On a Third Order Three Point Boundary Value Problem

(1)

Volume 2012, Article ID 513189,7pages 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. Khaldi}

2 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,[email protected]

Received 28 March 2012; Accepted 3 June 2012

Academic Editor: Rodica Costin

Copyrightq2012 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 poly-nomial 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 problemBVP:

ut ft, ut 0, 0< t <1, 1.1

u0 αu0, u1 βuη, u1 0, 1.2

whereη∈0,1, α, β∈R, f∈C0,1×R,R.

The parametersαand βare arbitrary in Rsuch that 12α2βη−2_{β /}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 thatft,0/0 and a generalized polynomial growth condition, that is, there exist two nonnegative functions

k, h∈L1₀_,₁_{such that}

_{ft, x}_≤_kt_|_x_|p_ht, _∀_{t, x}_∈₀_,₁_×_R_, _1.3

(2)

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 Kosmatov5, and Sun6have all considered third-order problems. Anderson7considered the three-point boundary value problem for1.1in the caset1 < t < t2 and the three-point conditionsut1 ut2 0, γut3 δut3 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 Ma10. More results can be found in11–13.

This paper is organized as follows. First, we list some preliminary materials to be used later. Then inSection 3, we present and prove our main results which consist in existence theorems. We end our work with some illustrating examples.

2. Preliminary Lemmas

LetE C0,1,R, with supremum norm||y||sup_t_∈_0,1|yt|, for ally∈E. Now, we state two preliminary results.

Lemma 2.1. Lety∈E. Ifζ12α2βη−2_{β /}0, then the three-point BVP

ut yt 0, 0< t <1,

u0 αu0, u1 βuη, u1 0 2.1

has a unique solution

ut −1

2

_t

0

t−s2_ysdsβ

ζ

t2₋₂_t₋₂_α η 0

η−sysds

1 ζ

1

0

1−s

t2

2

1s2α−2β tα2βη−s ysds.

2.2

Proof. Integratingut −ytover the interval0, t, we see that

ut −1

2

t

0

t−s2_ysds1

2At

2_Bt_C. _2.3

The constantsA,B, andCare given by the three-point boundary conditions1.2.

We define the integral operatorT :E → Eby

Tut − 1

2

_t

0

t−s2_f_{s, usds}β

ζ

t2₋₂_t₋₂_α η 0

η−sfs, usds

·1

ζ

1

0

1−s

t2

2

1s2α−2β tα2βη−s fs, usds.

(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 inE. By Ascoli-Arzela theorem, we prove thatT is a completely continuous operator. Now we cite the Leray-Schauder as nonlinear alternative.

Lemma 2.2see14. LetF be a Banach space andΩa bounded open subset ofF, 0 ∈ Ω. Let

T :Ω → F be a completely continuous operator. Then, either there existsx∈∂Ω,λ > 1 such that

Tx λx, or there exists a fixed pointx∗∈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

functionsk, h∈L1₀_,₁_{such that}

_{ft, x}_≤_kt_|_x_|p_ht, _∀_{t, x}_∈₀_,₁_×_R_, _3.1

1β12α ζ

1

ζ

2|1α|βη1β

1

0

1−sksds < 1

2, 3.2

1β12α ζ

1

ζ

2|1α|βη1β

1

0

1−shsds < 1

2. 3.3

Then the BVP1.1-1.2has at least one nontrivial solutionu∗∈C0,1,R.

Proof. Setting

M

1β12α ζ

1

ζ

2|1α|βη1β

1

0

1−sksds,

N

1β12α ζ

1

ζ

2|1α|βη1β

1

0

1−shsds,

3.4

by hypothesis3.2, we haveM < 1/2. Sinceft,0_/0, then there exists an intervalσ, τ⊂

0,1such that minσ≤t≤r|ft,0|>0, thenN >0. LetmM/N,Ω {u∈C0,1:||u||<1}.

Assume thatu∈∂Ω, λ >1 suchTuλu, then

λ u Tu max

0≤t≤1|Tut| ≤M u

p_N,

3.5

as||u||1, then

λ≤MN. 3.6

First, ifm≤1, thenλ≤2N <1; hence,λ <1, and this contradicts the fact thatλ >1.

ByLemma 2.2, we deduce thatThas a fixed pointu∗∈Ω, and then the BVP1.1-1.2has a

nontrivial solutionu∗∈C0,1,R.

(4)

Let us define the following notation:

a

1β12α ζ

1

ζ

2|1α|βη1β

. 3.7

Theorem 3.2. Under the conditions ofTheorem 3.1p ∈Rand if one of the following conditions

is satisfied

1there existn >1 andr >1 such that

1

0

knsds

1/n

<

1q1/q

2a

1

n

1

q1

, 3.8

1

0

hrsds

1/r

< 1l

1/l

2a

1

r

1

l 1

, 3.9

2there exist two constantsμ, τ >−1 such that

ks<

μ1μ2 2a s

μ_, _3.10

hs< τ1τ2

2a s

τ_, _3.11

meas

s∈0,1, ks<

μ1μ2 2a s

μ

>0,

meas

s∈0,1, hs< τ1τ2

2a s

τ

>0,

3.12

3the functionsksandhssatisfy

ks< 1

a, 3.13

hs< 1

a, 3.14

meas

s∈0,1, ks< 1 a

>0,

meas

s∈0,1, hs< 1 a

>0,

(5)

4the functionft, xsatisfies

ω1 lim

|x| → ∞sup maxt∈0,1

_{ft, x}

|x|p <

1

2a, 3.16

ω2 lim

|x| → ∞sup maxt∈0,1ft, x<

1

2a. 3.17

Then the BVP1.1-1.2has at least one nontrivial solutionu∗∈C0,1,R.

Proof. LetMand N be defined as in the proof ofTheorem 3.1. To proveTheorem 3.2, we

only need to prove thatM <1/2 andN <1/2.

1By using H ¨older inequality, we get

M≤a

1

0

knsds

1/n₁

0

1−sqds

1/q

. 3.18

Integrating, using3.8, and remarking thata >1, we arrive atM <1/2. Using H ¨older inequality a second time, we get

N≤a

1

0

hrsds

1/r₁

0

1−slds

1/l

. 3.19

Integrating and then using3.9, we arrive atN <1/2.

2Taking into account3.10, it yields

M <

μ1μ2 2a

a

1

0

1−ssμds 1

2. 3.20

On the other hand, using3.11, we obtain

N < τ1τ2

2a

a

1

0

1−ssμds 1

2. 3.21

3Using the same reasoning as in the proof of the second statement, we prove the third statement.

4From the conditionω1lim|x| → ∞sup maxt∈0,1|ft, x|/|x|p, we deduce that there

existsc1 > 0 such that for |x| > c1, we get|ft, x| ≤ ω1ε|x|p, for allε > 0. Choosing

εω1, then|ft, x| ≤2ω1|x|p. Now from the conditionω2 lim|x| → ∞sup maxt∈0,1|ft, x|,

we deduce that there existsc2>0 such that for|x|> c2, we have

(6)

choosingεω2, then|ft, x| ≤2ω2, for allx∈R\−c2, c2; consequently,

_{ft, x}_≤2ω1|x|p2ω2, ∀x∈R\−c, c, 3.23

wherec maxc1, c2, settingR max{|ft, x|:t, x∈0,1×−c, c}, then for allt, x∈

0,1×R, we get|ft, x| ≤kt|x|pht, wherekt 2ω1andht 2ω2. Using3.16, we

obtain

kt 2ω1 < 1

a, 3.24

then from3.13, we getM <1/2. Using3.14, we arrive at

ht 2ω2 <

1

a, 3.25

thenN <1/2. Now applying the third statement, we achieve the proof ofTheorem 3.2.

Example 3.3. Consider the three-point BVP,

u1

7u

5

√₂

t3

2 cost

arcsint

5 0, 0< t <1,

u0 −1 2u

₀_, _u₁ ₋1

2u

1

3

, u1 0.

3.26

We have

ft, x x

5

7

√₂

t3

2 cost

arcsint

5 , αβ− 1 2, η

1 3, ζ

2

3. 3.27

So,

_{ft, x}_≤ 1 7

√₂

t3

2 cost |x|

5arcsint

5 kt|x|

(7)

Applying the first statement ofTheorem 3.2forp5, nr2, to get

1

0

k2sds

1/2

⎛

⎝ 1

49

₁

0

√₂

s3

2 coss

2

ds

⎞ ⎠

1/2

0.1488

<

1q1/q

2a 0.16496,

1

0

h2sds

1/2

1 25

₁

0

arcsins2ds

1/2

0.13673

< 1l

1/l

2a 0.16496.

3.29

Then the BVP3.26has at least one nontrivial solutionu∗inC0,1.

References

1 A. Guezane-Lakoud and S. Kelaiaia, “Solvability of a nonlinear boundary value problem,” The Journal

of Nonlinear Sciences and Applications, vol. 4, no. 4, pp. 247–261, 2011.

2 J. R. Graef and B. Yang, “Positive solutions of a nonlinear third order eigenvalue problem,” Dynamic

Systems and Applications, vol. 15, no. 1, pp. 97–110, 2006.

3 J. R. Graef and B. Yang, “Existence and nonexistence of positive solutions of a nonlinear third order boundary value problem,” in The 8th Colloquium on the Qualitative Theory of Diﬀerential Equations, vol.

8 of Proceedings of The 8th Colloquium on the Qualitative Theory of Diﬀerential Equations, no. 9, pp. 1–13,

Electronic Journal of Qualitative Theory of Diﬀerential Equations, Szeged, Hungary, 2008.

4 L.-J. Guo, J.-P. Sun, and Y.-H. Zhao, “Existence of positive solutions for nonlinear third-order three-point boundary value problems,” Nonlinear Analysis, vol. 68, no. 10, pp. 3151–3158, 2008.

5 B. Hopkins and N. Kosmatov, “Third-order boundary value problems with sign-changing solutions,”

Nonlinear Analysis, vol. 67, no. 1, pp. 126–137, 2007.

6 Y. Sun, “Positive solutions of singular third-order three-point boundary value problem,” Journal of

Mathematical Analysis and Applications, vol. 306, no. 2, pp. 589–603, 2005.

7 D. R. Anderson, “Green’s function for a third-order generalized right focal problem,” Journal of

Mathematical Analysis and Applications, vol. 288, no. 1, pp. 1–14, 2003.

8 R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, vol. 436 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. 9 R. P. Agarwal, D. O’Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral

Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

10 R. Ma, “A survey on nonlocal boundary value problems,” Applied Mathematics E-Notes, vol. 7, pp. 257–279, 2007.

11 D. R. Anderson and J. M. Davis, “Multiple solutions and eigenvalues for third-order right focal boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 135– 157, 2002.

12 D. Anderson and R. Avery, “Multiple positive solutions to third-order discrete focal boundary value problem,” Computers & Mathematics with Applications, vol. 42, no. 3–5, pp. 333–340, 2001.

13 S. Li, “Positive solutions of nonlinear singular third-order two-point boundary value problem,”

(8)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of

Optimization

Journal of

Combinatorics

International Journal of Hindawi Publishing Corporation

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

Journal of

On a Third Order Three Point Boundary Value Problem

Research Article