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 ft, 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.
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
yL1 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
where
Ht, s Gt, s t
1− 01sgsds
1 0 Gr, sgrdr, Gt, s ⎧ ⎨ ⎩
s1−t if0≤s≤t≤1, t1−s if0≤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 1
0Gs, rσrdr
ds
Substituting in3.6we have
yt 1
0
Gt, sσsdst
1 0gs
1
0Gs, 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,R
such that
ft, x−ft, x≤lt|x−x| ∀x, x∈R, t∈0,1 3.13
holds. If
lL1
gL1lL1
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
Nyt 1
0
Ht, sfs, ysds, 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
yt−Nyt≤ 1
0
|Ht, s|fs, ys−fs, ysds
≤
1
0
|Gt, s|lsys−ysds
1 g∗
1
0
lsys−ysgr 1
0
|Gr, s|ds dr.
Therefore
Ny−Ny∞≤ 1
4
lL1
gL1lL1 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,R
andα∈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
Sy∈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−Nyt≤ 1
0
|Ht, s|fs, yms
−fs, ysds. 3.20
SincefisL1-Carath´eodory andg∈L10,1,R,then
Nym
−Ny∞≤ 1
4f
·, ym·
−f·, y·L1
gL1
4g∗ f
·, ym·
−f·, y·L1.
3.21
Hence
Nym
−Ny∞−→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}
Lety∈Bq. Then for eacht∈0,1, we have
Nyt 1
0
Ht, sfs, ysds. 3.23
ByH2we have for eacht∈0,1
Nyt≤ 1
0
|Ht, s|fs, ysds
≤ 1
4qL1qαpL1
gL1
4g∗ qL1qαpL1
.
3.24
Then for eachy∈Bqwe have
Ny ∞≤
1
4qL1qαpL1
gL1
4g∗ qL1qαpL1
:. 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
Nyτ2−N
yτ1≤
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
psysα
ds1
4qL1 g
L1
4g∗ qL1 g
L1
4g∗ 1
0
psysα
ds. 3.27
Then
y∞≤ 1
4pL1y α
∞
1
4qL1 gL1
4g∗ qL1 gL1
4g∗ pL1y α
Ify∞>1,we have
y1∞−α≤ 1
4p
1
4qL1 gL1
4g∗ qL1 gL1
4g∗ pL1. 3.29
Thus
y ∞≤
1
4p
1
4qL1 g
L1
4g∗ qL1 g
L1
4g∗ pL1
1/1−α
:ψ∗. 3.30
Hence
y∞≤max1, 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, sfs, 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, sfs, ysds, t∈0,1. 3.35
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
ft, 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 , gL1
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 5et12yt
1/3
1yt , a.e. t∈0,1,
y0 0, y1
1
0
s2ysds.
4.4
Set
ft, y5et12y
1/3
1y ,
t, y∈0,1×R. 4.5
We can easily show that conditionsH1,H2are satisfied with
α 1
3, pt 10e
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.
References
1 B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,”Boundary Value Problems, vol. 2009, Article ID 708576, 11 pages, 2009.
2 A. Belarbi and M. Benchohra, “Existence results for nonlinear boundary-value problems with integral boundary conditions,”Electronic Journal of Differential Equations, vol. 2005, no. 06, p. 10, 2005.
3 A. Belarbi, M. Benchohra, and A. Ouahab, “Multiple positive solutions for nonlinear boundary value problems with integral boundary conditions,”Archivum Mathematicum, vol. 44, no. 1, pp. 1–7, 2008.
4 M. Benchohra, S. Hamani, and J. J. Nieto, “The method of upper and lower solutions for second order differential inclusions with integral boundary conditions,”The Rocky Mountain Journal of Mathematics, vol. 40, no. 1, pp. 13–26, 2010.
5 G. Infante, “Nonlocal boundary value problems with two nonlinear boundary conditions,” Communications in Applied Analysis, vol. 12, no. 3, pp. 279–288, 2008.
6 A. Lomtatidze and L. Malaguti, “On a nonlocal boundary value problem for second order nonlinear singular differential equations,”Georgian Mathematical Journal, vol. 7, no. 1, pp. 133–154, 2000.
7 J. R. L. Webb, “Positive solutions of some higher order nonlocal boundary value problems,”Electronic Journal of Qualitative Theory of Differential Equations, no. 29, 15 pages, 2009.
8 J. R. L. Webb, “A unified approach to nonlocal boundary value problems,” inDynamic Systems and Applications. Vol. 5, pp. 510–515, Dynamic, Atlanta, Ga, USA, 2008.
9 J. R. L. Webb and G. Infante, “Positive solutions of nonlocal boundary value problems: a unified approach,”Journal of the London Mathematical Society, vol. 74, no. 3, pp. 673–693, 2006.
10 S. A. Brykalov, “A second order nonlinear problem with two-point and integral boundary conditions,”Georgian Mathematical Journal, vol. 1, pp. 243–249, 1994.
11 M. Denche and A. L. Marhoune, “High order mixed-type differential equations with weighted integral boundary conditions,”Electronic Journal of Differential Equations, vol. 2000, no. 60, 10 pages, 2000.
12 I. Kiguradze, “Boundary value problems for systems of ordinary differential equations,”Journal of Soviet Mathematics, vol. 43, no. 2, pp. 2259–2339, 1988.
13 A. M. Krall, “The adjoint of a differential operator with integral boundary conditions,”Proceedings of the American Mathematical Society, vol. 16, pp. 738–742, 1965.
14 R. Ma, “A survey on nonlocal boundary value problems,”Applied Mathematics E-Notes, vol. 7, pp. 257–279, 2007.
