Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 724854, 8 pages
http://dx.doi.org/10.1155/2013/724854
Research Article
An Existence Result for Nonlocal Impulsive Second-Order
Cauchy Problems with Finite Delay
Fang Li and Huiwen Wang
School of Mathematics, Yunnan Normal University, Kunming 650092, China
Correspondence should be addressed to Fang Li; fangli860@gmail.com
Received 1 October 2012; Revised 7 December 2012; Accepted 9 December 2012
Academic Editor: Toka Diagana
Copyright © 2013 F. Li and H. Wang. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We deal with the existence of mild solutions of a class of nonlocal impulsive second-order functional differential equations with
finite delay in a real Banach space X. An existence result on the mild solution is obtained by using the theory of the measures of
noncompactness. An example is presented.
1. Introduction
The Cauchy problem for various delay equations in Banach
spaces has been receiving more and more attention during
the past decades (see, e.g., [1–5]).
The literature concerning second- and higher-order ordinary functional differential equations is very extensive. We
only mention the works [1, 6–15], which are directly related
to this work.
On the other hand, the impulsive conditions have advantages over traditional initial value problems because they can
be used to model phenomena that cannot be modeled by
traditional initial value problems, such as the dynamics of
populations subject to abrupt changes (harvesting, diseases,
etc.) (see [16–27] and references therein). For this reason,
the theory of impulsive differential equations has become
an important area of investigation in recent years. Partial
differential equations of first and second order with impulses
have been studied by Rogovchenko [26], Liu [25], Cardinali
and Rubbioni [19], Liang et al. [24], Henrı́quez and Vásquez
[1], Hernández et al., [21–23], Arthi and Balachandran [17],
and so forth.
Moreover, we consider the nonlocal condition 𝑥(0) =
𝑔(𝑥) + 𝑥0 , where 𝑔 is a mapping from some space of functions
so that it constitutes a nonlocal condition (see [24, 28–30]
and the references therein), where it is demonstrated that
nonlocal conditions have better effects in applications than
traditional initial value problems.
In this paper, we pay our attention to the investigation
of the existence of mild solutions to the following impulsive
second-order functional differential equations with finite
delay in a real Banach space 𝑋:
𝑑2
𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝑓 (𝑡, 𝑥𝑡 , 𝑥 (𝑡)) ,
𝑑𝑡2
(1)
𝑡 ∈ (0, 𝑇] , 𝑡 ≠ 𝑡𝑘 , 𝑘 = 1, 2, . . . , 𝑝,
𝑥 (𝑡) = 𝑔 (𝑥) (𝑡) + 𝜙 (𝑡) ,
𝑡 ∈ [−𝑟, 0] ,
𝑥󸀠 (0) = 𝜉 ∈ 𝑋,
Δ𝑥 (𝑡𝑘 ) = 𝐼𝑘 (𝑥 (𝑡𝑘 )) ,
𝑘 = 1, 2, . . . , 𝑝,
(2)
(3)
(4)
where 𝐴 is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators {𝐶(𝑡)}𝑡∈R
on 𝑋. 𝑓, 𝑔 are given functions to be specified later. 𝜙 ∈
𝐶([−𝑟, 0], 𝑋), where 𝐶([𝑎, 𝑏], 𝑋) denotes the space of all
continuous functions from [𝑎, 𝑏] to 𝑋.
The impulsive moments {𝑡𝑘 } are given such that 0 = 𝑡0 <
𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑝 < 𝑡𝑝+1 = 𝑇, 𝐼𝑘 : 𝑋 → 𝑋 (𝑘 = 1, 2, . . . , 𝑝)
are appropriate functions, Δ𝑥(𝑡𝑘 ) represents the jump of a
function 𝑥 at 𝑡𝑘 , which is defined by Δ𝑥(𝑡𝑘 ) = 𝑥(𝑡𝑘+ ) − 𝑥(𝑡𝑘− ),
where 𝑥(𝑡𝑘+ ) and 𝑥(𝑡𝑘− ) are, respectively, the right and the left
limits of 𝑥 at 𝑡𝑘 .
For any continuous function 𝑥 defined on the interval
[−𝑟, 𝑇] and any 𝑡 ∈ [0, 𝑇], we denote by 𝑥𝑡 the element of
𝐶([−𝑟, 0], 𝑋) defined by 𝑥𝑡 (𝜃) = 𝑥(𝑡 + 𝜃) for 𝜃 ∈ [−𝑟, 0].
2
Abstract and Applied Analysis
In this paper, motivated by above works, we study (1)–(4)
in 𝑋 and obtain the existence theorem based on theory on
measures of noncompactness without the assumptions that
the nonlinearity 𝑓 satisfies a Lipschitz type condition and the
cosine family of bounded linear operators {𝐶(𝑡)}𝑡∈R generated
by 𝐴 is compact.
The following properties are well known [6, 7, 11, 12]:
(i) 𝐶 (𝑡) 𝑥 ∈ 𝐷 (𝐴) , 𝐶 (𝑡) 𝐴𝑥 = 𝐴𝐶 (𝑡) 𝑥
for 𝑥 ∈ 𝐷 (𝐴) , 𝑡 ∈ R;
(ii) 𝑆 (𝑡) 𝑥 ∈ 𝐷 (𝐴) , 𝑆 (𝑡) 𝐴𝑥 = 𝐴𝑆 (𝑡) 𝑥
for 𝑥 ∈ 𝐷 (𝐴) , 𝑡 ∈ R;
2. Preliminaries
𝑡
Throughout this paper, we set 𝐽 = [0, 𝑇], a compact interval
in R. We denote by 𝑋 a Banach space with norm ‖ ⋅ ‖, by 𝐿(𝑋)
the Banach space of all linear and bounded operators on 𝑋.
We abbreviate ‖𝑢‖𝐿1 (𝐽,R+ ) with ‖𝑢‖𝐿1 , for any 𝑢 ∈ 𝐿1 (𝐽, R+ ).
Let
(iii) ∫ 𝑆 (𝑠) 𝑥𝑑𝑠 ∈ 𝐷 (𝐴) ,
𝑃𝐶 (𝐽, 𝑋) := {𝑥 : 𝐽 󳨀→ 𝑋; 𝑥 (𝑡) is continuous at 𝑡 ≠ 𝑡𝑘 ,
(iv) 𝐶 (𝑡) 𝑥 − 𝑥 = ∫ 𝑆 (𝑠) 𝐴𝑥𝑑𝑠
the right limit 𝑥 (𝑡𝑘+ ) exists for 𝑘 = 1, 2, . . . , 𝑝} .
(5)
It is easy to check that 𝑃𝐶(𝐽, 𝑋) is a Banach space with the
norm
𝑡∈𝐽
for any 𝑥 ∈ 𝑃𝐶 (𝐽, 𝑋) .
(6)
for 𝑥 ∈ 𝑋, 𝑡 ∈ R;
0
𝑡
for 𝑥 ∈ 𝐷 (𝐴) , 𝑡 ∈ R.
(11)
For more details on strongly continuous cosine and sine
families, we refer the reader to [6, 7, 11, 12].
Next, we recall that the Hausdorff measure of noncompactness 𝜒(⋅) on each bounded subset Ω of Banach space 𝑌 is
defined by
𝜒 (Ω) = inf {𝜀 > 0; Ω has a finite 𝜀-net in 𝑋} .
(12)
This measure of noncompactness satisfies some basic
properties as follows.
We let 𝐽0 = (𝑡0 , 𝑡1 ], 𝐽1 = (𝑡1 , 𝑡2 ], . . . , 𝐽𝑝 = (𝑡𝑝 , 𝑡𝑝+1 ].
For B ⊆ 𝑃𝐶(𝐽, 𝑋), we denote by B|𝐽𝑖 the set
B|𝐽𝑖 = {𝑥 ∈ 𝐶 ([𝑡𝑖 , 𝑡𝑖+1 ] , 𝑋) ; 𝑥 (𝑡𝑖 ) = 𝑣 (𝑡𝑖+ ) , 𝑥 (𝑡) = 𝑣 (𝑡) ,
𝑡 ∈ 𝐽𝑖 , 𝑣 ∈ B}
Lemma 1 (see [31]). Let 𝑌 be a real Banach space, and let
𝐵, 𝐶 ⊆ 𝑌 be bounded. Then
(1) 𝜒(𝐵) = 0 if and only if 𝐵 is precompact;
(7)
𝑖 = 0, 1, 2, . . . , 𝑝.
A family {𝐶(𝑡)}𝑡∈R in 𝐿(𝑋) is called a cosine function on
𝑋 if
(i) 𝐶(0) = 𝐼 is the identity operator in 𝑋;
(iii) The map 𝑡 → 𝐶(𝑡)𝑥 is strongly continuous for each
𝑥 ∈ 𝑋.
The associated sine function is the family {𝑆(𝑡)}𝑡∈R of operators defined by
𝑡
𝑆 (𝑡) 𝑥 = ∫ 𝐶 (𝑠) 𝑥 𝑑𝑠,
for 𝑥 ∈ 𝑋, 𝑡 ∈ R.
0
(8)
One can define the infinitesimal generator 𝐴 of 𝐶(⋅) by
−2
𝐷 (𝐴) = {𝑥 ∈ 𝑋; lim 2𝑡 (𝐶 (𝑡) 𝑥 − 𝑥) ∈ 𝑋}
𝑡→0
𝐴𝑥 = lim 2𝑡−2 (𝐶 (𝑡) 𝑥 − 𝑥) ,
𝑡→0
(9)
𝑥 ∈ 𝐷 (𝐴) .
In this paper, we assume there exist positive constants 𝑀
and 𝑁 such that
‖𝑆 (𝑡)‖ ≤ 𝑁
for every 𝑡 ∈ 𝐽.
(2) 𝜒(𝐵) = 𝜒(𝐵) = 𝜒(𝑐𝑜𝑛𝑣𝐵), where 𝐵 and 𝑐𝑜𝑛𝑣𝐵 mean
the closure and convex hull of 𝐵, respectively;
(3) 𝜒(𝐵) ≤ 𝜒(𝐶) if 𝐵 ⊆ 𝐶;
(4) 𝜒(𝐵 ∪ 𝐶) ≤ max{𝜒(𝐵), 𝜒(𝐶)};
(5) 𝜒(𝐵 + 𝐶) ≤ 𝜒(𝐵) + 𝜒(𝐶), where 𝐵 + 𝐶 = {𝑥 + 𝑦; 𝑥 ∈
𝐵, 𝑦 ∈ 𝐶};
(ii) 𝐶(𝑡 + 𝑠) + 𝐶(𝑡 − 𝑠) = 2𝐶(𝑡)𝐶(𝑠) for all 𝑠, 𝑡 ∈ R;
‖𝐶 (𝑡)‖ ≤ 𝑀,
𝑡
𝐴 ∫ 𝑆 (𝑠) 𝑥𝑑𝑠 = 𝐶 (𝑡) 𝑥 − 𝑥
0
left continuous at 𝑡 = 𝑡𝑘 , and
‖𝑥‖𝑃𝐶 = sup ‖𝑥 (𝑡)‖ ,
0
(10)
(6) 𝜒(𝛼𝐵) = |𝛼|𝜒(𝐵), for any 𝛼 ∈ R;
(7) let 𝑍 be a Banach space and 𝑄 : 𝐷(𝑄) ⊆ 𝑌 → 𝑍 Lipschitz continuous with constant 𝜈. Then 𝜒(𝑄𝐵) ≤ 𝜈 ⋅
𝜒(𝐵) for all 𝐵 ⊆ 𝐷(𝑄) being bounded.
Proposition 2 (see [32], Page 125). Let Ω be a bounded set
for a real Banach space 𝑋. Then, for every 𝜀 > 0 there exists a
sequence {𝑥𝑛 }∞
𝑛=1 in Ω such that
∞
𝜒 (Ω) ≤ 2𝜒 ({𝑥𝑛 }𝑛=1 ) + 𝜀.
(13)
In the sequel, we make use of the following formulation
of Theorem 4.2.2 of [33] obtained by using Theorem 2 of [34].
1
Proposition 3. Let {𝑓𝑛 }∞
𝑛=1 be a sequence in 𝐿 (𝐽, 𝑋) such that
1
there exist 𝑣, 𝑞 ∈ 𝐿 + ([0, 𝑇]) with the properties:
(i) sup𝑛∈N ‖𝑓𝑛 (𝑡)‖ ≤ 𝑣(𝑡), 𝑎.𝑒. 𝑡 ∈ 𝐽;
(ii) 𝜒({𝑓𝑛 }∞
𝑛=1 ) ≤ 𝑞(𝑡), 𝑎.𝑒. 𝑡 ∈ 𝐽.
Abstract and Applied Analysis
3
Then, for every 𝑡 ∈ 𝐽, we have
𝑡
∞
0
𝑛=1
𝜒 ({∫ 𝑆 (𝑡 − 𝑠) 𝑓𝑛 (𝑠) 𝑑𝑠}
(H2) 𝐼𝑘 : 𝑋 → 𝑋 are compact operators and there
exist positive constants 𝑀1 , 𝑀2 such that
𝑡
) ≤ 2𝑁 ∫ 𝑞 (𝑠) 𝑑𝑠,
0
(14)
󵄩
󵄩󵄩
󵄩󵄩𝐼𝑘 (𝑥)󵄩󵄩󵄩 ≤ 𝑀1 ‖𝑥‖ + 𝑀2 ,
where 𝑁 is from (10).
A continuous map 𝑄 : 𝑊 ⊆ 𝑌 → 𝑌 is said to be a 𝜒contraction if there exists a positive constant 𝜈 < 1 such that
𝜒(𝑄𝐶) ≤ 𝜈 ⋅ 𝜒(𝐶) for any bounded closed subset 𝐶 ⊆ 𝑊.
Theorem 4 (see [31] (Darbo-Sadovskii)). If 𝑈 ⊆ 𝑌 is bounded
closed and convex, the continuous map F : 𝑈 → 𝑈 is a 𝜒contraction, then the map F has at least one fixed point in 𝑈.
Definition 5. A function 𝑥 : [−𝑟, 𝑇] → 𝑋 is called a mild
solution of system (1)–(4) if 𝑥0 = 𝑔(𝑥) + 𝜙, 𝑥|𝐽 ∈ 𝑃𝐶(𝐽, 𝑋)
and
(H3) 𝑔 : 𝐶([−𝑟, 0], 𝑋) → 𝑋 is a compact operator
and there exists a constant 𝑁1 > 0 such that
󵄩
󵄩󵄩
󵄩󵄩𝑔 (𝑥)󵄩󵄩󵄩[−𝑟,0] ≤ 𝑁1
for all 𝑥 ∈ 𝐶 ([−𝑟, 0] , 𝑋) ,
(21)
where ‖𝑔(𝑥)‖[−𝑟,0] = sup𝑡∈[−𝑟,0] ‖𝑔(𝑥)(𝑡)‖.
𝑇
(H4) There exists 𝑀∗ ∈ (0, 1) such that 8𝑁 ∫0 𝜂(𝑠)𝑑𝑠 <
𝑀∗ .
Theorem 7. Assume that (H1)–(H4) are satisfied, then there
exists a mild solution of (1)–(4) on [−𝑟, 𝑇] provided that
𝑝𝑀𝑀1 < 1.
𝑥 (𝑡) = 𝐶 (𝑡) (𝜙 (0) + 𝑔 (𝑥) (0)) + 𝑆 (𝑡) 𝜉
𝑡
+ ∫ 𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠)) 𝑑𝑠
(15)
0
+ ∑ 𝐶 (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘 )) ,
for any 𝑥 ∈ 𝑋, 𝑘 = 1, 2, . . . , 𝑝.
(20)
𝑡 ∈ 𝐽.
Proof. Define the operator Λ : F(𝑇) → F(𝑇) in the following way:
0<𝑡𝑘 <𝑡
Remark 6. A mild solution of (1)–(4) satisfies (2) and (4).
However, a mild solution may be not differentiable at zero.
3. Existence Result and Proof
In this section, we study the existence of mild solutions for
the system (1)–(4).
Let F(𝑇) stand for the space
F (𝑇) = {𝑥 : [−𝑟, 𝑇] → 𝑋; 𝑥|𝐽 ∈ 𝑃𝐶 (𝐽, 𝑋) ,
𝑥0 ∈ 𝐶 ([−𝑟, 0] , 𝑋)}
(16)
endowed with norm
‖𝑥‖F(𝑇) = sup ‖𝑥 (𝑡)‖ + sup ‖𝑥 (𝑡)‖ .
𝑡∈[−𝑟,0]
𝑡∈𝐽
(17)
We will require the following assumptions.
(H1) (i) 𝑓 : 𝐽×𝐶([−𝑟, 0], 𝑋)×𝑋 → 𝑋 satisfies 𝑓(⋅, 𝑣, 𝑤) :
𝐽 → 𝑋 is measurable for all (𝑣, 𝑤) ∈ 𝐶([−𝑟, 0], 𝑋) × 𝑋 and
𝑓(𝑡, ⋅, ⋅) : 𝐶([−𝑟, 0], 𝑋) × 𝑋 → 𝑋 is continuous for a.e. 𝑡 ∈ 𝐽,
and there exists a function 𝜇(⋅) ∈ 𝐿1 (𝐽, R+ ) such that
󵄩󵄩󵄩𝑓 (𝑡, 𝑣, 𝑤)󵄩󵄩󵄩 ≤ 𝜇 (𝑡) (1 + ‖𝑤‖)
(18)
󵄩
󵄩
for almost all 𝑡 ∈ 𝐽;
(ii) there exists a function 𝜂 ∈ 𝐿1 (𝐽, R+ ) such that for any
bounded sets 𝐷1 ⊂ 𝐶([−𝑟, 0], 𝑋), 𝐷2 ⊂ 𝑋
𝜒 (𝑓 (𝑡, 𝐷1 , 𝐷2 )) ≤ 𝜂 (𝑡) ( sup 𝜒 (𝐷1 (𝜃)) + 𝜒 (𝐷2 )) ,
𝜃∈[−𝑟,0]
a.e. 𝑡 ∈ 𝐽.
(19)
𝑔 (𝑥) (𝑡) + 𝜙 (𝑡) ,
𝑡 ∈ [−𝑟, 0] ,
{
{
{
{
{
𝐶 (𝑡) (𝑔 (𝑥) (0) + 𝜙 (0)) + 𝑆 (𝑡) 𝜉
{
{
{
{ 𝑡
(Λ𝑥) (𝑡) = {+ ∫ 𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥 , 𝑥 (𝑠)) 𝑑𝑠
{
𝑠
{
{
0
{
{
{
{
𝑡 ∈ 𝐽.
{+ ∑ 𝐶 (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘 )) ,
{ 0<𝑡𝑘 <𝑡
(22)
It is clear that the operator Λ is well defined, and the fixed
point of Λ is the mild solution of problems (1)–(4).
The operator Λ can be written in the form Λ = Λ 1 + Λ 2 ,
where the operators Λ 1 , Λ 2 are defined as follows:
𝑔 (𝑥) (𝑡) + 𝜙 (𝑡) ,
𝑡 ∈ [−𝑟, 0] ,
{
{
{
(Λ 1 𝑥) (𝑡) = {𝐶 (𝑡) (𝑔 (𝑥) (0) + 𝜙 (0))
{
{
𝑡 ∈ 𝐽,
{+𝑆 (𝑡) 𝜉,
0,
𝑡 ∈ [−𝑟, 0] ,
{
{
{
𝑡
{
{
{
(Λ 2 𝑥) (𝑡) = {∫0 𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠)) 𝑑𝑠
{
{
{
{
{+ ∑ 𝐶 (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘 )) , 𝑡 ∈ 𝐽.
{ 0<𝑡𝑘 <𝑡
(23)
Obviously, under the assumptions of 𝑔, Λ 1 is continuous.
For 𝑡 ∈ 𝐽, we can prove that Λ 2 is continuous.
Indeed, let {𝑥𝑛 }𝑛∈N be a sequence such that 𝑥𝑛 → 𝑥 in
F(𝑇) as 𝑛 → ∞. Since 𝑓 satisfies (H1)(i), for almost every
𝑡 ∈ 𝐽, we get
𝑓 (𝑡, 𝑥𝑡𝑛 , 𝑥𝑛 (𝑡)) 󳨀→ 𝑓 (𝑡, 𝑥𝑡 , 𝑥 (𝑡)) ,
as 𝑛 󳨀→ ∞.
(24)
4
Abstract and Applied Analysis
Noting that 𝑥𝑛 → 𝑥 in F(𝑇), we can see that there exists
𝜀 > 0 such that ‖𝑥𝑛 − 𝑥‖F(𝑇) ≤ 𝜀 for 𝑛 sufficiently large.
Therefore, we have
󵄩󵄩
󵄩
𝑛 𝑛
󵄩󵄩𝑓 (𝑡, 𝑥𝑡 , 𝑥 (𝑡)) − 𝑓 (𝑡, 𝑥𝑡 , 𝑥 (𝑡))󵄩󵄩󵄩
󵄩
󵄩
≤ 𝜇 (𝑡) (1 + 󵄩󵄩󵄩𝑥𝑛 (𝑡)󵄩󵄩󵄩) + 𝜇 (𝑡) (1 + ‖𝑥 (𝑡)‖)
󵄩
󵄩
≤ 2𝜇 (𝑡) + 𝜇 (𝑡) 󵄩󵄩󵄩𝑥𝑛 (𝑡) − 𝑥 (𝑡)󵄩󵄩󵄩 + 2𝜇 (𝑡) ‖𝑥 (𝑡)‖
For 𝑡 ∈ 𝐽, 𝑥 ∈ 𝐵𝜌 , we have
󵄩
󵄩 󵄩
󵄩
‖(Λ𝑥) (𝑡)‖ ≤ 󵄩󵄩󵄩𝐶 (𝑡) (𝑔 (𝑥) (0) + 𝜙 (0))󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑆 (𝑡) 𝜉󵄩󵄩󵄩
𝑡
󵄩
󵄩
+ ∫ 󵄩󵄩󵄩𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠))󵄩󵄩󵄩 𝑑𝑠
0
󵄩
󵄩
+ ∑ 󵄩󵄩󵄩𝐶 (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘 ))󵄩󵄩󵄩
(25)
0<𝑡𝑘 <𝑡
󵄩
󵄩
󵄩
󵄩
≤ 𝑀 (𝑁1 + 󵄩󵄩󵄩𝜙 (0)󵄩󵄩󵄩 + ∑ 󵄩󵄩󵄩𝐼𝑘 (𝑥 (𝑡𝑘 ))󵄩󵄩󵄩)
≤ 2𝜇 (𝑡) + 𝜇 (𝑡) 𝜀 + 2𝜇 (𝑡) ‖𝑥‖F(𝑇) .
0<𝑡𝑘 <𝑡
It follows from the Lebesgue’s dominated convergence theorem that
𝑡
󵄩 󵄩
+𝑁 (󵄩󵄩󵄩𝜉󵄩󵄩󵄩 +∫ 𝜇 (𝑠) (1+𝑒𝐿𝑠 𝑒−𝐿𝑠 ‖𝑥 (𝑠)‖) 𝑑𝑠)
0
󵄩
󵄩
= ℓ + 𝑀 ∑ 󵄩󵄩󵄩𝐼𝑘 (𝑥 (𝑡𝑘 ))󵄩󵄩󵄩
𝑡
󵄩
󵄩
∫ 󵄩󵄩󵄩𝑆 (𝑡 − 𝑠) [𝑓 (𝑠, 𝑥𝑠𝑛 , 𝑥𝑛 (𝑠)) − 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠))]󵄩󵄩󵄩 𝑑𝑠
0
𝑡
󵄩
󵄩
≤ 𝑁 ∫ 󵄩󵄩󵄩𝑓 (𝑠, 𝑥𝑠𝑛 , 𝑥𝑛 (𝑠)) − 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠))󵄩󵄩󵄩 𝑑𝑠
0
󳨀→ 0,
0<𝑡𝑘 <𝑡
𝑡
+ 𝑁 ∫ 𝜇 (𝑠) 𝑒𝐿𝑠 𝑒−𝐿𝑠 ‖𝑥 (𝑠)‖ 𝑑𝑠,
(26)
0
as 𝑛 󳨀→ ∞.
(34)
then
Moreover, noting that (H2), we obtain that
󵄩
󵄩
lim 󵄩󵄩Λ 2 𝑥𝑛 − Λ 2 𝑥󵄩󵄩󵄩F(𝑇) = 0.
𝑛 → ∞󵄩
(27)
This shows that Λ 2 is continuous. Therefore, Λ is continuous.
Let us introduce in the space F(𝑇) the equivalent norm
defined as
−𝐿𝑡
‖𝑥‖∗ = sup ‖𝑥 (𝑡)‖ + sup (𝑒
𝑡∈𝐽
𝑡∈[−𝑟,0]
‖𝑥 (𝑡)‖) ,
󵄩
󵄩
𝑒−𝐿𝑡 ‖(Λ𝑥) (𝑡)‖ ≤ 𝑒−𝐿𝑡 [ℓ + 𝑀 ∑ 󵄩󵄩󵄩𝐼𝑘 (𝑥 (𝑡𝑘 ))󵄩󵄩󵄩
0<𝑡𝑘 <𝑡
𝑡
+𝑁 ∫ 𝜇 (𝑠) 𝑒𝐿𝑠 𝑒−𝐿𝑠 ‖𝑥 (𝑠)‖ 𝑑𝑠]
0
(28)
≤ ℓ + 𝜌𝑝𝑀𝑀1 + 𝑝𝑀𝑀2
𝑡
where 𝐿 > 0 is a constant chosen so that
+ 𝑁 ∫ 𝜇 (𝑠) 𝑒−𝐿(𝑡−𝑠) 𝑑𝑠 ⋅ ‖𝑥‖∗ ,
0
𝑡
𝑁sup ∫ 𝜇 (𝑠) 𝑒−𝐿(𝑡−𝑠) 𝑑𝑠 < 1.
𝑡∈𝐽
0
therefore,
1
Noting that for any 𝜓 ∈ 𝐿 (𝐽, 𝑋), we have
𝑡
0
sup (𝑒−𝐿𝑡 ‖(Λ𝑥) (𝑡)‖)
𝑡∈𝐽
−𝐿(𝑡−𝑠)
lim sup ∫ 𝑒
𝐿 → +∞ 𝑡∈𝐽
𝜓 (𝑠) 𝑑𝑠 = 0,
(30)
𝐵𝜌 = {𝑥 ∈ F (𝑇) ; ‖𝑥‖∗ ≤ 𝜌} ,
1 − 𝑝𝑀𝑀1
(36)
+ [𝑝𝑀𝑀1 + sup (𝑁 ∫ 𝜇 (𝑠) 𝑒−𝐿(𝑡−𝑠) 𝑑𝑠)] 𝜌.
𝑡∈𝐽
(31)
where 𝜌 is a constant chosen so that
󵄩 󵄩
𝑁1 + 󵄩󵄩󵄩𝜙󵄩󵄩󵄩[−𝑟,0] + ℓ + 𝑝𝑀𝑀2
≤ ℓ + 𝑝𝑀𝑀2
𝑡
so, we can take the appropriate 𝐿 to satisfy (29).
Consider the set
𝜌≥
(35)
(29)
0
It results that
‖Λ𝑥‖∗ = sup ‖(Λ𝑥) (𝑡)‖ + sup (𝑒−𝐿𝑡 ‖(Λ𝑥) (𝑡)‖)
𝑡∈𝐽
𝑡∈[−𝑟,0]
> 0,
󵄩 󵄩
≤ 𝑁1 + 󵄩󵄩󵄩𝜙󵄩󵄩󵄩[−𝑟,0] + ℓ + 𝑝𝑀𝑀2
(32)
𝑡
+ (𝑝𝑀𝑀1 + 𝑁sup ∫ 𝜇 (𝑠) 𝑒−𝐿(𝑡−𝑠) 𝑑𝑠) 𝜌.
where ℓ := 𝑀(𝑁1 + ‖𝜙(0)‖) + 𝑁(‖𝜉‖ + ‖𝜇‖𝐿1 ) and ‖𝜙‖[−𝑟,0] =
sup𝑡∈[−𝑟,0] ‖𝜙(𝑡)‖.
Now, if 𝑡 ∈ [−𝑟, 0], 𝑥 ∈ 𝐵𝜌 , then
󵄩
󵄩
󵄩 󵄩
‖(Λ𝑥) (𝑡)‖ = 󵄩󵄩󵄩𝑔 (𝑥) (𝑡) + 𝜙 (𝑡)󵄩󵄩󵄩 ≤ 𝑁1 + 󵄩󵄩󵄩𝜙󵄩󵄩󵄩[−𝑟,0] .
(37)
(33)
𝑡∈𝐽
0
Let 𝐿 → +∞, we obtain
󵄩 󵄩
‖Λ𝑥‖∗ ≤ 𝑁1 + 󵄩󵄩󵄩𝜙󵄩󵄩󵄩[−𝑟,0] + ℓ + 𝑝𝑀𝑀2 + 𝑝𝑀𝑀1 𝜌 ≤ 𝜌. (38)
Hence for some positive number 𝜌, Λ𝐵𝜌 ⊂ 𝐵𝜌 .
Abstract and Applied Analysis
5
Using the strong continuity of {𝐶(𝑡)}𝑡∈R and the compactness condition on the operators 𝐼𝑘 , for 𝜀 > 0, there exists 𝛿 > 0
such that
󵄩
󵄩󵄩
󵄩󵄩(𝐶 (𝑡 + ℎ) − 𝐶 (𝑡)) 𝐼𝑘 (𝑥)󵄩󵄩󵄩 ≤ 𝜀,
𝑥 ∈ 𝐵𝜌 ,
(39)
𝑡 ∈ 𝐽, 𝑘 = 1, 2, . . . , 𝑝,
when |ℎ| < 𝛿. If 𝑡 ∈ [𝑡𝑘 , 𝑡𝑘+1 ] and ℎ < 𝛿 such that 𝑡 + ℎ ∈
[𝑡𝑘 , 𝑡𝑘+1 ], then
̃ ⊂ 𝑃𝐶(𝐽, 𝑋), by applying
For every bounded subset Ω
Proposition 2, for any 𝜀 > 0 there exists a sequence {𝑥𝑛 }∞
𝑛=1 ⊂
̃ such that
Ω
̃ ≤ 2𝜒𝑝𝑐 (Λ 2 {𝑥𝑛 }) + 𝜀,
𝜒𝑝𝑐 (Λ 2 Ω)
noting that the definition of 𝜒𝑝𝑐 , we have
̃ ≤ 2 max 𝜒𝑖 (Λ 2 {𝑥𝑛 } | ) + 𝜀.
𝜒𝑝𝑐 (Λ 2 Ω)
𝐽𝑖
𝑖=0,1,...,𝑝
󵄩󵄩
󵄩󵄩
󵄩󵄩
󵄩
󵄩󵄩 ∑ (𝐶 (𝑡 + ℎ − 𝑡 ) − 𝐶 (𝑡 − 𝑡 )) 𝐼 (𝑥 (𝑡 ))󵄩󵄩󵄩
󵄩󵄩
𝑘
𝑘
𝑘
𝑘 󵄩
󵄩󵄩
󵄩󵄩0<𝑡𝑘 <𝑡
󵄩󵄩
󵄩
(44)
(45)
Then, noting the equicontinuity of Λ 2 |𝐽𝑖 , 𝑖 = 0, 1, . . . , 𝑝, we
can apply Lemmas 2.1 and 2.2 of [35] to obtain
𝑝
󵄩
󵄩
≤ ∑ 󵄩󵄩󵄩(𝐶 (𝑡 + ℎ − 𝑡𝑘 ) − 𝐶 (𝑡 − 𝑡𝑘 )) 𝐼𝑘 (𝑥 (𝑡𝑘 ))󵄩󵄩󵄩 ≤ 𝑝𝜀.
𝑘=1
𝜒𝑖 (Λ 2 {𝑥𝑛 } |𝐽𝑖 ) = sup𝜒 (Λ 2 {𝑥𝑛 } (𝑡)) .
𝑡∈𝐽𝑖
(40)
For 𝑥 ∈ 𝐵𝜌 , by the hypothesis (H1)(i) and (40), we get
󵄩󵄩
󵄩
󵄩󵄩(Λ 2 𝑥) (𝑡 + ℎ) − (Λ 2 𝑥) (𝑡)󵄩󵄩󵄩
󵄩󵄩 𝑡+ℎ
󵄩
≤ 󵄩󵄩󵄩󵄩∫ 𝑆 (𝑡 + ℎ − 𝑠) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠)) 𝑑𝑠
󵄩󵄩 0
(46)
Then from (45) and (46), we have
̃ ≤ 2 max (sup𝜒 (Λ 2 {𝑥𝑛 } (𝑡)))
𝜒𝑝𝑐 (Λ 2 Ω)
𝑖=0,1,...,𝑝
𝑡∈𝐽𝑖
(47)
+ 𝜀 = 2sup𝜒 (Λ 2 {𝑥𝑛 } (𝑡)) + 𝜀.
𝑡∈𝐽
󵄩󵄩
󵄩
− ∫ 𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠)) 𝑑𝑠󵄩󵄩󵄩 + 𝑝𝜀
󵄩󵄩
0
𝑡
For every bounded subset Ω ⊂ F(𝑇), we have
𝑡
󵄩
󵄩
≤ 𝑝𝜀 + ∫ 󵄩󵄩󵄩(𝑆 (𝑡 + ℎ − 𝑠) − 𝑆 (𝑡 − 𝑠)) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠))󵄩󵄩󵄩 𝑑𝑠
0
+∫
𝑡+ℎ
𝑡
+ sup 𝜒 ((Λ 2 Ω) (𝑡)) = 𝜒𝑝𝑐 ((Λ 2 Ω) |𝑃𝐶(𝐽,𝑋) ) ,
󵄩󵄩
󵄩
󵄩󵄩𝑆 (𝑡 + ℎ − 𝑠) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠))󵄩󵄩󵄩 𝑑𝑠
𝑡
𝑡+ℎ
0
𝑡
≤ 𝑝𝜀 + [𝑀ℎ ∫ 𝜇 (𝑠) 𝑑𝑠 + 𝑁 ∫
𝑡∈[−𝑟,0]
(48)
𝜇 (𝑠) 𝑑𝑠] ⋅ (1 + 𝜌) .
(41)
As ℎ → 0 and 𝜀 → 0, the right-hand side of the inequality
above tends to zero independent of 𝑥, so Λ 2 maps bounded
sets into equicontinuous sets.
For bounded set 𝐵 ⊂ 𝑃𝐶(𝐽, 𝑋), we consider the map
𝜒𝑝𝑐 (𝐵) = max 𝜒𝑖 (𝐵|𝐽𝑖 ) ,
𝑖=0,1,...,𝑝
(42)
where 𝜒𝑖 is the Hausdorff measure of noncompactness on the
Banach space 𝐶(𝐽𝑖 , 𝑋) and 𝐵|𝐽𝑖 is defined in (7).
Furthermore, we define the Hausdorff measure of noncompactness 𝜒F on F(𝑇) as follows:
𝜒F (Y ) := 𝜒𝑝𝑐 (Y |𝑃𝐶(𝐽,𝑋) ) + sup 𝜒 (Y (𝑡)) ,
𝑡∈[−𝑟,0]
𝜒F (Λ 2 Ω) = 𝜒𝑝𝑐 ((Λ 2 Ω) |𝑃𝐶(𝐽,𝑋) )
Y ⊂ F (𝑇) .
(43)
moreover, by applying Proposition 2, for any 𝜀 > 0 there exists
a sequence {𝑦𝑛 }∞
𝑛=1 ⊂ Ω such that
𝜒F (Λ 2 Ω) ≤ 2𝜒F (Λ 2 {𝑦𝑛 }) + 𝜀
= 2𝜒𝑝𝑐 ((Λ 2 {𝑦𝑛 }) |𝑃𝐶(𝐽,𝑋) ) + 𝜀.
(49)
Combining with (48) and (49), we have
𝜒F (Λ 2 Ω) = 𝜒𝑝𝑐 ((Λ 2 Ω) |𝑃𝐶(𝐽,𝑋) )
≤ 2𝜒𝑝𝑐 ((Λ 2 {𝑦𝑛 }) |𝑃𝐶(𝐽,𝑋) ) + 𝜀.
(50)
Using the induction of (45)–(47) above, we can see
𝜒F (Λ 2 Ω) = 𝜒𝑝𝑐 ((Λ 2 Ω) |𝑃𝐶(𝐽,𝑋) )
≤ 2sup𝜒 (Λ 2 {𝑦𝑛 } (𝑡) |𝑡∈𝐽 ) + 𝜀.
𝑡∈𝐽
(51)
6
Abstract and Applied Analysis
Thus, from (51), (H2) and Proposition 3 and (3) in Lemma 1,
we can see
𝜒F (Λ 2 Ω) ≤ 2sup𝜒 (Λ 2 {𝑦𝑛 } (𝑡) |𝑡∈𝐽 ) + 𝜀
Combining with (H3), we have
𝜒F (ΛΩ) ≤ 𝜒F (Λ 1 Ω) + 𝜒F (Λ 2 Ω)
𝑡∈𝐽
𝑇
≤ 8𝑁 ∫ 𝜂 (𝑠) 𝑑𝑠 ⋅ 𝜒F (Ω) < 𝑀∗ 𝜒F (Ω) ,
𝑡
0
= 2sup [𝜒 ({ ∫ 𝑆 (𝑡−𝑠)𝑓 (𝑠, 𝑦̃𝑛𝑠 , 𝑦̃𝑛 (𝑠)) 𝑑𝑠
0
𝑡∈𝐽
+ ∑ 𝐶 (𝑡−𝑡𝑘 ) 𝐼𝑘 (𝑦̃𝑛 (𝑡𝑘 ))})]+𝜀
0<𝑡𝑘 <𝑡
𝑡
≤ 2sup [2𝑁 ∫ 𝜂 (𝑠) ( sup 𝜒 ({𝑦𝑛 (𝑠 + 𝜃)})
0
𝑡∈𝐽
𝜃∈[−𝑟,0]
+𝜒 ({𝑦̃𝑛 (𝑠)}) ) 𝑑𝑠] + 𝜀,
(52)
where 𝑦̃𝑛 (𝑡) := 𝑦𝑛 (𝑡)|𝑡∈𝐽 .
Noting that
hence Λ is a 𝜒F -contraction on F(𝑇). According to Theorem 4, the operator Λ has at least one fixed point 𝑥 in 𝐵𝜌 .
This completes the proof.
Next, we establish a condition that guarantee that a mild
solution satisfies (3).
Proposition 8. Assume that the hypotheses of Theorem 7 are
fulfilled and that 𝜙(0) + 𝑔(𝑥)(0) ∈ 𝐷(𝐴). If 𝑥(⋅) is a mild
solution of (1)–(4), then condition (3) holds.
𝑡
Proof. Clearly, (1/𝑡) ∫0 𝑆(𝑡 − 𝑠)𝑓(𝑠, 𝑥𝑠 , 𝑥(𝑠))𝑑𝑠 → 0 as 𝑡 → 0.
Moreover, noting that 𝜙(0)+𝑔(𝑥)(0) ∈ 𝐷(𝐴) and (11), we have
𝐶(⋅)(𝜙(0) + 𝑔(𝑥)(0)) is of class 𝐶1 . Therefore, we can see that
lim
sup 𝜒 ({𝑦𝑛 (𝑠 + 𝜃)}) ≤ sup 𝜒 ({𝑦𝑛 (𝜃)})
𝜃∈[−𝑟,0]
𝑡 → 0+
𝜃∈[−𝑟,0]
𝑥 (𝑡) − 𝑥 (0)
𝑡
1
= lim+ [ (𝐶 (𝑡) − 𝐼) (𝜙 (0) + 𝑔 (𝑥) (0)) + 𝑆 (𝑡) 𝜉
𝑡→0 𝑡
+ sup𝜒 ({𝑦̃𝑛 (𝑠)})
𝑠∈𝐽
≤ sup 𝜒 (Ω (𝜃))
4. Application
𝜃∈[−𝑟,0]
In this section, we consider an application of the theory
developed in Section 3 to the study of an impulsive partial
differential equation with unbounded delay.
+ 𝜒𝑝𝑐 (Ω) = 𝜒F (Ω) ,
≤ 𝜒𝑝𝑐 (Ω) .
(54)
Thus, by (52), we see
𝑡
𝜒F (Λ 2 Ω) ≤ 2sup [2𝑁 ∫ 𝜂 (𝑠) ( sup 𝜒 ({𝑦𝑛 (𝑠 + 𝜃)})
0
0
which shows the assertion.
𝑠∈𝐽
𝑡∈𝐽
+ ∫ 𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠 , 𝑥 (𝑠)) 𝑑𝑠] = 𝜉,
(53)
+ sup𝜒 (Ω (𝑠))
𝜒 ({𝑦̃𝑛 (𝑠)}) ≤ 𝜒 (Ω (𝑠))
Example 9. 𝑋 = 𝐿2 ([0, 𝜋]), 𝐴 : 𝐷(𝐴) ⊆ 𝑋 → 𝑋 is the map
defined by 𝐴𝜗 = 𝜗󸀠󸀠 with domain 𝐷(𝐴) = {𝜗 ∈ 𝑋 : 𝜗󸀠󸀠 ∈
𝑋, 𝜗(0) = 𝜗(𝜋) = 0}.
We consider the following integrodifferential model:
𝜕2
𝜕2
󵄨
󵄨
𝑣
𝜉)
=
𝑣 (𝑡, 𝜉) + sin 󵄨󵄨󵄨𝑣 (𝑡, 𝜉)󵄨󵄨󵄨
(𝑡,
𝜕𝑡2
𝜕𝜉2
𝜃∈[−𝑟,0]
+𝜒 ({𝑦̃𝑛 (𝑠)}) ) 𝑑𝑠] + 𝜀
𝑡
󵄨
󵄨󵄨
𝑡
󵄨𝑣 (𝜃, 𝜉)󵄨󵄨󵄨
+ 𝑡2 ∫ 𝛾 (𝜃 − 𝑡) ⋅ cos ( 󵄨
) 𝑑𝜃,
𝑡
𝑡−𝑟
≤ 2sup (4𝑁 ∫ 𝜂 (𝑠) 𝑑𝑠) ⋅ 𝜒F (Ω) + 𝜀
𝑣 (𝑡, 𝜋) = 𝑣 (𝑡, 0) = 0,
0
𝑡∈𝐽
(58)
𝑡
≤ sup 𝜒 (Ω (𝜃))
𝜃∈[−𝑟,0]
(57)
𝜋
𝑇
𝑣 (𝜃, 𝜉) = 𝑣0 (𝜃, 𝜉) + ∫ 𝑐 (𝜉, 𝑠) sin (1 + 𝑣 (𝜃, 𝑠)) 𝑑𝑠,
= 8𝑁 ∫ 𝜂 (𝑠) 𝑑𝑠 ⋅ 𝜒F (Ω) + 𝜀.
0
0
(55)
Since 𝜀 is arbitrary, we can obtain
𝑇
𝜒F (Λ 2 Ω) ≤ 8𝑁 ∫ 𝜂 (𝑠) 𝑑𝑠 ⋅ 𝜒F (Ω) .
0
− 𝑟 ≤ 𝜃 ≤ 0,
𝜋
(56)
𝜕
𝑣 (0, 𝜉) = 𝜔 (𝜉) ,
𝜕𝑡
Δ𝑣 (𝑡𝑘 , 𝜉) = ∫ 𝜌𝑘 (𝜉, 𝑦) 𝑑𝑦 ⋅ cos2 (𝑣 (𝑡𝑘 , 𝜉)) ,
0
1 ≤ 𝑘 ≤ 𝑝,
(59)
Abstract and Applied Analysis
7
where 𝑡 ∈ [0, 𝑇], 𝑟 > 0, 𝜉 ∈ [0, 𝜋], 0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑝 < 𝑇,
𝜔 ∈ 𝑋 and 𝑣𝑡 (𝜃, 𝜉) = 𝑣(𝑡 + 𝜃, 𝜉). 𝛾 : [−𝑟, 0] → R and 𝑐(𝜉, 𝑠),
𝜌𝑘 (𝜉, 𝑧) ∈ 𝐿2 ([0, 𝜋] × [0, 𝜋], R) satisfy the following assumptions.
(1) The function 𝛾 : [−𝑟, 0] → R is a continuous func0
tion and ∫−𝑟 |𝛾(𝜃)|𝑑𝜃 < ∞.
(2) The function 𝑐(𝜉, 𝑠), 𝜉, 𝑠 ∈ [0, 𝜋] is measurable and there exists a constant 𝑁1 such that
𝜋
𝜋
(𝜋 ∫0 ∫0 𝑐2 (𝜉, 𝑠)𝑑𝑠𝑑𝜉)
1/2
≤ 𝑁1 .
(3) For every 𝑘 = 1, 2, . . . , 𝑝, the function 𝜌𝑘 (𝜉, 𝑧), 𝑧 ∈
[0, 𝜋], is measurable and there exists a constant 𝑁
such that
𝜋
𝜋
0
0
2
1/2
(∫ (∫ 𝜌𝑘 (𝜉, 𝑧)𝑑𝑧) 𝑑𝜉)
≤ 𝑁.
(60)
𝐴𝑢 = 𝑢󸀠󸀠 .
0
󵄨
󵄨
󵄩
󵄩󵄩
2
󵄩󵄩𝑓 (𝑡, 𝜑, 𝑥 (𝑡))󵄩󵄩󵄩 ≤ ‖𝑥 (𝑡)‖ + 𝑡 ∫ 󵄨󵄨󵄨𝛾 (𝜃)󵄨󵄨󵄨 𝑑𝜃
−𝑟
(63)
≤ 𝜇 (𝑡) (1 + ‖𝑥 (𝑡)‖) ,
0
where 𝜇(𝑡) := max{1, 𝑡2 ∫−𝑟 |𝛾(𝜃)|𝑑𝜃}.
For any 𝑥1 , 𝑥2 ∈ 𝑋, 𝜑, 𝜑̃ ∈ 𝐶([−𝑟, 0], 𝑋),
󵄩󵄩
̃ 𝑥2 (𝑡)) (𝜉)󵄩󵄩󵄩󵄩
󵄩󵄩𝑓 (𝑡, 𝜑, 𝑥1 (𝑡)) (𝜉) − 𝑓 (𝑡, 𝜑,
0
󵄩
󵄨󵄩
󵄩
󵄩
󵄨
≤ 󵄩󵄩󵄩𝑥1 (𝑡) − 𝑥2 (𝑡)󵄩󵄩󵄩 + 𝑡 ∫ 󵄨󵄨󵄨𝛾 (𝜃)󵄨󵄨󵄨 󵄩󵄩󵄩𝜑 (𝜃) (𝜉) − 𝜑̃ (𝜃) (𝜉)󵄩󵄩󵄩 𝑑𝜃.
−𝑟
(64)
Therefore, for any bounded sets 𝐷1 ⊂ 𝐶([−𝑟, 0], 𝑋), 𝐷2 ⊂
𝑋, we have
0
󵄨
󵄨
𝜒 (𝑓 (𝑡, 𝐷1 , 𝐷2 )) ≤ 𝜒 (𝐷2 ) + 𝑡 ∫ 󵄨󵄨󵄨𝛾 (𝜃)󵄨󵄨󵄨 𝜒 (𝐷1 (𝜃)) 𝑑𝜃
−𝑟
To treat the above problem, we define
𝐷 (𝐴) = 𝐻2 ([0, 𝜋]) ∩ 𝐻01 ([0, 𝜋]) ,
For 𝑡 ∈ [0, 𝑇], we can see
0
󵄨
󵄨
≤ 𝜒 (𝐷2 ) + 𝑡 sup 𝜒 (𝐷1 (𝜃)) ∫ 󵄨󵄨󵄨𝛾 (𝜃)󵄨󵄨󵄨 𝑑𝜃
−𝑟
−𝑟≤𝜃≤0
(61)
≤ 𝜂 (𝑡) ( sup 𝜒 (𝐷1 (𝜃)) + 𝜒 (𝐷2 )) ,
𝐴 is the infinitesimal generator of a strongly continuous
cosine function {𝐶(𝑡)}𝑡∈R on 𝑋. Moreover, 𝐴 has a discrete
spectrum, the eigenvalues are −𝑛2 , 𝑛 ∈ N, with the corresponding normalized eigenvectors 𝜔𝑛 (𝑥) = √(2/𝜋) sin(𝑛𝑥);
the set {𝜔𝑛 ; 𝑛 ∈ N} is an orthonormal basis of 𝑋 and the
following properties hold.
2
(a) If 𝜔 ∈ 𝐷(𝐴), then 𝐴𝜔 = − ∑∞
𝑛=1 𝑛 ⟨𝜔, 𝜔𝑛 ⟩𝜔𝑛 .
−𝑟≤𝜃≤0
a.e. 𝑡 ∈ [0, 𝑇] ,
(65)
0
where 𝜂(𝑡) := max{1, 𝑡 ∫−𝑟 |𝛾(𝜃)|𝑑𝜃}.
For 𝑥 ∈ 𝑋,
󵄩
󵄩󵄩
󵄩󵄩𝐼𝑘 (𝑥)󵄩󵄩󵄩 ≤ 𝑁 (1 + ‖𝑥‖) ,
𝑘 = 1, 2, . . . , 𝑝.
(66)
(b) For each 𝜔 ∈ 𝑋, 𝐶(𝑡)𝜔 = ∑∞
𝑛=1 cos(𝑛𝑡)⟨𝜔, 𝜔𝑛 ⟩𝜔𝑛 and
(sin(𝑛𝑡)/𝑛)⟨𝜔,
𝜔𝑛 ⟩𝜔𝑛 . Consequently,
𝑆(𝑡)𝜔 = ∑∞
𝑛=1
‖𝐶(𝑡)‖ = ‖𝑆(𝑡)‖ ≤ 1 for 𝑡 ∈ R, and {𝑆(𝑡)} is compact
for every 𝑡 ∈ R.
̃∗ ∈ (0, 1)
Suppose further that there exists a constant 𝑀
𝑇
∗
̃ and 𝑝𝑁 < 1, then (59) has at least
such that 8 ∫0 𝜂(𝑠)𝑑𝑠 < 𝑀
a mild solution by Theorem 7.
For 𝜉 ∈ [0, 𝜋] and 𝜑 ∈ 𝐶([−𝑟, 0], 𝑋), we set
Acknowledgments
This work was partly supported by the NSF of China
(11201413), the NSF of Yunnan Province (2009ZC054M), the
Educational Commission of Yunnan Province (2012Z010)
and the Foundation of Key Program of Yunnan Normal
University.
𝑥 (𝑡) (𝜉) = 𝑣 (𝑡, 𝜉) ,
𝜙 (𝜃) (𝜉) = 𝑣0 (𝜃, 𝜉) ,
𝜃 ∈ [−𝑟, 0] ,
𝜋
𝑔 (𝜑 (𝜃)) (𝜉) = ∫ 𝑐 (𝜉, 𝑠) sin (1 + 𝜑 (𝜃) (𝑠)) 𝑑𝑠,
0
󵄨
󵄨
𝑓 (𝑡, 𝜑, 𝑥 (𝑡)) (𝜉) = sin 󵄨󵄨󵄨𝑥 (𝑡) (𝜉)󵄨󵄨󵄨
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(62)
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