Electronic Journal of Differential Equations, Vol. 2009(2009), No. 22, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS FOR DIFFERENTIAL
INCLUSIONS ON CLOSED MOVING CONSTRAINTS IN
BANACH SPACES
ADEL MAHMOUD GOMAA
Abstract. In this paper, we prove the existence of solutions to a multivalued
differential equation with moving constraints. We use a weak compactness
type condition expressed in terms of a strong measure of noncompactness.
1. Introduction
In this paper we study the existence of solutions to the multivalued differential
equation with moving constraints
ẋ(t) ∈ F t, x(t) a.e. on I,
x(t) ∈ Γ(t) ∀t ∈ [0, T ],
(1.1)
x(0) = x0 ∈ Γ(0).
Where F : [0, T ] × E → Pck (E)(Pck is the family of nonempty convex compact
subsets of E) and Γ : [0, T ] → Pf (E), (Pf (E) is the family of closed subsets of E).
Problem (1.1) has been studied by many authors; see for example [6, 2, 25, 8, 26]
when F is lower semicontinuous, and [14, 6] when F is upper semicontinuous with
Γ is independent of t. For Γ depending on t, we refer to [2, 20, 5]. In [16] we
consider the differential inclusions ẋ(t) ∈ A(t)x(t) + F (t, x(t)), x(0) = x0 where
{A(t) : 0 ≤ t ≤ T } is a family of densely defined closed linear operators generating
a continuous evolution operator S(t, s) and F is a multivalued function with closed
convex values in Banach spaces. there, we show how that this results can be used
in abstract control problems. Also in [17] we consider the Cauchy problem
ẋ(t) = f t, x(t) , t ∈ [0, T ]
x(0) = x0 ,
where f : [0, T ] × E → E and E is a Banach space. In [11, 12], we study nonlinear
differential equations. In [10] we study some differential inclusions with delay and
their topological properties. Much work has been done in the study of topological
properties of solution for differential inclusions; see [1, 3, 7, 21, 19, 15, 18].
In this paper we to prove the existence of solutions to (1.1) by using a measure
of strong noncompactness, γ, (see the next section). Since the Kuratowksi measure
2000 Mathematics Subject Classification. 32F27, 32C35, 35N15.
Key words and phrases. Differential inclusions; moving constraints; existence solutions.
c
2009
Texas State University - San Marcos.
Submitted December 1, 2008. Published January 27, 2009.
1
2
A. M. GOMAA
EJDE-2009/22
of noncompactness and the ball measure of noncomactness are measures of strong
noncomactness and we can construct many measures such γ as in [9], in this paper
Theorem 3.1 is a generalization of results for example Szufla [26] and IbrahimGomaa [20]. In Theorem 3.2, the assumption on noncompactness is weaker than
that of Benabdellah-Castaing and Ibrahim [5].
2. Preliminaries
Let E be a Banach space, E ∗ its topological dual space, Ew the Banach space E
endowed with the weak topology, B(0, 1) unit ball of E, I = [0, T ], (T > 0), and λ
be the Lebesgue measure on I . Consider B is the family of all bounded subsets of
E and C(I, E) is the space of all weakly continuous functions from I to E endowed
with the topology of weak uniform convergence.
Definition 2.1. By a measure of strong noncompactness, γ, we will understand a
function γ : B → R+ such that, for all U, V ∈ B,
(M1) U ⊂ V =⇒ γ(U ) ≤ γ(V ),
(M2) γ(U ∪ V ) ≤ max(γ(U ), γ(V )),
(M3) γ(convU ) = γ(U ),
(M4) γ(U + V ) ≤ γ(U ) + γ(V ),
(M5) γ(cU ) = |c|γ(U ), c ∈ R,
(M6) γ(U ) = 0 ⇐⇒ U is relatively compact in E,
(M7) γ(U ∪ {x}) = γ(U ), x ∈ E.
Definition 2.2. For any nonempty bounded subset U of E the weak measure of
noncompactness, β, and the Kuratowski’s measure of noncompactness, α, is defined
as:
α(U ) = inf{ε > 0 : U admits a finite number of sets with diameter less than ε.}
For the properties of β and α we refer the reader to [4, 13].
Definition 2.3. By a Kamke function we mean a function w : I × R+ → R+ such
that:
(i) w satisfies the Caratheodry 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 u̇(t) ≤ w t, u(t) a.e. on [0, c] and such that u(0) = 0.
Lemma 2.4 ([22, 4]). If γ : B → R+ satisfies conditions (M2), (M4), (M6), then,
for any nonempty U ∈ B,
γ(U ) ≤ γ(B(0, 1))α(U )
Lemma 2.5 ([24, 23]). If γ is a measure of weak (strong) noncompactness and
A ⊂ C(I, E) be a family of strongly equicontinuous functions, then γ(A(I)) =
sup{γ(A(t)) : t ∈ I}.
3. Main Results
Theorem 3.1. Let Γ : I → Pf (E) be a set-valued function such that its graph, G,
is left closed and F : I × E → Pck (E) be a scalarly measurable set-valued function
such that for any t ∈ I, F (t, .) is upper semicontinuous on E. Suppose that F
satisfies the following conditions:
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EXISTENCE SOLUTIONS
3
(A1) For each ε > 0, there exists a closed subset Iε of I with λ(I − Iε ) < ε such
that for any nonempty bounded subset Z of E, one has
γ(F (J × Z)) ≤ sup w(t, γ(Z))
t∈J
for any compact subset J of Iε ;
(A2) there is µ ∈ L1 (I, R+ ), such that kF (t, x)k < µ(t)(1 + kxk), for all (t, x) ∈
G;
(A3) for each (t, x) ∈ ([0, T [×E) ∩ G and ε > 0 there is (tε , xε ) ∈ G such that
0 < tε − t < ε and that
Z
xε − x ∈
tε
F (s, x)ds + (tε − t)εB(0, 1).
t
Then, for any x0 ∈ Γ(0), there is a solution for (1.1).
Proof. Let (εn )n∈N be a decreasing sequence in ]0, 1] with εn = 0. By [5, Proposition
6.1], there exist m > 1, a sequence (θn )n∈N of right continuous functions θn : I → I
such that θn (0) = 0, θn (T ) = T and θn (t) ∈ [t − εn , t], and a sequence (xn )n∈N from
I to E with
Rt
(i) for all t ∈ I, xn (t) = x0 + 0 ẋn (s) ds, where ẋn ∈ L1 (I, E);
(ii) for all t ∈ I, xn (θn (t)) ∈ Γ(θn (t));
(iii) ẋn (t) ∈ F (t, xn (θn (t)) + εn B(0, 1) a.e on I;
(iv) kẋn (t)k ≤ mµ(t) + 1, a.e on I.
From (iv) the sequence (xn ) is equicontinuous in C(I, E). For each t ∈ I, set
A(t) = {xn (t) : n ∈ N} and ρ(t) = γ(A(t)).
We claim that (xn )n∈N is relatively compact in the space C(I, E). So we will show
that ρ ≡ 0. Since for each (t, τ ) ∈ I × I, we have
γ{(xn )(τ ) : n ∈ N} ≤ γ{(xn )(t) : n ∈ N} + γ{(xn )(τ ) − (xn )(t) : n ∈ N}
and
γ{(xn )(t) : n ∈ N} ≤ γ{(xn )(τ ) : n ∈ N} + γ{(xn )(t) − (xn )(τ ) : n ∈ N},
then, from Lemma 2.4,
|ρ(τ ) − ρ(t)| ≤ γ(B(0, 1))α({xn (t) − xn (τ ) : n ∈ N}),
which implies
Z
|ρ(τ ) − ρ(t)| ≤ 2γ(B(0, 1))|
τ
(mµ(s) + 1) ds|.
t
It follows that ρ is an absolutely continuous and hence differentiable a.e. on I. Let
ε
,
ε > 0. Since εn → 0 as n → ∞, then we can find n0 ∈ N such that T εn < γ(B(0,1))
for all n ≥ n0 . Now let (t, τ ) ∈ I × I with t ≤ τ . In view of Condition (iii) and
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A. M. GOMAA
EJDE-2009/22
properties of γ ((M4), (M7)), we have
Z
ρ(τ ) − ρ(t) ≤ γ(
τ
ẋn (s) ds : n ∈ N )
Z τ
≤ γ(∪n∈N
F (s, θn (s))ds) + γ({εn B(0, 1)(τ − t) : n ∈ N})
t
Z τ
= γ(∪n∈N
F (s, θn (s))ds) + γ({εn B(0, 1)(τ − t) : n ≥ n0 })
t
Z τ
ε
≤ γ(∪n∈N
F (s, θn (s))ds) +
γ(B(0, 1))
γ(B(0, 1))
t
Z τ
≤ γ(∪n∈N
F (s, θn (s))ds) + ε.
t
t
Thus,
τ
Z
ρ(τ ) − ρ(t) ≤ γ(∪n∈N
F (s, θn (s))ds).
(3.1)
t
Since ρ is continuous and w is Caratheodory we can find a closed subset Iε of I,
δ > 0, η > 0 (η < δ) and for s1 , s2 ∈ Iε ; r1 , r2 ∈ [0, 2T ] such that if |s1 − s2 | < δ,
|r1 −r2 | < δ, then |w(s1 , r1 )−w(s2 , r2 )| < ε and if |s1 −s2 | < η, then |ρ(s1 )−ρ(s2 )| <
δ
2 . Consider the following partition, of [t, τ ], t = t0 < t1 < · · · < tm = τ such that
ti −ti−1 < η for i = 1, . . . , n. From Condition (A1) we can find a closed subset Jε of I
such that λ(I − Jε ) < ε and that for any compact subset K of Jε and any bounded
subset Z of E, γ(f (K × Z)) ≤ sups∈K w(s, γ(Z)). Let Ti = Jε ∩ [ti−1 , ti ] ∩ Iε ,
P = ∪m
i=1 Ti = [t, τ ] ∩ Jε ∩ Iε , Q = [t, τ ] − P and Ai = {xn (θn (t)) : n ∈ N, t ∈ Ti },
i = 1, . . . , m. In view of the mean value theorem, properties of γ ((M3), (M5)) and
Condition (A1), this implies
Z
γ(∪n∈N
Z
m
X
F (s, xn (θn (s)))ds) ≤ γ(
∪n∈N
P
i=1
m
X
≤ γ(
≤
≤
F (s, xn (θn (s)))ds)
Ti
λ(Ti )(convF (Ti × Ai )))
i=1
m
X
λ(Ti )γ(convF (Ti × Ai ))
i=1
m
X
λ(Ti ) sup w(si , γ(Ai )).
si ∈Ti
i=1
Now we have
γ(Ai ) = γ({xn (θn (s)) : n ∈ N, s ∈ Ti })
≤ γ({xn (s) : n ∈ N, s ∈ Ti }) + γ({xn (θn (s)) − xn (s) : n ∈ N, s ∈ Ti })
Z θn (s)
≤ γ({xn (s) : n ∈ N, s ∈ Ti }) + γ({
ẋn (r)dr : n ∈ N, s ∈ Ti }).
s
EJDE-2009/22
EXISTENCE SOLUTIONS
5
From Lemma 2.4 we know that
n Z θn (s)
o
γ
ẋn (r)dr : n ∈ N, s ∈ Ti
s
≤ γ(B(0, 1))α
n Z
θn (s)
ẋn (r)dr : n ∈ N, s ∈ Ti
o
.
s
Also limn→∞ |θn (s) − s| = 0. So, γ(Ai ) = γ({xn (s) : n ∈ N, s ∈ Ti }) + 2δ . Applying
Lemma 2.5, we get γ(Ai ) = supξi ∈Ti ρ(ξi ) + 2δ . Since w and ρ are continuous on
the closed subsets of Ti , then
Z
m
X
δ
γ ∪n∈N
F (s, xn (θn (s)))ds ≤
λ(Ti ) sup w( si , sup ρ(ξi ) +
2
si ∈Ti
ξi ∈Ti
P
i=1
≤
m
X
λ(Ti )w qi , ρ(ξi ) +
i=1
δ
,
2
where qi and ξi are elements of Ti . Moreover, for all s ∈ Ti , we have
δ
δ
δ
δ
|ρ(s) − ρ(ξi ) + | ≤ |ρ(s) − ρ(ξi )| + < + = δ.
2
2
2 2
This implies |w(s, ρ(s)) − w(qi , ρ(ξi ) + 2δ )| < ε for all s ∈ Ti . Consequently,
R
λ(Ti )w(qi , ρ(ξi ) + 2δ ) ≤ Ti w(s, ρ(s)) ds + ελ(Ti ). So,
Z
m Z
X
w(s, ρ(s))ds + ελ(Ti )
γ ∪n∈N
F (s, xn (θn (s)))ds ≤
P
Ti
Zi=1
=
w(s, ρ(s))ds + ελ(P )
ZPτ
≤
w(s, ρ(s))ds + ε(τ − t).
t
On the other hand,
Z
Z
γ(∪n∈N
F (s, xn (θn (s)))ds) ≤ 2mγ(B(0, 1))
µ(s)(1 + kxn (θn (s))k)ds.
Q
Q
As λ(Q) < 2ε and since ε is arbitrary, then
Z τ
Z
γ(∪n∈N
F (s, xn (θn (s)))ds) ≤
t
τ
w(s, ρ(s))ds,
(3.2)
t
Thus, from two relations (3.1), (3.2),
ρ̇(t) ≤ w(s, ρ(s))
a.e. onI.
ρ(0) = 0 and w is a Kamke function, then ρ is identically equal to zero. It follows
that (xn ) is relatively compact in C(I, E). Since, for all t ∈ I,
Z t
γ({xn (θn (t)) : n ∈ N}) ≤ γ({xn (t) : n ∈ N}) + γ(
(mµ(s) + 1)ds)B(0, 1)
θn (t)
and since limn→∞ |θn (t) − t| = 0, the set Ã(t) := {xn (θn (t)) : n ∈ N} is relatively
compact in E. By our assumption F (t, .) is upper semicontinuous, it follows that
F (t, A(t)) is compact for all t ∈ I, Furthermore, we have
ẋn (t) ∈ F (t, A(t)) + εn B(0, 1),
∀n ∈ N, ∀t ∈ I.
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A. M. GOMAA
EJDE-2009/22
Since ẋn is uniformly integrable, by [5, Theorem 5.4], the sequence ẋn is relatively
σ(L1 (I, E), L∞ (I, E)) compact. Therefore there are xnk ∈ C(I, E), g ∈ L1 (I, E)
and a subsequence (xnk ) of (xn ) such that (xnk ) converges to x in C(I, E) and
(xnk
˙ ) converges to g in L1 (I, E) for σ(L1 (I, E), L∞ (I, E)), with
Z t
x(t) = x0 +
g(s)ds, ∀t ∈ I.
0
Thus g = ẋ. Clearly for all t ∈ I, limn→∞ xnk (θn (t)) = x(t), and x(t) ∈ Γ(t), for
all t ∈ I. Finally, in virtue [5, Theorem 5.6, Remark 6.3] and the property (iii) we
obtain
ẋ(t) ∈ F (t, x(t)) a.e. onI.
Theorem 3.2. Let Γ : I → Pf (E) be a set-valued function with closed graph, G,
and F : G → Pck (E) be a set-valued function such that for any t ∈ I, F (t, .) is
upper semicontinuous on E. Assume that F satisfies the following conditions:
(A1’) For each ε > 0, there exists a closed subset Iε of I with λ(I − Iε ) < ε such
that for any nonempty bounded subset Z of E, one has
γ(F (G ∩ (I × Z))) ≤ sup w(t, γ(Z)),
t∈I
for any compact subset J of Iε ;
(A2’) there is a positive number c such that
kF (t, x)k < c(1 + kxk), ∀(t, x) ∈ G;
(A3’) for each (t, x) ∈ ([0, T [×E) ∩ G and for any ε > 0 there is (tε , xε ) ∈ G such
that 0 < tε − t < ε and
xε − x
∈ F (t, x) + εB(0, 1).
tε − t
Then, for each x0 ∈ Γ(0), there is a solution of (1.1).
Proof. Let Aε ([0, τ ]) (ε > 0, τ ∈ I) be the set of all points (x, θ) where θ : [0, τ ] →
[0, τ ] is an increasing right continuous function with θ(0) = 0, θ(τ ) = τ and for all
t ∈]0, τ [, θ(t) ∈ [t − ε, t] and x : [0, τ ] → E is such that:
Rt
(i) for all t ∈ [0, τ ], x(t) =x0 + 0 ẋ(s) ds, where ẋ ∈ L1 (I, E);
(ii) for all t ∈ [0, τ ], x θ(t) ∈ Γ(θ(t));
(iii) for all t ∈ [0, τ ], ẋ(t) ∈ F t, x(θ(t) + εB(0, 1), a.e.
Let ε ∈]0, 1] and (θ, x) ∈ Aε ([0, τ ]). Then by (A2’) and the fact that, for all
t ∈ [0, τ ], θ(t) ∈ [t − ε, t], we have
Z θ(t)
Z t
kx(θ(t))k ≤ kx0 k +
kẋ(s)k ds ≤ kx0 k +
kẋ(s)k ds
0
0
Z t
≤ kx0 k + εT +
c(1 + kx(θ((s))k) ds.
0
By Gronwall’s lemma, we obtain kx(θ(t))k ≤ (kx0 k + T )ecT which gives us
kx(θ(t))k + 1 ≤ (1 + kx0 k + T )ecT .
Consequently we get for all t ∈ [0, τ ],
F (t, x(θ(t))) ⊆ pcB(0, 1)
(3.3)
EJDE-2009/22
EXISTENCE SOLUTIONS
7
S
where p = (1 + kx0 k + T )ecT . Let Aε = τ ∈I Aε ([0, τ ]). Obviously Aε 6= ∅.
Partially order Aε such that for any (θi , xi ) ∈ Aε ([0, τi ]) ⊆ Aε (i = 1, 2) (θ1 , x1 ) ≤
(θ2 , x2 ) ⇐⇒ τ1 ≤ τ2 , θ1 = θ2 |[0,τ1 ] and x1 = x2 |[0,τ2 ] . Let C be a subset of Aε such
that each two elements of it are comparable that is there exists a subset N0 ⊆ N
such that C = {(θj , xj ) : j ∈ N0 } ⊆ Aε and each (θn , xn ), (θm , xn ) ∈ C we have
(θn , xn ) ≤ (θm , xm ) or (θm , xm ) ≤ (θn , xn ). Now we prove that C has an upper
bound. Let τ = supj∈N0 τj . Also let θ : [0, τ ] → [0, τ ] is such that, for each
j ∈ N0 , θ|[0,τj ] = θj and x : [0, τ [→ E with x|[0,τj ] = xj , for each j ∈ N0 . Let
{τkn } be increasing sequence in N0 such that τ = supn∈N τkn and for any n, m ∈
N, m < n we have ẋkn = ẋkm a.e. on [0, τkn ]. Now we can define ẋ : [0, τ [→ E
by, for any n ∈ N, ẋ(t) = ẋkn (t) a.e. on [0, τkn ]. From (3.3) ẋ is measurable
and kẋ(t)k ≤ pc + ε ≤ pc + 1.R We claim that x, ẋ can be extend to [0, τ ]. Now
for all t ∈ [0, τ [, x(t) = x0 + ]0,t] ẋ(s) ds, for all t ∈ [0, τ [, ẋ(θ(t)) ∈ Γ(θ(t)) and
R
ẋ(t) ∈ F (t, ẋ(θ(t))) + εB(0, 1) a.e. on [0, τ [. If x0 (t) = x0 + ]0,t[ ẋ(s) ds for all
R
t ∈ [0, τ ] then, for (t, t0 ) ∈ [0, τ [×[0, τ [, we have kx0 (t) − x0 (t0 )k ≤ [t,t0 [ (ps + 1) ds.
R
R
Then x∗ := limt→τ −0 (x0 + ]0,t[ ẋ(s) ds) = limn→∞ (x0 + ]0,τk [ ẋ(s) ds) exists. Since
n
x0 (τkn ) ∈ Γ(τkn ) and G is left closed, then (τ,Rx∗ ) ∈ G and hence the result. Let
x∗ = x(τ ) and ẋ(τ ) = 0. Then x(τ ) = x0 + ]0,τ ] ẋ(s) ds, x∗ = x(τ ) ∈ Γ(τ ) and
ẋ(t) ∈ F (t, x(θ(t))) + εB(0, 1) a.e. on [0, τ ]. Consequently we can extend (θ, x)
to [0, τ ] such that (θ, x) belongs to Aε ([0, τ ]) and it is an upper bound for C. By
Zorn’s lemma (Aε , ≤) has a maximal element (θε , xε ) ∈ Aε ([0, τε ]). We shall prove
that τε = T . Let τε < T . If δε > 0 such that δε < inf(ε, T − τε ). Then by (A3’)
there exists (t̂, x̂) ∈ G such that 0 < t̂ − τε ≤ δε and
x̂ − xε
∈ F (τε , xε (τε )) + εB(0, 1).
t̂ − τε
Let ŷ ∈ F (τε , xε (τε )) + εB(0, 1) such that x̂ − xε (τε ) = (t̂ − τε )ŷ. If θ̂ : [0, t̂] → [0, t̂]
and x̃ : [0, t̂] → E are defined as:

(

θε if t ∈ [0, τε ]
xε if t ∈ [0, τε ]
x̃(t) =
θ̂(t) = τε if t ∈]τε , t̂]

x̂ if t ∈ [τε , t̂]

t̂
if t = t̂,
Then it is easy to check that [5, p. 10.25] (θ̂, x̃) ∈ Aε ([0, t̂]) and (θε , xε ) < (θ̂, x̃).
This contradicts the fact that (θε , xε ) is maximal. Now there exist p > 1, (from
(3.3)) a sequence (θn )n∈N of right continuous functions (θ)n : I → I such that
θn (0) = 0, θn (T ) = T and θn (t) ∈ [t − εn , t], if we have decreasing sequence (εn )
ε
such that 0 < εn ≤ 1 εn → 0 as n → ∞ and T εn < γ(B(0,1))
, for all n ≥ n0 we can
define a sequence (xn ) of approximated solutions as the follows:
Rt
∀t ∈ I, xn (t) = x0 + 0 ẋn (s) ds, where ẋn ∈ L1 (I, E). (θn (t), xn (θn (t))) ∈ G.
ẋn (t) ∈ F t, xn (θn (t) + εn B(0, 1), a.e on I. kẋn (t)k ≤ pc + 1, a.e on I.
By the same arguments used in the proof of Theorem 3.1 we can prove that the
sequence (xn ) converges to an absolutely continuous function x which is a solution
for problem (1.1).
4. Conclusion
Let us remark that, if we replace γ in (A1’) by α, the condition
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A. M. GOMAA
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(A4) For each ε > 0, there exists a closed subset Iε of I with λ(I − Iε ) < ε such
that for almost all t ∈ Iε and for any nonempty bounded subset Z of E,
one has
inf α(F (G ∩ (([t − δ, t] ∩ I) × Z))) ≤ w(t, α(Z))
δ>0
implies Condition (A1’) and the converse is not true. Indeed Let ε > 0. Since w is
Caratheodory function, we can find a closed subset Iε of I with λ(I − Iε ) < ε such
that w is continuous on Iε and Condition (A4) holds on Iε . Let Z be a nonempty
bounded subset of E. It follows from (A4) that, for any τ > 0 and any t ∈ Iε , there
exists a δτ,t such that α(F (G ∩ (([t − δτ,t , t] ∩ I) × Z))) ≤ w(t, α(Z)) + τ . Let τ be
arbitrary but fixed, J be a compact subset of Iε . The collection {(t − δ2t , t + δ2t ) : t ∈
J} is an open cover for J. By compactness of J, there exist t01 , t02 . . . , t0n such that
δt0
δt0
δt0
δt0
δt0
δt0
J ⊆ ∪ni=1 (t0i − 2i , t0i + 2i ) ⊆ ∪ni=1 [t0i − 2i , t0i + 2i ]. Now if Ji = J ∩ [t0i − 2i , t0i + 2i ]
and ti = maxJi , 1 ≤ i ≤ n, then there exist t1 , t2 . . . tn ∈ J such that Ji ⊆ [ti −δti , ti ]
and J ⊆ ∪ni=1 [ti − δti , ti ]. This implies that,
α(F (G ∩ (J × Z))) ≤ α(∪ni=1 F (G ∩ (([ti − δti , ti ] ∩ I) × Z)))
≤ max α(F (G ∩ (([ti − δti , ti ] ∩ I) × Z)))
1≤i≤n
≤ max w(ti , α(Z)) + τ ≤ max w(t, α(Z)) + τ
t∈J
1≤i≤n
Since τ is arbitrary, Condition (A1’) holds. To show that the converse is not true
we give an example. Let f : [0, 1] × B(0, 1) → E be the single valued function
defined by f (t, x) = k(t)x, where k : [0, 1] → R,
(
1
if t is irrational
k(t) =
2
1/t if t is rational
Let also w(t, s) = k(t)s, for all (t, s) ∈ I × R+ . Clearly, w is a Kamke function. Let
ε > 0 and choose a closed subset Iε of I such that λ(I − Iε ) < ε and k is continuous
on Iε . Then for any compact subset J of Iε and any bounded subset Z of E,
α(f (G ∩ (J × Z))) ≤ α(f (J × Z)) = α ∪t∈J,x∈Z f {(t, x)}
= α ∪t∈J k(t)Z = sup k(t)α(Z)
t∈J
= sup w t, α(Z) .
t∈J
Then Condition (A1’) holds as the measure γ replaced by the measure α. Butfor
each t ∈ (0, 1) and each nonempty subset Z of E we have α f ([t − δ, t] × Z) =
α(Z)
α ∪s∈[t−δ,t] k(s)Z = α(Z) · sups∈[t−δ,t] k(s) = (t−δ)
2 . Thus, inf δ>0 α(F (([t −
δ, t] ∩ I) × Z)) = α(Z)
t2 . So if t is irrational then inf δ>0 α(F (([t − δ, t] ∩ I) × Z)) =
α(Z)
>
α(Z)
=
k(t)α(Z)
= w(t, α(Z)). Then (A4) does not hold and consequently
t2
Theorem 3.2 is a generalization of the following theorem.
Theorem 4.1 (Benabdellah-Castaing and Ibrahim [5]). Let F and Γ be as in
Theorem 3.2 except F satisfies Condition (A4) instead of (A1’). Then, for any
x0 ∈ Γ(0), there is a solution for (1.1).
EJDE-2009/22
EXISTENCE SOLUTIONS
9
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10
A. M. GOMAA
EJDE-2009/22
Adel Mahmoud Gomaa
Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt
E-mail address: gomaa5@hotmail.com