Yong-Hui Zhou,Yun-Rui Yang,Li Liu

(School of Math.and Physics,Lanzhou Jiaotong University, Lanzhou 730070,Gansu,PR China)

MULTIPLE POSITIVE SOLUTIONS FOR A FOURTH-ORDER NONLINEAR
EIGENVALUE PROBLEM??

Yong-Hui Zhou,Yun-Rui Yang?,Li Liu

(School of Math.and Physics,Lanzhou Jiaotong University, Lanzhou 730070,Gansu,PR China)

In this paper,by using the Guo-Krasnoselskii’s fixed-point theorem,we establish the existence and multiplicity of positive solutions for a fourth-order nonlinear eigenvalue problem.The corresponding examples are also included to demonstrate the results we obtained.

positive solutions;eigenvalue problem;fixed point

2000 Mathematics Subject Classification 34B15

Ann.of Appl.Math.

32:4(2016),418-428

1　Introduction

In the past decades,an increasing interest in the existence and multiplicity of positive solutions for boundary value problems has been evolved by using some fixedpoint theorems,for example,by the Krasnoselskii’s fixed-point theorem,Ma[1]and Li[2]respectively established the existence and multiplicity of positive solutions for some fourth-order boundary value problems.Zhong[3]established the existence of at least one positive solution for the following four-point boundary value problem

In 2015,Wu[4]obtained some new results on the existence of at least one positive solution for the following fourth-order three-point nonlinear eigenvalue problem

Bai[5]obtained the existence of triple positive solutions via the Leggett-Williams fixed-point theorem[6].There are other meaningful investigated results on the existence of positive solutions for some types of nonlinear differential equations,one can be referred to[1-4,7,8].But to the best of our knowledge,there are not many results on the existence of multiple positive solutions for fourth-order nonlinear eigenvalue problems with multi-points boundary value condition.

Based on the fact,our purpose in this paper is to investigate the existence and multiplicity of positive solutions for the following fourth-order three-point eigenvalue problem

where λ is a positive parameter,a,b,c,d are nonnegative constants satisfying ad+bc+ac＞0,b?aξ≥0,h(t)∈C[0,1],f∈C([0,1]×[0,+∞)×(?∞,0],[0,+∞)).

This paper is organized as follows.In Section 2,we introduce some preliminaries. In Section 3,we state and prove our main results on the existence and multiplicity of positive solutions for(1.1).At the same time,the corresponding examples are also included to demonstrate the results we obtained.

2　Preliminaries

For convenience,we first state some definitions and preliminary results which we need.Throughout this paper,we make the following assumptions:

(H1)f∈C([0,1]×[0,+∞)×(?∞,0],[0,+∞))is continuous;

(H2)h(t)∈C([0,1]),h(t)≤0 for all t∈[0,ξ],h(t)≥0 for all t∈[ξ,1],where;and h(t)?0 for any subinterval of[0,1].

Denote

and

where Δ=ad+bc+ac(1?ξ)＞0.

Our main results in this paper mainly depend on the following Guo-Krasnoselskii’s fixed-point theorem.

Theorem 2.1[9]Let E be a Banach space,K ?E be a cone in E.Assume that ?1and ?2are bounded open subsets of E with 0∈?1andand T:is a completely continuous operator such that either

(i)∥Tu∥≥∥u∥,u∈K∩??1and∥T u∥≤∥u∥,u∈K∩??2;or

(ii)∥T u∥≤∥u∥,u∈K∩??1and∥Tu∥≥∥u∥,u∈K∩??2holds,then T has a fixed point in

Let C[0,1]be endowed with the maximum norm

and C2[0,1]be endowed with the norm

Let G1(t,s)and G2(t,s)be the Green’s functions of the following boundary value problems

respectively.In particular,

It is easy to check that

Define a cone K in C2[0,1]by

Define an integral operator T:K→C2[0,1]by

Therefore,

and

It is easy to check that

Obviously,u(t)is a solution for the BVP(1.1)if and only if u(t)is a fixed point of the operator T.

Lemma 2.1 Assume that(H1)and(H2)hold.If b≥aξ,then T:K?→K is completely continuous.

Proof Denote

where

Next,for each t∈[0,1],we consider the following two cases:

Case 1 When t∈[0,ξ],for any u∈K,from(2.12),(H1),(H2)and b≥aξ,we have

Case 2 When t∈[ξ,1],for any u∈K,from(2.12),(H1),(H2)and b≥aξ,we have

From(2.13)and(2.14),we get

Moreover,for any u∈K,from(2.13),(2.14),(H1),(H2),and b≥aξ,we have

and

On the other hand,

Consequently T:K?→K.Furthermore,it is not difficult to check that the operator T is completely continuous by Arzela-Ascoli theorem.This completes the proof.

Lemma 2.2 Suppose that(H1)and(H2)hold.If b≥aξ,then for the operator T:K?→K,the following conclusions hold:

Proof We only prove(iii)and(iv),since the proofs of(i)and(ii)are similar to those of(iii)and(iv)respectively.

For any t∈[ξ,1],u∈K,∥u∥2=R,we getBy (2.3),(2.5)and(2.8),we have

By(2.5)and(2.9),we get

Case 1 For 0≤s≤t≤1,we have

Case 2 For 0≤t≤s≤1,we have

3　Main Results

Theorem 3.1 Suppose(H1)and(H2)hold.Furthermore,f satisfies either

Proof Let E=C2[0,1],?1={u∈E:∥u∥2＜r},?2={u∈E:∥u∥2＜R}, where 0＜r＜R.By Lemma 2.1,we know that the operator T:is completely continuous,then the condition(i)or(ii)of Lemma 2.2 is satisfied. Applying Theorem 2.1,it follows that T has a fixed pointThus u0is the solution for the BVP(1.1)and satisfies

Therefore,G1(t,s)≥εs(1?s)for any s∈[0,1].

By(2.8),we have

That is u0(t)＞0.This completes the proof.

If

Corollary 3.1Suppose(H1)and(H2)hold.Suppose further that f satisfies either

(i)f0=0,f∞=∞(superlinear);or(ii)f0=∞,f∞=0(sublinear), then the BVP(1.1)has at least one positive solution.

Example 3.1 Consider the following boundary value problem

In order to discuss the multiplicity of positive solutions for the BVP(1.1),we further assume

(H3)f(t,u,v)＞0,for any t∈[0,1]and|u|+|v|＞0.

Theorem 3.2 Suppose(H1)and(H2)hold.If one of the following two conditions holds:

Proof(i)Let E=C2[0,1],?1={u∈E:∥u∥2＜ r},?2={u∈E:∥u∥2＜R}.By the condition(i)or(iii)of Lemma 2.2,we get∥T u∥2≤∥u∥2when u∈K∩??1or u∈K∩??2.Choose ?3={u∈E:∥u∥2＜R0}such thatBy(2.10),we know thatfor anyand u∈K∩??3.Thus,it follows the following two cases:

Case 1 For 0≤s≤t≤1,we have

Case 2 For 0≤t≤s≤1,we have

Therefore,∥Tu∥2＞∥u∥2,u∈K∩??3.By Theorem 2.1,the BVP(1.1)has two positive solutionsTherefore

Together with Theorem 3.1,it follows that the BVP(1.1)has two distinct positive solutions u1and u2.

(ii)Let E=C2[0,1],?1={u∈E:∥u∥2＜r},?2={u∈E:∥u∥2＜R}.By the condition(ii)or(iv)of Lemma 2.2,we get∥Tu∥2≥∥u∥2when u∈K∩??1or u∈K∩??2.Choose ?3={u∈E:∥u∥2＜R0}such thatFor any t∈[ξ,1],u∈K∩??3,then|u(t)|+|u′′(t)|≤R0.By combining(2.5)with (2.8)and(2.9),we have

Applying the Theorem 3.1 again,the operator T has two fixed points u1∈satisfying(3.2),which are the two different positive solutions for the BVP(1.1).The proof is completed.

Corollary 3.2 Suppose(H1)-(H3)hold.Suppose further that f and λ satisfies either

(i)f0=f∞=0,or(ii)f0=f∞=∞,, then BVP(1.1)has at least two positive solutions.

Example 3.2 Consider the following boundary value problem

Let R0=1 such that|u|+|v|≤1,thenHence,if,then Theorem 3.2(ii)guarantees the existence of two positive solutions for the BVP(3.3).

References

[1]R.Ma,Multiple positive solutions for a semipositone fourth-order boundary value problem,J.Math.Anal.Appl.,33(2003),217-227.

[2]Y.Li,Existence and multiplicity of positive solutions for fourth-order three-point boundary value problem,Comput.Math.Appl.,26(2003),109-116.

[3]Y.L.Zhong,S.H.Chen and C.P.Wang,Existence results for a fourth-order differential equation with a four-point boundary condition,Appl.Math.Lett.,21(2008),465-470.

[4]H.Wu,Positive solutions to fourth-order three-point nonlinear eigenvalue problem, Ann.Math.Anal.Appl.,31(2015),96-104.

[5]Ch.Bai,Triple positive solutions of three-point boundary value problems of fourthorder differential equations,Comput.Math.Appl.,56(2008),1364-1371.

[6]R.I.Leggett and L.R.Williams,Multiple positive fixed points of nonlinear operators on ordered Banach spaces,Indiana Univ.Math.J.,28(1979),673-688.

[7]B.Liu,Positive solutions of fourth-order three-point boundary value problem Comput. Math.Appl.,148(2001),313-322.

[8]B.Liu,Positive solutions of fourth-order two-point boundary value problems,Appl. Math.Comput.,148(2004),407-420.

[9]D.Guo and V.Lskshmikanthan,Nonlinear Problems in Abstract Cones,Academic Press,San Diego,1988.

(edited by Mengxin He)

?Supported by the NSF of China(11301241),Science and Technology Plan Foundation of Gansu Province(145RJYA250),and Young Scientist Foundation of Lanzhou Jiaotong University(2011029).

?Manuscript June 12,2016;Revised September 25,2016

?.E-mail:lily1979101@163.com