The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces
In this paper, by using the new fixed point theorem of O’Regan and Precup and noncompact measure,
 the existence of solutions of second order multivalued boundary-value problem in Banach spaces and
 the existence of a mild solution for impulsive neutral functional differential inclus...
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| citation_txt | The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces / L. Wei, Jiang Zhu // Нелінійні коливання. — 2008. — Т. 11, № 2. — С. 191-207. — Бібліогр.: 13 назв. — англ. |
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| description | In this paper, by using the new fixed point theorem of O’Regan and Precup and noncompact measure,
the existence of solutions of second order multivalued boundary-value problem in Banach spaces and
the existence of a mild solution for impulsive neutral functional differential inclusions in Banach spaces
are studied. The compactuess conditions and the upper semicontinuity conditions of multivalued integral
operators are weaken in this paper.
З використанням нової теореми О’Регана та Прекупа, а також некомпактної мiри доведено iснування розв’язкiв багатозначної граничної задачi другого порядку в банахових просторах. Вивчається iснування помiрного розв’язку диференцiальних включень з iмпульсною дiєю
та нейтральним функцiоналом у банахових просторах. У роботi послаблено умови компактностi та верхньої напiвнеперервностi на багатозначнi iнтегральнi оператори.
|
| first_indexed | 2025-12-07T18:03:48Z |
| format | Article |
| fulltext |
UDC 517 . 9
A SECOND ORDER MULTIVALUED BOUNDARY-VALUE
PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL
DIFFERENTIAL INCLUSIONS IN BANACH SPACES*
БАГАТОЗНАЧНА ГРАНИЧНА ЗАДАЧА ДРУГОГО ПОРЯДКУ
ТА ДИФЕРЕНЦIАЛЬНI ВКЛЮЧЕННЯ З IМПУЛЬСНОЮ ДIЄЮ
ТА НЕЙТРАЛЬНИМ ФУНКЦIОНАЛОМ
Lei Wei, Jiang Zhu
School Math. Sci., Xuzhou Normal Univ.
Xuzhou 221116, P. R. China
e-mail: jzhuccy@yahoo.com.cn
In this paper, by using the new fixed point theorem of O’Regan and Precup and noncompact measure,
the existence of solutions of second order multivalued boundary-value problem in Banach spaces and
the existence of a mild solution for impulsive neutral functional differential inclusions in Banach spaces
are studied. The compactuess conditions and the upper semicontinuity conditions of multivalued integral
operators are weaken in this paper.
З використанням нової теореми О’Регана та Прекупа, а також некомпактної мiри доведе-
но iснування розв’язкiв багатозначної граничної задачi другого порядку в банахових просто-
рах. Вивчається iснування помiрного розв’язку диференцiальних включень з iмпульсною дiєю
та нейтральним функцiоналом у банахових просторах. У роботi послаблено умови компактно-
стi та верхньої напiвнеперервностi на багатозначнi iнтегральнi оператори.
1. Introduction. Differential inclusions is an important branch of the general theory of di-
fferential equations and has numerous applications. The problem of existence of solutions of
differential inclusions has been studied by many authors, see [1 – 8]. The main tool used by
these authors is the Leray – Schauder alternative theorem for set-valued mapping. However, in
the Leray – Schauder alternative theorem, the multivalued operator must be upper semiconti-
nuous and compact. In this paper, we will use the new fixed point theorem obtained by O’Regan
and Precup [9] and a noncompact measure to study the existence of solutions of a second order
multivalued boundary-value problem in Banach spaces and the existence of a mild solution for
impulsive neutral functional differential inclusions in Banach spaces. The compactness conditi-
ons and upper semicontinuity conditions on multivalued integral operators are weaken in this
paper.
In this paper, we denote by E a real Banach space, ‖ · ‖ is the norm in E. In the following,
we denote
K(E) = {A ⊂ E : A is nonempty and compact},
P (E) = {A ⊂ E : A is nonempty and closed},
C(E) = {A ⊂ E : A is nonempty and convex},
∗ This work is supported by Natural Science Foundation of the EDJP (05KGD110225), JSQLGC and National
Natural Science Foundation 10671167, 10771212, China.
c© Lei Wei, Jiang Zhu, 2008
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11 , N◦ 2 191
192 LEI WEI, JIANG ZHU
‖A‖ = sup{‖x‖ : x ∈ A}. Let F : D ⊂ E → 2E\∅ be a set-value mapping; ∀A ⊂ E
F−1(A) = {x ∈ D : F (x)
⋂
A 6= ∅}, graph (F ) = {(x, y) : x ∈ E, y ∈ F (x)} is said to the
graph of F.
Definition 1.1 [5]. Let X, Y be metric spaces. F : D ⊂ X → 2Y \∅ is said to be upper
semicontinuous (short as u.s.c.) if F−1(A) is closed in X wherever A ⊂ Y is closed.
Definition 1.2 [5]. Let (Ω,A) be a measurable space, F : Ω → 2X\∅ is said to be measurable
if F−1(B) ∈ A for every open subset B ⊂ E.
Lemma 1.1 [5]. Let J = [0, a] ⊂ R and F : J × E → 2E\{∅} be compact values. If F (t, ·)
is u.s.c. and F (·, x) has a strongly measurable selection, then there exists w(·) ∈ F (·, v(·)) for any
v ∈ C[J,E].
Lemma 1.2 [10]. Let C ⊂ L1([a, b], E) be separable. If there exists h ∈ L1[a, b] such that
‖u(t)‖ ≤ h(t) for a.e. t ∈ [a, b] and every u ∈ C, then
α
b∫
a
u(t)dt : u ∈ C
≤ 2
b∫
a
α(C(t)) dt.
Lemma 1.3 [5]. Assume that F : E → K(E) is u.s.c., if A ⊂ E is compact, then F (A) is
compact.
Lemma 1.4 [11] (Ascoli – Arzela). H ∈ C[T,E] is a relatively compact set if and only if H is
equicontinuous and for any t ∈ T, H(t) is relatively compact in E.
Lemma 1.5 [11] (Mazur). Let (E, ‖ · ‖) be a normed space, {xn}n∈N ⊂ E, x0 ∈ E and
w − lim
n→∞
xn = x0. Then for any ε > 0 there exist n ∈ N, αi ≥ 0, i = 1, 2, . . . , n,
n∑
i=1
αi = 1,
such that ‖x0 −
n∑
i=1
αixi‖ < ε.
Lemma 1.6 [9]. Let D be a closed, convex subset of a Banach space E and N : D → 2D.
Assume graph (N) is closed, and for any compact set A, N(A) is relatively compact. If there exists
x0 such that
M ⊂ D,M = co ({x0} ∪N(M)) and M = C with C ⊂ M countable ⇒ M is compact,
then N has a fixed point in D.
2. Boundary-value problem for differential inclusions. In this section, we prove the existence
of a C1-solution of the following second order multivalued boundary-value problem in Banach
spaces:
x′′(t) ∈ F (t, x(t), x′(t)) a.e. t ∈ [0, 1],
ax(0)− bx′(0) = x0, (2.1)
cx(1) + dx′(1) = x1,
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 193
where a ≥ 0, b ≥ 0, c ≥ 0, d ≥ 0, ad + bc > 0, x0, x1 ∈ E. Let
Gx =
{
w ∈ L1([0, 1], E) : w(·) ∈ F (·, x(·), x′(·)), x ∈ C1([0, 1], E)
}
,
where w(·) is a strongly measurable selection of F (·, x(·), x′(·)). Let
Jw(t) = h(t) +
1∫
0
g(t, s)w(s) ds, w ∈ Gx, N = J ◦G,
where g is the Green function with respect to inclusions (2.1), and h(t) is a solution of
x′′(t) = 0 a.e. t ∈ [0, 1],
ax(0)− bx′(0) = x0,
cx(1) + dx′(1) = x1.
Let a0 = max{|g(t, s)|, |gt(t, s)| : t, s ∈ [0, 1]}, and for any R > 0, UR = {x ∈ C1[T,E] :
‖x‖1 ≤ R}, where ‖x‖1 = max{‖x‖, ‖x′‖}.
We first make some assumptions about the multivalued map F : T × E → 2E .
(C1) F (·, x, y) has a strongly measurable selection for any x, y ∈ E;
(C2) F (t, ·, ·) is u.s.c., for a.e. t ∈ [0, 1];
(C3) for any r > 0 there exists lr ∈ L1[T,R+] such that ‖F (t, x, y)‖ ≤ lr(t) if ‖x‖ ≤ r and
‖y‖ ≤ r;
(C4) lim supρ→∞
a0
ρ
∫ 1
0
lρ(t)dt < 1;
(C5) for any R > 0 there exists w : T × [0, 2R] → R+ such that for any bounded sets
A,B ⊂ UR, the inequality α(F (s,A, B)) ≤ w(smax{α(A)), α(B)}) holds, and
ϕ(t) ≤ 2
1∫
0
(|g(t, s)|+ |gt(t, s)|)w(s, ϕ(s)) ds
has a unique nonnegative continuous zero solution.
Lemma 2.1. If F : [0, 1] × E × E → CK(E) satisfies (C1), (C2) and (C3), then N : UR →
→ C(C1[T,E]) has closed graph, and N(B) is relatively compact for any compact set B.
Proof. (a) We prove Nx 6= ∅ for any x ∈ UR. By Lemma 1.1 we know that F (·, x(·), x′(·))
has a strongly measurable selection. From (C3), F (·, x(·), x′(·)) has a Bochner selection, i.e.,
Nx 6= ∅.
(b) Since F has convex values, clearly N has convex values.
(c) Suppose xn → x, vn → v, n → ∞, and
vn(·) = h(·) +
1∫
0
g(·, s)wn(s) ds,
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
194 LEI WEI, JIANG ZHU
where
vn ∈ N(xn), wn(·) ∈ F (·, xn(·), x′n(·)).
It follows from Exercise 9.6 in [5] that {wn : n ≥ 1} has a weakly convergent subsequence in
L1[T,E]. Assume wnk
⇀ w, by Mazur Theorem we have
w ∈ co
( ∞⋂
m=1
∞⋃
k=m
{wnk
}
)
⊂ co
( ∞⋂
m=1
∞⋃
k=m
{
F (·, xnk
(·), x′nk
(·))
})
⊂ F (·, x(·), x′(·)).
Since
vnk
(·) = h(·) +
1∫
0
g(·, s)wnk
(s) ds → v, nk → ∞,
we have
v(·) = h(·) +
1∫
0
g(·, s)w(s) ds.
Thus N is a closed graph operator.
(d) We prove that N maps any compact set M ⊂ UR into a relatively compact set. For
this aim, it is enough to prove that {vn}n≥1 ⊂ N(M) has a convergent subsequence, that is,
{vn}n≥1 ⊂ N(M) is relatively compact. Suppose
vn(·) = h(·) +
1∫
0
g(·, s)wn(s) ds
where
vn ∈ N(xn), wn(·) ∈ F (·, xn(·), x′n(·)), xn ∈ M.
Since M is a compact subset of C1[T,E], M is bounded, hence by (C3), there exists k ∈
∈ L1[0, 1] such that ‖wn(s)‖ ≤ k(s) for a.e. s ∈ [0, 1], and then {vn : n ≥ 1} is bounded. Then
we have
α({vn(t) : n ≥ 1}) = α
h(t) +
1∫
0
g(t, s)wn(s)ds : n ≥ 1
=
= α
1∫
0
g(t, s)wn(s)ds : n ≥ 1
≤
≤ 2
1∫
0
α({g(t, s)wn(s) : n ≥ 1})ds ≤
≤ 2 max{|g(t, s)| : s ∈ [0, 1]}
1∫
0
α({wn(s) : n ≥ 1})ds.
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 195
It follows from the compactness of M that M(t),M ′(t) are compact for any t ∈ [0, 1]. Since
F (s, ·, ·) is u.s.c. and has compact values, we know that F (s,M(s),M ′(s)) is relatively compact
by Lemma 1.3. Thus, for any s ∈ [0, 1], α({wn(s) : n ≥ 1}) = 0. This shows that α({vn(s) :
n ≥ 1}) = 0. Similarly, from
α({v′n(t) : n ≥ 1}) = α
h′(t) +
1∫
0
gt(t, s)wn(s)ds : n ≥ 1
=
= α
1∫
0
gt(t, s)wn(s)ds : n ≥ 1
≤
≤ 2
1∫
0
α({gt(t, s)wn(s) : n ≥ 1})ds ≤
≤ 2 max{|gt(t, s)| : s ∈ [0, 1]}
1∫
0
α({wn(s) : n ≥ 1})ds,
we have that α({v′n(s) : n ≥ 1}) = 0. Thus for any t ∈ [0, 1], {vn(t)}n≥1 and {v′n(t)}n≥1 are
relatively compact. From
‖v′n(t1)− v′n(t2)‖ ≤ ‖h′(t1)− h′(t2)‖+
∥∥∥∥∥∥
1∫
0
gt(t1, s)wn(s) ds−
1∫
0
gt(t2, s)wn(s) ds
∥∥∥∥∥∥ ≤
≤ ‖h′(t1)− h′(t2)‖+
1∫
0
|gt(t1, s)− gt(t2, s)| ‖wn(s)‖ ds ≤
≤ ‖h′(t1)− h′(t2)‖+
1∫
0
|gt(t1, s)− gt(t2, s)|k(s)ds, (2.2)
we know that {v′n(·)}n≥1 is equicontinuous. By Theorem 1.2.7 in [10], N(M) is relatively compact.
Lemma 2.2. Assume that F : T × E × E → 2E satisfies (C3) and (C5), then N satisfies
M ⊂ U,M = co ({x0} ∪N(M)) and M = C with C ⊂ M countable ⇒ M is compact.
Proof. Suppose M ⊂ U, M = co ({x0} ∪N(M)) and M = C with C ⊂ M countable. Let
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
196 LEI WEI, JIANG ZHU
x ∈ M and v ∈ N(x), v = h +
∫ 1
0 g(·, s)w(s)ds, where w(·) ∈ F (·, x(·), x′(·)). Thus
‖v(t1)− v(t2)‖ ≤ ‖h(t1)− h(t2)‖+
∥∥∥∥∥∥
1∫
0
g(t1, s)w(s) ds−
1∫
0
g(t2, s)w(s) ds
∥∥∥∥∥∥ ≤
≤ ‖h(t1)− h(t2)‖+
1∫
0
|g(t1, s)− g(t2, s)| ‖w(s)‖ds ≤
≤ ‖h(t1)− h(t2)‖+
1∫
0
|g(t1, s)− g(t2, s)|k(s)ds.
This shows that N(M) is equicontinuous. Similarly, we can prove that N ′(M) = {u′|u ∈
∈ N(M)} is equicontinuous (see for example (2.2)). Since M = co ({x0}
⋃
N(M)), we have
that M, M ′ are equicontinuous. Since M is bounded, from Theorem 1.2.2 in [10] we know
that α(M(t)) and α(M ′(t)) are continuous. Since M = co ({x0}
⋃
N(M)) and C ⊂ M is
countable, there exists V = {vn : n ≥ 1} ⊂ N(M) such that C ⊂ co ({x0}
⋃
V ), where
vn(·) = h(·) +
∫ 1
0 g(·, s)wn(s) ds and wn(·) ∈ F (·, xn(·), x′n(·)), xn ∈ M. Therefore
α(M(t)) = α(C(t)) = α(C(t)) ≤ α (co ({x0} ∪ V ) (t)) =
= α(V (t)) = α
1∫
0
g(t, s)wn(s) ds : n ≥ 1
≤
≤ 2
1∫
0
α({g(t, s)wn(s) : n ≥ 1})ds.
From
α({g(t, s)wn(s) : n ≥ 1}) ≤ |g(t, s)|α(F (s,M(s),M ′(s)) ≤
≤ |g(t, s)|w(s,max{α(M(s)), α(M ′(s))}),
we know
α(M(t)) ≤ 2
1∫
0
|g(t, s)|w(s,max{α(M(s)), α(M ′(s))})ds.
Similarly, we have
α(M ′(t)) ≤ 2
1∫
0
|gt(t, s)|w(s,max{α(M(s)), α(M ′(s))})ds.
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 197
Hence
max{α(M(t)), α(M ′(t))} ≤ 2
1∫
0
(|g(t, s)|+ |gt(t, s)|)w(s,max{α(M(s)), α(M ′(s))})ds.
By (C5), we know that for any t ∈ [0, 1],
α(M(t)) = 0, α(M ′(t)) = 0.
From Theorem 1.2.4 and Theorem 1.2.6 in [10], we have
α0(M) = 0, α0(M ′) = 0, i.e., α1(M) = 0,
where α0 denotes the noncompact measure in C([0, 1], E) and α1 denotes the noncompact
measure in C1([0, 1], E). Hence M is relatively compact, i.e., M is compact.
Lemma 2.3. Assume that F : T × E × E → 2E satisfies (C3) and (C4). Then there exists
UR ⊂ C1([0, 1], E) such that N : UR → 2UR .
Proof. For any x ∈ C1[T,E], let
h(·) +
1∫
0
g(·, s)w(s)ds ∈ N(x), w(·) ∈ F (·, x(·), x′(·)).
We know
‖N(x)‖0 ≤ ‖h‖1 + a0 sup
1∫
0
‖w(s)‖ds : w(·) ∈ F (·, x(·), x′(·))
,
‖N(x)′‖0 ≤ ‖h‖1 + a0 sup
1∫
0
‖w(s)‖ds : w(·) ∈ F (·, x(·), x′(·))
,
where ‖ ·‖0 denotes the norm in C([0, 1], E) and ‖ ·‖1 denotes the norm in C1([0, 1], E). Denote
‖h‖1 = r, we have
‖N(x)‖1 ≤ r + a0 sup
1∫
0
‖w(s)‖ds : w(·) ∈ F (·, x(·), x′(·))
.
Assume ‖x‖1 ≤ ρ, from (C4) we know that for ρ enough large, we have
‖N(x)‖1
ρ
≤ r
ρ
+
a0 sup
{∫ 1
0 ‖w(x)‖ds : w(·) ∈ F (·, x(·), x′(·))
}
ρ
≤
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
198 LEI WEI, JIANG ZHU
≤ r
ρ
+
a0
∫ 1
0 lρ(t)dt
ρ
< 1.
This implies that there exists R > 0 such that N(x) ⊂ BR if ‖x‖1 ≤ R.
From Lemmas 2.1, 2.2, 2.3, and 1.6, we have the following theorem.
Theorem 2.1. Assume that F : T × E × E → CK(E) satisfies (C1) – (C5). Then inclusions
(2.1.) has at least a C1-solution.
3. Impulsive neutral functional differential inclusion. We first recall that a family {C(t) : t ∈
∈ R} of bounded linear operators in the Banach space E is called a strongly continuous cosine
family iff
(i) C(0) = I (I is the identity operator in E);
(ii) C(t + s) + C(t− s) = 2C(t)C(s), s, t ∈ R;
(iii) C(t)y is continuous in t on R for each fixed y ∈ E.
If C(t), t ∈ R, is a strongly continuous cosine family in E, then the strongly continuous sine
family S(t), t ∈ R, is the one parameter family of operators in E defined by
S(t)y =
t∫
0
C(s)yds, y ∈ E, t ∈ R.
The infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R} is the
operator A : E → E defined by
Ay =
d2
dt2
C(t)y
∣∣∣∣
t=0
.
In this section, we study the following initial value problem by using the theory of strongly
continuous cosine and sine families:
d
dt
[y′(t)− f(t, yt)] ∈ Ay(t) + F (t, yt),
t ∈ J = [0, b], t 6= tk,
4y |t=tk= Ik(y(t−k )), (3.1)
4y′ |t=tk= Ik(y(t−k )), k = 1, 2, . . . ,m,
y0 = φ, y′(0) = η,
where A is an infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R},
F : J×E → 2E , 0 = t0 < t1 < t2 < · · · < tm < tm+1 = b, f : J×E → E, φ ∈ C([−r, 0], E),
Ik, Ik ∈ C[E,E], k = 0, 1, . . . ,m.
In order to define the concept of a mild solution of the problem, we consider the space
Ω = {y : [−r, b] → E|yk ∈ C(Jk, E), k = 0, 1, . . . ,m and there exist y(t−k ) and y(t+k ), with
y(t−k ) = y(tk), k = 1, 2, . . . ,m, y(t) = φ(t) ∀t ∈ [−r, 0]}, which is a Banach space with the
norm
‖y‖Ω = sup{‖y‖Jk
: k = 0, 1, . . . ,m},
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 199
‖y‖Jk
= sup{‖y(t)|t ∈ Jk‖,
where yk is the restriction of y to Jk = (tk, tk+1], k = 1, . . . ,m, and J0 = [0, t1]. Define yt as
yt(s) = y(t + s), s ∈ [−r, 0], for any y ∈ Ω and for any t ∈ J. Let 4y|t=tk = y(t+k )− y(t−k ) and
4′y|t=tk = y′(t+k )− y′(t−k ).
Definition 3.1. y ∈ C([−r, b]\{t1, t2, . . . , tm}, E) is said to be a mild solution of (3.1.), if
∆y |t=tk= Ik(y(t−k )) and ∆y′ |t=tk= Ik(y(t−k )), k = 1, 2, . . . ,m, and there exists a v ∈ L1[J,E]
such that v(t) ∈ F (t, yt) a.e. on J, and
y(t) = C(t)φ(0) + S(t)[η − f(0, φ)] +
t∫
0
C(t− s)f(s, ys)ds +
t∫
0
S(t− s)v(s)ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))], t ∈ J.
Define
SF,y = {v ∈ L1(T,E) : v(t) ∈ F (t, yt) a.e. t ∈ J},
Ny =
h ∈ Ω : h(t) =
φ(t), if t ∈ [−r, 0],
C(t)φ(0) + S(t)[η − f(0, φ)]+
+
∫ t
0
C(t− s)f(s, ys)ds +
∫ t
0
S(t− s)v(s)ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))], if t ∈ J.
For the proof of our next main result, we will use the following assumptions:
(H1) A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R}
which is bounded (i.e., there exists M0 > 0 such that ‖C(t)‖ ≤ M0 ∀t ∈ R).
(H2) F : J ×E → CP (E) is a Caratheodory map, that is, F (·, x) has a strongly measurable
selection for any x ∈ E, F (t, ·) is u.s.c., for a.e. t ∈ [0, 1], and for any a bounded set X ⊂ Ω,
there exists k1 ∈ L1[J,R+] such that α(F (s,Xs)) ≤ k1(s)α(Xs).
(H3) f(t, u) is continuous in the second variable, and there exist p1, p2 ∈ L1[J,R+] such
that ‖f(t, u)‖ ≤ p1(t)p3(‖u‖Ω) + p2(t), with p3 : J → R+ being nondecreasing, and there
exists k2 ∈ L1([0, b], R+) such that α(f(s,A)) ≤ k2(s)α(A).
(H4) Let Ik, Ik ∈ C(E,E) and there exist dk, dk such that ‖Ik(x)‖ ≤ dk, ‖Ik(x)‖ ≤ dk for
each x ∈ E.
(H5) lim supρ→∞
∫ b
0
M0p3(ρ)p1(t) + M0blρ(t)
ρ
dt < 1.
Lemma 3.1 [4]. Let I be a real compact interval and E be a real Banach space, for all u ∈
∈ C[I, E], F (·, u) be measurable, F (t, ·) upper semicontinuous for a.e. t ∈ I. If Γ : L1[I, E] →
→ C[I, E] is a linear mapping, then
Γ ◦ SF : C[I, E] → BPC(C[I, E]), a.e. y 7→ (Γ ◦ SF )(y) = Γ(SF,y)
has closed graph.
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200 LEI WEI, JIANG ZHU
Theorem 3.1. If the hypotheses (H1), (H2), (H3), (H4), (H5) are satisfied, then (3.1.) at least
has a mild solution.
Proof. Clearly the fixed point of the operator N is a mild solution of (3.1.). Since F has
closed values, (H4) and Lemma 1.1 in [12] imply SF,y 6= ∅.
Step 1. We prove that Ny is convex for any y ∈ C([−r, b], E). Indeed, for any h1, h2 ∈ Ny,
there exist v1, v2 ∈ SF,y such that
h1(t) =
φ(t), if t ∈ T0,
C(t)φ(0) + S(t)[η − f(0, φ)]+
+
∫ t
0
C(t− s)f(s, ys)ds +
∫ t
0
S(t− s)v1(s)ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))], if t ∈ J,
h2(t) =
φ(t), if t ∈ T0,
C(t)φ(0) + S(t)[η − f(0, φ)]+
+
∫ t
0
C(t− s)f(s, ys)ds +
∫ t
0
S(t− s)v2(s)ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))], if t ∈ J.
For any λ ∈ [0, 1], then
(λh1 + (1− λ)h2)(t) =
=
φ(t), if t ∈ T0,
C(t)φ(0) + S(t)[η − f(0, φ)]+
+
∫ t
0
C(t− s)f(s, ys)ds +
∫ t
0
S(t− s)(λv1(s) + (1− λ)v2(s))ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))], if t ∈ J.
Since F has convex values, SF,y also has convex values, so λh1 + (1− λ)h2 ∈ Ny.
Step 2. We prove that N is a bounded operator. For each bounded set U ⊂ Ω, let R =
= sup{‖u‖Ω : u ∈ U}. Next, we show that N(U) is bounded in Ω. By the definition of N,
we only need to show that N(U) is bounded on [0, b]. For any h ∈ N(U), there exist y ∈ U,
v ∈ SF,y such that
h(t) = C(t)φ(0) + S(t)[η − f(0, φ)] +
t∫
0
C(t− s)f(s, ys)ds +
s∫
0
S(t, s)v(s)ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))], if t ∈ J.
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 201
Hence
‖h(t)‖ ≤ ‖C(t)‖ ‖φ(0)‖+ ‖S(t)‖ ‖η − f(0, φ)‖+
t∫
0
‖C(t− s)f(s, ys)‖ds+
+
t∫
0
‖S(t− s)v(s)‖ds + ‖
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))]‖ ≤
≤ M0‖φ‖0 + M0b‖η − f(0, φ)‖+ M0
t∫
0
(p3(R)p1(s) + p2(s))ds+
+
∥∥∥∥∥∥
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))]
∥∥∥∥∥∥ =
= M0‖φ‖+ M0b‖η − f(0, φ)‖+ M0p3(R)‖p1‖L1 + M0‖p2‖L1+
+
m∑
k=1
[dk + (b− tk)dk].
Step 3. We prove that N maps a bounded set into an equicontinuous set of Ω. Assume that
U ⊂ Ω is a bounded set and there exists M1 > 0 such that ‖y‖ ≤ M1 for any y ∈ U. By step 2
we know that there exists M2 > 0 such that ‖v‖ ≤ M2 for any y ∈ U and any v ∈ N(U). Let
y ∈ U, h ∈ Ny, so there exists v ∈ SF,y such that
h(t) = C(t)φ(0) + S(t)[η − f(0, φ)]+
+
t∫
0
C(t− s)f(s, ys)ds +
t∫
0
S(t− s)v(s)ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))].
If γ1 < γ2, and γ1, γ2 ∈ Jk, we have
‖h(γ2)− h(γ1)‖ ≤ ‖C(γ2)− C(γ1)‖ ‖φ(0)‖+ ‖S(γ2)− S(γ1)‖ |‖[η − f(0, φ)]‖+
+
∥∥∥∥∥∥
γ2∫
0
[C(γ2 − s)− C(γ1 − s)]f(s, ys)ds
∥∥∥∥∥∥+
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202 LEI WEI, JIANG ZHU
+
∥∥∥∥∥∥
γ2∫
γ1
C(γ1 − s)f(s, ys)ds
∥∥∥∥∥∥+
∥∥∥∥∥∥
γ2∫
0
(S(γ2 − s)− S(γ1 − s))v(s)ds
∥∥∥∥∥∥+
+
∥∥∥∥∥∥
γ2∫
γ1
S(γ1 − s)v(s)ds
∥∥∥∥∥∥+
∥∥∥∥∥∥
∑
0<ti<γ1
(γ2 − γ1)Ii(y(t−i ))
∥∥∥∥∥∥ .
This implies that
‖h(γ2)− h(γ1)‖ ≤ ‖C(γ2)− C(γ1)‖ |‖φ(0)‖+ ‖S(γ2)− S(γ1)‖ ‖[η − f(0, φ)]‖+
+
∥∥∥∥∥∥
γ2∫
0
[C(γ2 − s)− C(γ1 − s)][p1(s)p3(M1) + p2(s)]ds
∥∥∥∥∥∥+
+
∥∥∥∥∥∥
γ2∫
γ1
C(γ1 − s)[p1(s)p3(M1) + p2(s)]ds
∥∥∥∥∥∥+
+
∥∥∥∥∥∥
γ2∫
0
(S(γ2 − s)− S(γ1 − s))M2ds
∥∥∥∥∥∥+
+
∑
0<ti<γ1
(γ2 − γ1)di +
∥∥∥∥∥∥
γ2∫
γ1
S(γ1 − s)M2ds
∥∥∥∥∥∥ . (3.2)
As γ2 − γ1 → 0, the right-hand side of (3.2.) tends to zero. Equicontinuity for the cases γ1 <
< γ2 ≤ 0 is γ1 ≤ 0 ≤ γ2 is obvious. Thus N(U) is equicontinuous.
Step 4. We prove that N(D) is relatively compact for each compact set D ⊂ Ω. We only
need to show that for any {hn : n ≥ 1} ⊂ N(D), which has a convergent subsequence, i.e.,
{hn : n ≥ 1} ⊂ N(D) is relatively compact. Assume
hn(t) = C(t)φ(0) + S(t)[η − f(0, φ)] +
t∫
0
C(t− s)f(s, yns)ds +
t∫
0
S(t− s)vn(s)ds+
+
t∫
0
S(t− s)vn(s)ds +
∑
0<tk<t
[
Ik(yn(t−k )) + (t− tk)Ik(yn(t−k ))
]
,
where hn ∈ N(yn), yn ∈ D, vn ∈ SF,yn . From the conclusion in step 3 and Ascoli – Arzela
Theorem, we only need to prove that {hn(t) : n ≥ 1} is relatively compact for any t ∈ J. By
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 203
Lemma 1.2, (H2) and (H4), we know
α({hn(t) : n ≥ 1}) ≤ α(C(t)φ(0)) + α(S(t)[η − f(0, φ)])+
+ α
t∫
0
C(t− s)f(s, yns)ds : n ≥ 1
+
+ α
t∫
0
S(t− s)vn(s)ds : n ≥ 1
+
+ α
∑
0<tk<t
[Ik(yn(t−k )) + (t− tk)Ik(yn(t−k ))] : n ≥ 1
≤
≤ 2
t∫
0
α({C(t− s)f(s, yns) : n ≥ 1})ds+
+ 2
t∫
0
α({S(t− s)vn(s) : n ≥ 1})ds =
= 2
t∫
0
α({S(t− s)vn(s) : n ≥ 1})ds ≤
≤ 2
t∫
0
M0bα({yn(s) : n ≥ 1})k1(s)ds = 0.
Step 5. We prove that N has closed graph. Assume yn ∈ Ω, yn → y, hn ∈ Nyn, hn → h,
we will show h ∈ Ny, i.e., we only need to prove that there exists v ∈ SF,y such that
h(t) = C(t)φ(0) + S(t)[η − f(0, φ)] +
t∫
0
C(t− s)f(s, ys)ds+
+
t∫
0
S(t− s)v(s)ds +
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))]. (3.3)
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204 LEI WEI, JIANG ZHU
Assume vn ∈ SF,yn is such that
hn(t) = C(t)φ(0) + S(t)[η − f(0, φ)] +
t∫
0
C(t− s)f(s, yns)ds+
+
t∫
0
S(t− s)vn(s)ds +
∑
0<tk<t
[Ik(yn(t−k )) + (t− tk)Ik(yn(t−k ))].
Define Γ : L1[J,E] → C[J,E] by
Γ(v)(t) =
t∫
0
S(t− s)v(s)ds.
Then Γ is a linear bounded operator and hence Γ ◦ SF : Ω → Ω has closed graph from
Lemma 3.1. Hence
hn(t)− C(t)φ(0)− S(t)[η − f(0, φ)]−
t∫
0
C(t− s)f(s, yns)ds−
−
∑
0<tk<t
[Ik(yn(t−k )) + (t− tk)Ik(yn(t−k ))] −→
−→ h(t)− C(t)φ(0)− S(t)[η − f(0, φ)]−
t∫
0
C(t− s)f(s, ys)ds−
−
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))], n → ∞.
Therefore, there exists v ∈ SF,y such that
h(t)− C(t)φ(0)− S(t)[η − f(0, φ)]−
t∫
0
C(t− s)f(s, ys)ds−
−
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))] =
t∫
0
S(t− s)v(s)ds.
That is (3.3) holds.
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 205
Step 6. We prove that there exists U = {u ∈ Ω : ‖u‖Ω < R} such that N(U) ⊂ U. Indeed,
for any y ∈ Ω and h ∈ N(y), let
h(t) = C(t)φ(0) + S(t)[η − f(0, φ)] +
t∫
0
C(t− s)f(s, ys)ds +
t∫
0
S(t− s)v(s)ds+
+
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))],
here v ∈ SF,y. We know
‖h(t)‖ ≤ ‖C(t)φ(0)‖+ ‖S(t)[η − f(0, φ)]‖+
∥∥∥∥∥∥
t∫
0
C(t− s)f(s, ys)ds+
+
t∫
0
S(t− s)v(s)ds
∥∥∥∥∥∥+
∥∥∥∥∥∥
∑
0<tk<t
[Ik(y(t−k )) + (t− tk)Ik(y(t−k ))]
∥∥∥∥∥∥ .
Therefore, for any ‖y‖Ω ≤ ρ,
‖N(y)‖Ω ≤ M0‖φ‖+ M0b‖[η − f(0, φ)]‖+ M0
t∫
0
‖f(s, ys)‖ds+
+ M0b
t∫
0
‖v(s)‖ds +
m∑
k=1
[dk + (b− tk)dk] ≤
≤ N0 + M0
b∫
0
p3(ρ)p1(s)ds + M0b
b∫
0
lρ(s)ds,
where N0 = M0‖φ(0)‖+M0b‖η− f(0, φ‖+
m∑
k=1
[dk +(b− tk)dk]+M0
∫ b
0
p2(s)ds. Thus, for any
‖y‖Ω ≤ ρ, we know
‖N(y)‖Ω
ρ
≤ N0
ρ
+
b∫
0
M0p3(ρ)p1(s) + M0blρ(s)
ρ
ds.
From the condition (H5), there exists a large ρ such that
‖N(y)‖Ω
ρ
≤ N0
ρ
+
b∫
0
M0p3(ρ)p1(s) + M0blρ(s)
ρ
ds < 1.
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206 LEI WEI, JIANG ZHU
This implies that there exists R > 0 such that N(U) ⊂ U.
Step 7. For the U defined in step 6, if M ⊂ U,M ⊂ co ({0}
⋃
N(M)), M = C and C ⊂ M
is countable, then clearly there exists H = {hn : n ≥ 1} ⊂ N(M)
hn(t) = C(t)φ(0) + S(t)[η − f(0, φ)] +
t∫
0
C(t− s)f(s, yns)ds +
t∫
0
S(t− s)vn(s)ds+
+
∑
0<tk<t
[
Ik(yn(t−k )) + (t− tk)Ik(yn(t−k ))
]
,
where vn ∈ SF,yn and yn ∈ M are such that C ⊂ co ({0}
⋃
H). Thus
α(M(t)) = α(C(t)) = α(C(t)) ≤ α(co ({0}
⋃
H)(t)) = α(H(t)).
If t ∈ [−r, 0], clearly α(M(t)) = 0.
When t ∈ J0, we have
α(M(t)) = α(M(t)) ≤ α(H(t)) ≤
≤ 2M0
t∫
0
k2(s)α(M(s))ds + 2M0b
t∫
0
k(s)α(M(s))ds ≤
≤ 2(M0 + aM0b)
t∫
0
(k2(s) + k1(s))α(M(s))ds. (3.4)
From step 3, we know that N(U) is equicontinuous. From M ⊂ co ({0}
⋃
N(M)), M is equi-
continuous. Theorem 1.2.2 in [10] implies that α(M(t)) is continuous. By Gronwall Lemma in
[13], (3.4) implies α(M(t)) = 0 for t ∈ J0. Thus we know α(yn(t)) = 0 for any t ∈ J0,
moreover α(I1(yn(t1))) = 0, α(I1(yn(t1))) = 0. When t ∈ J1,
α(M(t)) = α(M(t)) ≤ α(H(t)) ≤
≤ 2M0
t∫
0
k2(s)α(M(s))ds + 2M0b
t∫
0
k(s)α(M(s))ds+
+ α({[I1(yn(t−1 )) + (t− tk)I1(yn(t−1 1))] : n ≥ 1}) ≤
≤ 2(M0 + aM0b)
t∫
0
(k2(s) + k1(s))α(M(s))ds + α({I1(yn(t1)) : n ≥ 1})+
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A SECOND ORDER MULTIVALUED BOUNDARY-VALUE PROBLEM AND IMPULSIVE NEUTRAL FUNCTIONAL . . . 207
+ bα({I1(yn(t1)) : n ≥ 1}) =
= 2(M0 + aM0b)
t∫
0
(k2(s) + k1(s))α(M(s)) ds.
Again using Gronwall Lemma, we know α(M(t)) = 0 for any t ∈ J1, thus α(I2(yn(t2))) = 0,
α(I2(yn(t2))) = 0. Similarly, from the continuouty property of Ik, Ik and Gronwall Lemma,
we have α(M(t)) = 0 for any t ∈ Jk : k = 1, 2, 3, . . . ,m. Clearly α(M(t)) = 0 holds for any
t ∈ [−r, b]. So α(M) = 0 by Theorem 1.2.4 in [10]. Hence M is compact in Ω. As a consequence
of Lemma 1.6 we deduce that N has a fixed point which is a mild solution of problem (3.1).
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Banach space // Ibid. — 2003. — 258. — P. 37 – 49.
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ons // Ibid. — 2000. — 244. — P. 594 – 612.
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Received 10.04.06,
after revision — 24.04.07
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 2
|
| id | nasplib_isofts_kiev_ua-123456789-178574 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T18:03:48Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wei, L. Zhu, Jiang 2021-02-27T17:38:04Z 2021-02-27T17:38:04Z 2008 The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces / L. Wei, Jiang Zhu // Нелінійні коливання. — 2008. — Т. 11, № 2. — С. 191-207. — Бібліогр.: 13 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178574 517.9 In this paper, by using the new fixed point theorem of O’Regan and Precup and noncompact measure,
 the existence of solutions of second order multivalued boundary-value problem in Banach spaces and
 the existence of a mild solution for impulsive neutral functional differential inclusions in Banach spaces
 are studied. The compactuess conditions and the upper semicontinuity conditions of multivalued integral
 operators are weaken in this paper. З використанням нової теореми О’Регана та Прекупа, а також некомпактної мiри доведено iснування розв’язкiв багатозначної граничної задачi другого порядку в банахових просторах. Вивчається iснування помiрного розв’язку диференцiальних включень з iмпульсною дiєю
 та нейтральним функцiоналом у банахових просторах. У роботi послаблено умови компактностi та верхньої напiвнеперервностi на багатозначнi iнтегральнi оператори. This work is supported by Natural Science Foundation of the EDJP (05KGD110225), JSQLGC and National
 Natural Science Foundation 10671167, 10771212, China. en Інститут математики НАН України Нелінійні коливання The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces Багатозначна гранична задача другого порядку та диференцiальнi включення з iмпульсною дiєю та нейтральним функцiоналом Article published earlier |
| spellingShingle | The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces Wei, L. Zhu, Jiang |
| title | The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces |
| title_alt | Багатозначна гранична задача другого порядку та диференцiальнi включення з iмпульсною дiєю та нейтральним функцiоналом |
| title_full | The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces |
| title_fullStr | The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces |
| title_full_unstemmed | The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces |
| title_short | The second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in Banach spaces |
| title_sort | second order multi-valued boundary value problem and impulsive neutral functional differential inclusions in banach spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/178574 |
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