Existence results for third order impulsive functional differential inclusions with multiplier p(t)
In this paper, we study the existence of solutions for third order impulsive functional differential inclusions with multiplier p(t). Two new results are obtained by suitable fixed point theorem combined with multivalued analysis theory.
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irk-123456789-1755802021-02-02T01:28:50Z Existence results for third order impulsive functional differential inclusions with multiplier p(t) Guobing Ye Jianhua Shen Jianli Li In this paper, we study the existence of solutions for third order impulsive functional differential inclusions with multiplier p(t). Two new results are obtained by suitable fixed point theorem combined with multivalued analysis theory. Вивчається питання iснування розв’язкiв для функцiонально-диференцiальних включень третього порядку з iмпульсною дiєю та мультиплiкатором p(t). Отримано два нових результати за допомогою придатної теореми про нерухому точку та результатiв з аналiзу багатозначних функцiй. 2012 Article Existence results for third order impulsive functional differential inclusions with multiplier p(t) / Guobing Ye, Jianhua Shen, Jianli Li // Нелінійні коливання. — 2012. — Т. 15, № 1. — С. 25-35. — Бібліогр.: 11 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175580 517.9 en Нелінійні коливання Інститут математики НАН України |
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In this paper, we study the existence of solutions for third order impulsive functional differential inclusions with multiplier p(t). Two new results are obtained by suitable fixed point theorem combined with multivalued analysis theory. |
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Article |
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Guobing Ye Jianhua Shen Jianli Li |
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Guobing Ye Jianhua Shen Jianli Li Existence results for third order impulsive functional differential inclusions with multiplier p(t) Нелінійні коливання |
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Guobing Ye Jianhua Shen Jianli Li |
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Guobing Ye |
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Existence results for third order impulsive functional differential inclusions with multiplier p(t) |
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Existence results for third order impulsive functional differential inclusions with multiplier p(t) |
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Existence results for third order impulsive functional differential inclusions with multiplier p(t) |
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Existence results for third order impulsive functional differential inclusions with multiplier p(t) |
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Existence results for third order impulsive functional differential inclusions with multiplier p(t) |
| title_sort |
existence results for third order impulsive functional differential inclusions with multiplier p(t) |
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Інститут математики НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/175580 |
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Existence results for third order impulsive functional differential inclusions with multiplier p(t) / Guobing Ye, Jianhua Shen, Jianli Li // Нелінійні коливання. — 2012. — Т. 15, № 1. — С. 25-35. — Бібліогр.: 11 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT guobingye existenceresultsforthirdorderimpulsivefunctionaldifferentialinclusionswithmultiplierpt AT jianhuashen existenceresultsforthirdorderimpulsivefunctionaldifferentialinclusionswithmultiplierpt AT jianlili existenceresultsforthirdorderimpulsivefunctionaldifferentialinclusionswithmultiplierpt |
| first_indexed |
2025-07-15T12:53:33Z |
| last_indexed |
2025-07-15T12:53:33Z |
| _version_ |
1837717534945574912 |
| fulltext |
UDC 517.9
EXISTENCE RESULTS FOR THIRD ORDER IMPULSIVE FUNCTIONAL
DIFFERENTIAL INCLUSIONS WITH MULTIPLIER p(t)*
ПРО IСНУВАННЯ РОЗВ’ЯЗКIВ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ ВКЛЮЧЕНЬ
ТРЕТЬОГО ПОРЯДКУ З МУЛЬТИПЛIКАТОРОМ p(t)
Guobing Ye
College Sci. Hunan Univ. Technology
Zhuzhou Hunan 412008, China
e-mail: yeguobing19@sina.com
Jianhua Shen
College Sci. Hangzhou Normal Univ.
Hangzhou Zhejang 310036, China
Jianli Li
College Math. and Comput. Sci., Hunan Normal Univ.
Changsha Hunan 410081, China
In this paper, we study the existence of solutions for third order impulsive functional differential inclusions
with multiplier p(t). Two new results are obtained by suitable fixed point theorem combined with multi-
valued analysis theory.
Вивчається питання iснування розв’язкiв для функцiонально-диференцiальних включень тре-
тього порядку з iмпульсною дiєю та мультиплiкатором p(t). Отримано два нових результати
за допомогою придатної теореми про нерухому точку та результатiв з аналiзу багатозначних
функцiй.
1. Introduction. This paper is concerned with the existence of solutions for third order impulsive
functional differential inclusions
(p(t)u′)′′(t) ∈ F (t, ut), t ∈ [0, T ], t 6= tk, k = 1, . . . ,m,
∆u(i)(tk) = Iik(u(tk)), i = 0, 1, 2, k = 1, . . . ,m, (1.1)
u(t) = φ(t), t ∈ [−r, 0], u(i)(0) = ηi, i = 1, 2,
where 0 = t0 < t1 < . . . < tm < tm+1 = T, F : [0, T ] ×D → P(Rn) is a multivalued map,
D = {ψ : [−r, 0] → Rn; ψ is continuous everywhere except for a finite number of points t at
which ψ(t
−
) and ψ(t
+
) exist with ψ(t
−
) = ψ(t)}, P(Rn) is the family of all nonempty subsets
of Rn, Iik ∈ C(Rn,Rn), i = 0, 1, 2, k = 1, 2, . . . ,m, φ ∈ D, p ∈ C1([0, T ],R+) and p′′ exists.
* Supported by the NNSF of China (no. 10871062) and the NNP of Hunan University of Technology
(no. 2011HZX14).
c© Guobing Ye, Jianhua Shen, Jianli Li, 2012
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 1 25
26 GUOBING YE, JIANHUA SHEN, JIANLI LI
u(i) : [0, T ] → Rn which is piecewise continuous in [0, T ] with points of discontinuity of the first
kind at the points tk ∈ [0, T ], i.e., there exist the limits u(i)(t+k ) < ∞ and u(i)(t−k ) = u(i)(tk) <
< ∞, u′′′ : [0, T ] → Rn, and ∆u(i)(tk) = u(i)(t+k )− u(i)(tk), i = 0, 1, 2, k = 1, 2, . . . ,m.
For any continuous function u defined on [−r, T ]\{t1, . . . , tm} and any t ∈ [0, T ],we denote
by ut the element ofD defined by ut(θ) = u(t+θ), θ ∈ [−r, 0].Here ut(·) represents the history
of the state from t− r, up to the present time t.
The theory of impulsive functional differential equations and inclusions has become more
important in recent years in some mathematical models of real phenomena, especially in control,
biological or medical domains.
The reason for this applicability arises from the fact that impulsive differential problems
are an appropriate model for describing processes which at certain moments change their state
rapidly (in a mathematical simulation it is opportune to assume that the changes of the state are
instantaneous and given by jumps) and which cannot be described using classical differential
problems. And functional differential inclusions are well known as differential inclusions with
memory, expressing the fact that the velocity of the system depends not only on the state of the
system at given instant but also on the history of the trajectory up to that instant. In addition,
this theory is interesting in itself since it exhibits several new phenomena such as rhythmical
beating, merging of solutions and non-continuity of solutions.
In recent years, M. Benchohra et al. [1] and Y. K Chang et al. [2] have investigated the
existence of solutions for impulsive functional differential inclusions
(p(t)y′(t))′ ∈ F (t, yt), t ∈ [0, T ], t 6= tk, k = 1, . . . ,m,
∆y|t=tk = Ik(y(t−k )), k = 1, . . . ,m,
(1.2)
∆y′|t=tk = Jk(y(t−k )), k = 1, . . . ,m,
y(t) = φ(t), t ∈ [−r, 0], y′(0) = η,
with p(t) = 1 and (1.2), respectively. Motivated by the above mentioned work, here we want to
derive the existence of solutions of (1.1).
2. Preliminaries. In this section, we introduce notations, definitions, and preliminary facts
from [1 – 10] which are used throughout this paper.
Let (X, d) be a metric space and N : X → P(X) be a multivalued map. We use the notati-
ons P (X) = {Y ∈ P(X) : Y 6= ∅}, Pcl(X) = {Y ∈ P(X) : Y closed}, Pb(X) = {Y ∈
∈ P(X) : Y bounded}, Pc(X) = {Y ∈ P(X) : Y convex}, and Pcp(X) = {Y ∈ P(X) :
Y compact}.
Definition 2.1. A multivalued map N : [0, T ] → Pcl(X) is said to be measurable if for each
x ∈ X the function g : [0, T ] → R+, defined by g(t) = Ed(x,N(t)) = inf{d(x, z) : z ∈ N(t)},
belongs to L1([0, T ],R).
Lemma 2.1 [1]. LetHd : P (X)×P (X) → R+∪{∞} byHd(A,B) = max{supa∈AEd(a,B),
supb∈B Ed(b, A)}. Then (Pb, cl(X), Hd) is a metric space and (Pcl(X), Hd) is a complete metric
space.
Definition 2.2. Let X be a nonempty closed subset of Rn, and N : X → P(Rn) be a
multivalued map with nonempty closed values. N is lower semicontinuous (l.s.c.) on X if the
set {x ∈ X : N(x) ∩ C 6= ∅} is open for each open set C in Rn.
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 1
EXISTENCE RESULTS FOR THIRD ORDER IMPULSIVE . . . 27
Definition 2.3. Let W be a subset of [0, T ] ×D. W is L ⊗ B measurable if W belongs to the
σ-algebra generated by all sets of the form J × D where J is Lebesgue measurable in [0, T ] and
D is Borel measurable in D.
Definition 2.4. A subset U of L1([0, T ],Rn) is decomposable if for each u, v ∈ U and J ⊂
⊂ [0, T ] measurable the function uχJ + vχ[0,T ]\J ∈ U , where χ stands for the characteristic
function.
Definition 2.5. Let X be a separable metric space and N : X → P(L1([0, T ],Rn)) be a
multivalued operator. We say N has property (BC) if
(1) N is (l.s.c.);
(2) N has nonempty closed and decomposable values.
In order to define the solution of (1.1), we consider the following spaces: PC = {u :
[0, T ] → Rn|uk ∈ C((tk, tk+1],Rn), k = 0, . . . ,m, and there exist u(t−k ) and u(t+k ) with u(t−k ) =
= u(tk), k = 0, . . . ,m},which is a Banach space with the norm ‖u‖PC = max{‖uk‖(tk,tk+1], k =
= 0, . . . ,m}, where uk is the restriction of u to (tk, tk+1], k = 0, . . . ,m.
Lemma 2.2 [1]. Let Ω = D ∪ PC. Then Ω is a Banach space with norm ‖u‖Ω = max{‖u‖D,
‖u‖PC}.
Definition 2.6. Let F : [0, T ] ×D → P(Rn) be a multivalued map with nonempty compact
values. Assign to F the multivalued operator F : Ω → P(L1([0, T ],Rn)) by letting F(u) =
= {v ∈ L1([0, T ],Rn) : v(t) ∈ F (t, ut) for a.e. t ∈ [0, T ]}. The operator F is called the
Niemytzki operator associated to F.
Definition 2.7. A function u ∈ Ω is said to be a solution of (1.1) if u satisfies (1.1).
Definition 2.8. A multivalued operator N : X → Pcl(X) is called
(1) γ-Lipschitz if and only if there exists γ > 0 such that
Hd(N(x), N(y)) ≤ γ d(x, y) for each x, y ∈ X,
(2) contraction if and only if it is γ-Lipschitz with γ < 1.
Definition 2.9. Let F : [0, T ] ×D → P(Rn) be a multivalued map with nonempty compact
values, where for D and P(Rn) we refer to (1.1). We say F is of lower semicontinuous type (l.s.c.
type) if its associated Niemytzki operator F is l.s.c. and has nonempty closed and decomposable
values.
Definition 2.10. The multivalued map N has a fixed point if there exists x ∈ X such that
x ∈ N(x). The set of fixed points of the multivalued map N will be denoted by Fix N.
Definition 2.11. For a function u : [−r, T ] → Rn, the set SF, u = {v ∈ L1([0, T ],Rn) : v(t) ∈
F (t, ut)} is known as the set of selection functions.
Definition 2.12. F has a measurable selection if there exists a measurable function(single -
valued) h : [0, T ] → Rn such that h(t) ∈ SF, u for each t ∈ [0, T ].
Lemma 2.3 [5]. Let X be a separable metric space and N : X → P(L1([0, T ],Rn)) be a
multivalued operator which has property (BC). Then N has a continuous selection, i.e., there
exists a continuous function f : X → L1([0, T ],Rn) such that f(x) ∈ N(x) for each x ∈ X.
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 1
28 GUOBING YE, JIANHUA SHEN, JIANLI LI
Lemma 2.4 [6]. Let X be a normed linear space with S ⊂ X convex and 0 ∈ S. Assume
H : S → S is a completely continuous operator. If the set ε(H) = {x ∈ S : x = λH(x) for
some λ ∈ (0, 1)} is bounded, then H has at least one fixed point in S.
Lemma 2.5 [7]. Let (X, d) be a complete metric space. If N : X → Pcl(X) is a contraction,
then Fix N 6= ∅.
Lemma 2.6 [8]. E ⊆ Ω is a relatively compact set if and only if E ⊆ Ω is uniformly bounded
and equicontinuous on each Jk, k = 0, . . . , p, where J0 = [−r, 0], Jk = (tk, tk+1], k = 0, . . . , p.
3. Main result. Let us introduce the following conditions for later use:
(H1) F : [0, T ]×D → P(Rn) has the property that F (·, ψ) : [0, T ] → Pcp(Rn) is measurable
for each ψ ∈ D, where for D and P(Rn) we refer to (1.1).
(H2) There exist nonnegative constants cik, i = 0, 1, 2, k = 1, . . . ,m, such that |Iik(u(tk))−
−Iik(v(tk))| ≤ cik|u(tk)− v(tk)|, Iik(0) = 0, i = 0, 1, 2, k = 1, . . . ,m, and for all u, v ∈ Ω.
(H3) There exists a function l ∈ L1([0, T ],R+) such that Hd(F (t, ψ), F (t, ϕ)) ≤ l(t)‖ψ −
−ϕ‖D, for a.e. t ∈ [0, T ] and any ψ,ϕ ∈ D, and Ed(0, F (t, 0)) ≤ l(t) for a.e. t ∈ [0, T ], where
for F and D we refer to (1.1).
(H4) Let F : [0, T ] × D → P(Rn) be a nonempty and compact valued multivalued map,
where for D and P(Rn) we refer to (1.1), such that (t, ψ) 7→ F (t, ψ) is L × B measurable, and
ψ 7→ F (t, ψ) is l.s.c. for a.e. t ∈ [0, T ].
(H5) There exists a function M ∈ L1([0, T ],R+) such that ‖F (t, ψ)‖ = sup{|v(t)| : v(t) ∈
∈ F (t, ψ)} ≤ M(t) for each t ∈ [0, T ], where for F we refer to (1.1).
Lemma 3.1 [11]. Let F : [0, T ] × D → P(Rn) be a multivalued map with nonempty and
compact values, where for D and P(Rn) we refer to (1.1). Assume (H4) and (H5) hold. Then F
is of l.s.c. type.
Theorem 3.1. Assume that (H1), (H2) and (H3) are satisfied. Then (1.1) has at least one
solution on [−r, T ], provided
γ =
T 2
p0
‖l‖L1 +
m∑
k=1
{
c0k +
T − tk
p0
p(tk)c1k +
(T − tk)2
p0
[
p′(tk)c1k + p(tk)c2k
]}
< 1,
where p0 = min{p(t) : t ∈ [0, T ]}.
Proof. We transform the problem (1.1) into a fixed point problem. Consider the multivalued
map G : Ω → P(Ω), defined by G(u) = {g ∈ Ω}, where
g(t) =
φ(t), t ∈ [−r, 0],
φ(0) + p(0)η1
∫ t
0
ds
p(s)
+ [p′(0)η1 + p(0)η2]
∫ t
0
sds
p(s)
+
+
∫ t
0
ds
p(s)
∫ s
0
(s− τ)h(τ)dτ +
∑
0<tk<t
{
I0k(u(tk))+
+p(tk)I1k(u(tk))
∫ t
tk
ds
p(s)
+
[
p′(tk)I1k(u(tk)) +
+ p(tk)I2k(u(tk))]
∫ t
tk
(s− tk)ds
p(s)
}
, t ∈ [0, T ] and h ∈ SF,u.
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 1
EXISTENCE RESULTS FOR THIRD ORDER IMPULSIVE . . . 29
It is clear that the fixed points of G are solutions of (1.1). For each u ∈ Ω, the set SF,u is
nonempty since by (H1), F has a measurable selection [16].
We shall show that G satisfies the assumptions of Lemma 2.5. The proof will be given in two
steps.
Step 1. G(u) ⊆ Pcl(Ω) for each u ∈ Ω.
Indeed, let {un} ⊆ G(u) such that un → u∗. Then there exists hn ∈ SF,u such that for each
t ∈ [0, T ],
un(t) = φ(0) + p(0)η1
t∫
0
ds
p(s)
+ [p′(0)η1 + p(0)η2]
t∫
0
sds
p(s)
+
t∫
0
ds
p(s)
s∫
0
(s− τ)hn(τ)dτ+
+
∑
0<tk<t
{
I0k(u(tk)) + p(tk)I1k(u(tk))
t∫
tk
ds
p(s)
+
+
[
p′(tk)I1k(u(tk)) + p(tk)I2k(u(tk))
] t∫
tk
(s− tk)ds
p(s)
}
.
Since F (0, ψ) has compact values and (H3) holds, we may pass to a subsequence if necessary to
get that hn converges to h in L1([0, T ],Rn) and hence h ∈ SF, u. Then for each t ∈ [0, T ],
un(t) → u∗(t) = φ(0) + p(0)η1
t∫
0
ds
p(s)
+
[
p′(0)η1 + p(0)η2
] t∫
0
sds
p(s)
+
+
t∫
0
ds
p(s)
∫ s
0
(s− τ)h(τ)dτ +
∑
0<tk<t
{
I0k(u(tk)) + p(tk)I1k(u(tk))
t∫
tk
ds
p(s)
+
+
[
p′(tk)I1k(u(tk)) + p(tk)I2k(u(tk))
] t∫
tk
(s− tk)ds
p(s)
}
.
So u∗ ∈ G(u), and in particular, G(u) ⊆ Pcl(Ω).
Step 2. It can be shown that there exists γ < 1 such that Hd(G(u), G(u)) ≤ γ‖u − u‖Ω for
all u, u ∈ Ω.
Let u, u ∈ Ω and g ∈ G(u). Then there exists h(t) ∈ F (t, ut) such that for each t ∈ [0, T ],
g(t) = φ(0) + p(0)η1
t∫
0
ds
p(s)
+ [p′(0)η1 + p(0)η2]
t∫
0
sds
p(s)
+
t∫
0
ds
p(s)
s∫
0
(s− τ)h(τ)dτ+
+
∑
0<tk<t
{
I0k(u(tk)) + p(tk)I1k(u(tk))
t∫
tk
ds
p(s)
+
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 1
30 GUOBING YE, JIANHUA SHEN, JIANLI LI
+
[
p′(tk)I1k(u(tk)) + p(tk)I2k(u(tk))
] t∫
tk
(s− tk)ds
p(s)
}
.
From (H3) it follows that, for each t ∈ [0, T ],
Hd(.F (t, ut), F (t, ut)). ≤ l(t)‖ut − ut‖D.
Hence there exists ω(t) ∈ F (t, ut) such that
|h(t)− ω(t)| ≤ l(t)‖ut − ut‖D, t ∈ [0, T ].
Consider U : [0, T ] → P(Rn), given by U(t) = {ω(t) : |h(t) − ω(t)| ≤ l(t)‖ut − ut‖D}.
Since the multivalued operator V (t) = U(t)∩F (t, ut) is measurable [16], there exists a function
h(t), which is a measurable selection for V. So, h(t) ∈ F (t, ut) and |h(t)−h(t)| ≤ l(t)‖ut−ut‖D,
for each t ∈ [0, T ].
We define, for each t ∈ [0, T ],
g(t) = φ(0) + p(0)η1
t∫
0
ds
p(s)
+
[
p′(0)η1 + p(0)η2
] t∫
0
sds
p(s)
+
t∫
0
ds
p(s)
s∫
0
(s− τ)h(τ)dτ+
+
∑
0<tk<t
{
I0k(u(tk)) + p(tk)I1k(u(tk))
t∫
tk
ds
p(s)
+
+
[
p′(tk)I1k(u(tk)) + p(tk)I2k(u(tk))
] t∫
tk
(s− tk)ds
p(s)
}
.
Then we have
|g(t)− g(t)| ≤
t∫
0
ds
p(s)
s∫
0
(s− τ)|h(τ)− h(τ)|dτ +
∑
0<tk<t
{
|I0k(u(tk))− I0k(u(tk))|+
+ p(tk)|I1k(u(tk))− I1k(u(tk))|
t∫
tk
ds
p(s)
+
+
[
p′(tk)|I1k(u(tk))− I1k(u(tk))|+ p(tk)|I2k(u(tk))− I2k(u(tk))|
]
×
×
t∫
tk
(s− tk)ds
p(s)
}
≤ T
p0
t∫
0
(t− s)|h(s)− h(s)|ds+
∑
0<tk<t
{
c0k|u(tk)− u(tk)|+
+
T − tk
p0
p(tk)c1k|u(tk)− u(tk)|+
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EXISTENCE RESULTS FOR THIRD ORDER IMPULSIVE . . . 31
+
(T − tk)2
p0
[
p′(tk)c1k|u(tk)− u(tk)|+ p(tk)c2k|u(tk)− u(tk)|
]}
≤
≤ T 2
p0
t∫
0
|h(s)− h(s)|ds+
∑
0<tk<t
{
c0k +
T − tk
p0
p(tk)c1k+
+
(T − tk)2
p0
[
p′(tk)c1k + p(tk)c2k
]}
|u(tk)− u(tk)| ≤
≤ T 2
p0
t∫
0
l(s)‖us − us‖Dds+
m∑
k=1
{
c0k +
T − tk
p0
p(tk)c1k+
+
(T − tk)2
p0
[
p′(tk)c1k + p(tk)c2k
]}
|u(tk)− u(tk)| ≤
≤ T 2
p0
t∫
0
l(s) ds‖u− u‖Ω +
m∑
k=1
{
c0k +
T − tk
p0
p(tk)c1k+
+
(T − tk)2
p0
[
p′(tk)c1k + p(tk)c2k
]}
‖u− u‖Ω ≤
≤
{
T 2
p0
‖l‖L1 +
m∑
k=1
[
c0k +
T − tk
p0
p(tk)c1k +
(T − tk)2
p0
(
p′(tk)c1k + p(tk)c2k
)]}
‖u− u‖Ω.
So ‖g(t) − g(t)‖Ω ≤ γ‖u − u‖Ω. By an analogous reasoning, obtained by interchanging the
roles of u and ū, it follows that Hd(G(u), G(ū)) ≤ γ‖u− ū‖Ω. Therefore, G is a contraction. By
Lemma 2.5, G has a fixed point which is a solution of (1.1).
Theorem 3.2. In addition to (H4) and (H5), assume that the following condition holds:
(H6). There exist constants dik, i = 0, 1, 2, k = 1, . . . ,m, such that |Iik(u(tk))| ≤ dik|u(tk)|
for each u ∈ Ω. Then (1.1) has at least one solution on [−r, T ], provided
γ =
m∑
k=1
{
d0k +
T − tk
p0
p(tk)d1k +
(T − tk)2
p0
[
p′(tk)d1k + p(tk)d2k
]}
< 1.
Proof. Note that (H4), (H5), and Lemma 3.1 imply that F is of l.s.c. type. Then, from Lem-
ma 2.3, there exists a continuous function f : Ω → L1([0, T ],Rn) such that f(u) ∈ F(u) for
each u ∈ Ω.
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32 GUOBING YE, JIANHUA SHEN, JIANLI LI
We consider the equation
(p(t)u′)′′(t) = f(u)(t), t ∈ [0, T ], t 6= tk, k = 1, . . . ,m,
∆u(i)(tk) = Iik(u(tk)), i = 0, 1, 2, k = 1, . . . ,m, (3.1)
u(t) = φ(t), t ∈ [−r, 0], u(i)(0) = ηi, i = 1, 2,
It is clear that if u ∈ Ω is a solution of (3.1), then u is a solution of (1.1). Transform the
problem (3.1) into a fixed point problem. Consider the operator J : Ω → Ω, defined by
J(u)(t) =
φ(t), t ∈ [−r, 0],
φ(0) + p(0)η1
∫ t
0
ds
p(s)
+ [p′(0)η1 + p(0)η2]
∫ t
0
sds
p(s)
+
+
∫ t
0
ds
p(s)
∫ s
0
(s− τ)f(u)(τ)dτ +
∑
0<tk<t
{
I0k(u(tk))+
+p(tk)I1k(u(tk))
∫ t
tk
ds
p(s)
+
[
p′(tk)I1k(u(tk))+
+p(tk)I2k(u(tk))
]∫ t
tk
(s− tk)ds
p(s)
}
, t ∈ [0, T ].
We shall show that J satisfies all assumptions of Lemma 2.4. The proof will be given in four
steps.
Step 1. J is continuous.
Since the functions f and Iik are continuous, this conclusion can be easily obtained.
Step 2. J maps arbitrary bounded subset of Ω into one bounded set in Ω.
Let Ba = {u ∈ Ω : ‖u‖Ω ≤ a} be arbitrary bounded subset of Ω and u ∈ Ba, there exists
f ∈ F(u) such that for t ∈ [0, T ],
J(u)(t) = φ(0) + p(0)η1
t∫
0
ds
p(s)
+ [p′(0)η1 + p(0)η2]
t∫
0
sds
p(s)
+
t∫
0
ds
p(s)
s∫
0
(s− τ)f(u)(τ)dτ+
+
∑
0<tk<t
{
I0k(u(tk)) + p(tk)I1k(u(tk))
t∫
tk
ds
p(s)
+
+
[
p′(tk)I1k(u(tk)) + p(tk)I2k(u(tk))
] t∫
tk
(s− tk)ds
p(s)
}
. (3.2)
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EXISTENCE RESULTS FOR THIRD ORDER IMPULSIVE . . . 33
From (H5) and (H6), we get for each t ∈ [0, T ],
|J(u)(t)| ≤ |φ(0)|+ p(0)|η1|
T
p0
+
[
p′(0)|η1|+ p(0)|η2|
] T 2
p0
+
T 2
p0
T∫
0
|f(u)(s)| ds+
+
m∑
k=1
{
|I0k(u(tk))|+
T − tk
p0
p(tk)|I1k(u(tk))|+
+
(T − tk)2
p0
[
p′(tk)|I1k(u(tk))|+ p(tk)|I2k(u(tk))|
]}
≤
≤ |φ(0)|+ p(0)|η1|
T
p0
+
[
p′(0)|η1|+ p(0)|η2|
] T 2
p0
+
T 2
p0
T∫
0
M(s) ds+
+
m∑
k=1
{
d0k +
T − tk
p0
p(tk)d1k +
(T − tk)2
p0
[
p′(tk)d1k + p(tk)d2k
]}
|u(tk)| ≤
≤ |φ(0)|+ p(0)|η1|
T
p0
+
[
p′(0)|η1|+ p(0)|η2|
] T 2
p0
+
T 2
p0
‖M‖L1+
+
m∑
k=1
{
d0k +
T − tk
p0
p(tk)d1k +
(T − tk)2
p0
[
p′(tk)d1k + p(tk)d2k
]}
‖u‖Ω.
Then, for each u ∈ Ba, we have
‖J(u)‖Ω ≤ |φ(0)|+ p(0)|η1|
T
p0
+
[
p′(0)|η1|+ p(0)|η2|+ ‖M‖L1
] T 2
p0
+ γ‖u‖Ω ≤
≤ |φ(0)|+ p(0)|η1|
T
p0
+
[
p′(0)|η1|+ p(0)|η2|+ ‖M‖L1
] T 2
p0
+ γa. (3.3)
Therefore, J(Ba) is bounded.
Step 3. J maps arbitrary bounded set into one equicontinuous set in Ω.
Let ρ1, ρ2 ∈ (tk, tk+1], k = 1, . . . ,m, ρ1 < ρ2, and u ∈ Ba be arbitrary bounded subset of
Ω. By (3.2), we get
|J(u)(ρ2)− J(u)(ρ1)| ≤ p(0)|η1|
ρ2∫
ρ1
ds
p(s)
+
[
p′(0)|η1|+ p(0)|η2|
] ρ2∫
ρ1
sds
p(s)
+
+
ρ2∫
ρ1
ds
p(s)
s∫
0
(s− τ)|f(u)(τ)| dτ +
∑
0<tk<ρ2
{
p(tk)|I1k(u(tk))|
ρ2∫
ρ1
ds
p(s)
+
ISSN 1562-3076. Нелiнiйнi коливання, 2012, т . 15, N◦ 1
34 GUOBING YE, JIANHUA SHEN, JIANLI LI
+
[
p′(tk)|I1k(u(tk))|+ p(tk)|I2k(u(tk))|
] ρ2∫
ρ1
(s− tk)ds
p(s)
}
≤
≤
{
p(0)|η1|+ T
[
p′(0)|η1|+ p(0)|η2|
]} ρ2∫
ρ1
ds
p(s)
+ T
ρ2∫
ρ1
ds
p(s)
T∫
0
M(τ)dτ+
+
∑
0<tk<ρ2
{
p(tk)d1k|u(tk)|+ T
[
p′(tk)d1k|u(tk)|+ p(tk)d2k|u(tk)|
]} ρ2∫
ρ1
ds
p(s)
≤
≤
{
p(0)|η1|+ T
[
p′(0)|η1|+ p(0)|η2|
]} ρ2∫
ρ1
ds
p(s)
+ T‖M‖L1
ρ2∫
ρ1
ds
p(s)
+
+ ‖u‖Ω
m∑
k=1
{
p(tk)d1k + T
[
p′(tk)d1k + p(tk)d2k
]} ρ2∫
ρ1
ds
p(s)
≤
≤
{
p(0)|η1|+ T
[
p′(0)|η1|+ p(0)|η2|+ ‖M‖L1
]} ρ2∫
ρ1
ds
p(s)
+
+ a
m∑
k=1
{
p(tk)d1k + T
[
p′(tk)d1k + p(tk)d2k
]} ρ2∫
ρ1
ds
p(s)
.
According to the complete continuity of integrable function M, the right-hand side of the
above inequality tends to zero as ρ2 → ρ1. The consequence for the cases ρ1, ρ2 ∈ (0, t1] and
[−r, 0] is obvious. Then J(Ba) is one equicontinuous set in Ω.
As a consequence of Step 1 to Step 3 together with Lemma 2.6 and the Ascoli – Arzela
theorem, we conclude that J : Ω → Ω is completely continuous.
Step 4. The set ε(J) = {u ∈ Ω : u = λJ(u), for some 0 < λ < 1} is bounded.
For each u ∈ ε(J), by (3.3), we have
‖u‖Ω = λ‖J(u)‖Ω ≤ ‖J(u)‖Ω ≤ |φ(0)|+ p(0)|η1|
T
p0
+
[
p′(0)|η1|+ p(0)|η2|+ ‖M‖L1
] T 2
p0
+
+
m∑
k=1
{
d0k +
T − tk
p0
p(tk)d1k +
(T − tk)2
p0
[
p′(tk)d1k + p(tk)d2k
]}
‖u‖Ω =
= |φ(0)|+ p(0)|η1|
T
p0
+
[
p′(0)|η1|+ p(0)|η2|+ ‖M‖L1
] T 2
p0
+ γ‖u‖Ω.
Then
‖u‖Ω ≤
|φ(0)|+ p(0)|η1| Tp0 + [p′(0)|η1|+ p(0)|η2|+ ‖M‖L1 |] T 2
p0
1− γ
,
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EXISTENCE RESULTS FOR THIRD ORDER IMPULSIVE . . . 35
i.e., ε(J) is bounded.
In view of Lemma 2.4, we deduce that J has a fixed point which in turn is a solution of (1.1).
1. Benchohra M., Ouahab A. Initial and boundary value problems for second order impulsive functional dif-
ferential inclusions // Electron. J. Qual. Theory Different. Equat. — 2003. — 3. — P. 1 – 10.
2. Yong-Kui Chang, Wan-Tong Li. Existence results for second order impulsive differential inclusions // J. Math.
Anal. and Appl. — 2005. — 301. — P. 477 – 490.
3. Wei L., Zhu J. A second-order multivalued boundary-value problem and impulsive neutral functional di-
fferential inclusions in Banach spaces // Nonlinear Oscillations. — 2008. — 11, № 2. — P. 200 – 218.
4. Belarbi A., Benchohra M., Ouahab A. Nonconvex-valued impulsive functional differential inclusions with
variable times // Nonlinear Oscillations. — 2007. — 10, № 4. — P. 447 – 463.
5. Bressan A., Colombo G. Existence and selections of maps with decomposable values // Stud. Math. — 1988.
— 90. — P. 69 – 86.
6. Smart D. R. Fixed point theorems. — Cambridge: Cambridge Univ. Press, 1974.
7. Covitz H., Nadler S. B. (Jr.) Multivalued contraction mappings in generalized metric spaces // Isr. J. Math. —
1970. — 8. — P. 5 – 11.
8. Fu X. L., Yan B. Q., Liu Y. S. Theory of impulsive differential system. — Beijing: Sci. Press, 2005.
9. Deimling K. Multivalued differential equations. — Berlin: De Gruyter, 1992.
10. Castaing C., Valadier M. Convex analysis and measurable multifunctions // Lect. Notes Math. — Berlin:
Springer, 1977. — 580.
11. Frigon M. Théorémes d’existence de solutions d’inclusions différentielles / Eds A. Granas, M. Frigon //
Topological Methods in Differential Equations and Inclusions (NATO Adv. Sci. Inst. Ser. C Math. Phys.
Sci). — Dordrecht: Kluwer Academic, 1995. — 472. — P. 51 – 87.
Received 02.09.10,
after revision — 16.02.11
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