About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 489353, 12 pages
http://dx.doi.org/10.1155/2012/489353
Research Article

Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation

1Faculty of Science, Alexandria University, Alexandria, Egypt
2Faculty of Science, Garyounis University, Benghazi, Libya

Received 13 October 2011; Accepted 5 December 2011

Academic Editor: IstvΓ‘nΒ GyΓΆri

Copyright Β© 2012 A. M. A. El-Sayed 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

We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equation π‘₯ξ…žξ…ž(𝑑)=𝑓(𝑑,π‘₯(πœ™(𝑑))), π‘‘βˆˆ(0,1), with the nonlocal condition βˆ‘π‘šπ‘˜=1π‘Žπ‘˜π‘₯(πœπ‘˜)=π‘₯0, π‘₯ξ…žβˆ‘(0)+𝑛𝑗=1𝑏𝑗π‘₯ξ…ž(πœ‚π‘—)=π‘₯1, where πœπ‘˜βˆˆ(π‘Ž,𝑑)βŠ‚(0,1), πœ‚π‘—βˆˆ(𝑐,𝑒)βŠ‚(0,1), and π‘₯0,π‘₯1>0. As an application the integral and the nonlocal conditions βˆ«π‘‘π‘Žπ‘₯(𝑑)𝑑𝑑=π‘₯0, π‘₯ξ…ž(0)+π‘₯(𝑒)βˆ’π‘₯(𝑐)=π‘₯1 will be considered.

1. Introduction

The nonlocal boundary value problems of ordinary differential equations arise in a variety of different areas of applied mathematics and physics.

The study of nonlocal boundary value problems was initiated by Il’in and Moiseev [1, 2]. Since then, the non-local boundary value problems have been studied by several authors. The reader is referred to [3–22] and references therein.

In most of all these papers, the authors assume that the function π‘“βˆΆ[0,1]×𝑅+→𝑅+ is continuous. They all assume thatlimπ‘₯β†’βˆžπ‘“(π‘₯)π‘₯=0or∞,limπ‘₯β†’0𝑓(π‘₯)π‘₯=0or∞.(1.1) These assumptions are restrictive, and there are many functions that do not satisfy these assumptions.

Here we assume that the function π‘“βˆΆ[0,1]×𝑅+→𝑅+ is measurable in π‘‘βˆˆ[0,1] for all π‘₯βˆˆπ‘…+ and continuous in π‘₯βˆˆπ‘…+ for almost all π‘‘βˆˆ[0,1] is and there exists an integrable function π‘ŽβˆˆπΏ1[0,1] and a constant 𝑏>0 such that||||≀||||[]𝑓(𝑑,π‘₯)π‘Ž(𝑑)+𝑏|π‘₯|,βˆ€(𝑑,π‘₯)∈0,1×𝐷.(1.2) Our aim here is to study the existence of at least one monotonic positive solution for the nonlocal problem of the second-order functional differential equationπ‘₯ξ…žξ…ž(𝑑)=𝑓(𝑑,π‘₯(πœ™(𝑑))),π‘‘βˆˆ(0,1),(1.3) with the nonlocal conditionπ‘šξ“π‘˜=1π‘Žπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έ=π‘₯0,π‘₯ξ…ž(0)+𝑛𝑗=1𝑏𝑗π‘₯ξ…žξ€·πœ‚π‘—ξ€Έ=π‘₯1,(1.4) where πœπ‘˜βˆˆ(π‘Ž,𝑑)βŠ‚(0,1),πœ‚π‘—βˆˆ(𝑐,𝑒)βŠ‚(0,1),andπ‘₯0,π‘₯1>0.

As an application, the problem with the integral and nonlocal conditionsξ€œπ‘‘π‘Žπ‘₯(𝑑)𝑑𝑑=π‘₯0,π‘₯ξ…ž(0)+π‘₯(𝑒)βˆ’π‘₯(𝑐)=π‘₯1,(1.5) is studied.

It must be noticed that the nonlocal conditionsπ‘₯(𝜏)=π‘₯0,𝜏∈(π‘Ž,𝑑),π‘₯ξ…ž(0)+π‘₯ξ…ž(πœ‚)=π‘₯1,πœ‚βˆˆ(𝑐,𝑒),π‘šξ“π‘˜=1π‘Žπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έ=0,πœπ‘˜βˆˆ(π‘Ž,𝑑),π‘₯ξ…ž(0)+𝑛𝑗=1𝑏𝑗π‘₯ξ…žξ€·πœ‚π‘—ξ€Έ=0,πœ‚π‘—ξ€œβˆˆ(𝑐,𝑒),π‘‘π‘Žπ‘₯(𝑑)𝑑𝑑=0,π‘₯ξ…ž(0)+π‘₯(𝑒)=π‘₯(𝑐)(1.6) are special cases of our the nonlocal and integral conditions.

2. Integral Equation Representation

Consider the functional differential equation (1.3) with the nonlocal condition (1.4) with the following assumptions.(i)π‘“βˆΆ[0,1]×𝑅+→𝑅+ is measurable in π‘‘βˆˆ[0,1] for all π‘₯βˆˆπ‘…+ and continuous in π‘₯βˆˆπ‘…+ for almost all π‘‘βˆˆ[0,1] and there exists an integrable function π‘ŽβˆˆπΏ1[0,1], and a constant 𝑏>0 such that ||||≀||||[]𝑓(𝑑,π‘₯)π‘Ž(𝑑)+𝑏|π‘₯|,βˆ€(𝑑,π‘₯)∈0,1×𝐷.(2.1)(ii)πœ™βˆΆ(0,1)β†’(0,1) is continuous.(iii)βˆ‘π‘<1/(3βˆ’π΅),𝐡=(𝑛𝑗=1𝑏𝑗+1)βˆ’1.(iv)π‘šξ“π‘˜=1π‘Žπ‘˜>0,βˆ€π‘˜=1,2,…,π‘š,𝑛𝑗=1𝑏𝑗>0,βˆ€π‘—=1,2,…,𝑛.(2.2) Now, we have the following Lemma.

Lemma 2.1. The solution of the nonlocal problem (1.3)-(1.4) can be expressed by the integral equation ξƒ―π‘₯π‘₯(𝑑)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°ξƒ©βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+π΅π‘‘βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(2.3) where βˆ‘π΄=(π‘šπ‘˜=1π‘Žπ‘˜)βˆ’1βˆ‘,𝐡=(𝑛𝑗=1𝑏𝑗+1)βˆ’1.

Proof. Integrating (1.3), we get π‘₯ξ…ž(𝑑)=π‘₯ξ…ž(ξ€œ0)+𝑑0𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(2.4) Integrating (2.4), we obtain π‘₯(𝑑)=π‘₯(0)+π‘₯ξ…ž(ξ€œ0)𝑑+𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(2.5) Let 𝑑=πœπ‘˜, in (2.5), we get π‘šξ“π‘˜=1π‘Žπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έ=π‘›ξ“π‘˜=1π‘Žπ‘˜π‘₯(0)+π‘›ξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξ…ž(0)+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(2.6) and we deduce that ξƒ―π‘₯π‘₯(0)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξ…ž(0)βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°ξƒ©βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,𝐴=π‘šξ“π‘˜=1π‘Žπ‘˜ξƒͺβˆ’1.(2.7) Substitute from (2.7) into (2.5), we obtain ξƒ―π‘₯π‘₯(𝑑)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+π‘₯ξ…žξƒ©(0)π‘‘βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜ξƒͺ+ξ€œπ‘‘0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(2.8) Let 𝑑=πœ‚π‘—, in (2.4), we obtain 𝑛𝑗=1𝑏𝑗π‘₯ξ…žξ€·πœ‚π‘—ξ€Έ=𝑛𝑗=1𝑏𝑗π‘₯ξ…ž(0)+𝑛𝑗=1π‘π‘—ξ€œπœ‚π‘—0π‘₯𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,1βˆ’π‘₯ξ…ž(0)=π‘₯ξ…ž(0)𝑛𝑗=1𝑏𝑗+𝑛𝑗=1π‘π‘—ξ€œπœ‚π‘—0𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(2.9) and we deduce that π‘₯ξ…žξƒ©π‘₯(0)=𝐡1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒͺ𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,𝐡=𝑛𝑗=1𝑏𝑗ξƒͺ+1βˆ’1.(2.10) Substitute from (2.10) into (2.8), we obtain ξƒ―π‘₯π‘₯(𝑑)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°ξƒ©βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+π΅π‘‘βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°,+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(2.11) which proves that the solution of the nonlocal problem (1.3)-(1.4) can be expressed by the integral equation (2.3).

3. Existence of Solution

We study here the existence of at least one monotonic nondecreasing solution π‘₯∈𝐢[0,1] for the integral equation (2.3).

Theorem 3.1. Assume that (i)–(iv) are satisfied. Then the nonlocal problem (1.3)-(1.4) has at least one solution π‘₯∈𝐢[0,1].

Proof. Define the subset π‘„π‘ŸβŠ‚πΆ(0,1) by π‘„π‘Ÿ={π‘₯∈𝐢∢|π‘₯(𝑑)|β‰€π‘Ÿ,π‘Ÿ=(𝐴π‘₯0+𝐡π‘₯1+(3βˆ’π΅)β€–π‘Žβ€–)/(1βˆ’(3βˆ’π΅)𝑏),π‘Ÿ>0}. Clear the set π‘„π‘Ÿ which is nonempty, closed, and convex.
Let 𝐻 be an operator defined byξƒ―π‘₯(𝐻π‘₯)(𝑑)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°ξƒ©βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑s+π΅π‘‘βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(3.1)
Let π‘₯βˆˆπ‘„π‘Ÿ, then||||ξƒ―π‘₯(𝐻π‘₯)(𝑑)≀𝐴0+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έ||||ξƒ°ξƒ©βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+π΅π‘‘βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1+𝑛𝑗=1π‘π‘—ξ€œπœ‚π‘—0||||ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑0||||ξƒ―π‘₯(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠≀𝐴0+π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œ10ξ€Ί||||||||ξ€»ξƒ°ξƒ―π‘₯π‘Ž(𝑠)+𝑏π‘₯(πœ™(𝑠))𝑑𝑠+𝐡1+𝑛𝑗=1π‘π‘—ξ€œ10ξ€Ί||||||||ξ€»ξƒ°+ξ€œπ‘Ž(𝑠)+𝑏π‘₯(πœ™(𝑠))𝑑𝑠10ξ€Ί||π‘Ž||||π‘₯||ξ€»(𝑠)+𝑏(πœ™(𝑠))𝑑𝑠≀𝐴π‘₯0+β€–π‘Žβ€–+𝑏supπ‘‘βˆˆπΌ||||π‘₯(πœ™(𝑑))+𝐡π‘₯1+𝐡𝑛𝑗=1π‘π‘—β€–π‘Žβ€–+𝑏𝐡𝑛𝑗=1𝑏𝑗supπ‘‘βˆˆπΌ||||+π‘₯(πœ™(𝑑))β€–π‘Žβ€–+𝑏supπ‘‘βˆˆπΌ||||π‘₯(πœ™(𝑑))≀𝐴π‘₯0+𝐡π‘₯1+2β€–π‘Žβ€–+2𝑏‖π‘₯β€–+(1βˆ’π΅)β€–π‘Žβ€–+𝑏(1βˆ’π΅)β€–π‘₯‖≀𝐴π‘₯0+𝐡π‘₯1+(3βˆ’π΅)β€–π‘Žβ€–+(3βˆ’π΅)π‘π‘Ÿβ‰€π‘Ÿ,(3.2) then π»βˆΆπ‘„π‘Ÿβ†’π‘„π‘Ÿ and {𝐻π‘₯(𝑑)} is uniformly bounded in π‘„π‘Ÿ.
Also for 𝑑1,𝑑2∈[0,1] such that 𝑑1<𝑑2, we have𝑑(𝐻π‘₯)2ξ€Έξ€·π‘‘βˆ’(𝐻π‘₯)1𝑑=𝐡2βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑑)))𝑑𝑠𝑑20𝑑2ξ€Έξƒ©π‘‘βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑑)))π‘‘π‘ βˆ’π΅1βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°βˆ’ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑑)))𝑑𝑠𝑑10𝑑1ξ€Έξ€·π‘‘βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑑)))𝑑𝑠=𝐡2βˆ’π‘‘1ξ€Έξƒ―π‘₯1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(s,π‘₯(πœ™(𝑑)))𝑑𝑠𝑑10𝑑2βˆ’π‘‘1ξ€Έ+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑑)))𝑑𝑠𝑑2𝑑1𝑑2ξ€Έβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑑)))𝑑𝑠.(3.3) Then ||𝑑(𝐻π‘₯)2ξ€Έξ€·π‘‘βˆ’(𝐻π‘₯)1ξ€Έ||||𝑑≀𝐡2βˆ’π‘‘1||ξƒ―π‘₯1+𝑛𝑗=1π‘π‘—ξ€œπœ‚π‘—0ξ€Ί||||||||ξ€»ξƒ°+||π‘‘π‘Ž(𝑠)+𝑏π‘₯(πœ™(𝑠))𝑑𝑠2βˆ’π‘‘1||ξ€œπ‘‘10ξ€Ί||||||||ξ€»+ξ€œπ‘Ž(𝑠)+𝑏π‘₯(πœ™(𝑠))𝑑𝑠𝑑2𝑑1𝑑2||π‘Ž||||π‘₯||ξ€»||π‘‘βˆ’π‘ ξ€Έξ€Ί(𝑠)+𝑏(πœ™(𝑠))𝑑𝑠≀𝐡2βˆ’π‘‘1||π‘₯1+𝑛𝑗=1𝑏𝑗[]+||π‘‘β€–π‘Žβ€–+π‘π‘Ÿ2βˆ’π‘‘1||[]+ξ€œβ€–π‘Žβ€–+π‘π‘Ÿπ‘‘2𝑑1ξ€Ίπ‘‘β€–π‘Žβ€–π‘‘π‘ +π‘π‘Ÿ2βˆ’π‘‘1ξ€».(3.4) The above inequality shows that ||𝑑(𝐻π‘₯)2ξ€Έβˆ’ξ€·π‘‘(𝐻π‘₯)1ξ€Έ||⟢0as𝑑2βŸΆπ‘‘1.(3.5) Therefore {𝐻π‘₯(𝑑)} is equicontinuous. By the ArzelΓ -Ascoli theorem, {𝐻π‘₯(𝑑)} is relatively compact.
Since all conditions of the Schauder theorem hold, then 𝐻 has a fixed point in π‘„π‘Ÿ which proves the existence of at least one solution π‘₯∈𝐢[0,1] of the integral equation (2.3), wherelim𝑑→0+ξƒ―π‘₯π‘₯(𝑑)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))π‘‘π‘ βˆ’π΅π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜ξƒ―π‘₯1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠=π‘₯(0),lim𝑑→1βˆ’ξƒ―π‘₯π‘₯(𝑑)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°ξƒ©βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+𝐡1βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠10(1βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠=π‘₯(1).(3.6) To complete the proof, we prove that the integral equation (2.3) satisfies nonlocal problem (1.3)-(1.4). Differentiating (2.3), we get π‘₯ξ…žξƒ―π‘₯(𝑑)=𝐡1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑0π‘₯𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(3.7)ξ…žξ…ž(𝑑)=𝑓(𝑑,π‘₯(πœ™(𝑑))).(3.8) Let 𝑑=πœπ‘˜ in (2.3), we obtain π‘₯ξ€·πœπ‘˜ξ€Έξƒ―π‘₯=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°+ξ€œβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))π‘‘π‘ πœπ‘˜0ξ€·πœπ‘˜ξ€Έβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(3.9) which proves π‘šξ“π‘˜=1π‘Žπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έ=π‘₯0.(3.10) Also let 𝑑=πœ‚π‘— in (3.7), we obtain π‘₯ξ…žξ€·πœ‚π‘—ξ€Έξƒ―π‘₯=𝐡1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))π‘‘π‘ πœ‚π‘—0𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(3.11) then 𝑛𝑗=1𝑏𝑗π‘₯ξ…žξ€·πœ‚π‘—ξ€Έ=𝐡𝑛𝑗=1𝑏𝑗π‘₯1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑛𝑗=1π‘π‘—ξ€œπœ‚π‘—0𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(3.12) Let 𝑑=0 in (3.7), we obtain π‘₯ξ…žξƒ―π‘₯(0)=𝐡1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°.𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠(3.13) Adding (3.12) and (3.13), we obtain π‘₯ξ…ž(0)+𝑛𝑗=1𝑏𝑗π‘₯ξ…žξ€·πœ‚π‘—ξ€Έ=π‘₯1.(3.14) This implies that there exists at least one solution π‘₯∈𝐢[0,1] of the nonlocal problem (1.3) and (1.4). This completes the proof.

Corollary 3.2. The solution of the problem (1.3)-(1.4) is monotonic nondecreasing.

Proof. Let𝑑1<𝑑2, we deduce from (2.3) that π‘₯𝑑1ξ€Έξƒ―π‘₯=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°ξƒ©π‘‘βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+𝐡1βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑10𝑑1ξ€Έξƒ―π‘₯βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠<𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°ξƒ©π‘‘βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+𝐡2βˆ’π΄π‘šξ“π‘˜=1π‘Žπ‘˜πœπ‘˜π‘₯ξƒͺξƒ―1βˆ’π‘›ξ“π‘—=1π‘π‘—ξ€œπœ‚π‘—0ξƒ°+ξ€œπ‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑20𝑑2ξ€Έξ€·π‘‘βˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠=π‘₯2ξ€Έ,(3.15) which proves that the solution π‘₯ of the problem (1.3)-(1.4) is monotonic nondecreasing.

3.1. Positive Solution

Let 𝑏𝑗=0,𝑗=1,2,…𝑛 and π‘₯1=0, then the nonlocal problem condition (1.4) will beπ‘šξ“π‘˜=1π‘Žπ‘˜π‘₯ξ€·πœπ‘˜ξ€Έ=π‘₯0,π‘₯ξ…ž(0)=0.(3.16)

Theorem 3.3. Let the assumptions (i)–(iv) of Theorem 3.1 be satisfied. Then the solution of the nonlocal problem (1.3)–(3.16) is positive π‘‘βˆˆ[𝑑,1].

Proof. Let 𝑏𝑗=0,𝑗=1,2,…𝑛 and π‘₯1=0 in the integral equation (2.3) and the nonlocal condition (1.4), then the solution of the nonlocal problem (1.3)–(3.16) will be given by the integral equation ξƒ―π‘₯π‘₯(𝑑)=𝐴0βˆ’π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξƒ°+ξ€œβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(3.17) where βˆ‘π΄=(π‘šπ‘˜=1π‘Žπ‘˜)βˆ’1.
Let π‘‘βˆˆ[𝑑,1], thenξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έξ€œβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠≀𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,πœπ‘˜β‰€π‘‘,π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))π‘‘π‘ β‰€π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπ‘‘0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(3.18) Multiplying by βˆ‘π΄=(π‘šπ‘˜=1π‘Žπ‘˜)βˆ’1, we obtain π΄π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπœπ‘˜0ξ€·πœπ‘˜ξ€Έβˆ’π‘ π‘“(𝑠,π‘₯(πœ™(𝑠)))π‘‘π‘ β‰€π΄π‘šξ“π‘˜=1π‘Žπ‘˜ξ€œπ‘‘0=ξ€œ(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠,(3.19) and the solution π‘₯ of the nonlocal problem (1.3) and (3.16), given by the integral equation (3.17), is positive for π‘‘βˆˆ[𝑑,1]. This complete the proof.

Example 3.4. Consider the nonlocal problem of the second-order functional differential equation (1.3) with two-point boundary condition π‘₯ξ…ž(0)=0,π‘₯(πœ‚)=π‘₯0,πœ‚βˆˆ(π‘Ž,𝑑)βŠ‚(0,1).(3.20) Applying our results here, we deduce that the two-point boundary value problem (1.3)–(3.20) has at least one monotonic nondecreasing solution π‘₯∈𝐢[0,1] represented by the integral equation π‘₯(𝑑)=π‘₯0βˆ’ξ€œπœ‚0(ξ€œπœ‚βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠+𝑑0(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(3.21) This the solution is positive with 𝑑>πœ‚.

4. Nonlocal Integral Condition

Let π‘₯∈𝐢[0,1] be the solution of the nonlocal problem (1.3) and (1.4).

Let π‘Žπ‘˜=π‘‘π‘˜βˆ’π‘‘π‘˜βˆ’1,πœπ‘˜βˆˆ(π‘‘π‘˜βˆ’1,π‘‘π‘˜)βŠ‚(π‘Ž,𝑑)βŠ‚(0,1) and let 𝑏𝑗=πœ‰π‘—βˆ’πœ‰π‘—βˆ’1,πœ‚π‘—βˆˆ(πœ‰π‘—βˆ’1,πœ‰π‘—)βŠ‚(𝑐,𝑒)βŠ‚(0,1), thenπ‘šξ“π‘˜=1ξ€·π‘‘π‘˜βˆ’π‘‘π‘˜βˆ’1ξ€Έπ‘₯ξ€·πœπ‘˜ξ€Έ=π‘₯0,π‘₯ξ…ž(0)+𝑛𝑗=1ξ€·πœ‰π‘—βˆ’πœ‰π‘—βˆ’1ξ€Έπ‘₯ξ…žξ€·πœ‚π‘—ξ€Έ=π‘₯1.(4.1) From the continuity of the solution π‘₯ of the nonlocal problem (1.3) and (1.4), we obtainlimπ‘šπ‘šβ†’βˆžξ“π‘˜=1ξ€·π‘‘π‘˜βˆ’π‘‘π‘˜βˆ’1ξ€Έπ‘₯ξ€·πœπ‘˜ξ€Έ=ξ€œπ‘‘π‘Žπ‘₯π‘₯(𝑠)𝑑𝑠,ξ…ž(0)+limπ‘›π‘›β†’βˆžξ“π‘—=1ξ€·πœ‰π‘—βˆ’πœ‰π‘—βˆ’1ξ€Έπ‘₯ξ…žξ€·πœ‚π‘—ξ€Έ=π‘₯ξ…žξ€œ(0)+𝑒𝑐π‘₯ξ…ž(𝑠)𝑑𝑠,(4.2) and the nonlocal condition (1.4) transformed to the integral conditionξ€œπ‘‘π‘Žπ‘₯(𝑠)𝑑𝑠=π‘₯0,π‘₯ξ…ž(0)+π‘₯(𝑒)βˆ’π‘₯(𝑐)=π‘₯1,(4.3) and the solution of the integral equation (2.3) will beπ‘₯(𝑑)=(π‘‘βˆ’π‘Ž)βˆ’1ξ‚»π‘₯0βˆ’ξ€œπ‘‘π‘Žξ€œπ‘‘0ξ‚Ό+(π‘‘βˆ’π‘ )𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑𝑑((π‘βˆ’π‘)+1)βˆ’1ξ‚»π‘₯(π‘‘βˆ’1)1βˆ’ξ€œπ‘’π‘ξ€œπ‘‘0𝑓+ξ€œ(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠𝑑𝑑𝑑0𝑓(𝑠,π‘₯(πœ™(𝑠)))𝑑𝑠.(4.4) Now, we have the following theorem.

Theorem 4.1. Let the assumptions (i)–(iv) of Theorem 3.1 be satisfied. Then the nonlocal problem π‘₯ξ…žξ…ž,ξ€œ(𝑑)=𝑓(𝑑,π‘₯(πœ™(𝑑))),π‘‘βˆˆ(0,1)π‘‘π‘Žπ‘₯(𝑠)𝑑𝑠=π‘₯0,π‘₯ξ…ž(0)+π‘₯(𝑒)βˆ’π‘₯(𝑐)=π‘₯1(4.5) has at least one monotonic nondecreasing solution π‘₯∈𝐢[0,1] represented by (4.4).

References

  1. V. A. Il'in and E. I. Moiseev, β€œA nonlocal boundary value problem of the first kind for the Sturm-Liouville operator in differential and difference interpretations,” Differentsial'nye Uravneniya, vol. 23, no. 7, pp. 1198–1207, 1987.
  2. V. A. Il'in and E. I. Moiseev, β€œA nonlocal boundary value problem of the second kind for the Sturm-Liouville operator,” Differentsial'nye Uravneniya, vol. 23, no. 8, pp. 1422–1431, 1987.
  3. Y. An, β€œExistence of solutions for a three-point boundary value problem at resonance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 8, pp. 1633–1643, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. R. F. Curtain and A. J. Pritchand, Functional Analysis in Modern Applied Mathematics, Academic Press, 1977.
  5. P. W. Eloe and Y. Gao, β€œThe method of quasilinearization and a three-point boundary value problem,” Journal of the Korean Mathematical Society, vol. 39, no. 2, pp. 319–330, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. A. M. A. El-Sayed and Kh. W. Elkadeky, β€œCaratheodory theorem for a nonlocal problem of the differential equation x=f(t,x),” Alexandria Journal of Mathematics, vol. 1, no. 2, pp. 8–14, 2010.
  7. Y. Feng and S. Liu, β€œExistence, multiplicity and uniqueness results for a second order m-point boundary value problem,” Bulletin of the Korean Mathematical Society, vol. 41, no. 3, pp. 483–492, 2004. View at Publisher Β· View at Google Scholar
  8. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
  9. C. P. Gupta, β€œSolvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation,” Journal of Mathematical Analysis and Applications, vol. 168, no. 2, pp. 540–551, 1992. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. Y. Guo, Y. Ji, and J. Zhang, β€œThree positive solutions for a nonlinear nth-order m-point boundary value problem,” Nonlinear Analysis: Theory, Methods and Applications, vol. 68, no. 11, pp. 3485–3492, 2008. View at Publisher Β· View at Google Scholar
  11. G. Infante and J. R. L. Webb, β€œPositive solutions of some nonlocal boundary value problems,” Abstract and Applied Analysis, vol. 2003, no. 18, pp. 1047–1060, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1970.
  13. F. Li, M. Jia, X. Liu, C. Li, and G. Li, β€œExistence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 8, pp. 2381–2388, 2008. View at Publisher Β· View at Google Scholar
  14. R. Liang, J. Peng, and J. Shen, β€œPositive solutions to a generalized second order three-point boundary value problem,” Applied Mathematics and Computation, vol. 196, no. 2, pp. 931–940, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. B. Liu, β€œPositive solutions of a nonlinear three-point boundary value problem,” Computers & Mathematics with Applications. An International Journal, vol. 44, no. 1-2, pp. 201–211, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. X. Liu, J. Qiu, and Y. Guo, β€œThree positive solutions for second-order m-point boundary value problems,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 733–742, 2004. View at Publisher Β· View at Google Scholar
  17. R. Ma, β€œPositive solutions of a nonlinear three-point boundary-value problem,” Electronic Journal of Differential Equations, vol. 34, pp. 1–8, 1999. View at Zentralblatt MATH
  18. R. Ma, β€œMultiplicity of positive solutions for second-order three-point boundary value problems,” Computers & Mathematics with Applications, vol. 40, no. 2-3, pp. 193–204, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  19. R. Ma, β€œPositive solutions for second-order three-point boundary value problems,” Applied Mathematics Letters, vol. 14, no. 1, pp. 1–5, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. R. Ma and N. Castaneda, β€œExistence of solutions of nonlinear m-point boundary-value problems,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 556–567, 2001. View at Publisher Β· View at Google Scholar
  21. S. K. Ntouyas, β€œNonlocal initial and boundary value problems: a survey,” in Handbook of Differential Equations: Ordinary Differential Equations. Vol. II, A. Canada, P. Drabek, and A. Fonda, Eds., pp. 461–557, Elsevier, Amsterdam, The Netherlands, 2005.
  22. Y. Sun and X. Zhang, β€œExistence of symmetric positive solutions for an m-point boundary value problem,” Boundary Value Problems, vol. 2007, Article ID 79090, 14 pages, 2007. View at Publisher Β· View at Google Scholar