Existence of Solution and Positive Solution for a Nonlinear Third-Order -Point BVP

Volume 2010, Article ID 126192,8pages doi:10.1155/2010/126192

Research Article

Existence of Solution and Positive Solution for

a Nonlinear Third-Order

m

-Point BVP

Jian-Ping Sun and Fan-Xia Jin

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

Correspondence should be addressed to Jian-Ping Sun,[email protected]

Received 5 November 2010; Accepted 14 December 2010

Academic Editor: Tomonari Suzuki

Copyrightq2010 J.-P. Sun and F.-X. Jin. 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.

In this paper, we are concerned with the following nonlinear third-orderm-point boundary value problem:ut ft, ut, ut 0,t∈0,1,u0 A,u1−m_i−2₁ aiuξi B,u0 C. Some

existence criteria of solution and positive solution are established by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.

1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves, or gravity-driven flows and so on 1.

Recently, third-order two-point or three-point boundary value problemsBVPshave received much attention from many authors; see 2–10 and the references therein. In particular, Yao10employed the Leray-Schauder fixed point theorem to prove the existence of solution and positive solution for the BVP

ut ft, ut, ut0, t∈0,1,

u0 A, u1 B, u0 C. 1.1

worth mentioning that Jin and Lu12studied some third-order differential equation with the followingm-point boundary conditions:

u0 0, u1

m−2

i1

aiuξi, u0 0. 1.2

The main tool used was Mawhin’s continuation theorem.

Motivated greatly by 10,12, in this paper, we investigate the following nonlinear third-orderm-point BVP:

ut ft, ut, ut0, t∈0,1,

u0 A, u1−

m−2

i1

aiuξi B, u0 C.

1.3

Throughout, we always assume thatai ≥0i1,2, . . . , m−2and 0< ξ1< ξ2<· · ·< ξm−2<1. The purpose of this paper is to consider the local properties off on some bounded sets and establish some existence criteria of solution and positive solution for the BVP1.3by using the Schauder fixed point theorem. An example is also included to illustrate the importance of the results obtained.

2. Main Results

Lemma 2.1. Letm_i−₁2ai/1. Then, for anyv∈C0,1, the BVP

ut vt, t∈0,1,

u0 A, u1−

m−2

i1

aiuξi B

2.1

has a unique solution

ut B−

_m−2 i1 ai

₁ ξivsds

1−m_i−₁2ai

tA−

₁

0

Gt, svsds, t∈0,1, 2.2

where

Gt, s

⎧ ⎨ ⎩

s, 0≤s≤t≤1,

t, 0≤t≤s≤1. 2.3

Proof. Ifuis a solution of the BVP2.1, then we may suppose that

ut MtN−

₁

0

By the boundary conditions in2.1, we know that

M B−

m−2 i1 ai

₁ ξivsds

1−m_i−₁2ai

, NA. 2.5

Therefore, the unique solution of the BVP2.1

ut B−

m−2 i1 ai

₁ ξivsds

1−m_i−₁2ai

tA−

₁

0

Gt, svsds, t∈0,1. 2.6

In the remainder of this paper, we always assume thatm_i−₁2ai/1. For convenience,

we denote

R −∞,∞, R 0,∞, R− −∞,0,

σ 2

1−m_i−₁2ai

2m_i−₁2ai1− _m−2

i1 ai

, ηmax

|A|, B 1−m_i−₁2ai

,|C|σ

. 2.7

The following theorem guarantees the existence of solution for the BVP1.3.

Theorem 2.2. Assume that f : 0,1×R× R → R is continuous and there existc > 0 and

1/4≤k≤1/2such that

maxft, u, v:t∈0,1, |u| ≤4ηc, |v| ≤σk4ηc≤σ4k−1ηkc. 2.8

Then the BVP1.3has one solutionu0satisfying

|u0t| ≤4ηc, t∈0,1, _u

0t≤σk

4ηc, t∈0,1. 2.9

Proof. LetEC0,1×C0,1be equipped with the normu, vmax{u_∞,v_∞/σk}, whereu_∞maxt∈0,1|ut|. ThenEis a Banach space.

Letvt ut,t∈0,1. Then the BVP1.3is equivalent to the following system:

ut vt, t∈0,1,

vt −ft, ut, vt, t∈0,1,

u0 A, u1−

m−2

i1

aiuξi B, v0 C.

Furthermore, it is easy to know that the system2.10is equivalent to the following system:

ut B−

_m−2 i1 ai

1 ξivsds

1−m_i−₁2ai

tA−

₁

0

Gt, svsds, t∈0,1,

vt C−

_t

0

fs, us, vsds, t∈0,1.

2.11

Now, if we define an operatorF:E → Eby

Fu, v F1u, v, F2u, v, 2.12

where

F1u, vt

B−mi−12ai ₁

ξivsds

1−m_i−₁2ai

tA−

1

0

Gt, svsds, t∈0,1,

F2u, vt C− _t

0

fs, us, vsds, t∈0,1,

2.13

then it is easy to see thatF :E → Eis completely continuous and the system2.11and so the BVP1.3is equivalent to the fixed point equation

Fu, v u, v, u, v∈E. 2.14

LetV {u, v∈E:u, v ≤4ηc}. ThenVis a closed convex subset ofE. Suppose thatu, v∈V. Thenu∞≤4ηcandv∞≤σk4ηc. So,

|ut| ≤4ηc, t∈0,1, 2.15

|vt| ≤σk4ηc, t∈0,1, 2.16

which implies that

_{ft, ut, vt≤σ}

From2.16and 1/4≤k≤1/2, we have

F1u, v_∞≤max t∈0,1

⎛

⎝

₁₋Bm−2 i1 ai

m−2 i1 ai

₁ ξivsds

1−m_i−₁2ai

⎞

⎠_t|A|max

t∈0,1 ₁

0

Gt, s|vs|ds

≤ B 1−m_i−₁2ai

σk4ηc mi−12ai

1−m_i−₁2ai

|A|σk

4ηc 2

≤2η2

_m−2

i1 ai1− _m−2

i1 ai

21−m_i−₁2ai

σk4ηc

4η

k1 2

kc.

2.18

On the other hand, it follows from2.17that

F2u, v∞_tmax_∈0,1

C−

_t

0

fs, us, vsds

≤ |C| 1 0

_{fs, us, vsds}

≤σησ4k−1ηkc σk4ηc.

2.19

In view of2.18and2.19, we know that

F1u, v, F2u, vmax

F1u, v∞,F2u, v_σk∞

≤4ηc,

2.20

which shows thatF : V → V. Then it follows from the Schauder fixed point theorem that F has a fixed point u0, v0 ∈ V. In other words, the BVP 1.3has one solution u0, which satisfies

|u0t| ≤4ηc, t∈0,1, _u

0t≤σk

4ηc, t∈0,1. 2.21

Theorem 2.3. Assume thatA ≥ 0,B ≥ 0,C ≤ 0,m_i−₁2ai < 1,f : 0,1×R×R− → R is

continuous, and there existc >0and1/4≤k≤1/2such that

maxft, u, v:t∈0,1, 0≤u≤4ηc, −σk4ηc≤v≤0≤σ4k−1ηkc. 2.22

Then the BVP1.3has one solutionu0satisfying

0≤u0t≤4ηc, t∈0,1,

−σk4ηc≤u0t≤0, t∈0,1.

2.23

Proof. Let

f1t, u, v ⎧ ⎨ ⎩

ft, u, v, t, u, v∈0,1×R×R−,

ft, u,0, t, u, v∈0,1×R×R,

f2t, u, v ⎧ ⎨ ⎩

f1t, u, v t, u, v∈0,1×R×R,

f1t,0, v t, u, v∈0,1×R−×R.

2.24

Thenf2:0,1×R×R → Ris continuous and

maxf2t, u, v:t∈0,1, |u| ≤4ηc, |v| ≤σk

4ηc

maxft, u, v:t∈0,1, 0≤u≤4ηc, −σk4ηc≤v≤0 ≤σ4k−1ηkc.

2.25

Consider the BVP

ut f2

t, ut, ut0, t∈0,1,

u0 A, u1−

m−2

i1

aiuξi B, u0 C.

2.26

ByTheorem 2.2, we know that the BVP2.26has one solutionu0satisfying

|u0t| ≤4ηc, t∈0,1, _u

0t≤σk

4ηc, t∈0,1. 2.27

SinceC≤0, we get

u0t C− _t

0 f2

s, u0s, u0s

In view of2.28andu₀1−_im−₁2aiu₀ξi B, we have

u0t≥u01≥ B 1−m_i−₁2ai

≥0, t∈0,1, 2.29

which implies that

u0t≥u00 A≥0, t∈0,1. 2.30

It follows from2.28,2.30, and the definition off2that

f2

t, u0t, u0t

ft, u0t, u0t

, t∈0,1. 2.31

Therefore,u0is a solution of the BVP1.3and satisfies

0≤u0t≤4ηc, t∈0,1,

−σk4ηc≤u₀t≤0, t∈0,1. 2.32

Corollary 2.4. Assume that all the conditions ofTheorem 2.3are fulfilled. Then the BVP1.3has one positive solution if one of the following conditions is satisfied:

iAB >0;

iiC <0;

iiift,0,0≡/0,t∈0,1.

Proof. Since it is easy to prove Casesiiandiii, we only prove Case i. It follows from

Theorem 2.3that the BVP1.3has a solutionu0, which satisfies

0≤u0t≤4ηc, t∈0,1,

−σk4ηc≤u₀t≤0, t∈0,1. 2.33

Suppose thatAB >0. Then for anyt∈0,1, we have

u0t

B−m_i−₁2ai 1

ξiu0sds

1−m_i−₁2ai

tA−

₁

0

Gt, su0sds

≥ Bt

1−m_i−₁2ai

A

≥BtA ≥BAt

>0,

2.34

Example 2.5. Consider the BVP

ut ft, ut, ut0, t∈0,1,

u0 1, u1−1 2u

1 2

0, u0 −1,

2.35

whereft, u, v u2_{/189 1−tv}2_{/141/9,t, u, v∈0,1×R} ×R−.

A simple calculation shows thatσ2/3 andη3/2. Thus, if we choosek1/3 and c 1, then all the conditions ofTheorem 2.3andiofCorollary 2.4are fulfilled. It follows

fromCorollary 2.4that the BVP2.35has a positive solution.

Acknowledgment

This paper was supported by the National Natural Science Foundation of China10801068.

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