Existence and Topological Properties of Solution Sets for Differential Inclusions with Delay

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Adel Mahmoud Gomaa

Existence and Topological Properties of Solution Sets for Differential Inclusions with Delay

Abstract. We consider the problem ˙x(t) ∈ A(t)x(t) + F (t, θtx)) a.e. on [0, b], x = κ on [−d, 0] in a Banach space E, where κ belongs to the Banach space, C_E([−d, 0]), of all continuous functions from [−d, 0] into E. A multifunction F from [0, b]× CE([−d, 0]) into the set, Pf c(E), of all nonempty closed convex subsets of E is weakly sequentially hemi-continuous, θtx(s) = x(t + s) for all s∈ [−d, 0] and {A(t) : 0 6 t 6 b} is a family of densely defined closed linear operators generating a continuous evolution operatorS(t, s). Under a generalization of the compactness assumptions, we prove an existence result and give some topological properties of our solution sets that generalizes earlier theorems by Papageorgiou, Rolewicz, Deimling, Frankowska and Cichon..

2000 Mathematics Subject Classification: 46E30, 28A20, 60B12, 34K25, 34K20.

Key words and phrases: Differential inclusions, mutifunctions, measures of noncompactness, delay.

1. Introduction. Differential inclusions appear in the study of nonsmooth Hamiltonian systems and nonsmooth optimal control problems as in Clark [7]. In ad- dition, the works of de Korvin-Kleyle [27], Papageorgiou ([26], [27]), Salinetti-Wets ([35], [36]) and Yovits-Foulk-Rosi [38] illustrate the importance of the multifunctions in various applied fields, like optimization ([35], [36]), mathematical economics ([26], [27]) and in the analysis of uncertain information system ([27], [38]). In this paper we will deal with the differential inclusion

(P )

˙x(t) ∈ A(t)x(t) + F (t, θtx), t∈ [0, b]

x =K on [−d, 0],

where b, d > 0 and K belongs to the Banach space, C^E([−d, 0]), of all continuous functions from [−d, 0] into E. F : [0, b] × CE([−d, 0]) → E is weakly sequentially hemi-continuous, θtx(s) = x(t + s) for all s ∈ [−d, 0] and {A(t) : 0 6 t 6 b} is a

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family of densely defined closed linear operators generating a continuous evolution operator S(t, s).

Our purpose is to prove an existence theorem for integral solution of the problem (P ). Moreover, we give some topological properties for our solution set, S( K), of all integral solutions for (P ). The problem (P ) was investigated, without delay, by many authors ([10], [11], [30], [34], [6], [13] for instance). In this case when A(t) = 0, S(t, s) = id and a mild solution is a Carath´eodory one, we have a generalization to the existence theorems of Deimling [10], Ibrahim-Gomaa [18], Kisielewicz [20], Papageorgiou [30] and [32]. As A(t) 6= 0 our results extend that of [11], [30], [34] and [6]. Also in [14] we give a generalization to recent results, to the Cauchy problem

˙x(t) = f t, x(t)

, t∈ [0, T ] x(0) = x0,

where f : [0, T ] × E −→ E and E is a Banach space, by using weak and strong measures of noncompactness, and in [12] we study the solution set for the inclusion

¨x (t) ∈ F (t, x(t), ˙x(t)) under three boundary conditions x(0) = x0, x(η) = x(T ) where 0 < η < T . Moreover much work has been done to study the topological properties of the solution set for the differential inclusions see ([1], [2], [5], [19], [17], and [16] for instance). Papageorgiou, [31] , consider the problem x(t) ∈ p(t)+Rt

0k(t, s)extF (s, x(s)) ds where extF (t, s) denotes the set of extremal points of F (s, x(s)), this problem arise in the study of control system, also Papageorgiou, [28]

, study semilinear evolution inclusions and their use in optimal problems. Volterra integral inclusions of the type x(t) ∈ p(t) +Rt

0k(t, s)F (s, x(s))ds has been studied by Papageorgiou, [29], in a separable Banach space E with p(.), x(.) ∈ C([0, b], E), K(t, s) be a function from [0, b]×[0, b] into the set of all continuous, linear operators from E into E (L(E)) for all 0 ≤ s ≤ t ≤ b, kk(t, s)k ≤ C < ∞ and F is a multifunction on [0, b] × E with nonempty closed valued in E. Under certain conditions on F and K existence theorems are proved and continuity properties of the solution set are studied. Moreover, S. Hu - V. Lakshmikantham and N. S. Papageorgiou, [15], study semilinear evolution inclusions in which the linear, closed and densely defined operator which generates the strongly continuous semigroup depends on parameter.

2. Preliminaries. A multivalued function F on a Banach space E into the set of nonempty closed subsets, Pf(E), of E is said to be upper semicontinuous if and only if for all closed subset A of E F⁻(A) = {x ∈ E : F (x) ∩ A 6= ∅} is closed in E and F is called w −w sequentially upper semicontinuous if every weakly closed subset A of E F⁻(A) is weakly sequentially closed. Such a multivalued function is called upper hemi-continuous ( resp. weakly upper hemi-continuous) if and only if for any x^∗ ∈ E^∗, c ∈ IR {x ∈ E : supx∈F (x)(x^∗, x) < c} is open in E (resp. in Ew ), where Ew is the Banach space E with the weak topology and E^∗is the dual space. Furthermore F is called weakly sequentially upper hemi- continuous if and only if for any x^∗ ∈ E^∗ the function h : Ew −→ IR defined by h(x) = sup_{x∈F (x)}(x^∗, x) is sequentially upper hemi-continuous. For other properties of the multivalued function we refer to [21], [8], and [4] for instance.

The following lemmas is necessary in the proof our main results.

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Lemma 2.1 ([24], [22]). If γ is a measure of weak (strong) noncompactness and A ⊂ C^w(I, E) is a family of strongly equicontinuous functions, then γ(A(I)) = sup{γ(A(t)) : t ∈ I}.

Lemma 2.2 ([6]) Let Y and E be Banach spaces and let F : E → P^{f c}(Y ) be weakly sequentially upper hemi-continuous. If there exist a ∈ L¹(I, IR), (xn)_n∈IN ⊂ C(I, E) and (yn)n∈IN∪{0}⊂ L¹(I, E) such that kF (x)k ≤ a(t) for all x ∈ C(I, E), xn(t) → x0(t) weakly a.e. on I, yn → y⁰ weakly in L¹(I, E) and yn(t) ∈ F (xn(t)) a.e. on I, then y0(t) ∈ F (x0(t)) a.e. on I.

Lemma 2.3 [6] Let F : I× E −→ P^{f c}(E) and, for each t ∈ I, let F (t, .) be weakly sequentially upper hemi-continuous. If there exists a ∈ L¹(I, IR) such that, for each (t, x) ∈ I × E, kF (t, x)k ≤ a(t) and, for each x ∈ E, F (., x) has a measurable selection, then for each y ∈ C(I, E) there exists at least one measurable (and integrable) selection σ(.) of F (., y(.)).

In this paper we consider E is a Banach space, I = [0, b] and Pf c(E) is the family of nonempty closed convex subsets of E and we shall use some steps that used by Szufla [37] . For any nonempty bounded subset Z of E, the Kuratowski measure of noncompactness, α, and the Hausdorff measure of noncompactness, α^∗, are defined as:

α(Z) = inf{ε > 0 : Z admits a finite number of sets with diameter < ε}.

α^∗(Z) = inf{ε > 0 : Z admits a finite number of balls with radius < ε}.

For the properties of α and α^∗we refer to [3] and [9] for instance.

By a Kamke function we mean a function w : I × IR⁺ −→ IR⁺ such that:

(i) w satisfies the Carath´eodory conditions, (ii) for all t ∈ I; w(t, 0) = 0,

(iii) for any c ∈ (0, b], u ≡ 0 is the only absolutely continuous function on [0, c] which satisfies ˙u(t) ≤ w(t, u(t)) a.e. on [0, c] and such that u(0) = 0.

Let F : I → 2^E− {∅} be measurable and integrable bounded with weakly compact values. The set of all integrable selections of F, S ¹_F, is weakly compact in the Banach space, L¹(I, E), of Lebesque Bochner integrable functions f : I → E endowed with the usual norm [4]. Let L(E) be the algebra of all continuous, linear operators from E to E. If S : I × I → L(E) is such that S(t, 0)x0 is a solution of the problem

(i) ˙x(t) = A(t)x x(0) = x0

where {A(t) : t ∈ I} is a family of densely defined, closed, linear operators on E.

A continuous function x : [−d, b] → E is called an integral solution of the problem (P ) if

x =K on [−d, 0] and x(t) = S(t, 0)K(0) + Z t

0 S(t, s)f(s)ds for all t ∈ I, since f(s) ∈ F (s, θsx) and f ∈ L¹(I, E). S (.,.) is called a fundamental solution of (i) or the family {A(t) : t ∈ I} is a generator of S (.,.) see [34] and [33].

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3. Main Results.

Theorem 3.1 Let β be either a Kuratowski measure of noncompactness or a Haus- dorff measure of noncompactness; 0 ≤ M < ∞ : w a Kamke function and let F be a multifunction from [0, b] × CE([−d, 0]) into the set, Pf c(E), of all nonempty closed convex subsets of E such that

(F1) for each ε > 0, there exists a closed subset Iε of I with λ(I − Iε) < ε such that for any nonempty bounded subset A of CE([−d, 0]) and for each closed subset J ⊆ I^ε, one has

β(F (J× A)) ≤ sup

t∈Jw(t, β(A(0)))

(F2) kF (t, K)k ≤ c(t)(1+ k K(0)k for each K ∈ CE([−d, 0]) and for some c ∈ L¹(I, IR) a.e. on I,

(F3) F(., K) has a measurable selection, for each K ∈ CE([−d, 0])

(F4) for each t ∈ I, F (t, .) is weakly sequentially upper hemi-continuous. Further, let {A(t) : t ∈ I} is a generator of a fundamental solution S : I × I −→ L(E) such that

a)S(t, t) = id, t ∈ I, id is the identity function on E;

b) S(t, s)S(s, r) = S(t, r), t, s, r ∈ I;

c) S is continuous;

d)kS(t, s)k ≤ M, t, s ∈ I;

e) for each s∈ I, S(., s) is uniformly continuous.

Then, for each K ∈ CE([−d, 0]), problem (P ) has an integral solution. and the solution set of all integral solutions of (P ), S(K), is compact.

Proof First we drive a priori bound for the integral solutions of problem (P ) on I. If x is such a solution, then we have x equal to K on [−d, 0] and x(t) = S(t, 0)K(0) +Rt

0S(t, s)f(s)ds for all t ∈ I with f(s) ∈ F (s, θ^sx) and f ∈ L¹(I, E).

So, for each t ∈ I,

kx(t)k ≤ kS(t, 0)kkK(0)k + Z t

0 kS(t, s)kkf(s)k ds

≤ MkK(0)k + Z t

0

M c(t)(1 +kx(s)k) ds

≤ MkK(0)k + Mkck + Z t

0

M c(t)kx(s)k ds.

if M1 = (MkK(0)k + kck)e^M^kck, then kx(t)k ≤ M¹. Put ϕ(t) = c(t)(1 + M1). So we may assume without any loss of generality kF1(t, x(t))k ≤ ϕ(t) a.e. on I since, otherwise, with BM₁ = {x ∈ E : kx(t)k ≤ M¹}, we can replace F by F⁰ which is defined by

F⁰(t, x(t)) =

( F1(t, x(t)) if x ∈ BM1

F1(t,^M¹_kxk^.x(t)) if x /∈ B^M1.

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For arbitrary n ∈ IN, define Φ1: [−d,_n^b] × E → E by Φ1(t, x) =

K(t) if t ∈ [−d, 0]

K(0) + nt(x − K(0)) if t ∈ [0,n^b] and define F1 : [0,_n^b] × E → Pf c(E) by F1(t, x) = F (t, θ^b

n(Φ1(., x))).Thus, from Lemma 2.3, for each v ∈ C([0,_n^b], E) we can find at least one integrable selection σ of F1(., v(.)). Consequently we can define a multivalued function G : BM1 ⊆ C([0,_n^b], E) −→ 2^C([0,ⁿ^b^],E)by

(Gx)(t) = S(t, 0)K(0) +Z t

0 S(t, s)F (t, θ_n^b(Φ1(., x(s))))ds,

for each x ∈ BM1 , Gx 6= ∅. Since S is continuous we can define a function ξ : L¹([0,_n^b], E) −→ C([0,_n^b], E) by ξ(f)(t) = S(t, 0)K(0) +Rt

0S(t, s)f(s) ds. If we set V = {f ∈ L¹([0,_n^b], E) : kfk ≤ ϕ(t) a. e. on [0,_n^b]}, then V is uniformly integrable in L¹([0,_n^b], E) and, since S(., s) is uniformly continuous, ξ(V ) = {x ∈ C([0,_n^b], E) : x(t) = S(0, t)K(0)+Rt

0S(t, s)f(s)ds , f ∈ V } is nonempty equicontinuous subset of C([0,_n^b], E) and so, convξ(V ) is nonempty convex closed equibounded and equicontinuous subset of C([0,_n^b], E).

Let (xm, ym) ∈ Graph G such that x^m → x, y^m → y in C([0,n^b], E) ym : I → C([0,_n^b], E) is given by ym(t) = S(t, 0)K(0) +Rt

0S(t, s)f^m(s) ds, fm ∈ L¹([0,_n^b], E), fm(s) ∈ F1(s, xm(s)) and

fm(t) =

( S(t, 0)K(0) if 0 ≤ t ≤ nm^b

S(t, 0)K(0) +Rt−nm^b

0 S(t, s)f^m(s)ds if _nm^b ≤ t ≤n^b. Thus

m→∞lim kξ(f^m) − fmk = lim

m→∞ sup

t∈[0,n^b]kξ(f^m)(t) − fm(t)k

≤ lim_m_→∞( sup

t∈[0,nm^b ]kξ(f^m)(t) − fm(t)k + sup

t∈[nm^b ,_n^b]kξ(xⁿ)(t) − fm(t)k)

≤ lim_m

→∞( sup

t∈[0,nm^b ]

Z t

0 kS(t, s)f^mkds + sup

t∈[nm^b ,_n^b]

k Z t

0 S(t, s)f^mds

− Z t−nm^b

0 S(t, s)f^mdsk)

≤ lim_m_→∞( sup

t∈[0,nm^b ]

Z t 0

M ϕ(s) ds + sup

t∈[nm^b ,_n^b]

Z t t−nm^b

kS(t, s)f^mkds)

≤ lim_m

→∞( sup

t∈[0,_nm^b ]

Z t 0

M ϕ(s) ds + sup

t∈[_nm^b ,_n^b]

Z t t−_nm^b

M ϕ(s)) = 0.

Obviously the sets H := {fm: m ∈ IN} and G := {ξ(fm) : m ∈ IN} are equicontinuous. Let ρ(t) := β(H(t)), t ∈ [0,_n^b]. Then ρ(0) = 0. We claim that ρ is

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differentiable a.e. on [0,_n^b]. Since kfm− ξ(f^m)k → 0 as m → ∞ so, from Lemma 2.1, β((Id − ξ)H) = 0 which given that

β({f^m: m ∈ IN}) = β({ξ(fm) : m ∈ IN}).

Since for all t, τ ∈ [0,_n^b],

β{ξ(f^m)(τ): m ∈ IN} ≤ β{ξ(fm)(t): m ∈ IN}+β{ξ(fm)(τ) − ξ(fm)(t):m ∈ IN}

and

β{ξ(f^m)(t):m ∈ IN} ≤ β{ξ(fm)(τ):m ∈ IN}+β{ξ(fm)(t) − ξ(fm)(τ): m ∈ IN},

then| ρ(τ) − ρ(t)| ≤ 2β(B(0, 1))Rτ

tMϕ(s) ds. therefore ρ is absolutely continuous function and thus it is differentiable a.e. on [0,_n^b]. Let (t, τ) ∈ [0,n^b] × [0,n^b] such that t ≤ τ. Since ρ is continuous and w is Caratheodory we can find a closed subset Iε of [0,_n^b], δ > 0, η > 0 (η < δ) and for s1, s2 ∈ I^ε; r1, r2 ∈ [0,^2bn] such that if

· · · < t^r= τ such that ti− tⁱ−1 < η for i = 1,· · · , r. Let Aⁱ = {x(s) : x ∈ H, s ∈ [ti−1, ti] ∩ Iε}. Let Z be a bounded subset of E and A = {θ_n^b(Φ1(., x)) : x ∈ Z}.

Thus, for each t∈ [0,_n^b], β(F1({t} × Z)) = β(F ({t} × A)). From Condition (1) we can find a closed subset Jε of [0,_n^b] such that λ(I − Jε) < ε and that for any compact subset C of Jε β(F1(C × Z)) = β(F (C × A)) ≤ sups∈Cw(s, β(Z)). Let Ti = Jε∩ [tⁱ−1, ti] ∩ Iε, P = Pm

i=1Ti = [t, τ] ∩ Jε∩ I^ε and Q = [t, τ] − P. Thus Rτ

t F1(s, H(s)) ds ⊂R

PF1(s, H(s)) ds +R

QF1(s, H(s)). In virtue of Lemma 2.1 and from the continuity of ρ, we have

β(Ai) = sup{β(H(s)) : s ∈ [ti−1, ti] ∩ Iε},

and by the mean value theorem we obtain

Z

P

F1(s, H(s)) ds ⊂ Xm i=1

Z

Ti

F1(s, H(s)) ds ⊂ Xn i=1

λ(Ti)convF1(Ti× Aⁱ)

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Now, we have

β(

Z

P

F1(s, H(s)) ds) ≤ Xm i=1

λ(Ti)β(F1(Ti× Aⁱ))

≤ Xm i=1

λ(Ti) sup

si∈Ti

w(si, β(Ai))

= Xm i=1

λ(Ti)w(qi, ρ(pi)); (qi, pi∈ Tⁱ)

≤ Xm i=1

Z

Ti

w(s, ρ(s)) ds + ελ(Ti)

= Z

P

w(s, ρ(s)) ds + ελ(P )

≤ Z τ

t

w(s, ρ(s)) ds + ε(τ− t).

Moreover, we get β(R

QF1(s, H(s)) ds) ≤ 2R

Qϕ(s) ds.As λ(Q) < 2ε and since ε is arbitrary, then

(1) β(

Z τ t

F1(s, H(s)) ds) ≤Z τ t

w(s, ρ(s)) ds.

On the other hand, we have

(2) β(ξ(H)(τ ))≤ β(ξ(H)(t)) + β(

Z τ t

F1(s, H(s)) ds).

By relations (1) and (2) we get

ρ(τ )− ρ(t) ≤ β(

Z τ t

F1(s, H(s)) ds) ≤Z τ t

w(s, ρ(s)) ds.

Therefore ˙ρ(t) ≤ w(t, ρ(t)) a.e. on [0,_n^b]. Since ρ(0) = 0 and w is a Kamke function, then ρ ≡ 0. Thus the weak closure of (fm)_m∈IN is weakly compact and so we can suppose that the sequence (fm)m∈IN converges to a continuous function x1 such that x1 = K on [−d, 0] and for each t ∈ [0,n^b]

x1(t) = S(t, 0)K(0) +Z t

0 S(t, s)l1(s)ds where l1(s) ∈ F (s, θ_n^b(Φ1(., x1(t)))) a.e. on [0,_n^b].

Now, by the mathematical induction for some k ∈ {2, 3, ...n}, we can assume that there exists the function x_k−1 such that x_k−1 = K on [−d, 0] and for each t∈ [0,^(k−1)bn ]

xk−1(t) = S(t, 0)K(0) +Z t

0 S(t, s)l^k−1(s)ds

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lk−1(s) ∈ F (t, θ^(k−1)b

n Φk−1(., xk−1(s))) a.e. and lk−1∈ L¹([0,(k − 1)b n ], E) also let Φk: [−d,^kb_n] × E → E be such that

Φk(t, x) =

( xk−1(t) if t ∈ [−d,^(k−1)b_n ]

x_k−1(^(k^−1)b_n ) + n(t − ^(k^−1)b_n )(x − xk−1(^(k^−1)b_n )) if t ∈ [^(k^−1)b_n ,^kb_n].

Arguing as in above, for the multifunction Fk : [^(k^−1)b_n ,^kb_n] × E → Pf c(E) which is defined by Fk(t, x) = F (t, θ^kb

n(Φk(., x))), we have a continuous function xk defined on [^(k−1)b_n ,^kb_n] by

xk(t) = S(t,(k − 1)b

n )xk−1((k − 1)b n ) +Z t

(k−1)b n

S(t, s)l^k(s)ds

where lk (s) ∈ F (s, θ^kb_n(Φk(., xk(s)))) a.e on [^(k^−1)b_n ,^kb_n] and lk ∈ L¹([^(k^−1)b_n ,^kb_n], E).

Moreover, if we put k⁰= k − 1, then for each t ∈ [^k_n⁰^b,^kb_n] we have

xk⁰(k⁰b

n ) = S(k⁰b

n , 0)K(0) + Z ^{k0 b}_n

0 S(k⁰b

n , s)lk⁰(s)ds hence

xk(t) = S(t,k⁰b n )S(k⁰b

n , 0)K(0) + Z ^{k0 b}_n

0 S(t,k⁰b n )S(k⁰b

n , s)lk⁰(s)ds +Z t

k0 b n

S(t, s)l^k(s)ds

= S(t, 0)K(0) +Z ^{k0 b}_n

0 S(t, s)l^k⁰(s)ds +Z t

k0 b n

S(t, s)l^k(s)ds

= S(t, 0)K(0) +Z t

0 S(t, s)g^k(s)ds, where

gk(t) =

( lk⁰(t) if t ∈ [0,^k_n⁰^b] lk(t) if t ∈ [^k_n⁰^b,^kb_n].

Consequently, for all n ∈ IN, we have a continuous function vⁿ such that vn = K 0n [-d,0] and for each t ∈ I is defined by

vn(t) = S(t, 0)K(0) +Z t

0 S(t, s)hⁿ(s)ds, with

hn(t) ∈ F (t, θ^kb_nΦk(., vn(t))) a.e. on I.

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where t ∈ [^(k^−1)b_n ,^kb_n] ⊂ I, for k ∈ {1, 2, 3, ...n}. Now we claim that the set L = {vⁿ: n ∈ IN} is an equicontinuous set . So let t1, t2 ∈ I with t¹< t2. Then

k vⁿ(t1) − vn(t2) k≤k S(t1, 0)− S(t², 0)kk K(0) k +Z t1

0 k S(t¹, s)− S(t², s)kk hⁿ(s) k ds +Z t2

t1

k S(t², s)kk hⁿ(s) k ds

≤ k S(t¹, 0)− S(t², 0)kk K(0) k +Z t1

0 k S(t¹, s)− S(t², s)kk ϕ(s) k ds + M Z t2

t1

k ϕ(s) k ds

and, since vn = K on [-d,0], this shows that L is equicontinuous in CE[−d, b].

Moreover the set β(L(t)) = β({vn(t) : n ∈ IN}) is such that β(L(0)) = 0 and, by the same as above, we get β(L(t)) = 0 for all t ∈ I. Thus by Ascoli’s theorem we may the sequence {vn : n ∈ IN} converges uniformly to a function v ∈ CE([−d, b]) such that y = K on [-d,0]. But β({hn(t) : n ∈ IN}) = 0 and so {hn(t) : n ∈ IN} is relatively compact. Create a new multivalued function F(t) = conv{hn(t) :n ∈IN}.

Thus F(t) is nonempty convex and compact. Now we can say that the set δ¹_F = {l ∈ L¹(I, E) : l(t) ∈ F(t)} is nonempty convex and weakly compact. By Eberlein- Smulian Theorem there exists a subsequence (hˇ nk) of (hn) such that hnk→ l weakly, l∈ δF¹. Thus vn tends weakly to S(t, 0)K(0) +Rt

0S(t, s)l(s)ds. Moreover since, for each n ∈ IN, vn ∈ C^E([−d, b]), vn converges uniformly to v on each compact subset of [−d, b] and v is uniformly continuous on [-d,0]; also for each t ∈ I, there exists n > _d^b with t ∈ [^(k^−1)b_n ,^kb_n] for k ∈ {1, 2, ...n − 1} so, as k⁰= k − 1,

k θ^kb_nΦk(., vn(t)) − θtvk

≤ sup

s∈[−d,−n^b]

[k Φk(kb

n + s, vn(t)) − v(kb n + s) k + k v(kb

n + s) − v(t + s) k]

+ sup

s∈[−n^b,0]

[

k vⁿ(k⁰b n ) + n(b

n+ s)(vn(t) − vn(k⁰b

n )) − v(kb

n + s) k + k v(kb

n + s) − v(t + s) k]

≤ sup

s∈[−d,−_n^b]

[k vⁿ(kb

n + s) − v(kb n + s) k + k v(kb

n + s) − v(t + s) k]

+ sup

s∈[−n^b,0]

[

b k (vⁿ(t) − vn(k⁰b

n )) k + k vn(k⁰b

n ) − v(kb n + s) k + k v(kb

n + s) − v(t + s) k

] → 0 as n → ∞.

Thus, from Lemma 2.2, we conclude that the solution set , S(K), of integral solutions of (P ) is nonempty. Next if {vn: n ∈ IN} is a sequence of S(K), then argu-

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ing as in the proof above we can show that, for each t ∈ I, β({vn(t) : n ∈ IN}) = 0.

Thus this sequence has a convergent subsequence, so S(K) is compact.

If we replace in Theorem 3.1 β by a measure of weak noncompactness, then for each K ∈ CE([−d, 0]) the solution set of all integral solutions of Problem (P ), S(K), is nonempty weakly compact subset of C([−d, b], E). Moreover we can define the multifunction S : CE([−d, 0]) → 2C([−d,b],E) such that, for each K ∈ C^E([−d, 0]), S(K) is the solution set of problem (P ).

In the following theorem we assume:

C(H) {Hn : n ∈ IN} be a sequence of multifunctions from I ×CE([−d, 0]) into the set, Pf c(E), of nonempty closed convex subsets of E such that

(1) H(t, K) = ∩ⁿ⁼n=1^∞Hn(t, K),

(2) Hn+1(t, K) ⊂ Hn(t, K), for all n ∈ IN,

(3) if h is the Hausdorff distance, then limn→∞h(Hn(t, K), H(t, K)) = 0, (4) for some C > 0 k Hn(t, K) k≤ C,

(5) for n ∈ IN, Hn satisfies conditions F1, F3 and F4of Theorem 3.1.

Theorem 3.2 If E is a separable Banach space, H(t, .) is weakly sequentially upper hemi-continuous, H(., K) has a measurable selection and hypotheses C(H) hold, then for each K ∈ CE([−d, 0]) SH(K) = ∩ⁿ⁼n=1^∞SHn(K).

Proof Thanks to our assumptions, we obtain the solution set SH(K) is nonempty and for any n ∈ IN, S^H(K) ⊆ S^Hn(K), so S^H(K) ⊆ ∩^n=∞n=1 SHn(K). Conversely, let v∈ ∩^n=∞n=1 SHn(K). Thus there exists hⁿsuch that, for each t ∈ I, hⁿ(t) ∈ Hⁿ(t, θtv) and v(t) = S(t, 0)K(0)+Rt

0S(t, s)hⁿ(s)ds. From condition C(H)(3) we have hn(t) ∈ H(t, θtv)+ εn(t)B1a.e. on I, where εn(t) = h(Hn(t, θtv), H(t, θtv))→ 0 as n → ∞ and B1is the closed unit ball in E. By condition C(H)(4), the sequence {hⁿ: n ∈ IN}

is uniformly bounded. We consider a subsequence {hnk(t) : nk ∈ IN} and we can passing to convex combination of hnk(t) denoted by ˜hnk(t). Thus ˜hnk(t) → l(t) ∈ E and moreover ˜hnk(t) ∈P

m≥nγm(H(t, θtv) + εm(t)B1) a.e. on I, where γm(t) ≥ 0 and also P_m_≥nγm = 1. At this point, we let nk → ∞ and since H has convex values, so l(t) ∈ H(t, θtv) and hence the result.

Theorem 3.3 The multifunction S is upper semicontinuous and both the multifunctions St : CE([−d, 0]) → 2^E, defined by St(K) = {v(t) : v ∈ S(K)} and that S_K: I → 2^E which is defined by S_K(t) = {v(t) : v ∈ S(K)} is upper semicontinuous and has compact values. Further, the set ∪t∈IS_K(t) is compact in E.

Proof For each closed subset Z of CE([−d, b]), to show that S is upper semicontinuous, we claim that A={K ∈ C^E([−d, 0]) : SK∩Z 6= ∅} is sequentially closed in C^E([−d, 0]).

Let {Kn : n ∈ IN} ⊂ A such that Kn → K. Then SKn ∩ Z 6= ∅ and hence there exists vn ∈ SKn ∩ Z , where vⁿ(t) = S(t, 0)Kn(0) +Rt

0S(t, s)gⁿ(s)ds,with gn(s) ∈ F (s, θsvn) a.e. on I and gn(.) ∈ L¹(I, E). Now, for each t ∈ I, we have

β({vⁿ(t) : n ∈ IN}) ≤ Mβ({Kn(0) : n ∈ IN}) + Mβ({Z t 0

gn(s)ds : n ∈ IN}).

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But β({Kn(0) : n ∈ IN}) = 0, where Kn→ K. Thus

β({vⁿ(t) : n ∈ IN}) ≤ Mβ({Z t 0

gn(s)ds : n ∈ IN}).

Arguing as in the proof of Theorem 3.1 we have β({vn(t) : n ∈ IN}) = 0. Now since the sequence {vn(t) : n ∈ IN} is equicontinuous, so from Arzela-Ascoli theorem we can find a subsequence (vnk) converges to v0 in CE([−d, b]). Let vnk(t) = S(t, 0)Knk(0)+Rt

0S(t, s)gⁿk(s)ds, where gnk(s) ∈ F (s, θsvnk) a.e. on I and gnk(.) ∈ L¹(I, E) . Then we can write gnk = K on [−d, 0] and

gnk(t) =

( S(t, 0)K(0) if 0 ≤ t ≤ _n^b_k

S(t, 0)K(0) +Rt−nm^b

0 S(t, s)gⁿ^k(s)ds if _n^b_k ≤ t ≤ b.

As in the proof of Theorem 3.1 we obtain β({gnk(t) : nk ∈ IN}) = 0 for t ∈ I, so gnk→ g⁰∈ L¹(I, E) and from Lemma 2.2 g0(t) ∈ F (t, θtv0). Thus

v0(t) = S(t, 0)K(0) +Z t

0 S(t, s)g⁰(s)ds

and consequently A={K ∈ C^E([−d, 0]) : SK ∩ Z 6= ∅} is sequentially closed in CE([−d, 0]) thus S is upper semicontinuous. Further, by the same arguments we can show that P = {K ∈ CE([−d, 0]) : St(K) ∩ Z 6= ∅}is closed so, St(K) is upper semicontinuous. Since S(K) is compact, then both SK and Sthas compact values.

Lastly the set Q={t∈ I : SK(t) ∩ Z} is closed, then from Berge’s Theorem [4] ∪t∈I

S_K(t) is compact in E.

Now we consider the following control problem (Q)

˙x(t) ∈ A(t)x(t) + F (t, θ^tx) x= K ∈ Z

minimise ω(x(b))

where Z is a compact subset of CE([−d, 0]) and ω : E −→ IR is lower semicontinuous.

We say that Problem (P ) has an optimal solution if there exist K0 ∈ Z and v ∈ S(K⁰) such that ω(v(b)) = inf{ω(x(b)) : x ∈ S(K0)}.

Theorem 3.4 Under the assumptions of Theorem 3.1, Problem (Q) has an optimal solution.

Proof IfK⁰∈ Z ⊆ C^E([−d, 0]), then there exists a continuous function v ∈ S(K0) and so, v(b) ∈ Sb(K0). But Sb is upper semicontinuous and has compact values thus, from Berge0s Theorem [4], Sb(Z) is compact and so, ω has its minimum b0

on Sb(Z). Thus there exists K1 ∈ Z such that v⁰ ∈ S^b(K1), where ω(v0) = b0

v0 ∈ S^b(Z), thus v0 ∈ SK1(b) which means that v0 = v(b) for some v ∈ S(K1).

Therefore ω(v(b)) = inf{ω(x(b)) : x ∈ S(K1)}.

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Adel Mahmoud Gomaa Helwan University

Department of Mathematics, Faculty of Science, Egypt E-mail: gomaa5@hotmail.com

(Received: 14.05.2008)