nonlinear impulsive partial functional differential inclusions with state dependent delay and multivalued jumps

Contents lists available atScienceDirectNonlinear Analysis: Hybrid Systems journal homepage:www.elsevier.com/locate/nahs Nonlinear impulsive partial functional differential inclusions wi

Trang 1

Contents lists available atScienceDirect

Nonlinear Analysis: Hybrid Systems

journal homepage:www.elsevier.com/locate/nahs

Nonlinear impulsive partial functional differential inclusions with

state-dependent delay and multivalued jumps

Nadjet Abadaa, Mouffak Benchohrab,∗, Hadda Hammouchec

aDépartement de Mathématiques, Université Mentouri de Constantine, Algérie

bLaboratoire de Mathématiques, Université de Sidi Bel Abbès, BP 89, 22000 Sidi Bel Abbès, Algérie

cDépartement de Mathématiques, Université Kasdi Merbah de Ouargla, Algérie

a r t i c l e i n f o

Article history:

Received 29 January 2009

Accepted 19 May 2010

Keywords:

Partial differential inclusions

Impulses

Multivalued jumps

State-dependent delay

Integral solution

Semigroup

a b s t r a c t

In this paper, we shall establish sufficient conditions for the existence of integral solutions for some nondensely defined impulsive semilinear functional differential inclusions with state-dependent delay in separable Banach spaces We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators

1 Introduction

In this paper, we shall be concerned with the existence of integral solutions defined on a compact real interval for first order impulsive semilinear functional inclusions with state-dependent delay in a separable Banach space of the form:

y0(t) ∈Ay(t) +F(t,yρ(t,y t)), t∈J= [0,b] , (1.1)

where F :J×D→P(E)is a given multivalued map with nonempty convex compact values,Dis the phase space defined axiomatically (see Section2) which contains the mappings from(−∞,0]into E,φ ∈D,0=t0<t1< · · · <t m<t m+ 1=

b, I k:D →P(E),k=1,2, ,m are bounded valued multivalued maps,P(E)is the collection of all nonempty subsets

of E, ρ :I×D→ (−∞,b], A:D(A) ⊂E→E is a nondensely defined closed linear operator on E, and E a real separable

Banach space with norm| | For any function y defined on(−∞,b] \ {t1,t2, ,t m}and any t ∈ J, we denote by y tthe element ofDdefined by

y t(θ) =y(t+ θ), θ ∈ (−∞,0]

In recent years, impulsive differential and partial differential equations have been the object of much investigation because they can describe various models of real processes and phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics We refer the reader to the monographs of Bainov and Simeonov [1], Benchohra

∗Corresponding author Fax: +213 48 54 43 44.

E-mail addresses:n65abada@yahoo.fr (N Abada), benchohra@univ-sba.dz (M Benchohra), h.hammouche@yahoo.fr (H Hammouche).

doi:10.1016/j.nahs.2010.05.008

Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state-dependent delay and

Trang 2

et al [2], Lakshmikantham et al [3], and Samoilenko and Perestyuk [4] where numerous properties of their solutions are studied, and a detailed bibliography is given Semilinear functional differential equations and inclusions with or without

impulses have been extensively studied where the operator A generates a C0-semigroup Existence and uniqueness, among other things, are derived; see the books of Ahmed [5,6], Benchohra et al [7], Heikkila and Lakshmikantam [8], Kamenski

et al [9] and the papers by Ahmed [10,11], Liu [12] and Rogovchenko [13,14] In [15] Abada et al have studied the controllability of a class of impulsive semilinear functional differential inclusions in Fréchet spaces by means of the extrapolation method [16,17], and in [18] the existence of mild and extremal mild solutions for first-order semilinear densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions has been considered To the best of our knowledge, there are very few results for impulsive evolution inclusions with multivalued jump operators; see [18–20]

The notion of the phase spaceDplays an important role in the study of both qualitative and quantitative theory A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [21] (see also [22,23]) For a detailed discussion on this topic we refer the reader to the book by Hino et al [24] For the case where the impulses are

absent (i.e I k=0,k=1, ,m), an extensive theory has been developed for the problem(1.1)–(1.3) We refer to Belmekki

et al [25], Corduneanu and Lakshmikantham [26], Hale and Kato [21], Hino et al [24], Lakshmikantham et al [27] and Shin [28] The literature related to ordinary and partial functional differential equations with delay for whichρ(t, ψ) =t is very

extensive; see for instance the books [29–32] and the papers therein On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations and for this reason the study of this type

of equations has received great attention in the last year, see, for instance, [33,34] and the references therein The literature related to partial functional differential equations with state-dependent delay is limited (see [35–37])

This paper is organized as follows In Section2, we will recall briefly some basic definitions and preliminary facts which will be used throughout the following sections In Section3we give some examples of operators with nondense domain In Section4, we prove existence of integral solutions for problem(1.1)–(1.3) Our approach will be based for the existence of integral solutions, on a fixed point theorem of Dhage [38] for the sum of a contraction map and a completely continuous map

In Section5we present some examples of phase spaces Finally in Section6we give an example to illustrate the abstract theory The results of the present paper extend to a nondensely defined operator some ones considered in the previous literature

2 Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper For

ψ ∈Dthe norm ofψis defined by

k ψkD =sup{| ψ(θ)| : θ ∈ (−∞,0]}

Also B(E)denotes the Banach space of bounded linear operators from E into E, with norm

kNkB(E)=sup{|N(y)| : |y| =1}

L1(J,E)denotes the Banach space of measurable functions y:J−→E which are Bochner integrable normed by

kykL1 =

Z b

0

|y(t)|dt.

To consider the define the solution of problem (1.1)–(1.3), it is convenient to introduce some additional concepts and notations Consider the following space

Bb=

n

y: (−∞,b] →E,y k∈C(J k,E)and there exist y(t k−),y(t k+)with y(t k) =y(t k−),y(t) = φ(t),t ≤0

o

where y k is the restriction of y to J k= (t k,t k+ 1] ,k=0, ,m Letk · kbbe the semi-norm inBbdefined by

kykb= ky0kD+sup{|y(s)| :0≤s≤b} , y∈Bb.

In this work, we will employ an axiomatic definition for the phase spaceDwhich is similar to those introduced in [24] Specifically,Dwill be a linear space of functions mapping(−∞,0]into E endowed with a semi normk kD, and satisfies the following axioms introduced at first by Hale and Kato in [21]:

(A1) There exist a positive constant H and functions K(·), M(·) :R+→R+with K continuous and M locally bounded, such that for any b>0, if y: (−∞,b] →E, y∈D, and y(·)is continuous on[0,b], then for every t∈ [0,b]the following conditions hold:

(i) y tis inD;

(ii)|y(t)| ≤Hky tkD;

(iii) ky tkD≤K(t)sup{|y(s)| :0≤s≤t} +M(t)ky0kD, and H,K and M are independent of y(·)

(A2) The spaceDis complete

Trang 3

In what follows we use the following notations

K b=sup{K(t) : t ∈J} and M b=sup{M(t) : t∈J}

Definition 2.1 ([ 39 ]) Let E be a Banach space An integrated semigroup is a family of operators(S(t))t≥ 0of bounded linear

operators S(t)on E with the following properties:

(i) S(0) =0;

(ii) t →S(t)is strongly continuous;

(iii) S(s)S(t) = Rs

0(S(t+r) −S(r))dr,for all t,s≥0.

Definition 2.2 An integrated semigroup(S(t))t≥ 0is called exponential bounded, if there exists constant M≥0 andω ∈R such that

|S(t)| ≤Meωt, for t≥0.

Moreover(S(t))t≥ 0is called nondegenerate if S(t)y=0, for all t≥0, implies y=0.

Definition 2.3 An operator A is called a generator of an integrated semigroup, if there existsω ∈R such that(ω, +∞) ⊂ ρ(A), and there exists a strongly continuous exponentially bounded family(S(t))t≥ 0of linear bounded operators such that

S(0) =0 and(λI−A)− 1= λ R∞

0 e− λt S(t)dt for allλ > ω.

If A is the generator of an integrated semigroup(S(t))t≥ 0which is locally Lipschitz, then from [39], S(·)y is continuously

differentiable if and only if y∈D(A) In particular S0(t)y:= d

and(S0(t))t≥ 0is a C0semigroup on D(A) Here and hereafter, we assume that A satisfies the Hille–Yosida condition, that is, there exists M>0 andω ∈R such that(ω, ∞) ⊂ ρ(A)and

sup{ (λI− ω)n| (λI−A)−n| : λ > ω, n∈N} ≤M.

Note that, since A satisfies the Hille–Yosida condition,

kS0(t)kB(E)≤Meωt, t ≥0,

where M andω are the constants considered in the Hille–Yosida condition (see [40]) Let(S(t))t≥ 0, be the integrated

semigroup generated by A.

Consider the Cauchy Problem

y0(t) =Ay(t) +f(t), t∈ [0,b] , y(0) =y0∈E. (2.1) Then we have the following

Theorem 2.1 ([ 40 ]) Let f : [0,b] →E be a continuous function Then for y0∈D(A), there exists a unique continuous function

(i) Rt

0y(s)ds∈D(A)for t∈ [0,b] ,

(ii) y(t) =y0+AR0t y(s)ds+ R0t f(s)ds,t ∈ [0,b] ,

(iii) |y(t)| ≤Meωt |y0| + Rt

0e−ωs|f(s)|ds ,t∈ [0,b]

y(t) =S0(t)y0+ d

dt

Z t

0

S(t−s)f(s)ds, t≥0. (2.2)

Let Bλ= λR(λ,A) := λ(λI−A)− 1 Then [40] for all y∈D(A),Bλy→y asλ → ∞ Also from the Hille–Yosida condition

(with n=1) it easy to see that limλ→∞|Bλy| ≤M|y|, since

|Bλ| = | λ(λI−A)− 1| ≤ Mλ

λ − ω . Thus limλ→∞|Bλ| ≤M Also if y is given by(2.2), then

y(t) =S0(t)y0+ lim

λ→∞

Z t

0

S0(t−s)Bλf(s)ds, t≥0. (2.3)

Trang 4

We finish this section, with notations, definitions, and some results from multivalued analysis Let(X,d)be a metric space.

We use the notations:

P cl(X) = {Y ∈P(X) :Y closed} , P bd(X) = {Y∈P(X) :Y bounded}

P cv(X) = {Y ∈P(X) :Y convex} , P cp(X) = {Y ∈P(X) :Y compact}

Consider H d:P(X) ×P(X) →R+S{∞}given by

H d(A,B) =max{sup

a∈A

d(a,B),sup

b∈B

d(A,b)},

where d(A,b) =infa∈A d(a,b),d(a,B) =infb∈B d(a,b) Then(P bd,cl(X),H d)is a metric space and(P cl(X),H d)is a generalized metric space (see [41])

A multivalued map N:J→P cl(X)is said to be measurable if, for each x∈X , the function Y :J→R defined by

Y(t) =d(x,N(t)) =inf{d(x,z) :z∈N(t)},

is measurable

Definition 2.4 A measurable multivalued function F : J → P bd,cl(X)is said to be integrably bounded if there exists a functionw ∈L1(J,R+)such thatk vk ≤ w(t)a.e t∈J for allv ∈F(t).

A multivalued map F :X→P(X)is convex (closed) valued if F(x)is convex (closed) for all x∈X F is bounded on bounded

sets if F(B) = Sx∈B F(x)is bounded in X for all B∈P b(X)i.e supx∈B{sup{|y| :y∈F(x)}} < ∞ F is upper semi-continuous (u.s.c for short) on X if for each x0∈X the set F(x0)is nonempty, closed subset of X , and for each open setUof X containing

F(x0), there exists an open neighborhoodVof x0such that F(V) ⊆ U.G is said to be completely continuous if F(B)is

relatively compact for every B∈P bd(X) If the multivalued map F is completely continuous with nonempty compact valued, then G is u.s.c if and only if F has closed graph i.e x n→x∗,y n→y∗,y n∈G(x∗)imply y∗∈G(x∗).

Definition 2.5 A multivalued map F:J×D→P(E)is said to be Carathéodory if

(i) t7−→F(t,u)is measurable for each u∈D

(ii) u7−→F(t,u)is u.s.c for almost each t∈J.

Definition 2.6 A multivalued operator N :J→P cl(X)is called

(a) contraction if and only if there existsγ >0 such that

H d(N(x),N(y)) ≤ γd(x,y), for each x,y∈X,

withγ <1,

(b) N has a fixed point if there exists x∈X such that x∈N(x).

For more details on multivalued maps and the proof of the known results cited in this section we refer interested reader to the books of Deimling [42], Gorniewicz [43] and Hu and Papageorgiou [44]

3 Examples of operators with nondense domain

In this section we shall present examples of linear operators with nondense domain satisfying the Hille–Yosida estimate More details can be found in the paper by Da Prato and Sinestrari [45]

Example 3.1 Let E=C([0,1] ,R)and the operator A:D(A) →E defined by Ay=y0, where

D(A) = {y∈C1((0,1),R) :y(0) =0}

Then

D(A) = {y∈C((0,1),R) :y(0) =0} 6=E.

Example 3.2 Let E=C([0,1] ,R)and the operator A:D(A) →E defined by Ay=y00, where

D(A) = {y∈C2((0,1),R) :y(0) =y(1) =0}

Then

D(A) = {y∈C((0,1),R) :y(0) =y(1) =0} 6=E.

Trang 5

Example 3.3 Let us set for someα ∈ (0,1)

0([0,1] ,R) =

y: [0,1] →R:y(0) =0 and sup

0 ≤t<s≤ 1

|y(t) −y(s)|

|t−s|α < ∞

and the operator A:D(A) →E defined by Ay= −y0, where

D(A) = {y∈C1+α((0,1),R) :y(0) =y0(0) =0}

Then

D(A) =hα

0((0,1),R) =

y: [0,1] →R:lim

δ→ 0 sup

0 <|t−s|≤ δ

|y(t) −y(s)|

|t−s|α =0

6=E. Here

C1+α([0,1] ,R) = {y: [0,1] →R:y0∈Cα([0,1] ,R)}.

The elements of hα((0,1),R)are called little Holder functions and it can be proved that the closure of C1((0,1),R)in

Cα((0,1),R)is hα((0,1),R)(see [46] Theorem 5.3)

Example 3.4 LetΩ ⊂ Rnbe a bounded open set with regular boundaryΓ and define E = C(Ω,R)and the operator

D(A) = {y∈C(Ω,R) :y=0 onΓ;∆y∈C(Ω,R)}.

Here∆is the Laplacian in the sense of distributions onΩ In this case we have

D(A) = {y∈C(Ω,R) :y=0 onΓ} 6=E.

4 Existence of integral solutions

Now, we are able to state and prove our main theorem for the initial value problem(1.1)–(1.3) Before starting and proving this one, we give the definition of the integral solution

Definition 4.1 We say that y : (−∞,T] → E is an integral solution of(1.1)–(1.3)if y(t) = φ(t)for all t ∈ (−∞,0],

the restriction of y(·)to the interval[0,b]is continuous, and there existv(·) ∈ L1(J k,E) andIk ∈ I k(y t k), such that v(t) ∈F(t,yρ(t,y t))a.e t∈ [0,b], and y satisfies the integral equation,

(i) y(t) = φ(0) +ARt

0y(s)ds+ Rt

0v(s)ds+ P

0 <t k<tIk , t ∈J. (ii)Rt

0y(s)ds∈D(A)for t∈J.

From the definition it follows that y(t) ∈D(A), for each t≥0, in particularφ(0) ∈D(A) Moreover, from [39,40] y satisfies

the following variation of constants formula:

y(t) =S0(t)φ(0) + d

dt

Z t

0

0 <t k<t

S0(t−t k)Ik t ≥0. (4.1)

We notice also that, if y satisfies(4.1), then

y(t) =S0(t)φ(0) + lim

λ→∞

Z t

0

S0(t−s)Bλv(s)ds+ X

0 <t k<t

S0(t−t k)Ik, t ≥0. The key tool in our approach is the following form of the fixed point theorem of Dhage [38,47]

Theorem 4.1 Let X be a Banach space,A:X→P cl,cv,bd(X)andB:X→P cp,cv(X), two multivalued operators satisfying

Then either

(i) The operator inclusionλx∈Ax+Bx has a solution forλ =1, or

(ii) the setE = {u∈X|u∈ λAu+ λBu,0≤ λ ≤1}is unbounded.

SetD˜ be the space of functions inDwhich have values in D(A).We always assume thatρ : I×D → (−∞,b]is continuous Additionally, we introduce following hypotheses:

Trang 6

(Hφ) The function t → φtis continuous fromR(ρ−) = {ρ(s, ϕ) : (s, ϕ) ∈J×D, ρ(s, ϕ) ≤0}intoDand there exists a

continuous and bounded function Lφ:R(ρ−) → (0, ∞)such thatk φtkD ≤Lφ(t)kφkDfor every t∈R(ρ−)

(H1) A satisfies Hille–Yosida condition;

(H2) There exist constants c k>0,k=1, ,m with Meωb K bPm

k= 1c k<1 such that

H d(I k(y),I k(x)) ≤c k|y−x| for each x,y∈ ˜D.

(H3) The multivalued map F:J×D→P cp,cv(E)is Carathéodory;

(H4) the operator S0(t)is compact in D(A)wherever t>0;

(H5) There exist a function p∈L1(J,R+)and a continuous nondecreasing functionψ : [0, ∞) → (0, ∞)such that

kF(t,x)kP ≤p(t)ψ(kxkD) for a.e t∈J and each x∈D,

withR0be−ωs p(s)ds< ∞,

lim sup

u→+∞

(M b+Lφ+MK

b)kφkD+K b

u

c∗

1+c∗

2ψ K b u+ (M b+Lφ+MK b)kφkD

Rb

0 e− ωs p(s)ds >1 (4.2) where

1−Meωb K b

m

P

k= 1

c k

+ M b+Lφ+MK

m

X

k= 1

|I k(0)| +c k M b+Lφ+MK

b k φkD

and

1−Meωb K b

m

P

k= 1

c k

The next result is a consequence of the phase space axioms

Lemma 4.1 ([ 36 ], Lemma 2.1) If y: (−∞,b] →E is a function such that y0= φand y|J ∈PC(J:D(A)), then

ky skD ≤ (M a+Lφ)kφkD+K asup{ky(θ)k; θ ∈ [0,max{0,s}]} , s∈R(ρ−) ∪J,

t∈ R (ρ − )Lφ(t), M a=supt∈J M(t)and K a=supt∈J K(t).

Remark 4.1 We remark that condition(Hφ)is satisfied by functions which are continuous and bounded In fact, if the space

Dsatisfies axiom C2in [24] then there exists a constant L> 0 such thatk φkD ≤L sup{k φ(θ)k : θ ∈ [−∞,0]}for every

φ ∈Dthat is continuous and bounded, (see [24] Proposition 7.1.1) for details Consequently

k φtkD ≤L

sup

θ≤ 0

k φ(θ)k

k φkD

k φkD, for everyφ ∈D\ {0}

Theorem 4.2 Assume that(Hφ)and(H1)–(H5)hold If φ(0) ∈ D(A), then the problem(1.1)–(1.3)has at least one integral

Proof Transform the problem(1.1)–(1.3)into a fixed point problem Set

Ω =PC(−∞,b] ,D(A) ,

and consider the multivalued operator N:Ω →P(Ω)defined by

N(y) = {h∈Ω}

with

h(t) =







φ(t), if t≤0,

S0(t)φ(0) + d

dt

Z t

0

0 <t k<t

S0(t−t k)Ik, v ∈S F,yρ( s,ys),Ik∈I k(y(t k−)) if t∈J.

Trang 7

Forφ ∈Ddefine the functione φ : (−∞,b] →E such that:

e φ(t) =

φ(t), if t≤0

S0(t)φ(0), if t∈J.

Thene φ0= φ For each x∈Bb with x(0) =0, we denote by x the function defined by

x(t) =

0, t∈ (−∞,0] ,

x(t), t∈J.

If y(.)satisfies(4.1), we can decompose it as y(t) = e φ(t) +x(t),0≤t≤b, which implies y t= e φt+x t, for every 0≤t ≤b

and the function x(.)satisfies

x(t) = d

dt

Z t

0

0 <t k<t

S0(t−t k)Ik t∈J, wherev(s) ∈S F,xρ( s,xs+e φs) +e φρ(s,xs+e φs)andIk∈I k x t k+ e φt k

Let

B0= {x∈Bb: x0=0∈D}

For any x∈Bb0we have

kxkb= kx0kD+sup{|x(s)| :0≤s≤b} =sup{|x(s)| :0≤s≤b}

Thus(Bb0, k · kb)is a Banach space We define the two multivalued operatorsA,B : B0b → P(Bb0)byA(x) := {h ∈

B0} ,B(x) := {h∈B0}with

h(t) =







0, if t ∈ (−∞,0];

X

0 <t k<t

S0(t−t k)Ik, Ik∈I k x t k+ e φt k

, if t ∈J, and

h(t) =







d

dt

Z t

0

S(t−s)v(s)ds, v(s) ∈S F,xρ( s,xs+e φs)+eφρ(s,xs+eφs) if t ∈J.

Obviously to prove that the multivalued operator N has a fixed point is reduced that the operator inclusion x∈A(x)+B(x) has one, so it turns to show that the multivalued operatorsAand B satisfy all conditions ofTheorem 4.1 For better readability, we break the proof into a sequence of steps

Let x1,x2∈B0 Then for t∈J

0 <t k<t

S0(t−t k)I k(x1t

k+ e φt k), X

0 <t k<t

S0(t−t k)I k(x2t

k+ e φt k)

!

≤ Meωb X

0 ≤t k≤t

I k(x1t

k) −I k(x2t

k)

≤ Meωb

m

X

k= 1

c kkx1t

k−x2t

kkD

≤ Meωb K b

m

X

k= 1

c kkx1−x2kD. Hence by (H2)Ais a contraction

This will be given in several claims

The operatorBis equivalent to the compositionL◦S F on L1(J,E), whereL:L1(J,E) →Bb0is the continuous operator defined by

L(v)(t) = d

dt

Z t

0

S(t−s)v(s)ds, t ∈J.

Trang 8

Then, it suffices to show that L◦ S F has compact values on B0 Let x ∈ B0 arbitrary andvn a sequence such that

vn(t) ∈S F,xρ( t,xt+e φt) +e φρ(t,xt+eφt), a.e t∈J Since F(t,xρ(t,x t+e φt)+e φρ(t,x t+e φt))is compact, we may pass to a subsequence Suppose thatvn→ vin L1w(J,E)(the space endowed with the weak topology), wherev(t) ∈F(t,xρ(t,x t+e φt)+ e φρ(t,x t+e φt)), a.e t∈J.

An application of Mazur’s theorem [48] implies thatvnconverges strongly tovand hencev(t) ∈S F,xρ( t,xt+e φt) +e φρ(t,xt+eφt) From

the continuity ofL, it follows thatLvn(t) →Lv(t)pointwise on J as n → ∞ In order to show that the convergence is uniform, we first show that{Lvn}is an equicontinuous sequence Letτ1, τ2∈J, then we have:

|L(vn(τ1)) −L(vn(τ2))| =

d

dt

Z τ 1 0

S(τ1−s)vn(s)ds− d

dt

Z τ 2 0

S(τ2−s)vn(s)ds

≤

lim

λ→∞

Z τ 1 0

[S0(τ1−s) −S0(τ2−s)]Bλvn(s)ds

+

lim

λ→∞

Z τ 2

τ 1

S0(τ2−s)Bλvn(s)ds

Asτ1 → τ2, the right hand-side of the above inequality tends to zero Since S0(t)is a strongly continuous operator and

the compactness of S0(t),t > 0, implies the uniform continuity (see [5,49]) Hence{Lvn}is equi-continuous, and an application of Arzéla–Ascoli theorem implies that there exists a subsequence which is uniformly convergent Then we have

Lvn j →Lv ∈ (L◦S F)(x)as j7→ ∞, and so(L◦S F)(x)is compact ThereforeBis a compact valued multivalued operator

onB0

Claim 2:B(x)is convex for each z∈D0

Let h1,h2∈B(x), then there existv1, v2∈S F,xρ( t,xt+e φt)+eφρ(t,xt+eφt)such that, for each t ∈J we have

h i(t) =







0, if t ∈ (−∞,0],

d

dt

Z t

0

S(t−s)vi(s)ds if t ∈J,





 , i=1,2. Let 0≤ δ ≤1 Then, for each t ∈J, we have

(δh1+ (1− δ)h2)(t) =







0, if t∈ (−∞,0], d

dt

Z t

0

S(t−s)[δv1(s) + (1− δ)v2(s)]ds if t∈J.







Since F has convex values, one has

δh1+ (1− δ)h2∈B(x).

Let B q= {x∈B0: kxkb≤q} ,q>0 a bounded set inB0 It is equivalent to show that there exists a positive constant l such that for each x∈B qwe havekB(x)kb≤l So choose x∈B q, then fromLemma 4.1it follows that For each h∈B(x),

and each x∈B q, there existsv ∈S F,xρ( t,xt+e φt)+eφρ(t,xt+eφt)such that

h(t) = d

dt

Z t

0

S(t−s)v(s)ds. From (A) we have

kxρ(t,x t+e φt)+ e φρ(t,x t+e φt)kD ≤K b q+ (M b+Lφ)kφkD+K b M| φ(0)| =q∗.

Then by (H6) we have

|h(t)| ≤Meωbψ(q∗) Z t

0

e−ωs p(s)ds:=l. This further, implies that

khkB0 ≤l.

HenceB(B)is bounded

Let B q be, as above, a bounded set and h∈B(x)for some x∈B Then, there existsv ∈S F,xρ( t,xt+e φt) +e φρ(t,xt+eφt)such that

h(t) = d

dt

Z t

0

S(t−s)v(s)ds, t∈J.

Trang 9

Letτ1, τ2∈J\ {t1,t2, ,t m} , τ1< τ2 Thus if >0, we have

|h(τ2) −h(τ1)| ≤

lim

λ→∞

Z τ 1 −

0

[S0(τ2−s) −S0(τ1−s)]Bλv(s)ds

+

lim

λ→∞

Z τ 1

τ 1 −

[S0(τ2−s) −S0(τ1−s)]Bλv(s)ds

+

lim

λ→∞

Z τ 2

τ 1

S0(τ2−s)Bλv(s)ds

≤ ψ(q∗)

Z τ 1 −

0

kS0(τ2−s) −S0(τ1−s)kB(E)p(s)ds

+ ψ(q∗)

Z τ 1

τ 1 −

kS0(τ2−s) −S0(τ1−s)kB(E)p(s)ds+Meωbψ(q∗)

Z τ 2

τ 1

e−ωs p(s)ds.

Asτ1 → τ2and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since S0(t)is

a strongly continuous operator and the compactness of S0(t)for t > 0 implies the uniform continuity This proves the

equicontinuity for the case where t 6= t i,i = 1, ,m+1 It remains to examine the equicontinuity at t = t i First we

prove the equicontinuity at t=t i−, we have for some x∈B q, there existsv ∈S F,xρ( t,xt+e φt) +e φρ(t,xt+eφt)such that

h(t) = d

dt

Z t

0

S(t−s)v(s)ds, t∈J. Fixδ1>0 such that{t k,k6=i} ∩ [t i− δ1,t i+ δ1] = ∅ For 0< µ < δ1, we have

|h(t i− µ) −h(t i)| ≤ lim

λ→∞

Z t i− µ

0

S0(t i− µ −s) −S0(t i−s)Bλv(s ds+ Meωbψ(q∗) Z t i

t i− µ

e−ωs p(s)ds; which tends to zero asµ →0 Define

ˆ

h0(t) =h(t), t ∈ [0,t1]

and

ˆ

h i(t) =

h(t), if t ∈ (t i,t i+ 1]

h(t i+), if t =t i.

Next, we prove equicontinuity at t=t i+ Fixδ2>0 such that{t k,k6=i} ∩ [t i− δ2,t i+ δ2] = ∅ Then

ˆ

h(t i) =

Z t i

0

T(t i−s)v(s)ds. For 0< µ < δ2, we have

| ˆh(t i+ µ) − ˆh(t i)| ≤ lim

λ→∞

Z t i

0

S0(t i+ µ −s) −S0(t i−s)Bλv(s ds+Meωbψ(q∗) Z t i

+ µ

t i

e−ωs p(s)ds; The right hand-side tends to zero asµ →0 The equicontinuity for the casesτ1 < τ2 ≤0 andτ1≤0≤ τ2follows from the uniform continuity ofφon the interval(−∞,0] As a consequence of Claims 1–3 together with Arzelá–Ascoli theorem

it suffices to show thatBmaps B into a precompact set in E.

Let 0<t <b be fixed and letbe a real number satisfying 0< <t For x∈B q, we define

h(t) =S0() lim

λ→∞

Z t−

0

S0(t−s− )Bλv(s)ds, wherev ∈S F,xρ( t,xt+e φt)+eφρ(t,xt+e φt) Since

lim

λ→∞

Z t−

0

S0(t−s− )Bλv(s)ds

≤Meωbψ(q∗)

Z t−

0

e−ωs p(s)ds, the set

lim

λ→∞

Z t−

0

S0(t−s− )Bλv(s)ds: v ∈S F,xρ( t,xt+e φt) +e φρ(t,xt+eφt).x∈B q

is bounded Since S0(t)is a compact operator for t>0, the set

H(t) = {h(t) :h∈B(x)}

Trang 10

is precompact in E for every,0< <t Moreover, for every h∈B(x)we have

|h(t) −h(t)| ≤Meωbψ(q∗) Z t

−

t

e−ωs p(s)ds.

Therefore, there are precompact sets arbitrarily close to the set H(t) = {h(t) :h∈B(x)} Hence the set H(t) = {h(t) :h∈

B(B q)}is precompact in E Hence the operatorBis totally bounded

Step 3: A priori bounds.

Now it remains to show that the set

E = x∈B0:x∈ λA(x) + λB(x)for some 0< λ <1

is bounded Let x∈E, then there existv ∈S F,xρ( t,xt+e φt) +e φρ(t,xt+eφt)andIk∈I k x t k+ e φt k

such that for each t ∈J,

x(t) = λd

dt

Z t

0

S(t−s)v(s) + λ X

0 <t k<t

S0(t−t k)Ik.

This implies by (H2), (H5) that, for each t ∈J, we have

|x(t)| ≤ λMeωt

Z t

0

e−ωs p(s)ψ(kxρ(s,x s+e φs)+ e φρ(s,x s+e φs)kD)ds+ λMeωt

m

X

k= 1

I k x t k+ e φt k

≤ λMeωt

Z t

0

e−ωs

p(s)ψ K b|x(s)| + (M b+Lφ+MK

b)kφkD

ds

+ λMeωt

m

X

k= 1

I k x t k+ e φt k −I k(0) + λMeωt

m

X

k= 1

|I k(0)|

≤ λMeωt

Z t

0

e−ωs p(s)ψ K b|x(s)| + (M b+Lφ+MK

b)kφkD

ds

+ λMeωt

m

X

k= 1

|I k(0)| + λMeωt

m

X

k= 1

c k K b|x(s)| + (M b+Lφ+MK

b)kφkD

≤c∗eωt+Meωt

"

Z t

0

e−ωs p(s)ψ K b|x(s)| + (M b+Lφ+MK

b)kφkD

ds+K b

m

X

k= 1

c k|x(t)|

# Hence from(4.3)–(4.5)we have

(M b+Lφ+MK

b)kφkD+K b|x(s)| ≤c1∗+c2∗

Z t

0

e−ωs p(s)ψ K b|x(t)| + (M b+Lφ+MK

b)kφkD

ds. Thus

(M b+Lφ+MK

b)kφkD+K bkxkB0

c∗

1+c∗

2ψ K bkxkB0+ (M b+Lφ+MK b)kφkD

Rb

0e− ωs p(s)ds

From(4.2)it follows that there exists a constant R > 0 such that for each x ∈ E withkxkB0 > R the condition(4.6)is violated HencekxkB0 ≤R for each x∈E, which means that the setEis bounded As a consequence ofTheorem 4.1,A+B

has a fixed point x∗on the interval(−∞,b], so y∗=x∗+ e φis a fixed point of the operator N which is the mild solution of

problem(1.1)–(1.3)

5 Examples of phase spaces

In this section we give some usual phase spaces

Let g: (−∞,0] → [1, ∞)be a continuous, nondecreasing function with g(0) =1, which satisfies the conditions (g-1), (g-2) of [24] This means that the function

−∞ <θ≤−t

g(t+ θ)

g(θ)

is locally bounded for t ≥0 and that limθ→−∞g(θ) = ∞.

We said thatφ : [−∞,0] →E is normalized piecewise continuous, ifφis left continuous and the restriction ofφto any interval[−r,0]is piecewise continuous

Please cite this article in press as: N Abada, et al., Nonlinear impulsive partial functional differential inclusions with state- dependent delay and< /small>

Trang...

Trang

Định dạng
Số trang	13
Dung lượng	409,79 KB