1.Introduction FangLiandHuiwenWang AnExistenceResultforNonlocalImpulsiveSecond-OrderCauchyProblemswithFiniteDelay ResearchArticle

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Volume 2013, Article ID 724854,8pages 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; [email protected] 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 spaceX. 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 ordi- nary 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 advan- tages 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], Henr´ıquez and V´asquez [1], Hern´andez et al., [21–23], Arthi and Balachandran [17], and so forth.

Moreover, we consider the nonlocal condition 𝑥(0) = 𝑔(𝑥)+𝑥₀, where𝑔is 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 space𝑋:

𝑑²

𝑑𝑡²𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝑓 (𝑡, 𝑥_𝑡, 𝑥 (𝑡)) , 𝑡 ∈ (0, 𝑇] , 𝑡 ̸= 𝑡_𝑘, 𝑘 = 1, 2, . . . , 𝑝,

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𝑥 (𝑡) = 𝑔 (𝑥) (𝑡) + 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] , (2)

𝑥^󸀠(0) = 𝜉 ∈ 𝑋, (3)

Δ𝑥 (𝑡_𝑘) = 𝐼_𝑘(𝑥 (𝑡_𝑘)) , 𝑘 = 1, 2, . . . , 𝑝, (4) where 𝐴 is the infinitesimal generator of a strongly continuous cosine family of bounded linear operators{𝐶(𝑡)}_𝑡∈R on 𝑋. 𝑓, 𝑔are 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]to𝑋.

The impulsive moments{𝑡_𝑘}are given such that0 = 𝑡₀<

𝑡₁ < ⋅ ⋅ ⋅ < 𝑡_𝑝 < 𝑡_𝑝+1 = 𝑇,𝐼_𝑘 : 𝑋 → 𝑋 (𝑘 = 1, 2, . . . , 𝑝) are appropriate functions,Δ𝑥(𝑡_𝑘)represents the jump of a function𝑥at𝑡_𝑘, which is defined byΔ𝑥(𝑡_𝑘) = 𝑥(𝑡⁺_𝑘) − 𝑥(𝑡⁻_𝑘), where𝑥(𝑡⁺_𝑘)and𝑥(𝑡⁻_𝑘)are, respectively, the right and the left limits of𝑥at𝑡_𝑘.

For any continuous function𝑥 defined on the interval [−𝑟, 𝑇]and any𝑡 ∈ [0, 𝑇], we denote by 𝑥_𝑡the element of 𝐶([−𝑟, 0], 𝑋)defined by𝑥_𝑡(𝜃) = 𝑥(𝑡 + 𝜃)for𝜃 ∈ [−𝑟, 0].

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In this paper, motivated by above works, we study (1)–(4) in𝑋and obtain the existence theorem based on theory on measures of noncompactness without the assumptions that the nonlinearity𝑓satisfies a Lipschitz type condition and the cosine family of bounded linear operators{𝐶(𝑡)}_𝑡∈Rgenerated by𝐴is compact.

2. Preliminaries

Throughout this paper, we set𝐽 = [0, 𝑇], a compact interval inR. We denote by𝑋a Banach space with norm‖ ⋅ ‖, by𝐿(𝑋) the Banach space of all linear and bounded operators on𝑋.

We abbreviate‖𝑢‖_𝐿¹_(𝐽,R⁺₎with‖𝑢‖_𝐿¹, for any𝑢 ∈ 𝐿¹(𝐽,R⁺).

Let

𝑃𝐶 (𝐽, 𝑋) := {𝑥 : 𝐽 󳨀→ 𝑋; 𝑥 (𝑡) is continuous at𝑡 ̸= 𝑡_𝑘, left continuous at𝑡 = 𝑡_𝑘, and

the right limit𝑥 (𝑡⁺_𝑘)exists for𝑘=1, 2, . . . , 𝑝} . (5) It is easy to check that𝑃𝐶(𝐽, 𝑋)is a Banach space with the norm

‖𝑥‖_𝑃𝐶=sup

𝑡∈𝐽 ‖𝑥 (𝑡)‖ , for any 𝑥 ∈ 𝑃𝐶 (𝐽, 𝑋) . (6) We let𝐽₀= (𝑡₀, 𝑡₁],𝐽₁= (𝑡₁, 𝑡₂], . . . , 𝐽_𝑝= (𝑡_𝑝, 𝑡_𝑝+1].

ForB⊆ 𝑃𝐶(𝐽, 𝑋), we denote byB|_𝐽_𝑖the set

B|_𝐽_𝑖 = { 𝑥 ∈ 𝐶 ([𝑡_𝑖, 𝑡_𝑖+1] , 𝑋) ; 𝑥 (𝑡_𝑖) = 𝑣 (𝑡⁺_𝑖) , 𝑥 (𝑡) = 𝑣 (𝑡) , 𝑡 ∈ 𝐽_𝑖, 𝑣 ∈B}

(7) 𝑖 = 0, 1, 2, . . . , 𝑝.

A family{𝐶(𝑡)}_𝑡∈Rin𝐿(𝑋)is called a cosine function on 𝑋if

(i)𝐶(0) = 𝐼is the identity operator in𝑋;

(ii)𝐶(𝑡 + 𝑠) + 𝐶(𝑡 − 𝑠) = 2𝐶(𝑡)𝐶(𝑠)for all𝑠, 𝑡 ∈R;

(iii) The map𝑡 → 𝐶(𝑡)𝑥is strongly continuous for each 𝑥 ∈ 𝑋.

The associated sine function is the family{𝑆(𝑡)}_𝑡∈Rof operators defined by

𝑆 (𝑡) 𝑥 = ∫^𝑡

0𝐶 (𝑠) 𝑥 𝑑𝑠, for𝑥 ∈ 𝑋, 𝑡 ∈R. (8) One can define the infinitesimal generator𝐴of𝐶(⋅)by

𝐷 (𝐴) = {𝑥 ∈ 𝑋; lim

𝑡 → 02𝑡⁻²(𝐶 (𝑡) 𝑥 − 𝑥) ∈ 𝑋}

𝐴𝑥 =lim

𝑡 → 02𝑡⁻²(𝐶 (𝑡) 𝑥 − 𝑥) , 𝑥 ∈ 𝐷 (𝐴) .

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In this paper, we assume there exist positive constants𝑀 and𝑁such that

‖𝐶 (𝑡)‖ ≤ 𝑀, ‖𝑆 (𝑡)‖ ≤ 𝑁 for every𝑡 ∈ 𝐽. (10)

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 for𝑥 ∈ 𝐷 (𝐴) , 𝑡 ∈R;

(iii) ∫^𝑡

0𝑆 (𝑠) 𝑥𝑑𝑠 ∈ 𝐷 (𝐴) , 𝐴 ∫^𝑡

0𝑆 (𝑠) 𝑥𝑑𝑠 = 𝐶 (𝑡) 𝑥 − 𝑥 for 𝑥 ∈ 𝑋, 𝑡 ∈R;

(iv)𝐶 (𝑡) 𝑥 − 𝑥 = ∫^𝑡

0𝑆 (𝑠) 𝐴𝑥𝑑𝑠 for𝑥 ∈ 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 noncom- pactness𝜒(⋅)on each bounded subsetΩof Banach space𝑌is defined by

𝜒 (Ω) =inf{𝜀 > 0; Ωhas a finite𝜀-net in𝑋} . (12) This measure of noncompactness satisfies some basic properties as follows.

Lemma 1 (see [31]). Let 𝑌be a real Banach space, and let 𝐵, 𝐶 ⊆ 𝑌be bounded. Then

(1)𝜒(𝐵) = 0if and only if𝐵is precompact;

(2)𝜒(𝐵) = 𝜒(𝐵) = 𝜒(𝑐𝑜𝑛𝑣𝐵), where𝐵and𝑐𝑜𝑛𝑣𝐵mean the closure and convex hull of𝐵, respectively;

(3)𝜒(𝐵) ≤ 𝜒(𝐶)if𝐵 ⊆ 𝐶;

(4)𝜒(𝐵 ∪ 𝐶) ≤max{𝜒(𝐵), 𝜒(𝐶)};

(5)𝜒(𝐵 + 𝐶) ≤ 𝜒(𝐵) + 𝜒(𝐶), where𝐵 + 𝐶 = {𝑥 + 𝑦; 𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶};

(6)𝜒(𝛼𝐵) = |𝛼|𝜒(𝐵), for any𝛼 ∈R;

(7)let𝑍be a Banach space and𝑄 : 𝐷(𝑄) ⊆ 𝑌 → 𝑍Lip- schitz continuous with constant𝜈. Then𝜒(𝑄𝐵) ≤ 𝜈 ⋅ 𝜒(𝐵)for all𝐵 ⊆ 𝐷(𝑄)being bounded.

Proposition 2 (see [32], Page 125). LetΩbe a bounded set for a real Banach space𝑋. Then, for every𝜀 > 0there exists a sequence{𝑥_𝑛}^∞_𝑛=1inΩsuch that

𝜒 (Ω) ≤ 2𝜒 ({ 𝑥_𝑛}^∞_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].

Proposition 3. Let{𝑓_𝑛}^∞_𝑛=1be a sequence in𝐿¹(𝐽, 𝑋)such that there exist𝑣, 𝑞 ∈ 𝐿¹₊([0, 𝑇])with the properties:

(i) sup_𝑛∈N‖𝑓_𝑛(𝑡)‖ ≤ 𝑣(𝑡), 𝑎.𝑒. 𝑡 ∈ 𝐽;

(ii)𝜒({𝑓_𝑛}^∞_𝑛=1) ≤ 𝑞(𝑡), 𝑎.𝑒. 𝑡 ∈ 𝐽.

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Then, for every𝑡 ∈ 𝐽, we have 𝜒 ({∫^𝑡

0𝑆 (𝑡 − 𝑠) 𝑓_𝑛(𝑠) 𝑑𝑠}^∞

𝑛=1) ≤ 2𝑁 ∫^𝑡

0𝑞 (𝑠) 𝑑𝑠, (14) where𝑁is from(10).

A continuous map𝑄 : 𝑊 ⊆ 𝑌 → 𝑌is said to be a𝜒- contraction if there exists a positive constant𝜈 < 1such that 𝜒(𝑄𝐶) ≤ 𝜈 ⋅ 𝜒(𝐶)for any bounded closed subset𝐶 ⊆ 𝑊.

Theorem 4 (see [31] (Darbo-Sadovskii)). If𝑈 ⊆ 𝑌is bounded closed and convex, the continuous mapF : 𝑈 → 𝑈is a𝜒- contraction, then the mapFhas at least one fixed point in𝑈.

Definition 5. A function𝑥 : [−𝑟, 𝑇] → 𝑋is called a mild solution of system (1)–(4) if𝑥₀ = 𝑔(𝑥) + 𝜙,𝑥|_𝐽 ∈ 𝑃𝐶(𝐽, 𝑋) and

𝑥 (𝑡) = 𝐶 (𝑡) (𝜙 (0) + 𝑔 (𝑥) (0)) + 𝑆 (𝑡) 𝜉 + ∫^𝑡

0𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥_𝑠, 𝑥 (𝑠)) 𝑑𝑠 + ∑

0<𝑡_𝑘<𝑡𝐶 (𝑡 − 𝑡_𝑘) 𝐼_𝑘(𝑥 (𝑡_𝑘)) , 𝑡 ∈ 𝐽.

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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).

LetF(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

𝑡∈[−𝑟,0]‖𝑥 (𝑡)‖ +sup

𝑡∈𝐽 ‖𝑥 (𝑡)‖ . (17) We will require the following assumptions.

(H1) (i)𝑓 : 𝐽×𝐶([−𝑟, 0], 𝑋)×𝑋 → 𝑋satisfies𝑓(⋅, 𝑣, 𝑤) : 𝐽 → 𝑋is measurable for all(𝑣, 𝑤) ∈ 𝐶([−𝑟, 0], 𝑋) × 𝑋and 𝑓(𝑡, ⋅, ⋅) : 𝐶([−𝑟, 0], 𝑋) × 𝑋 → 𝑋is continuous for a.e.𝑡 ∈ 𝐽, and there exists a function𝜇(⋅) ∈ 𝐿¹(𝐽,R⁺)such that

󵄩󵄩󵄩󵄩𝑓(𝑡,𝑣,𝑤)󵄩󵄩󵄩󵄩 ≤ 𝜇(𝑡)(1 + ‖𝑤‖) (18) for almost all𝑡 ∈ 𝐽;

(ii) there exists a function𝜂 ∈ 𝐿¹(𝐽,R⁺)such that for any bounded sets𝐷₁⊂ 𝐶([−𝑟, 0], 𝑋),𝐷₂⊂ 𝑋

𝜒 (𝑓 (𝑡, 𝐷₁, 𝐷₂)) ≤ 𝜂 (𝑡) ( sup

𝜃∈[−𝑟,0]𝜒 (𝐷₁(𝜃)) + 𝜒 (𝐷₂)) , a.e. 𝑡 ∈ 𝐽.

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(H2)𝐼_𝑘 : 𝑋 → 𝑋are compact operators and there exist positive constants𝑀₁, 𝑀₂such that

󵄩󵄩󵄩󵄩𝐼^𝑘(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝑀¹‖𝑥‖ + 𝑀₂, for any𝑥 ∈ 𝑋, 𝑘 = 1, 2, . . . , 𝑝.

(20) (H3)𝑔 : 𝐶([−𝑟, 0], 𝑋) → 𝑋is a compact operator and there exists a constant𝑁₁> 0such that

󵄩󵄩󵄩󵄩𝑔(𝑥)󵄩󵄩󵄩󵄩[−𝑟,0]≤ 𝑁₁ for all𝑥 ∈ 𝐶 ([−𝑟, 0] , 𝑋) , (21) where‖𝑔(𝑥)‖_[−𝑟,0]=sup_{𝑡∈[−𝑟,0]}‖𝑔(𝑥)(𝑡)‖.

(H4) There exists𝑀^∗ ∈ (0, 1)such that8𝑁 ∫₀^𝑇𝜂(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.

Proof. Define the operatorΛ :F(𝑇) → F(𝑇)in the following way:

(Λ𝑥) (𝑡) = {{ {{ {{ {{ {{ {{ {{ {{ {{ {

𝑔 (𝑥) (𝑡) + 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] , 𝐶 (𝑡) (𝑔 (𝑥) (0) + 𝜙 (0)) + 𝑆 (𝑡) 𝜉

+ ∫^𝑡

0𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥𝑠, 𝑥 (𝑠)) 𝑑𝑠 + ∑

0<𝑡𝑘<𝑡

𝐶 (𝑡 − 𝑡_𝑘) 𝐼_𝑘(𝑥 (𝑡_𝑘)) , 𝑡 ∈ 𝐽.

(22) It is clear that the operatorΛis well defined, and the fixed point ofΛis the mild solution of problems (1)–(4).

The operatorΛcan be written in the formΛ = Λ₁+ Λ₂, where the operatorsΛ₁, Λ₂are defined as follows:

(Λ₁𝑥) (𝑡) = {{ {{ {{ {

𝑔 (𝑥) (𝑡) + 𝜙 (𝑡) , 𝑡 ∈ [−𝑟, 0] , 𝐶 (𝑡) (𝑔 (𝑥) (0) + 𝜙 (0))

+𝑆 (𝑡) 𝜉, 𝑡 ∈ 𝐽,

(Λ₂𝑥) (𝑡) = {{ {{ {{ {{ {{ {{ {

0, 𝑡 ∈ [−𝑟, 0] ,

∫^𝑡

0𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥_𝑠, 𝑥 (𝑠)) 𝑑𝑠 + ∑

0<𝑡𝑘<𝑡

𝐶 (𝑡 − 𝑡_𝑘) 𝐼_𝑘(𝑥 (𝑡_𝑘)) , 𝑡 ∈ 𝐽.

(23) Obviously, under the assumptions of𝑔,Λ₁is continuous.

For𝑡 ∈ 𝐽, we can prove thatΛ₂is continuous.

Indeed, let{𝑥^𝑛}_𝑛∈Nbe a sequence such that𝑥^𝑛 → 𝑥in F(𝑇)as𝑛 → ∞. Since𝑓satisfies (H1)(i), for almost every 𝑡 ∈ 𝐽, we get

𝑓 (𝑡, 𝑥^𝑛_𝑡, 𝑥^𝑛(𝑡)) 󳨀→ 𝑓 (𝑡, 𝑥_𝑡, 𝑥 (𝑡)) , as𝑛 󳨀→ ∞. (24)

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Noting that𝑥^𝑛 → 𝑥inF(𝑇), we can see that there exists 𝜀 > 0such that‖𝑥^𝑛− 𝑥‖_F(𝑇) ≤ 𝜀 for 𝑛sufficiently large.

Therefore, we have

󵄩󵄩󵄩󵄩𝑓(𝑡,𝑥^𝑛^𝑡, 𝑥^𝑛(𝑡)) − 𝑓 (𝑡, 𝑥_𝑡, 𝑥 (𝑡))󵄩󵄩󵄩󵄩

≤ 𝜇 (𝑡) (1 + 󵄩󵄩󵄩󵄩𝑥^𝑛(𝑡)󵄩󵄩󵄩󵄩) + 𝜇 (𝑡) (1 + ‖𝑥 (𝑡)‖)

≤ 2𝜇 (𝑡) + 𝜇 (𝑡) 󵄩󵄩󵄩󵄩𝑥^𝑛(𝑡) − 𝑥 (𝑡)󵄩󵄩󵄩󵄩 + 2𝜇 (𝑡) ‖𝑥 (𝑡)‖

≤ 2𝜇 (𝑡) + 𝜇 (𝑡) 𝜀 + 2𝜇 (𝑡) ‖𝑥‖_F(𝑇).

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It follows from the Lebesgue’s dominated convergence theorem that

∫^𝑡

0󵄩󵄩󵄩󵄩𝑆(𝑡 − 𝑠)[𝑓(𝑠,𝑥^𝑛^𝑠, 𝑥^𝑛(𝑠)) − 𝑓 (𝑠, 𝑥_𝑠, 𝑥 (𝑠))]󵄩󵄩󵄩󵄩 𝑑𝑠

≤ 𝑁 ∫^𝑡

0󵄩󵄩󵄩󵄩𝑓(𝑠,𝑥^𝑛𝑠, 𝑥^𝑛(𝑠)) − 𝑓 (𝑠, 𝑥_𝑠, 𝑥 (𝑠))󵄩󵄩󵄩󵄩 𝑑𝑠

󳨀→ 0, as𝑛 󳨀→ ∞.

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Moreover, noting that (H2), we obtain that

𝑛 → ∞lim󵄩󵄩󵄩󵄩Λ²𝑥^𝑛− Λ₂𝑥󵄩󵄩󵄩󵄩F(𝑇) = 0. (27) This shows thatΛ₂is continuous. Therefore,Λis continuous.

Let us introduce in the spaceF(𝑇)the equivalent norm defined as

‖𝑥‖_∗= sup

𝑡∈[−𝑟,0]‖𝑥 (𝑡)‖ +sup

𝑡∈𝐽 (𝑒^−𝐿𝑡‖𝑥 (𝑡)‖) , (28) where𝐿 > 0is a constant chosen so that

𝑁sup𝑡∈𝐽 ∫^𝑡

0𝜇 (𝑠) 𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠 < 1. (29) Noting that for any𝜓 ∈ 𝐿¹(𝐽, 𝑋), we have

𝐿 → +∞lim sup

𝑡∈𝐽 ∫^𝑡

0𝑒^{−𝐿(𝑡−𝑠)}𝜓 (𝑠) 𝑑𝑠 = 0, (30) so, we can take the appropriate𝐿to satisfy (29).

Consider the set

𝐵_𝜌= { 𝑥 ∈F(𝑇) ; ‖𝑥‖_∗≤ 𝜌} , (31) where𝜌is a constant chosen so that

𝜌 ≥ 𝑁₁+ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩[−𝑟,0]+ ℓ + 𝑝𝑀𝑀₂

1 − 𝑝𝑀𝑀₁ > 0, (32)

whereℓ := 𝑀(𝑁₁+ ‖𝜙(0)‖) + 𝑁(‖𝜉‖ + ‖𝜇‖_𝐿1)and‖𝜙‖_[−𝑟,0]= sup_{𝑡∈[−𝑟,0]}‖𝜙(𝑡)‖.

Now, if𝑡 ∈ [−𝑟, 0],𝑥 ∈ 𝐵_𝜌, then

‖(Λ𝑥) (𝑡)‖ = 󵄩󵄩󵄩󵄩𝑔 (𝑥) (𝑡) + 𝜙 (𝑡)󵄩󵄩󵄩󵄩 ≤ 𝑁¹+ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩[−𝑟,0]. (33)

For𝑡 ∈ 𝐽,𝑥 ∈ 𝐵_𝜌, we have

‖(Λ𝑥) (𝑡)‖ ≤ 󵄩󵄩󵄩󵄩𝐶 (𝑡) (𝑔 (𝑥) (0) + 𝜙 (0))󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑆(𝑡)𝜉󵄩󵄩󵄩󵄩

+ ∫^𝑡

0󵄩󵄩󵄩󵄩𝑆(𝑡 − 𝑠)𝑓(𝑠,𝑥^𝑠, 𝑥 (𝑠))󵄩󵄩󵄩󵄩 𝑑𝑠 + ∑

0<𝑡𝑘<𝑡󵄩󵄩󵄩󵄩𝐶(𝑡 − 𝑡^𝑘) 𝐼_𝑘(𝑥 (𝑡_𝑘))󵄩󵄩󵄩󵄩

≤ 𝑀 (𝑁₁+󵄩󵄩󵄩󵄩𝜙 (0)󵄩󵄩󵄩󵄩+ ∑

0<𝑡_𝑘<𝑡󵄩󵄩󵄩󵄩𝐼^𝑘(𝑥 (𝑡_𝑘))󵄩󵄩󵄩󵄩) +𝑁 (󵄩󵄩󵄩󵄩𝜉󵄩󵄩󵄩󵄩+∫₀^𝑡𝜇 (𝑠) (1+𝑒^𝐿𝑠𝑒^−𝐿𝑠‖𝑥 (𝑠)‖) 𝑑𝑠)

= ℓ + 𝑀 ∑

0<𝑡_𝑘<𝑡󵄩󵄩󵄩󵄩𝐼^𝑘(𝑥 (𝑡_𝑘))󵄩󵄩󵄩󵄩

+ 𝑁 ∫^𝑡

0𝜇 (𝑠) 𝑒^𝐿𝑠𝑒^−𝐿𝑠‖𝑥 (𝑠)‖ 𝑑𝑠,

(34) then

𝑒^−𝐿𝑡‖(Λ𝑥) (𝑡)‖ ≤ 𝑒^−𝐿𝑡[ℓ + 𝑀 ∑

0<𝑡𝑘<𝑡󵄩󵄩󵄩󵄩𝐼^𝑘(𝑥 (𝑡_𝑘))󵄩󵄩󵄩󵄩

+𝑁 ∫^𝑡

0𝜇 (𝑠) 𝑒^𝐿𝑠𝑒^−𝐿𝑠‖𝑥 (𝑠)‖ 𝑑𝑠]

≤ ℓ + 𝜌𝑝𝑀𝑀₁+ 𝑝𝑀𝑀₂ + 𝑁 ∫^𝑡

0𝜇 (𝑠) 𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠 ⋅ ‖𝑥‖_∗,

(35) therefore,

sup𝑡∈𝐽 (𝑒^−𝐿𝑡‖(Λ𝑥) (𝑡)‖)

≤ ℓ + 𝑝𝑀𝑀₂ + [𝑝𝑀𝑀₁+sup

𝑡∈𝐽 (𝑁 ∫^𝑡

0𝜇 (𝑠) 𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠)] 𝜌.

(36)

It results that

‖Λ𝑥‖_∗ = sup

𝑡∈[−𝑟,0]‖(Λ𝑥) (𝑡)‖ +sup

𝑡∈𝐽 (𝑒^−𝐿𝑡‖(Λ𝑥) (𝑡)‖)

≤ 𝑁₁+ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩[−𝑟,0]+ ℓ + 𝑝𝑀𝑀₂ + (𝑝𝑀𝑀₁+ 𝑁sup

𝑡∈𝐽 ∫^𝑡

0𝜇 (𝑠) 𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠) 𝜌.

(37)

Let𝐿 → +∞, we obtain

‖Λ𝑥‖_∗≤ 𝑁₁+ 󵄩󵄩󵄩󵄩𝜙󵄩󵄩󵄩󵄩[−𝑟,0]+ ℓ + 𝑝𝑀𝑀₂+ 𝑝𝑀𝑀₁𝜌 ≤ 𝜌. (38) Hence for some positive number𝜌,Λ𝐵_𝜌⊂ 𝐵_𝜌.

(5)

Using the strong continuity of{𝐶(𝑡)}_𝑡∈Rand the compactness condition on the operators𝐼_𝑘, for𝜀 > 0, there exists𝛿 > 0 such that

󵄩󵄩󵄩󵄩(𝐶(𝑡 + ℎ) − 𝐶(𝑡))𝐼^𝑘(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝜀, 𝑥 ∈ 𝐵^𝜌,

𝑡 ∈ 𝐽, 𝑘 = 1, 2, . . . , 𝑝, (39) when|ℎ| < 𝛿. If𝑡 ∈ [𝑡_𝑘, 𝑡_𝑘+1]andℎ < 𝛿such that𝑡 + ℎ ∈ [𝑡_𝑘, 𝑡_𝑘+1], then

󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

󵄩󵄩󵄩 ∑

0<𝑡_𝑘<𝑡(𝐶 (𝑡 + ℎ − 𝑡_𝑘) − 𝐶 (𝑡 − 𝑡_𝑘)) 𝐼_𝑘(𝑥 (𝑡_𝑘))󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩

≤∑^𝑝

𝑘=1󵄩󵄩󵄩󵄩(𝐶(𝑡 + ℎ − 𝑡^𝑘) − 𝐶 (𝑡 − 𝑡_𝑘)) 𝐼_𝑘(𝑥 (𝑡_𝑘))󵄩󵄩󵄩󵄩 ≤ 𝑝𝜀.

(40) For𝑥 ∈ 𝐵_𝜌, by the hypothesis (H1)(i) and (40), we get

󵄩󵄩󵄩󵄩(Λ²𝑥) (𝑡 + ℎ) − (Λ₂𝑥) (𝑡)󵄩󵄩󵄩󵄩

≤󵄩󵄩󵄩󵄩

󵄩󵄩󵄩󵄩󵄩∫^𝑡+ℎ

0 𝑆 (𝑡 + ℎ − 𝑠) 𝑓 (𝑠, 𝑥_𝑠, 𝑥 (𝑠)) 𝑑𝑠

− ∫^𝑡

0𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥_𝑠, 𝑥 (𝑠)) 𝑑𝑠󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 + 𝑝𝜀

≤ 𝑝𝜀 + ∫^𝑡

0󵄩󵄩󵄩󵄩(𝑆(𝑡 + ℎ − 𝑠) − 𝑆(𝑡 − 𝑠))𝑓(𝑠,𝑥^𝑠, 𝑥 (𝑠))󵄩󵄩󵄩󵄩 𝑑𝑠 + ∫^𝑡+ℎ

𝑡 󵄩󵄩󵄩󵄩𝑆(𝑡 + ℎ − 𝑠)𝑓(𝑠,𝑥^𝑠, 𝑥 (𝑠))󵄩󵄩󵄩󵄩 𝑑𝑠

≤ 𝑝𝜀 + [𝑀ℎ ∫^𝑡

0𝜇 (𝑠) 𝑑𝑠 + 𝑁 ∫^𝑡+ℎ

𝑡 𝜇 (𝑠) 𝑑𝑠] ⋅ (1 + 𝜌) . (41) Asℎ → 0and𝜀 → 0, the right-hand side of the inequality above tends to zero independent of𝑥, soΛ₂maps bounded sets into equicontinuous sets.

For bounded set𝐵 ⊂ 𝑃𝐶(𝐽, 𝑋), we consider the map 𝜒_𝑝𝑐(𝐵) = max

𝑖=0,1,...,𝑝𝜒_𝑖(𝐵|_𝐽_𝑖) , (42) where𝜒_𝑖is the Hausdorff measure of noncompactness on the Banach space𝐶(𝐽_𝑖, 𝑋)and𝐵|_𝐽_𝑖is defined in (7).

Furthermore, we define the Hausdorff measure of non- compactness𝜒_FonF(𝑇)as follows:

𝜒_F(Y) := 𝜒_𝑝𝑐(Y|_{𝑃𝐶(𝐽,𝑋)}) + sup

𝑡∈[−𝑟,0]𝜒 (Y(𝑡)) , Y ⊂F(𝑇) . (43)

For every bounded subset Ω ⊂ 𝑃𝐶(𝐽, 𝑋), by applying̃ Proposition2, for any𝜀 > 0there exists a sequence{𝑥_𝑛}^∞_𝑛=1⊂ Ω̃such that

𝜒_𝑝𝑐(Λ₂Ω) ≤ 2𝜒̃ _𝑝𝑐(Λ₂{𝑥_𝑛}) + 𝜀, (44) noting that the definition of𝜒_𝑝𝑐, we have

𝜒_𝑝𝑐(Λ₂Ω) ≤ 2̃ max

𝑖=0,1,...,𝑝𝜒_𝑖(Λ₂{𝑥_𝑛} |_𝐽_𝑖) + 𝜀. (45) Then, noting the equicontinuity ofΛ₂|_𝐽_𝑖, 𝑖 = 0, 1, . . . , 𝑝, we can apply Lemmas 2.1 and 2.2 of [35] to obtain

𝜒_𝑖(Λ₂{𝑥_𝑛} |_𝐽_𝑖) =sup

𝑡∈𝐽_𝑖

𝜒 (Λ₂{𝑥_𝑛} (𝑡)) . (46)

Then from (45) and (46), we have

𝜒_𝑝𝑐(Λ₂Ω) ≤ 2̃ max

𝑖=0,1,...,𝑝(sup

𝑡∈𝐽𝑖

𝜒 (Λ₂{𝑥_𝑛} (𝑡))) + 𝜀 = 2sup

𝑡∈𝐽𝜒 (Λ₂{𝑥_𝑛} (𝑡)) + 𝜀.

(47)

For every bounded subsetΩ ⊂F(𝑇), we have 𝜒_F(Λ₂Ω) = 𝜒_𝑝𝑐((Λ₂Ω) |_{𝑃𝐶(𝐽,𝑋)})

+ sup

𝑡∈[−𝑟,0]𝜒 ((Λ₂Ω) (𝑡)) = 𝜒_𝑝𝑐((Λ₂Ω) |_{𝑃𝐶(𝐽,𝑋)}) , (48) moreover, by applying Proposition2, for any𝜀 > 0there exists a sequence{𝑦_𝑛}^∞_𝑛=1⊂ Ωsuch that

𝜒_F(Λ₂Ω) ≤ 2𝜒_F(Λ₂{𝑦_𝑛}) + 𝜀

= 2𝜒_𝑝𝑐((Λ₂{𝑦_𝑛}) |_{𝑃𝐶(𝐽,𝑋)}) + 𝜀. (49) Combining with (48) and (49), we have

𝜒_F(Λ₂Ω) = 𝜒_𝑝𝑐((Λ₂Ω) |_{𝑃𝐶(𝐽,𝑋)})

≤ 2𝜒_𝑝𝑐((Λ₂{𝑦_𝑛}) |_{𝑃𝐶(𝐽,𝑋)}) + 𝜀. (50)

Using the induction of (45)–(47) above, we can see 𝜒_F(Λ₂Ω) = 𝜒_𝑝𝑐((Λ₂Ω) |_{𝑃𝐶(𝐽,𝑋)})

≤ 2sup

𝑡∈𝐽𝜒 (Λ₂{𝑦_𝑛} (𝑡) |_𝑡∈𝐽) + 𝜀. (51)

(6)

Thus, from (51), (H2) and Proposition3and (3) in Lemma1, we can see

𝜒_F(Λ₂Ω) ≤ 2sup

𝑡∈𝐽𝜒 (Λ₂{𝑦_𝑛} (𝑡) |_𝑡∈𝐽) + 𝜀

= 2sup

𝑡∈𝐽 [𝜒 ({ ∫^𝑡

0𝑆 (𝑡−𝑠)𝑓 (𝑠, ̃𝑦𝑛𝑠, ̃𝑦_𝑛(𝑠)) 𝑑𝑠 + ∑

0<𝑡_𝑘<𝑡𝐶 (𝑡−𝑡_𝑘) 𝐼_𝑘( ̃𝑦_𝑛(𝑡_𝑘))})]+𝜀

≤ 2sup

𝑡∈𝐽 [2𝑁 ∫^𝑡

0𝜂 (𝑠) ( sup

𝜃∈[−𝑟,0]𝜒 ({ 𝑦_𝑛(𝑠 + 𝜃)}) +𝜒 ({ ̃𝑦_𝑛(𝑠)}) ) 𝑑𝑠] + 𝜀,

(52) where ̃𝑦_𝑛(𝑡) := 𝑦_𝑛(𝑡)|_𝑡∈𝐽.

Noting that

𝜃∈[−𝑟,0]sup 𝜒 ({ 𝑦_𝑛(𝑠 + 𝜃)}) ≤ sup

𝜃∈[−𝑟,0]𝜒 ({ 𝑦_𝑛(𝜃)}) +sup

𝑠∈𝐽𝜒 ({ ̃𝑦_𝑛(𝑠)})

≤ sup

𝜃∈[−𝑟,0]𝜒 (Ω (𝜃)) +sup

𝑠∈𝐽𝜒 (Ω (𝑠))

≤ sup

𝜃∈[−𝑟,0]𝜒 (Ω (𝜃)) + 𝜒_𝑝𝑐(Ω) = 𝜒_F(Ω) ,

(53)

𝜒 ({ ̃𝑦_𝑛(𝑠)}) ≤𝜒 (Ω (𝑠))

≤ 𝜒_𝑝𝑐(Ω) . (54)

Thus, by (52), we see 𝜒_F(Λ₂Ω) ≤ 2sup

𝑡∈𝐽 [2𝑁 ∫^𝑡

0𝜂 (𝑠) ( sup

𝜃∈[−𝑟,0]𝜒 ({ 𝑦_𝑛(𝑠 + 𝜃)}) +𝜒 ({ ̃𝑦_𝑛(𝑠)}) ) 𝑑𝑠] + 𝜀

≤ 2sup

𝑡∈𝐽 (4𝑁 ∫^𝑡

0𝜂 (𝑠) 𝑑𝑠) ⋅ 𝜒_F(Ω) + 𝜀

= 8𝑁 ∫^𝑇

0 𝜂 (𝑠) 𝑑𝑠 ⋅ 𝜒_F(Ω) + 𝜀.

(55) Since𝜀is arbitrary, we can obtain

𝜒_F(Λ₂Ω) ≤ 8𝑁 ∫^𝑇

0 𝜂 (𝑠) 𝑑𝑠 ⋅ 𝜒_F(Ω) . (56)

Combining with (H3), we have 𝜒_F(ΛΩ) ≤ 𝜒_F(Λ₁Ω) + 𝜒_F(Λ₂Ω)

≤ 8𝑁 ∫^𝑇

0 𝜂 (𝑠) 𝑑𝑠 ⋅ 𝜒F(Ω) < 𝑀^∗𝜒_F(Ω) , (57) henceΛ is a𝜒_F-contraction on F(𝑇). According to Theo- rem4, the operator Λhas at least one fixed point𝑥in𝐵_𝜌. This completes the proof.

Next, we establish a condition that guarantee that a mild solution satisfies (3).

Proposition 8. Assume that the hypotheses of Theorem7are fulfilled and that 𝜙(0) + 𝑔(𝑥)(0) ∈ 𝐷(𝐴). If𝑥(⋅) is a mild solution of (1)–(4), then condition(3)holds.

Proof. Clearly,(1/𝑡) ∫₀^𝑡𝑆(𝑡−𝑠)𝑓(𝑠, 𝑥_𝑠, 𝑥(𝑠))𝑑𝑠 → 0as𝑡 → 0.

Moreover, noting that𝜙(0)+𝑔(𝑥)(0) ∈ 𝐷(𝐴)and (11), we have 𝐶(⋅)(𝜙(0) + 𝑔(𝑥)(0))is of class𝐶¹. Therefore, we can see that

𝑡 → 0lim⁺

𝑥 (𝑡) − 𝑥 (0) 𝑡

= lim

𝑡 → 0⁺

1

𝑡 [ (𝐶 (𝑡) − 𝐼) (𝜙 (0) + 𝑔 (𝑥) (0)) + 𝑆 (𝑡) 𝜉 + ∫^𝑡

0𝑆 (𝑡 − 𝑠) 𝑓 (𝑠, 𝑥_𝑠, 𝑥 (𝑠)) 𝑑𝑠] = 𝜉,

(58)

which shows the assertion.

4. Application

In this section, we consider an application of the theory developed in Section3to the study of an impulsive partial differential equation with unbounded delay.

Example 9. 𝑋 = 𝐿²([0, 𝜋]),𝐴 : 𝐷(𝐴) ⊆ 𝑋 → 𝑋is the map defined by𝐴𝜗 = 𝜗^󸀠󸀠with domain𝐷(𝐴) = {𝜗 ∈ 𝑋 : 𝜗^󸀠󸀠 ∈ 𝑋, 𝜗(0) = 𝜗(𝜋) = 0}.

We consider the following integrodifferential model:

𝜕²

𝜕𝑡²𝑣 (𝑡, 𝜉) = 𝜕²

𝜕𝜉²𝑣 (𝑡, 𝜉) +sin󵄨󵄨󵄨󵄨𝑣(𝑡,𝜉)󵄨󵄨󵄨󵄨

+ 𝑡²∫^𝑡

𝑡−𝑟𝛾 (𝜃 − 𝑡) ⋅cos(󵄨󵄨󵄨󵄨𝑣(𝜃,𝜉)󵄨󵄨󵄨󵄨

𝑡 ) 𝑑𝜃, 𝑣 (𝑡, 𝜋) = 𝑣 (𝑡, 0) = 0, 𝑣 (𝜃, 𝜉) = 𝑣₀(𝜃, 𝜉) + ∫^𝜋

0 𝑐 (𝜉, 𝑠)sin(1 + 𝑣 (𝜃, 𝑠)) 𝑑𝑠,

− 𝑟 ≤ 𝜃 ≤ 0, 𝜕

𝜕𝑡𝑣 (0, 𝜉) = 𝜔 (𝜉) , Δ𝑣 (𝑡_𝑘, 𝜉) = ∫^𝜋

0 𝜌_𝑘(𝜉, 𝑦) 𝑑𝑦 ⋅cos²(𝑣 (𝑡_𝑘, 𝜉)) , 1 ≤ 𝑘 ≤ 𝑝, (59)

(7)

where𝑡 ∈ [0, 𝑇],𝑟 > 0,𝜉 ∈ [0, 𝜋],0 < 𝑡₁ < 𝑡₂< ⋅ ⋅ ⋅ < 𝑡_𝑝 < 𝑇, 𝜔 ∈ 𝑋and𝑣_𝑡(𝜃, 𝜉) = 𝑣(𝑡 + 𝜃, 𝜉).𝛾 : [−𝑟, 0] → R and𝑐(𝜉, 𝑠), 𝜌_𝑘(𝜉, 𝑧) ∈ 𝐿²([0, 𝜋] × [0, 𝜋],R)satisfy the following assumptions.

(1) The function𝛾 : [−𝑟, 0] → R is a continuous func- tion and∫_−𝑟⁰ |𝛾(𝜃)|𝑑𝜃 < ∞.

(2) The function 𝑐(𝜉, 𝑠), 𝜉, 𝑠 ∈ [0, 𝜋] is measurable and there exists a constant 𝑁₁ such that (𝜋 ∫₀^𝜋∫₀^𝜋𝑐²(𝜉, 𝑠)𝑑𝑠𝑑𝜉)^1/2≤ 𝑁₁.

(3) For every𝑘 = 1, 2, . . . , 𝑝, the function𝜌_𝑘(𝜉, 𝑧), 𝑧 ∈ [0, 𝜋], is measurable and there exists a constant𝑁 such that

(∫^𝜋

0 (∫^𝜋

0 𝜌_𝑘(𝜉, 𝑧)𝑑𝑧)²𝑑𝜉)

1/2

≤ 𝑁. (60)

To treat the above problem, we define 𝐷 (𝐴) = 𝐻²([0, 𝜋]) ∩ 𝐻₀¹([0, 𝜋]) ,

𝐴𝑢 = 𝑢^󸀠󸀠. (61)

𝐴 is the infinitesimal generator of a strongly continuous cosine function{𝐶(𝑡)}_𝑡∈R on𝑋. Moreover,𝐴has a discrete spectrum, the eigenvalues are −𝑛², 𝑛 ∈ N, with the corre- sponding normalized eigenvectors𝜔_𝑛(𝑥) = √(2/𝜋)sin(𝑛𝑥);

the set{𝜔_𝑛; 𝑛 ∈ N} is an orthonormal basis of𝑋 and the following properties hold.

(a) If𝜔 ∈ 𝐷(𝐴), then𝐴𝜔 = − ∑^∞_𝑛=1𝑛²⟨𝜔, 𝜔_𝑛⟩𝜔_𝑛.

(b) For each𝜔 ∈ 𝑋,𝐶(𝑡)𝜔 = ∑^∞_𝑛=1cos(𝑛𝑡)⟨𝜔, 𝜔_𝑛⟩𝜔_𝑛and 𝑆(𝑡)𝜔 = ∑^∞_𝑛=1(sin(𝑛𝑡)/𝑛)⟨𝜔, 𝜔_𝑛⟩𝜔_𝑛. Consequently,

‖𝐶(𝑡)‖ = ‖𝑆(𝑡)‖ ≤ 1for𝑡 ∈ R, and{𝑆(𝑡)}is compact for every𝑡 ∈R.

For𝜉 ∈ [0, 𝜋]and𝜑 ∈ 𝐶([−𝑟, 0], 𝑋), we set 𝑥 (𝑡) (𝜉) = 𝑣 (𝑡, 𝜉) ,

𝜙 (𝜃) (𝜉) = 𝑣0(𝜃, 𝜉) , 𝜃 ∈ [−𝑟, 0] , 𝑔 (𝜑 (𝜃)) (𝜉) = ∫^𝜋

0 𝑐 (𝜉, 𝑠)sin(1 + 𝜑 (𝜃) (𝑠)) 𝑑𝑠, 𝑓 (𝑡, 𝜑, 𝑥 (𝑡)) (𝜉) =sin󵄨󵄨󵄨󵄨𝑥(𝑡)(𝜉)󵄨󵄨󵄨󵄨

+ 𝑡²∫⁰

−𝑟𝛾 (𝜃) ⋅cos(󵄨󵄨󵄨󵄨𝜑(𝜃)(𝜉)󵄨󵄨󵄨󵄨

𝑡 ) 𝑑𝜃,

𝐼_𝑘(𝑥 (𝑡_𝑘)) (𝜉) = ∫^𝜋

0 𝜌_𝑘(𝜉, 𝑦) 𝑑𝑦 ⋅cos²(𝑥 (𝑡_𝑘) (𝜉)) .

(62) Then the above equation (59) can be reformulated as the abstract (1)–(4).

For𝑡 ∈ [0, 𝑇], we can see

󵄩󵄩󵄩󵄩𝑓(𝑡,𝜑,𝑥(𝑡))󵄩󵄩󵄩󵄩 ≤ ‖𝑥(𝑡)‖ + 𝑡²∫⁰

−𝑟󵄨󵄨󵄨󵄨𝛾(𝜃)󵄨󵄨󵄨󵄨𝑑𝜃

≤ 𝜇 (𝑡) (1 + ‖𝑥 (𝑡)‖) ,

(63)

where𝜇(𝑡) :=max{1, 𝑡²∫_−𝑟⁰ |𝛾(𝜃)|𝑑𝜃}.

For any𝑥₁, 𝑥₂∈ 𝑋,𝜑, ̃𝜑 ∈ 𝐶([−𝑟, 0], 𝑋),

󵄩󵄩󵄩󵄩𝑓(𝑡,𝜑,𝑥¹(𝑡)) (𝜉) − 𝑓 (𝑡, ̃𝜑, 𝑥₂(𝑡)) (𝜉)󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩𝑥¹(𝑡) − 𝑥₂(𝑡)󵄩󵄩󵄩󵄩 + 𝑡 ∫_−𝑟⁰ 󵄨󵄨󵄨󵄨𝛾(𝜃)󵄨󵄨󵄨󵄨󵄩󵄩󵄩󵄩𝜑(𝜃)(𝜉) − ̃𝜑(𝜃)(𝜉)󵄩󵄩󵄩󵄩𝑑𝜃.

(64) Therefore, for any bounded sets𝐷₁⊂ 𝐶([−𝑟, 0], 𝑋),𝐷₂⊂ 𝑋, we have

𝜒 (𝑓 (𝑡, 𝐷₁, 𝐷₂)) ≤ 𝜒 (𝐷₂) + 𝑡 ∫⁰

−𝑟󵄨󵄨󵄨󵄨𝛾(𝜃)󵄨󵄨󵄨󵄨𝜒(𝐷¹(𝜃)) 𝑑𝜃

≤ 𝜒 (𝐷₂) + 𝑡 sup

−𝑟≤𝜃≤0𝜒 (𝐷₁(𝜃)) ∫⁰

−𝑟󵄨󵄨󵄨󵄨𝛾(𝜃)󵄨󵄨󵄨󵄨𝑑𝜃

≤ 𝜂 (𝑡) ( sup

−𝑟≤𝜃≤0𝜒 (𝐷₁(𝜃)) + 𝜒 (𝐷₂)) , a.e.𝑡 ∈ [0, 𝑇] ,

(65) where𝜂(𝑡) :=max{1, 𝑡 ∫_−𝑟⁰ |𝛾(𝜃)|𝑑𝜃}.

For𝑥 ∈ 𝑋,

󵄩󵄩󵄩󵄩𝐼^𝑘(𝑥)󵄩󵄩󵄩󵄩 ≤ 𝑁 (1 + ‖𝑥‖) , 𝑘 = 1, 2, . . . , 𝑝. (66) Suppose further that there exists a constant𝑀̃^∗ ∈ (0, 1) such that8 ∫₀^𝑇𝜂(𝑠)𝑑𝑠 < ̃𝑀^∗and𝑝𝑁 < 1, then (59) has at least a mild solution by Theorem7.

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.

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1.Introduction FangLiandHuiwenWang AnExistenceResultforNonlocalImpulsiveSecond-OrderCauchyProblemswithFiniteDelay ResearchArticle

Research Article

An Existence Result for Nonlocal Impulsive Second-Order Cauchy Problems with Finite Delay

Fang Li and Huiwen Wang

1. Introduction

2. Preliminaries

3. Existence Result and Proof

4. Application

Acknowledgments

References

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Optimization

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