Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 513189, 7 pages
doi:10.1155/2012/513189
Research Article
On a Third-Order Three-Point Boundary
Value Problem
A. Guezane-Lakoud,1 N. Hamidane,1 and R. Khaldi2
1
Laboratory of Advanced Materials, Faculty of Sciences, Badji Mokhtar-Annaba University,
P.O. Box 12, 23000 Annaba, Algeria
2
Laboratory LASEA, Faculty of Sciences, Badji Mokhtar-Annaba University, P.O. Box 12,
23000 Annaba, Algeria
Correspondence should be addressed to N. Hamidane, nhamidane@yahoo.com
Received 28 March 2012; Accepted 3 June 2012
Academic Editor: Rodica Costin
Copyright q 2012 A. Guezane-Lakoud 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.
We consider a third-order three-point boundary value problem. We introduce a generalized polynomial growth condition to obtain the existence of a nontrivial solution by using Leray-Schauder
nonlinear alternative, then we give an example to illustrate our results.
1. Introduction
In this work, we study the existence of nontrivial solution for the following third-order three
point boundary value problem BVP:
u t ft, ut 0,
u0 αu 0,
0 < t < 1,
u1 βu η ,
u 1 0,
1.1
1.2
where η ∈ 0, 1, α, β ∈ R, f ∈ C0, 1 × R, R.
The parameters α and β are arbitrary in R such that 1 2α 2βη − 2β / 0. Our aim
is to give new conditions on the nonlinearity of f, then using Leray-Schauder nonlinear
alternative, we establish the existence of nontrivial solution. We only assume that ft, 0 /
0
and a generalized polynomial growth condition, that is, there exist two nonnegative functions
k, h ∈ L1 0, 1 such that
ft, x ≤ kt|x|p ht, ∀t, x ∈ 0, 1 × R,
1.3
where p ∈ R , so our conditions are new and more general than 1.
2
International Journal of Mathematics and Mathematical Sciences
Such problems arise in the study of the equilibrium states of a heated bar. Very recently,
there have been several papers on third-order boundary value problems. Graef and Yang
2, 3, Guo et al. 4, Hopkins and Kosmatov 5, and Sun 6 have all considered third-order
problems. Anderson 7 considered the three-point boundary value problem for 1.1 in the
case t1 < t < t2 and the three-point conditions ut1 u t2 0, γut3 δu t3 0; using
the Krasnoselskii and Leggett-Williams fixed-point theorems, the existence of solutions to the
nonlinear problem is proved. Excellent surveys of theoretical results can be found in Agarwal
8, Agarwal et al. 9, and Ma 10. More results can be found in 11–13.
This paper is organized as follows. First, we list some preliminary materials to be used
later. Then in Section 3, we present and prove our main results which consist in existence
theorems. We end our work with some illustrating examples.
2. Preliminary Lemmas
Let E C0, 1, R, with supremum norm ||y|| supt∈0,1 |yt|, for all y ∈ E. Now, we state
two preliminary results.
Lemma 2.1. Let y ∈ E. If ζ 1 2α 2βη − 2β /
0, then the three-point BVP
u t yt 0,
u0 αu 0,
0 < t < 1,
u1 βu η ,
u 1 0
2.1
has a unique solution
t
η
β 2
t − 2t − 2α
η − s ysds
ζ
0
0
1
t2 1
1 s 2α − 2β t α 2βη − s ysds.
1 − s
ζ 0
2
ut −
1
2
t − s2 ysds 2.2
Proof. Integrating u t −yt over the interval 0, t, we see that
ut −
1
2
t
1
t − s2 ysds At2 Bt C.
2
0
2.3
The constants A, B, and C are given by the three-point boundary conditions 1.2.
We define the integral operator T : E → E by
t
η
β 2
t − 2t − 2α
η − s fs, usds
t − s fs, usds ζ
0
0
t2 1 1
·
1 s 2α − 2β t α 2βη − s fs, usds.
1 − s
ζ 0
2
1
T ut −
2
2
2.4
International Journal of Mathematics and Mathematical Sciences
3
By Lemma 2.1, the BVP 1.1-1.2 has a solution if and only if the operator T has
a fixed point in E. By Ascoli-Arzela theorem, we prove that T is a completely continuous
operator. Now we cite the Leray-Schauder as nonlinear alternative.
Lemma 2.2 see 14. Let F be a Banach space and Ω a bounded open subset of F, 0 ∈ Ω. Let
T : Ω → F be a completely continuous operator. Then, either there exists x ∈ ∂Ω, λ > 1 such that
T x λx, or there exists a fixed point x∗ ∈ Ω.
3. Main Results
In this section, we present and prove our main results.
Theorem 3.1. It is assumed that ft, 0 / 0, ζ / 0, p ∈ R , and there exist two nonnegative
1
functions k, h ∈ L 0, 1 such that
ft, x ≤ kt|x|p ht,
∀t, x ∈ 0, 1 × R,
1
1 2α 1 1
1 β
1 − sksds < ,
ζ 2|1 α| β η 1 β
ζ
2
0
1
1 2α 1 1
2|1 α| βη 1 β
1 β
1 − shsds < .
ζ
ζ
2
0
3.1
3.2
3.3
Then the BVP 1.1-1.2 has at least one nontrivial solution u∗ ∈ C0, 1, R.
Proof. Setting
1
1 2α 1 M 1 β
1 − sksds,
ζ 2|1 α| β η 1 β
ζ
0
1
1 2α 1 2|1 α| βη 1 β
N 1 β
1 − shsds,
ζ
ζ
0
3.4
by hypothesis 3.2, we have M < 1/2. Since ft, 0 /
0, then there exists an interval σ, τ ⊂
0, 1 such that minσ≤t≤r |ft, 0| > 0, then N > 0. Let m M/N, Ω {u ∈ C0, 1 : ||u|| < 1}.
Assume that u ∈ ∂Ω, λ > 1 such T u λu, then
λu T u max|T ut| ≤ Mup N,
3.5
λ ≤ M N.
3.6
0≤t≤1
as ||u|| 1, then
First, if m ≤ 1, then λ ≤ 2N < 1; hence, λ < 1, and this contradicts the fact that λ > 1.
By Lemma 2.2, we deduce that T has a fixed point u∗ ∈ Ω, and then the BVP 1.1-1.2 has a
nontrivial solution u∗ ∈ C0, 1, R.
Second, if m ≥ 1, then λ ≤ 2M < 1. By arguing as above, we complete the proof.
4
International Journal of Mathematics and Mathematical Sciences
Let us define the following notation:
a
1 2α 1 2|1 α| βη 1 β .
1 β
ζ
ζ
3.7
Theorem 3.2. Under the conditions of Theorem 3.1 p ∈ R and if one of the following conditions
is satisfied
1 there exist n > 1 and r > 1 such that
1
1/n
1/q 1q
1 1
1 ,
<
2a
n q
k sds
n
0
1
1/r
h sds
r
<
0
1 l1/l
2a
1 1
1 ,
r l
3.8
3.9
2 there exist two constants μ, τ > −1 such that
μ1 μ2 μ
s ,
ks <
2a
τ 1τ 2 τ
s ,
2a
μ1 μ2 μ
meas s ∈ 0, 1, ks <
> 0,
s
2a
τ 1τ 2 τ
meas s ∈ 0, 1, hs <
s > 0,
2a
hs <
3.10
3.11
3.12
3 the functions ks and hs satisfy
ks <
1
,
a
3.13
hs <
1
,
a
3.14
1
> 0,
meas s ∈ 0, 1, ks <
a
1
meas s ∈ 0, 1, hs <
> 0,
a
3.15
International Journal of Mathematics and Mathematical Sciences
5
4 the function ft, x satisfies
ft, x
1
,
2a
3.16
1
.
ω2 lim sup max ft, x <
|x| → ∞
t∈0,1
2a
3.17
ω1 lim sup max
|x| → ∞
t∈0,1
|x|
p
<
Then the BVP 1.1-1.2 has at least one nontrivial solution u∗ ∈ C0, 1, R.
Proof. Let M and N be defined as in the proof of Theorem 3.1. To prove Theorem 3.2, we
only need to prove that M < 1/2 and N < 1/2.
1 By using Hölder inequality, we get
1
M≤a
1/n 1
k sds
n
0
1/q
q
1 − s ds
.
3.18
0
Integrating, using 3.8, and remarking that a > 1, we arrive at M < 1/2.
Using Hölder inequality a second time, we get
1
N≤a
1/r 1
h sds
r
0
1/l
l
1 − s ds
.
3.19
0
Integrating and then using 3.9, we arrive at N < 1/2.
2 Taking into account 3.10, it yields
1
μ1 μ2
1
μ
a 1 − ss ds .
M<
2a
2
0
3.20
On the other hand, using 3.11, we obtain
τ 1τ 2
N<
2a
1
1
μ
a 1 − ss ds .
2
0
3.21
3 Using the same reasoning as in the proof of the second statement, we prove the
third statement.
4 From the condition ω1 lim|x| → ∞ sup maxt∈0,1 |ft, x|/|x|p , we deduce that there
exists c1 > 0 such that for |x| > c1 , we get |ft, x| ≤ ω1 ε|x|p , for all ε > 0. Choosing
ε ω1 , then |ft, x| ≤ 2ω1 |x|p . Now from the condition ω2 lim|x| → ∞ sup maxt∈0,1 |ft, x|,
we deduce that there exists c2 > 0 such that for |x| > c2 , we have
ft, x ≤ ω2 ε,
∀x ∈ R \ −c2 , c2 ,
3.22
6
International Journal of Mathematics and Mathematical Sciences
choosing ε ω2 , then |ft, x| ≤ 2ω2 , for all x ∈ R\ − c2 , c2 ; consequently,
ft, x ≤ 2ω1 |x|p 2ω2 ,
∀x ∈ R \ −c, c,
3.23
where c maxc1 , c2 , setting R max{|ft, x| : t, x ∈ 0, 1 × −c, c}, then for all t, x ∈
0, 1 × R, we get |ft, x| ≤ kt|x|p ht, where kt 2ω1 and ht 2ω2 . Using 3.16, we
obtain
kt 2ω1 <
1
,
a
3.24
then from 3.13, we get M < 1/2. Using 3.14, we arrive at
ht 2ω2 <
1
,
a
3.25
then N < 1/2. Now applying the third statement, we achieve the proof of Theorem 3.2.
Example 3.3. Consider the three-point BVP,
1
u u5
7
√
2
t3
cos t
2
arcsin t
0,
5
1
1
,
u1 − u
2
3
1
u0 − u 0,
2
0 < t < 1,
3.26
u 1 0.
We have
x5
ft, x 7
√
2
t3
cos t
2
arcsin t
,
5
1
1
2
αβ− ,η ,ζ .
2
3
3
3.27
So,
ft, x ≤ 1
7
√
2
t3
arcsin t
cos t |x|5 kt|x|5 ht.
2
5
3.28
International Journal of Mathematics and Mathematical Sciences
7
Applying the first statement of Theorem 3.2 for p 5, n r 2, to get
1
1/2
k2 sds
0
⎛
1
⎝
49
1 √
2
0
2
s3
cos s
2
⎞1/2
ds⎠
0.1488
1
0
1/2
h2 sds
1/q
1q
0.16496,
<
2a
1
1/2
1
2
0.13673
arcsin s ds
25 0
<
3.29
1 l1/l
0.16496.
2a
Then the BVP 3.26 has at least one nontrivial solution u∗ in C0, 1.
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