Abstract Semilinear Evolution Equations with Convex-Power Condensing Operators

Abstract

By using the techniques of convex-power condensing operators and fixed point theorems, we investigate the existence of mild solutions to nonlocal impulsive semilinear differential equations. Two examples are also given to illustrate our main results.

1. Introduction

This paper is concerned with the existence of mild solutions for the following impulsive semilinear differential equations with nonlocal conditions 𝑢^′(𝑡)=𝐴𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,𝑇],𝑡≠𝑡_𝑖,𝑢(0)(𝑢),𝑡=𝑔Δ𝑢_𝑖=𝐼_𝑖𝑢𝑡_𝑖,𝑖=1,2,…,𝑝,0<𝑡₁<𝑡₂<⋯<𝑡_𝑝<𝑇, (1) where 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 is the infinitesimal generator of strongly continuous semigroup 𝑆(𝑡) for 𝑡>0 in a real Banach space 𝑋andΔ𝑢(𝑡_𝑖)=𝑢(𝑡⁺_𝑖)−𝑢(𝑡⁻_𝑖) constitutes an impulsive condition. 𝑓 and 𝑔 are 𝑋-valued functions to be given later.

As far as we know, the first paper dealing with abstract nonlocal initial value problems for semilinear differential equations is due to [1]. Because nonlocal conditions have better effect in the applications than the classical initial ones, many authors have studied the following type of semilinear differential equations under various conditions on 𝑆(𝑡),𝑓, and 𝑔: 𝑢^′(𝑡)(𝑡)((𝑡[],𝑢(0)(𝑢).=𝐴𝑢+𝑓𝑡,𝑢)),𝑡∈0,𝑇=𝑔 (2)

For instance, Byszewski and Lakshmikantham [2] proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when 𝑓 and 𝑔 satisfy Lipschitz type conditions. In [3], Ntouyas and Tsamatos studied the case with compactness conditions. Byszewski and Akca [4] established the existence of solution to functionaldifferential equation when the semigroup is compact and 𝑔 is convex and compact on a given ball. Subsequently, Benchohra and Ntouyas [5] discussed the second-order differential equation under compact conditions. The fully nonlinear case was considered by Aizicovici and McKibben [6], Aizicovici and Lee [7], Aizicovici and Staicu [8], García-Falset [9], Paicu and Vrabie [10], Obukhovski and Zecca [11], and Xue [12, 13].

Recently, the theory of impulsive differential inclusions has become an important object of investigation because of its wide applicability in biology, medicine, mechanics, and control theory and in more and more fields. Cardinali and Rubbioni [14] studied the multivalued impulsive semilinear differential equation by means of the Hausdorff measure of noncompactness. Liang et al. [15] investigated the nonlocal impulsive problems under the assumptions that 𝑔 is compact, Lipschitz, and 𝑔 is not compact and not Lipschitz, respectively. All these studies are motivated by the practical interests of nonlocal impulsive Cauchy problems. For a more detailed bibliography and exposition on this subject, we refer to [14–18].

The present paper is motivated by the following facts. Firstly, the approach used in [9, 12, 13, 19, 20] relies on the assumption that the coefficient 𝑙 of the function 𝑓 about the measure of noncompactness satisfies a strong inequality, which is difficult to be verified in applications. Secondly, in [21], it seems that authors have considered the inequality restriction on coefficient function 𝑙(𝑡) of 𝑓 may be relaxed for impulsive nonlocal differential equations. However, in fact, they only solve the classical initial value problems 𝑢(0)=𝑢₀ rather than the nonlocal initial problems 𝑢(0)=𝑢₀+𝑔(𝑢). For more details, one can refer to the proof of Theorem 3.1 in [21] (see the inequalities (3.3) and (3.4) in page 5 and the estimations of the measure of noncompactness in page 6 and page 7 of [21]).

Therefore, we will continue to discuss the impulsive nonlocal differential equations under more general assumptions. Throughout this work, we mainly use the property of convex-power condensing operators and fixed point theorems to obtain the main result (Theorem 10). Indeed, the fixed point theorem about the convex-power condensing operators is an extension for Darbo-Sadovskii’s fixed point theorem. But the former seems more effective than the latter at times for some problems. For example, in [22] we ever applied the former to study the nonlocal Cauchy problem and obtained more general and interesting existence results. Based on the results obtained, we discuss the impulsive nonlocal differential equations. Fortunately, applying the techniques of convex-power condensing operators and fixed point theorems solves the difficulty involved by coefficient restriction that is, the constraint condition for the coefficient function 𝑙(𝑡) of 𝑓 is unnecessary (see Theorem 10). Therefore, our results generalize and improve many previous ones in this field, such as [9, 12, 13, 19, 20].

The outline of this paper is as follows. In Section 2, we recall some concepts and facts about the measure of noncompactness, fixed point theorems, and impulsive semilinear differential equations. In Section 3, we obtain the existence results of (1) when 𝑔 is compact in 𝑃𝐶([0,𝑇];𝑋). In Section 4, we discuss the existence result of (1) when 𝑔 is Lipschitz continuous, while Section 5 contains two illustrating examples.

2. Preliminaries

Let 𝐸 be a real Banach space, we introduce the Hausdorff measure of noncompactness 𝛼 defined on each bounded subset Ω of 𝐸 by 𝛼(Ω)𝑥=inf𝑟>0;therearefinitepoints₁,𝑥₂,…,𝑥_𝑛∈𝐸withΩ⊂^𝑛_𝑖=1𝐵𝑥_𝑖,𝑟. (3)

Now we recall some basic properties of the Hausdorff measure of noncompactness.

Lemma 1. For all bounded subsets Ω,Ω₁, andΩ₂ of 𝐸, the following properties are satisfied:(1)Ω is precompact if and only if 𝛼(Ω)=0;(2)𝛼(Ω)=𝛼(Ω)=𝛼(coΩ), where Ω and coΩ mean the closure and convex hull of Ω, respectively;(3)𝛼(Ω₁)≤𝛼(Ω₂) when Ω₁⊂Ω₂;(4)𝛼(Ω₁⋃Ω₂)≤max{𝛼(Ω₁),𝛼(Ω₂)}; (5)𝛼(𝜆Ω)=|𝜆|𝛼(Ω), for any 𝜆∈𝑅;(6)𝛼(Ω₁+Ω₂)≤𝛼(Ω₁)+𝛼(Ω₂), where Ω₁+Ω₂={𝑥+𝑦;𝑥∈Ω₁,𝑦∈Ω₂};(7)if {𝑊_𝑛}^+∞_𝑛=1 is a decreasing sequence of nonempty bounded closed subsets of 𝐸 and lim_𝑛→∞𝛼(𝑊_𝑛)=0, then ∩^+∞_𝑛=1𝑊_𝑛 is nonempty and compact in 𝐸.

The map 𝑄∶𝐷⊂𝐸→𝐸 is said to be 𝛼-condensing if for every bounded and not relatively compact 𝐵⊂𝐷, we have 𝛼(𝑄𝐵)<𝛼(𝐵) (see [23]).

Lemma 2 (see  [9]: Darbo-Sadovskii). If 𝐷⊂𝐸 is bounded closed and convex, the continuous map 𝑄∶𝐷→𝐷 is 𝛼-condensing, then the map 𝑄 has at least one fixed point in 𝐷.

In the sequel, we will continue to generalize the definition of condensing operator. First of all, we give some notations.

Let 𝐷⊂𝐸 be bounded closed and convex, the map 𝑄∶𝐷→𝐷, and 𝑥₀∈𝐷 for every 𝐵⊂𝐷, set 𝑄^(1,𝑥0)(𝐵)(𝐵),𝑄=𝑄_(𝑛,𝑥0)𝐵=𝑄co𝑄^{(𝑛−1,𝑥0)}𝐵,𝑥₀,𝑛=2,3,…, (4) where co means the closure of the convex hull.

Now we give the definition of a kind of new operator.

Definition 3. Let 𝐷⊂𝐸 be bounded closed and convex, the map 𝑄∶𝐷→𝐷 is said to be 𝛼-convex-power condensing if there exist 𝑥₀∈𝐷,𝑛₀∈𝑁 and for every bounded and not relatively compact 𝐵⊂𝐷, we have 𝛼𝑄^{(𝑛0,𝑥0)}(𝐵)<𝛼(𝐵). (5) From this definition, if 𝛼(𝑄^{(𝑛0,𝑥0)}(𝐵))=𝛼(𝐵), one obtains 𝐵⊂𝐸 as relatively compact.

Subsequently, we give the fixed point theorem about the convex-power condensing operator.

Lemma 4 (see [23]). If 𝐷⊂𝐸 is bounded closed and convex, the continuous map 𝑄∶𝐷→𝐷 is 𝛼-convex-power condensing, then the map 𝑄 has at least one fixed point in 𝐷.

Throughout this paper, let (𝑋,‖⋅‖) be a real Banach space. We denote by 𝐶([0,𝑇];𝑋) the Banach space of all continuous functions from [0,𝑇] to 𝑋 with the norm ‖𝑢‖= sup {‖𝑢(𝑡)‖,𝑡∈[0,𝑇]} and by 𝐿¹([0,𝑇];𝑋) the Banach space of all 𝑋-valued Bochner integrable functions defined on [0,𝑇] with the norm ‖𝑢‖₁=∫^𝑇₀‖𝑢(𝑡)‖𝑑𝑡. Let 𝑃𝐶([0,𝑇];𝑋)={𝑢∶𝑢is a function from [0,𝑇] into𝑋such that 𝑢(𝑡)is continuous at𝑡≠𝑡_𝑖and the left continuous at 𝑡=𝑡_𝑖and the right limit𝑢(𝑡⁺_𝑖)exists for𝑖=1,2,...,𝑝}. It is easy to check that 𝑃𝐶([0,𝑇];𝑋) is a Banach space with the norm ‖𝑢‖_𝑃𝐶=sup{‖𝑢(𝑡)‖,𝑡∈[0,𝑇]} and 𝐶([0,𝑇];𝑋)⊆𝑃𝐶([0,𝑇];𝑋)⊆𝐿¹([0,𝑇];𝑋). Moreover, we denote 𝛽 by the Hausdorff measure of noncompactness of 𝑋, denote 𝛽_𝑐 by the Hausdorff measure of noncompactness of 𝐶([0,𝑇];𝑋) and denote 𝛽_𝑝𝑐 by the Hausdorff measure of noncompactness of 𝑃𝐶([0,𝑇];𝑋).

Throughout this work, we suppose the following

(𝐻_𝐴) The linear operator 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 generates an equicontinuous 𝐶₀-semigroup {𝑆(𝑡)∶𝑡≥0}. Hence, there exists a positive number 𝑀 such that ‖𝑆(𝑡)‖≤𝑀.

For further information about the theory of semigroup of operators, we may refer to some classic books, such as [24–26].

To discuss the problem (1), we also need the following lemma.

Lemma 5. If  𝑊⊆𝑃𝐶([0,𝑇];𝑋) is bounded, then one has sup_{[]𝑡∈0,𝑇}𝛽(𝑊(𝑡))≤𝛽_𝑝𝑐(𝑊), (6) where 𝑊(𝑡)={𝑢(𝑡);𝑢∈𝑊}⊂𝑋.

Lemma 6 (see [27]). If  𝑊⊆𝐶([0,𝑇];𝑋) is bounded, then for all 𝑡∈[0,𝑇], 𝛽(𝑊(𝑡))≤𝛽_𝑐(𝑊). (7) Furthermore, if  𝑊 is equicontinuous on [0,𝑇], then 𝛽(𝑊(𝑡)) is continuous on [0,𝑇] and 𝛽_𝑐(𝑊)=sup𝛽(𝑊(𝑡))∶𝑡∈[0,𝑇]. (8)

Since 𝐶₀-semigroup 𝑆(𝑡) is said to be equicontinuous, the following lemma is easily checked.

Lemma 7. If the semigroup 𝑆(𝑡) is equicontinuous and 𝑤∈𝐿¹([0,𝑇];𝑅⁺), then the set {∫^𝑡₀𝑆(𝑡−𝑠)𝑢(𝑠)𝑑𝑠,‖𝑢(𝑠)‖≤𝑤(𝑠) for a.e. 𝑠∈[0,𝑇]} is equicontinuous for 𝑡∈[0,𝑇].

Definition 8. A function 𝑢∈𝑃𝐶([0,𝑇];𝑋) is said to be a mild solution of the nonlocal problem (1), if it satisfies 𝑢(𝑡)=𝑆(𝑡)𝑔(𝑢)+^𝑡₀𝑆()𝑓((𝑠+𝑡−𝑠𝑠,𝑢))𝑑𝑠_{0<𝑡𝑖<𝑡}𝑆𝑡−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖,0≤𝑡≤𝑇. (9)

In addition, let 𝑟 be a finite positive constant, and set 𝐵_𝑟∶={𝑥∈𝑋∶‖𝑥‖≤𝑟} and 𝑊_𝑟∶={𝑢∈𝐶([0,𝑇];𝑋)∶𝑢(𝑡)∈𝐵_𝑟,forall𝑡∈[0,𝑇]}.

3. 𝑔 Is Compact

In this section, we state and prove the existence theorems for the nonlocal impulsive problem (1). First, we give the following hypotheses:

(𝐻_𝑓)(1)𝑓∶[0,𝑇]×𝑋→𝑋 is a Carathéodory function; that is, for all 𝑥∈𝑋,𝑓(⋅,𝑥)∶[0,𝑇]→𝑋 is measurable and for a.e. 𝑡∈[0,𝑇],𝑓(𝑡,⋅)∶𝑋→𝑋 is continuous;(2) for finite positive constant 𝑟>0, there exists a function 𝛼_𝑟∈𝐿¹(0,𝑇;𝑅) such that 𝑓(𝑡,𝑥)≤𝛼_𝑟(𝑡) (10) for a.e. 𝑡∈[0,𝑇] and 𝑥∈𝐵_𝑟;(3) there exists a function 𝑙(𝑡)∈𝐿¹(0,𝑇;𝑅) such that 𝛽𝑓(𝑡,𝐷)≤𝑙(𝑡)𝛽(𝐷) (11) for a.e. 𝑡∈[0,𝑇] and every bounded subset 𝐷⊂𝐵_𝑟;

(𝐻_𝑔) 𝑔∶𝑃𝐶([0,𝑇];𝑋)→𝑋 is a continuous and compact mapping; furthermore, there exists a positive number 𝑁₁ such that ‖𝑔(𝑢)‖≤𝑁₁, for any 𝑢∈𝑊_𝑟;

(𝐻_𝐼) 𝐼_𝑖∶𝑋→𝑋 is a continuous and compact mapping for every 𝑖=1,2,…,𝑝;

(𝐻_𝑟) 𝑀(𝑁₁+‖𝛼_𝑟‖_𝐿1+sup_{𝑢∈𝑊𝑟}∑^𝑝_𝑖=1‖𝐼_𝑖(𝑢(𝑡_𝑖))‖)≤𝑟.

Remark 9. The mapping 𝑓 is said to be 𝐿¹-Carathéodory if the assumption (𝐻_𝑓)(1)(2) is satisfied.

Theorem 10. If the hypotheses (𝐻_𝐴), (𝐻_𝑓)(1)(2)(3), (𝐻_𝑔),(𝐻_𝐼), and (𝐻_𝑟) are satisfied, then the nonlocal problem (1) has at least one mild solution on [0,𝑇].

To prove the above theorem, we need the following lemma.

Lemma 11. If the condition (𝐻_𝑟) holds, then for arbitrary bounded set 𝐵⊂𝑊_𝑟, we have 𝛽^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,𝐵(𝑠))𝑑𝑠≤4𝑀^𝑡₀𝛽𝑓(𝑠,𝐵(𝑠))𝑑𝑠,𝑡∈[0,𝑇]. (12)

This proof is quite similar to that of Lemma 3.1 in [20]; we omit it.

Proof of Theorem 10. We consider the operator 𝑄∶𝑃𝐶([0,𝑇];𝑋)→𝑃𝐶([0,𝑇];𝑋) defined by (𝑄𝑢)(𝑡)=𝑆(𝑡)𝑔(𝑢)+^𝑡₀𝑆()𝑓((𝑠+𝑡−𝑠𝑠,𝑢))𝑑𝑠_{0<𝑡𝑖<𝑡}𝑆𝑡−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖,0≤𝑡≤𝑇. (13)
It is easy to see that the fixed points of 𝑄 are the mild solutions of nonlocal impulsive semilinear differential equation (1). Subsequently, we shall prove that 𝑄 has a fixed point by using Lemma 4.
We shall first prove that 𝑄 is continuous on 𝑃𝐶([0,𝑇];𝑋). In fact, let {𝑢_𝑛}^+∞_𝑛=1⊂𝑃𝐶([0,𝑇];𝑋) be an arbitrary sequence satisfying lim_𝑛→+∞𝑢_𝑛=𝑢 in 𝑃𝐶([0,𝑇];𝑋). It follows from Definition 8 that 𝑄𝑢_𝑛−𝑄𝑢≤𝑀𝑔𝑢_𝑛(𝑢+−𝑔)_𝑇0𝑓𝑠,𝑢_𝑛(𝑠((𝑠+)−𝑓𝑠,𝑢))𝑑𝑠_𝑝𝑖=1𝐼_𝑖𝑢_𝑛𝑡_𝑖−𝐼_𝑖𝑢𝑡_𝑖. (14) According to the continuity of 𝑓 in its second argument, for each 𝑠∈[0,𝑇], we have the following: lim_𝑛→+∞𝑓𝑠,𝑢_𝑛(𝑠)=𝑓(𝑠,𝑢(𝑠)). (15) In addition, 𝑔 and 𝐼_𝑖 are all continuous for each 𝑖=1,2,…,𝑝, and hence, the Lebesgue dominated convergence theorem implies lim_𝑛→+∞𝑄𝑢_𝑛−𝑄𝑢=0. (16) Namely, 𝑄 is continuous on 𝑃𝐶([0,𝑇];𝑋).
Subsequently, we claim that 𝑄𝑊_𝑟⊆𝑊_𝑟. Actually, by (𝐻_𝑟), we obtain ‖(𝑄𝑢)(𝑡)‖≤𝑆(𝑡)𝑔(𝑢)+^𝑡₀𝑆()𝑓((𝑠+𝑡−𝑠𝑠,𝑢))𝑑𝑠_{0<𝑡𝑖<𝑡}𝑆𝑡−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖≤𝑀𝑁₁+𝛼_𝑟𝐿1+sup_{𝑢∈𝑊𝑟}^𝑝_𝑖=1𝐼_𝑖𝑢𝑡_𝑖≤𝑟, (17) for any 𝑢∈𝑊_𝑟⊆𝑃𝐶([0,𝑇];𝑋). Thus, 𝑄𝑊_𝑟⊆𝑊_𝑟.
Now we demonstrate that 𝑄𝑊_𝑟 is equicontinuous for any 𝑡>0. Let 𝑡∈(0,𝑇],𝜀>0. Since 𝑔 is compact, 𝑔(𝑊_𝑟) is relatively compact; that is, there is a finite family {𝑥_𝑗}^𝑚_𝑗=1⊂𝑔(𝑊_𝑟) such that for any 𝑢∈𝑊_𝑟, there exists some 𝑗∈{1,2,…,𝑚} such that 2𝑀𝑔(𝑢)−𝑥_𝑗≤𝜀5. (18) On the other hand, as 𝑆(𝑡) is equicontinuous at 𝑡>0, we can choose 𝛿₁∈(0,𝑡) such that 𝑆𝑡+𝛿^′−𝑆(𝑡)𝑥_𝑗≤𝜀5_𝑝𝑖=1,𝑆𝑡+𝛿^′−𝑡_𝑖−𝑆𝑡−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖≤𝜀5, (19) for each 𝛿^′∈𝑅,|𝛿^′|<𝛿₁, uniformly for 𝑢∈𝑊_𝑟, and 𝑗∈{1,2,…,𝑚}. By (𝐻_𝑓(2)), it can be obtained that there exists 𝛿₂∈(0,𝑡) such that 𝑀^𝑡+𝛿′_𝑡𝑓((𝑠𝜀𝑠,𝑢))𝑑𝑠≤5, (20) for each 𝛿^′∈𝑅,|𝛿^′|<𝛿₂, uniformly for 𝑢∈𝑊_𝑟. Furthermore, by Lemma 7, we get that there exists 𝛿₃∈(0,𝑡) such that ^𝑡₀𝑆𝑡+𝛿^′()𝑓((𝑠𝜀−𝑠−𝑆𝑡−𝑠𝑠,𝑢))𝑑𝑠≤5, (21) for each 𝛿^′∈𝑅,|𝛿^′|<𝛿₃, uniformly for 𝑢∈𝑊_𝑟. Thus, there exists ℎ=min{𝛿₁,𝛿₂,𝛿₃} such that (𝑄𝑢)𝑡+ℎ^′−(𝑡≤𝑆𝑄𝑢)()𝑡+ℎ^′(𝑡)𝑔(𝑢+−𝑆)_𝑝𝑖=1𝑆𝑡+ℎ^′−𝑡_𝑖−𝑆𝑡−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑡₀𝑆𝑡+ℎ^′−𝑠−𝑆(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠+^𝑡+ℎ′_𝑡𝑆𝑡+ℎ^′𝑓((𝑠≤𝑆−𝑠𝑠,𝑢))𝑑𝑠𝑡+ℎ^′−𝑆(𝑡)𝑔(𝑢)−𝑥_𝑗+𝑆𝑡+ℎ^′−𝑆(𝑡)𝑥_𝑗+^𝑝_𝑖=1𝑆𝑡+ℎ^′−𝑡_𝑖−𝑆𝑡−𝑡_𝑖𝐼_𝑖𝑢𝑡_𝑖+^𝑡₀𝑆𝑡+ℎ^′()𝑓((𝑠+−𝑠−𝑆𝑡−𝑠𝑠,𝑢))𝑑𝑠_{𝑡+ℎ′𝑡}𝑆𝑡+ℎ^′𝑓((𝑠𝑔(𝑢)−𝑠𝑠,𝑢))𝑑𝑠≤2𝑀−𝑥_𝑗+𝜀5+𝜀5+𝜀5+𝑀^𝑡+ℎ′_𝑡𝑓((𝑠≤𝜀𝑠,𝑢))𝑑𝑠5+𝜀5+𝜀5+𝜀5+𝜀5=𝜀, (22) for each ℎ^′∈𝑅,|ℎ^′|<ℎ, uniformly for 𝑢∈𝑊_𝑟. Therefore, 𝑄𝑊_𝑟 is equicontinuous at 𝑡>0.
Similarly, we can conclude that 𝑄𝑊_𝑟 is also equicontinuous at 𝑡=0. Thus, 𝑄𝑊_𝑟 is equicontinuous on [0,𝑇].
Set 𝑊=co(𝑄𝑊_𝑟). It is obvious that 𝑊 is equicontinuous on [0,𝑇] and 𝑄 maps 𝑊 into itself.
Next, we shall prove that 𝑄∶𝑊→𝑊 is a convex-power condensing operator. Take 𝑥₀∈𝑊; by the definition of convex-power condensing operator, we shall show that there exists a positive integral 𝑛₀ such that 𝛽_𝑝𝑐𝑄^{(𝑛0,𝑥0)}𝐵<𝛽_𝑝𝑐(𝐵) (23) if 𝐵⊂𝑊 is not relatively compact. In fact, by using the conditions (𝐻_𝑔) and (𝐻_𝐼), we get from Lemma 11 that 𝛽𝑄^(1,𝑥0)𝐵𝑡)𝑡𝑆(𝑡)𝑔(𝐵)(=𝛽((𝑄𝐵)())≤𝛽+𝛽^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,𝐵(𝑠))𝑑𝑠+𝛽^𝑝_𝑖=1𝑆𝑡−𝑡_𝑖𝐼_𝑖𝐵𝑡_𝑖=4𝑀^𝑡₀𝛽𝑓(𝑠,𝐵(𝑠))𝑑𝑠≤4𝑀^𝑡₀𝑙(𝑠)𝛽(𝐵(𝑠))𝑑𝑠≤4𝑀^𝑡₀𝑙(𝑠)𝑑𝑠𝛽_𝑝𝑐(𝐵). (24) Since 𝑙(𝑡)∈𝐿¹(0,𝑇;𝑅⁺), there exists a continuous function 𝜔∶[0,𝑇]→𝑅¹ such that for any 0<𝜀<1, ^𝑇₀|𝑙(𝑠)−𝜔(𝑠)|𝑑𝑠<𝜀. (25) Then 𝛽𝑄^(1,𝑥0)𝐵(𝑡)≤4𝑀^𝑡₀|𝑙(𝑠)−𝜔(𝑠)|𝑑𝑠+^𝑡₀|𝜔(𝑠)|𝑑𝑠𝛽_𝑝𝑐(𝐵)≤4𝑀𝜀+𝑁₂𝑡𝛽_𝑝𝑐(𝐵), (26) where 𝑁₂=max{|𝜔(𝑡)|∶𝑡∈[0,𝑇]}. Hence, 𝛽𝑄^(2,𝑥0)𝐵𝑡)𝑆(𝑡)𝑔(≤𝛽co𝑄^(1,𝑥0)𝐵,𝑥₀+𝛽^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,co𝑄^(1,𝑥0)𝐵(𝑠),𝑥₀(𝑠)𝑑𝑠+𝛽^𝑝_𝑖=1𝑆𝑡−𝑡_𝑖𝐼_𝑖co𝑄^(1,𝑥0)𝐵𝑡_𝑖,𝑥₀𝑡_𝑖=𝛽^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,co𝑄^(1,𝑥0)𝐵(𝑠),𝑥₀(𝑠)𝑑𝑠≤4𝑀^𝑡₀𝛽𝑓𝑠,co𝑄^(1,𝑥0)𝐵(𝑠),𝑥₀(𝑠)𝑑𝑠≤4𝑀^𝑡₀𝑙(𝑠)𝛽𝑄^(1,𝑥0)𝐵(𝑠)𝑑𝑠≤4𝑀^𝑡₀(|𝑙(𝑠)−𝜔(𝑠)|+|𝜔(𝑠)|)4𝑀𝜀+𝑁₂𝑠𝛽_𝑝𝑐(𝐵)𝑑𝑠≤4²𝑀²𝜀𝜀+𝑁₂𝑡+𝑁₂𝑡𝜀+𝑁₂𝑡²2𝛽_𝑝𝑐(𝐵)=4²𝑀²𝜀²+𝐶¹₂𝜀𝑁₂𝑡+𝑁₂𝑡²𝛽2!^𝑝𝑐(𝐵). (27) Thus, 𝛽𝑄^(3,𝑥0)𝐵𝑡)𝑆(𝑡)𝑔(≤𝛽co𝑄^(2,𝑥0)𝐵,𝑥₀+𝛽^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,co𝑄^(2,𝑥0)𝐵(𝑠),𝑥₀(𝑠)𝑑𝑠+𝛽^𝑝_𝑖=1𝑆𝑡−𝑡_𝑖𝐼_𝑖co𝑄^(2,𝑥0)𝐵𝑡_𝑖,𝑥₀𝑡_𝑖=𝛽^𝑡₀𝑆(𝑡−𝑠)𝑓𝑠,co𝑄^(2,𝑥0)𝐵(𝑠),𝑥₀(𝑠)𝑑𝑠≤4𝑀^𝑡₀𝛽𝑓𝑠,co𝑄^(2,𝑥0)𝐵(𝑠),𝑥₀(𝑠)𝑑𝑠≤4𝑀^𝑡₀𝑙(𝑠)𝛽𝑄^(2,𝑥0)𝐵(𝑠)𝑑𝑠≤4𝑀^𝑡₀(|𝑙(𝑠)−𝜔(𝑠)|+|𝜔(𝑠)|)×4²𝑀²𝜀²+𝐶¹₂𝜀𝑁₂𝑠+𝑁₂𝑠²2!𝑑𝑠𝛽_𝑝𝑐(𝐵)≤4³𝑀³𝜀𝜀²+𝐶¹₂𝜀𝑁₂𝑡+𝑁₂𝑡²2!+𝑁₂^𝑡₀𝜀²+𝐶¹₂𝜀𝑁₂𝑠+𝑁₂𝑠²𝛽2!𝑑𝑠_𝑝𝑐(𝐵)=4³𝑀³𝜀³+𝐶¹₃𝜀²𝑁₂𝑡+𝐶²₃𝜀𝑁₂𝑡²+𝑁2!₂𝑡³𝛽3!^𝑝𝑐(𝐵), (28) and hence, by the method of mathematical induction, for any positive integer 𝑛 and 𝑡∈[0,𝑇], we obtain 𝛽𝑄^(𝑛,𝑥0)𝐵(𝑡)≤4^𝑛𝑀^𝑛𝜀^𝑛+𝐶¹_𝑛𝜀^𝑛−1𝑁₂𝑡+𝐶²_𝑛𝜀^𝑛−2𝑁₂𝑡²𝑁2!+⋯+²𝑡^𝑛𝛽𝑛!^𝑝𝑐(𝐵). (29) Therefore, for any positive integer 𝑛, we have 𝛽_𝑝𝑐𝑄^(𝑛,𝑥0)𝐵≤4^𝑛𝑀^𝑛𝜀^𝑛+𝐶¹_𝑛𝜀^𝑛−1𝑁₂𝑇+𝐶²_𝑛𝜀^𝑛−2𝑁₂𝑇²𝑁2!+⋯+²𝑇^𝑛𝛽𝑛!^𝑝𝑐(𝐵). (30) Since lim_𝑛→+∞[𝜀^𝑛−1𝑛(𝑛/(𝑛−1))^𝑛−1]^1/𝑛=𝜀, it follows from the Stirling Formula (see [28]) that 𝛽_𝑝𝑐𝑄^(𝑛,𝑥0)𝐵1≤𝑜𝑛^𝑠4^𝑛𝑀^𝑛𝛽_𝑝𝑐(𝐵),𝑛⟶∞,∀𝑠>1, (31) and hence, there exists sufficiently large positive integer 𝑛₀ such that 𝛽_𝑝𝑐𝑄^{(𝑛0,𝑥0)}𝐵<𝛽_𝑝𝑐(𝐵), (32) which shows that 𝑄∶𝑊→𝑊 is a convex-power condensing operator. From Lemma 4, we get that 𝑄 has at least one fixed point in 𝑊; that is, (1) has at least one mild solution 𝑢∈𝑊. This completes the proof.

Remark 12. The technique of constructing convex-power condensing operator plays a key role in the proof of Theorem 10, which enables us to get rid of the strict inequality restriction on the coefficient function 𝑙(𝑡) of 𝑓. However, in many previous articles, such as [9, 12, 13, 19, 20], the authors had to impose a strong inequality condition on the integrable function 𝑙(𝑡), as they used Darbo-Sadovskii’s fixed point theorem only. Thus, our result extends and complements those obtained in [9, 12, 13, 19, 20] and has more broad applications.

Remark 13. If we use the following assumption instead of (𝐻_𝑓)(3):
(𝐻_𝑓)(3^′) there exists a constant 𝑙>0 such that 𝛽𝑓(𝑡,𝐷)≤𝑙𝛽(𝐷) (33) for a.e. 𝑡∈[0,𝑇] and every bounded subset 𝐷⊂𝐵_𝑟,

we may use the same method to obtain 𝛽_𝑝𝑐𝑄^(𝑛,𝑥0)𝐵≤4^𝑛𝑀^𝑛𝑙^𝑛𝑇^𝑛𝛽𝑛!^𝑝𝑐(𝐵), (34) for any 𝐵⊂𝑊. Thus, there exists a large enough positive integral 𝑛₀ such that 4^𝑛0𝑀^𝑛0𝑙^𝑛0𝑇^𝑛0𝑛₀!<1; (35) namely, 𝛽_𝑝𝑐𝑄^(𝑛,𝑥0)𝐵≤𝛽_𝑝𝑐(𝐵),𝐵⊂𝑊. (36) Therefore, we can get the following consequence.

Theorem 14. If the hypotheses (𝐻_𝐴), (𝐻_𝑓)(1)(2)(3^′), (𝐻_𝑔),(𝐻_𝐼), and (𝐻_𝑟) are satisfied, then the nonlocal problem (1) has at least one mild solution on [0,𝑇].

4. 𝑔 Is Lipschitz Continuous

In this section, by applying the proof of Theorem 10 and Darbo-Sadovskii’s fixed point theorem, we give the existence of mild solutions of the problem (1) when the nonlocal condition 𝑔 is Lipscitz continuous in 𝑃𝐶([0,𝑇];𝑋).

We give the following hypotheses:

(𝐻^′_𝑔) there exists a constant 𝑘>0 such that 𝑔(𝑢)−𝑔(𝑣)≤𝑘‖𝑢−𝑣‖,for𝑢,𝑣∈𝑃𝐶([0,𝑇];𝑋); (37)

(𝐻^′_𝐼) 𝐼_𝑖∶𝑋→𝑋 is Lipschitz continuous with Lipschitz constant 𝑘_𝑖, for 𝑖=1,2,…,𝑝.

Theorem 15. If the hypotheses (𝐻_𝐴), (𝐻_𝑓)(1)(2)(3), (𝐻^′_𝑔), (𝐻^′_𝐼), and (𝐻_𝑟) are satisfied, then the nonlocal problem (1) has at least one mild solution on [0,𝑇] provided that 𝑀(𝑘+∑^𝑝_𝑖=1𝑘_𝑖+4‖𝑙‖_𝐿¹)<1.

Proof of Theorem 15. Given 𝑥∈𝑊, let's first consider the following Cauchy initial problem: 𝑢^′(𝑡)=𝐴𝑢(𝑡)+𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,𝑇],𝑡≠𝑡_𝑖,Δ𝑢𝑡_𝑖=𝐼_𝑖𝑥𝑡_𝑖,𝑖=1,2,…,𝑝,0<𝑡₁<𝑡₂<⋯<𝑡_𝑝𝑢(0)(𝑥).<𝑇,=𝑔 (38) From the proof of Theorem 10, we can easily see that there exists at least one mild solution to (38). Define 𝐺∶𝑊→𝑊 by that 𝐺𝑥 is the mild solution to (38). Then (𝐺𝑥)(𝑡)=𝑆(𝑡)𝑔(𝑥)+_{0<𝑡𝑖<𝑡}𝑆𝑡−𝑡_𝑖𝐼_𝑖𝑥𝑡_𝑖+^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,(𝐺𝑥)(𝑠))𝑑𝑠. (39) Now, we will show that 𝐺 is 𝛽_𝑝𝑐-condensing on 𝑊. According to Lemma 11, for any bounded subset 𝐵⊂𝑊, we deduce 𝛽𝑡𝑆(𝑡)𝑔(𝐵)((𝐺𝐵)())≤𝛽+_{0<𝑡𝑖<𝑡}𝑆𝑡−𝑡_𝑖𝐼_𝑖𝐵𝑡_𝑖+𝛽^𝑡₀𝑆(𝑡−𝑠)𝑓(𝑠,(𝐺𝐵)(𝑠))𝑑𝑠≤𝑀𝑘𝛽_𝑝𝑐(𝐵)+𝑀^𝑝_𝑖=1𝑘_𝑖𝛽_𝑝𝑐(𝐵)+4𝑀^𝑡₀𝛽𝑓(𝑠,(𝐺𝐵)(𝑠))𝑑𝑠≤𝑀𝑘+^𝑝_𝑖=1𝑘_𝑖𝛽_𝑝𝑐(𝐵)+4𝑀^𝑡₀𝑙(𝑠)𝛽((𝐺𝐵)(𝑠))𝑑𝑠≤𝑀𝑘+^𝑝_𝑖=1𝑘_𝑖𝛽_𝑝𝑐(𝐵)+4𝑀‖𝑙‖_𝐿¹𝛽_𝑝𝑐(𝐺𝐵), (40) which implies that 𝛽_𝑝𝑐()≤𝑀𝐺𝐵𝑘+∑^𝑝_𝑖=1𝑘_𝑖1−4𝑀‖𝑙‖_𝐿¹𝛽_𝑝𝑐(𝐵). (41) In addition, since 𝑀(𝑘+∑^𝑝_𝑖=1𝑘_𝑖+4‖𝑙‖_𝐿¹)<1, it follows that the mapping 𝐾 is a 𝛽_𝑝𝑐-condensing operator on 𝑊. In view of Lemma 2, the mapping 𝐺 has at least one fixed point in 𝑊, which produces a mild solution for the nonlocal impulsive problem (1).

Remark 16. Similarly, one can show that the conclusion of Theorem 15 remains valid provided that hypothesis (𝐻_𝑓)(3) is replaced by condition (𝐻_𝑓)(3^′).

Remark 17. In Theorem 15, we do not assume the compactness of nonlocal item 𝑔. Under the Lipschitz assumption, we make full use of the conclusion of Theorem 10, the properties of noncompact measure and the technique of fixed point to deal with the solution operator 𝐺.

Remark 18. Recently, the existence results for fractional differential equations have been widely studied in many papers. For more details on this theory one can refer to [29, 30] and references therein. It should be pointed out that the techniques and ideas in this paper can also be used to study fractional equations. In the future, we will also try to investigate to nonlocal controllability of impulsive differential equations by applying the similar techniques, methods, and compactness conditions. Further discussions on this topic will be in our consequent papers.

5. Examples

In this section, we shall give two examples to illustrate Theorems 10 and 15.

Example 1. Consider the following semilinear parabolic system: 𝜕𝑢()()𝑢()((,[]𝜕𝑡𝑡,𝑥=−𝐴𝑥,𝐷𝑡,𝑥+𝐹𝑡,𝑢𝑡,𝑥))𝑡∈0,𝑇,𝑥∈Ω,𝑡≠𝑡_𝑖,𝐷^𝛼𝑢()[]|𝛼|𝑢𝑡𝑡,𝑥=0,𝑡∈0,𝑇,𝑥∈𝜕Ωfor≤𝑚,_+𝑖,𝑥−𝑢𝑡⁻_𝑖,𝑥=𝐼_𝑖𝑢𝑡_𝑖𝑢()=,𝑥,𝑖=1,2,…,𝑝,0,𝑥_Ω^𝑇₀Γ𝑡,𝑥,𝑥^′,𝑢𝑡,𝑥^′𝑑𝑡𝑑𝑥^′,𝑥∈Ω, (42) where Ω is a bounded domain in 𝑅^𝑛(𝑛≥1) with smooth boundary 𝜕Ω,𝐴(𝑥,𝐷)𝑢=Σ_{|𝛼|≤2𝑚}𝑎_𝛼(𝑥)𝐷^𝛼𝑢 is strongly elliptic, 𝐹∶[0,𝑇]×𝑅→𝑅, and Γ∶[0,𝑇]×Ω×Ω×𝑅→𝑅.

Let 𝑋=𝐿²(Ω) and define the operator 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 by 𝐷(𝐴)=𝐻^2𝑚(Ω)∩𝐻^𝑚₀(Ω),𝐴𝑢=−𝐴(𝑥,𝐷)𝑢. (43) Then the operator 𝐴 is an infinitesimal generator of an equicontinuous 𝐶₀-semigroup 𝑆(𝑡) on 𝑋 (see [26]).

Suppose that the function Γ∶[0,𝑇]×Ω×Ω×𝑅→𝑅 satisfies the following conditions:(i)the Carathodory condition, that is, Γ(𝑡,𝑥,𝑥^′,𝑟), is a continuous function about 𝑟 for a.e. (𝑡,𝑥,𝑥^′)∈[0,𝑇]×Ω×Ω;Γ(𝑡,𝑥,𝑥^′,𝑟) is measurable about (𝑡,𝑥,𝑥^′) for each fixed 𝑟∈𝑅;(ii)|Γ(𝑡,𝑥,𝑥^′,𝑟)−Γ(𝑡,𝑥,𝑥^′,𝑟)|≤𝜇_𝑘(𝑡,𝑥,𝑥,𝑥^′) for all (𝑡,𝑥,𝑥^′,𝑟),(𝑡,𝑥,𝑥^′,𝑟)∈[0,𝑇]×Ω×Ω×𝑅 with |𝑟|≤𝑘, where 𝜇_𝑘∈𝐿¹([0,𝑇]×Ω×Ω×𝑅;𝑅⁺) satisfies lim_𝑥→𝑥∫_Ω∫^𝑇₀𝜇_𝑘(𝑡,𝑥,𝑥,𝑥^′)𝑑𝑡𝑑𝑥^′=0, uniformly in 𝑥∈Ω;(iii)|Γ(𝑡,𝑥,𝑥^′,𝑟)|≤(𝛿/𝑇𝑚(Ω))|𝑟|+Φ(𝑡,𝑥,𝑥^′) for all 𝑟∈𝑅, where Φ∈𝐿²([0,𝑇]×Ω×Ω;𝑅⁺) and 𝛿>0.We assume the following.(1)𝑓∶[0,𝑇]×𝑋→𝑋 is defined by 𝑓(𝑡,𝑧)(𝑥)=𝐹(𝑡,𝑧(𝑥)),𝑥∈Ω. (44) Moreover, for given 𝑟>0, there exist two integrable functions 𝜙_𝑟,𝜓∶[0,𝑇]→𝑅 such that ‖𝑓(𝑡,𝑧)‖≤𝜙_𝑟(𝑡) and 𝛽(𝑓(𝑡,𝐷))≤𝜓(𝑡)𝛽(𝐷) for a.e. 𝑡∈[0,𝑇],𝑧∈𝐵_𝑟 and every bounded subset 𝐷⊂𝐵_𝑟;(2)𝑔∶𝐶([0,𝑇];𝑋)→𝑋 is defined by 𝑔(𝑢)(𝑥)=_Ω^𝑇₀Γ𝑡,𝑥,𝑥^′,𝑢𝑡,𝑥^′𝑑𝑡𝑑𝑥^′,𝑥∈Ω. (45) From Theorem 4.2 in [31], we get directly that 𝑔 is well defined and is a completely continuous operator by the above conditions about the function Γ.(3)𝐼_𝑖∶𝑋→𝑋 is a continuous and compact function for each 𝑖=1,2,…,𝑝,defined by 𝐼_𝑖(𝑢)(𝑥)=𝐼_𝑖(𝑢(𝑥)). (46)

Let us observe that the problem (42) may be reformulated as the abstract problem (1) under the above conditions. By using Theorem 10, the problem (42) has at least one mild solution 𝑢∈𝐶([0,𝑇];𝐿²(Ω)) provided that the hypothesis (𝐻_𝑟) holds.

Example 2. Consider the following partial differential system: 𝜕𝑢()=𝜕𝜕𝑡𝑡,𝑥𝑢()((,[]𝜕𝑥𝑡,𝑥+𝐹𝑡,𝑢𝑡,𝑥))𝑡∈0,𝑇,𝑥∈Ω,𝑡≠𝑡_𝑖,𝑢()[]𝑢𝑡𝑡,𝑥=0,𝑡∈0,𝑇,𝑥∈𝜕Ω,_+𝑖,𝑥−𝑢𝑡⁻_𝑖,𝑥=𝐼_𝑖𝑢𝑡_𝑖𝑢()=,𝑥,𝑖=1,2,…,𝑝,0,𝑥^𝑞_𝑗=1𝑐_𝑗𝑢𝑠_𝑗,𝑥, (47) where Ω is a bounded domain in 𝑅^𝑛(𝑛≥1) with smooth boundary 𝜕Ω,𝐹∶[0,𝑇]×𝑅→𝑅, and 𝑐_𝑗 and 𝑠_𝑗 both are given real numbers for 𝑗=1,2,…,𝑞.

Let 𝑋=𝐶(Ω), and define the operator 𝐴∶𝐷(𝐴)⊆𝑋→𝑋 by 𝐷(𝐴)=𝑣∈𝑋∶𝑣^′∈𝑋,𝑣(𝑥)=0,𝑥∈𝜕Ω,𝐴𝑣=𝑣^′. (48)

As is known to all, the operator 𝐴 is an infinitesimal generator of the semigroup 𝑆(𝑡) defined by 𝑆(𝑡)𝑥(𝑠)=𝑥(𝑡+𝑠) for each 𝑥∈𝑋. Here, 𝑆(𝑡) is equicontinuous but is not compact.

We now suppose the following.(1)𝑓∶[0,𝑇]×𝑋→𝑋 is defined by 𝑓𝑡,𝑦(𝑥)=𝐹𝑡,𝑦(𝑥),𝑥∈Ω. (49) Moreover, for given 𝑟>0, there exist two integrable functions 𝜑_𝑟,𝜔∶[0,𝑇]→𝑅 such that ‖𝑓(𝑡,𝑦)‖≤𝜑_𝑟(𝑡) and 𝛽(𝑓(𝑡,𝐷))≤𝜛(𝑡)𝛽(𝐷) for a.e. 𝑡∈[0,𝑇],𝑦∈𝐵_𝑟, and every bounded subset 𝐷⊂𝐵_𝑟;(2)𝑔∶𝑃𝐶([0,𝑇];𝑋)→𝑋 is defined by 𝑔(𝑢)(𝑥)=^𝑞_𝑗=1𝑐_𝑗𝑢𝑠_𝑗,𝑥,0<𝑠₁<𝑠₂<⋯<𝑠_𝑞<𝑇,𝑥∈Ω. (50) Then 𝑔 is Lipschitz continuous with constant 𝑘=∑^𝑞_𝑗=1|𝑐_𝑗|; that is, the assumption 𝐻^′_𝑔 is satisfied.(3)𝐼_𝑖∶𝑋→𝑋 is a continuous function for each 𝑖=1,2,…,𝑝, defined by 𝐼_𝑖(𝑢)(𝑥)=𝐼_𝑖(𝑢(𝑥)). (51) Here we take 𝐼_𝑖(𝑢(𝑥))=(𝛼_𝑖|𝑢(𝑥)|+𝑡_𝑖)⁻¹,𝛼_𝑖>0,𝑖=1,2,…,𝑝,0<𝑡₁<𝑡₂<⋯<𝑡_𝑞<𝑇,𝑥∈Ω. Then 𝐼_𝑖 is Lipschitz continuous with constant 𝑘_𝑖=𝛼_𝑖/𝑡²_𝑖, 𝑖=1,2,…,𝑝; that is, the assumption 𝐻^′_𝐼 is satisfied.

Let us observe that (47) may be rewritten as the abstract problem (1) under the above conditions. If the following inequality 𝑀^𝑞_𝑗=1𝑐_𝑗+^𝑝_𝑖=1𝛼_𝑖𝑡²_𝑖+4‖𝜛‖_𝐿¹<1 (52) holds, then according to Theorem 15, the impulsive problem (47) has at least one mild solution in 𝑃𝐶([0,𝑇];𝑋).

Acknowledgments

The authors would like to thank the referees for their careful reading and their valuable comments and suggestions to improve their results. This research is supported by the Natural Science Foundation of China (11201410 and 11271316), the Natural Science Foundation of Jiangsu Province (BK2012260), and the Natural Science Foundation of Jiangsu Education Committee (10KJB110012).