Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article
!A Letter to Editor for this article has been published. To view the article details, please click the β€˜Letter to the Editor’ tab above.

Research Article | Open Access

Volume 2009 |Article ID 258090 | 16 pages | https://doi.org/10.1155/2009/258090

MultiPoint BVPs for Second-Order Functional Differential Equations with Impulses

Academic Editor: Fernando Lobo Pereira
Received14 Apr 2009
Accepted10 Jun 2009
Published12 Aug 2009

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: βˆ’π‘’ξ…žξ…ž[](𝑑)=𝑓(𝑑,𝑒(𝑑),𝑒(πœƒ(𝑑))),π‘‘βˆˆπ½=0,1,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έ=πΌπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€Έ,π‘˜=1,…,π‘š,𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)=𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)=𝑑𝑒(πœ‰),(1.1) where π‘“βˆˆπΆ(𝐽×𝐑2,𝐑),    0β‰€πœƒ(𝑑)≀𝑑,π‘‘βˆˆπ½,πœƒβˆˆπΆ(𝐽),π‘Žβ‰₯0,𝑏β‰₯0,0≀𝑐≀1,0≀𝑑≀1, 0<πœ‚,πœ‰<1. 0<𝑑1<𝑑2<β‹―<π‘‘π‘š<1, 𝑓 is continuous everywhere except at {π‘‘π‘˜}×𝑅2; 𝑓(𝑑+π‘˜,β‹…,β‹…), and 𝑓(π‘‘βˆ’π‘˜,β‹…,β‹…) exist with 𝑓(π‘‘βˆ’π‘˜,β‹…,β‹…)=𝑓(π‘‘π‘˜,β‹…,β‹…); πΌπ‘˜βˆˆπΆ(𝑅,𝑅),Ξ”π‘’ξ…ž(π‘‘π‘˜)=π‘’ξ…ž(𝑑+π‘˜)βˆ’π‘’ξ…ž(π‘‘βˆ’π‘˜). Denote π½βˆ’=𝐽⧡{𝑑𝑖,𝑖=1,2,…,π‘š}. Let 𝑃𝐢(𝐽,𝑅)={π‘’βˆΆπ½β†’π‘…;𝑒(𝑑)|π½βˆ’ is continuous, 𝑒(𝑑+π‘˜) and 𝑒(π‘‘βˆ’π‘˜) exist with 𝑒(π‘‘βˆ’π‘˜)=𝑒(π‘‘π‘˜),π‘˜=1,2,…,π‘š}; 𝑃𝐢1(𝐽,𝑅)={π‘’βˆΆπ½β†’π‘…;𝑒(𝑑)|π½βˆ’ is continuous differentiable, π‘’ξ…ž(𝑑+π‘˜) and π‘’ξ…ž(π‘‘βˆ’π‘˜) exist with π‘’ξ…ž(π‘‘βˆ’π‘˜)=π‘’ξ…ž(π‘‘π‘˜),π‘˜=1,2,…,π‘š}. Let 𝐸=𝑃𝐢1(𝐽,𝑅)∩𝐢2(𝐽,𝑅). A function π‘’βˆˆπΈ 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 [1–3]. 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, 4–6] for ordinary differential equations, [7–11] 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, 16–21]. These works motivate that we study the multipoint boundary value problems for the impulsive functional differential equation (1.1).

We also note that when πΌπ‘˜=0 and πœƒ(𝑑)=𝑑, 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, 16–18] 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)  π‘Žβ‰₯0,𝑏β‰₯0,0≀𝑐≀1,0≀𝑑≀1,0<πœ‚,πœ‰<1,π‘Ž+𝑐>0,𝑏+𝑑>0.

For any given function π‘”βˆˆπΈ, we denote 𝐴𝑔=max𝑔(0)βˆ’π‘Žπ‘”ξ…ž(0)βˆ’π‘π‘”(πœ‚),π‘Žπœ‹+𝑐sinπœ‹πœ‚π‘”(1)+π‘π‘”ξ…ž(1)βˆ’π‘‘π‘”(πœ‰)ξ‚Ό,π΅π‘πœ‹+𝑑sinπœ‹πœ‰π‘”ξ€½π΄=max𝑔,0,𝑐𝑔(𝑑)=𝐡𝑔sin(πœ‹π‘‘),π‘Ÿ=πœ‹2.(2.1) We now present main results of this section.

Theorem 2.1. Assume thatβ€‰β€‰π‘’βˆˆπΈ satisfies βˆ’π‘’ξ…žξ…ž(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))≀0,π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š,𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)≀𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)≀𝑑𝑒(πœ‰),(2.2) where π‘Žβ‰₯0,𝑏β‰₯0, 0≀𝑐≀1,0≀𝑑≀1,0<πœ‚, β€‰πœ‰<1, πΏπ‘˜β‰₯0 and constants 𝑀,𝑁 satisfy 𝑀>0,𝑁β‰₯0,𝑀+𝑁2+π‘šξ“π‘˜=1πΏπ‘˜β‰€1.(2.3) Then 𝑒(𝑑)≀0 for π‘‘βˆˆπ½.

Proof. Suppose, to the contrary, that 𝑒(𝑑)>0 for some π‘‘βˆˆπ½.
If 𝑒(1)=maxπ‘‘βˆˆπ½π‘’(𝑑)>0, then π‘’ξ…ž(1)β‰₯0, 𝑒(1)β‰₯𝑒(πœ‰), and 𝑑𝑒(πœ‰)≀𝑒(1)≀𝑒(1)+π‘π‘’ξ…ž(1)≀𝑑𝑒(πœ‰).(2.4) So 𝑑=1 and 𝑒(πœ‰) is a maximum value.
If 𝑒(0)=maxπ‘‘βˆˆπ½π‘’(𝑑)>0, then π‘’ξ…ž(0)≀0, 𝑒(0)β‰₯𝑒(πœ‚), and 𝑐𝑒(πœ‚)≀𝑒(0)≀𝑒(0)βˆ’π‘π‘’ξ…ž(0)≀𝑐𝑒(πœ‚).(2.5) So 𝑐=1 and 𝑒(πœ‚) is a maximum value.
Therefore, there is a 𝜌∈(0,1) such that 𝑒(𝜌)=maxπ‘‘βˆˆπ½π‘’(𝑑)>0,π‘’ξ…žξ€·πœŒ+≀0.(2.6)
Suppose that 𝑒(𝑑)β‰₯0 for π‘‘βˆˆπ½. From the first inequality of (2.2), we obtain that π‘’ξ…žξ…ž(𝑑)β‰₯0 for π‘‘βˆˆπ½. Hence 𝑒(0)=maxπ‘‘βˆˆπ½π‘’(𝑑)or𝑒(1)=maxπ‘‘βˆˆπ½π‘’(𝑑).(2.7)
If π‘’ξ…ž(0)β‰₯0, then π‘’ξ…žξ…ž(𝑑)β‰₯0, π‘‘βˆˆ(𝑑𝑖,𝑑𝑖+1], it is easy to obtain that 𝑒(𝑑) is nondecreasing. Since 𝑒(1)≀𝑑𝑒(πœ‰)≀𝑒(1), it follows that 𝑒(𝑑)≑𝐾 (𝐾>0) for π‘‘βˆˆ[πœ‰,1]. From the first inequality of (2.2), we have that when π‘‘βˆˆ[πœ‰,1],0<𝑀𝐾≀𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))β‰€π‘’ξ…žξ…ž(𝑑)=0,(2.8) which is a contradiction.
If π‘’ξ…ž(0)≀0, then 𝑒(0)=maxπ‘‘βˆˆπ½π‘’(𝑑)>0, or 𝑒(1)=maxπ‘‘βˆˆπ½π‘’(𝑑)>0. If 𝑒(0)=maxπ‘‘βˆˆπ½π‘’(𝑑)>0, then 𝑒(𝑑)≑𝐾 (𝐾>0) for π‘‘βˆˆ[0,πœ‚]. If 𝑒(1)=maxπ‘‘βˆˆπ½π‘’(𝑑)>0, then 𝑒(𝑑)≑𝐾 for π‘‘βˆˆ[πœ‰,1].
From the first inequality of (2.2), we have that when π‘‘βˆˆ[πœ‰,1],0<𝑀𝐾≀𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))β‰€π‘’ξ…žξ…ž(𝑑)=0,(2.9) which is a contradiction.
Suppose that there exist 𝑑1,𝑑2∈𝐽 such that 𝑒(𝑑1)>0 and 𝑒(𝑑2)<0. We consider two possible cases.Case 1 (𝑒(0)>0). Since 𝑒(𝑑2)<0, there is πœ…>0,πœ€>0 such that 𝑒(πœ…)=0, 𝑒(𝑑)β‰₯0 for π‘‘βˆˆ[0,πœ…) and 𝑒(𝑑)<0 for all π‘‘βˆˆ(πœ…,πœ…+πœ€]. It is easy to obtain that π‘’ξ…žξ…ž(𝑑)β‰₯0 for π‘‘βˆˆ[0,πœ…]. If π‘‘βˆ—<πœ…, then 0<𝑀𝑒(π‘‘βˆ—)β‰€π‘’ξ…žξ…ž(π‘‘βˆ—)≀0, a contradiction. Hence π‘‘βˆ—>πœ…+πœ€. Let π‘‘βˆ—βˆˆ[0,π‘‘βˆ—) such that 𝑒(π‘‘βˆ—)=minπ‘‘βˆˆ[0,π‘‘βˆ—)𝑒(𝑑), then 𝑒(π‘‘βˆ—)<0. From the first inequality of (2.2), we have π‘’ξ…žξ…žξ€·π‘‘(𝑑)β‰₯(𝑀+𝑁)π‘’βˆ—ξ€Έξ€Ί,π‘‘βˆˆ0,π‘‘βˆ—ξ€Έ,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š.(2.10) Integrating the above inequality from 𝑠(π‘‘βˆ—β‰€π‘ β‰€π‘‘βˆ—) to π‘‘βˆ—, we obtain π‘’ξ…žξ€·π‘‘βˆ—ξ€Έβˆ’π‘’ξ…ž(𝑑𝑠)β‰₯βˆ—ξ€Έ(ξ€·π‘‘βˆ’π‘ π‘€+𝑁)π‘’βˆ—ξ€Έ+𝑠<π‘‘π‘˜<π‘‘βˆ—πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έβ‰₯ξ€·π‘‘βˆ—ξ€Έξ€·π‘‘βˆ’π‘ (𝑀+𝑁)π‘’βˆ—ξ€Έ+π‘šξ“π‘˜=1πΏπ‘˜π‘’ξ€·π‘‘βˆ—ξ€Έ.(2.11) Hence βˆ’π‘’ξ…ž[𝑑(𝑠)β‰₯βˆ—ξ€Έβˆ’π‘ (𝑀+𝑁)+π‘šξ“π‘˜=1πΏπ‘˜π‘’ξ€·π‘‘βˆ—ξ€Έ,π‘‘βˆ—β‰€π‘ β‰€π‘‘βˆ—,(2.12) and then integrate from π‘‘βˆ— to π‘‘βˆ— to obtain ξ€·π‘‘βˆ’π‘’βˆ—ξ€Έξ€·π‘‘<π‘’βˆ—ξ€Έξ€·π‘‘βˆ’π‘’βˆ—ξ€Έβ‰€ξ€œπ‘‘βˆ—π‘‘βˆ—ξ€·π‘ βˆ’π‘‘βˆ—ξ€Έξ€·π‘‘(𝑀+𝑁)π‘’βˆ—ξ€Έπ‘‘π‘ βˆ’π‘šξ“π‘˜=1πΏπ‘˜π‘’ξ€·π‘‘βˆ—ξ€Έξƒ©β‰€βˆ’π‘€+𝑁2ξ€·π‘‘βˆ—βˆ’π‘‘βˆ—ξ€Έ2+π‘šξ“π‘˜=1πΏπ‘˜ξƒͺπ‘’ξ€·π‘‘βˆ—ξ€Έξƒ©β‰€βˆ’π‘€+𝑁2+π‘šξ“π‘˜=1πΏπ‘˜ξƒͺπ‘’ξ€·π‘‘βˆ—ξ€Έ.(2.13) From (2.3), we have that 𝑒(π‘‘βˆ—)>0. This is a contradiction.Case 2 (𝑒(0)≀0). Let π‘‘βˆ—βˆˆ[0,π‘‘βˆ—) such that 𝑒(π‘‘βˆ—)=minπ‘‘βˆˆ[0,π‘‘βˆ—)𝑒(𝑑)≀0. From the first inequality of (2.2), we have π‘’ξ…žξ…žξ€·π‘‘(𝑑)β‰₯(𝑀+𝑁)π‘’βˆ—ξ€Έξ€Ί,π‘‘βˆˆ0,π‘‘βˆ—ξ€Έ,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š.(2.14) The rest proof is similar to that of Case 1. The proof is complete.

Theorem 2.2. Assume that (𝐻) holds and π‘’βˆˆπΈ satisfies βˆ’π‘’ξ…žξ…žξ€Ί(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))+(𝑀+π‘Ÿ)𝑐𝑒(𝑑)+𝑁𝑐𝑒(πœƒ(𝑑))≀0,π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+πΏπ‘˜π‘π‘’ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š,(2.15) where constants 𝑀,𝑁 satisfy (2.3), and πΏπ‘˜β‰₯0, then 𝑒(𝑑)≀0 for π‘‘βˆˆπ½.

Proof. Assume that 𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)≀𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)≀𝑑𝑒(πœ‰), then 𝑐𝑒(𝑑)≑0. By Theorem 2.1, 𝑒(𝑑)≀0.
Assume that 𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)≀𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)>𝑑𝑒(πœ‰), then
𝑐𝑒(𝑑)=sin(πœ‹π‘‘)ξ€·π‘πœ‹+𝑑sin(πœ‹πœ‰)𝑒(1)+π‘π‘’ξ…ž(ξ€Έ1)βˆ’π‘‘π‘’(πœ‰).(2.16)
Put 𝑦(𝑑)=𝑒(𝑑)+𝑐𝑒(𝑑),π‘‘βˆˆπ½, then 𝑦(𝑑)β‰₯𝑒(𝑑) for all π‘‘βˆˆπ½, and
π‘¦ξ…ž(𝑑)=π‘’ξ…ž(𝑑)+πœ‹cos(πœ‹π‘‘)ξ€·π‘πœ‹+𝑑sin(πœ‹πœ‰)𝑒(1)+π‘π‘’ξ…žξ€Έπ‘¦(1)βˆ’π‘‘π‘’(πœ‰),π‘‘βˆˆπ½,ξ…žξ…ž(𝑑)=π‘’ξ…žξ…ž(𝑑)βˆ’π‘Ÿπ‘π‘’(𝑑),π‘‘βˆˆπ½.(2.17) Hence 𝑦(0)=𝑒(0),𝑦(1)=𝑒(1),𝑦(πœ‰)=𝑒(πœ‰)+sin(πœ‹πœ‰)ξ€·π‘πœ‹+𝑑sin(πœ‹πœ‰)𝑒(1)+π‘π‘’ξ…ž(ξ€Έ,𝑦1)βˆ’π‘‘π‘’(πœ‰)ξ…ž(0)=π‘’ξ…ž(πœ‹0)+ξ€·π‘πœ‹+𝑑sin(πœ‹πœ‰)𝑒(1)+π‘π‘’ξ…ž(ξ€Έ,𝑦1)βˆ’π‘‘π‘’(πœ‰)ξ…ž(1)=π‘’ξ…ž(πœ‹1)βˆ’ξ€·π‘πœ‹+𝑑sin(πœ‹πœ‰)𝑒(1)+π‘π‘’ξ…ž(ξ€Έ,1)βˆ’π‘‘π‘’(πœ‰)βˆ’π‘¦ξ…žξ…ž(𝑑)+𝑀𝑦(𝑑)+𝑁𝑦(πœƒ(𝑑))=βˆ’π‘’ξ…žξ…žξ€Ί(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))+(𝑀+π‘Ÿ)𝑐𝑒(𝑑)+𝑁𝑐𝑒(πœƒ(𝑑))≀0,𝑦(0)βˆ’π‘Žπ‘¦ξ…ž(0)=𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)βˆ’π‘Žπœ‹ξ€·π‘πœ‹+𝑑sin(πœ‹πœ‰)𝑒(1)+π‘π‘’ξ…žξ€Έ(1)βˆ’π‘‘π‘’(πœ‰)≀𝑐𝑒(πœ‚)≀𝑐𝑦(πœ‚),𝑦(1)+π‘π‘¦ξ…ž(1)βˆ’π‘‘π‘¦(πœ‰)=𝑒(1)+π‘π‘’ξ…ž(1)βˆ’π‘‘π‘’(πœ‰)βˆ’π‘πœ‹ξ€·π‘πœ‹+𝑑sin(πœ‹πœ‰)𝑒(1)+π‘π‘’ξ…žξ€Έβˆ’(1)βˆ’π‘‘π‘’(πœ‰)𝑑sin(πœ‹πœ‰)ξ€·π‘πœ‹+𝑑sinπœ‹πœ‰π‘’(1)+π‘π‘’ξ…žξ€Έ(1)βˆ’π‘‘π‘’(πœ‰)≀0,Ξ”π‘¦ξ…žξ€·π‘‘π‘˜ξ€Έ=Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€ΈΞ”π‘ξ…žπ‘’ξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+πΏπ‘˜π‘π‘’ξ€·π‘‘π‘˜ξ€Έ=πΏπ‘˜π‘¦ξ€·π‘‘π‘˜ξ€Έ.(2.18) By Theorem 2.1, 𝑦(𝑑)≀0 for all π‘‘βˆˆπ½, which implies that 𝑒(𝑑)≀0 for π‘‘βˆˆπ½.
Assume that 𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)>𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)≀𝑑𝑒(πœ‰), then
𝑐𝑒(𝑑)=sinπœ‹π‘‘ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…žξ€Έ(0)βˆ’π‘π‘’(πœ‚).(2.19) Put 𝑦(𝑑)=𝑒(𝑑)+𝑐𝑒(𝑑),π‘‘βˆˆπ½, then 𝑦(𝑑)β‰₯𝑒(𝑑) for all π‘‘βˆˆπ½, and π‘¦ξ…ž(𝑑)=π‘’ξ…ž(𝑑)+πœ‹cos(πœ‹π‘‘)ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…žξ€Έπ‘¦(0)βˆ’π‘π‘’(πœ‚),π‘‘βˆˆπ½,ξ…žξ…ž(𝑑)=π‘’ξ…žξ…ž(𝑑)βˆ’π‘Ÿπ‘π‘’(𝑑),π‘‘βˆˆπ½.(2.20) Hence 𝑦(0)=𝑒(0),𝑦(1)=𝑒(1),𝑦(πœ‚)=𝑒(πœ‚)+sin(πœ‹πœ‚)ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…ž(ξ€Έ,𝑦0)βˆ’π‘π‘’(πœ‚)ξ…ž(0)=π‘’ξ…ž(πœ‹0)+ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…ž(ξ€Έ,𝑦0)βˆ’π‘π‘’(πœ‚)ξ…ž(1)=π‘’ξ…ž(πœ‹1)βˆ’ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…ž(ξ€Έ,0)βˆ’π‘π‘’(πœ‚)βˆ’π‘¦ξ…žξ…ž(𝑑)+𝑀𝑦(𝑑)+𝑁𝑦(πœƒ(𝑑))=βˆ’π‘’ξ…žξ…žξ€Ί(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))+(𝑀+π‘Ÿ)𝑐𝑒(𝑑)+𝑁𝑐𝑒(πœƒ(𝑑))≀0,𝑦(0)βˆ’π‘Žπ‘¦ξ…ž(0)βˆ’π‘π‘¦(πœ‚)=𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)βˆ’π‘π‘’(πœ‚)βˆ’π‘Žπœ‹ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…žξ€Έβˆ’(0)βˆ’π‘π‘’(πœ‚)𝑐sin(πœ‹πœ‚)ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…ž(ξ€Έ0)βˆ’π‘π‘’(πœ‚)≀0,𝑦(1)+π‘π‘¦ξ…ž(1)=𝑒(1)+π‘π‘’ξ…ž(1)βˆ’π‘πœ‹ξ€·π‘Žπœ‹+𝑐sin(πœ‹πœ‚)𝑒(0)βˆ’π‘Žπ‘’ξ…žξ€Έ(0)βˆ’π‘π‘’(πœ‚)≀𝑑𝑒(πœ‰)≀𝑑𝑦(πœ‰),Ξ”π‘¦ξ…žξ€·π‘‘π‘˜ξ€Έ=Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έ+Ξ”π‘ξ…žπ‘’ξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+πΏπ‘˜π‘π‘’ξ€·π‘‘π‘˜ξ€Έ=πΏπ‘˜π‘¦ξ€·π‘‘π‘˜ξ€Έ.(2.21)
By Theorem 2.1, 𝑦(𝑑)≀0 for all π‘‘βˆˆπ½, which implies that 𝑒(𝑑)≀0 for π‘‘βˆˆπ½.
Assume that 𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)>𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)>𝑑𝑒(πœ‰), then 𝑐𝑒(𝑑)=𝐴𝑒sin(πœ‹π‘‘).
Put 𝑦(𝑑)=𝑒(𝑑)+𝑐𝑒(𝑑),π‘‘βˆˆπ½, then 𝑦(𝑑)β‰₯𝑒(𝑑) for all π‘‘βˆˆπ½, and π‘¦ξ…ž(𝑑)=π‘’ξ…ž(𝑑)+π΄π‘’π‘¦πœ‹cos(πœ‹π‘‘),π‘‘βˆˆπ½,ξ…žξ…ž(𝑑)=π‘’ξ…žξ…ž(𝑑)βˆ’π‘Ÿπ‘π‘’(𝑑),π‘‘βˆˆπ½.(2.22) Hence 𝑦𝑦(0)=𝑒(0),𝑦(1)=𝑒(1),(πœ‚)=𝑒(πœ‚)+𝐴𝑒sin(πœ‹πœ‚),𝑦(πœ‰)=𝑒(πœ‰)+𝐴𝑒𝑦sin(πœ‹πœ‰),ξ…ž(0)=π‘’ξ…ž(0)+π΄π‘’πœ‹,π‘¦ξ…ž(1)=π‘’ξ…ž(1)βˆ’π΄π‘’πœ‹,βˆ’π‘¦ξ…žξ…ž(𝑑)+𝑀𝑦(𝑑)+𝑁𝑦(πœƒ(𝑑))=βˆ’π‘’ξ…žξ…žξ€Ί(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))+(𝑀+π‘Ÿ)𝑐𝑒(𝑑)+𝑁𝑐𝑒(πœƒ(𝑑))≀0,𝑦(0)βˆ’π‘Žπ‘¦ξ…ž(0)βˆ’π‘π‘¦(πœ‚)=𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)βˆ’π‘π‘’(πœ‚)βˆ’π‘Žπ΄π‘’πœ‹βˆ’π‘π΄π‘’sin(πœ‹πœ‚)≀0,𝑦(1)+π‘π‘¦ξ…ž(1)βˆ’π‘‘π‘¦(πœ‰)=𝑒(1)+π‘π‘’ξ…ž(1)βˆ’π‘‘π‘’(πœ‰)βˆ’π‘π΄π‘’πœ‹βˆ’π‘‘π΄π‘’sin(πœ‹πœ‰)≀0,Ξ”π‘¦ξ…žξ€·π‘‘π‘˜ξ€Έ=Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έ+Ξ”π‘ξ…žπ‘’ξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+πΏπ‘˜π‘π‘’ξ€·π‘‘π‘˜ξ€Έ=πΏπ‘˜π‘¦ξ€·π‘‘π‘˜ξ€Έ.(2.23)
By Theorem 2.1, 𝑦(𝑑)≀0 for all π‘‘βˆˆπ½, which implies that 𝑒(𝑑)≀0 for π‘‘βˆˆπ½. The proof is complete.

3. Linear Problem

In this section, we consider the linear boundary value problem βˆ’π‘’ξ…žξ…ž(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))=𝜎(𝑑),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έ=πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š,𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)=𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)=𝑑𝑒(πœ‰).(3.1)

Theorem 3.1. Assume that (𝐻) holds, 𝜎∈𝐢(𝐽), π‘’π‘˜βˆˆπ‘…, and constants 𝑀,𝑁 satisfy (2.3) with ξ‚΅π‘Žπœ‡=(1+2𝑏)+12(π‘Ž+𝑏+1)8ξ‚€1+2π‘ξ‚π‘Ž+𝑏+12ξ‚Άξ‚΅(𝑀+𝑁)+1+(1+𝑏)2ξ‚Άπ‘Ž+𝑏+1π‘šξ“π‘˜=1πΏπ‘˜<1.(3.2) Further suppose that there exist 𝛼,π›½βˆˆπΈ such that (β„Ž1)𝛼≀𝛽 on 𝐽,(β„Ž2)βˆ’π›Όξ…žξ…žξ€Ί(𝑑)+𝑀𝛼(𝑑)+𝑁𝛼(πœƒ(𝑑))+(𝑀+π‘Ÿ)𝑐𝛼(𝑑)+𝑁𝑐𝛼(πœƒ(𝑑))β‰€πœŽ(𝑑),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›Όξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π›Όξ€·π‘‘π‘˜ξ€Έ+πΏπ‘˜π‘π›Όξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š,(3.3)(β„Ž3)βˆ’π›½ξ…žξ…žξ€Ί(𝑑)+𝑀𝛽(𝑑)+𝑁𝛽(πœƒ(𝑑))βˆ’(𝑀+π‘Ÿ)π‘βˆ’π›½(𝑑)+π‘π‘βˆ’π›½ξ€»(πœƒ(𝑑))β‰₯𝜎(𝑑),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›½ξ…žξ€·π‘‘π‘˜ξ€Έβ‰€πΏπ‘˜π›½ξ€·π‘‘π‘˜ξ€Έβˆ’πΏπ‘˜π‘βˆ’π›½ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š.(3.4)Then the boundary value problem (3.1) has one unique solution 𝑒(𝑑) and 𝛼≀𝑒≀𝛽 for π‘‘βˆˆπ½.

Proof. We first show that the solution of (3.1) is unique. Let 𝑒1,𝑒2 be the solution of (3.1) and set 𝑣=𝑒1βˆ’π‘’2. Thus, βˆ’π‘£ξ…žξ…ž(𝑑)+𝑀𝑣(𝑑)+𝑁𝑣(πœƒ(𝑑))=0,π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘£ξ…žξ€·π‘‘π‘˜ξ€Έ=πΏπ‘˜π‘£ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š,𝑣(0)βˆ’π‘Žπ‘£ξ…ž(0)=𝑐𝑣(πœ‚),𝑣(1)+π‘π‘£ξ…ž(1)=𝑑𝑣(πœ‰).(3.5) By Theorem 2.1, we have that 𝑣≀0 for π‘‘βˆˆπ½, that is, 𝑒1≀𝑒2 on 𝐽. Similarly, one can obtain that 𝑒2≀𝑒1 on 𝐽. Hence 𝑒1=𝑒2.
Next, we prove that if 𝑒 is a solution of (3.1), then 𝛼≀𝑒≀𝛽. Let 𝑝=π›Όβˆ’π‘’. From boundary conditions, we have that 𝑐𝛼(𝑑)=𝑐𝑝(𝑑) for all π‘‘βˆˆπ½. From (β„Ž2) and (3.1), we have
βˆ’π‘ξ…žξ…žξ€Ί(𝑑)+𝑀𝑝(𝑑)+𝑁𝑝(πœƒ(𝑑))+(𝑀+π‘Ÿ)𝑐𝑝(𝑑)+𝑁𝑐𝑝(πœƒ(𝑑))≀0,π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘ξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘ξ€·π‘‘π‘˜ξ€Έ+πΏπ‘˜π‘π‘ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š.(3.6) By Theorem 2.1, we have that 𝑝=π›Όβˆ’π‘’β‰€0 on 𝐽. Analogously, 𝑒≀𝛽 on 𝐽.
Finally, we show that the boundary value problem (3.1) has a solution by five steps as follows.
Step 1. Let 𝛼(𝑑)=𝛼(𝑑)+𝑐𝛼(𝑑),𝛽(𝑑)=𝛽(𝑑)βˆ’π‘βˆ’π›½(𝑑). We claim that(1)βˆ’π›Όξ…žξ…ž(𝑑)+𝑀𝛼(𝑑)+𝑁𝛼(πœƒ(𝑑))+(𝑀+π‘Ÿ)𝑐𝛼(𝑑)+𝑁𝑐𝛼(πœƒ(𝑑))β‰€πœŽ(𝑑)forπ‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›Όξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π›Όξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š,(3.7)(2)βˆ’π›½ξ…žξ…ž(𝑑)+𝑀𝛽(𝑑)+𝑁𝛽(πœƒ(𝑑))βˆ’(𝑀+π‘Ÿ)π‘βˆ’π›½(𝑑)+π‘π‘βˆ’π›½ξ‚„(πœƒ(𝑑))β‰₯𝜎(𝑑)forπ‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›½ξ…žξ€·π‘‘π‘˜ξ€Έβ‰€πΏπ‘˜π›½ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š,(3.8)(3)𝛼(𝑑)≀𝛼(𝑑)≀𝛽(𝑑)≀𝛽(𝑑) for π‘‘βˆˆπ½.
From (β„Ž2) and (β„Ž3), we have
βˆ’π›Όξ…žξ…ž(𝑑)+𝑀𝛼(𝑑)+𝑁𝛼(πœƒ(𝑑))β‰€πœŽ(𝑑),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›Όξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π›Όξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜βˆ’,π‘˜=1,…,π‘š.(3.9)π›½ξ…žξ…ž(𝑑)+𝑀𝛽(𝑑)+𝑁𝛽(πœƒ(𝑑))β‰₯𝜎(𝑑),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›½ξ…žξ€·π‘‘π‘˜ξ€Έβ‰€πΏπ‘˜π›½ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š,(3.10)𝛼(0)βˆ’π‘Žπ›Όξ…ž(0)βˆ’π‘π›Ό(πœ‚)=𝛼(0)βˆ’π‘Žπ›Όξ…ž(0)βˆ’π‘π›Ό(πœ‚)βˆ’(π‘Žπœ‹+𝑐sin(πœ‹πœ‚))𝐡𝛼≀0,(3.11)𝛼(1)+π‘π›Όξ…ž(1)βˆ’π‘‘π›Ό(πœ‰)=𝛼(1)+π‘π›Όξ…ž(0)βˆ’π‘‘π›Ό(πœ‰)βˆ’(π‘πœ‹+𝑑sin(πœ‹πœ‰))π΅π›Όβˆ’ξ‚ƒβ‰€0,(3.12)𝛽(0)βˆ’π‘Žπ›½ξ…ž(0)βˆ’π‘ξ‚„π›½(πœ‚)=βˆ’π›½(0)+π‘Žπ›½ξ…ž(0)+𝑐𝛽(πœ‚)βˆ’(π‘Žπœ‹+𝑐sin(πœ‹πœ‚))π΅βˆ’π›½βˆ’ξ‚ƒβ‰€0,(3.13)𝛽(1)+π‘π›½ξ…ž(1)βˆ’π‘‘ξ‚„π›½(πœ‰)=βˆ’π›½(1)βˆ’π‘π›½ξ…ž(0)+𝑑𝛽(πœ‰)βˆ’(π‘πœ‹+𝑑sin(πœ‹πœ‰))π΅βˆ’π›½β‰€0.(3.14) From (3.9)–(3.14), we obtain that 𝑐𝛼(𝑑)=π‘βˆ’π›½(𝑑)≑0,π‘‘βˆˆπ½. Combining (3.9) and (3.10), we obtain that ( 1) and ( 2) hold.
It is easy to see that 𝛼≀𝛼,𝛽≀𝛽 on 𝐽. We show that 𝛼≀𝛽 on 𝐽. Let 𝑝=π›Όβˆ’π›½, then 𝑝(𝑑)=𝛼(𝑑)βˆ’π›½(𝑑)+𝑐𝛼(𝑑)+π‘βˆ’π›½(𝑑). From (3.9)–(3.14), we have
βˆ’π‘ξ…žξ…ž(𝑑)+𝑀𝑝(𝑑)+𝑁𝑝(πœƒ(𝑑))≀0,π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘ξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š,𝑝(0)βˆ’π‘Žπ‘ξ…ž(0)βˆ’π‘π‘(πœ‚)=𝛼(0)βˆ’π‘Žπ›Όξ…ž(0)βˆ’π‘π›Ό(πœ‚)βˆ’(π‘Žπœ‹+𝑐sin𝐼(πœ‹πœ‚))π΅π›Όβˆ’π›½(0)+π‘Žπ›½ξ…ž(0)+𝑐𝛽(πœ‚)βˆ’(π‘Žπœ‹+𝑐sin(πœ‹πœ‚))π΅βˆ’π›½β‰€0,𝑝(1)+π‘π‘ξ…ž(1)βˆ’π‘‘π‘(πœ‰)=𝛼(1)+π‘π›Όξ…ž(1)βˆ’π‘‘π›Ό(πœ‰)βˆ’(π‘πœ‹+𝑑sin(πœ‹πœ‰))π΅π›Όβˆ’π›½(1)βˆ’π‘π›½ξ…ž(1)+𝑑𝛽(πœ‚)βˆ’(π‘πœ‹+𝑑sin(πœ‹πœ‰))π΅βˆ’π›½β‰€0,Ξ”π‘ξ…žξ€·π‘‘π‘˜ξ€Έ=Ξ”π›Όξ…žξ€·π‘‘π‘˜ξ€Έβˆ’Ξ”π›½ξ…žξ€·π‘‘π‘˜ξ€Έ+Ξ”π‘ξ…žπ‘Žξ€·π‘‘π‘˜ξ€Έ+Ξ”π‘ξ…žβˆ’π›½ξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜ξ€·π›Όξ€·π‘‘π‘˜ξ€Έξ€·π‘‘βˆ’π›½π‘˜ξ€Έξ€Έ+πΏπ‘˜ξ€·π‘π›Όξ€·π‘‘π‘˜ξ€Έ+π‘βˆ’π›½ξ€·π‘‘π‘˜ξ€Έξ€Έ=πΏπ‘˜π‘ξ€·π‘‘π‘˜ξ€Έ.(3.15) By Theorem 2.1, we have that 𝑝≀0 on 𝐽, that is, 𝛼≀𝛽 on 𝐽. So ( 3) holds.
Step 2. Consider the boundary value problem βˆ’π‘’ξ…žξ…ž(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))=𝜎(𝑑),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έ=πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š,𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)=πœ†,𝑒(1)+π‘π‘’ξ…ž(1)=𝛿,(3.16) where πœ†βˆˆπ‘,π›Ώβˆˆπ‘. We show that the boundary value problem (3.16) has one unique solution 𝑒(𝑑,πœ†,𝛿).
It is easy to check that the boundary value problem (3.16) is equivalent to the integral equation:
𝑒(𝑑)=𝛿𝑑+(1βˆ’π‘‘)πœ†+π‘πœ†+π‘Žπ›Ώ+ξ€œπ‘Ž+𝑏+110[]+𝐺(𝑑,𝑠)𝜎(𝑠)βˆ’π‘€π‘’(𝑠)βˆ’π‘π‘’(πœƒ(𝑠))𝑑𝑠0<π‘‘π‘˜<π‘‘ξ€·π‘‘βˆ’π‘‘π‘˜πΏξ€Έξ€Ίπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜ξ€»βˆ’1π‘Ž+𝑏+1(𝑑+𝑏)π‘šξ“π‘˜=1ξ€Ίξ€·1βˆ’π‘‘π‘˜ξ€ΈπΏ+π‘ξ€»ξ€Ίπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜ξ€»,(3.17) where 1𝐺(𝑑,𝑠)=ξƒ―π‘Ž+𝑏+1(π‘Ž+𝑑)(1+π‘βˆ’π‘ ),0≀𝑑≀𝑠≀1,(π‘Ž+𝑠)(1+π‘βˆ’π‘‘),0≀𝑠≀𝑑≀1.(3.18)
It is easy to see that 𝑃𝐢(𝐽,𝑅) with norm ‖𝑒‖=maxπ‘‘βˆˆπ½|𝑒(𝑑)| is a Banach space. Define a mapping Ξ¦βˆΆπ‘ƒπΆ(𝐽,𝑅)→𝑃𝐢(𝐽,𝑅) by
(Φ𝑒)(𝑑)=𝛿𝑑+(1βˆ’π‘‘)πœ†+π‘πœ†+π‘Žπ›Ώ+ξ€œπ‘Ž+𝑏+110[]+𝐺(𝑑,𝑠)𝜎(𝑠)βˆ’π‘€π‘’(𝑠)βˆ’π‘π‘’(πœƒ(𝑠))𝑑𝑠0<π‘‘π‘˜<π‘‘ξ€·π‘‘βˆ’π‘‘π‘˜πΏξ€Έξ€Ίπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜ξ€»βˆ’1π‘Ž+𝑏+1(𝑑+𝑏)π‘šξ“π‘˜=1ξ€Ίξ€·1βˆ’π‘‘π‘˜ξ€ΈπΏ+π‘ξ€»ξ€Ίπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜ξ€».(3.19) For any π‘₯,π‘¦βˆˆπ‘ƒπΆ(𝐽,𝑅), we have ||||β‰€ξ€œ(Ξ¦π‘₯)(𝑑)βˆ’(Φ𝑦)(𝑑)10[]𝐺(𝑑,𝑠)𝑀(𝑦(𝑠)βˆ’π‘₯(𝑠))+𝑁(𝑦(πœƒ(𝑠))βˆ’π‘₯(πœƒ(𝑠)))𝑑𝑠+1+(1+𝑏)2ξ‚Άπ‘Ž+𝑏+1π‘šξ“π‘˜=1πΏπ‘˜β‰€ξ€œβ€–π‘₯βˆ’π‘¦β€–10𝐺(𝑑,𝑠)𝑑𝑠‖π‘₯βˆ’π‘¦β€–(𝑀+𝑁)+1+(1+𝑏)2ξ‚Άπ‘Ž+𝑏+1π‘šξ“π‘˜=1πΏπ‘˜β€–π‘₯βˆ’π‘¦β€–.(3.20) Since maxπ‘‘βˆˆπ½ξ€œ10𝐺(𝑑,𝑠)𝑑𝑠=π‘Ž(1+2𝑏)+12(π‘Ž+𝑏+1)8ξ‚€1+2π‘ξ‚π‘Ž+𝑏+12,(3.21) the inequality (3.2) implies that Ξ¦βˆΆπ‘ƒπΆ(𝐽)→𝑃𝐢(𝐽) is a contraction mapping. Thus there exists a unique π‘’βˆˆπ‘ƒπΆ(𝐽) such that Φ𝑒=𝑒. The boundary value problem (3.16) has a unique solution.
Step 3. We show that for any π‘‘βˆˆπ½, the unique solution 𝑒(𝑑,πœ†,𝛿) of the boundary value problem (3.16) is continuous in πœ† and 𝛿. Let 𝑒(𝑑,πœ†π‘–,𝛿𝑖),𝑖=1,2, be the solution of βˆ’π‘’ξ…žξ…ž(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))=𝜎(𝑑),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έ=πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜,π‘˜=1,…,π‘š,𝑒(0)βˆ’π‘Žπ‘’ξ…ž(0)=πœ†π‘–,𝑒(1)+π‘π‘’ξ…ž(1)=𝛿𝑖,𝑖=1,2.(3.22) Then 𝑒𝑑,πœ†π‘–,𝛿𝑖=𝛿𝑖𝑑+(1βˆ’π‘‘)πœ†π‘–+π‘πœ†π‘–+π‘Žπ›Ώπ‘–+ξ€œπ‘Ž+𝑏+110𝐺(𝑑,𝑠)𝜎(𝑠)βˆ’π‘€π‘’π‘ ,πœ†π‘–,π›Ώπ‘–ξ€Έξ€·βˆ’π‘π‘’πœƒ(𝑠),πœ†π‘–,𝛿𝑖+𝑑𝑠0<π‘‘π‘˜<π‘‘ξ€·π‘‘βˆ’π‘‘π‘˜πΏξ€Έξ€Ίπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜ξ€»βˆ’1Γ—π‘Ž+𝑏+1(𝑑+𝑏)π‘šξ“π‘˜=1ξ€Ίξ€·1βˆ’π‘‘π‘˜ξ€ΈπΏ+π‘ξ€»ξ€Ίπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ+π‘’π‘˜ξ€»,𝑖=1,2.(3.23)
From (3.23), we have that
‖‖𝑒𝑑,πœ†1,𝛿1ξ€Έξ€·βˆ’π‘’π‘‘,πœ†2,𝛿2‖‖≀||πœ†1βˆ’πœ†2||+||𝛿1βˆ’π›Ώ2||‖‖𝑒+(𝑀+𝑁)𝑑,πœ†1,𝛿1ξ€Έξ€·βˆ’π‘’π‘‘,πœ†2,𝛿2ξ€Έβ€–β€–maxπ‘‘βˆˆπ½ξ€œ10𝐺(𝑑,𝑠)𝑑𝑠+‖𝑒𝑑,πœ†1,𝛿1ξ€Έξ€·βˆ’π‘’π‘‘,πœ†2,𝛿2ξ€Έβ€–ξ‚΅1+(1+𝑏)2ξ‚Άπ‘Ž+𝑏+1π‘šξ“π‘˜=1πΏπ‘˜β‰€||πœ†β€–π‘₯βˆ’π‘¦β€–1βˆ’πœ†2||+||𝛿1βˆ’π›Ώ2||‖‖𝑒+πœ‡π‘‘,πœ†1,𝛿1ξ€Έξ€·βˆ’π‘’π‘‘,πœ†2,𝛿2ξ€Έβ€–β€–.(3.24)
Hence
‖‖𝑒(𝑑,πœ†1,𝛿1ξ€·)βˆ’π‘’π‘‘,πœ†2,𝛿2ξ€Έβ€–β€–0≀1ξ€·||πœ†1βˆ’πœ‡1βˆ’πœ†2||+||𝛿1βˆ’π›Ώ2||ξ€Έ.(3.25)
Step 4. We show that 𝛼(𝑑)≀𝑒(𝑑,πœ†,𝛿)≀𝛽(𝑑)(3.26) for any π‘‘βˆˆπ½, πœ†βˆˆ[𝑐𝛼(πœ‚),𝑐𝛽(πœ‚)], and π›Ώβˆˆ[𝑑𝛼(πœ‰),𝑑𝛽(πœ‰)], where 𝑒(𝑑,πœ†,𝛿) is unique solution of the boundary value problem (3.16).
Let π‘š(𝑑)=𝛼(𝑑)βˆ’π‘’(𝑑,πœ†,𝛿). From (3.9), (3.11), (3.12), and (3.16), we have that π‘š(0)βˆ’π‘Žπ‘šξ…ž(0)β‰€π‘π‘š(πœ‚),π‘š(1)+π‘π‘šξ…ž(1)β‰€π‘‘π‘š(πœ‰), and
βˆ’π‘šξ…žξ…ž(𝑑)+π‘€π‘š(𝑑)+π‘π‘š(πœƒ(𝑑))=βˆ’π›Όξ…žξ…ž(𝑑)+𝑀𝛼(𝑑)+𝑁𝛼(πœƒ(𝑑))+π‘’ξ…žξ…ž(𝑑,πœ†)βˆ’π‘€π‘’(𝑑,πœ†,𝛿)βˆ’π‘π‘’(πœƒ(𝑑),πœ†,𝛿)β‰€πœŽ(𝑑)βˆ’πœŽ(𝑑)≀0,Ξ”π‘šξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘’ξ€·π‘‘π‘˜ξ€Έ.(3.27) By Theorem 2.1, we obtain that π‘šβ‰€0 on 𝐽, that is, 𝛼(𝑑)≀𝑒(𝑑,πœ†,𝛿) on 𝐽. Similarly, 𝑒(𝑑,πœ†,𝛿)≀𝛽(𝑑) on 𝐽.
Step 5. Let 𝐷=[𝑐𝛼(πœ‚),𝑐𝛽(πœ‚)]Γ—[𝑑𝛼(πœ‰),𝑑𝛽(πœ‰)]. Define a mapping πΉβˆΆπ·β†’π‘2 by 𝐹(πœ†,𝛿)=(𝑒(πœ‚,πœ†,𝛿),𝑒(πœ‰,πœ†,𝛿)),(3.28) where 𝑒(𝑑,πœ†,𝛿) is unique solution of the boundary value problem (3.16). From Step 4, we have 𝐹(𝐷)βŠ‚π·.(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 (πœ†0,𝛿0)∈𝐷 such that π‘’ξ€·πœ‚,πœ†0,𝛿0ξ€Έ=πœ†0ξ€·,π‘’πœ‰,πœ†0,𝛿0ξ€Έ=𝛿0.(3.30) Obviously, 𝑒(𝑑,πœ†0,𝛿0) is unique solution of the boundary value problem (3.1). This completes the proof.

4. Main Results

Let π‘€βˆˆπ‘, π‘βˆˆπ‘. We first give the following definition.

Definition 4.1. A function π›ΌβˆˆπΈ is called a lower solution of the boundary value problem (1.2) if βˆ’π›Όξ…žξ…ž(𝑑)+(𝑀+π‘Ÿ)𝑐𝛼(𝑑)+𝑁𝑐𝛼(πœƒ(𝑑))≀𝑓(𝑑,𝛼(𝑑),𝛼(πœƒ(𝑑))),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›Όξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΌπ‘˜ξ€·π›Όξ€·π‘‘π‘˜ξ€Έξ€Έ+πΏπ‘˜π‘π›Όξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š.(4.1)

Definition 4.2. A function π›½βˆˆπΈ is called an upper solution of the boundary value problem (1.2) if βˆ’π›½ξ…žξ…ž(𝑑)βˆ’(𝑀+π‘Ÿ)π‘βˆ’π›½(𝑑)βˆ’π‘π‘βˆ’π›½(πœƒ(𝑑))β‰₯𝑓(𝑑,𝛽(𝑑),𝛽(πœƒ(𝑑)))π‘‘βˆˆπ½,π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›½ξ…žξ€·π‘‘π‘˜ξ€Έβ‰€πΌπ‘˜π›½ξ€·π‘‘π‘˜ξ€Έβˆ’πΏπ‘˜π‘βˆ’π›½ξ€·π‘‘π‘˜ξ€Έ,π‘˜=1,…,π‘š.(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(𝑑)≀𝛽(𝑑) on 𝐽,(𝐻2)the constants 𝑀,𝑁 in definition of upper and lower solutions satisfy (2.3), (3.2), and 𝑓(𝑑,π‘₯,𝑦)βˆ’π‘“π‘‘,π‘₯,𝑦β‰₯βˆ’π‘€π‘₯βˆ’π‘₯ξ€Έξ€·βˆ’π‘π‘¦βˆ’π‘¦ξ€Έ,πΌπ‘˜(π‘₯)βˆ’πΌπ‘˜(𝑦)β‰₯πΏπ‘˜(π‘₯βˆ’π‘¦),π‘₯≀𝑦,(4.3) for 𝛼(𝑑)≀π‘₯≀π‘₯≀𝛽(𝑑),𝛼(πœƒ(𝑑))≀𝑦≀𝑦≀𝛽(πœƒ(𝑑)),π‘‘βˆˆπ½.
Then, there exist monotone sequences {𝛼𝑛},{𝛽𝑛} with 𝛼0=𝛼,𝛽0=𝛽 such that limπ‘›β†’βˆžπ›Όπ‘›(𝑑)=𝜌(𝑑),limπ‘›β†’βˆžπ›½π‘›(𝑑)=𝜚(𝑑) uniformly on 𝐽, and 𝜌,𝜚 are the minimal and the maximal solutions of (1.2), respectively, such that
𝛼0≀𝛼1≀𝛼2β‰€β‹―π›Όπ‘›β‰€πœŒβ‰€π‘₯β‰€πœšβ‰€π›½π‘›β‰€β‹―β‰€π›½2≀𝛽1≀𝛽0(4.4) on 𝐽, where π‘₯ is any solution of (1.1) such that 𝛼(𝑑)≀π‘₯(𝑑)≀𝛽(𝑑) on 𝐽.

Proof. Let [𝛼,𝛽]={π‘’βˆˆπΈβˆΆπ›Ό(𝑑)≀𝑒(𝑑)≀𝛽(𝑑),π‘‘βˆˆπ½}. For any π›Ύβˆˆ[𝛼,𝛽], we consider the boundary value problem βˆ’π‘’ξ…žξ…ž(𝑑)+𝑀𝑒(𝑑)+𝑁𝑒(πœƒ(𝑑))=𝑓(𝑑,𝛾(𝑑),𝛾(πœƒ(𝑑)))+𝑀𝛾(𝑑)+𝑁𝛾(πœƒ(𝑑)),π‘‘βˆˆπ½,Ξ”π‘’ξ…žξ€·π‘‘π‘˜ξ€Έ=πΌπ‘˜ξ€·π›Ύξ€·π‘‘π‘˜ξ€Έξ€Έβˆ’πΏπ‘˜ξ€·π‘’ξ€·π‘‘π‘˜ξ€Έξ€·π‘‘βˆ’π›Ύπ‘˜ξ€Έξ€Έ,π‘˜=1,…,π‘š.𝑒(0)βˆ’π‘Žπ‘₯ξ…ž(0)=𝑐𝑒(πœ‚),𝑒(1)+π‘π‘’ξ…ž(1)=𝑑𝑒(πœ‰).(4.5)
Since 𝛼 is a lower solution of (1.2), from (𝐻2), we have that
βˆ’π›Όξ…žξ…ž(𝑑)+𝑀𝛼(𝑑)+𝑁𝛼(πœƒ(𝑑))≀𝑓(𝑑,𝛼(𝑑),𝛼(πœƒ(𝑑)))+𝑀𝛼(𝑑)+𝑁𝛼(πœƒ(𝑑))βˆ’(𝑀+π‘Ÿ)𝑐𝛼(𝑑)βˆ’π‘π‘π›Ό(πœƒ(𝑑))≀𝑓(𝑑,𝛾(𝑑),𝛾(πœƒ(𝑑)))+𝑀𝛾(𝑑)+𝑁𝛾(πœƒ(𝑑))βˆ’(𝑀+π‘Ÿ)𝑐𝛼(𝑑)βˆ’π‘π‘π›Ό(πœƒ(𝑑)),Ξ”π›Όξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΌπ‘˜ξ€·π›Όξ€·π‘‘π‘˜ξ€Έξ€Έ+πΏπ‘˜π‘π›Όξ€·π‘‘π‘˜ξ€Έβ‰₯πΌπ‘˜ξ€·π›Ύξ€·π‘‘π‘˜ξ€Έξ€Έ+πΏπ‘˜π›Όξ€·π‘‘π‘˜ξ€Έβˆ’πΏπ‘˜π›Ύξ€·π‘‘π‘˜ξ€Έ+πΏπ‘˜π‘π›Όξ€·π‘‘π‘˜ξ€Έ.(4.6)
Similarly, we have that
βˆ’π›½ξ…žξ…ž(𝑑)+𝑀𝛽(𝑑)+𝑁𝛽(πœƒ(𝑑))β‰₯𝑓(𝑑,𝛾(𝑑),𝛾(πœƒ(𝑑)))+𝑀𝛾(𝑑)+𝑁𝛾(πœƒ(𝑑))+(𝑀+π‘Ÿ)π‘βˆ’π›½(𝑑)+π‘π‘βˆ’π›½(πœƒ(𝑑)),Ξ”π›½ξ…žξ€·π‘‘π‘˜ξ€Έβ‰€πΌπ‘˜ξ€·π›½ξ€·π‘‘π‘˜ξ€Έξ€Έβˆ’πΏπ‘˜π‘βˆ’π›½ξ€·π‘‘π‘˜ξ€Έβ‰€πΌπ‘˜ξ€·π›Ύξ€·π‘‘π‘˜ξ€Έξ€Έ+πΏπ‘˜π›½ξ€·π‘‘π‘˜ξ€Έβˆ’πΏπ‘˜π›Ύξ€·π‘‘π‘˜ξ€Έβˆ’πΏπ‘˜π‘βˆ’π›½ξ€·π‘‘π‘˜ξ€Έ.(4.7)
By Theorem 3.1, the boundary value problem (4.5) has a unique solution π‘’βˆˆ[𝛼,𝛽]. We define an operator Ξ¨ by 𝑒=Ψ𝛾, then Ξ¨ is an operator from [𝛼,𝛽] to [𝛼,𝛽].
We will show that
(a)𝛼≀Ψ𝛼,Ψ𝛽≀𝛽,(b)Ξ¨ is nondecreasing in [𝛼,𝛽].
From Ξ¨π›Όβˆˆ[𝛼,𝛽] and Ξ¨π›½βˆˆ[𝛼,𝛽], we have that (a) holds. To prove (b), we show that Ψ𝜈1β‰€Ξ¨πœˆ2 if π›Όβ‰€πœˆ1β‰€πœˆ2≀𝛽.
Let πœˆβˆ—1=Ψ𝜈1,πœˆβˆ—2=Ψ𝜈2 ,and 𝑝=πœˆβˆ—1βˆ’πœˆβˆ—2, then by (𝐻2) and boundary conditions, we have that
βˆ’π‘ξ…žξ…žξ€·(𝑑)+𝑀𝑝(𝑑)+𝑁𝑝(πœƒ(𝑑))=𝑓𝑑,𝜈1(𝑑),𝜈1ξ€Έ(πœƒ(𝑑))+π‘€πœˆ1(𝑑)+π‘πœˆ1ξ€·(πœƒ(𝑑))βˆ’π‘“π‘‘,𝜈2(𝑑),𝜈2ξ€Έ(πœƒ(𝑑))βˆ’π‘€πœˆ2(𝑑)βˆ’π‘πœˆ2(πœƒ(𝑑))≀0,Ξ”π‘ξ…žξ€·π‘‘π‘˜ξ€Έβ‰₯πΏπ‘˜π‘ξ€·π‘‘π‘˜ξ€Έ,𝑝(0)βˆ’π‘Žπ‘ξ…ž(0)=𝑐𝑝(πœ‚),𝑝(1)+π‘π‘’ξ…ž(1)=𝑑𝑝(πœ‰).(4.8) By Theorem 2.1, 𝑝(𝑑)≀0 on 𝐽, which implies that Ψ𝜈1β‰€Ξ¨πœˆ2.
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
𝛼0≀𝛼1≀𝛼2≀⋯≀𝛼𝑛≀𝛽𝑛≀⋯≀𝛽2≀𝛽1≀𝛽0(4.9) on π‘‘βˆˆπ½, and each 𝛼𝑛,π›½π‘›βˆˆπΈ satisfies βˆ’π›Όπ‘›ξ…žξ…ž(𝑑)+𝑀𝛼𝑛(𝑑)+𝑁𝛼𝑛(πœƒ(𝑑))=𝑓𝑑,π›Όπ‘›βˆ’1(𝑑),π›Όπ‘›βˆ’1ξ€Έ(πœƒ(𝑑))+π‘€π›Όπ‘›βˆ’1(𝑑)+π‘π›Όπ‘›βˆ’1(πœƒ(𝑑)),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›Όξ…žπ‘›ξ€·π‘‘π‘˜ξ€Έ=πΌπ‘˜ξ€·π›Όπ‘›βˆ’1ξ€·π‘‘π‘˜ξ€Έξ€Έ+πΏπ‘˜ξ€·π›Όπ‘›ξ€·π‘‘π‘˜ξ€Έβˆ’π›Όπ‘›βˆ’1ξ€·π‘‘π‘˜π›Όξ€Έξ€Έ,π‘˜=1,2,…,π‘š,𝑛(0)βˆ’π‘Žπ›Όξ…žπ‘›(0)=𝑐𝛼𝑛(πœ‚),𝛼𝑛(1)+π‘π›Όξ…žπ‘›(1)=𝑑𝛼𝑛(πœ‰),βˆ’π›½π‘›ξ…žξ…ž(𝑑)+𝑀𝛽𝑛(𝑑)+𝑁𝛽𝑛(πœƒ(𝑑))=𝑓𝑑,π›½π‘›βˆ’1(𝑑),π›½π‘›βˆ’1ξ€Έ(πœƒ(𝑑))+π‘€π›½π‘›βˆ’1(𝑑)+π‘π›½π‘›βˆ’1(πœƒ(𝑑)),π‘‘βˆˆπ½,π‘‘β‰ π‘‘π‘˜,Ξ”π›½π‘›ξ€·π‘‘π‘˜ξ€Έ=πΌπ‘˜ξ€·π›½π‘›βˆ’1ξ€·π‘‘π‘˜ξ€Έξ€Έ+πΏπ‘˜ξ€·π›½π‘›ξ€·π‘‘π‘˜ξ€Έβˆ’π›½π‘›βˆ’1ξ€·π‘‘π‘˜π›½ξ€Έξ€Έ,π‘˜=1,2,…,π‘š,𝑛(0)βˆ’π‘Žπ›½ξ…žπ‘›(0)=𝑐𝛽𝑛(πœ‚),𝛽𝑛(1)+π‘π›½ξ…žπ‘›(1)=𝑑𝛽𝑛(πœ‰).(4.10) Therefore, there exist 𝜌,𝜚 such that such that limπ‘›β†’βˆžπ›Όπ‘›(𝑑)=𝜌(𝑑), limπ‘›β†’βˆžπ›½π‘›(𝑑)=𝜚(𝑑) uniformly on 𝐽. Clearly, 𝜌,𝜚 are solutions of (1.1).
Finally, we prove that if π‘₯∈[𝛼0,𝛽0] is any solution of (1.1), then 𝜌(𝑑)≀π‘₯(𝑑)β‰€πœš(𝑑) on 𝐽. To this end, we assume, without loss of generality, that 𝛼𝑛(𝑑)≀π‘₯(𝑑)≀𝛽𝑛(𝑑) for some 𝑛. Since 𝛼0(𝑑)≀π‘₯(𝑑)≀𝛽0(𝑑), from property (b), we can obtain
𝛼𝑛+1(𝑑)≀π‘₯(𝑑)≀𝛽𝑛+1(𝑑),π‘‘βˆˆπ½.(4.11) Hence, we can conclude that 𝛼𝑛(𝑑)≀π‘₯(𝑑)≀𝛽𝑛(𝑑),βˆ€π‘›.(4.12) Passing to the limit as π‘›β†’βˆž, we obtain 𝜌(𝑑)≀π‘₯(𝑑)β‰€πœš(𝑑),π‘‘βˆˆπ½.(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, β€œImproved generalized quasilinearization (GQL) method,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24, no. 11, pp. 1627–1637, 1995. View at: Google Scholar | Zentralblatt MATH | 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, β€œA 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. M. Sockol and A. S. Vatsala, β€œA 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, β€œA 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 | Zentralblatt MATH | MathSciNet
  7. D. Jiang and J. Wei, β€œMonotone 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 | Zentralblatt MATH | MathSciNet
  8. T. Jankowski, β€œAdvanced differential equations with nonlinear boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 490–503, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. J. J. Nieto and R. RodrΓ­guez-LΓ³pez, β€œExistence 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 | Zentralblatt MATH | MathSciNet
  10. J. J. Nieto and R. RodrΓ­guez-LΓ³pez, β€œRemarks 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. Z. He and X. He, β€œPeriodic 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. F. Zhang, A. Zhao, and J. Yan, β€œMonotone 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 | Zentralblatt MATH | MathSciNet
  13. Q. Yao, β€œMonotone 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. T. Jankowski, β€œOrdinary 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 | Zentralblatt MATH | MathSciNet
  15. W. Ding, J. Mi, and M. Han, β€œPeriodic 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. G. Infante and J. R. L. Webb, β€œThree-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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. P. W. Eloe and L. Zhang, β€œComparison 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  18. S. A. Marano, β€œA 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. T. Jankowski, β€œSolvability 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. Y. Liu, β€œNon-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 | MathSciNet
  21. Y. Liu, β€œPositive 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 Site | Google Scholar | MathSciNet

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.


More related articles

1116Β Views | 338Β Downloads | 0Β Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles