Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces

Abstract

This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach space 𝐸∶𝑢^′(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡)), 𝑡∈[0,𝑎], 𝑡≠𝑡_𝑘, Δ𝑢|_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑢(𝑡_𝑘)), 0<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑎, 𝑢(0)=𝑢₀, where 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 is a closed linear operator, and 𝑓∶[0,𝑎]×𝐸×𝐸→𝐸 is a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearity 𝑓, some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.

1. Introduction and Main Results

Differential equations involving impulse effects occur in many applications: physics, population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, and so forth. Therefore, it has been an object of intensive investigation in recent years; see, for instance, the monographs [1–5]. Correspondingly, the existence of mild solutions of impulsive evolution differential equations has also been studied by several authors; see [6–8]. However, the theory still remains to be developed.

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions for the initial value problem (IVP) of first-order nonlinear impulsive evolution equations 𝑢^′(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑢∣_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑢(𝑡_𝑘)),𝑘=1,2,…,𝑚,𝑢(0)=𝑢₀ (1.1) in an ordered Banach space 𝐸, where 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 is a closed linear operator, −𝐴 generates a 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) in 𝐸, 𝑓∶𝐽×𝐸×𝐸→𝐸 is a nonlinear mapping, 𝐽=[0,𝑎],𝑎>0 is a constant, 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑎, 𝐼_𝑘∶𝐸→𝐸 is an impulsive function, 𝑘=1,2,…,𝑚, 𝑢₀∈𝐸, and 𝐺𝑢(𝑡)=^𝑡₀𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠 (1.2) is a Volterra integral operator with integral kernel 𝐾∈𝐶(Δ,ℝ⁺),Δ={(𝑡,𝑠)∣0≤𝑠≤𝑡≤𝑎}, Δ𝑢∣_{𝑡=𝑡𝑘} denotes the jump of 𝑢(𝑡) at 𝑡=𝑡_𝑘, that is, Δ𝑢∣_{𝑡=𝑡𝑘}=𝑢(𝑡⁺_𝑘)−𝑢(𝑡⁻_𝑘), where 𝑢(𝑡⁺_𝑘) and 𝑢(𝑡⁻_𝑘) represent the right and left limits of 𝑢(𝑡) at 𝑡=𝑡_𝑘, respectively. Let PC(𝐽,𝐸)={𝑢∶𝐽→𝐸∣𝑢(𝑡) is continuous at 𝑡≠𝑡_𝑘, and left continuous at 𝑡=𝑡_𝑘, and 𝑢(𝑡⁺_𝑘) exists, 𝑘=1,2,…,𝑚}. Evidently, PC(𝐽,𝐸) is a Banach space with norm ‖𝑢‖_PC=sup_𝑡∈𝐽‖𝑢(𝑡)‖. Let 𝐽^′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑚}. An abstract function 𝑢∈PC(𝐽,𝐸)∩𝐶¹(𝐽^′,𝐸)∩𝐶(𝐽^′,𝐸₁) (𝐸₁ is a Banach space with norm ‖𝑥‖₁=‖𝑥‖+‖𝐴𝑥‖) is called a solution of IVP(1.1), if 𝑢(𝑡) satisfies all the equalities of (1.1).

Let 𝐸 be an ordered Banach space with norm ‖⋅‖ and partial order ≤, whose positive cone 𝑃={𝑥∈𝐸∣𝑥≥0} is normal with a normal constant 𝑁. If an abstract function 𝑣∈PC(𝐽,𝐸)∩𝐶¹(𝐽^′,𝐸)∩𝐶(𝐽^′,𝐸₁) satisfies 𝑣^′(𝑡)+𝐴𝑣(𝑡)≤𝑓(𝑡,𝑣(𝑡),𝐺𝑣(𝑡)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑣∣_{𝑡=𝑡𝑘}≤𝐼_𝑘(𝑣(𝑡_𝑘)),𝑘=1,2,…,𝑚,𝑣(0)≤𝑢₀, (1.3) we call it a lower solution of IVP(1.1). If all the inequalities of (1.3) are inverse, we call it an upper solution of IVP(1.1).

In 1999, Liu [6], by means of the semigroup theory, has proved the existence and uniqueness of mild solutions for IVP(1.1) when 𝑓=𝑓(𝑡,𝑢). He demands that the nonlinear term 𝑓 and the impulsive function 𝐼_𝑘 satisfy the following conditions: ‖𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)‖≤𝐶‖𝑢−𝑣‖,𝑡∈𝐽,𝑢,𝑣∈𝐸,‖𝐼_𝑘(𝑢)−𝐼_𝑘(𝑣)‖≤ℎ_𝑘‖𝑢−𝑣‖,𝑢,𝑣∈𝐸, (1.4)(1.5) where 𝐶 and ℎ_𝑘(𝑘=1,2,…,𝑚) are positive constants and satisfy 𝑀^∗(𝐶𝑎+^𝑚_𝑘=1ℎ_𝑘)<1, (1.6) where 𝑀^∗=max_𝑡∈𝐽‖𝑇(𝑡)‖. Inequality (1.6) is a strongly restricted condition, and it is difficult to satisfy in applications.

Recently, Anguraj and Arjunan [7], under similar assumptions of [6], have obtained a unique mild solution for IVP(1.1) when 𝑓=𝑓(𝑡,𝑢,𝐺𝑢,𝑆𝑢). Cardinali and Rubbioni [8] have discussed the existence of mild solutions for the impulsive evolution differential inclusions under the measure of noncompactness conditions on every bounded set 𝐷⊂𝐸. However the assumptions in these papers are also difficult to satisfy in applications.

The purpose of this paper is to improve and extend the above mentioned results. We will delete the Lipschitz condition (1.5) for impulsive function 𝐼_𝑘 and the restriction condition (1.6) and improve condition (1.4) for nonlinear term 𝑓. Our main results are as follows.

Theorem 1.1. Let 𝐸 be an ordered Banach space, whose positive cone 𝑃 is normal, 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 be a closed linear operator, −𝐴 generate a positive 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0), 𝑓∈𝐶(𝐽×𝐸×𝐸,𝐸), and 𝐼_𝑘∈𝐶(𝐸,𝐸), 𝑘=1,2,…,𝑚. If IVP(1.1) has a lower solution 𝑣₀ and an upper solution 𝑤₀ with 𝑣₀≤𝑤₀ and the following conditions are satisfied: (𝐻₁)there exists a positive constant 𝐶 such that 𝑓(𝑡,𝑥₂,𝑦₂)−𝑓(𝑡,𝑥₁,𝑦₁)≥−𝐶(𝑥₂−𝑥₁), (1.7) for any 𝑡∈𝐽, 𝑣₀(𝑡)≤𝑥₁≤𝑥₂≤𝑤₀(𝑡), and 𝐺𝑣₀(𝑡)≤𝑦₁≤𝑦₂≤𝐺𝑤₀(𝑡),(𝐻₂)for any 𝑥₁,𝑥₂∈𝐸 with 𝑣₀(𝑡_𝑘)≤𝑥₁≤𝑥₂≤𝑤₀(𝑡_𝑘), 𝑘=1,2,…,𝑚, one has 𝐼_𝑘(𝑥₁)≤𝐼_𝑘(𝑥₂), (1.8)(𝐻₃)there exists a positive constant 𝐿 such that 𝛼({𝑓(𝑡,𝑥_𝑛,𝑦_𝑛)})≤𝐿(𝛼({𝑥_𝑛})+𝛼({𝑦_𝑛})), (1.9) for any 𝑡∈𝐽, and increasing or decreasing monotonic sequences {𝑥_𝑛}⊂[𝑣₀(𝑡),𝑤₀(𝑡)] and {𝑦_𝑛}∈[𝐺𝑣₀(𝑡),𝐺𝑤₀(𝑡)].Then IVP(1.1) has minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

Clearly, condition (𝐻₃) greatly improves the measure of noncompactness condition in [8]. Therefore, Theorem 1.1 greatly improves the main results in [6–8]. In Theorem 1.1, if Banach space 𝐸 is weakly sequentially complete, condition (𝐻₃) holds automatically; see [9, Theorem 2.2]. Hence, from Theorem 1.1, we have the following.

Corollary 1.2. Let 𝐸 be an ordered and weakly sequentially complete Banach space, whose positive cone 𝑃 is normal, 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 be a closed linear operator, −𝐴 generate a positive 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0), 𝑓∈𝐶(𝐽×𝐸×𝐸,𝐸), and 𝐼_𝑘∈𝐶(𝐸,𝐸), 𝑘=1,2,…,𝑚. If IVP(1.1) has a lower solution 𝑣₀ and an upper solution 𝑤₀ with 𝑣₀≤𝑤₀, and the conditions (𝐻₁) and (𝐻₂) are satisfied, then IVP(1.1) has minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

The proof of Theorem 1.1 will be shown in the next section. In Section 2, we also discuss the uniqueness of mild solutions for IVP(1.1) between the lower solution and upper solution (see Theorem 2.4).

2. Proof of the Main Results

Let 𝐶(𝐽,𝐸) denote the Banach space of all continuous 𝐸-value functions on interval 𝐽 with norm ‖𝑢‖_𝐶=max_𝑡∈𝐽‖𝑢(𝑡)‖ and let 𝐶¹(𝐽,𝐸) denote the Banach space of all continuously differentiable 𝐸-value functions on interval 𝐽 with norm ‖𝑢‖_𝐶¹=max{‖𝑢‖_𝐶,‖𝑢^′‖_𝐶}. Consider the initial value problem (IVP) of linear evolution equation without impulse 𝑢^′(𝑡)+𝐴𝑢(𝑡)=ℎ(𝑡),𝑡∈𝐽,𝑢(0)=𝑢₀. (2.1) It is well-known [10, chapter 4, Theorem 2.9], when 𝑢₀∈𝐷(𝐴) and ℎ∈𝐶¹(𝐽,𝐸), IVP(2.1) has a classical solution 𝑢∈𝐶¹(𝐽,𝐸)∩𝐶(𝐽,𝐸₁) expressed by 𝑢(𝑡)=𝑇(𝑡)𝑢₀+^𝑡₀𝑇(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,𝑡∈𝐽. (2.2) Generally, when 𝑢₀∈𝐸 and ℎ∈𝐶(𝐽,𝐸), the function 𝑢 given by (2.2) belongs to 𝐶(𝐽,𝐸) and it is called a mild solution of IVP(2.1).

Let us start by defining what we mean by a mild solution of problem 𝑢^′(𝑡)+𝐴𝑢(𝑡)=ℎ(𝑡),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑢∣_{𝑡=𝑡𝑘}=𝐼_𝑘(𝑢(𝑡_𝑘)),𝑘=1,2,…,𝑚,𝑢(0)=𝑢₀. (2.3)

Definition 2.1. A function 𝑢∈PC(𝐽,𝐸) is called a mild solution of IVP(2.3), if 𝑢 is a solution of integral equation 𝑢(𝑡)=𝑇(𝑡)𝑢₀+^𝑡₀𝑇(𝑡−𝑠)ℎ(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑇(𝑡−𝑡_𝑘)𝐼_𝑘(𝑢(𝑡_𝑘)),𝑡∈𝐽. (2.4)

To prove Theorem 1.1, for any ℎ∈PC(𝐽,𝐸), we consider the linear initial value problem (LIVP) of impulsive evolution equation 𝑢^′(𝑡)+𝐴𝑢(𝑡)+𝐶𝑢(𝑡)=ℎ(𝑡),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑢∣_{𝑡=𝑡𝑘}=𝑦_𝑘,𝑘=1,2,…,𝑚,𝑢(0)=𝑥, (2.5) where 𝐶≥0, 𝑥∈𝐸, and 𝑦_𝑘∈𝐸, 𝑘=1,2,…,𝑚.

Lemma 2.2. For any ℎ∈PC(𝐽,𝐸), 𝑥∈𝐸, and 𝑦_𝑘∈𝐸,𝑘=1,2,…,𝑚, LIVP(2.5) has a unique mild solution 𝑢∈PC(𝐽,𝐸) given by 𝑢(𝑡)=𝑆(𝑡)𝑥+^𝑡₀𝑆(𝑡−𝑠)ℎ(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝑦_𝑘,𝑡∈𝐽, (2.6) where 𝑆(𝑡)=𝑒^−𝐶𝑡𝑇(𝑡)(𝑡≥0) is a 𝐶₀-semigroup generated by −(𝐴+𝐶𝐼).

Proof. Let 𝑦₀=0. If 𝑢∈PC(𝐽,𝐸) is a mild solution of LIVP(2.5), then the restriction of 𝑢 on (𝑡_𝑘−1,𝑡_𝑘] satisfies the initial value problem of linear evolution equation without impulse 𝑢^′(𝑡)+𝐴𝑢(𝑡)+𝐶𝑢(𝑡)=ℎ(𝑡),𝑡_𝑘−1<𝑡≤𝑡_𝑘,𝑢(𝑡⁺_𝑘−1)=𝑢(𝑡_𝑘−1)+𝑦_𝑘−1. (2.7) Hence, on (𝑡_𝑘−1,𝑡_𝑘], 𝑢(𝑡) can be expressed by 𝑢(𝑡)=𝑆(𝑡−𝑡_𝑘−1)𝑢(𝑡_𝑘−1)+𝑆(𝑡−𝑡_𝑘−1)𝑦_𝑘−1+^𝑡_𝑡𝑘−1𝑆(𝑡−𝑠)ℎ(𝑠)𝑑𝑠. (2.8) Iterating successively in the above equality with 𝑢(𝑡_𝑗) for 𝑗=𝑘−1,𝑘−2,…,1,0, we see that 𝑢 satisfies (2.6).
Inversely, we can verify directly that the function 𝑢∈PC(𝐽,𝐸) defined by (2.6) satisfies all the equalities of LIVP(2.5).

Let 𝛼(⋅) denote the Kuratowskii measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [11]. For any 𝐵⊂𝐶(𝐽,𝐸) and 𝑡∈𝐽, set 𝐵(𝑡)={𝑢(𝑡)∣𝑢∈𝐵}⊂𝐸. If 𝐵 is bounded in 𝐶(𝐽,𝐸), then 𝐵(𝑡) is bounded in 𝐸, and 𝛼(𝐵(𝑡))≤𝛼(𝐵). In the proof of Theorem 1.1 we need the following lemma.

Lemma 2.3. Let 𝐵={𝑢_𝑛}⊂PC(𝐽,𝐸) be a bounded and countable set. Then 𝛼(𝐵(𝑡)) is the Lebesgue integrable on 𝐽, and 𝛼({_𝐽𝑢_𝑛(𝑡)𝑑𝑡})≤2_𝐽𝛼(𝐵(𝑡))𝑑𝑡. (2.9)

This lemma can be found in [12].

Evidently, PC(𝐽,𝐸) is also an ordered Banach space with the partial order “≤” reduced by the positive function cone 𝐾_PC={𝑢∈PC(𝐽,𝐸)∣𝑢(𝑡)≥0,𝑡∈𝐽}. 𝐾_PC is also normal with the same normal constant 𝑁. For 𝑣,𝑤∈PC(𝐽,𝐸) with 𝑣≤𝑤, we use [𝑣,𝑤] to denote the ordered interval {𝑢∈PC(𝐽,𝐸)∣𝑣≤𝑢≤𝑤} in PC(𝐽,𝐸), and use [𝑣(𝑡),𝑤(𝑡)] to denote the ordered interval {𝑥∈𝐸∣𝑣(𝑡)≤𝑥≤𝑤(𝑡)} in 𝐸.

Proof of Theorem 1.1. Let 𝐷=[𝑣₀,𝑤₀]. We define a mapping 𝑄∶𝐷→PC(𝐽,𝐸) by (𝑄𝑢)(𝑡)=𝑆(𝑡)𝑢₀+^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))+𝐶𝑢(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑢(𝑡_𝑘)),𝑡∈𝐽. (2.10) Clearly, 𝑄∶𝐷→PC(𝐽,𝐸) is continuous. By Lemma 2.2, the mild solution of IVP(1.1) is equivalent to the fixed point of operator 𝑄. By assumptions (𝐻₁) and (𝐻₂), 𝑄 is increasing in 𝐷 and maps any bounded set in 𝐷 into a bounded set.
We show that 𝑣₀≤𝑄𝑣₀,𝑄𝑤₀≤𝑤₀. Let ℎ(𝑡)≜𝑣^′₀(𝑡)+𝐴𝑣₀(𝑡)+𝐶𝑣₀(𝑡). By the definition of the lower solution, we easily see that ℎ∈PC(𝐽,𝐸) and ℎ(𝑡)≤𝑓(𝑡,𝑣₀(𝑡),𝐺𝑣₀(𝑡))+𝐶𝑣₀(𝑡) for 𝑡∈𝐽^′. Because 𝑣₀(𝑡) is a solution of LIVP(2.5) for 𝑥=𝑣₀(0) and 𝑦_𝑘=Δ𝑣₀∣_{𝑡=𝑡𝑘},𝑘=1,2,…,𝑚, by Lemma 2.2 and the positivity of operator 𝑆(𝑡)(𝑡≥0), we have𝑣₀(𝑡)=𝑆(𝑡)𝑣₀(0)+^𝑡₀𝑆(𝑡−𝑠)ℎ(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)Δ𝑣₀∣_{𝑡=𝑡𝑘}≤𝑆(𝑡)𝑢₀+^𝑡₀𝑆(𝑡−𝑠)ℎ(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑣₀(𝑡_𝑘))=𝑆(𝑡)𝑢₀+^𝑘_𝑗=1^𝑡𝑗_𝑡𝑗−1𝑆(𝑡−𝑠)ℎ(𝑠)𝑑𝑠+^𝑡_𝑡𝑘𝑆(𝑡−𝑠)ℎ(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑣₀(𝑡_𝑘))≤𝑆(𝑡)𝑢₀+^𝑘_𝑗=1^𝑡𝑗_𝑡𝑗−1𝑆(𝑡−𝑠)(𝑓(𝑠,𝑣₀(𝑠),𝐺𝑣₀(𝑠))+𝐶𝑣₀+(𝑠))𝑑𝑠_𝑡𝑡𝑘𝑆(𝑡−𝑠)(𝑓(𝑠,𝑣₀(𝑠),𝐺𝑣₀(𝑠))+𝐶𝑣₀(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑣₀(𝑡_𝑘))=𝑆(𝑡)𝑢₀+^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑣₀(𝑠),𝐺𝑣₀(𝑠))+𝐶𝑣₀(𝑠))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑆(𝑡−𝑡_𝑘)𝐼_𝑘(𝑣₀(𝑡_𝑘))=(𝑄𝑣₀)(𝑡),𝑡∈𝐽, (2.11) namely, 𝑣₀≤𝑄𝑣₀. Similarly, it can be shown that 𝑄𝑤₀≤𝑤₀. Combining these facts with the increasing property of 𝑄 in 𝐷, we see that 𝑄 maps 𝐷 into itself, and 𝑄∶𝐷→𝐷 is a continuously increasing operator.
Now, we define two sequences {𝑣_𝑛} and {𝑤_𝑛} in 𝐷 by the iterative scheme𝑣_𝑛=𝑄𝑣_𝑛−1,𝑤_𝑛=𝑄𝑤_𝑛−1,𝑛=1,2,…. (2.12) Then from the monotonicity of 𝑄, it follows that 𝑣₀≤𝑣₁≤⋯≤𝑣_𝑛≤⋯≤𝑤_𝑛≤⋯≤𝑤₁≤𝑤₀. (2.13) Next, we will show that {𝑣_𝑛} and {𝑤_𝑛} are uniformly convergent on 𝐽.
For convenience, we denote 𝑀=max_𝑡∈𝐽‖𝑆(𝑡)‖,𝐾₀=max_{(𝑡,𝑠)∈Δ}𝐾(𝑡,𝑠). Let 𝐽₁=[0,𝑡₁], 𝐽_𝑘=(𝑡_𝑘−1,𝑡_𝑘], 𝑘=2,3,…,𝑚+1 and let 𝐵={𝑣_𝑛∣𝑛∈ℕ} and 𝐵₀={𝑣_𝑛−1∣𝑛∈ℕ}. From 𝐵₀=𝐵∪{𝑣₀}, it follows that 𝛼(𝐵₀(𝑡))=𝛼(𝐵(𝑡)) for 𝑡∈𝐽. Let𝜑(𝑡)∶=𝛼(𝐵(𝑡))=𝛼(𝐵₀(𝑡)),𝑡∈𝐽. (2.14) By Lemma 2.3, 𝜑(𝑡) is Lebesgue integrable on 𝐽. Going from 𝐽₁ to 𝐽_𝑚+1 interval by interval we show that 𝜑(𝑡)≡0 on 𝐽.
For 𝑡∈𝐽, there exists a 𝐽_𝑘 such that 𝑡∈𝐽_𝑘. By (1.2) and Lemma 2.3, we have that𝛼(𝐺(𝐵₀)(𝑡))=𝛼({^𝑡₀𝐾(𝑡,𝑠)𝑣_𝑛−1≤(𝑠)𝑑𝑠∣𝑛∈ℕ})_{𝑘−1𝑗=1}𝛼({^𝑡𝑗_𝑡𝑗−1𝐾(𝑡,𝑠)𝑣_𝑛−1(𝑠)𝑑𝑠∣𝑛∈ℕ})+𝛼({^𝑡_𝑡𝑘−1𝐾(𝑡,𝑠)𝑣_𝑛−1(𝑠)𝑑𝑠∣𝑛∈ℕ})≤2𝐾₀(^𝑘−1_𝑗=1^𝑡𝑗_𝑡𝑗−1𝛼(𝐵₀(𝑠))𝑑𝑠+^𝑡_𝑡𝑘−1𝛼(𝐵₀(𝑠))𝑑𝑠)=2𝐾₀^𝑡₀𝜑(𝑠)𝑑𝑠, (2.15) and therefore, ^𝑡₀𝛼(𝐺(𝐵₀)(𝑠))𝑑𝑠≤2𝑎𝐾₀^𝑡₀𝜑(𝑠)𝑑𝑠. (2.16)
For 𝑡∈𝐽₁, from (2.10), (2.16), Lemma 2.3 and assumption (𝐻₃), we have𝜑(𝑡)=𝛼(𝐵(𝑡))=𝛼(𝑄(𝐵₀(𝑡)))=𝛼({^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑣_𝑛−1(𝑠),𝐺𝑣_𝑛−1(𝑠))+𝐶𝑣_𝑛−1(𝑠))𝑑𝑠∣𝑛∈ℕ})≤2^𝑡₀𝛼({𝑆(𝑡−𝑠)(𝑓(𝑠,𝑣_𝑛−1(𝑠),𝐺𝑣_𝑛−1(𝑠))+𝐶𝑣_𝑛−1(𝑠))∣𝑛∈ℕ})𝑑𝑠≤2𝑀^𝑡₀(𝐿(𝛼(𝐵₀(𝑠))+𝛼(𝐺(𝐵₀)(𝑠)))+𝐶𝛼(𝐵₀(𝑠)))𝑑𝑠=2𝑀(𝐿+𝐶+2𝑎𝐾₀𝐿)^𝑡₀𝜑(𝑠)𝑑𝑠. (2.17) By this and the Gronwall-Bellman inequality, we obtain that 𝜑(𝑡)≡0 on 𝐽₁. In particular, 𝛼(𝐵₀(𝑡₁))=𝜑(𝑡₁)=0, this means that 𝐵₀(𝑡₁) is precompact in 𝐸. Combining this with the continuity of 𝐼₁, it follows that 𝐼₁(𝐵₀(𝑡₁)) is precompact in 𝐸, and 𝛼(𝐼₁(𝐵₀(𝑡₁)))=0.
Now, for 𝑡∈𝐽₂, by (2.10) and the above argument for 𝑡∈𝐽₁, we have𝜑(𝑡)=𝛼(𝐵(𝑡))=𝛼(𝑄(𝐵₀)(𝑡))≤𝛼({^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑣_𝑛−1(𝑠),𝐺𝑣_𝑛−1(𝑠))+𝐶𝑣_𝑛−1(𝑠))𝑑𝑠∣𝑛∈ℕ})+𝛼({𝑆(𝑡−𝑡₁)𝐼₁(𝑣_𝑛−1(𝑡₁))∣𝑛∈ℕ})≤2𝑀(𝐿+𝐶+2𝑎𝐾₀𝐿)^𝑡₀𝜑(𝑠)𝑑𝑠+𝑀𝛼(𝐼₁(𝐵₀(𝑡₁)))=2𝑀(𝐿+𝐶+2𝑎𝐾₀𝐿)^𝑡_𝑡1𝜑(𝑠)𝑑𝑠. (2.18) Again by the Gronwall-Bellman inequality, we obtain that 𝜑(𝑡)≡0 on 𝐽₂, from which we obtain that 𝛼(𝐵₀(𝑡₂))=0 and 𝛼(𝐼₂(𝐵₀(𝑡₂)))=0.
Continuing such a process interval by interval up to 𝐽_𝑚+1, we can prove that 𝜑(𝑡)≡0 on every 𝐽_𝑘, 𝑘=1,2,…,𝑚+1. This means that {𝑣_𝑛(𝑡)} is precompact in 𝐸 for every 𝑡∈𝐽_𝑘. Hence {𝑣_𝑛(𝑡)} has a convergent subsequence in 𝐸. Combining this fact with the monotonicity (2.13), we easily prove that {𝑣_𝑛(𝑡)} itself is convergent in 𝐸, that is, there exists 𝑢(𝑡)∈𝐸 such that 𝑣_𝑛(𝑡)→𝑢(𝑡) as 𝑛→∞ for every 𝑡∈𝐽_𝑘. On the other hand, for any 𝑡∈𝐽_𝑘, we have𝑣_𝑛(𝑡)=𝑆(𝑡)𝑢₀+^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑣_𝑛−1(𝑠),𝐺𝑣_𝑛−1(𝑠))+𝐶𝑣_𝑛−1(𝑠))𝑑𝑠+_{0<𝑡𝑖<𝑡}𝑆(𝑡−𝑡_𝑖)𝐼_𝑖(𝑣_𝑛−1(𝑡_𝑖)). (2.19) Let 𝑛→∞, then by the Lebesgue-dominated convergence theorem, for 𝑡∈𝐽_𝑘, we have 𝑢(𝑡)=𝑆(𝑡)𝑢₀+^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))+𝐶𝑢(𝑠))𝑑𝑠+_{0<𝑡𝑖<𝑡}𝑆(𝑡−𝑡_𝑖)𝐼_𝑖(𝑢(𝑡_𝑖)), (2.20) and 𝑢∈𝐶(𝐽_𝑘,𝐸). Therefore, for any 𝑡∈𝐽, we have 𝑢(𝑡)=𝑆(𝑡)𝑢₀+^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))+𝐶𝑢(𝑠))𝑑𝑠+_{0<𝑡𝑖<𝑡}𝑆(𝑡−𝑡_𝑖)𝐼_𝑖(𝑢(𝑡_𝑖)). (2.21) Namely, 𝑢∈PC(𝐽,𝐸), and 𝑢=𝑄𝑢. Similarly, we can prove that there exists 𝑢∈PC(𝐽,𝐸) such that 𝑢=𝑄𝑢. By the monotonicity of operator 𝑄, it is easy to prove that 𝑢 and 𝑢 are the minimal and maximal fixed points of 𝑄 in [𝑣₀,𝑤₀], and they are the minimal and maximal mild solutions of IVP(1.1) in [𝑣₀,𝑤₀], respectively.

Now we discuss the uniqueness of mild solutions for IVP(1.1) in [𝑣₀,𝑤₀]. If we replace assumption (𝐻₃) by the following assumption:(𝐻₄)there exist positive constants 𝐶₁ and 𝐶₂ such that 𝑓(𝑡,𝑥₂,𝑦₂)−𝑓(𝑡,𝑥₁,𝑦₁)≤𝐶₁(𝑥₂−𝑥₁)+𝐶₂(𝑦₂−𝑦₁), (2.22) for any 𝑡∈𝐽, 𝑣₀(𝑡)≤𝑥₁≤𝑥₂≤𝑤₀(𝑡),𝐺𝑣₀(𝑡)≤𝑦₁≤𝑦₂≤𝐺𝑤₀(𝑡), and we have the following existence result.

Theorem 2.4. Let 𝐸 be an ordered Banach space, whose positive cone 𝑃 is normal, 𝐴 be a closed linear operator in 𝐸, −𝐴 generate a positive 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0), 𝑓∈𝐶(𝐽×𝐸×𝐸,𝐸) and 𝐼_𝑘∈𝐶(𝐸,𝐸), 𝑘=1,2,…,𝑚. If IVP(1.1) has a lower solution 𝑣₀ and an upper solution 𝑤₀ with 𝑣₀≤𝑤₀, such that the assumptions (𝐻₁),(𝐻₂), and (𝐻₄) are satisfied, then IVP(1.1) has a unique mild solution between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ or 𝑤₀.

Proof of Theorem 2.4. We firstly prove that (𝐻₁) and (𝐻₄) imply (𝐻₃). For 𝑡∈𝐽, let {𝑥_𝑛}⊂[𝑣₀(𝑡),𝑤₀(𝑡)] and {𝑦_𝑛}⊂[𝐺𝑣₀(𝑡),𝐺𝑤₀(𝑡)] be two increasing sequences. For 𝑚,𝑛∈ℕ with 𝑚>𝑛, by (𝐻₁) and (𝐻₄), we have 𝜃≤(𝑓(𝑡,𝑥_𝑚,𝑦_𝑚)−𝑓(𝑡,𝑥_𝑛,𝑦_𝑛))+𝐶(𝑥_𝑚−𝑥_𝑛)≤(𝐶₁+𝐶)(𝑥_𝑚−𝑥_𝑛)+𝐶₂(𝑦_𝑚−𝑦_𝑛). (2.23) By the normality of cone 𝑃, we have ‖𝑓(𝑡,𝑥_𝑚,𝑦_𝑚)−𝑓(𝑡,𝑥_𝑛,𝑦_𝑛)‖≤(𝐶+𝑁𝐶+𝑁𝐶₁)‖𝑥_𝑚−𝑥_𝑛‖+𝑁𝐶₂‖𝑦_𝑚−𝑦_𝑛‖. (2.24) From this inequality and the definition of the measure of noncompactness, it follows that 𝛼({𝑓(𝑡,𝑥_𝑛,𝑦_𝑛)})≤(𝐶+𝑁𝐶+𝑁𝐶₁)𝛼(𝑥_𝑛)+𝑁𝐶₂𝛼(𝑦_𝑛)≤𝐿(𝛼({𝑥_𝑛})+𝛼({𝑦_𝑛})), (2.25) where 𝐿=𝐶+𝑁𝐶+𝑁𝐶₁+𝑁𝐶₂. If {𝑥_𝑛} and {𝑦_𝑛} are two decreasing sequences, the above inequality is also valid. Hence (𝐻₃) holds.
Therefore, by Theorem 1.1, IVP(1.1) has minimal solution 𝑢 and maximal solution 𝑢 in [𝑣₀,𝑤₀]. By the proof of Theorem 1.1, (2.10)–(2.13) are valid. Going from 𝐽₁ to 𝐽_𝑚+1 interval by interval we show that 𝑢≡𝑢 on every 𝐽_𝑘.
For 𝑡∈𝐽₁, by (2.10) and assumption (𝐻₄), we have𝜃≤𝑢(𝑡)−𝑢(𝑡)=𝑄𝑢(𝑡)−𝑄𝑢(𝑡)=^𝑡₀𝑆(𝑡−𝑠)(𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))−𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))+𝐶(𝑢(𝑠)−𝑢≤(𝑠)))𝑑𝑠_𝑡0𝑆(𝑡−𝑠)((𝐶+𝐶₁)(𝑢(𝑠)−𝑢(𝑠))+𝐶₂(𝐺𝑢(𝑠)−𝐺𝑢(𝑠)))𝑑𝑠≤𝑀(𝐶+𝐶₁+𝑎𝐶₂𝐾₀)^𝑡₀(𝑢(𝑠)−𝑢(𝑠))𝑑𝑠. (2.26) From this inequality and the normality of cone 𝑃, it follows that ‖𝑢(𝑡)−𝑢(𝑡)‖≤𝑁𝑀(𝐶+𝐶₁+𝑎𝐶₂𝐾₀)^𝑡₀‖𝑢(𝑠)−𝑢(𝑠)‖𝑑𝑠. (2.27) By the Gronwall-Bellman inequality, we obtain that 𝑢(𝑡)≡𝑢(𝑡) on 𝐽₁.
For 𝑡∈𝐽₂, since 𝐼₁(𝑢(𝑡₁))=𝐼₁(𝑢(𝑡₁)), using (2.10) and the same argument as above for 𝑡∈𝐽₁, we can prove that‖𝑢(𝑡)−𝑢(𝑡)‖≤𝑁𝑀(𝐶+𝐶₁+𝑎𝐶₂𝐾₀)^𝑡₀‖𝑢(𝑠)−𝑢(𝑠)‖𝑑𝑠=𝑁𝑀(𝐶+𝐶₁+𝑎𝐶₂𝐾₀)^𝑡_𝑡1‖𝑢(𝑠)−𝑢(𝑠)‖𝑑𝑠. (2.28) Again, by the Gronwall-Bellman inequality, we obtain that 𝑢(𝑡)≡𝑢(𝑡) on 𝐽₂.
Continuing such a process interval by interval up to 𝐽_𝑚+1, we see that 𝑢(𝑡)≡𝑢(𝑡) over the whole 𝐽. Hence, 𝑢^∗∶=𝑢=𝑢 is the unique mild solution of IVP(1.1) in [𝑣₀,𝑤₀], which can be obtained by a monotone iterative procedure starting from 𝑣₀ or 𝑤₀.

Acknowledgments

The author is very grateful to the reviewers for their helpful comments and suggestions. The paper was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and the Project of NWNU-KJCXGC-3-47.