Volume 2011, Article ID 260309,9pages doi:10.1155/2011/260309
Research Article
Second-Order Boundary Value Problem with Integral Boundary Conditions
Mouffak Benchohra,
1Juan J. Nieto,
2and Abdelghani Ouahab
11Department of Mathematics, University of Sidi Bel Abbes, BP 89, 2000 Sidi Bel Abbe, Algeria
2Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Correspondence should be addressed to Mouffak Benchohra,[email protected] Received 28 May 2010; Revised 1 August 2010; Accepted 1 October 2010
Academic Editor: Gennaro Infante
Copyrightq2011 Mouffak Benchohra 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.
The nonlinear alternative of the Leray Schauder type and the Banach contraction principle are used to investigate the existence of solutions for second-order differential equations with integral boundary conditions. The compactness of solutions set is also investigated.
1. Introduction
This paper is concerned with the existence of solutions for the second-order boundary value problem
−yt f t, yt
, a.e. t∈0,1,
y0 0, y1
1
0
gsysds, 1.1
wheref:0,1×R → Ris a given function andg:0,1 → R is an integrable function.
Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers1–9and the references therein. Moreover, boundary value problems with integral boundary conditions have been studied by a number of authors, for example 10–14.
The goal of this paper is to give existence and uniqueness results for the problem 1.1.
Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative15.
2. Preliminaries
In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. LetAC10,1,Rbe the space of differentiable functionsy : 0,1 → R,whose first derivative,y, is absolutely continuous.
We takeC0,1,Rto be the Banach space of all continuous functions from0,1into Rwith the norm
y
∞supyt: 0≤t≤1
, 2.1
and we letL10,1,Rdenote the Banach space of functionsy:0,1 → Rthat are Lebesgue integrable with norm
y
L1 1
0
ytdt. 2.2
Definition 2.1. A mapf :0,1×R → Ris said to beL1-Carath´eodory if it→ft, uis measurable for eachu∈R,
iiu→ft, uis continuous for almost eacht∈0,1, iiifor everyr >0 there existshr ∈L10,1,Rsuch that
ft, u≤hrt for a.e. t∈0,1and all |u| ≤r. 2.3
3. Existence and Uniqueness Results
Definition 3.1. A functiony∈AC10,1,Ris said to be a solution of1.1ifysatisfies1.1.
In what follows one assumes that g∗ 01sgsds /1. One needs the following auxiliary result.
Lemma 3.2. . Letσ:L10,1,R. Then the function defined by
yt 1
0
Ht, sσsds 3.1
is the unique solution of the boundary value problem
−yt σt, a.e. t∈0,1,
y0 0, y1
1
0
gsysds, 3.2
where
Ht, s Gt, s t
1− 01sgsds 1
0
Gr, sgrdr,
Gt, s
⎧⎨
⎩
s1−t if 0≤s≤t≤1, t1−s if 0≤t≤s≤1.
3.3
Proof. Letybe a solution of the problem3.2. Then integratingly, we obtain
yt y0 ty0− t
0
t−sσsds, y1 y0−
1
0
1−sσsds.
3.4
Hence
yt 1
0
tgsysds
1
0
t1−sσsds− t
0
t−sσsds, 3.5
yt 1
0
tgsysds
1
0
Gt, sσsds, 3.6
where
Gt, s
⎧⎨
⎩
s1−t if 0≤s≤t≤1,
t1−s if 0≤t≤s≤1. 3.7
Now, multiply3.6bygand integrate over0,1, to get 1
0
gsysds
1
0
gs
s 1
0
gryrdr
1
0
Gs, rσrdr
ds
1
0
sgs 1
0
gsysds
1
0
gs 1
0
Gs, rσrdr
ds.
3.8
Thus,
1
0
gsysds
1
0gs 01Gs, rσrdr ds
1− 01sgsds . 3.9
Substituting in3.6we have
yt 1
0
Gt, sσsdst 01gs 01Gs, rσrdr ds
1− 01sgsds . 3.10
Therefore
yt 1
0
Ht, sσsds. 3.11
Setg∗|1−g∗|.Note that
|Gt, s| ≤ 1
4 fort, s∈0,1×0,1. 3.12
Our first result reads
Theorem 3.3. Assume thatfis anL1-Carath´eodory function and the following hypothesis A1There existsl∈L10,1,Rsuch that
ft, x−ft, x≤lt|x−x| ∀x, x∈R, t∈0,1 3.13 holds. If
lL1g
L1lL1
g∗ <4, 3.14
then the BVP1.1has a unique solution.
Proof. Transform problem 1.1 into a fixed-point problem. Consider the operator N : C0,1,R → C0,1,Rdefined by
N y
t 1
0
Ht, sf s, ys
ds, t∈0,1. 3.15
We will show thatNis a contraction. Indeed, considery, y ∈C0,1,R.Then we have for eacht∈0,1
N y
t−N y
t≤ 1
0
|Ht, s|f s, ys
−f
s, ysds
≤ 1
0
|Gt, s|lsys−ysds 1
g∗ 1
0
lsys−ysgr1
0
|Gr, s|ds dr.
3.16
Therefore
N y
−N y
∞≤ 1 4
lL1g
L1lL1
g∗
y−y
∞, 3.17
showing that,Nis a contraction and hence it has a unique fixed point which is a solution to 1.1. The proof is completed.
We now present an existence result for problem1.1.
Theorem 3.4. Suppose that hypotheses
H1The functionf:0,1×R → Ris anL1-Carath´eodory,
H2There exist functionsp, q∈L10,1,Randα∈0,1such that
ft, u≤pt|u|αqt for eacht, u∈0,1×R, 3.18
are satisfied. Then the BVP1.1has at least one solution. Moreover the solution set
S
y∈C0,1,R:y solution of the problem1.1
3.19
is compact.
Proof. Transform the BVP 1.1 into a fixed-point problem. Consider the operator N as defined in Theorem 3.3. We will show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof will be given in several steps.
Step 1N is continuous. Let{ym}be a sequence such thatym → yinC0,1,R.Then N
yn
t−N y
t≤ 1
0
|Ht, s|f
s, yms
−f
s, ysds. 3.20
SincefisL1-Carath´eodory andg∈L10,1,R,then N
ym
−N y
∞≤ 1 4f
·, ym·
−f
·, y·
L1
g
L1
4g∗ f
·, ym·
−f
·, y·
L1.
3.21
Hence
N ym
−N y
∞−→0 asm−→ ∞. 3.22
Step 2Nmaps bounded sets into bounded sets inC0,1,R. Indeed, it is enough to show that there exists a positive constantsuch that for eachy∈Bq{y∈C0,1,R:y∞≤q}
one hasNy∞≤.
Lety∈Bq. Then for eacht∈0,1, we have
N y
t 1
0
Ht, sf s, ys
ds. 3.23
ByH2we have for eacht∈0,1
N y
t≤ 1
0
|Ht, s|f
s, ysds
≤ 1 4q
L1qαp
L1
g
L1
4g∗ q
L1qαp
L1
.
3.24
Then for eachy∈Bqwe have
Ny
∞≤ 1 4q
L1qαp
L1
g
L1
4g∗ q
L1qαp
L1
:. 3.25
Step 3Nmaps bounded set into equicontinuous sets ofC0,1,R. Letτ1, τ2 ∈0,1,τ1 <
τ2andBqbe a bounded set ofC0,1,Ras inStep 2. Lety∈Bqandt∈0,1we have
N y
τ2−N y
τ1≤ 1
0
|Hτ2, s−Hτ1, s|qsdsqα 1
0
|Hτ2, s−Hτ1, s|psds.
3.26
Asτ2 → τ1the right-hand side of the above inequality tends to zero. ThenNBqis equicontinuous. As a consequence of Steps1to3together with the Arzela-Ascoli theorem we can conclude thatN:C0,1,R → C0,1,Ris completely continuous.
Step 4A priori bounds on solutions. Lety γNyfor some 0 < γ < 1. This implies by H2that for eacht∈0,1we have
yt≤ 1 4
1
0
psysαds1 4q
L1g
L1
4g∗ q
L1g
L1
4g∗ 1
0
psysαds. 3.27
Then
y∞≤ 1 4p
L1yα∞ 1 4q
L1g
L1
4g∗ q
L1g
L1
4g∗ p
L1yα∞. 3.28
Ify∞>1,we have y1−α
∞ ≤ 1
4p 1 4q
L1g
L1
4g∗ q
L1g
L1
4g∗ p
L1. 3.29
Thus
y
∞≤ 1
4p1 4q
L1g
L1
4g∗ q
L1g
L1
4g∗ p
L1
1/1−α
:ψ∗. 3.30
Hence
y
∞≤max 1, ψ∗
:M. 3.31
Set
U:
y∈C0,1,R:y
∞< M1
, 3.32
and consider the operatorN : U → C0,1,R.From the choice ofU, there is noy ∈ ∂U such thaty γNyfor someγ ∈ 0,1.As a consequence of the nonlinear alternative of Leray-Schauder type15, we deduce thatNhas a fixed pointyinUwhich is a solution of the problem1.1.
Now, prove thatSis compact. Let{ym}m≥1be a sequence inS, then
ymt 1
0
Ht, sf
s, yms
ds, m≥1, t∈0,1. 3.33
As in Steps3and4we can easily prove that there existsM >0 such that ym
∞< M, ∀m≥1, 3.34
and the set{ym:m≥1}is equicontinuous inC0,1,R,hence by Arzela-Ascoli theorem we can conclude that there exists a subsequence of{ym:m≥1}converging toyinC0,1,R.
Using that fast thatfis anL1-Carath´edory we can prove that
yt 1
0
Ht, sf s, ys
ds, t∈0,1. 3.35
ThusSis compact.
4. Examples
We present some examples to illustrate the applicability of our results.
Example 4.1. Consider the following BVP
−yt 1 5et1
1
1yt, a.e. t∈0,1,
y0 0, y1
1
0
s1
2 ysds.
4.1
Set
f t, y
1 5et1
1 1y,
t, y
∈0,1×R. 4.2
We can easily show that conditionsA1,3.14are satisfied with
lt 1
5et1, gt s1
2 , lL1 1−e−1
5e , g
L1 3
4, g∗ 5 12.
4.3
Hence, byTheorem 3.3, the BVP4.1has a unique solution on0,1.
Example 4.2. Consider the following BVP
−yt 5et12yt1/3
1yt , a.e. t∈0,1,
y0 0, y1
1
0
s2ysds.
4.4
Set
f t, y
5et12y1/3 1y ,
t, y
∈0,1×R. 4.5
We can easily show that conditionsH1,H2are satisfied with
α 1
3, pt 10et, qt 5et, t∈0,1. 4.6
Hence, by Theorem 3.4, the BVP 4.4 has at least one solution on 0,1. Moreover, its solutions set is compact.
Acknowledgment
The authors are grateful to the referees for their remarks.
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