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

Abstract

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], Henríquez 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: 𝑑²𝑑𝑡²𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓𝑡,𝑥_𝑡(𝑡),(],𝑥𝑡∈0,𝑇,𝑡≠𝑡_𝑘𝑥(𝑡)(𝑥𝑡)(𝑡)[],𝑥,𝑘=1,2,…,𝑝,(1)=𝑔)(+𝜙,𝑡∈−𝑟,0(2)_′(0)𝑡=𝜉∈𝑋,(3)Δ𝑥_𝑘=𝐼_𝑘𝑥𝑡_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, 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 {𝑡_𝑘} are given such that 0=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 function 𝑥 at 𝑡_𝑘, which is defined by Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘), where 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) are, respectively, the right and the left limits of 𝑥 at 𝑡_𝑘.

For any continuous function 𝑥 defined on the interval [−𝑟,𝑇] and any 𝑡∈[0,𝑇], we denote by 𝑥_𝑡 the element of 𝐶([−𝑟,0],𝑋) defined by 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃) for 𝜃∈[−𝑟,0].

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.

2. Preliminaries

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 𝑃𝐶(𝐽,𝑋)∶=𝑥∶𝐽⟶𝑋;𝑥(𝑡)iscontinuousat𝑡≠𝑡_𝑘,leftcontinuousat𝑡=𝑡_𝑘,andtherightlimit𝑥𝑡⁺_𝑘existsfor𝑘=1,2,…,𝑝.(5)

It is easy to check that 𝑃𝐶(𝐽,𝑋) is a Banach space with the norm ‖𝑥‖_𝑃𝐶=sup_𝑡∈𝐽‖𝑥(𝑡)‖,forany𝑥∈𝑃𝐶(𝐽,𝑋).(6)

We let 𝐽₀=(𝑡₀,𝑡₁], 𝐽₁=(𝑡₁,𝑡₂],…,𝐽_𝑝=(𝑡_𝑝,𝑡_𝑝+1].

For 𝔅⊆𝑃𝐶(𝐽,𝑋), we denote by 𝔅|_𝐽𝑖 the set 𝔅|_𝐽𝑖=𝑥∈𝐶𝑡_𝑖,𝑡_𝑖+1,𝑋;𝑥𝑡_𝑖=𝑣𝑡⁺_𝑖,𝑥(𝑡)=𝑣(𝑡),𝑡∈𝐽_𝑖,𝑣∈𝔅(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; (ii)𝐶(𝑡+𝑠)+𝐶(𝑡−𝑠)=2𝐶(𝑡)𝐶(𝑠) for all 𝑠,𝑡∈𝐑; (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 𝑆(𝑡)𝑥=^𝑡₀𝐶(𝑠)𝑥𝑑𝑠,for𝑥∈𝑋,𝑡∈𝐑.(8)

One can define the infinitesimal generator 𝐴 of 𝐶(⋅) by 𝐷(𝐴)=𝑥∈𝑋;lim_𝑡→02𝑡⁻²(𝐶(𝑡)𝑥−𝑥)∈𝑋𝐴𝑥=lim_𝑡→02𝑡⁻²(𝐶(𝑡)𝑥−𝑥),𝑥∈𝐷(𝐴).(9)

In this paper, we assume there exist positive constants 𝑀 and 𝑁 such that ‖𝐶(𝑡)‖≤𝑀,‖𝑆(𝑡)‖≤𝑁forevery𝑡∈𝐽.(10) The following properties are well known [6, 7, 11, 12]: (𝑡)(𝐴)(𝑡)(𝑡)𝑥(𝐴)(𝑡)(𝐴)(𝑡)(𝑡)𝑥(𝐴)(i)𝐶𝑥∈𝐷,𝐶𝐴𝑥=𝐴𝐶for𝑥∈𝐷,𝑡∈𝐑;(ii)𝑆𝑥∈𝐷,𝑆𝐴𝑥=𝐴𝑆for𝑥∈𝐷,𝑡∈𝐑;(iii)^𝑡₀𝑆(𝑠)(𝐴),𝐴𝑥𝑑𝑠∈𝐷_𝑡0𝑆(𝑠)(𝑡)(𝑡)𝑥𝑑𝑠=𝐶𝑥−𝑥for𝑥∈𝑋,𝑡∈𝐑;(iv)𝐶𝑥−𝑥=^𝑡₀𝑆(𝑠)𝐴𝑥𝑑𝑠for𝑥∈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;Ωhasafinite𝜀-netin𝑋}.(12)

This measure of noncompactness satisfies some basic properties as follows.

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; (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}; (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 {𝑥_𝑛}^∞_𝑛=1 in Ω such that 𝜒(Ω)≤2𝜒𝑥_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=1 be a sequence in 𝐿¹(𝐽,𝑋) such that there exist 𝑣,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. Then, for every 𝑡∈𝐽, we have 𝜒^𝑡₀𝑆(𝑡−𝑠)𝑓_𝑛(𝑠)𝑑𝑠^∞_𝑛=1≤2𝑁^𝑡₀𝑞(𝑠)𝑑𝑠,(14) where 𝑁 is from (10).

A continuous map 𝑄∶𝑊⊆𝑌→𝑌 is said to be a 𝜒-contraction 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 𝜒-contraction, 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 𝑥(𝑡)(𝑡)𝜙(0)(𝑥0)(𝑡)𝜉+=𝐶+𝑔)(+𝑆_𝑡0𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘,𝑡∈𝐽.(15)

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(),𝑥∈𝑃𝐶𝐽,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→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⊂𝑋𝜒𝑓𝑡,𝐷₁,𝐷₂≤𝜂(𝑡)sup_{[]𝜃∈−𝑟,0}𝜒𝐷₁(𝜃)+𝜒𝐷₂,a.e.𝑡∈𝐽.(19)  (H2) 𝐼_𝑘∶𝑋→𝑋 are compact operators and there exist positive constants 𝑀₁,𝑀₂ such that 𝐼_𝑘(𝑥)≤𝑀₁‖𝑥‖+𝑀₂,forany𝑥∈𝑋,𝑘=1,2,…,𝑝.(20)  (H3) 𝑔∶𝐶([−𝑟,0],𝑋)→𝑋 is a compact operator and there exists a constant 𝑁₁>0 such that 𝑔(𝑥)_[−𝑟,0]≤𝑁₁forall𝑥∈𝐶([−𝑟,0],𝑋),(21) where ‖𝑔(𝑥)‖_[−𝑟,0]=sup_{𝑡∈[−𝑟,0]}‖𝑔(𝑥)(𝑡)‖.  (H4) There exists 𝑀^∗∈(0,1) such that 8𝑁∫^𝑇₀𝜂(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 operator Λ∶𝔉(𝑇)→𝔉(𝑇) in the following way: (𝑡)=𝑔(𝑥𝑡)(𝑡)[],𝐶(𝑡)𝑔(𝑥0)(0)(𝑡)𝜉Λ𝑥)()(+𝜙,𝑡∈−𝑟,0)(+𝜙+𝑆+∫^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠+∑_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘,𝑡∈𝐽.(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=Λ₁+Λ₂, where the operators Λ₁,Λ₂ are defined as follows: Λ₁𝑥𝑡)=𝑔(𝑥𝑡)(𝑡)[],𝐶(𝑡)𝑔(𝑥0)(0)(𝑡)Λ()(+𝜙,𝑡∈−𝑟,0)(+𝜙+𝑆𝜉,𝑡∈𝐽,₂𝑥(𝑡)=[],∫0,𝑡∈−𝑟,0_𝑡0𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠+∑_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘,𝑡∈𝐽.(23)
Obviously, under the assumptions of 𝑔, Λ₁ is continuous. For 𝑡∈𝐽, we can prove that Λ₂ is continuous.
Indeed, let {𝑥^𝑛}_𝑛∈𝐍 be a sequence such that 𝑥^𝑛→𝑥 in 𝔉(𝑇) as 𝑛→∞. Since 𝑓 satisfies (H1)(i), for almost every 𝑡∈𝐽, we get 𝑓𝑡,𝑥^𝑛_𝑡,𝑥^𝑛(𝑡)→𝑓𝑡,𝑥_𝑡,𝑥(𝑡),as𝑛→∞.(24)
Noting that 𝑥^𝑛→𝑥 in 𝔉(𝑇), we can see that there exists 𝜀>0 such that ‖𝑥^𝑛−𝑥‖_𝔉(𝑇)≤𝜀 for 𝑛 sufficiently large. Therefore, we have 𝑓𝑡,𝑥^𝑛_𝑡,𝑥^𝑛(𝑡)−𝑓𝑡,𝑥_𝑡(𝑡(𝑡)𝑥,𝑥)≤𝜇1+^𝑛(𝑡(𝑡‖𝑥(𝑡(𝑡)(𝑡𝑥)+𝜇)(1+)‖)≤2𝜇+𝜇)^𝑛(𝑡)(𝑡(𝑡𝑥(𝑡(𝑡)(𝑡)(𝑡𝑥‖−𝑥)+2𝜇)‖)‖≤2𝜇+𝜇𝜀+2𝜇)‖_𝔉(𝑇).(25) It follows from the Lebesgue’s dominated convergence theorem that ^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,𝑥^𝑛_𝑠,𝑥^𝑛(𝑠)−𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠≤𝑁^𝑡₀𝑓𝑠,𝑥^𝑛_𝑠,𝑥^𝑛(𝑠)−𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠→0,as𝑛→∞.(26) Moreover, noting that (H2), we obtain that lim_𝑛→∞Λ₂𝑥^𝑛−Λ₂𝑥_𝔉(𝑇)=0.(27) This shows that Λ₂ is continuous. Therefore, Λ is continuous.
Let us introduce in the space 𝔉(𝑇) the equivalent norm defined as ‖𝑥‖_∗=sup_{[]𝑡∈−𝑟,0}‖𝑥(𝑡)‖+sup_𝑡∈𝐽𝑒^−𝐿𝑡‖𝑥(𝑡)‖,(28) where 𝐿>0 is a constant chosen so that 𝑁sup_𝑡∈𝐽^𝑡₀𝜇(𝑠)𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠<1.(29) Noting that for any 𝜓∈𝐿¹(𝐽,𝑋), we have lim_𝐿→+∞sup_𝑡∈𝐽^𝑡₀𝑒^{−𝐿(𝑡−𝑠)}𝜓(𝑠)𝑑𝑠=0,(30) so, we can take the appropriate 𝐿 to satisfy (29).
Consider the set 𝐵_𝜌=𝑥∈𝔉(𝑇);‖𝑥‖_∗≤𝜌,(31) where 𝜌 is a constant chosen so that 𝑁𝜌≥¹+𝜙_[−𝑟,0]+ℓ+𝑝𝑀𝑀₂1−𝑝𝑀𝑀₁>0,(32) where ℓ∶=𝑀(𝑁₁+‖𝜙(0)‖)+𝑁(‖𝜉‖+‖𝜇‖_𝐿1) and ‖𝜙‖_[−𝑟,0]=sup_{𝑡∈[−𝑟,0]}‖𝜙(𝑡)‖.
Now, if 𝑡∈[−𝑟,0], 𝑥∈𝐵_𝜌, then ‖(Λ𝑥)(𝑡)‖=𝑔(𝑥)(𝑡)+𝜙(𝑡)≤𝑁₁+𝜙_[−𝑟,0].(33) For 𝑡∈𝐽, 𝑥∈𝐵_𝜌, we have 𝑡≤𝐶(𝑡)𝑔(𝑥0)(0)+𝑆(𝑡)𝜉+‖(Λ𝑥)()‖)(+𝜙_𝑡0𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘≤𝑀𝑁₁+𝜙(0)+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+𝑁𝜉+^𝑡₀𝜇(𝑠)1+𝑒^𝐿𝑠𝑒^−𝐿𝑠‖𝑥(𝑠)‖𝑑𝑠=ℓ+𝑀_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+𝑁^𝑡₀𝜇(𝑠)𝑒^𝐿𝑠𝑒^−𝐿𝑠‖𝑥(𝑠)‖𝑑𝑠,(34) then 𝑒^−𝐿𝑡‖(Λ𝑥)(𝑡)‖≤𝑒^−𝐿𝑡ℓ+𝑀_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥𝑡_𝑘+𝑁^𝑡₀𝜇(𝑠)𝑒^𝐿𝑠𝑒^−𝐿𝑠‖𝑥(𝑠)‖𝑑𝑠≤ℓ+𝜌𝑝𝑀𝑀₁+𝑝𝑀𝑀₂+𝑁^𝑡₀𝜇(𝑠)𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠⋅‖𝑥‖_∗,(35) therefore, sup_𝑡∈𝐽𝑒^−𝐿𝑡‖(Λ𝑥)(𝑡)‖≤ℓ+𝑝𝑀𝑀₂+𝑝𝑀𝑀₁+sup_𝑡∈𝐽𝑁^𝑡₀𝜇(𝑠)𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠𝜌.(36) It results that ‖Λ𝑥‖_∗=sup_{[]𝑡∈−𝑟,0}‖(Λ𝑥)(𝑡)‖+sup_𝑡∈𝐽𝑒^−𝐿𝑡‖(Λ𝑥)(𝑡)‖≤𝑁₁+𝜙_[−𝑟,0]+ℓ+𝑝𝑀𝑀₂+𝑝𝑀𝑀₁+𝑁sup_𝑡∈𝐽^𝑡₀𝜇(𝑠)𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠𝜌.(37)
Let 𝐿→+∞, we obtain ‖Λ𝑥‖_∗≤𝑁₁+𝜙_[−𝑟,0]+ℓ+𝑝𝑀𝑀₂+𝑝𝑀𝑀₁𝜌≤𝜌.(38) Hence for some positive number 𝜌, Λ𝐵_𝜌⊂𝐵_𝜌.
Using the strong continuity of {𝐶(𝑡)}_𝑡∈𝐑 and the compactness condition on the operators 𝐼_𝑘, for 𝜀>0, there exists 𝛿>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. If 𝑡∈[𝑡_𝑘,𝑡_𝑘+1] and ℎ<𝛿 such that 𝑡+ℎ∈[𝑡_𝑘,𝑡_𝑘+1], then _{0<𝑡𝑘<𝑡}𝐶𝑡+ℎ−𝑡_𝑘−𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘≤^𝑝_𝑘=1𝐶𝑡+ℎ−𝑡_𝑘−𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑥𝑡_𝑘≤𝑝𝜀.(40)
For 𝑥∈𝐵_𝜌, by the hypothesis (H1)(i) and (40), we get Λ₂𝑥(𝑡+ℎ)−Λ₂𝑥(𝑡≤_0𝑡+ℎ)𝑆(𝑡+ℎ−𝑠)𝑓𝑠,𝑥_𝑠(𝑠)−,𝑥𝑑𝑠_𝑡0𝑆(𝑡−𝑠)𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠+𝑝𝜀≤𝑝𝜀+^𝑡₀(𝑆(𝑡+ℎ−𝑠)−𝑆(𝑡−𝑠))𝑓𝑠,𝑥_𝑠(𝑠)+,𝑥𝑑𝑠_𝑡𝑡+ℎ𝑆(𝑡+ℎ−𝑠)𝑓𝑠,𝑥_𝑠,𝑥(𝑠)𝑑𝑠≤𝑝𝜀+𝑀ℎ^𝑡₀𝜇(𝑠)𝑑𝑠+𝑁_𝑡^𝑡+ℎ𝜇(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 Λ₂ maps bounded sets into equicontinuous sets.
For bounded set 𝐵⊂𝑃𝐶(𝐽,𝑋), we consider the map 𝜒_𝑝𝑐(𝐵)=max_{𝑖=0,1,…,𝑝}𝜒_𝑖𝐵|_𝐽𝑖,(42) where 𝜒_𝑖 is the Hausdorff measure of noncompactness on the Banach space 𝐶(𝐽_𝑖,𝑋) and 𝐵|_𝐽𝑖 is defined in (7).
Furthermore, we define the Hausdorff measure of noncompactness 𝜒_𝔉 on 𝔉(𝑇) as follows: 𝜒_𝔉(𝕐)∶=𝜒_𝑝𝑐𝕐|_{𝑃𝐶(𝐽,𝑋)}+sup_{[]𝑡∈−𝑟,0}𝜒(𝕐(𝑡)),𝕐⊂𝔉(𝑇).(43)
For every bounded subset Ω⊂𝑃𝐶(𝐽,𝑋), by applying Proposition 2, for any 𝜀>0 there exists a sequence {𝑥_𝑛}^∞_𝑛=1⊂Ω such that 𝜒_𝑝𝑐Λ₂Ω≤2𝜒_𝑝𝑐Λ₂𝑥_𝑛+𝜀,(44) noting that the definition of 𝜒_𝑝𝑐, we have 𝜒_𝑝𝑐Λ₂Ω≤2max_{𝑖=0,1,…,𝑝}𝜒_𝑖Λ₂𝑥_𝑛|_𝐽𝑖+𝜀.(45) Then, noting the equicontinuity of Λ₂|_𝐽𝑖,  𝑖=0,1,…,𝑝, we can apply Lemmas 2.1 and 2.2 of [35] to obtain 𝜒_𝑖Λ₂𝑥_𝑛|_𝐽𝑖=sup_{𝑡∈𝐽𝑖}𝜒Λ₂𝑥_𝑛(𝑡).(46) Then from (45) and (46), we have 𝜒_𝑝𝑐Λ₂Ω≤2max_{𝑖=0,1,…,𝑝}sup_{𝑡∈𝐽𝑖}𝜒Λ₂𝑥_𝑛(𝑡)+𝜀=2sup_𝑡∈𝐽𝜒Λ₂𝑥_𝑛(𝑡)+𝜀.(47)
For every bounded subset Ω⊂𝔉(𝑇), we have 𝜒_𝔉Λ₂Ω=𝜒_𝑝𝑐Λ₂Ω|_{𝑃𝐶(𝐽,𝑋)}+sup_{[]𝑡∈−𝑟,0}𝜒Λ₂Ω(𝑡)=𝜒_𝑝𝑐Λ₂Ω|_{𝑃𝐶(𝐽,𝑋)},(48) moreover, by applying Proposition 2, for any 𝜀>0 there exists a sequence {𝑦_𝑛}^∞_𝑛=1⊂Ω such that 𝜒_𝔉Λ₂Ω≤2𝜒_𝔉Λ₂𝑦_𝑛+𝜀=2𝜒_𝑝𝑐Λ₂𝑦_𝑛|_{𝑃𝐶(𝐽,𝑋)}+𝜀.(49) Combining with (48) and (49), we have 𝜒_𝔉Λ₂Ω=𝜒_𝑝𝑐Λ₂Ω|_{𝑃𝐶(𝐽,𝑋)}≤2𝜒_𝑝𝑐Λ₂𝑦_𝑛|_{𝑃𝐶(𝐽,𝑋)}+𝜀.(50) Using the induction of (45)–(47) above, we can see 𝜒_𝔉Λ₂Ω=𝜒_𝑝𝑐Λ₂Ω|_{𝑃𝐶(𝐽,𝑋)}≤2sup_𝑡∈𝐽𝜒Λ₂𝑦_𝑛(𝑡)|_𝑡∈𝐽+𝜀.(51) Thus, from (51), (H2) and Proposition 3 and (3) in Lemma 1, we can see 𝜒_𝔉Λ₂Ω≤2sup_𝑡∈𝐽𝜒Λ₂𝑦_𝑛(𝑡)|_𝑡∈𝐽+𝜀=2sup_𝑡∈𝐽𝜒^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,𝑦_𝑛𝑠,𝑦_𝑛(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐶𝑡−𝑡_𝑘𝐼_𝑘𝑦_𝑛𝑡_𝑘+𝜀≤2sup_𝑡∈𝐽2𝑁^𝑡₀𝜂(𝑠)sup_{[]𝜃∈−𝑟,0}𝜒𝑦_𝑛(𝑦𝑠+𝜃)+𝜒_𝑛(𝑠)𝑑𝑠+𝜀,(52) where 𝑦_𝑛(𝑡)∶=𝑦_𝑛(𝑡)|_𝑡∈𝐽.
Noting that sup_{[]𝜃∈−𝑟,0}𝜒𝑦_𝑛(𝑠+𝜃)≤sup_{[]𝜃∈−𝑟,0}𝜒𝑦_𝑛(𝜃)+sup_𝑠∈𝐽𝜒𝑦_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),𝜒𝑦(53)_𝑛(𝑠)≤𝜒(Ω(𝑠))≤𝜒_𝑝𝑐(Ω).(54) Thus, by (52), we see 𝜒_𝔉Λ₂Ω≤2sup_𝑡∈𝐽2𝑁^𝑡₀𝜂(𝑠)sup_{[]𝜃∈−𝑟,0}𝜒𝑦_𝑛(𝑦𝑠+𝜃)+𝜒_𝑛(𝑠)𝑑𝑠+𝜀≤2sup_𝑡∈𝐽4𝑁^𝑡₀𝜂(𝑠)𝑑𝑠⋅𝜒_𝔉(Ω)+𝜀=8𝑁^𝑇₀𝜂(𝑠)𝑑𝑠⋅𝜒_𝔉(Ω)+𝜀.(55)
Since 𝜀 is arbitrary, we can obtain 𝜒_𝔉Λ₂Ω≤8𝑁^𝑇₀𝜂(𝑠)𝑑𝑠⋅𝜒_𝔉(Ω).(56)
Combining with (H3), we have 𝜒_𝔉(ΛΩ)≤𝜒_𝔉Λ₁Ω+𝜒_𝔉Λ₂Ω≤8𝑁^𝑇₀𝜂(𝑠)𝑑𝑠⋅𝜒_𝔉(Ω)<𝑀^∗𝜒_𝔉(Ω),(57) hence Λ is a 𝜒_𝔉-contraction on 𝔉(𝑇). According to Theorem 4, the operator Λ has at least one fixed point 𝑥 in 𝐵_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 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.

Proof. Clearly, (1/𝑡)∫^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,𝑥_𝑠,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_𝑡→0+𝑥(𝑡)−𝑥(0)𝑡=lim_𝑡→0+1𝑡𝐶(𝑡)𝜙(0)(𝑥0(𝑡)𝜉+(−𝐼)+𝑔)()+𝑆_𝑡0𝑆(𝑡−𝑠)𝑓𝑠,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 Section 3 to the study of an impulsive partial differential equation with unbounded delay.

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𝜕𝑡²𝑣()=𝜕𝑡,𝜉²𝜕𝜉²𝑣(𝑡,𝜉)+sin𝑣(𝑡,𝜉)+𝑡^2𝑡_𝑡−𝑟𝛾()𝑣(𝜃−𝑡⋅cos𝜃,𝜉)𝑡𝑣()()𝑣()𝑑𝜃,𝑡,𝜋=𝑣𝑡,0=0,𝜃,𝜉=𝑣₀(𝜃,𝜉)+^𝜋₀𝑐()((𝜕𝜉,𝑠sin1+𝑣𝜃,𝑠))𝑑𝑠,−𝑟≤𝜃≤0,𝑣()(𝜉),𝑡𝜕𝑡0,𝜉=𝜔Δ𝑣_𝑘,𝜉=^𝜋₀𝜌_𝑘𝜉,𝑦𝑑𝑦⋅cos²𝑣𝑡_𝑘,𝜉,1≤𝑘≤𝑝,(59) where 𝑡∈[0,𝑇], 𝑟>0, 𝜉∈[0,𝜋], 0<𝑡₁<𝑡₂<⋯<𝑡_𝑝<𝑇, 𝜔∈𝑋 and 𝑣_𝑡(𝜃,𝜉)=𝑣(𝑡+𝜃,𝜉). 𝛾∶[−𝑟,0]→𝐑 and 𝑐(𝜉,𝑠),𝜌_𝑘(𝜉,𝑧)∈𝐿²([0,𝜋]×[0,𝜋],𝐑) satisfy the following assumptions. (1)The function 𝛾∶[−𝑟,0]→𝐑 is a continuous function and ∫⁰_−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∫^𝜋₀∫^𝜋₀𝑐²(𝜉,𝑠)𝑑𝑠𝑑𝜉)^1/2≤𝑁₁. (3)For every 𝑘=1,2,…,𝑝, the function 𝜌_𝑘(𝜉,𝑧),𝑧∈[0,𝜋], is measurable and there exists a constant 𝑁 such that ^𝜋₀^𝜋₀𝜌_𝑘(𝜉,𝑧)𝑑𝑧²𝑑𝜉^1/2≤𝑁.(60)
To treat the above problem, we define 𝐷(𝐴)=𝐻²([0,𝜋])∩𝐻¹₀([0,𝜋]),𝐴𝑢=𝑢^′′.(61)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 𝜔_𝑛(𝑥)=(2/𝜋)sin(𝑛𝑥); the set {𝜔_𝑛;𝑛∈𝐍} is an orthonormal basis of 𝑋 and the following properties hold. (a)If 𝜔∈𝐷(𝐴), then 𝐴𝜔=−∑^∞_𝑛=1𝑛²⟨𝜔,𝜔_𝑛⟩𝜔_𝑛. (b)For each 𝜔∈𝑋, 𝐶(𝑡)𝜔=∑^∞_𝑛=1cos(𝑛𝑡)⟨𝜔,𝜔_𝑛⟩𝜔_𝑛 and 𝑆(𝑡)𝜔=∑^∞_𝑛=1(sin(𝑛𝑡)/𝑛)⟨𝜔,𝜔_𝑛⟩𝜔_𝑛. Consequently, ‖𝐶(𝑡)‖=‖𝑆(𝑡)‖≤1 for 𝑡∈𝐑, and {𝑆(𝑡)} is compact for every 𝑡∈𝐑.
For 𝜉∈[0,𝜋] and 𝜑∈𝐶([−𝑟,0],𝑋), we set 𝑥(𝑡𝜉)(),𝜙(𝜃𝜉))(=𝑣𝑡,𝜉)(=𝑣₀()[],𝑔𝜑(𝜃)(𝜉)=𝜃,𝜉,𝜃∈−𝑟,0_𝜋0𝑐()(𝜃𝑠)𝑓(𝑡)(𝜉)𝑥(𝑡𝜉𝜉,𝑠sin1+𝜑)(𝑑𝑠,𝑡,𝜑,𝑥=sin)()+𝑡²⁰_−𝑟𝛾(𝜃)𝜑(𝜃𝜉⋅cos)()𝑡𝐼𝑑𝜃,_𝑘𝑥𝑡_𝑘(𝜉)=^𝜋₀𝜌_𝑘𝜉,𝑦𝑑𝑦⋅cos²𝑥𝑡_𝑘(𝜉).(62) Then the above equation (59) can be reformulated as the abstract (1)–(4).
For 𝑡∈[0,𝑇], we can see 𝑓𝑡,𝜑,𝑥(𝑡)≤‖𝑥(𝑡)‖+𝑡²⁰_−𝑟𝛾(𝜃(𝑡‖𝑥(𝑡,)𝑑𝜃≤𝜇)(1+)‖)(63) where 𝜇(𝑡)∶=max{1,𝑡²∫⁰_−𝑟|𝛾(𝜃)|𝑑𝜃}.
For any 𝑥₁,𝑥₂∈𝑋,  𝜑,𝜑∈𝐶([−𝑟,0],𝑋), 𝑓𝑡,𝜑,𝑥₁(𝑡)(𝜉)−𝑓𝑡,𝜑,𝑥₂(𝑡)(𝜉≤𝑥₁)(𝑡)−𝑥₂(𝑡)+𝑡⁰_−𝑟𝛾(𝜃)𝜑(𝜃)(𝜉)−𝜑(𝜃)(𝜉)𝑑𝜃.(64)
Therefore, for any bounded sets 𝐷₁⊂𝐶([−𝑟,0],𝑋), 𝐷₂⊂𝑋, we have 𝜒𝑓𝑡,𝐷₁,𝐷₂≤𝜒𝐷₂+𝑡⁰_−𝑟𝛾(𝜃)𝜒𝐷₁(𝜃)𝐷𝑑𝜃≤𝜒₂+𝑡sup_{−𝑟≤𝜃≤0}𝜒𝐷₁(𝜃)⁰_−𝑟𝛾(𝜃(𝑡))𝑑𝜃≤𝜂sup_{−𝑟≤𝜃≤0}𝜒𝐷₁(𝜃)+𝜒𝐷₂,a.e.𝑡∈[0,𝑇],(65) where 𝜂(𝑡)∶=max{1,𝑡∫⁰_−𝑟|𝛾(𝜃)|𝑑𝜃}.
For 𝑥∈𝑋, 𝐼_𝑘(𝑥)≤𝑁(1+‖𝑥‖),𝑘=1,2,…,𝑝.(66)
Suppose further that there exists a constant 𝑀^∗∈(0,1) such that 8∫^𝑇₀𝜂(𝑠)𝑑𝑠<𝑀^∗ and 𝑝𝑁<1, then (59) has at least a mild solution by Theorem 7.

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.