Infinite Boundary Value Problems for Second-Order Nonlinear Impulsive Differential Equations with Supremum

Abstract

We investigate the infinite boundary value problems for second-order impulsive differential equations with supremum by establishing a new comparison result and using the lower and upper solution method, and obtain the existence results for their maximal and minimal solutions.

1. Introduction

Differential equations with supremum are used modelling different real processes, and have been receiving much attention in recent years (see [1, 2]). In the theory of automatic regulation, for example, they are used in describing the system for regulation of the voltage of generator with constant current: 𝑇₀𝑢^′(𝑡)+𝑢(𝑡)+𝑞max_{𝑠∈[𝑡−ℎ,𝑡]}𝑢(𝑠)=𝑓(𝑡) (see [1]). If the equation is impulsive, periodic boundary value problem for first-order differential equation with supremum on finite domain was studied in [2], and on infinite domain, infinite boundary value problem for the same equation was investigated in [3]. Such equations with supremum are about first-order in the previous literature [1–3], but little is about second-order. Motivated by [2–5], we discuss in this paper the existence of maximal and minimal solutions of the system (IBVP): 𝑥^′′(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑥^′(𝑡),sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑥^′(𝑠)),𝑡≠𝑡_𝑘,𝑡∈𝐽,𝑘=1,2,…,Δ𝑥|_{𝑡=𝑡𝑘}=𝑎_𝑘𝑥^′(𝑡_𝑘),𝑘=1,2,…,Δ𝑥^′|_{𝑡=𝑡𝑘}=𝐼_𝑘̃(𝑥(𝑡_𝑘),𝑥^′(𝑡_𝑘)),𝑘=1,2,…,𝑥(0)=𝑥₀,𝑥^′(0)=𝑥^′(∞),𝑥^′(𝑡)=𝑥^′(0),𝑡∈[−ℎ,0], (1.1) where 𝐽=[0,+∞);𝑓∈𝐶[𝐽×𝑅×𝑅×𝑅,𝑅]; 𝑎_𝑘,ℎ∈𝑅₊, 𝐼_𝑘̃∈𝐶[𝑅×𝑅,𝑅], 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯<+∞, and 𝑡_𝑘→+∞ as 𝑘→+∞,𝑘=1,2,…;𝑥₀∈𝑅 and ∑^∞_𝑘=1𝑎_𝑘 is convergent, 𝑥^′(∞)=lim_𝑡→+∞𝑥^′(𝑡);Δ𝑥|_{𝑡=𝑡𝑘}=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘) denotes the jump of 𝑥(𝑡) at 𝑡=𝑡_𝑘, where 𝑥(𝑡⁺_𝑘) and 𝑥(𝑡⁻_𝑘) resent the right-hand and left-hand limit of 𝑥(𝑡) at 𝑡=𝑡_𝑘, respectively. Δ𝑥^′|_{𝑡=𝑡𝑘} has similar meaning for 𝑥^′(𝑡). Denote 𝐽₀=[0,𝑡₁], 𝐽₁=(𝑡₁,𝑡₂],…,𝐽_𝑘=(𝑡_𝑘,𝑡_𝑘+1]…, 𝐽^′=𝐽⧵{𝑡₁,𝑡₂,…,𝑡_𝑘…}, 𝐽_ℎ=[−ℎ,+∞).

Let 𝑃𝐶[𝐽_ℎ,𝑅]={𝑥∶𝐽_ℎ→𝑅∣𝑥(𝑡)=𝑥(0), for 𝑡∈[−ℎ,0]; 𝑥(𝑡) is continuous at 𝑡∈𝐽^′, left continuous at 𝑡=𝑡_𝑘, and each 𝑥(𝑡⁺_𝑘) exists, for 𝑘=1,2,…}, PC¹[𝐽_ℎ,𝑅]={𝑥∶𝐽_ℎ→𝑅∣𝑥^′(𝑡)=𝑥^′(0), for 𝑡∈[−ℎ,0]; 𝑥^′(𝑡) is continuous at 𝑡≠𝑡_𝑘, 𝑥(𝑡⁻_𝑘), 𝑥(𝑡⁺_𝑘), 𝑥^′(𝑡⁺_𝑘) and 𝑥^′(𝑡⁻_𝑘) exist, and 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘), 𝑘=1,2,…}, 𝐵𝑃𝐶[𝐽_ℎ,𝑅]={𝑥∈𝑃𝐶[𝐽_ℎ,𝑅]|sup_𝑡∈𝐽‖𝑥(𝑡)‖<+∞}, 𝑇𝑃𝐶[𝐽_ℎ,𝑅]={𝑥∈𝐵𝑃𝐶[𝐽_ℎ,𝑅]∣lim_𝑡→+∞𝑥(𝑡)=𝑥(∞)exists}, 𝐵𝑃𝐶¹[𝐽_ℎ,𝑅]={𝑥∈PC¹[𝐽_ℎ,𝑅]|sup_𝑡∈𝐽‖𝑥^′(𝑡)‖<+∞}, and TPC¹[𝐽_ℎ,𝑅]={𝑥∈𝐵𝑃𝐶¹[𝐽_ℎ,𝑅]∣lim_𝑡→+∞𝑥^′(𝑡)=𝑥^′(∞)exists}.

We get from [4] that 𝑥^′₋(𝑡_𝑘)=𝑥^′(𝑡⁻_𝑘). In the following, 𝑥^′(𝑡_𝑘) is understood as 𝑥^′₋(𝑡_𝑘). Evidently, 𝐵𝑃𝐶[𝐽_ℎ,𝑅] equipped with the norm ‖𝑥‖_𝐵=sup_𝑡∈𝐽‖𝑥(𝑡)‖ is a Banach space and 𝑇𝑃𝐶[𝐽_ℎ,𝑅]⊂𝐵𝑃𝐶[𝐽_ℎ,𝑅].

We say 𝑥∈TPC¹[𝐽_ℎ,𝑅]∩𝐶²[𝐽^′,𝑅] is a solution of IBVP(1.1), if it is satisfies (1.1).

In Section 2, we prove the existence result of minimal and maximal solutions for first-order impulsive differential equations which nonlinearly involve the operator 𝐵, that is, Theorem 2.5. In special case of IBVP(2.1) where 𝑓=𝑓(𝑡,𝑢(𝑡),sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑢(𝑠)) and 𝐼_𝑘̃=𝐼_𝑘̃(𝑢(𝑡_𝑘)), the infinite boundary value problems for first-order impulsive differential equations were studied in [3]. In Section 3, by applying Theorem 2.5, the main result (Theorem 3.1) of this paper is obtained, that is the existence theorem of minimal and maximal solutions of IBVP(1.1).

2. Result for First-Order Impulsive Differential Equation with Nonlinear Operator Terms

Consider the existence of solutions for the following first-order impulsive differential equations: 𝑦^′(𝑡)=𝑓(𝑡,(𝐵𝑦)(𝑡),𝑦(𝑡),sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑦(𝑠)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑦|_{𝑡=𝑡𝑘}=𝐼_𝑘̃((𝐵𝑦)(𝑡_𝑘),𝑦(𝑡_𝑘)),𝑘=1,2,…,𝑦(0)=𝑦(∞),𝑦(𝑡)=𝑦(0),𝑡∈[−ℎ,0], (2.1) where 𝑓,𝐼_𝑘̃(𝑘=1,2,…) are the same as IBVP (1.1), and (𝐵𝑦)(𝑡)=𝑥₀+∫^𝑡₀𝑦(𝑠)𝑑𝑠+∑_{0<𝑡𝑘<𝑡}𝑎_𝑘𝑦(𝑡_𝑘).

Lemma 2.1 (Comparison Result). Let 𝑥∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅]. Assume that there exist 𝑎,𝑏,𝑐∈𝐶[𝐽,𝑅₊]∩𝐿¹(𝐽), 𝑡𝑎∈𝐿¹(𝐽), 𝑏≢0, constants 𝐿_𝑘≥0,𝑘=1,2,…, and ∑^∞_𝑘=1𝐿_𝑘<∞ such that 𝑥^′(𝑡)≥−𝑎(𝑡)(𝐷𝑥)(𝑡)−𝑏(𝑡)𝑥(𝑡)−𝑐(𝑡)sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑥(𝑠),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑥|_{𝑡=𝑡𝑘}≥−𝐿_𝑘𝑥(𝑡_𝑘),𝑘=1,2,…,𝑥(0)≥𝑥(∞),𝑥(𝑡)=𝑥(0),𝑡∈[−ℎ,0]. (2.2) Then 𝑥(𝑡)≥0 for 𝑡∈𝐽_ℎ provided that 𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}{^∞₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^∞_𝑘=1𝐿_𝑘}≤1, (2.3) where (𝐷𝑥)(𝑡)=∫^𝑡₀𝑥(𝑠)𝑑𝑠+∑_{0<𝑡𝑘<𝑡}𝑎_𝑘𝑥(𝑡_𝑘).

Proof. Set 𝑚(𝑡)=𝑥(𝑡)𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}, then we have from (2.2) that 𝑚^′(𝑡)≥−𝑎(𝑡)𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[^𝑡₀𝑚(𝑠)𝑒^{−∫𝑠0𝑏(𝜏)𝑑𝜏}𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑎_𝑘𝑚(𝑡_𝑘)𝑒^{−∫𝑡𝑘0𝑏(𝜏)𝑑𝜏}]−𝑐(𝑡)𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑚(𝑠)𝑒^{−∫𝑠0𝑏(𝜏)𝑑𝜏},𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑚|_{𝑡=𝑡𝑘}≥−𝐿_𝑘𝑚(𝑡_𝑘),𝑘=1,2,…,𝑚(0)≥𝑚(∞)𝑒^{−∫∞0𝑏(𝜏)𝑑𝜏},𝑚(𝑡)=𝑚(0)𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏},𝑡∈[−ℎ,0]. (2.4)
We claim that 𝑚(𝑡)≥0 for 𝑡∈𝐽, moreover 𝑚(𝑡)≥0 for 𝑡∈𝐽_ℎ. Otherwise, we will consider two cases.
Case 1. 𝑚(𝑡)≤0 for 𝑡∈𝐽, and there exists 𝑡^∗₁∈𝐽 such that 𝑚(𝑡^∗₁)<0. Case 2. there exist 𝑡^∗₁,𝑡^∗₂∈𝐽 such that 𝑚(𝑡^∗₁)<0, 𝑚(𝑡^∗₂)>0.
In Case 1, we see from (2.4) that 𝑚^′(𝑡)≥0 for 𝑡∈𝐽, 𝑡≠𝑡_𝑘. On the other hand 𝑚(𝑡⁺_𝑘)=𝑚(𝑡_𝑘)+Δ𝑚|_{𝑡=𝑡𝑘}≥𝑚(𝑡_𝑘), thus 𝑚(𝑡) is increasing on 𝐽, and 𝑚(0)≤𝑚(𝑡^∗₁)<0,𝑚(0)≤𝑚(∞)≤𝑚(0)𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}. Hence 𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}≤1, which is a contradiction.
In Case 2, denote sup_𝑡∈𝐽𝑚(𝑡)=𝜆, then 𝜆>0, and it is clear that sup_{𝑡∈𝐽ℎ}𝑚(𝑡)=sup_𝑡∈𝐽𝑚(𝑡)=𝜆. Then we have either that (𝑎): there exists some 𝐽_𝑖 such that 𝑚(𝑡^∗₀)=𝜆 for some 𝑡^∗₀∈𝐽_𝑖, or 𝑚(𝑡⁺_𝑖)=𝜆, or that (𝑏)∶𝑚(∞)=𝜆.
In subcase (𝑎), we only discuss the case of 𝑚(𝑡^∗₀)=𝜆 for 𝑡^∗₀∈𝐽_𝑖, since the discussion of the case of 𝑚(𝑡⁺_𝑖)=𝜆 is similar.
If there exists some 𝐽_𝑖 such that 𝑚(𝑡^∗₀)=𝜆 for 𝑡^∗₀∈𝐽_𝑖, then we have from (2.4) that for 𝑡∈𝐽, 𝑡≠𝑡_𝑘, 𝑚^′(𝑡)≥𝜆𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[−𝑐(𝑡)−𝑎(𝑡)(𝑡+∑^∞_𝑘=1𝑎_𝑘)]. (2.5) For any integer 𝑙≥𝑖, the calculus fundamental principle implies that 𝑚(𝑡_𝑙+1)−𝑚(𝑡^∗₀)=^𝑡𝑙+1_𝑡∗0𝑚^′(𝑡)𝑑𝑡+^𝑙_𝑘=𝑖+1Δ𝑚|_{𝑡=𝑡𝑘}≥−𝜆^𝑡𝑙+1_𝑡∗0𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡−𝜆^𝑙_𝑘=𝑖+1𝐿_𝑘. (2.6) Let 𝑙→+∞, we have 𝑚(∞)−𝜆≥−𝜆^∞_𝑡∗0𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡−𝜆^∞_𝑘=𝑖+1𝐿_𝑘. (2.7) This means that 𝑚(∞)≥𝜆{1−^∞_𝑡∗0𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡−^∞_𝑘=𝑖+1𝐿_𝑘}. (2.8) From (2.3), we have 𝑚(∞)≥0 and 𝑚(0)≥𝑚(∞)𝑒^{−∫∞0𝑏(𝜏)𝑑𝜏}≥0. Therefore 0<𝑡^∗₁<+∞. Without loss of generality, we assume that 𝑡^∗₁∈𝐽_𝑗.
If 𝑡^∗₀<𝑡^∗₁, then 𝑖≤𝑗. Hence using the same method as is used above, we have−𝜆>𝑚(𝑡^∗₁)−𝑚(𝑡^∗₀)≥−𝜆{^𝑡∗1_𝑡∗0𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^𝑗_𝑘=𝑖+1𝐿_𝑘}≥−𝜆{^∞₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^∞_𝑘=1𝐿_𝑘}, (2.9) hence, ^∞₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^∞_𝑘=1𝐿_𝑘>1, (2.10) which is a contradiction to (2.3).
If 𝑡^∗₀>𝑡^∗₁, then 𝑖≥𝑗. Similar argument shows that−𝑚(0)>𝑚(𝑡^∗₁)−𝑚(0)≥−𝜆{^𝑡∗1₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^𝑗_𝑘=1𝐿_𝑘}, (2.11) which, noticing 𝑚(0)≥𝑚(∞)𝑒^{−∫∞0𝑏(𝜏)𝑑𝜏}, implies that −𝑚(∞)>−𝜆𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}{^𝑡∗1₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^𝑗_𝑘=1𝐿_𝑘}. (2.12) Adding (2.8) and (2.12), we show that 1<^∞_𝑡∗0𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘+)]𝑑𝑡_{∞𝑘=𝑖+1}𝐿_𝑘+𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}{^𝑡∗1₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^𝑗_𝑘=1𝐿_𝑘}<𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}{^∞₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^∞_𝑘=1𝐿_𝑘}, (2.13) which also contradicts (2.3).
In subcase (𝑏), 𝑚(∞)=𝜆, then it follows from (2.12) that−𝜆>−𝜆𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}{^∞₀𝑒^{∫𝑡0𝑏(𝜏)𝑑𝜏}[𝑐(𝑡)+𝑎(𝑡)(𝑡+^∞_𝑘=1𝑎_𝑘)]𝑑𝑡+^∞_𝑘=1𝐿_𝑘}. (2.14) This also leads to a contradiction with (2.3).
Therefore, the Case 2 is also impossible. Then, we conclude that 𝑚(𝑡)≥0 on 𝐽_ℎ, and hence 𝑥(𝑡)≥0 on 𝐽_ℎ. The proof is complete.

We first consider the following linear impulsive differential equations: 𝑦^′(𝑡)=𝑓(𝑡,𝐵𝜂(𝑡),𝜂(𝑡),sup_{𝑠∈[𝑡−ℎ,𝑡]}𝜂(𝑠))−𝑎(𝑡)((𝐵𝑦)(𝑡)−(𝐵𝜂)(𝑡))−𝑏(𝑡)(𝑦(𝑡)−𝜂(𝑡))−𝑐(𝑡)(sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑦(𝑠)−sup_{𝑠∈[𝑡−ℎ,𝑡]}𝜂(𝑠)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑦|_{𝑡=𝑡𝑘}=𝐼_𝑘̃((𝐵𝜂)(𝑡_𝑘),𝜂(𝑡_𝑘))−𝐿_𝑘(𝑦(𝑡_𝑘)−𝜂(𝑡_𝑘)),𝑘=1,2,…,𝑦(0)=𝑦(∞),𝑦(𝑡)=𝑦(0),𝑡∈[−ℎ,0]. (2.15)

Let us list some conditions for convenience.(𝐻₁)There exist 𝑝,𝑞,𝑙∈𝐶[𝐽,𝑅₊]∩𝐿¹(𝐽),𝑡𝑝∈𝐿¹(𝐽), such that |𝑓(𝑡,𝑢,𝑣,𝑤)|≤𝑝(𝑡)|𝑢|+𝑞(𝑡)|𝑣|+𝑙(𝑡)|𝑤|,𝑢,𝑣,𝑤∈𝑅,𝑡∈𝐽. (2.16)(𝐻₂)There exist 𝐿_𝑘≥0,𝑘=1,2,…, such that ∑^∞_𝑘=1𝐿_𝑘 is convergent and |𝐼_𝑘(𝑢,𝑣)|≤𝐿_𝑘|𝑣|,𝑢,𝑣∈𝑅,∀𝑡∈𝐽,𝑘=1,2,…. (2.17)

Lemma 2.2. Let 𝑎,𝑏,𝑐,𝑡𝑎∈𝐶[𝐽,𝑅₊]∩𝐿¹(𝐽), 𝑏≢0, 𝐿_𝑘≥0, 𝑘=1,2,…, with ∑^∞_𝑘=1𝐿_𝑘<∞, and assume also that conditions (𝐻₁) and (𝐻₂) hold. Then for any 𝜂∈𝐵𝑃𝐶[𝐽_ℎ,𝑅], 𝑦∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅] is a solution of the linear impulsive differential equations (2.15) if and only if 𝑦∈𝐵𝑃𝐶[𝐽_ℎ,𝑅] is a solution of the following impulsive integral equation: 𝑦(𝑡)=𝑒^{−∫𝑡0𝑏(𝜏)𝑑𝜏}×{(𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1)⁻¹[^∞₀𝑒^{∫s0𝑏(𝜏)𝑑𝜏}[𝑓(𝑠,𝐵𝜂(𝑠),𝜂(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝜂(𝑟))−𝑎(𝑠)((𝐵𝑦)(𝑠)−(𝐵𝜂)(𝑠))+𝑏(𝑠)𝜂(𝑠)−𝑐(𝑠)(sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑦(𝑟)−sup_{𝑟∈[𝑠−ℎ,𝑠]}+𝜂(𝑟))]𝑑𝑠_∞𝑘=1𝑒^{∫𝑡𝑘0𝑏(𝜏)𝑑𝜏}[𝐼_𝑘̃((𝐵𝜂)(𝑡_𝑘),𝜂(𝑡_𝑘))−𝐿_𝑘(𝑦(𝑡_𝑘)−𝜂(𝑡_𝑘+))]]_𝑡0𝑒^{∫𝑠0𝑏(𝜏)𝑑𝜏}[𝑓(𝑠,𝐵𝜂(𝑠),𝜂(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝜂(𝑟))−𝑎(𝑠)((𝐵𝑦)(𝑟)−(𝐵𝜂)(𝑟))+𝑏(𝑠)𝜂(𝑠)−𝑐(𝑠)(sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑦(𝑟)−sup_{𝑟∈[𝑠−ℎ,𝑠]}+𝜂(𝑟))]𝑑𝑠}_{0<𝑡𝑘<𝑡}𝑒^{∫𝑡𝑘𝑡𝑏(𝜏)𝑑𝜏}[𝐼_𝑘̃((𝐵𝜂)(𝑡_𝑘),𝜂(𝑡_𝑘))−𝐿_𝑘(𝑦(𝑡_𝑘)−𝜂(𝑡_𝑘))],∀𝑡∈𝐽, (2.18) with the initial condition 𝑦(𝑡)=𝑦(0), for 𝑡∈[−ℎ,0].

Proof. By the definition of 𝐵, we have |𝐵𝜂(𝑠)|≤|𝑥₀|+(𝑠+∑^∞_𝑘=1𝑎_𝑘)‖𝜂‖_𝐵. Together with (𝐻₁), (𝐻₂), we have |𝑓(𝑠,𝐵𝜂(𝑠),𝜂(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝜂(𝑟))−𝑎(𝑠)((𝐵𝑦)(𝑠)−(𝐵𝜂)(𝑠))+𝑏(𝑠)𝜂(𝑠)−𝑐(𝑠)(sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑦(𝑟)−sup_{𝑟∈[𝑠−ℎ,𝑠]}𝜂(𝑟))|≤[𝑝(𝑠)(𝑠+^∞_𝑘=1𝑎_𝑘)+𝑞(𝑠)+𝑙(𝑠)+𝑎(𝑠)(𝑠+^∞_𝑘=1𝑎_𝑘)+𝑏(𝑠)+𝑐(𝑠)]‖𝜂‖_𝐵+[𝑐(𝑠)+𝑎(𝑠)(𝑠+^∞_𝑘=1𝑎_𝑘)]‖𝑦‖_𝐵+(𝑝(𝑠)+2𝑎(𝑠))|𝑥₀|𝐼̃|=∶𝑀(𝑠),_𝑘((𝐵𝜂)(𝑡_𝑘),𝜂(𝑡_𝑘))−𝐿_𝑘(𝑦(𝑡_𝑘)−𝜂(𝑡_𝑘))|≤(𝐿_𝑘+𝐿_𝑘)‖𝜂‖_𝐵+𝐿_𝑘‖𝑦‖_𝐵=∶𝑁_𝑘, (2.19) which, noticing 𝑀(𝑠)∈𝐿¹(𝐽) and ∑^∞_𝑘=1𝑁_𝑘<∞, implies that the right hand of (2.18) is well defined. Moreover, we show by direct computation that 𝑦∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅] is a solution of (2.15).
We next prove the uniqueness of solution. Let 𝑦₁,𝑦₂ be any two solutions of (2.15), and 𝑦=𝑦₁−𝑦₂, then we have 𝑦^′(𝑡)=𝑦^′₁(𝑡)−𝑦^′₂(𝑡)=−𝑎(𝑡)(𝐷𝑦)(𝑡)−𝑏(𝑡)𝑦(𝑡)−𝑐(𝑡)sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑦(𝑠),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑦|_{𝑡=𝑡𝑘}=−𝐿_𝑘𝑦(𝑡_𝑘),𝑘=1,2,…,𝑦(0)=𝑦(∞),𝑦(𝑡)=𝑦(0),𝑡∈[−ℎ,0]. (2.20) Hence Lemma 2.1 implies that 𝑦≥0, that is, 𝑦₁≥𝑦₂. Similar argument shows that 𝑦₁≤𝑦₂. Therefore 𝑦₁=𝑦₂. We complete the proof.

Lemma 2.3. Let (𝐻₁) and (𝐻₂) be satisfied. Assume further that 1𝑔=𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1[(2𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1)^∞₀[𝑐(𝑠)+𝑎(𝑠)(𝑠+^∞_𝑘=1𝑎_𝑘+)]𝑑𝑠_∞𝑘=1𝐿_𝑘(𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}+𝑒^{∫𝑡𝑘0𝑏(𝜏)𝑑𝜏}−1)]<1, (2.21) then the integral equation (2.18) possesses a unique solution 𝑦∈𝐵𝑃𝐶[𝐽_ℎ,𝑅].

Proof. For any 𝜂∈𝐵𝑃𝐶[𝐽_ℎ,𝑅], we define the operator 𝑇 by (𝑇𝑦)(𝑡) being the right hand of (2.18) and (𝑇𝑦)(0)=(𝑇𝑦)(𝑡), 𝑡∈[−ℎ,0]. By virtue of (𝐻₁), (𝐻₂), it is obvious that 𝑇∶𝐵𝑃𝐶[𝐽_ℎ,𝑅]→𝐵𝑃𝐶[𝐽_ℎ,𝑅]. Then for any 𝑦₁,𝑦₂∈𝐵𝑃𝐶[𝐽_ℎ,𝑅], we have |(𝐵𝑦₂)(𝑠)−(𝐵𝑦₁)(𝑠)|≤(𝑠+^∞_𝑘=1𝑎_𝑘)‖𝑦₁−𝑦₂‖_𝐵. (2.22) Moreover, |(𝑇𝑦₁)(𝑡)−(𝑇𝑦₂≤1)(𝑡)|𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1[|^∞₀𝑒^{∫𝑠0𝑏(𝜏)𝑑𝜏}[𝑎(𝑠)(𝐵𝑦₂(𝑠)−𝐵𝑦₁(𝑠))+𝑐(𝑠)sup_{𝑟∈[𝑠−ℎ,𝑠]}(𝑦₂(𝑟)−𝑦₁+(𝑟))]𝑑𝑠|_∞𝑘=1𝑒^{∫𝑡𝑘0𝑏(𝜏)𝑑𝜏}𝐿_𝑘|𝑦₂(𝑡_𝑘)−𝑦₁(𝑡_𝑘)|]+|^𝑡₀𝑒^{−∫𝑡𝑠𝑏(𝜏)𝑑𝜏}[𝑎(𝑠)((𝐵𝑦₂)(𝑠)−(𝐵𝑦₁)(𝑠))+𝑐(𝑠)sup_{𝑟∈[𝑠−ℎ,𝑠]}(𝑦₂(𝑟)−𝑦₁(𝑟))]𝑑𝑠|+_{0<𝑡𝑘<𝑡}𝑒^{−∫𝑡𝑡𝑘𝑏(𝜏)𝑑𝜏}𝐿_𝑘|𝑦₂(𝑡_𝑘)−𝑦₁(𝑡_𝑘≤𝑒)|_{∫∞0𝑏(𝜏)𝑑𝜏}𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1^∞0[𝑎(𝑠)|(𝐵𝑦₂)(𝑠)−(𝐵𝑦₁)(𝑠)|+𝑐(𝑠)sup_{𝑟∈[𝑠−ℎ,𝑠]}|𝑦₂(𝑟)−𝑦₁+1(𝑟)|]𝑑𝑠𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1^∞_𝑘=1𝑒^{∫𝑡𝑘0𝑏(𝜏)𝑑𝜏}𝐿_𝑘|𝑦₂(𝑡_𝑘)−𝑦₁(𝑡_𝑘+)|_∞0[𝑎(𝑠)|(𝐵𝑦₂)(𝑠)−(𝐵𝑦₁)(𝑠)|+𝑐(𝑠)sup_{𝑟∈[𝑠−ℎ,𝑠]}|𝑦₂(𝑟)−𝑦₁+(𝑟)|]𝑑𝑠_∞𝑘=1𝐿_𝑘|𝑦₂(𝑡_𝑘)−𝑦₁(𝑡_𝑘≤1)|𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1[(2𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1)^∞₀[𝑐(𝑠)+𝑎(𝑠)(𝑠+^∞_𝑘=1𝑎_𝑘+)]𝑑𝑠_∞𝑘=1𝐿_𝑘(𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}+𝑒^{∫𝑡𝑘0𝑏(𝜏)𝑑𝜏}−1)]‖𝑦₁−𝑦₂‖_𝐵. (2.23) Thus, ‖𝑇𝑦₁−𝑇𝑦₂‖_𝐵≤𝑔‖𝑦₁−𝑦₂‖_𝐵. Hence, Banach's fixed point theorem implies that 𝑇 has a unique fixed point, that is, a unique solution of (2.18).
For any 𝜂∈𝐵𝑃𝐶[𝐽_ℎ,𝑅], define an operator 𝐴 by 𝐴∶(𝐴𝜂)(𝑡)= the right hand of (2.18) on 𝐽, and (𝐴𝜂)(𝑡)=(𝐴𝜂)(0) for 𝑡∈[−ℎ,0].

Lemmas 2.2 and 2.3 immediately yield the following result.

Lemma 2.4. 𝑦∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅] is a solution of (2.1) if and only if 𝑦∈𝐵𝑃𝐶[𝐽_ℎ,𝑅] is a fixed point of 𝐴.

Let us list some conditions for convenience.

(𝐻₃) There exist the upper and lower solutions of (2.1), that is, 𝑢₀,𝑣₀∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅], satisfying 𝑢₀(𝑡)≤𝑣₀(𝑡), 𝑢^′₀(𝑡)≤𝑓(𝑡,(𝐵𝑢₀)(𝑡),𝑢₀(𝑡),sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑢₀(𝑠)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑢₀|_{𝑡=𝑡𝑘}≤𝐼_𝑘̃((𝐵𝑢₀)(𝑡_𝑘),𝑢₀(𝑡_𝑘𝑢)),𝑘=1,2,…,₀(0)≤𝑢₀(∞),𝑢₀(𝑡)=𝑢₀(0),𝑡∈[−ℎ,0], (2.24) and 𝑣₀(𝑡) satisfies inverse inequalities above.

Define the sets [𝑢₀,𝑣₀]={𝑢∈𝑃𝐶[𝐽_ℎ,𝑅]∶𝑢₀(𝑡)≤𝑢(𝑡)≤𝑣₀(𝑡),𝑡∈𝐽_ℎ}, Ω={(𝑡,𝑥,𝑦,𝑧)∶𝑡∈𝐽,(𝐵𝑢₀)(𝑡)≤𝑥(𝑡)≤(𝐵𝑣₀)(𝑡), 𝑢₀(𝑡)≤𝑦(𝑡)≤𝑣₀(𝑡), sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑢₀(𝑠)≤𝑧(𝑡)≤sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑣₀(𝑠)}.

(𝐻₄) There exist 𝑎,𝑏,𝑐,𝑡𝑎∈𝐶[𝐽,𝑅₊]∩𝐿¹(𝐽) with 𝑏≢0, such that 𝑓(𝑡,𝑥,𝑦,𝑧)−𝑓(𝑡,𝑥,𝑦,𝑧)≥−𝑎(𝑡)(𝑥−𝑥)−𝑏(𝑡)(𝑦−𝑦)−𝑐(𝑡)(𝑧−𝐼̃𝑧),∀𝑡∈𝐽,_𝑘(𝑥,𝑦)−𝐼_𝑘̃(𝑥,𝑦)≥−𝐿_𝑘(𝑦−𝑦),𝑘=1,2,…, (2.25) where (𝑡,𝑥,𝑦,𝑧),(𝑡,𝑥,𝑦,𝑧)∈Ω,𝑥≤𝑥,𝑦≤𝑦,𝑧≤𝑧.

Theorem 2.5. Assume that conditions (𝐻₁)-(𝐻₄), (2.3), and (2.21) hold. Then (2.1) has minimal and maximal solutions 𝑢_∗,𝑣^∗∈[𝑢₀,𝑣₀]; moreover, the iterative sequences {𝑣_𝑛(𝑡)} and {𝑢_𝑛(𝑡)} converge uniformly on each 𝐽_𝑘 to 𝑣^∗(𝑡) and 𝑢_∗(𝑡), where 𝑢_𝑛(𝑡)=𝐴𝑢_𝑛−1(𝑡),𝑣_𝑛(𝑡)=𝐴𝑣_𝑛−1𝑢(𝑡),∀𝑡∈𝐽,_𝑛(𝑡)=𝑢_𝑛(0),𝑣_𝑛(𝑡)=𝑣_𝑛(0),𝑡∈[−ℎ,0],𝑛=1,2,…. (2.26)

Proof. Firstly, the proof of Lemma 2.2 implies that the operator 𝐴 is well defined.

Next, we will show that 𝑢₀≤𝐴𝑢₀,𝐴𝑣₀≤𝑣₀ and 𝐴 is nondecreasing in [𝑢₀,𝑣₀].

Indeed, for any 𝜂∈[𝑢₀,𝑣₀], we have by Lemmas 2.2 and 2.3 that 𝐴𝜂∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅] is a unique solution of (2.15), together with (2.26), we deduce that 𝑢_𝑛(𝑡)=𝑒^{−∫𝑡0𝑏(𝑠)𝑑𝑠}{(𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1)⁻¹×[^∞₀𝑒^{∫𝑠0𝑏(𝜏)𝑑𝜏}[𝑓(𝑠,(𝐵𝑢_𝑛−1)(𝑠),𝑢_𝑛−1(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1(𝑟))−𝑎(𝑠)((𝐵𝑢_𝑛)(𝑠)−(𝐵𝑢_𝑛−1)(𝑠))+𝑏(𝑠)𝑢_𝑛−1(𝑠)−𝑐(𝑠)(sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛(𝑟)−sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1+(𝑟))]𝑑𝑠_∞𝑘=1𝑒^{∫𝑡𝑘0𝑏(𝜏)𝑑𝜏}[𝐼_𝑘̃((𝐵𝑢_𝑛−1)(𝑡_𝑘),𝑢_𝑛−1(𝑡_𝑘))−𝐿_𝑘(𝑢_𝑛(𝑡_𝑘)−𝑢_𝑛−1(𝑡_𝑘+))]]_𝑡0𝑒^{∫𝑠0𝑏(𝜏)𝑑𝜏}[𝑓(𝑠,(𝐵𝑢_𝑛−1)(𝑠),𝑢_𝑛−1(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1(𝑟))−𝑎(𝑠)((𝐵𝑢_𝑛)(𝑠)−(𝐵𝑢_𝑛−1)(𝑠))+𝑏(𝑠)𝑢_𝑛−1(𝑠)−𝑐(𝑠)(sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛(𝑟)−sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1+(𝑟))]𝑑𝑠}_{0<𝑡𝑘<𝑡}𝑒^{∫𝑡𝑘𝑡𝑏(𝜏)𝑑𝜏}[𝐼_𝑘̃((𝐵𝑢_𝑛−1)(𝑡_𝑘),𝑢_𝑛−1(𝑡_𝑘))−𝐿_𝑘(𝑢_𝑛(𝑡_𝑘)−𝑢_𝑛−1(𝑡_𝑘))]. (2.27) Let 𝑢₁−𝑢₀=𝑢, then by (𝐻₃), (2.15), and the definition of 𝐴, we have 𝑢^′(𝑡)=𝑢^′₁(𝑡)−𝑢^′₀(𝑡)≥−𝑎(𝑡)(𝐷𝑢)(𝑡)−𝑏(𝑡)𝑢(𝑡)−𝑐(𝑡)sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑢(𝑠),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑢|_{𝑡=𝑡𝑘}≥−𝐿_𝑘𝑢(𝑡_𝑘),𝑘=1,2,…,𝑢(0)≥𝑢(∞),𝑢(𝑡)=𝑢(0),𝑡∈[−ℎ,0]. (2.28) This implies by Lemma 2.1 that 𝑢(𝑡)≥0, that is, 𝐴𝑢₀=𝑢₁≥𝑢₀. Analogously, we get 𝐴𝑣₀≤𝑣₀. Similar argument by the facts that 𝐴𝜂 is a solution of (2.15) and (𝐻₄), shows that 𝐴 is nondecreasing. Moreover, together with (2.26), we have 𝑢₀(𝑡)≤𝑢₁(𝑡)≤⋯≤𝑢_𝑛(𝑡)≤⋯≤𝑣_𝑛(𝑡)≤⋯≤𝑣₁(𝑡)≤𝑣₀(𝑡),𝑡∈𝐽_ℎ. (2.29) Therefore it follows from (2.29) that lim_𝑛→∞𝑢_𝑛(𝑡)=𝑢_∗(𝑡),𝑡∈𝐽_ℎ, (2.30) and then there exists a constant 𝐿^∗>0, such that ‖𝑢_𝑛‖_𝐵≤𝐿^∗,𝑛=1,2,…. Hence, for 𝑠∈𝐽, by (𝐻₁), (𝐻₂), we have |𝑓(𝑠,(𝐵𝑢_𝑛−1)(𝑠),𝑢_𝑛−1(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1(𝑟))−𝑎(𝑠)((𝐵𝑢_𝑛)(𝑠)−(𝐵𝑢_𝑛−1)(𝑠))+𝑏(𝑠)𝑢_𝑛−1(𝑠)−𝑐(𝑠)(sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛(𝑟)−sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1(𝑟))|≤[(𝑝(𝑠)+2𝑎(𝑠))(𝑠+^∞_𝑘=1𝑎_𝑘)+𝑞(𝑠)+𝑙(𝑠)+𝑏(𝑠)+2𝑐(𝑠)]𝐿^∗+(2𝑎(𝑠)+𝑝(𝑠))|𝑥₀|𝐼̃|,_𝑘((𝐵𝑢_𝑛−1)(𝑡_𝑘),𝑢_𝑛−1(𝑡_𝑘))−𝐿_𝑘(𝑢_𝑛(𝑡_𝑘)−𝑢_𝑛−1(𝑡_𝑘))|≤𝐿^∗(𝐿_𝑘+2𝐿_𝑘),𝑛,𝑘=1,2,…. (2.31) Hence it follows from (2.26), (2.31) that {𝑢_𝑛(𝑡)} is equicontinuous on each 𝐽_𝑘. So in view of (2.30), an application of 𝐴𝑟𝑧𝑒𝑙𝑎-𝐴𝑠𝑐𝑜𝑙𝑖^′𝑠 theorem and diagonal method implies that there exists a subsequence {𝑢_𝑛𝑖(𝑡)}⊂{𝑢_𝑛(𝑡)} such that {𝑢_𝑛𝑖(𝑡)} converges uniformly on each 𝐽_𝑘 to 𝑢_∗(𝑡). Then the whole sequence {𝑢_𝑛(𝑡)} converges uniformly on each 𝐽_𝑘 to 𝑢_∗(𝑡). Thus 𝑢_∗(𝑡)∈𝑃𝐶[𝐽_ℎ,𝑅], and the fact that ‖𝑢_𝑛‖_𝐵≤𝐿^∗ implies ‖𝑢_∗‖_𝐵≤𝐿^∗. Hence 𝑢_∗∈𝐵𝑃𝐶[𝐽_ℎ,𝑅]. In view of (2.30), the continuity of 𝑓 and 𝐼_𝑘̃ gives that 𝑓(𝑠,(𝐵𝑢_𝑛−1)(𝑠),𝑢_𝑛−1(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1(𝑟))−𝑎(𝑠)(𝑢_𝑛(𝑠)−𝑢_𝑛−1(𝑠))+𝑏(𝑠)𝑢_𝑛−1(𝑠)−𝑐(𝑠)(sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛(𝑟)−sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_𝑛−1(𝑟))→𝑓(𝑠,(𝐵𝑢_∗)(𝑠),𝑢_∗(𝑠),sup_{𝑟∈[𝑠−ℎ,𝑠]}𝑢_∗(𝑟))+𝑏(𝑠)𝑢_∗𝐼̃(𝑠),𝑠∈𝐽,𝑛→∞,_𝑘((𝐵𝑢_𝑛−1)(𝑡_𝑘),𝑢_𝑛−1(𝑡_𝑘))−𝐿_𝑘(𝑢_𝑛(𝑡_𝑘)−𝑢_𝑛−1(𝑡_𝑘→𝐼̃))_𝑘((𝐵𝑢_∗)(𝑡_𝑘),𝑢_∗(𝑡_𝑘)),𝑛→∞,𝑘=1,2,…. (2.32) By the facts that 𝑎(𝑠),𝑏(𝑠),𝑐(𝑠),𝑠𝑎(𝑠),𝑝(𝑠),𝑞(𝑠),𝑙(𝑠),𝑠𝑝(𝑠)∈𝐿¹(𝐽), and ∑^∞_𝑘=1𝐿^∗(𝐿_𝑘+2𝐿_𝑘) is convergent, observing (2.27) and taking limits as 𝑛→∞, the dominated convergence theorem yields that 𝑢_∗(𝑡)=(𝑒^{∫∞0𝑏(𝜏)𝑑𝜏}−1)⁻¹×{^∞₀𝑒^{∫𝑠𝑡𝑏(𝜏)𝑑𝜏}[𝑓(𝑠,(𝐵𝑢_∗)(𝑠),𝑢_∗(𝑠),sup_{𝑟∈[𝑡−ℎ,𝑡]}𝑢_∗(𝑟))+𝑏(𝑠)𝑢_∗+(𝑠)]𝑑𝑠_∞𝑘=1𝑒^{∫𝑡𝑘𝑡𝑏(𝜏)𝑑𝜏}𝐼_𝑘̃((𝐵𝑢_∗)(𝑡_𝑘),𝑢_∗(𝑡_𝑘+))}_𝑡0𝑒^{∫𝑠𝑡𝑏(𝜏)𝑑𝜏}[𝑓(𝑠,(𝐵𝑢_∗)(𝑠),𝑢_∗(𝑠),sup_{𝑟∈[𝑡−ℎ,𝑡]}𝑢_∗(𝑟))+𝑏(𝑠)𝑢_∗(𝑠)]𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑒^{∫𝑡𝑘𝑡𝑏(𝜏)𝑑𝜏}𝐼_𝑘̃((𝐵𝑢_∗)(𝑡_𝑘),𝑢_∗(𝑡_𝑘)),𝑡∈𝐽, (2.33) that is, 𝑢_∗(𝑡)=𝐴𝑢_∗(𝑡),𝑢_∗(𝑡) is a fixed point of 𝐴. It is easy to check that 𝑢_∗(𝑡)∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅]. Therefore we conclude by Lemma 2.4 that 𝑢_∗(𝑡) is a solution of (2.1).

Similarly, we can show that {𝑣_𝑛(𝑡)} converges uniformly on each 𝐽_𝑘 to 𝑣^∗(𝑡), and 𝑣^∗∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅] is also a solution of (2.1).

Clearly, 𝑢_∗,𝑣^∗∈[𝑢₀,𝑣₀]. Using a standard method, we can show that 𝑢_∗,𝑣^∗ is the minimal and maximal solutions of (2.1) in [𝑢₀,𝑣₀].

Remark 2.6. Theorem of [3] is a special case of Theorem 2.5 in this paper, where 𝑓 and 𝐼_𝑘̃ did not involve the operator 𝐵. Hence Theorem 2.5 in this paper extends and improves the result of [3].

Remark 2.7. In system 2.1, if the interval is finite [0,𝑚], then the conditions of (𝐻₁), (𝐻₂) can be deleted. Thus Theorem 2.5 in this paper extends and improves the result of [2].

3. Main Result for Second-Order Impulsive Differential Equation

Let us list other conditions for convenience.

(𝐻^′₃) There exist 𝑦₀,𝑧₀∈TPC¹[𝐽_ℎ,𝑅]∩𝐶²[𝐽^′,𝑅], and 𝑦₀(𝑡)≤𝑧₀(𝑡),𝑦^′₀(𝑡)≤𝑧^′₀(𝑡) such that 𝑧₀^′′(𝑡)≥𝑓(𝑡,𝑧₀(𝑡),𝑧^′₀(𝑡),sup_{𝑟∈[𝑡−ℎ,𝑡]}𝑧^′₀(𝑟)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑧₀|_{𝑡=𝑡𝑘}=𝑎_𝑘𝑧^′₀(𝑡_𝑘),𝑘=1,2,…,Δ𝑧^′₀|_{𝑡=𝑡𝑘}≥𝐼_𝑘̃(𝑧₀(𝑡_𝑘),𝑧^′₀(𝑡_𝑘𝑧)),𝑘=1,2,…,₀(0)=𝑥₀,𝑧^′₀(0)≥𝑧^′₀𝑧(∞),_′0(0)=𝑧^′₀(𝑡),𝑡∈[−ℎ,0], (3.1) and 𝑦₀(𝑡) satisfies inverse inequalities above.

(𝐻^′₄) There exist 𝑎,𝑏,𝑐,𝑡𝑎∈𝐶[𝐽,𝑅₊]∩𝐿¹(𝐽) with 𝑏≢0, such that 𝑓(𝑡,𝑥,𝑦,𝑧)−𝑓(𝑡,𝑥,𝑦,𝑧)≥−𝑎(𝑡)(𝑥−𝑥)−𝑏(𝑡)(𝑦−𝑦)−𝑐(𝑡)(𝑧−𝐼̃𝑧),∀𝑡∈𝐽,_𝑘(𝑥,𝑦)≥𝐼_𝑘̃(𝑥,𝑦),𝑘=1,2,…, (3.2) where (𝑡,𝑥,𝑦,𝑧),(𝑡,𝑥,𝑦,𝑧)∈Ω^′, 𝑥≤𝑥,𝑦≤𝑦,𝑧≤𝑧, Ω^′={(𝑡,𝑥,𝑦,𝑧)∶𝑡∈𝐽, 𝑦₀(𝑡)≤𝑥(𝑡)≤𝑧₀(𝑡), 𝑦^′₀(𝑠)≤𝑦(𝑡)≤𝑧^′₀(𝑠), sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑦^′₀(𝑠)≤𝑧(𝑡)≤sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑧^′₀(𝑠)}.

Theorem 3.1. Assume that conditions (𝐻₁), (𝐻₂), (𝐻^′₃), (𝐻^′₄) and (2.3), (2.21) hold. Then IBVP(1.1) has minimal and maximal solutions 𝑦_∗,𝑧^∗∈TPC¹[𝐽_ℎ,𝑅]∩𝐶²[𝐽^′,𝑅].

Proof. Let 𝑥^′(𝑡)=𝑦(𝑡). Then IBVP(1.1) is equivalent to the following system: 𝑥^′(𝑡)=𝑦(𝑡),𝑦^′(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑦(𝑡),sup_{𝑠∈[𝑡−ℎ,𝑡]}𝑦(𝑠)),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑥|_{𝑡=𝑡𝑘}=𝑎_𝑘𝑦(𝑡_𝑘),Δ𝑦|_{𝑡=𝑡𝑘}=𝐼_𝑘̃(𝑥(𝑡_𝑘),𝑦(𝑡_𝑘))𝑘=1,2,…,𝑥(0)=𝑥₀,𝑦(0)=𝑦(∞),𝑦(𝑡)=𝑦(0),𝑡∈[−ℎ,0]. (3.3) Clearly, the system 𝑥^′(𝑡)=𝑦(𝑡),𝑡∈𝐽,Δ𝑥|_{𝑡=𝑡𝑘}=𝑎_𝑘𝑦(𝑡_𝑘),𝑘=1,2,…,𝑥(0)=𝑥₀, (3.4) has a unique solution 𝑥∈𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅] and 𝑥(𝑡)=𝑥₀+∫^𝑡₀𝑦(𝑠)𝑑𝑠+∑_{0<𝑡𝑘<𝑡}𝑎_𝑘𝑦(𝑡_𝑘). Let (𝐵𝑦)(𝑡)=𝑥₀+^𝑡₀𝑦(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑎_𝑘𝑦(𝑡_𝑘), (3.5) we have 𝑥(𝑡)=(𝐵𝑦)(𝑡), and then IBVP (1.1) is transformed into first-order impulsive equations (2.1).
Let 𝑦^′₀(𝑡)=𝑢₀(𝑡),𝑧^′₀(𝑡)=𝑣₀(𝑡), we have 𝑢₀≤𝑣₀. By the condition (𝐻^′₃) and the definition of 𝐵, we get that 𝑦₀(𝑡)=(𝐵𝑢₀)(𝑡), 𝑧₀(𝑡)=(𝐵𝑣₀)(𝑡), and 𝑢₀,𝑣₀ satisfy (𝐻₃). By the condition (𝐻^′₄), it is easy to see that (𝐻₄) holds. Hence, it follows from Theorem 2.5 that (2.1) has minimal and maximal solutions 𝑢_∗,𝑣^∗∈𝑇𝑃𝐶[𝐽_ℎ,𝑅]∩𝐶¹[𝐽^′,𝑅].
Let 𝑦_∗(𝑡)=(𝐵𝑢_∗)(𝑡),𝑧^∗(𝑡)=(𝐵𝑣^∗)(𝑡), then 𝑦_∗,𝑧^∗∈TPC¹[𝐽_ℎ,𝑅]∩𝐶²[𝐽^′,𝑅]. It follows by simple calculation that 𝑦^′_∗(𝑡)=𝑢_∗(𝑡),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑦_∗|_{𝑡=𝑡𝑘}=𝑎_𝑘𝑢_∗(𝑡_𝑘𝑦),𝑘=1,2,…,_∗(0)=𝑥₀,(𝑧^∗)^′(𝑡)=𝑣^∗(𝑡),𝑡∈𝐽,𝑡≠𝑡_𝑘,Δ𝑧^∗|_{𝑡=𝑡𝑘}=𝑎_𝑘𝑣^∗(𝑡_𝑘𝑧),𝑘=1,2,…,_∗(0)=𝑥₀. (3.6)(3.7) The facts that 𝑢_∗,𝑣^∗ satisfies (2.1) and 𝑦_∗,𝑧^∗ satisfies (3.7) imply that 𝑦_∗,𝑧^∗∈TPC¹[𝐽_ℎ,𝑅]∩𝐶²[𝐽^′,𝑅] are solutions of IBVP(1.1).
Finally, it is easy to show that 𝑦_∗,𝑧^∗ are the minimal and maximal solutions of IBVP(1.1), respectively. We complete the proof.

Acknowledgments

The authors are grateful to the referees for their comments and suggestions. This research was partially supported by the Science Foundation of Shanxi Province of China (2006011013) and Technological Research and Development Foundation of Institutions of Higher Learning in Shanxi Province (200811053).