∞) then, we get
|x(t)| ≤ Z t
t−r
|a(u + r)||x(u)|du + e−R0ta(u+r)du
Z 0 −r |a(u + r)||x(u)|du + Z t 0
|a(s + r)|e−Rsta(u+r)du
Z s s−r |a(u + r)||x(u)|duds + X 0<tk<t |Ik(x(tk))|e −Rt tka(s+r)ds. Thus |x(t)| ≤ Z t t−r |a(u + r)| sup u∈[0,t] |x(u)|du + e−R0ta(u+r)du Z 0 −r |a(u + r)| sup u∈[0,t] |x(u)|du + Z t 0
|a(s + r)|e−Rsta(u+r)du
Z s s−r |a(u + r)| sup u∈[0,t] |x(u)|duds + X 0<tk<t ck sup u∈[0,t] |x(u)|. Hence |x(t)| ≤ α sup s∈(0,t) |x(s)| + e−R0ta(u+r)du Z 0 −r |a(u + r)| sup s∈(0,t) |x(s)| + X 0≤tk≤t ck sup s∈(0,t) |x(s)|. This implies that
sup t∈∈[−r,∞) |x(t)| ≤ 1 − ∞ X k=1 ck .
3.2
Stability via Krasnoselskii fixed point theorem
In this section we present the stability result of problem (3.0.1) via the following the-orem
Theorem 3.2.1. Let (H1) − (H2) and the following condition
(H4) there exist αk, βk ≥ 0, k = 1, . . . such that
are satisfied. If α + ∞ X k=1 αk < 1 and ∞ X k=1
βk < ∞. Then the problem (3.0.1) has unique
bounded solution. If βk = 0, k = 1, . . . the solution of problem (3.0.1) tends to zero as
t → ∞.
Proof. Let N : P Cb → P Cb be operator defined in theorem 3.1.1. N = A + B where
A, B : P Cb → P Cb by Bφ(t) = ψ(0)e−R0ta(s+r)ds+Rt t−ra(u + r)x(u)du −e−Rt 0a(u+r)duR0 −ra(u + r)x(u)du −Rt 0 a(s + r)e −Rt sa(u+r)duRs s−ra(u + r)x(u)duds if t ∈ [0, ∞), ψ(t) if t ∈ [−r, 0]. and Aφ(t) = X 0<tk<t Ik(φ(tk))e −Rt tka(s+r)ds , t ∈ [0, ∞) 0 if t ∈ [−r, 0].
Step 1 B is a contraction. Let φ, η ∈ P Cb then
|(Bφ)(t) − (Bη)(t)| ≤ Z t t−r
a(u + r)(φ(u) − η(u))du + Z t 0
a(s + r)e−Rsta(u+r)du
Z s
s−r
a(u + r)(φ(u) − η(u))duds ≤ Z t t−r
|a(u + r)||φ(u) − η(u)|du +
Z t
0
|a(s + r)|e−Rsta(u+r)du
Z s
s−r
|a(u + r)||φ(u) − η(u)|duds
Hence
kB(φ) − B(η)kb ≤ αkφ − ηkb, for all φ, η ∈ P Cb.
Step 2. A is continuous
Given φn→ φ in P Cb, then there exists M > 0 such that
29 and |(Aφn)(t) − (Aφ)(t)| ≤ X 0<tk<t |Ik(φn(tk)) − Ik(φ(tk))|. Since X αk< ∞, and X βk < ∞,
hence for every > 0, ∃k0, k0 ∈ N such that ∞ X k=k0 αk < 6M, ∞ X k=k0 βk < 6,
Using the fact limk→∞tk= ∞, ∃n0 ∈ N, such that for each k ≥ n0 ⇒ tk ≥ k0.
From (H4), we get k Aφn− Aφ kb ≤ X 0<tk≤tn0−1 |Ik(φn(tk)) − Ik(φ(tk))| + ∞ X k=n0 (2M αk+ 2βk) ≤ n0−1 X k=1 |Ik(φn(tk)) − Ik(φ(tk))| +2 3.
Used the fact that Ik are continuous functions, then we have
n0−1 X k=0 |Ik(φn(tk)) − Ik(φ(tk))| → 0 as n → ∞ Hence kAφn− Aφkb → 0 as n → ∞.
Step 3 From, (H4), we can easily prove that A sends bounded sets into bounded sets
in P Cb. We will now show that A(M ) is contained in a compact set
Step 4. A sends bounded sets in P Cb into almost equicontinuous sets of P Cb. Let r >
0, Br := {y ∈ P Cb : kyk∞≤ r} be a bounded set in P Cb, τ1, τ2 ∈ [0, ∞), τ1 < τ2,
and φ ∈ Br we have Aφ(τ1) = X 0≤tk≤τ1 Ik(φ(tk))e −Rτ1 tk a(s+r)ds
and Aφ(τ2) = X 0≤tk≤τ2 Ik(φ(tk))e −Rτ2 tk a(s+r)ds = X 0≤tk≤τ2 Ik(φ(tk))e −Rτ1 tk a(s+r)ds+ Rτ2 τ1a(s+r)ds . Then |Aφ(τ1) − Aφ(τ2)| ≤ 1 − e− Rτ2 τ1 a(s+r)ds X∞ k=1 (αkr + βk). Thus |Aφ(τ1) − Aφ(τ2)| → 0 as τ1 → τ2.
Step 5. We now show that the set A(B(0, r)) is equiconvergent, i.e. for every ε > 0, there exists T (ε) > 0 such that |Aφ(t) − Aφ(s)| ≤ ε for every t, s ≥ T () and each φ ∈ B(0, r). Letting φ ∈ B(0, r). Then
|Aφ(t) − Aφ(s)| ≤ X s≤tk<t |Ik(φ(tk))| Since ∞ X k=1 ck < ∞, ∞ X k=1
dk < ∞ , then there exist k0() ∈ N such that ∞ X k=k0() (αkr + βk) ≤ ε. Hence |Aφ(t) − Aφ(s)| ≤ ε, ∀ t ≥ k0().
Then A(B(0, r)) is equiconvergent. With Lemma 3.1.1 and Steps 2-4, we conclude that A is completely continuous.
Step 6. Now, we show that the set
M = {x ∈ P C(J, R) : x = λBx
λ
+ λA(x), λ ∈ (0, 1)} is bounded. Let x ∈ M then
Let t ∈ [0, ∞) then, we get |x(t)| ≤
Z t
t−r
|a(u + r)||x(u)|du + e−R0ta(u+r)du
Z 0 −r |a(u + r)||x(u)|du + Z t 0
|a(s + r)|e−Rsta(u+r)du
Z s s−r |a(u + r)||x(u)|duds + X 0<tk<t |Ik(x(tk))|e −Rt tka(s+r)ds.
31 Thus |x(t)| ≤ Z t t−r |a(u + r)| sup u∈[0,t] |x(u)|du + e−R0ta(u+r)du Z 0 −r |a(u + r)| sup u∈[0,t] |x(u)|du + Z t 0
|a(s + r)|e−Rsta(u+r)du
Z s s−r |a(u + r)| sup u∈[0,t] |x(u)|duds + X 0<tk<t αk sup u∈[0,t] |x(u)| + X 0≤tk≤t βk. Therefore |x(t)| ≤ α sup s∈(0,t) |x(s)| + e−R0ta(u+r)du Z 0 −r |a(u + r)| sup s∈(0,t) |x(s)| + X 0<tk<t αk sup s∈(0,t) |x(s)| + X 0<tk<t βk. Hence |x(t)| ≤ ∞ X k=1 βk 1 − α − ∞ X k=1 αk .
By theorem 1.5.4 the problem (3.0.1) has a bounded solution.
Let ψ be an initial condition and > 0 such that kψk∞ ≤ , then
|x(t)| ≤ + X 0≤tk≤t αk sup s∈(0,t) |x(s)|. So, |x(t)| ≤ 1 − ∞ X k=1 αk .
3.3
Perturbated problem
In this section, we will show prove the bounded of solution and zero asymptotically stability of the following problem:
x0(t) = −a(t)x(t − r) + f (t, xt), t ∈ J := [0, ∞), t 6= tk, k = 1, . . . , ∆xt=tk = Ik(x(t − k)), k = 1, . . . , x(t) = ψ(t), t ∈ [−r, 0] (3.3.1)
where f : [0, ∞) × C([−r, 0], R) → R be a continuous function and 0 = t0 < t1 < . . . <
tk < . . . , lim
k→∞tk = ∞.
Our main results are based on Krasnoslkii’s fixed point theorem 1.5.4.
Theorem 3.3.1. Suppose that (H1) and (H2) and the following condition
(H5) there exist p : [0, ∞) → R+ measurable, integrable function and positive number
M ≥ 0 such that
|f (t, x)| ≤ p(t)kxk∞ for all x ∈ C([−r, 0], R)
where
Z t
0
p(s)e−Rusa(l)dlds ≤ M, for all t ∈ [0, ∞).
hold. Then problem (3.3.1) has at least one solution and all this solution are bounded.
If in (H4) we have
∞
X
k=1
βk = 0, then for every small initial condition, the solution of
problem (3.3.1) tends to zero as t → ∞.
Proof. Consider N∗ : P Cb → P Cb by (N∗x)(t) = ψ(0)e−R0ta(s+r)ds+Rt t−ra(u + r)x(u)du −e−Rt 0a(u+r)duR0 −ra(u + r)x(u)du −Rt 0 a(s + r)e −Rt sa(u+r)duRs s−ra(u + r)x(u)duds + X 0<tk<t Ik(x(tk))e −Rt tka(s+r)ds+ Z t 0 f (s, xs)e− Rt sa(u+r)duds if t ∈ [0, ∞), ψ(t) if t ∈ [−r, 0].
Let A∗ : P Cb → P Cb be operator defined by
(A∗x)(t) = X 0<tk<t Ik(x(tk))e −Rt tka(s+r)ds+ Z t 0 f (s, xs)e− Rt sa(u+r)duds, t ∈ [0, ∞) 0 if t ∈ [−r, 0]. Then N∗(x) = A∗(x) + B(x), for each x ∈ P Cb,
where B is defined in theorem 3.2.1. As in theorem 3.2.1, we can prove that A∗
completely continuous and from (H1) the operator A is contractive. Also we can easily
prove that the set
M = {x ∈ P C(J, R) : x = λB∗
x λ
33 is bounded. Hence by theorem 1.5.4 then problem (3.3.1) has at least one solution and all this solutions are bounded. Now we show that for every small initial condition the solution x of problem (3.3.1) tends to zero as t → ∞.
Let t ∈ [0, ∞) then, we get |x(t)| ≤
Z t
t−r
|a(u + r)||x(u)|du + e−R0ta(u+r)du
Z 0 −r |a(u + r)||x(u)|du + Z t 0
|a(s + r)|e−Rsta(u+r)du
Z s s−r |a(u + r)||x(u)|duds + X 0<tk<t |Ik(x(tk))|e −Rt tka(s+r)ds+ Z t 0 |f (s, xs)|e− Rt sa(u+r)duds. Thus |x(t)| ≤ Z t t−r |a(u + r)| sup u∈[0,t] |x(u)|du + e−R0ta(u+r)du Z 0 −r |a(u + r)| sup u∈[0,t] |x(u)|du + Z t 0
|a(s + r)|e−Rsta(u+r)du
Z s s−r |a(u + r)| sup u∈[0,t] |x(u)|duds + X 0<tk<t αk sup u∈[0,t] |x(u)| + Z t 0 kxsk∞p(s)e− Rt sa(u+r)duds. Therefore µ(t) ≤ 1 1 − ∞ X k=1 αk + Z t 0 µ(s)p(s)e−Rsta(u+r)duds , where µ(t) = sup{|x(s)| : s ∈ [−r, t]}. By Gronwall lemma, we obtain
|x(t)| ≤ 1 − ∞ X k=1 αk 1 + M e M 1 − ∞ X k=1 αk .
Chapter 4
Impulsive differential equations on
the half-line
In [58], Precup, established the role of vector-valued metric convergence in the study of semilinear operator systems. In recent years, many authors studied the existence of solutions for system of differential equations by using the vector version fixed point theorem; see [12, 47–49, 59] and in the references therein.
In this paper we consider the following system of impulsive differential equations
x0(t) = f (t, x, y), t ∈ J := [0, ∞), t 6= tk, k = 1, . . . , y0(t) = g(t, x, y), t ∈ J, t 6= tk, k = 1, . . . , x(t+k) − x(t−k) = Ik(x(tk), y(tk)), k = 1, . . . ,
y(t+k) − y(t−k) = Ik(x(tk), y(tk)), k = 1, . . . ,
x(0) = x0,
y(0) = y0,
(4.0.1)
where x0, y0 ∈ R, f, g : J × R × R → R are a given functions, Ik, Ik ∈ C(R × R, R).
The notations x(t+k) = lim
h→0+x(tk+ h) and x(t
−
k) = lim
h→0+x(tk − h) stand for the right
and the left limits of the functions y at t = tk, respectively.
By using Perov’s and Krasnoselskii fixed point type theorems in generalized Banach spaces, we prove the existence, uniqueness and compactness of solution sets of above problem .
4.1
Uniqueness and continuous dependence on
ini-tial data
In order to define a solution for problem (4.0.1), consider the space of piecewise con-tinuous functions
P Cb = {y ∈ P C([0, ∞), R) : y is bounded}
where P C([0, ∞), R) = {y : [0, ∞) → R, yk ∈ C((tk, tk+1], R), k = 0, . . . , y(t−k) and y(t + k)
exist and satisfy y(tk) = y(t−k) for k = 1, . . .}.
P Cb is a Banach space with norm
kykb = sup{|y(t)| : t ∈ [0, ∞)}.
Definition 4.1.1. A function (x, y) ∈ P Cb(J, R) × P Cb(J, R) is said to be a solution
of (4.0.1) if and only if x(t) = x0+ Rt 0 f (s, x(s), y(s))ds + X 0<tk<t Ik(x(tk), y(tk)), t ∈ J, y(t) = y0+ Rt 0 g(s, x(s), y(s))ds + X 0<tk<t Ik(x(tk), y(tk)), t ∈ J.
In this section we assume the following conditions:
(H1) There exist functions li ∈ L1(J, R+), i = 1, . . . , 4, such that
|f (t, x, y) − f (s, x, y)| ≤ l1(t)|x − x| + l2(t)|y − y|, for all x, x, y, y ∈ R
and
|g(t, x, y) − g(s, x, y)| ≤ l3(t)|x − x| + l4(t)|y − y|, for all x, x, y, y ∈ R.
(H2) There exist constants a1k, a2k ≥ 0, k = 1, . . . , such that
|Ik(x, y) − Ik(x, y)| ≤ a1k|x − x| + a2k|y − y|, for all x, x, y, y ∈ R
and
∞
X
k=1
|Ik(0, 0)| < ∞.
(H3) There exist constants b1k, b2k ≥ 0, k = 1, . . . , such that
|Ik(x, y) − Ik(x, y)| ≤ b1k|x − x| + b2k|y − y|, for all x, x, y, y ∈ R
and
∞
X
k=1
37 We will use the Perov fixed point theorem to prove that a solution of problem (4.0.1) is bounded and tends to zero as t → ∞.
Theorem 4.1.1. Assume that (H1) − (H3) are satisfied and the matrix
M = kl1kL1 + ∞ X k=1 a1k kl2kL1 + ∞ X k=1 a2k kl3kL1 + ∞ X k=1 b1k kl4kL1 + ∞ X k=1 b2k ∈ M2×2(R+) (4.1.1) where ∞ X k=1 aik < ∞ and ∞ X k=1 bik < ∞, i = 1, 2,
converges to zero and f (·, 0, 0), g(·, 0, 0) ∈ L1(J, R). Then the problem (4.0.1) has
unique solution. If we add that and
∞ X k=1 a1k+ ∞ X k=1 a2k+ ∞ X k=1 b1k+ ∞ X k=1 b2k < 1,
the unique solution of (4.0.1) is bounded.
Proof. Consider the operator N : P C × P C → P C × P C defined by N (x, y) = (N1(x, y), N2(x, y)) where N1(x, y)(t) = x0+ Z t 0 f (s, x(s), y(s))ds + X 0<tk<t Ik(x(tk), y(tk)), t ∈ [0, ∞) and N2(x, y)(t) = y0+ Z t 0 g(s, x(s), y(s))ds + X 0<tk<t Ik(x(tk), y(tk)), t ∈ [0, ∞).
We show that the operator N was well defined. Given (x, y) ∈ P Cb × P Cb, t ∈
[0, ∞), then kN1(x, y)kb ≤ |x0| + Z t 0 |f (s, x(s), y(s))|ds + X 0<tk<t |Ik(x(tk), y(tk))| ≤ kl1kL1kxkb+ kl2kL1kykb+ X 0<tk<t (a1kkxkb+ a2kkykb) +kf (·, 0, 0)kL1 + X 0<tk<t (|Ik(0, 0)| + | ¯Ik(0, 0)|).
Similarly we have kN2(x, y)kb ≤ kl3kL1kxkb+ kl4kL1kykb + X 0<tk<t (b1kkxkb+ b2kkykb) +kg(·, 0, 0)kL1 + X 0<tk<t (|Ik(0, 0)| + | ¯Ik(0, 0)|). Thus kN1(x, y)kb kN1(x, y)kb ≤ kl1kL1 + ∞ X k=1 a1k kl2kL1 + ∞ X k=1 a2k kl3kL1 + ∞ X k=1 b1k kl4kL1 + ∞ X k=1 b1kkxk + b2k kxkb kykb + kf (., 0, 0)kL1 + ∞ X k=1 (kIk(0, 0)kb+ k ¯Ik(0, 0)kb) kg(., 0, 0)kL1 + ∞ X k=1 (|Ik(0, 0)| + | ¯Ik(0, 0)|) .
This implies that the operator N is well defined. Clearly, fixed points of the operator N are solutions of problem (4.0.1). We show that N is a contraction. Let (x, y), (x, y) ∈ P Cb× P Cb. Then (H1) and (H2) imply
|N1(x, y)(t) − N1(x, y)(t)| ≤ Z t 0 |f (s, x(s), y(s)) − f (s, x(s), y(s))|ds + ∞ X 0<tk<t |Ik(x(tk), y(tk)) − Ik(x(tk), y(tk))| ≤ Z t 0 (l1(s)|x(s) − x(s)| + l2(s)|y(s) − y(s)|) ds + X 0<tk<t (a1k|x(tk) − x(tk)| + a2k|y(tk) − y(tk)|). Thus kN1(x, y) − N1(x, y)kb ≤ (kl1kL1 + ∞ X k=1 a1k)kx − xkb + (kl2kL1 + ∞ X k=1 a2k)ky − ykb.
39 Similarly we have kN2(x, y) − N2(x, y)kb ≤ (kl3kL1 + ∞ X k=1 b1k)kx − xkb + (kl4kL1 + ∞ X k=1 b2k)ky − ykb. Therefore kN (x, y) − N (x, y)kb ≤ M kx − xkb ky − ykb
, for all (x, y), (x, y) ∈ P Cb× P Cb.
Hence, by Theorem 2.3.1, the operator N has at least one fixed point which is a solution of problem (4.0.1).
Now we show that the solution (x, y) is bounded. Let t ∈ [0, ∞). Then, we get
|x(t)| ≤ |x0| + Z t 0 |f (s, x(s), y(s))|ds + X 0<tk<t |Ik(x(tk), y(tk))| ≤ |x0| + Z t 0 (l1(s)|x| + l2(s)|y|)ds + ∞ X k=1 a1k|x(tk)| + ∞ X k=1 a2k|y(tk)| +kf (., 0, 0)kL1 + kg(., 0, 0)kL1+ ∞ X k=1 |Ik(0, 0) + ∞ X k=1 | ¯Ik(0, 0)| and |y(t)| ≤ |y0| + Z t 0 (l3(s)|x(s)| + l4(s)|y(s)|)ds + ∞ X k=1 b1k|x(tk)| + ∞ X k=1 b2k|y(tk)| +kf (., 0, 0)kL1 + kg(., 0, 0)kL1 + ∞ X k=1 |Ik(0, 0) + ∞ X k=1 | ¯Ik(0, 0)|. Thus |x(t)| + |y(t)| ≤ |x0| + |y0| + Z t 0 ((l1(s) + l3(s))|x(s)| + (l2(s) + l4(s))|y(s)|)ds + ( ∞ X k=1 a1k+ ∞ X k=1 a2k+ ∞ X k=1 b1k+ ∞ X k=1 b2k)(|x(tk)| + |y(tk)|) +2kf (., 0, 0)kL1 + 2kg(., 0, 0)kL1 + 2 ∞ X k=1 |Ik(0, 0) + 2 ∞ X k=1 | ¯Ik(0, 0)|.
Hence sup s∈(0,t) (|x(s)| + |y(s)|) ≤ |x0| + |y0| + Z t 0 (l1(s) + l3(s) + l2(s) + l4(s)) × sup s∈[0,t] (|x(s)| + |y(s)|)ds + ∞ X k=1 a1k+ ∞ X k=1 a2k+ ∞ X k=1 b1k+ ∞ X k=1 b2k ! sup s∈[0,t] (|x(tk)| + |y(tk)|) + 2kf (., 0, 0)kL1 + 2kg(., 0, 0)kL1 + 2 ∞ X k=1 |Ik(0, 0) + 2 ∞ X k=1 | ¯Ik(0, 0)|.
This implies that sup s∈(0,t) (|x(s)| + |y(s)|) ≤ α + Z t 0 l(s) sup s∈[0,t] (|x(s)| + |y(s)|)ds where α = |x0| + |y0| + 2kf (., 0, 0)kL1 + 2kg(., 0, 0)kL1 + 2 ∞ X k=1 |Ik(0, 0) + 2 ∞ X k=1 | ¯Ik(0, 0)| 1 − ∞ X k=1 a1k+ ∞ X k=1 a2k+ ∞ X k=1 b1k+ ∞ X k=1 b2k ! and l(s) = l1(s) + l2(s) + l3(s) + l4(s) 1 − ∞ X k=1 a1k+ ∞ X k=1 a2k+ ∞ X k=1 b1k + ∞ X k=1 b2k ! .
Using the Gronwall Inequality we obtain sup s∈[0,t] (|x(s)| + |y(s)|) ≤ α exp Z t 0 l(s)ds . Then kxkb+ kykb ≤ α exp Z ∞ 0 l(s)ds . This implies that the solution (x, y) is bounded.
For the next result we prove the continuous dependence of solutions on initial conditions.
Theorem 4.1.2. Assume the conditions (H1)−(H3) hold, that the matrix M defined in
(4.1.1) converges to zero, that Ik(0, 0) = ¯Ik(0, 0), k = 1, . . . and f (t, 0, 0) = g(t, 0, 0) =
0, t ∈ J. For every (x0, y0) ∈ R × R we denote by (x(t, x0), y(t, y0)) the solution of
41 Proof. Let (x0, y0), (¯x0, ¯y0) ∈ R×R. Then from Theorem 4.1.1, there exist (x(·, x0), y(·, y0)),
(¯x(·, ¯x0), ¯y(·, ¯y0)) ∈ P Cb × P Cb such that
x(t, x0) = x0+ Z t 0 f (s, x(s, x0), y(s, y0))ds + X 0<tk<t Ik(x(tk, x0), y(tk, y0)), t ∈ [0, ∞), y(t, y0) = y0+ Z t 0 g(s, x(s, x0), y(s, y0))ds + X 0<tk<t Ik(x(tk, x0), y(tk, y0)), t ∈ [0, ∞), x(t, ¯x0) = ¯x0+ Z t 0 f (s, x(s, ¯x0), y(s, ¯y0))ds + X 0<tk<t Ik(x(tk, ¯x0), y(tk, ¯y0)), t ∈ [0, ∞), and ¯ y(t, ¯y0) = ¯y0+ Z t 0 g(s, x(s, ¯x0), y(s, ¯y0))ds + X 0<tk<t Ik(x(tk, ¯x0), y(tk, ¯y0)), t ∈ [0, ∞).
Hence from the proof of Theorem 4.1.1 we deduce that kx(·, x0) − ¯x(·, ¯x0)kb+ ky(·, y0) − ¯y(·, ¯y0)kb ≤ |x0− ¯x0| + |y0− ¯y0| 1 − ∞ X k=1 a1k+ ∞ X k=1 a2k+ ∞ X k=1 b1k+ ∞ X k=1 b2k ! × exp Z ∞ 0 l(s)ds . Then kx(·, x0) − ¯x(·, ¯x0)kb+ ky(·, y0) − ¯y(·, ¯y0)kb → 0, as (x0, y0) → (¯x0, ¯y0).
4.2
Existence and compactness of solution sets
In this section we present an application of e Krasnoselskii’s type fixed point theorem to problem (4.0.1).
Theorem 4.2.1. Let (H1) be satisfied and the following conditions:
(H4) There exist αk, βk ≥ 0, k = 1, . . . , such that
(H5) There exist αk, βk ≥ 0, k = 1, . . . , such that
|Ik(x, y)| ≤ αk|x| + βk|y| + ck, for all (x, y) ∈ R × R.
If M∗ = kl1kL1 kl2kL1 kl3kL1 kl4kL1 ∈ M2×2(R+) (4.2.1)
converges to zero and
∞ X k=1 αk+ ∞ X k=1 αk+ ∞ X k=1 βk+ ∞ X k=1 βk< 1, ∞ X k=1 ck < ∞ and ∞ X k=1 ck < ∞,
then the problem (4.0.1) has at least one bounded solution.
Proof. Let N : P Cb × P Cb → P Cb × P Cb be the operator defined in Theorem 4.1.1.
N = A + B where A, B : P Cb× P Cb → P Cb× P Cb are defined by
B(x(t), y(t)) = (B1(x(t), y(t)), B2(x(t), y(t)), t ∈ J,
where B1(x, y) = x0+ Rt 0 f (s, x(s), y(s))ds B2(x, y)) = y0+ Rt 0g(s, x(s), y(s))ds and
A(x(t), y(t)) = (A1(x(t), y(t)), A2(x(t), y(t)), t ∈ J
where A1(x, y) = ∞ X 0<tk<t Ik(x(tk), y(tk)) A2(x, y)) = ∞ X 0<tk<t Ik(x(tk), y(tk)).
Step 1 B is a contraction. Let (x, y), (x, y) ∈ P Cb× P Cb. Then
|B1(x(t), y(t)) − B1(xt), y(t))| ≤ Z t 0 |f (s, x(s), y(s)) − f (s, x(s), y(s))| ≤ Z t 0 (l1(s)|x(s) − x(s)| + l2(s)|y(s) − y(s)|) ds. Hence kB1(x, y) − B1(x, y)kb ≤ kl1kL1kx − xkb + kl2kL1ky − ykb. Similarly we have kB2(x, y) − B2(x, y)kb ≤ kl3kL1kx − xkb + kl4kL1ky − ykb. Therefore kB(x, y) − B(x, y)kb ≤ kl1kL1 kl2kL1 kl3kL1 kl4kL1 kx − xkb ky − ykb .
43 Step 2. A is continuous
Given (xn, yn) → (x, y) in P Cb× P Cb, then there exists M, M
0
> 0 such that kxnkb ≤ M and kynkb ≤ M0 for every n ∈ N,
and |(A1xn)(t) − (A1x)(t)| ≤ X 0≤tk<t |Ik(xn, yn) − Ik(x, y)|. Since X αk< ∞, and X βk < ∞,
hence for every > 0, there exists k0 ∈ N such that ∞ X k=k0 αk< 6M, ∞ X k=k0 βk< 6M0.
Using the fact limk→∞tk = ∞, there exists n0 ∈ N, such that for each k ≥ n0 ⇒
tk ≥ k0. From (H3), we get k A1(xn, yn) − A1(x, y) kP Cb ≤ X 0≤tk≤tn0−1 |Ik(xn, yn) − Ik(x, y)| + ∞ X k=k0 (2M αk+ 2M0βk) ≤ k0−1 X k=1 |Ik(xn, yn) − Ik(x, y)| +2 3.
Using the fact that Ik are continuous functions, then we have
n0−1 X k=0 |Ik(xn, yn) − Ik(x, y)| → 0 as n → ∞. Hence kA1(xn, yn) − A1(x, y)kb → 0 as n → ∞. Similarly we have kA2(xn, yn) − A2(x, y)kb → 0 as n → ∞. Thus
Step 3 From (H3), we can easily prove that A maps bounded sets into bounded sets
in P C × P C.
We will now show that A(M ) is contained in a compact set.
Step 4. A maps bounded sets in P Cb× P Cb into almost equicontinuous sets of P Cb×
P Cb. Let r = (r1, r2) > 0, Br := {(x, y) ∈ P Cb × P Cb : k(x, y)k∞ ≤ r} be a
bounded set in P C × P C, τ1, τ2 ∈ [0, ∞), τ1 < τ2, and φ ∈ Br. We have
Aφ(τ1) = (A1φ(τ1), A2φ(τ1)) where A1φ(τ1) = P 0≤tk≤τ1Ik(φ1(tk), φ2(tk)) A2φ(τ1). = P0≤tk≤τ2Ik(φ1(tk), φ2(tk)) . Then |A1φ(τ2) − A1φ(τ1)| ≤ X τ1≤tk≤τ2 Ik(φ1(tk), φ2(tk)). Thus |A1φ(τ2) − A1φ(τ1)| → 0 as τ1 → τ2. Similarly we have |A2φ(τ2) − A2φ(τ1)| → 0 as τ1 → τ2. Thus |Aφ(τ1) − Aφ(τ2)| → 0 as τ1 → τ2.
Step 5. We now show that the set A(B(0, r)) is equiconvergent, i.e. for every ε > 0, there exists T (ε) > 0 such that kA(φ(t)) − A(φ(s))k ≤ ε for every t, s ≥ T () and
each φ ∈ B(0, r). Letting φ ∈ B(0, r), then for every > 0, there exists k0 ∈ N
such that ∞ X k=k0 αk< 2r1 , ∞ X k=k0 βk< 2r2 , |A1φ(t) − A1φ(s)| ≤ X s≤tk≤t Ik(φ1(tk), φ2(tk)) ≤ X s≤tk≤t (αkr1+ βkr2).
Then, for every s, t > k0, we get
|A1φ(t) − A1φ(s)| ≤ r1 ∞ X k=k0 αk+ r2 ∞ X k=k0 βk.
45 Therefore for all φ ∈ B(0, r) and s, t > k0 we have
|A1φ(t) − A1φ(s)| ≤ .
Similarly we can prove that there exists ¯k0 > 0 such that for all φ ∈ B(0, r) and
s, t > k0 we have
|A2φ(t) − A2φ(s)| ≤ .
Thus, for every (, ) > 0 there exists (k0, ¯k0) > 0 such that for all s, t > k0 and
s, t > ¯k0 we have
|Aφ(τ1) − Aφ(τ2)| ≤ (, ), for every φ ∈ B(0, r).
Step 6 Now, we show that the set
M = {(x, y) ∈ P C × P C; (x, y) = λB(x
λ, y
λ) + λA(x, y), λ ∈ (0, 1)} is bounded. Let (x, y) ∈ M then
|x(t)| ≤ |x0| + Z t 0 λ f s,x(s) λ , y(s) λ ds + X 0<tk<t |Ik(x(tk), y(tk))| ≤ |x0| + Z t 0 (l1(s)|x(s)| + l2(s)|y(s)|)ds + ∞ X k=1 αk|x(tk)| + ∞ X k=1 βk|y(tk)| + ∞ X k=1 ck |y(t)| ≤ |y0| + Z t 0 (l3(s)|x(s)| + l4(s)|y(s)|)ds + ∞ X k=1 αk|x(tk)| + ∞ X k=1 βk|y(tk)| + ∞ X k=1 ck. Thus |x(t)| + |y(t)| ≤ |x0| + |y0| + Z t 0 ((l1(s) + l3(s))|x(s)| + (l(s)2+ l(s)4)|y(s)|)ds + ( ∞ X k=1 αk+ ∞ X k=1 αk)|x(tk| + ( ∞ X k=1 βk+ ∞ X k=1 βk)|y(tk| + ∞ X k=1 ck+ ∞ X k=1 ck.
Hence sup s∈[0,t] (|x(s)| + |y(s)|) ≤ |x0| + |y0| + Z t 0 (l1(s) + l3(s) + l2(s) + l4(s)) × sup s∈[0,t] (|x(s)| + |y(s)|)ds + ( ∞ X k=1 αk+ ∞ X k=1 αk) sup s∈(0,t) |x(tk| + ( ∞ X k=1 βk+ ∞ X k=1 βk) sup s∈(0,t) |y(tk| + ∞ X k=1 ck+ ∞ X k=1 ck.
This implies that sup s∈(0,t) (|x(s)| + |y(s)|) ≤ β + Z t 0 l∗(s) sup s∈[0,t] (|x(s)| + |y(s)|)ds where β = |x0| + |y0| + ∞ X k=1 ck+ ∞ X k=1 ck 1 − ( ∞ X k=1 αk+ ∞ X k=1 αk+ ∞ X k=1 βk+ ∞ X k=1 βk) and l∗(s) = l1(s) + l3(s) + l2(s) + l4(s) 1 − ( ∞ X k=1 αk+ ∞ X k=1 αk+ ∞ X k=1 βk+ ∞ X k=1 βk) .
By the Gronwall Inequality, we have sup
s∈(0,t)
(|x(s)| + |y(s)|) ≤ βekl∗kL1. Then
kxkb ≤ βekl∗kL1, and kykb ≤ βekl∗kL1.
Hence from Theorem 2.3.2 the problem (4.0.1) has at least one solution.
By simple modification in the prove we can obtain the following result.
47
(H6) There exists p ∈ L1(J, R+), and let ψ : R+ → (0, ∞) be a continuous nondecreasing
function such that
|f (t, x, y)| ≤ p(t)ψ(|x| + |y|), for all (x, y) ∈ R × R and
|g(t, x, y)| ≤ p(t)ψ(|x| + |y|), for all (x, y) ∈ R × R. If ¯ M∗ = ∞ X k=1 a1k ∞ X k=1 a2k ∞ X k=1 b1k ∞ X k=1 b2k ∈ M2×2(R+) (4.2.2)
converges to zero, then the problem (4.0.1) has unique bounded solution.
By the nonlinear alternative in generalized Banach space we can also prove the following result.
Theorem 4.2.3. Assume that (H4) − (H6) hold. If
∞ X k=1 αk+ ∞ X k=1 αk+ ∞ X k=1 βk+ ∞ X k=1 βk < ∞, ∞ X k=1 ck< ∞ and ∞ X k=1 ck< ∞,
then the problem (4.0.1) has at least one solution. Moreover, the solution set S(x0, y0) = {(x, y) ∈ P Cb× P Cb : (x, y) is solution of(4.0.1)}
Chapter 5
Differential Inclusions
It is well known that in C([0, ∞), Rn), the distance between two trajectories of an
ordinary differential equation y0 = f (x, y) (with a Lipschitz continuous vector field)
is majorized by the distance between two initial points multiplied by an exponential function of time. The object of this chapter is to prove a Filippov type theorem and a Filippov-Wazewski type theorem for impulsive differential inclusions. More precisely, we consider the problem
y0 ∈ F (t, y(t)), a.e. t ∈ J := [0, b], t 6= tk, k = 1, . . . , m, (5.0.1)
y(t+k) − y(tk) = Ik(y(t−k)), k = 1, . . . , m, (5.0.2)
y(0) = y0, (5.0.3)
where F : J × Rn → P(Rn) is a multivalued map, y
0 ∈ Rn, and Ik ∈ C(Rn, Rn),
k = 1, 2, . . . , m. In the case where the impulses are absent (i.e., Ik ≡ 0, k = 1, 2, . . . , m)
and the problem (5.0.1)–(5.0.3) reduces to an autonomous control system, that is, F (t, y) := f (t, U ), where f is single map and U is a control set, a property of Filippov’s theorem. Some generalizations of Filippov’s theorem were considered by Frankowska [28] and Zhu [68].
5.1
Filippov’s Theorem
In order to define a solution of (5.0.1)–(5.0.3), we consider the space
P C(J, Rn) = {y :J → Rn| y(t) is continuous everywhere except for some tk at
which y(t−k) and y(t+k) exist and y(t−k) = y(tk), k = 1, 2, . . . , m }.
Clearly, P C(J, Rn) is a Banach space with the norm
kykP C = sup{|y(t)| : t ∈ J }.
We begin by defining what is meant by a solution of the problem (5.0.1)–(5.0.3). 49
