Adel Mahmoud Gomaa
On theorems for weak solutions of nonlinear differential equations with and without delay in
Banach spaces
Abstract. In the present work we give an existence theorem for bounded weak solution of the differential equation
˙x(t) = A(t)x(t) + f (t, x(t)), t≥ 0
where{A(t) : t ∈ IR+} is a family of linear operators from a Banach space E into itself, Br={x ∈ E : kxk ≤ r} and f : IR+× Br→ E is weakly-weakly continuous.
Furthermore, we give existence theorem for the differential equation with delay
˙x(t) = ˆA(t)x(t) + fd(t, θtx) if t∈ [0, T ],
where T, d > 0, CBr([−d, 0]) is the Banach space of continuous functions from [−d, 0]
into Br, fd: [0, T ]× CBr([−d, 0]) → E weakly-weakly continuous function, ˆA(t) : [0, T ]→ L(E) is strongly measurable and Bochner integrable operator on [0, T ] and θtx(s) = x(t + s) for all s∈ [−d, 0].
2000 Mathematics Subject Classification: 34A34, 34C29, 34K25, 34K20.
Key words and phrases: Nonlinear differential equations, weak solutions, measures of noncompactness, delay.
1. Introduction. Let E be a Banach space and Br = {x ∈ E : kxk ≤ r}. In Section 3 , we wish to investigate the problem
(P ) ˙x(t) = A(t)x(t) + f(t, x(t)), t ≥ 0,
where {A(t) : t ∈ IR+} is a family of linear operators from E into itself and f : IR+× Br → E is weakly-weakly continuous. Main result of this section generalize
many previous theorems. In fact, in the case A(t) = 0 we have, as a special case, some improvement to the existence theorem of Cramer-Lakshmikantham-Mitchell [11], Boundourides [4], Ibrahim-Gomaa [17], Szep [26] and Papageorgiou [25]. Cramer- Lakshmikantham-Mitchell [11] studied the special case of Problem (P ) in a nonre- flexive Banach space, Boundourides [4] and Papageorgiou [25], found weak solutions for the special case of Problem (P ) on a finite interval [0, T ] with 0 < T < ∞. Szep in [26] studied the special case of Problem (P ) in a reflexive Banach space, while we use in this paper more general compactness assumptions. Ibrahim-Gomaa [17]
proved the existence of weak solutions for the special case of Problem (P ) on a finite interval [0, T ]. Also in [15] we consider 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. If A(t) 6= 0 our result is a generalization to that of Cichon[9] , since we are able to reduce the compactness assumptions. Moreover Cichon [8], found a weak solution for the problem (P ) and Gomaa in [14] studied the nonlinear differential equation with delay.
Finally, we investigate the equation ˙x(t) = ˆA(t)x(t) + fd(t, θtx(.)) and so, we obtain similar result to that for the pbroblem (P ).
2. Preliminaries. Throughout the paper E denote a Banach space and E∗ is its dual space, L(E) is the space of linear operators from E into itself, λ is the Lebesgue measure on IR+, BE ia the family of all nonempty bounded subsets of E, RE is the family of all nonempty and relatively weakly compact subsets of E and h, i is the pairing between E and E∗. A function u : [a, b]→ E, is said to be Pettis integrable if for any measurable subset D of [a, b] there is an element vD in E such that hvD, fi =R
Dhu(s), fids for all f ∈ E∗, in this case we write vD =R
Du(s) ds while a function u : [a, b] → E is said to be Bochner integrable if there exists a sequence of countable-valued functions {un} converging almost everywhere on [a, b]
such that limn→∞Rb
a kun(s) − u(s)k ds =0. We note that every Bochner integrable function is Pettis integrable (see [16]). Let L(IR+, E) be the space of measurable functions u : IR+ −→ E, Bochner integrable on IR+. Further, let C(w)(IR+, E) be the space of all (weakly) continuous functions from IR+ to E endowed with the topology of almost uniform weak convergence and CE([−d, 0]) be the Banach space of continuous functions from the closed interval [−d, 0](d ≥ 0) into E.
A nonempty family K ⊂ RE is said to be a kernel if it satisfies the following conditions:
(i) A ∈ K =⇒ conv A∈ K, (ii) B 6= ∅, B ⊂ A =⇒ A ∈ K,
(iii) A subfamily of all weakly compact sets in K is closed in the family of all bounded and closed subsets of E with the topology generated by the Hausdorff distance.
A function γ : BE→ [0, ∞) is said to be a measure of weak noncompactness with the kernel K if it is subject to the following conditions:
(i) γ(A) = 0 ⇐⇒ A ∈ K,
(ii) γ(A) = γ(A), where A is weak closure of the set A, (iii) γ(conv A) = γ(A),
(iv) A, B ∈ BE, B⊂ A =⇒ γ(B) ≤ γ(A) [21, 3].
Denote by N a basis of neighbourhoods of zero in a locally convex space composed of closed convex sets. Let N0 = {rN : N ∈ N }. The following two definitions can be found in [6, 9].
A function p : N0 → [0, ∞) is said to be p− function if it satisfies the following conditions:
(i) X, Y ∈ N0, X⊂ Y =⇒ p(X) ≤ p(Y ),
(ii) for each ε > 0 there exists X∈ N0 such that p(X) < ε, (iii) p(X) > 0 whenever X /∈ K.
A function γ : BE → [0, ∞) is said to be (K, N , p)− measure of weak noncom- pactness if and only if
γ(U ) = inf{ε > 0 : ∃A ∈ K, X ∈ N0, U ⊂ A + X, p(X) ≤ ε}, for each U ∈ BE.
For any nonempty bounded subset Z of E we recall the definition of De Blasi’s measure of weak noncompactness
β(Z) = inf{ε>0:∃K, nonempty weakly compact subset of E, Z ⊆K + εB1}.
For the properties of β see [13, 3].
Each (K, N , p)−measure of weak noncompactness is a measure of weak noncom- pactness with the kernel K. De Blasi’s measure is (K, N , p) − measure of weak noncompactness [3, 6].
Put I := [0, T ], then 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.
Assume that M = M(IR+, E) is a Banach space of measurable functions x : IR+−→ E with kxk ∈ M(IR+, IR), kxkM= kkxkkM(IR+,IR), where
(1) M(IR+, IR)⊂ L(IR+, IR),
(2) M(IR+, IR) contains all essentially bounded function with compact support, (3) if x ∈ M(IR+, IR), y : IR+ −→ IR is measurable with |y| < |x|, then y ∈ M(IR+, IR) andkykM(IR+,IR)<kxkM(IR+,IR),
(4) if x ∈ M(IR+, IR), xn ∈ M(IR+, IR), |xn| ≤ |x| and limn→∞xn(t) = 0 a.e. on IR+, then limn→∞kxnkM(IR+,IR)= 0.
Let M0 denote the associate space to M [23].
We will need the following lemmas in the proof of the main results [9, 18, 3].
Lemma 2.1 Let γ be a (RE,N , p)−measure of weak noncompactness such that p(αX)
= αp(X) with X ∈ N0, α ∈ IR and N is composed of balanced sets. So, for each bounded subset U of E and for each A ∈ L(E), we have γ(AU )≤ |A|γ(U).
Lemma 2.2 In the setting of Lemma 2.1 if D :[a,b]→ L(E) is a continuous mapping, then
γ( [
t∈[a,b]
D(t)U )≤ sup
t∈[a,b]|D(t)|γ(U).
Lemma 2.3 If γ be a (RE,N , p)−measure of weak noncompactness such that p(αX)
= αp(X) with X ∈ N0, α∈ IR+ and for each X, Y ∈ N0 we find X + Y ∈ N0, then (M1) γ(U + V ) ≤ γ(U) + γ(V ),
(M2) γ(αU) = αγ(U),
(M3) γ(U) = 0 ⇐⇒ U is relatively compact in E, (M4) γ(U ∪ {x}) = γ(U), x ∈ E.
(M5) U ⊆ V =⇒ γ(U) 6 γ(V ), (M6) γ(convU) = γ(U).
Under the assumptions in Lemma 2.3 on the measure γ we give a result analogous to that proved by Ambrosetti in [2] and the other proved by Papageorgiou [25].
Lemma 2.4 Let V ⊆ C(I, E) be a bounded equicontinuous for the strong topology and V (J) = {x(t) : x ∈ V, t ∈ J}, where J is a subinterval of I. Then, under the assumptions in Lemma 2.3, γ(V (J)) = supt∈Jγ(V ({t})) = γ((J(s0))) for some s0 ∈ J.
Proof For each ε > 0 there exists a δ > 0 such that if|t−t0| < δ, then |x(t)−x(t0)| <
ε for all x∈ V. Let {ti}ni=0 be a partition of J such that |ti+1− ti| < δ, i=0,1,2,..,n- 1. Now, from the definition of γ, for each Ai ∈ RE we can find X∈ N0 such that V (ti) ⊂ Ai+ X, p(X) < ε + ε0. If θ∈ V (J), there exists x ∈ V such that θ = x(t).
Without loss of generality we consider t /∈ {ti}ni=0, then t ∈ (ti, ti+1) for some i = 0, 1, 2, ..., n− 1 and θ ∈ {x(ti) + (x(t) − x(ti)) : t ∈ (ti, ti+1)} ⊂ Ai+ X + εB1. From M6 in Lemma 2.3 and assumptions on γ γ(V (J)) ≤ γ(Sn
i=0Ai+ (X + εB1)), p(X + εB1) ≤ ε0+ ε(1 + p(B1)). Thus γ(V (J)) ≤ ε0+ ε(1 + (pB1)) by taking ε → 0, we have γ(V (J)) ≤ ε0 = supt∈Jγ(V (t)). On the other hand, since V (t) ⊆ V (J) for each t ∈ J, then we obtain supt∈Jγ(V (t)) ≤ γ(V (J)). Therefore γ(V (J)) =
supt∈Jγ(V (t)) = γ((J(s0))) for some s0 ∈ J.
If for each t ∈ IR+, A(t)∈ L(E) and ˙x(t) denotes the weak derivative of x at t, then we consider the differential equation
(1) ˙x(t) = A(t)x(t).
Let E be the direct sum of E0 and E1, where E0= {x0 ∈ E : ∃ a bounded weak solution x of (1) and x(0) = x0} is closed and has a closed complement E1. Let G ∈ C(IR+× IR+, E) be the Green function corresponding to (1):
(2) G(t, s) =
S(t)P S−1(s) if 0 ≤ s ≤ t
−S(t)(id − P )S−1(s) if 0 ≤ t ≤ s,
where S : IR+−→ L(E) is a solution of the differential equation
˙S(t) = A(t)S(t), S(0) = id, and P is the projection of E onto E0 while P (E1) = {0}.
In the following we shall consider the nonlinear differential equation, (P ) ˙x(t) = A(t)x(t) + f(t, x(t)) t ≥ 0.
This problem was studied by many authors ([9, 22, 14, 10, 20] for instance). As- sume, that f : IR+× Br −→ E is weakly-weakly continuous and A : IR+ −→ L(E) is strongly measurable and Bochner integrable on every subinterval J of IR+. More- over, we will deal with the differential equation
(Q) ˙x(t) = ˆA(t)x(t) + fd(t, θtx), t∈ I,
where fd : I × CE([−d, 0]) → E is weakly-weakly continuous function, θtx(s) = x(t + s) for all s∈ [−d, 0] and ˆA(t) : I→ L(E) is strongly measurable and Bochner integrable operator on I.
A continuous function x : [−d, T ] → E is called a weak solution of the problem (Q) if for some ϕ ∈ CE([−d, 0]),
x = ϕ on [−d, 0] and x(t) = G(t, 0)ϕ(0) + Z t
0 G(t, s)f(s, x(s))ds for all t ∈ I, where R is understood in the sense of the Pettis integral.
In [14] we proved the following two theorem:
Theorem 2.5 If we assume that, f : IR+× Br−→ E is weakly-weakly continuous function ; A : IR+ −→ L(E) is strongly measurable and Bochner integrable on every subinterval J of IR+; for each t ∈ IR+, G(t, .) ∈ M0 with kG(t, .)kM ≤ c for some c ∈ IR+; there exists a function m : IR+→ IR+ belongs to M0 such that kf(t, x)k ≤ m(t) for every (t, x) ∈ IR+× Br and ckmkM < r; G(t, .)f(s, x(.)) is Pettis integrable on each compact subset of IR+; for each T > 0 and for each ε > 0, there exists a closed subset Iε of [0, T ] with λ([0, T ] − Iε) < ε such that for any nonempty bounded subset U of E one has β(f(J × U)) ≤ supt∈Jw(t, β(U )), for any compact subset J of Iε, then, for each x0 ∈ E0 such that kx0k ≤ r−ckmkkG(t,0)kM, there exists at least one weak ( and bounded) solution of (P ).
Theorem 2.6 When in the setting of Theorem 2.5 we replace the function f by the function fd and the operator A by ˆA, then for each ϕ∈ CE([−d, 0]) such that kϕ(0)k ≤ r−ckmkkG(t,0)kM the problem (Q) has at least one weak solution.
3. Main Results. The aim of this paper is to prove the following two results:
Theorem 3.1 In Theorem 2.5 if we replace De Blasi’s measure β by a (RE,N , p)−
measure of weak noncompactness γ such that:
∀α ∈ IR ∀X, Y ∈ N0 αX∈ N0 X + Y ∈ N0moreover p(X + Y )≤ p(X) + p(Y ) and p(αX) = |α|p(X),
then for each x0∈ E0with kx0k ≤ r−ckmkkG(t,0)kM we have a weak (and bounded) solution of (P ).
Theorem 3.2 If we replace the function f by the function fd and the operator A by ˆA in the setting of Theorem 3.1, then for each ϕ ∈ CE([−d, 0]) such that kϕ(0)k ≤ r−ckmkkG(t,0)kM the problem (Q) has at least one weak solution.
Proof of Theorem 3.1. Let x0 ∈ E0 with kx0k ≤ r−ckmkd M. By the definition ofG and its properties ([23]) there exists d > 0 such that kG(t, 0)k ≤ d. Then G(t, 0)x0 is a solution of (1) and kG(t, 0)x0k ≤ dkx0k ≤ r − ckmkM. Put
S={x∈Cw(IR+, E) :kx(t)−x(τ)k≤r Z τ
t |A(s)|ds+
Z τ t
m(s)ds 0≤ t ≤ τ}.
S is almost equicotinuous closed and convex subset of Cw(IR+, E). Since G(t, .)f(s, x(.)) is Pettis integrable on each compact subset of IR+, then we can define a mapping φ by
φ(x)(t) =G(t, 0)x0+Z ∞
0 G(t, s)f(s, x(s)) ds for t ∈ IR+ for each x ∈ S and t ∈ IR+the integral is convergent:
kφ(x)(t)k ≤ dkx0k + ckmkM≤ r.
Since y = φ(x) is a weak solution of the equation ˙y(t) = A(t)y(t) + f(t, x(t)), then
kφ(x)(t) − φ(x)(τ)k ≤ Z τ
t kA(s)φ(x)(s) + f(s, x(s))k ds
≤ r Z τ
t |A(s)| ds + Z τ
t
m(s) ds
so φ : S → S. Moreover, it can be shown as in [8], φ is a continuous mapping from S into S . Let D ={xn : n = 0, 1, 2, · · · } with x0 as an arbitrary element of S and φ(xn) = xn+1. Thus D ⊂ S and, from (M4), γ(D) = γ(φ(D)). If G is the set of all limit points of the sequence (xn), then φ(G) = G. Put R(X) = conv φ(X) for X ⊂ S and consider Ω is the family of all subsets X of S such that G ⊂ X and R(X)⊂ X. Now S ∈ Ω and so Ω 6= ∅.
If we denote by V the intersection of all sets of the family Ω, the mapping t→ γ(φ(V )(t)) is absolutely continuous. If t ≥ 0 and ε > 0, then there exists T ≥ t
such that kmχ[T,∞[kM< 2cε. From the Scorza-Dragoni theorem there exists a closed subset Iε of I = [0, T ] such that λ(I − Iε) < δ and w is uniformly continuous on Iε× [0, 2T ].
There exists a closed subset Jε of I such that λ(I − Jε) < ε and that for any compact subset K of Jεand any bounded subset Z of E,
(3) γ(f (K × Z)) ≤ sup
s∈Kw(s, γ(Z)).
There exist δ(ε), η > 0 (η < δ) such that if s1, s2 ∈ Iε; r1, r2 ∈ [0, 2T ] with |s1− s2| < δ, |r1− r2| < δ, then |w(s1, r1) − w(s2, r2)| < ε and |s1− s2| < η implies
|γ(V (s1)) − γ(V (s2))| < δ. Fix τ such that 0 ≤ t ≤ τ ≤ T and consider the partition of [t, τ], t = t0 < t1 <· · · < tm = τ such that ti− ti−1 < η for i = 1,· · · , n. Let Ti= Jε∩ [ti−1, ti] ∩ Iε, P =Pm
i=1Ti= [t, τ] ∩ Jε∩ Iεand Q = [t, τ] − P. Since G(t, .) is uniformly continuous on P, we can find η0> 0 (η0 < δ) such that if r1, r2∈ P and
|r1− r2| < η0, then
kG(t, r1) − G(t, r2)k < ε and we can find si in Ti with
(4) sup
s∈Ti
kG(t, s)k = kG(t, si)k.
Further, we have
φ(V )(t) ⊂ G(t, 0)x0+Z
PG(t, s)f(s, V (s)) ds + Z
QG(t, s)f(s, V (s)) ds +Z ∞
T G(t, s)f(s, V (s)) ds.
By the mean value theorem for the Pettis-integral [11, 1, 24] we obtain Z
PG(t, s)f(s, V (s)) ds ⊂ Xm i=1
Z
Ti
G(t, s)f(s, V (s)) ds
⊂ Xn i=1
λ(Ti)conv{G(t, s)f(s, w):s ∈ Ti, w∈ V (s)}.
Let Si= {x(t) : x ∈ S, t ∈ Ti}. Hence, by Lemma 2.4:
(5) γ(Si) = sup{γ(S(t)) : t ∈ Ti} = γ(S(s0i)), for some s0i∈ Ti.
In view of (4), (5), (3), Lemma 2.1 and Lemma 2.4, we have
γ(
Z
PG(t, s)f(s, V (s)) ds) ≤ Xm i=1
λ(Ti)γ([
s∈Ti
{G(t, s)f(Ti× Si)})
≤ Xm i=1
λ(Ti) sup
s∈Ti
kG(t, s)kγ(f(Ti× Si))
≤ Xm i=1
λ(Ti)kG(t, si)k sup
ti∈Ti
w(ti, γ(Si))
= Xm i=1
λ(Ti)kG(t, si)kw(qi, γ((S)(s0))); qi ∈ Ti. Moreover, from the continuity of a Kamke function w on Iε × [0, 2T ],
|w(s, γ(S(s))) − w(qi, γ(S(s∗i)))| ≤ λ(P )ε0 for all s∗∈ Ti. So λ(Ti)kG(t, si)kw(qi, γ(S(s∗i))) ≤Z
Ti
kG(t, s)kw(s, γ(S(s))) ds +ε0λ(Ti) λ(P ) and
γ(
Z
PG(t, s)f(s, V (s)) ds) ≤ Xm i=1
(Z
Ti
kG(t, s)kw(s, γ(S(s))) ds +ε0λ(Ti) λ(P ) )
= Z
PkG(t, s)kw(s, γ(S(s))) ds + ε0. (6)
But
γ(φ(V )(t)) ≤ γ(
Z
PG(t, s)f(s, V (s)) ds) + γ(
Z
QG(t, s)f(s, V (s))ds) (7)
+ γ(Z ∞
T G(t, s)f(s, V (s)) ds) furthermore, we have
(8) γ(
Z
QG(t, s)f(s, V (s)) ds) ≤ γ(B1)Z
QkG(t, s)km(s) ds and
(9) γ(
Z ∞
T G(t, s)f(s, V (s)) ds) ≤ ckmχ[T,∞]kMγ(B(0, 1))≤ ε
so as λ(Q) < ε and since ε, ε0 are arbitrary positive numbers this implies, owing to the relations (7), (8), (9) and (6), that
γ(φ(V (t))) ≤ Z
PkG(t, s)kw(s, γ(S(s))) ds
≤ Z τ
t kG(t, s)kw(s, γ(S(s))) ds.
(10)
On the other hand, we have
(11) γ(φ(V )(τ ))≤ γ(φ(V )(t)) + γ(
Z τ
t G(t, s)f(s, V (s)) ds).
Define ρ(t) := γ(V (t)), from (10) and (11) we get
ρ(τ )− ρ(t) ≤ γ(
Z τ
t G(t, s)f(s, V (s)) ds) ≤ Z τ
t kG(t, s)kw(s, ρ(s)) ds.
Therefore ˙ρ(t) ≤ cw(t, ρ(t)) a.e. Since ρ(0) = 0 and w is a Kamke function, then ρ≡ 0 and so, ξ(V ) = sup{γ(V (t)) : t ∈ IR+} = 0. In view of Lemma 2.3, we conclude that the closed set V is conditionally compact in Cw(IR+, E) and so it is a convex and compact subset of Cw(IR+, E). From Schauder-Tichonov theorem, since φ is a continuous mapping from V to V, there is a fixed point y of φ such that y is the desired weak solution of (P ) and with supt∈IR+ky(t)k ≤ r.
We need now to recall some necessary definitions. A multivalued function F on a Banach space E into the set of nonempty closed subsets, Pf(E), of E 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 : supy∈F (x)(x∗, y) < c} is open in E (resp. in Ew ), where Ew is the Banach space E with the weak topology. 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) = supy∈F (x)(x∗, y) is sequentially upper hemi-continuous. For other properties of the multivalued function we refer to [19, 12, 5].
Lemma 3.3 ([7]) Let Y be a Banach spaces and Pf c(Y ) be the set of all closed and convex subsets of Y and let F : E → Pf c(Y ) be weakly sequentiallyupperhemicontinuous.
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.
Proof of Theorem 3.2. For each (t, x)∈ [−d,Tn] × E put Γ1(t, x) =
ϕ(t) if t ∈ [−d, 0]
ϕ(0) + nt(x− ϕ(0) if t ∈ [0,Tn],
where n is a positive integer. Let f1 : [0,Tn] × E → E be a function defined by f1(t, x) = fd(t, θT
n(Γ1(., x))). From Theorem 3.1 there is a continuous function u1
such that u1 = ϕ on [−d, 0] and for each t ∈ [0,Tn]
u1(t) = G(t, 0)ϕ(0) +Z t
0 G(t, s)f1(s, u1(s))ds with supt∈[0,T
n]ku1(t)k ≤ r.
If we put k0 = k − 1, then there exists a bounded function uk0for some k ∈ {2, 3, ...n} such that uk0 = ϕ on [−d, 0] and for each t ∈ [0,k0nT]
uk0(t) = G(t, 0)ϕ(0) +Z t
0 G(t, s)fk0(s, uk0(s))ds, where
fk0(t, x) = fd(t, θk0 T n
Γk0(., x)).
Let Γk : [−d,kTn ] × E → E be such that
Γk(t, x) =
( uk0(t) if t ∈ [−d,kn0T]
uk0(k0nT) + n(t − k0nT)(x − uk0(k0nT)) if t ∈ [kn0T,kTn ] . Thus if the function fk : [k0nT,kTn ] × E → E is defined by fk(t, x)
= fd(t, θkT
n (Γk(., x))), then we have a continuous function uk defined on [kn0T,kTn ] by
uk(t) = G(t,k0T
n )uk0(k0T n ) +Z t
k0 T n
G(t, s)fk(s, uk(s))ds.
Further, for 0 ≤ s ≤ r ≤ t, we obtain G(t, s)G(s, r) = G(t, r). Moreover for each t ∈ [kn0T,kTn ] we have
uk0(k0T
n ) = G(k0T
n , 0)ϕ(0) + Z k0 Tn
0 G(k0T
n , s)fk0(s, uk0(s))ds.
Hence
uk(t) = G(t,k0T n )G(k0T
n , 0)ϕ(0) + Z k0 Tn
0 G(t,k0T n )G(k0T
n , s)fk0(s, uk0(s))ds +Z t
k0 b n
G(t, s)fk(s, u(s))ds
= G(t, 0)ϕ(0) +Z k0bn
0 G(t, s)fk0(s, uk0(s))ds +Z t
k0 b n
G(t, s)fk(s, uk(s))ds
= G(t, 0)ϕ(0) +Z t
0 G(t, s)gk(s, uk(s))ds, where
gk(t, uk(t)) =
( fk0(t, uk0(t)) if t ∈ [0,kn0T] fk(t, uk(t)) if t ∈ [kn0T,kTn ] .
Consequently, for all n ∈ IN, we have a continuous bounded function vn such that vn= ϕ on [-d,0] and for each t ∈ I , vn is defined by
vn(t) = G(t, 0)ϕ(0) +Z t
0 G(t, s)fd(s, θkb
nΓk(., vn(s)))ds,
where k ∈ {1, 2, 3, ...n} is such that k0nT ≤ t ≤ kTn . Put L = {vn : n ∈ IN}. Let t1, t2
∈ I and t1< t2. Then
k vn(t1) − vn(t2) k
≤ k G(t1, 0)− G(t2, 0)kk ϕ(0) k +Z t1
0 k G(t1, s)− G(t2, s)kk hn(s, vn(s)) k ds +Z t2
t1
k G(t2, s)kk hn(s, vn(s)) k ds
≤ k G(t1, 0)− G(t2, 0)kk ϕ(0) k +Z t
0 k G(t1, s)− G(t2, s)kk m(s) k ds + c Z t2
t1 k m(s) k ds
and since vn = ϕ on [-d,0], thus L is equicontinuous in CE[−d, b]. Moreover γ(L(t)) = γ({vn(t) : n ∈ IN}), γ(L(0)) = 0 and thus γ(L(t)) = 0 for all t ∈ I. Now by Ascoli’s theorem the sequence {vn : n ∈ IN} converges weakly uniformly to a function v ∈ CE([−d, b]) with y = ϕ on [-d,0]. Moreover γ({hn(t) : n ∈ IN}) = 0 so {hn(t) : n ∈ IN} is relatively weakly compact. Create a multivalued function F(t) = conv{hn(t) : n ∈ IN}. Thus F(t) is nonempty convex and weakly compact.
The set δ1F= {l ∈ L1(I, E) : l(t) ∈ F(t)} is nonempty convex and weakly compact thus by Eberlein- ˇSmulian Theorem there exists a subsequence (hnk) of (hn) such that hnk→ l weakly, l ∈ δ1F. Thus vn tends weakly to G(t, 0)ϕ(0) +Rt
0G(t, s)l(s)ds.
Moreover for each n ∈ IN vn ∈ CE([−d, T ]) vn converges uniformly to v on each compact subset of [−d, T ] and v is uniformly continuous on [-d,0]. But for each t ∈ I there exists n > Td with t ∈ [k0nT,kTn ] for k ∈ {1, 2, ...n − 1} so
k θkTn Γk(., vn(t)) − θtvk
≤ sup
s∈[−d,−Tn]
[k Γk(kT
n + s, vn(t)) − v(kT n + s) k + k v(kT
n + s) − v(t + s) k]
+ sup
s∈[−Tn,0]
[
k vn(k0T
n ) + n(kT
n + s −k0T
n )(vn(t) − vn(k0T n )) − v(kT
n + s) k + k v(kT
n + s) − v(t + s) k]
≤ sup
s∈[−d,−Tn]
[k vn(kT
n + s) − v(kT n + s) k + k v(kT
n + s) − v(t + s) k]
+ sup
s∈[−Tn,0]
[
T k (vn(t) − vn(k0T
n )) k + k vn(k0T
n ) − v(kT n + s) k + k v(kT
n + s) − v(t + s) k
] → 0 as n → ∞.
Thus by Lemma 3.3 we conclude that v(.) is the desired weak solution of (Q) with
supt∈Ikv(t)k ≤ r.
Acknowledgement. The author is very grateful to Professor Mieczys law Cicho´n (Pozna´n, Poland) for his careful reading of the paper and for the valuable help.
References [1] A. Alexiewicz, Analiza funkcjonalna, Warszawa 1969.
[2] A. Ambrosetti, Una teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend Sem Mat Univ Padova, 39 (1967), 349-360.
[3] J. Bana`s and K. Goebel, Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics v. 60, Marcel Dekker Inc. New Yourk 1980.
[4] M. Boundourides, An existence theorem for ordinary differential equations in Banach spaces, Bull. Austral. Math. Soc. 22 (1980), 457-463.
[5] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics v 580. Springer Verlag 1977.
[6] M. Cicho´n, A point of view on measures of noncompactness, Demonstratio Math., 26 (1993), 767-777.
[7] M. Cicho´n, Differential inclusions and abstract control problems, Bull. Austral Math. Soc., 53 (1996), 109-122.
[8] M. Cicho´n, On bounded weak solutions of a nonlinear differential equation in Banach space, Functiones et Approximatio, 11 (1992), 27-35.
[9] M. Cicho´n, On measures of weak noncompactness, Publicationes Mathematicae Debrecen, 45 1-2 (1994), 93-102.
[10] M. Cicho´n, Trichotomy and bounded solutions of nonlinear differential equations, 119(3) (1994), 275-284.
[11] E. Cramer, V. Lakshmikantham and A.R. Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. 2 (1976), 169-177.
[12] F.S. De Blasi and J. Myjak, On continuous approximations for multifunctions, Pacific J.
Math. 123 (1986), 9-13.
[13] I.T. Gohberg, L.S. Goldenstein and A.S. Markus, Investigation of some properties of bounded linear operators in connection with their q-norms, U∼c en. Zap. Kishinevskogo Univ. 29 (1957), 29-36 [Russian].
[14] A.M. Gomaa, Existence solutions for differential equations with delay in Banach spaces, to appear in Math. & Phy. Soc.of Egypt.
[15] A.M. Gomaa, Weak and strong solutions for differential equations in Banach spaces, Chaos, Solitons & Fractals, 18 (4) (2003), 687-692.
[16] E. Hille and R.S. Phillips, Functional Analysis and Semigroups, Amer. Math. Soc. Providence R.I. 1957.
[17] A.G. Ibrahim and A.M. Gomaa, Strong and weak solutions for differential inclusions with moving constraints in Banach spaces, PU. M. A. 8(1) (1997), 53-65.
[18] V.I. Istratescu, On a measures of noncompactness, Bull. Math. Soc. Sci. Math., R. S.
Roumanie 16 (1972), 195-197.
[19] E. Klein and A. Thompson, Theory of correspondences, Wiley, New York 1984.
[20] S. Krzy`ska and I. Kubiaczyk On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces, Demonstratio Mathematica, 17 (2) (1999), 323-330.
[21] K. Kuratowski, Sur les espaces compl´etes, Fund. Math. 15 (1930), 301-309.
[22] M. Makowiak, On some bounded solutions of a nonlinear differential equations, Demonstra- tioMathematica, 15 (4) (1994), 801-808.
[23] J.L. Massera and J.J. Shaffer,Linear Differential Equations and Function Spaces, New York- London 1966.
[24] A.R. Michell and C. Smith,An existence theorem for weak solution of differential equations in Banach spaces, Nonlinear Equation in Abstract Spaces (V. Lakshmikantham, ed), (1978) 387-404.
[25] N.S. Papageorgiou, Weak solutions of differential equations in Banach spaces, Bull. Austral.
Math. Soc., 33 (1986), 407-418.
[26] A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar,6 (1971), 197-203.
Adel Mahmoud Gomaa Helwan University
Department of Mathematics, Faculty of Science, Egypt E-mail: gomaa5@hotmail.com
(Received: 14.05.2007)
