XIE DA-PENG,LIU YANGAND SUN MING-ZHE

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

(2.Department of Mathematics,College of Science,Yanbian University,Yanji,Jilin,133002)

Existence of Positive Solutions for Higher Order Boundary Value Problem on Time Scales*

XIE DA-PENG1,LIU YANG1AND SUN MING-ZHE2

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

(2.Department of Mathematics,College of Science,Yanbian University,Yanji,Jilin,133002)

Communicated by Li Yong

In this paper,we investigate the existence of positive solutions of a class higher order boundary value problems on time scales.The class of boundary value problems educes a four-point(or three-point or two-point)boundary value problems, for which some similar results are established.Our approach relies on the Krasnosel'skii f i xed point theorem.The result of this paper is new and extends previously known results.

higher order boundary value problem,positive solution,semipositone, on time scale,f i xed point

1 Introduction

In this paper,we study the existence of positive solutions of higher order boundary value problem(BVP)on time scales as follows:

We assume that

(H1)φ and φ are increasing nonconstant functions def i ned on[a,b]with φ(a)=φ(a)=0;

(H2)f is continuous and there exists M ≥0 such that

(H3)h and g are continuous and there exists M ≥0 such that

Lemma 1.1[1]Assume that

(1)x(t)is a bounded function on[a,b],i.e.,there exist c,C∈R such that

(2)φ(t)and φ(t)are increasing on[a,b];

(3)Riemann-Stieltjes integrals

Let α=φ(b)and β=φ(b).For any continuous solution x(t)of the BVP(1.1)-(1.2),by Lemma 1.1,there exist ξ,η∈(a,b)such that

If

and

then(1.2)reduces to

The existence of positive solutions of the BVP(1.1)-(1.3)has been studied by several authors when a=0,b=1 and n=2(see[2–4]).

If

and

then(1.2)reduces to

The existence of positive solutions of the BVP(1.1)-(1.4)has been studied when a=0, b=1 in[5–6].

If

then(1.2)reduces to

The existence of positive solutions of the BVP(1.1)-(1.5)has been studied when a=0, b=1 in[7–8].

In fact,(H2)implies that f is not necessarily nonnegative,monotone,superlinear and sublinear.And also this assumption implies that the BVP(1.1)-(1.2)is semipositive.The purpose of this paper is to establish the existence of positive solutions of the BVP(1.1)-(1.2) by using Krasnosel'skii fi xed point theorem in cones.

2 Related Lemmas

Lemma 2.1 If y(t)∈C[a,b],then the boundary value problem

has a unique solution

where

Proof. In fact,if x(t)is a solution of the problem(2.1),then we may suppose that

Since

we get

It follows from

that

Using this and

we have

Therefore,the problem (2.1)has a unique solution

where G(t,s)is de fi ned by(2.2).

Lemma 2.2 G(t,s)has the following properties:

(i)0≤G(t,s)≤k(s)for t,s∈[a,b],where

(ii)G(t,s)≥γ(t)k(s)for t,s∈[a,b],where

Proof. It is obvious that G(t,s)is nonnegative.Moreover,

So,(i)holds.

It is clear that(ii)holds for s=a or s=b.For s∈(a,b)and t∈[a,b],we have

Since

and

we obtain

Thus,(ii)holds.The proof is completed.

Remark 2.1 By simple computations,we get

and

For n≥4,by simple computations,we have

From this,we obtain

Thus,

Lemma 2.3 Let

Then

Proof.Since

we have

and

Hence,(i)and(ii)hold.

and

Thus,

and

Hence,

It follows from(2.3)and(2.4)that(iii)holds.

Lemma 2.4[9]Let E=(E,‖·‖)be a Banach space and P?E be a cone in E.Assume that Ω1and Ω2are open subsets of E with 0∈Ω1and?Ω2.Let A:P∩(Ω1)→P be continuous and completely continuous.In addition,suppose either

(1)‖Ax‖≤‖x‖,x∈P∩?Ω1,and‖Ax‖≥‖x‖,x∈P∩?Ω2, or

(2)‖Ax‖≥‖x‖,x∈P∩?Ω1,and‖Ax‖≤‖x‖,x∈P∩?Ω2.

Then A has a fi xed point in P∩(Ω1).

3 Main Result

Let E=C[a,b]be a real Banach space with the maximum norm,and de fi ne the cone P?E by

where γ(t)is as in Lemma 2.2.For brevity,we denote

Theorem 3.1 Suppose that(H1)–(H{3)hold.Assumethat there exi}st three positivenumbers r1,r2and C3withs u ch that

where α,β,C1,C2are a{s in(3.1)–(3.4),and}

Then the BVP(1.1)-(1.2)has at least one positive solution.

Proof. Let

Then by Lemma 2.3(i)and(ii),we have

and

Consider the following boundary value problem:

This problem is equivalent to the integral equation

We de fi ne the operator A as follows:

We claim that A(P)?P.In fact,for each x∈P and t∈[a,b],by Lemma 2.2(i)we have

Hence,

On the other hand,for any t∈[a,b],by(3.11)and Lemma 2.2(ii),we have

This implies that A(P)?P.Similar to the proof of Remark 2.1 in[2],it is easy to prove that the operator A:P∩(Ω2)→P is continuous and compact.

If x∈P∩?Ω1,we obtain

Then,by(3.7)and(3.12),we have

Thus,

which imply

Then by(3.14),(3.17)and Lem ma 2.2(i),we have

Therefore,

For x∈P∩?Ω2,we have

In view of(3.8)and(3.19),for

we have

which implies

Using this and Lemma 2.2(ii),for x∈P∩?Ω2,we have

Thus,

According to Lemma 2.4 and using the inequalities(3.18)and(3.22),we assert that the operator A has at least one fi xed point x*∈P∩(Ω2).This implies that the BVP(3.9) has at least one solution x*∈P with r2≤‖x*‖≤r1.

Let

We check that x*is a solution of the BVP(1.1)-(1.2).In fact,since Ax*=x*,we have

This shows that

In other words,x*is a solution of the BVP(1.1)-(1.2).Therefore,the BVP(1.1)-(1.2)has at least one solution x*satisfying x*+x0∈P and r2≤‖x*+x0‖≤r1.

Since

we have by Lemma 2.4(iii)that

which implies that x*is a positive solution of the BVP(1.1)-(1.2).

[1]Guo D J,Lakshmikantham V.Nonlinear Problems in Abstract Cones.Notes Rep.Math.Sci. Engrg.vol.5.Boston-Mass:Academic Press,1988.

[2]Bai Z B,Du Z J.Positive solutions for some second-order four-point boundary value problems. J.Math.Anal.Appl.,2007,330:34–50.

[3]Bai Z B,Ge W G,Wang Y F.Multiplicity results for some second-order four-point boundary value problems.Nonlinear Anal.,2005,60:491–500.

[4]Bai C Z,Xie D P,Liu Y,Wang C L.Positive solutions for second-order four-point boundary value problems with alternating coefficient.Nonlinear Anal.,2009,70:2014–2023.

[5]Eloe P W,Ahmad B.Positive solutions of a nonlinear nth order boundary value problems with nonlocal conditions.Appl.Math.Lett.,2005,18:521–527.

[6]Hao X A,Liu L S,Wu Y H.Positive solutions of nonlinear nth-order singular nonlocal boundary value problems.Boundary Value Problems,2007,2007:234–244.

[7]Eloe P W,Henderson J.Positive solutions for higher order ordinary dif f erential equations. Electron.J.Dif f erential Equations,1995,3:1–8.

[8]Xie D P,Bai C Z,Liu Y,Wang C L.Existence of positive solutions for nth-order boundary value problem with sign changing nonlinearity.Electron.J.Qual.Theory Dif f erential Equations, 2008,8:1–10.

[9]Guo D,Lakshmikantham V.Problem in Abstract Cones.New York:Academic Press,1988.

tion:34B18

A

1674-5647(2013)01-0001-13

*Received date:Jan.8,2010.

The NSF(11201109)of China,the NSF(10040606Q50)of Anhui Province,Excellent Talents Foundation(2012SQRL165)of University of Anhui Province and the NSF(2012kj09)of Heifei Normal University.