YAo QING-LIU

(Department of Applied Mathematics,Nanjing University of Finance and Economics, Nanjing,210003)

Communicated by Shi Shao-yun

A(k,n-k)Conjugate Boundary Value Problem with Semipositone Nonlinearity

YAo QING-LIU

(Department of Applied Mathematics,Nanjing University of Finance and Economics, Nanjing,210003)

Communicated by Shi Shao-yun

The existence of positive solution is proved for a(k,n-k)conjugate boundary value problem in which the nonlinearity may make negative values and may be singular with respect to the time variable.The main results of Agarwalet al.(Agarwal R P,Grace S R,O’Regan D.Semipositive higher-order diferential equations.Appl.Math.Letters,2004,14:201-207)are extended.The basic tools are the Hammerstein integral equation and the Krasnosel’skii’s cone expansion-compression technique.

higher order ordinary diferential equation,boundary value problem, semipositone nonlinearity,positive solution

1 Introduction

Letn≥2,1≤k≤n-1 be two positive integers andλ＞0 be a positive parameter.In this paper,we study the existence of positive solution to the following nonlinear(k,n-k) conjugate boundary value problem:

The solutionu?of the problem(P)is called positive ifu?(t)＞0 for 0＜t＜1.

For the functionf(t,x),we use the following assumptions:

(A1)f:(0,1)×[0,+∞)→(-∞,+∞)is continuous.

(A2)There exists a nonnegative functionh∈L1[0,1]∩C(0,1)such that

(A3)For eachr＞0,there exists a nonnegative functionjr∈L1[0,1]∩C(0,1)such that

The assumptions(A2)and(A3)show thatf(t,x)may be singular att=0 andt=1, and may not have any numerical lower bound.Therefore,the problem(P)is singular and semipositone.The problems of this type arise naturally in chemical reactor theory,see[1].

In applications,one is interested in showing the existence of positive solution for someλ.Whenh(t)≡M≥0,the problem(P)has been frequently investigated in recent years, for example,see[2-9]and the references therein.

In 2004,Agarwalet al.[8]established the following existence theorem of positive solution:Theorem 1.1([8],Theorem 2.3)Suppose that the following conditions are satisfed:

(a1)f:[0,1]×[0,+∞)→(-∞,+∞)is continuous and there exists a constant M＞0such that f(t,x)+M≥0for any(t,x)∈[0,1]×[0,+∞);

(a2)There exists a continuous and nondecreasing function ζ:[0,+∞)→[0,+∞)such that

and

(a5)There exists an ε with

such that

where

Then the problem(P)has at least one positive solution

In Theorem 1.1,G(t,s)is the Green function of the problem(P)withf(t,x)≡0.For the expression ofG(t,s),see Section 2.The functionh(t)≡Mis a constant and the nonlinearityf(t,x)is continuous on[0,1]×[0,+∞).

The purpose of this paper is to extend Theorem 1.1.In this paper,we study the problem (P)under the assumptions(A1)-(A3).Therefore,we allowh(t)to be an integral function on[0,1]andf(t,x)to be singular att=0 andt=1.

We apply the Anuradha’s substitution technique and the Krasnosel’skii’s cone expansioncompression method to the problem(P)(see[10-12]).By introducing two height functions and considering the integrals of the height functions,we establish a local existence theorem. Finally,we verify that the theorem extends the Theorem 1.1 and illustrate that our extend is true by an example.

2 Preliminaries

Firstly,we list some symbols used in this paper.

LetC[0,1]be the Banach space of all continuous functions on[0,1]equipped with the norm

Let the polynomials

Let the sets

ThenKis a cone of nonnegative functions inC[0,1].

LetG(t,s)be the Green function of the homogeneous linear(k,n-k)boundary value problem(P)withf(t,x)≡0.ThenG(t,s)has the exact expression

By[2],

Let

Then

Let the constants

If(A1)-(A3)hold,thenT:K→C[0,1]is well-defned andTu∈C[0,1].

Secondly,we need the following lemmas in order to prove the main results.

Let[c]?=max{0,c}.Foru∈K,defne the operatorTas follows:

Lemma 2.1Assume u∈Cn-1[0,1]∩Cn(0,1)such that

Then

Proof.See Lemma 2.1 in[8].

Proof.See Lemma 2.1 and Theorem 1 in[7].

Proof.It is easy to see that

By Lemma 2.2,we have

Lemma 2.4If(A1)-(A3)hold,then T:K→K is completely continuous.

Proof.T(K)?Kis derived from Lemma 2.1.The remainder is a standard argument,for example,see Step II in the proof of Theorem 2.2 in[12]or Step II in the proof of Theorem 1 in[13].

Lemma 2.5Ifˉu∈K is a fxed point of the operator T and‖ˉu‖＞λη,then u?(t)is a positive solution of the problem(P),where u?=ˉu-λw.

Proof.By the defnition ofη,we have

Since‖ˉu‖＞λη,one has

It shows that

By the equality andTˉu=ˉu,one has

Since

by the properties ofw(t),we get

This shows thatu?(t)is a solution of the problem(P).Since

the solutionu?(t)is positive.

3 Main Results

We use the following control functions:

In geometry,φ(t,r)is the maximum height off(t,[u-λw(t)]?)+h(t)on the set{t}×[rp(t),r],ψ(t,r)is the minimum height off(t,[u-λw(t)]?)+h(t)on the same set.If (A1)-(A3)hold,thenφ(t,r)andψ(t,r)are nonnegative integrable function on[0,1]for anyr＞0.

We obtain the following local existence results.

Theorem 3.1Assume that(A1)-(A3)hold and there exist two positive numbers r2＞r1＞λη such that one of the following conditions is satisfed:

Then the problem(P)has at least one positive solution u?such that

Proof.We only prove the case(b1).

Ifu∈??(r1),then

By the defnition ofφ(t,r1),we have

It follows

Ifu∈??(r2),then

By the defnition ofψ(t,r2),one has

It follows

by Lemma 2.5,we known thatu?=ˉu-λwis a positive solution of the problem(P).Further,

Corollary 3.1Assume that(A1)-(A3)hold and there exist two positive numbers r1and r2with r2＞r1＞λη such that one of the following conditions is satisfed:

Then the problem(P)has at least one positive solution u?such that

Proof.We only prove the case(c1).

By Lemma 2.3 and(c1),one has

By Theorem 3.1(b1),the proof is completed.

4 A New Extend

In this section,we demonstrate that Theorem 3.1 extends Theorem 1.1.

Proposition 4.1Theorem1.1is a special case of Theorem3.1(b1).

Proof.Letr1,r2,ζ(x),ξ(x)be as in Theorem 1.1.

Sinceζ(x)is nondecreasing on[0,+∞),by(a2)and(a3),one has,for 0＜t＜1,

It follows

Let

Then

Sinceh(t)≡M,by Lemma 2.2 in[8],one has

So

By(a5),if

then

Sinceξ(x)is nondecreasing on(0,+∞),by(a4),one has,forδ≤t≤1-δ,

By(a5),we get

Since(A1)-(A3)hold,Theorem 1.1 now can be derived from Theorem 3.1(b1).

Example 4.2Consider the(2,4-2)conjugate boundary value problem {

In this problem,

Since

But the problem has one positive solution for someλ＞0.

In fact,let

Since

one has

For any 0≤t≤1,one has

Sincew(t)≤ηq(t),ifλ≤192 andx≥30,then for 0≤t≤1,

Since

If 11.6830≤λ≤81.92,then

By Theorem 3.2(c2),the problem has a positive solutionu?such that

for anyλwith 11.6830≤λ≤81.92.

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A

1674-5647(2015)01-0051-11

10.13447/j.1674-5647.2015.01.06

Received date:Dec.18,2012.

Foundation item:The NSF(11071109)of China.

E-mail address:yaoqingliu2002@hotmail.com(Yao Q L).

2010 MR subject classifcation:34B15,35B18