Table of Contents Author Guidelines Submit a Manuscript
Letter to the Editor
Mathematical Problems in Engineering
Volume 2009, Article ID 258090, 16 pages
http://dx.doi.org/10.1155/2009/258090
Research Article

MultiPoint BVPs for Second-Order Functional Differential Equations with Impulses

1Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
2Department of Mathematics, Hunan First Normal University, Changsha, Hunan 410205, China
3College of Science, Zhejiang Forestry University, Hangzhou, Zhejiang 311300, China
4Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 14 April 2009; Accepted 10 June 2009

Academic Editor: Fernando Lobo Pereira

Copyright © 2009 Xuxin Yang et al. 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.

Abstract

This paper is concerned about the existence of extreme solutions of multipoint boundary value problem for a class of second-order impulsive functional differential equations. We introduce a new concept of lower and upper solutions. Then, by using the method of upper and lower solutions introduced and monotone iterative technique, we obtain the existence results of extreme solutions.

1. Introduction

In this paper, we consider the multipoint boundary value problems for the impulsive functional differential equation: 𝑢[](𝑡)=𝑓(𝑡,𝑢(𝑡),𝑢(𝜃(𝑡))),𝑡𝐽=0,1,𝑡𝑡𝑘,Δ𝑢𝑡𝑘=𝐼𝑘𝑢𝑡𝑘,𝑘=1,,𝑚,𝑢(0)𝑎𝑢(0)=𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)=𝑑𝑢(𝜉),(1.1) where 𝑓𝐶(𝐽×𝐑2,𝐑),    0𝜃(𝑡)𝑡,𝑡𝐽,𝜃𝐶(𝐽),𝑎0,𝑏0,0𝑐1,0𝑑1, 0<𝜂,𝜉<1. 0<𝑡1<𝑡2<<𝑡𝑚<1, 𝑓 is continuous everywhere except at {𝑡𝑘}×𝑅2; 𝑓(𝑡+𝑘,,), and 𝑓(𝑡𝑘,,) exist with 𝑓(𝑡𝑘,,)=𝑓(𝑡𝑘,,); 𝐼𝑘𝐶(𝑅,𝑅),Δ𝑢(𝑡𝑘)=𝑢(𝑡+𝑘)𝑢(𝑡𝑘). Denote 𝐽=𝐽{𝑡𝑖,𝑖=1,2,,𝑚}. Let 𝑃𝐶(𝐽,𝑅)={𝑢𝐽𝑅;𝑢(𝑡)|𝐽 is continuous, 𝑢(𝑡+𝑘) and 𝑢(𝑡𝑘) exist with 𝑢(𝑡𝑘)=𝑢(𝑡𝑘),𝑘=1,2,,𝑚}; 𝑃𝐶1(𝐽,𝑅)={𝑢𝐽𝑅;𝑢(𝑡)|𝐽 is continuous differentiable, 𝑢(𝑡+𝑘) and 𝑢(𝑡𝑘) exist with 𝑢(𝑡𝑘)=𝑢(𝑡𝑘),𝑘=1,2,,𝑚}. Let 𝐸=𝑃𝐶1(𝐽,𝑅)𝐶2(𝐽,𝑅). A function 𝑢𝐸 is called a solution of BVP(1.1) if it satisfies (1.1).

The method of upper and lower solutions combining monotone iterative technique offers an approach for obtaining approximate solutions of nonlinear differential equations [13]. There exist much literature devoted to the applications of this technique to general boundary value problems and periodic boundary value problems, for example, see [1, 46] for ordinary differential equations, [711] for functional differential equations, and [12] for differential equations with piecewise constant arguments. For the studies about some special boundary value problems, for example, Lidston boundary value problems and antiperiodic boundary value problems, one may see [13, 14] and the references cited therein.

Here, we hope to mention some papers where existence results of solutions of certain boundary value problems of impulsive differential equations were studied [11, 15] and certain multipoint boundary value problems also were studied [6, 1621]. These works motivate that we study the multipoint boundary value problems for the impulsive functional differential equation (1.1).

We also note that when 𝐼𝑘=0 and 𝜃(𝑡)=𝑡, the boundary value problem (1.1) reduces to multi-point boundary value problems for ordinary differential equations which have been studied in many papers, see, for example, [6, 1618] and the references cited therein. To our knowledge, only a few papers paid attention to multi-point boundary value problems for impulsive functional differential equations.

In this paper, we are concerned with the existence of extreme solutions for the boundary value problem (1.1). The paper is organized as follows. In Section 2, we establish two comparison principles. In Section 3, we consider a linear problem associated to (1.1) and then give a proof for the existence theorem. In Section 4, we first introduce a new concept of lower and upper solutions. By using the method of upper and lower solutions with a monotone iterative technique, we obtain the existence of extreme solutions for the boundary value problem (1.1).

2. Comparison Principles

In the following, we always assume that the following condition (𝐻) is satisfied:

(𝐻)𝑎0,𝑏0,0𝑐1,0𝑑1,0<𝜂,𝜉<1,𝑎+𝑐>0,𝑏+𝑑>0.

For any given function 𝑔𝐸, we denote 𝐴𝑔=max𝑔(0)𝑎𝑔(0)𝑐𝑔(𝜂),𝑎𝜋+𝑐sin𝜋𝜂𝑔(1)+𝑏𝑔(1)𝑑𝑔(𝜉),𝐵𝑏𝜋+𝑑sin𝜋𝜉𝑔𝐴=max𝑔,0,𝑐𝑔(𝑡)=𝐵𝑔sin(𝜋𝑡),𝑟=𝜋2.(2.1) We now present main results of this section.

Theorem 2.1. Assume that  𝑢𝐸 satisfies 𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))0,𝑡𝐽,𝑡𝑡𝑘,Δ𝑢𝑡𝑘𝐿𝑘𝑢𝑡𝑘,𝑘=1,,𝑚,𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉),(2.2) where 𝑎0,𝑏0, 0𝑐1,0𝑑1,0<𝜂,  𝜉<1, 𝐿𝑘0 and constants 𝑀,𝑁 satisfy 𝑀>0,𝑁0,𝑀+𝑁2+𝑚𝑘=1𝐿𝑘1.(2.3) Then 𝑢(𝑡)0 for 𝑡𝐽.

Proof. Suppose, to the contrary, that 𝑢(𝑡)>0 for some 𝑡𝐽.
If 𝑢(1)=max𝑡𝐽𝑢(𝑡)>0, then 𝑢(1)0, 𝑢(1)𝑢(𝜉), and 𝑑𝑢(𝜉)𝑢(1)𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉).(2.4) So 𝑑=1 and 𝑢(𝜉) is a maximum value.
If 𝑢(0)=max𝑡𝐽𝑢(𝑡)>0, then 𝑢(0)0, 𝑢(0)𝑢(𝜂), and 𝑐𝑢(𝜂)𝑢(0)𝑢(0)𝑏𝑢(0)𝑐𝑢(𝜂).(2.5) So 𝑐=1 and 𝑢(𝜂) is a maximum value.
Therefore, there is a 𝜌(0,1) such that 𝑢(𝜌)=max𝑡𝐽𝑢(𝑡)>0,𝑢𝜌+0.(2.6)
Suppose that 𝑢(𝑡)0 for 𝑡𝐽. From the first inequality of (2.2), we obtain that 𝑢(𝑡)0 for 𝑡𝐽. Hence 𝑢(0)=max𝑡𝐽𝑢(𝑡)or𝑢(1)=max𝑡𝐽𝑢(𝑡).(2.7)
If 𝑢(0)0, then 𝑢(𝑡)0, 𝑡(𝑡𝑖,𝑡𝑖+1], it is easy to obtain that 𝑢(𝑡) is nondecreasing. Since 𝑢(1)𝑑𝑢(𝜉)𝑢(1), it follows that 𝑢(𝑡)𝐾 (𝐾>0) for 𝑡[𝜉,1]. From the first inequality of (2.2), we have that when 𝑡[𝜉,1],0<𝑀𝐾𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))𝑢(𝑡)=0,(2.8) which is a contradiction.
If 𝑢(0)0, then 𝑢(0)=max𝑡𝐽𝑢(𝑡)>0, or 𝑢(1)=max𝑡𝐽𝑢(𝑡)>0. If 𝑢(0)=max𝑡𝐽𝑢(𝑡)>0, then 𝑢(𝑡)𝐾 (𝐾>0) for 𝑡[0,𝜂]. If 𝑢(1)=max𝑡𝐽𝑢(𝑡)>0, then 𝑢(𝑡)𝐾 for 𝑡[𝜉,1].
From the first inequality of (2.2), we have that when 𝑡[𝜉,1],0<𝑀𝐾𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))𝑢(𝑡)=0,(2.9) which is a contradiction.
Suppose that there exist 𝑡1,𝑡2𝐽 such that 𝑢(𝑡1)>0 and 𝑢(𝑡2)<0. We consider two possible cases.Case 1 (𝑢(0)>0). Since 𝑢(𝑡2)<0, there is 𝜅>0,𝜀>0 such that 𝑢(𝜅)=0, 𝑢(𝑡)0 for 𝑡[0,𝜅) and 𝑢(𝑡)<0 for all 𝑡(𝜅,𝜅+𝜀]. It is easy to obtain that 𝑢(𝑡)0 for 𝑡[0,𝜅]. If 𝑡<𝜅, then 0<𝑀𝑢(𝑡)𝑢(𝑡)0, a contradiction. Hence 𝑡>𝜅+𝜀. Let 𝑡[0,𝑡) such that 𝑢(𝑡)=min𝑡[0,𝑡)𝑢(𝑡), then 𝑢(𝑡)<0. From the first inequality of (2.2), we have 𝑢𝑡(𝑡)(𝑀+𝑁)𝑢,𝑡0,𝑡,𝑡𝑡𝑘,Δ𝑢𝑡𝑘𝐿𝑘𝑢𝑡𝑘,𝑘=1,,𝑚.(2.10) Integrating the above inequality from 𝑠(𝑡𝑠𝑡) to 𝑡, we obtain 𝑢𝑡𝑢(𝑡𝑠)(𝑡𝑠𝑀+𝑁)𝑢+𝑠<𝑡𝑘<𝑡𝐿𝑘𝑢𝑡𝑘𝑡𝑡𝑠(𝑀+𝑁)𝑢+𝑚𝑘=1𝐿𝑘𝑢𝑡.(2.11) Hence 𝑢[𝑡(𝑠)𝑠(𝑀+𝑁)+𝑚𝑘=1𝐿𝑘𝑢𝑡,𝑡𝑠𝑡,(2.12) and then integrate from 𝑡 to 𝑡 to obtain 𝑡𝑢𝑡<𝑢𝑡𝑢𝑡𝑡𝑠𝑡𝑡(𝑀+𝑁)𝑢𝑑𝑠𝑚𝑘=1𝐿𝑘𝑢𝑡𝑀+𝑁2𝑡𝑡2+𝑚𝑘=1𝐿𝑘𝑢𝑡𝑀+𝑁2+𝑚𝑘=1𝐿𝑘𝑢𝑡.(2.13) From (2.3), we have that 𝑢(𝑡)>0. This is a contradiction.Case 2 (𝑢(0)0). Let 𝑡[0,𝑡) such that 𝑢(𝑡)=min𝑡[0,𝑡)𝑢(𝑡)0. From the first inequality of (2.2), we have 𝑢𝑡(𝑡)(𝑀+𝑁)𝑢,𝑡0,𝑡,𝑡𝑡𝑘,Δ𝑢𝑡𝑘𝐿𝑘𝑢𝑡𝑘,𝑘=1,,𝑚.(2.14) The rest proof is similar to that of Case 1. The proof is complete.

Theorem 2.2. Assume that (𝐻) holds and 𝑢𝐸 satisfies 𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))+(𝑀+𝑟)𝑐𝑢(𝑡)+𝑁𝑐𝑢(𝜃(𝑡))0,𝑡𝐽,𝑡𝑡𝑘,Δ𝑢𝑡𝑘𝐿𝑘𝑢𝑡𝑘+𝐿𝑘𝑐𝑢𝑡𝑘,𝑘=1,,𝑚,(2.15) where constants 𝑀,𝑁 satisfy (2.3), and 𝐿𝑘0, then 𝑢(𝑡)0 for 𝑡𝐽.

Proof. Assume that 𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉), then 𝑐𝑢(𝑡)0. By Theorem 2.1, 𝑢(𝑡)0.
Assume that 𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)>𝑑𝑢(𝜉), then
𝑐𝑢(𝑡)=sin(𝜋𝑡)𝑏𝜋+𝑑sin(𝜋𝜉)𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉).(2.16)
Put 𝑦(𝑡)=𝑢(𝑡)+𝑐𝑢(𝑡),𝑡𝐽, then 𝑦(𝑡)𝑢(𝑡) for all 𝑡𝐽, and
𝑦(𝑡)=𝑢(𝑡)+𝜋cos(𝜋𝑡)𝑏𝜋+𝑑sin(𝜋𝜉)𝑢(1)+𝑏𝑢𝑦(1)𝑑𝑢(𝜉),𝑡𝐽,(𝑡)=𝑢(𝑡)𝑟𝑐𝑢(𝑡),𝑡𝐽.(2.17) Hence 𝑦(0)=𝑢(0),𝑦(1)=𝑢(1),𝑦(𝜉)=𝑢(𝜉)+sin(𝜋𝜉)𝑏𝜋+𝑑sin(𝜋𝜉)𝑢(1)+𝑏𝑢(,𝑦1)𝑑𝑢(𝜉)(0)=𝑢(𝜋0)+𝑏𝜋+𝑑sin(𝜋𝜉)𝑢(1)+𝑏𝑢(,𝑦1)𝑑𝑢(𝜉)(1)=𝑢(𝜋1)𝑏𝜋+𝑑sin(𝜋𝜉)𝑢(1)+𝑏𝑢(,1)𝑑𝑢(𝜉)𝑦(𝑡)+𝑀𝑦(𝑡)+𝑁𝑦(𝜃(𝑡))=𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))+(𝑀+𝑟)𝑐𝑢(𝑡)+𝑁𝑐𝑢(𝜃(𝑡))0,𝑦(0)𝑎𝑦(0)=𝑢(0)𝑎𝑢(0)𝑎𝜋𝑏𝜋+𝑑sin(𝜋𝜉)𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉)𝑐𝑢(𝜂)𝑐𝑦(𝜂),𝑦(1)+𝑏𝑦(1)𝑑𝑦(𝜉)=𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉)𝑏𝜋𝑏𝜋+𝑑sin(𝜋𝜉)𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉)𝑑sin(𝜋𝜉)𝑏𝜋+𝑑sin𝜋𝜉𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉)0,Δ𝑦𝑡𝑘=Δ𝑢𝑡𝑘Δ𝑐𝑢𝑡𝑘𝐿𝑘𝑢𝑡𝑘+𝐿𝑘𝑐𝑢𝑡𝑘=𝐿𝑘𝑦𝑡𝑘.(2.18) By Theorem 2.1, 𝑦(𝑡)0 for all 𝑡𝐽, which implies that 𝑢(𝑡)0 for 𝑡𝐽.
Assume that 𝑢(0)𝑎𝑢(0)>𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉), then
𝑐𝑢(𝑡)=sin𝜋𝑡𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂).(2.19) Put 𝑦(𝑡)=𝑢(𝑡)+𝑐𝑢(𝑡),𝑡𝐽, then 𝑦(𝑡)𝑢(𝑡) for all 𝑡𝐽, and 𝑦(𝑡)=𝑢(𝑡)+𝜋cos(𝜋𝑡)𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢𝑦(0)𝑐𝑢(𝜂),𝑡𝐽,(𝑡)=𝑢(𝑡)𝑟𝑐𝑢(𝑡),𝑡𝐽.(2.20) Hence 𝑦(0)=𝑢(0),𝑦(1)=𝑢(1),𝑦(𝜂)=𝑢(𝜂)+sin(𝜋𝜂)𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢(,𝑦0)𝑐𝑢(𝜂)(0)=𝑢(𝜋0)+𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢(,𝑦0)𝑐𝑢(𝜂)(1)=𝑢(𝜋1)𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢(,0)𝑐𝑢(𝜂)𝑦(𝑡)+𝑀𝑦(𝑡)+𝑁𝑦(𝜃(𝑡))=𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))+(𝑀+𝑟)𝑐𝑢(𝑡)+𝑁𝑐𝑢(𝜃(𝑡))0,𝑦(0)𝑎𝑦(0)𝑐𝑦(𝜂)=𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂)𝑎𝜋𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂)𝑐sin(𝜋𝜂)𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂)0,𝑦(1)+𝑏𝑦(1)=𝑢(1)+𝑏𝑢(1)𝑏𝜋𝑎𝜋+𝑐sin(𝜋𝜂)𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂)𝑑𝑢(𝜉)𝑑𝑦(𝜉),Δ𝑦𝑡𝑘=Δ𝑢𝑡𝑘+Δ𝑐𝑢𝑡𝑘𝐿𝑘𝑢𝑡𝑘+𝐿𝑘𝑐𝑢𝑡𝑘=𝐿𝑘𝑦𝑡𝑘.(2.21)
By Theorem 2.1, 𝑦(𝑡)0 for all 𝑡𝐽, which implies that 𝑢(𝑡)0 for 𝑡𝐽.
Assume that 𝑢(0)𝑎𝑢(0)>𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)>𝑑𝑢(𝜉), then 𝑐𝑢(𝑡)=𝐴𝑢sin(𝜋𝑡).
Put 𝑦(𝑡)=𝑢(𝑡)+𝑐𝑢(𝑡),𝑡𝐽, then 𝑦(𝑡)𝑢(𝑡) for all 𝑡𝐽, and 𝑦(𝑡)=𝑢(𝑡)+𝐴𝑢𝑦𝜋cos(𝜋𝑡),𝑡𝐽,(𝑡)=𝑢(𝑡)𝑟𝑐𝑢(𝑡),𝑡𝐽.(2.22) Hence 𝑦𝑦(0)=𝑢(0),𝑦(1)=𝑢(1),(𝜂)=𝑢(𝜂)+𝐴𝑢sin(𝜋𝜂),𝑦(𝜉)=𝑢(𝜉)+𝐴𝑢𝑦sin(𝜋𝜉),(0)=𝑢(0)+𝐴𝑢𝜋,𝑦(1)=𝑢(1)𝐴𝑢𝜋,𝑦(𝑡)+𝑀𝑦(𝑡)+𝑁𝑦(𝜃(𝑡))=𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))+(𝑀+𝑟)𝑐𝑢(𝑡)+𝑁𝑐𝑢(𝜃(𝑡))0,𝑦(0)𝑎𝑦(0)𝑐𝑦(𝜂)=𝑢(0)𝑎𝑢(0)𝑐𝑢(𝜂)𝑎𝐴𝑢𝜋𝑐𝐴𝑢sin(𝜋𝜂)0,𝑦(1)+𝑏𝑦(1)𝑑𝑦(𝜉)=𝑢(1)+𝑏𝑢(1)𝑑𝑢(𝜉)𝑏𝐴𝑢𝜋𝑑𝐴𝑢sin(𝜋𝜉)0,Δ𝑦𝑡𝑘=Δ𝑢𝑡𝑘+Δ𝑐𝑢𝑡𝑘𝐿𝑘𝑢𝑡𝑘+𝐿𝑘𝑐𝑢𝑡𝑘=𝐿𝑘𝑦𝑡𝑘.(2.23)
By Theorem 2.1, 𝑦(𝑡)0 for all 𝑡𝐽, which implies that 𝑢(𝑡)0 for 𝑡𝐽. The proof is complete.

3. Linear Problem

In this section, we consider the linear boundary value problem 𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))=𝜎(𝑡),𝑡𝐽,𝑡𝑡𝑘,Δ𝑢𝑡𝑘=𝐿𝑘𝑢𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚,𝑢(0)𝑎𝑢(0)=𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)=𝑑𝑢(𝜉).(3.1)

Theorem 3.1. Assume that (𝐻) holds, 𝜎𝐶(𝐽), 𝑒𝑘𝑅, and constants 𝑀,𝑁 satisfy (2.3) with 𝑎𝜇=(1+2𝑏)+12(𝑎+𝑏+1)81+2𝑏𝑎+𝑏+12(𝑀+𝑁)+1+(1+𝑏)2𝑎+𝑏+1𝑚𝑘=1𝐿𝑘<1.(3.2) Further suppose that there exist 𝛼,𝛽𝐸 such that (1)𝛼𝛽 on 𝐽,(2)𝛼(𝑡)+𝑀𝛼(𝑡)+𝑁𝛼(𝜃(𝑡))+(𝑀+𝑟)𝑐𝛼(𝑡)+𝑁𝑐𝛼(𝜃(𝑡))𝜎(𝑡),𝑡𝐽,𝑡𝑡𝑘,Δ𝛼𝑡𝑘𝐿𝑘𝛼𝑡𝑘+𝐿𝑘𝑐𝛼𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚,(3.3)(3)𝛽(𝑡)+𝑀𝛽(𝑡)+𝑁𝛽(𝜃(𝑡))(𝑀+𝑟)𝑐𝛽(𝑡)+𝑁𝑐𝛽(𝜃(𝑡))𝜎(𝑡),𝑡𝐽,𝑡𝑡𝑘,Δ𝛽𝑡𝑘𝐿𝑘𝛽𝑡𝑘𝐿𝑘𝑐𝛽𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚.(3.4)Then the boundary value problem (3.1) has one unique solution 𝑢(𝑡) and 𝛼𝑢𝛽 for 𝑡𝐽.

Proof. We first show that the solution of (3.1) is unique. Let 𝑢1,𝑢2 be the solution of (3.1) and set 𝑣=𝑢1𝑢2. Thus, 𝑣(𝑡)+𝑀𝑣(𝑡)+𝑁𝑣(𝜃(𝑡))=0,𝑡𝐽,𝑡𝑡𝑘,Δ𝑣𝑡𝑘=𝐿𝑘𝑣𝑡𝑘,𝑘=1,,𝑚,𝑣(0)𝑎𝑣(0)=𝑐𝑣(𝜂),𝑣(1)+𝑏𝑣(1)=𝑑𝑣(𝜉).(3.5) By Theorem 2.1, we have that 𝑣0 for 𝑡𝐽, that is, 𝑢1𝑢2 on 𝐽. Similarly, one can obtain that 𝑢2𝑢1 on 𝐽. Hence 𝑢1=𝑢2.
Next, we prove that if 𝑢 is a solution of (3.1), then 𝛼𝑢𝛽. Let 𝑝=𝛼𝑢. From boundary conditions, we have that 𝑐𝛼(𝑡)=𝑐𝑝(𝑡) for all 𝑡𝐽. From (2) and (3.1), we have
𝑝(𝑡)+𝑀𝑝(𝑡)+𝑁𝑝(𝜃(𝑡))+(𝑀+𝑟)𝑐𝑝(𝑡)+𝑁𝑐𝑝(𝜃(𝑡))0,𝑡𝐽,𝑡𝑡𝑘,Δ𝑝𝑡𝑘𝐿𝑘𝑝𝑡𝑘+𝐿𝑘𝑐𝑝𝑡𝑘,𝑘=1,,𝑚.(3.6) By Theorem 2.1, we have that 𝑝=𝛼𝑢0 on 𝐽. Analogously, 𝑢𝛽 on 𝐽.
Finally, we show that the boundary value problem (3.1) has a solution by five steps as follows.
Step 1. Let 𝛼(𝑡)=𝛼(𝑡)+𝑐𝛼(𝑡),𝛽(𝑡)=𝛽(𝑡)𝑐𝛽(𝑡). We claim that(1)𝛼(𝑡)+𝑀𝛼(𝑡)+𝑁𝛼(𝜃(𝑡))+(𝑀+𝑟)𝑐𝛼(𝑡)+𝑁𝑐𝛼(𝜃(𝑡))𝜎(𝑡)for𝑡𝐽,𝑡𝑡𝑘,Δ𝛼𝑡𝑘𝐿𝑘𝛼𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚,(3.7)(2)𝛽(𝑡)+𝑀𝛽(𝑡)+𝑁𝛽(𝜃(𝑡))(𝑀+𝑟)𝑐𝛽(𝑡)+𝑁𝑐𝛽(𝜃(𝑡))𝜎(𝑡)for𝑡𝐽,𝑡𝑡𝑘,Δ𝛽𝑡𝑘𝐿𝑘𝛽𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚,(3.8)(3)𝛼(𝑡)𝛼(𝑡)𝛽(𝑡)𝛽(𝑡) for 𝑡𝐽.
From (2) and (3), we have
𝛼(𝑡)+𝑀𝛼(𝑡)+𝑁𝛼(𝜃(𝑡))𝜎(𝑡),𝑡𝐽,𝑡𝑡𝑘,Δ𝛼𝑡𝑘𝐿𝑘𝛼𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚.(3.9)𝛽(𝑡)+𝑀𝛽(𝑡)+𝑁𝛽(𝜃(𝑡))𝜎(𝑡),𝑡𝐽,𝑡𝑡𝑘,Δ𝛽𝑡𝑘𝐿𝑘𝛽𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚,(3.10)𝛼(0)𝑎𝛼(0)𝑐𝛼(𝜂)=𝛼(0)𝑎𝛼(0)𝑐𝛼(𝜂)(𝑎𝜋+𝑐sin(𝜋𝜂))𝐵𝛼0,(3.11)𝛼(1)+𝑏𝛼(1)𝑑𝛼(𝜉)=𝛼(1)+𝑏𝛼(0)𝑑𝛼(𝜉)(𝑏𝜋+𝑑sin(𝜋𝜉))𝐵𝛼0,(3.12)𝛽(0)𝑎𝛽(0)𝑐𝛽(𝜂)=𝛽(0)+𝑎𝛽(0)+𝑐𝛽(𝜂)(𝑎𝜋+𝑐sin(𝜋𝜂))𝐵𝛽0,(3.13)𝛽(1)+𝑏𝛽(1)𝑑𝛽(𝜉)=𝛽(1)𝑏𝛽(0)+𝑑𝛽(𝜉)(𝑏𝜋+𝑑sin(𝜋𝜉))𝐵𝛽0.(3.14) From (3.9)–(3.14), we obtain that 𝑐𝛼(𝑡)=𝑐𝛽(𝑡)0,𝑡𝐽. Combining (3.9) and (3.10), we obtain that ( 1) and ( 2) hold.
It is easy to see that 𝛼𝛼,𝛽𝛽 on 𝐽. We show that 𝛼𝛽 on 𝐽. Let 𝑝=𝛼𝛽, then 𝑝(𝑡)=𝛼(𝑡)𝛽(𝑡)+𝑐𝛼(𝑡)+𝑐𝛽(𝑡). From (3.9)–(3.14), we have
𝑝(𝑡)+𝑀𝑝(𝑡)+𝑁𝑝(𝜃(𝑡))0,𝑡𝐽,𝑡𝑡𝑘,Δ𝑝𝑡𝑘𝐿𝑘𝑝𝑡𝑘,𝑘=1,,𝑚,𝑝(0)𝑎𝑝(0)𝑐𝑝(𝜂)=𝛼(0)𝑎𝛼(0)𝑐𝛼(𝜂)(𝑎𝜋+𝑐sin𝐼(𝜋𝜂))𝐵𝛼𝛽(0)+𝑎𝛽(0)+𝑐𝛽(𝜂)(𝑎𝜋+𝑐sin(𝜋𝜂))𝐵𝛽0,𝑝(1)+𝑏𝑝(1)𝑑𝑝(𝜉)=𝛼(1)+𝑏𝛼(1)𝑑𝛼(𝜉)(𝑏𝜋+𝑑sin(𝜋𝜉))𝐵𝛼𝛽(1)𝑏𝛽(1)+𝑑𝛽(𝜂)(𝑏𝜋+𝑑sin(𝜋𝜉))𝐵𝛽0,Δ𝑝𝑡𝑘=Δ𝛼𝑡𝑘Δ𝛽𝑡𝑘+Δ𝑐𝑎𝑡𝑘+Δ𝑐𝛽𝑡𝑘𝐿𝑘𝛼𝑡𝑘𝑡𝛽𝑘+𝐿𝑘𝑐𝛼𝑡𝑘+𝑐𝛽𝑡𝑘=𝐿𝑘𝑝𝑡𝑘.(3.15) By Theorem 2.1, we have that 𝑝0 on 𝐽, that is, 𝛼𝛽 on 𝐽. So ( 3) holds.
Step 2. Consider the boundary value problem 𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))=𝜎(𝑡),𝑡𝐽,𝑡𝑡𝑘,Δ𝑢𝑡𝑘=𝐿𝑘𝑢𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚,𝑢(0)𝑎𝑢(0)=𝜆,𝑢(1)+𝑏𝑢(1)=𝛿,(3.16) where 𝜆𝐑,𝛿𝐑. We show that the boundary value problem (3.16) has one unique solution 𝑢(𝑡,𝜆,𝛿).
It is easy to check that the boundary value problem (3.16) is equivalent to the integral equation:
𝑢(𝑡)=𝛿𝑡+(1𝑡)𝜆+𝑏𝜆+𝑎𝛿+𝑎+𝑏+110[]+𝐺(𝑡,𝑠)𝜎(𝑠)𝑀𝑢(𝑠)𝑁𝑢(𝜃(𝑠))𝑑𝑠0<𝑡𝑘<𝑡𝑡𝑡𝑘𝐿𝑘𝑢𝑡𝑘+𝑒𝑘1𝑎+𝑏+1(𝑡+𝑏)𝑚𝑘=11𝑡𝑘𝐿+𝑏𝑘𝑢𝑡𝑘+𝑒𝑘,(3.17) where 1𝐺(𝑡,𝑠)=𝑎+𝑏+1(𝑎+𝑡)(1+𝑏𝑠),0𝑡𝑠1,(𝑎+𝑠)(1+𝑏𝑡),0𝑠𝑡1.(3.18)
It is easy to see that 𝑃𝐶(𝐽,𝑅) with norm 𝑢=max𝑡𝐽|𝑢(𝑡)| is a Banach space. Define a mapping Φ𝑃𝐶(𝐽,𝑅)𝑃𝐶(𝐽,𝑅) by
(Φ𝑢)(𝑡)=𝛿𝑡+(1𝑡)𝜆+𝑏𝜆+𝑎𝛿+𝑎+𝑏+110[]+𝐺(𝑡,𝑠)𝜎(𝑠)𝑀𝑢(𝑠)𝑁𝑢(𝜃(𝑠))𝑑𝑠0<𝑡𝑘<𝑡𝑡𝑡𝑘𝐿𝑘𝑢𝑡𝑘+𝑒𝑘1𝑎+𝑏+1(𝑡+𝑏)𝑚𝑘=11𝑡𝑘𝐿+𝑏𝑘𝑢𝑡𝑘+𝑒𝑘.(3.19) For any 𝑥,𝑦𝑃𝐶(𝐽,𝑅), we have ||||(Φ𝑥)(𝑡)(Φ𝑦)(𝑡)10[]𝐺(𝑡,𝑠)𝑀(𝑦(𝑠)𝑥(𝑠))+𝑁(𝑦(𝜃(𝑠))𝑥(𝜃(𝑠)))𝑑𝑠+1+(1+𝑏)2𝑎+𝑏+1𝑚𝑘=1𝐿𝑘𝑥𝑦10𝐺(𝑡,𝑠)𝑑𝑠𝑥𝑦(𝑀+𝑁)+1+(1+𝑏)2𝑎+𝑏+1𝑚𝑘=1𝐿𝑘𝑥𝑦.(3.20) Since max𝑡𝐽10𝐺(𝑡,𝑠)𝑑𝑠=𝑎(1+2𝑏)+12(𝑎+𝑏+1)81+2𝑏𝑎+𝑏+12,(3.21) the inequality (3.2) implies that Φ𝑃𝐶(𝐽)𝑃𝐶(𝐽) is a contraction mapping. Thus there exists a unique 𝑢𝑃𝐶(𝐽) such that Φ𝑢=𝑢. The boundary value problem (3.16) has a unique solution.
Step 3. We show that for any 𝑡𝐽, the unique solution 𝑢(𝑡,𝜆,𝛿) of the boundary value problem (3.16) is continuous in 𝜆 and 𝛿. Let 𝑢(𝑡,𝜆𝑖,𝛿𝑖),𝑖=1,2, be the solution of 𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))=𝜎(𝑡),𝑡𝐽,𝑡𝑡𝑘,Δ𝑢𝑡𝑘=𝐿𝑘𝑢𝑡𝑘+𝑒𝑘,𝑘=1,,𝑚,𝑢(0)𝑎𝑢(0)=𝜆𝑖,𝑢(1)+𝑏𝑢(1)=𝛿𝑖,𝑖=1,2.(3.22) Then 𝑢𝑡,𝜆𝑖,𝛿𝑖=𝛿𝑖𝑡+(1𝑡)𝜆𝑖+𝑏𝜆𝑖+𝑎𝛿𝑖+𝑎+𝑏+110𝐺(𝑡,𝑠)𝜎(𝑠)𝑀𝑢𝑠,𝜆𝑖,𝛿𝑖𝑁𝑢𝜃(𝑠),𝜆𝑖,𝛿𝑖+𝑑𝑠0<𝑡𝑘<𝑡𝑡𝑡𝑘𝐿𝑘𝑢𝑡𝑘+𝑒𝑘1×𝑎+𝑏+1(𝑡+𝑏)𝑚𝑘=11𝑡𝑘𝐿+𝑏𝑘𝑢𝑡𝑘+𝑒𝑘,𝑖=1,2.(3.23)
From (3.23), we have that
𝑢𝑡,𝜆1,𝛿1𝑢𝑡,𝜆2,𝛿2||𝜆1𝜆2||+||𝛿1𝛿2||𝑢+(𝑀+𝑁)𝑡,𝜆1,𝛿1𝑢𝑡,𝜆2,𝛿2max𝑡𝐽10𝐺(𝑡,𝑠)𝑑𝑠+𝑢𝑡,𝜆1,𝛿1𝑢𝑡,𝜆2,𝛿21+(1+𝑏)2𝑎+𝑏+1𝑚𝑘=1𝐿𝑘||𝜆𝑥𝑦1𝜆2||+||𝛿1𝛿2||𝑢+𝜇𝑡,𝜆1,𝛿1𝑢𝑡,𝜆2,𝛿2.(3.24)
Hence
𝑢(𝑡,𝜆1,𝛿1)𝑢𝑡,𝜆2,𝛿201||𝜆1𝜇1𝜆2||+||𝛿1𝛿2||.(3.25)
Step 4. We show that 𝛼(𝑡)𝑢(𝑡,𝜆,𝛿)𝛽(𝑡)(3.26) for any 𝑡𝐽, 𝜆[𝑐𝛼(𝜂),𝑐𝛽(𝜂)], and 𝛿[𝑑𝛼(𝜉),𝑑𝛽(𝜉)], where 𝑢(𝑡,𝜆,𝛿) is unique solution of the boundary value problem (3.16).
Let 𝑚(𝑡)=𝛼(𝑡)𝑢(𝑡,𝜆,𝛿). From (3.9), (3.11), (3.12), and (3.16), we have that 𝑚(0)𝑎𝑚(0)𝑐𝑚(𝜂),𝑚(1)+𝑏𝑚(1)𝑑𝑚(𝜉), and
𝑚(𝑡)+𝑀𝑚(𝑡)+𝑁𝑚(𝜃(𝑡))=𝛼(𝑡)+𝑀𝛼(𝑡)+𝑁𝛼(𝜃(𝑡))+𝑢(𝑡,𝜆)𝑀𝑢(𝑡,𝜆,𝛿)𝑁𝑢(𝜃(𝑡),𝜆,𝛿)𝜎(𝑡)𝜎(𝑡)0,Δ𝑚𝑡𝑘𝐿𝑘𝑢𝑡𝑘.(3.27) By Theorem 2.1, we obtain that 𝑚0 on 𝐽, that is, 𝛼(𝑡)𝑢(𝑡,𝜆,𝛿) on 𝐽. Similarly, 𝑢(𝑡,𝜆,𝛿)𝛽(𝑡) on 𝐽.
Step 5. Let 𝐷=[𝑐𝛼(𝜂),𝑐𝛽(𝜂)]×[𝑑𝛼(𝜉),𝑑𝛽(𝜉)]. Define a mapping 𝐹𝐷𝐑2 by 𝐹(𝜆,𝛿)=(𝑢(𝜂,𝜆,𝛿),𝑢(𝜉,𝜆,𝛿)),(3.28) where 𝑢(𝑡,𝜆,𝛿) is unique solution of the boundary value problem (3.16). From Step 4, we have 𝐹(𝐷)𝐷.(3.29) Since 𝐷 is a compact convex set and 𝐹 is continuous, it follows by Schauder’s fixed point theorem that 𝐹 has a fixed point (𝜆0,𝛿0)𝐷 such that 𝑢𝜂,𝜆0,𝛿0=𝜆0,𝑢𝜉,𝜆0,𝛿0=𝛿0.(3.30) Obviously, 𝑢(𝑡,𝜆0,𝛿0) is unique solution of the boundary value problem (3.1). This completes the proof.

4. Main Results

Let 𝑀𝐑, 𝑁𝐑. We first give the following definition.

Definition 4.1. A function 𝛼𝐸 is called a lower solution of the boundary value problem (1.2) if 𝛼(𝑡)+(𝑀+𝑟)𝑐𝛼(𝑡)+𝑁𝑐𝛼(𝜃(𝑡))𝑓(𝑡,𝛼(𝑡),𝛼(𝜃(𝑡))),𝑡𝐽,𝑡𝑡𝑘,Δ𝛼𝑡𝑘𝐼𝑘𝛼𝑡𝑘+𝐿𝑘𝑐𝛼𝑡𝑘,𝑘=1,,𝑚.(4.1)

Definition 4.2. A function 𝛽𝐸 is called an upper solution of the boundary value problem (1.2) if 𝛽(𝑡)(𝑀+𝑟)𝑐𝛽(𝑡)𝑁𝑐𝛽(𝜃(𝑡))𝑓(𝑡,𝛽(𝑡),𝛽(𝜃(𝑡)))𝑡𝐽,𝑡𝐽,𝑡𝑡𝑘,Δ𝛽𝑡𝑘𝐼𝑘𝛽𝑡𝑘𝐿𝑘𝑐𝛽𝑡𝑘,𝑘=1,,𝑚.(4.2) Our main result is the following theorem.

Theorem 4.3. Assume that (𝐻)  holds. If the following conditions are satisfied: (𝐻1)𝛼,𝛽 are lower and upper solutions for boundary value problem (1.2) respectively, and 𝛼(𝑡)𝛽(𝑡) on 𝐽,(𝐻2)the constants 𝑀,𝑁 in definition of upper and lower solutions satisfy (2.3), (3.2), and 𝑓(𝑡,𝑥,𝑦)𝑓𝑡,𝑥,𝑦𝑀𝑥𝑥𝑁𝑦𝑦,𝐼𝑘(𝑥)𝐼𝑘(𝑦)𝐿𝑘(𝑥𝑦),𝑥𝑦,(4.3) for 𝛼(𝑡)𝑥𝑥𝛽(𝑡),𝛼(𝜃(𝑡))𝑦𝑦𝛽(𝜃(𝑡)),𝑡𝐽.
Then, there exist monotone sequences {𝛼𝑛},{𝛽𝑛} with 𝛼0=𝛼,𝛽0=𝛽 such that lim𝑛𝛼𝑛(𝑡)=𝜌(𝑡),lim𝑛𝛽𝑛(𝑡)=𝜚(𝑡) uniformly on 𝐽, and 𝜌,𝜚 are the minimal and the maximal solutions of (1.2), respectively, such that
𝛼0𝛼1𝛼2𝛼𝑛𝜌𝑥𝜚𝛽𝑛𝛽2𝛽1𝛽0(4.4) on 𝐽, where 𝑥 is any solution of (1.1) such that 𝛼(𝑡)𝑥(𝑡)𝛽(𝑡) on 𝐽.

Proof. Let [𝛼,𝛽]={𝑢𝐸𝛼(𝑡)𝑢(𝑡)𝛽(𝑡),𝑡𝐽}. For any 𝛾[𝛼,𝛽], we consider the boundary value problem 𝑢(𝑡)+𝑀𝑢(𝑡)+𝑁𝑢(𝜃(𝑡))=𝑓(𝑡,𝛾(𝑡),𝛾(𝜃(𝑡)))+𝑀𝛾(𝑡)+𝑁𝛾(𝜃(𝑡)),𝑡𝐽,Δ𝑢𝑡𝑘=𝐼𝑘𝛾𝑡𝑘𝐿𝑘𝑢𝑡𝑘𝑡𝛾𝑘,𝑘=1,,𝑚.𝑢(0)𝑎𝑥(0)=𝑐𝑢(𝜂),𝑢(1)+𝑏𝑢(1)=𝑑𝑢(𝜉).(4.5)
Since 𝛼 is a lower solution of (1.2), from (𝐻2), we have that
𝛼(𝑡)+𝑀𝛼(𝑡)+𝑁𝛼(𝜃(𝑡))𝑓(𝑡,𝛼(𝑡),𝛼(𝜃(𝑡)))+𝑀𝛼(𝑡)+𝑁𝛼(𝜃(𝑡))(𝑀+𝑟)𝑐𝛼(𝑡)𝑁𝑐𝛼(𝜃(𝑡))𝑓(𝑡,𝛾(𝑡),𝛾(𝜃(𝑡)))+𝑀𝛾(𝑡)+𝑁𝛾(𝜃(𝑡))(𝑀+𝑟)𝑐𝛼(𝑡)𝑁𝑐𝛼(𝜃(𝑡)),Δ𝛼𝑡𝑘𝐼𝑘𝛼𝑡𝑘+𝐿𝑘𝑐𝛼𝑡𝑘𝐼𝑘𝛾𝑡𝑘+𝐿𝑘𝛼𝑡𝑘𝐿𝑘𝛾𝑡𝑘+𝐿𝑘𝑐𝛼𝑡𝑘.(4.6)
Similarly, we have that
𝛽(𝑡)+𝑀𝛽(𝑡)+𝑁𝛽(𝜃(𝑡))𝑓(𝑡,𝛾(𝑡),𝛾(𝜃(𝑡)))+𝑀𝛾(𝑡)+𝑁𝛾(𝜃(𝑡))+(𝑀+𝑟)𝑐𝛽(𝑡)+𝑁𝑐𝛽(𝜃(𝑡)),Δ𝛽𝑡𝑘𝐼𝑘𝛽𝑡𝑘𝐿𝑘𝑐𝛽𝑡𝑘𝐼𝑘𝛾𝑡𝑘+𝐿𝑘𝛽𝑡𝑘𝐿𝑘𝛾𝑡𝑘𝐿𝑘𝑐𝛽𝑡𝑘.(4.7)
By Theorem 3.1, the boundary value problem (4.5) has a unique solution 𝑢[𝛼,𝛽]. We define an operator Ψ by 𝑢=Ψ𝛾, then Ψ is an operator from [𝛼,𝛽] to [𝛼,𝛽].
We will show that
(a)𝛼Ψ𝛼,Ψ𝛽𝛽,(b)Ψ is nondecreasing in [𝛼,𝛽].
From Ψ𝛼[𝛼,𝛽] and Ψ𝛽[𝛼,𝛽], we have that (a) holds. To prove (b), we show that Ψ𝜈1Ψ𝜈2 if 𝛼𝜈1𝜈2𝛽.
Let 𝜈1=Ψ𝜈1,𝜈2=Ψ𝜈2 ,and 𝑝=𝜈1𝜈2, then by (𝐻2) and boundary conditions, we have that
𝑝(𝑡)+𝑀𝑝(𝑡)+𝑁𝑝(𝜃(𝑡))=𝑓𝑡,𝜈1(𝑡),𝜈1(𝜃(𝑡))+𝑀𝜈1(𝑡)+𝑁𝜈1(𝜃(𝑡))𝑓𝑡,𝜈2(𝑡),𝜈2(𝜃(𝑡))𝑀𝜈2(𝑡)𝑁𝜈2(𝜃(𝑡))0,Δ𝑝𝑡𝑘𝐿𝑘𝑝𝑡𝑘,𝑝(0)𝑎𝑝(0)=𝑐𝑝(𝜂),𝑝(1)+𝑝𝑢(1)=𝑑𝑝(𝜉).(4.8) By Theorem 2.1, 𝑝(𝑡)0 on 𝐽, which implies that Ψ𝜈1Ψ𝜈2.
Define the sequences {𝛼𝑛},{𝛽𝑛} with 𝛼0=𝛼,𝛽0=𝛽 such that 𝛼𝑛+1=Ψ𝛼𝑛,𝛽𝑛+1=Ψ𝛽𝑛 for 𝑛=0,1,2, From (a) and (b), we have
𝛼0𝛼1𝛼2𝛼𝑛𝛽𝑛𝛽2𝛽1𝛽0(4.9) on 𝑡𝐽, and each 𝛼𝑛,𝛽𝑛𝐸 satisfies 𝛼𝑛(𝑡)+𝑀𝛼𝑛(𝑡)+𝑁𝛼𝑛(𝜃(𝑡))=𝑓𝑡,𝛼𝑛1(𝑡),𝛼𝑛1(𝜃(𝑡))+𝑀𝛼𝑛1(𝑡)+𝑁𝛼𝑛1(𝜃(𝑡)),𝑡𝐽,𝑡𝑡𝑘,Δ𝛼𝑛𝑡𝑘=𝐼𝑘𝛼𝑛1𝑡𝑘+𝐿𝑘𝛼𝑛𝑡𝑘𝛼𝑛1𝑡𝑘𝛼,𝑘=1,2,,𝑚,𝑛(0)𝑎𝛼𝑛(0)=𝑐𝛼𝑛(𝜂),𝛼𝑛(1)+𝑏𝛼𝑛(1)=𝑑𝛼𝑛(𝜉),𝛽𝑛(𝑡)+𝑀𝛽𝑛(𝑡)+𝑁𝛽𝑛(𝜃(𝑡))=𝑓𝑡,𝛽𝑛1(𝑡),𝛽𝑛1(𝜃(𝑡))+𝑀𝛽𝑛1(𝑡)+𝑁𝛽𝑛1(𝜃(𝑡)),𝑡𝐽,𝑡𝑡𝑘,Δ𝛽𝑛𝑡𝑘=𝐼𝑘𝛽𝑛1𝑡𝑘+𝐿𝑘𝛽𝑛𝑡𝑘𝛽𝑛1𝑡𝑘𝛽,𝑘=1,2,,𝑚,𝑛(0)𝑎𝛽𝑛(0)=𝑐𝛽𝑛(𝜂),𝛽𝑛(1)+𝑏𝛽𝑛(1)=𝑑𝛽𝑛(𝜉).(4.10) Therefore, there exist 𝜌,𝜚 such that such that lim𝑛𝛼𝑛(𝑡)=𝜌(𝑡), lim𝑛𝛽𝑛(𝑡)=𝜚(𝑡) uniformly on 𝐽. Clearly, 𝜌,𝜚 are solutions of (1.1).
Finally, we prove that if 𝑥[𝛼0,𝛽0] is any solution of (1.1), then 𝜌(𝑡)𝑥(𝑡)𝜚(𝑡) on 𝐽. To this end, we assume, without loss of generality, that 𝛼𝑛(𝑡)𝑥(𝑡)𝛽𝑛(𝑡) for some 𝑛. Since 𝛼0(𝑡)𝑥(𝑡)𝛽0(𝑡), from property (b), we can obtain
𝛼𝑛+1(𝑡)𝑥(𝑡)𝛽𝑛+1(𝑡),𝑡𝐽.(4.11) Hence, we can conclude that 𝛼𝑛(𝑡)𝑥(𝑡)𝛽𝑛(𝑡),𝑛.(4.12) Passing to the limit as 𝑛, we obtain 𝜌(𝑡)𝑥(𝑡)𝜚(𝑡),𝑡𝐽.(4.13) This completes the proof.

Acknowledgments

This work is supported by the NNSF of China (10571050;10871062) and Hunan Provincial Natural Science Foundation of China (NO:09JJ3010), and Science Research Fund of Hunan provincial Education Department (No: 06C052 ).

References

  1. G. S. Ladde, V. Lakshmikantham, and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, vol. 27 of Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, Pitman Advanced Publishing Program, London, UK, 1985. View at MathSciNet
  2. V. Lakshmikantham, S. Leela, and F. A. McRae, “Improved generalized quasilinearization (GQL) method,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24, no. 11, pp. 1627–1637, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. Henderson, Boundary Value Problems for Functional-Differential Equations, World Scientific, River Edge, NJ, USA, 1995. View at MathSciNet
  4. D. Jiang, M. Fan, and A. Wan, “A monotone method for constructing extremal solutions to second-order periodic boundary value problems,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 189–197, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Sockol and A. S. Vatsala, “A unified exhaustive study of monotone iterative method for initial value problems,” Nonlinear Studies, vol. 8, pp. 429–438, 2004. View at Google Scholar
  6. C. P. Gupta, “A Dirichlet type multi-point boundary value problem for second order ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 26, no. 5, pp. 925–931, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Jiang and J. Wei, “Monotone method for first- and second-order periodic boundary value problems and periodic solutions of functional differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 50, no. 7, pp. 885–898, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. Jankowski, “Advanced differential equations with nonlinear boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 490–503, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. J. Nieto and R. Rodríguez-López, “Existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions,” Computers & Mathematics with Applications, vol. 40, no. 4-5, pp. 433–442, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. J. Nieto and R. Rodríguez-López, “Remarks on periodic boundary value problems for functional differential equations,” Journal of Computational and Applied Mathematics, vol. 158, no. 2, pp. 339–353, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z. He and X. He, “Periodic boundary value problems for first order impulsive integro-differential equations of mixed type,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 8–20, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. Zhang, A. Zhao, and J. Yan, “Monotone iterative method for differential equations with piecewise constant arguments,” Indian Journal of Pure and Applied Mathematics, vol. 31, no. 1, pp. 69–75, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. Q. Yao, “Monotone iterative technique and positive solutions of Lidstone boundary value problems,” Applied Mathematics and Computation, vol. 138, no. 1, pp. 1–9, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. T. Jankowski, “Ordinary differential equations with nonlinear boundary conditions of anti-periodic type,” Computers & Mathematics with Applications, vol. 47, no. 8-9, pp. 1429–1436, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W. Ding, J. Mi, and M. Han, “Periodic boundary value problems for the first order impulsive functional differential equations,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 433–446, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. G. Infante and J. R. L. Webb, “Three-point boundary value problems with solutions that change sign,” Journal of Integral Equations and Applications, vol. 15, no. 1, pp. 37–57, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. P. W. Eloe and L. Zhang, “Comparison of Green's functions for a family of multipoint boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 246, no. 1, pp. 296–307, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. A. Marano, “A remark on a second-order three-point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 183, no. 3, pp. 518–522, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. T. Jankowski, “Solvability of three point boundary value problems for second order differential equations with deviating arguments,” Journal of Mathematical Analysis and Applications, vol. 312, no. 2, pp. 620–636, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. Liu, “Non-homogeneous boundary-value problems of higher order differential equations with p-Laplacian,” Electronic Journal of Differential Equations, vol. 2008, no. 20, pp. 1–43, 2008. View at Google Scholar · View at MathSciNet
  21. Y. Liu, “Positive solutions of mixed type multi-point non-homogeneous BVPs for p-Laplacian equations,” Applied Mathematics and Computation, vol. 206, no. 2, pp. 796–805, 2008. View at Publisher · View at Google Scholar · View at MathSciNet