Multi-point Boundary Value Problems for Nonlinear Fourth-order Dif f erential Equations with All Order Derivatives

YANG LIU

(Department of Mathematics,Hefei Normal University,Hefei,230061)

Communicated by Shi Shao-yun

Multi-point Boundary Value Problems for Nonlinear Fourth-order Dif f erential Equations with All Order Derivatives

YANG LIU

(Department of Mathematics,Hefei Normal University,Hefei,230061)

Communicated by Shi Shao-yun

By using f i xed point theorem,multiple positive solutions for some fourthorder multi-point boundary value problems with nonlinearity depending on all order derivatives are obtained.The associated Green's functions are also given.

multi-point boundary value problem,positive solution,cone,f i xed point

1 Introduction

In this paper,we are interested in the positive solution for fourth-order nonlinear dif f erential equation

subject to the boundary conditions

or

where 0＜ξ1＜ξ2＜···＜ξm-2＜1,0＜βi＜1,i=1,2,···,m-2,and f∈C([0,1]×R4,[0,+∞)).

It is well known that the boundary value problems of nonlinear dif f erential equations arise in a large number of problems in physics,biology and chemistry.For example,thedeformations of an elastic beam in the equilibrium state can be described as a boundary value problem of some fourth-order dif f erential equations.Owing to its importance in application, the existence of positive solutions for nonlinear second-order or high-order boundary value problems have been studied by many authors(see[1–15]).

When it comes to positive solutions of nonlinear fourth-order boundary value problems, the dif f erent two point boundary value problems are considered by many authors(see[16–24]).Few paper deals with the multi-point cases.Furthermore,for nonlinear fourth-order equations,many results were established under the case that the nonlinear term does not depend on the f i rst,second and third order derivatives in[16–23].Few paper deals with the positive solutions under the situation that all order derivatives are involved in the nonlinear term explicitly(see[25–27]).In fact,the derivatives are of great importance in the problem in some cases.For example,this is the case in the linear elastic beam equation(Euler-Bernoulli equation)

where u(t)is the deformation function,L is the length of the beam,f(t)is the load density, E is the Young's modulus of elasticity and I is the moment of inertia of the cross-section of the beam.In this problem,the physical meaning of the derivatives of the function u(t)is as follows:u(4)(t)is the load density stif f ness,u′′′(t)is the shear force stif f ness,u′′(t)is the bending moment stif f ness and the u′(t)is the slope(see[24]).If the payload depends on the shear force stif f ness,bending moment stif f ness or the slope,the derivatives of the unknown function are involved in the nonlinear term explicitly.

The goal of the present paper is to study the fourth-order multi-point boundary value problems(1.1)-(1.2)and(1.1)-(1.3),in which all order derivatives are involved in the nonlinear term explicitly.In this sense,the problem studied in this paper are more general than before.In order to overcome the difficulty of the derivatives that appear,our main technique is to transfer the problem into an equivalent operator equation by constructing the associate Green's function and apply a f i xed point theorem due to[28].In this paper, multiple monotone positive solutions for the problems(1.1)-(1.2)and(1.1)-(1.3)are established.The results extend the study for fourth-order boundary value problems of nonlinear ordinary dif f erential equations.

2 Preliminaries and Lemmas

In this section,some preliminaries and lemmas used later are presented.

Def i nition 2.1The map α is said to be a nonnegative continuous convex functional on a cone P of a real Banach space E provided that α:P→[0,+∞)is continuous and

Def i nition 2.2The map β is said to be a nonnegative continuous concave functional on a cone P of a real Banach space E provided that β:P→[0,+∞)is continuous and

Let γ,θ be nonnegative continuous convex functionals on P,α be a nonnegative continuous concave functional on P,and ψ be a nonnegative continuous functional on P.Then for positive numbers a,b,c and d,we def i ne the following convex sets:

and a closed set

Lemma 2.1Let P be a cone in a Banach space E,γ and θ be nonnegative continuous convex functionals on P,α be a nonnegative continuous concave functional on P,and ψ be a nonnegative continuous functional on P satisfying

such that for some positive numbers l and d,

(S1){x∈P(γ,θ,α,b,c,d)|α(x)＞b}/=? and α(Tx)＞b for x∈P(γ,θ,α,b,c,d);

(S2)α(Tx)＞b for x∈P(γ,α,b,d)with θ(Tx)＞c;

(S3)0/∈R(γ,ψ,a,d)and ψ(Tx)＜a for x∈R(γ,ψ,a,d)with ψ(x)=a.

Then T has at least three f i xed points x1,x2,such that

3 Positive Solutions for the Problem(1.1)-(1.2)

We consider the fourth-order m-point boundary value problem

where 0＜ξ1＜ξ2＜···＜ξm-2＜1,0＜βi＜1,i=1,2,···,m-2,and

Lemma 3.1Let ξ0=0,ξm-1=1,β0=βm-1=0,and y(t)∈C[0,1].The problem (3.1)-(3.2)has the unique solution

where

for i=1,2,···,m-1.

Proof.Let G(t,s)be the Green's function of the problem x(4)(t)=0 with the boundary condition(3.2).We can suppose

where ai,bi(i=0,1,2,3)are unknown coefficients.Considering the properties of Green's function and the boundary condition(3.2),we have

A straightforward calculation shows that

These give the explicit expression of the Green's function and the proof of Lemma 3.1 is completed.

Lemma 3.2The Green's function G(t,s)satisf i es

Proof.For ξi-1≤s≤ξi,i=1,2,···,m-1,

Then

which induces that G(t,s)is decreasing on t.By a simple computation,we see

This ensures that

Lemma 3.3If x(t)∈C3[0,1],

and furthermore,x(4)(t)≥0 and there exists a t0such that x(4)(t0)＞0,then x(t)has the following properties:

where

are positive constants.

Proof.Since

x′′′(t)is increasing on[0,1].Considering

we have

Thus x′′(t)is decreasing on[0,1].Considering this together with the boundary condition

we conclude that

Then x(t)is concave on[0,1].Taking into account that

we get

(1)From the concavity of x(t),we have

Multiplying both sides with βiand considering the boundary condition,we have

Thus

(2)By using the mean-value theorem together with the concavity of x(t),we have

Multiplying both sides with βiand considering the boundary condition,we have

Comparing(3.3)with(3.5)yields that

(3)By

and

we get

By

and

we get

Consequently,

The proof of Lemma 3.3 is completed.

Remark 3.1Lemma 3.3 ensures that

{

Let the Banach space E=C3[0,1]be endowed with the norm

Def i ne the cone P?E by {

Let the nonnegative continuous concave functional α,the nonnegative continuous convex functionals γ,θ and the nonnegative continuous functional ψ be def i ned on the cone by

By Lemma 3.3,the functionals def i ned above satisfy

Denote

Assume that there exist constants a,b,d＞0 with a＜b＜λd such that

Theorem 3.1Assume that there exist constants a,b,d＞0 with a＜b＜d such that (A1)–(A3)are fulf i lled.Then the problem(1.1)-(1.2)has at least three positive solutions x1, x2,x3satisfying

Proof.The problem(1.1)-(1.2)has a solution x=x(t)if and only if x solves the operator equation

Then

Thus

This ensures that the condition(S1)of Lemma 2.1 is fulf i lled.

For x∈P(γ,θ,α,b,c,d),we have

From(A2),we see

Hence,by the def i nitions of α and the cone P,we can get

which means

By(3.4)and b＜λd,we have

for all x∈P(γ,α,b,d)with

Thus,the condition(S2)of Lemma 2.1 holds.

We show that(S3)also holds.We see that

Suppose that

with

Then by(A3),

which ensures that the condition(S3)of Lemma 2.1 is fulf i lled.Thus,an application of Lemma 2.1 implies that the fourth-order m-point boundary value problem(1.1)-(1.2)has at least three positive concave and decreasing solutions x1,x2,x3with the properties that

4 Positive Solutions for the Problem(1.1)-(1.3)

Lemma 4.1The Green's function of the problem x(4)(t)=0 with the boundary condition (1.3)is

Lemma 4.2H(t,s)≥0,t,s∈[0,1].

Proof.For ξi-1≤s≤ξi,i=1,2,···,m-1,

Then

which implies that H(t,s)is increasing on t.The fact that

ensures that

Lemma 4.3If x(t)∈C3[0,1],

and there exists a t0such that x(4)(t0)＞0,then

where

are positive constants.

The proof of Lemma 4.3 is analogous to Lemma 3.3 and omitted here.

Remark 4.1We see that{

Let the Banach space E=C3[0,1]be endowed with the norm Def i ne the cone P1?E by

{

Let the nonnegative continuous concave functional α,the nonnegative continuous convex functionals γ,θ and the nonnegative continuous functional ψ be def i ned on the cone by

By Lemma 4.3,the functionals def i ned above satisfy

Denote

Assume that there exist constants a,b,d＞0 with a＜b＜λ1d such that

Theorem 4.1Assume that there exist constants a,b,d＞0 with a＜b＜d such that (A4)–(A6)are fulf i lled.Then the problem(1.1)-(1.3)has at least three positive solutions x1, x2,x3with the properties that

The proof of Theorem 4.1 is analogous to Lemma 3.1 and omitted here.

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A

1674-5647(2013)02-0108-13

Received date:Oct.13,2010.

The NSF(10040606Q50)of Anhui Province,China.

E-mail address:xjiangfeng@163.com(Yang L).

34B10,34B15