Existence of Positive Solutions for Singular Fourth Order Coupled System with Sign Changing Nonlinear Terms

XIA TIAN,GAO JING-LU AND CAO CHUN-LING

(School of Mathematics,Jilin University,Changchun,130012)

Communicated by Gao Wen-jie

Existence of Positive Solutions for Singular Fourth Order Coupled System with Sign Changing Nonlinear Terms

XIA TIAN,GAO JING-LU AND CAO CHUN-LING*

(School of Mathematics,Jilin University,Changchun,130012)

Communicated by Gao Wen-jie

This paper deals with the existence of positive solutions to a singular fourth order coupled system with integral boundary conditions.Since the nonlinear terms f,g may change sign or be singular at t=0 or t=1,the authors make a priori estimates to overcome some difficulties and apply Guo-Krasnoselskii f i xed point theorem to prove the existence of solutions of the system under suitable assumptions. Finally,some examples to illustrate the main results are given.

f i xed point theorem,positive solution,integral boundary condition,

1 Introduction

In this paper,we investigate the existence of positive solutions to the following singular fourth order system of ordinary dif f erential equations with integral boundary conditions:

where f,g,ωi,gi(s)and hi(s),i=1,2,satisfy

(H1)f,g∈C[[0,1]×[0,+∞)×[0,+∞)×(-∞,0]×(-∞,0],(-∞,+∞)];

and are probably singular at t=0 or t=1;

(H3)gi,hi∈L1[0,1]are nonnegative,andμi,νi∈(0,1),where

In recent years,boundary val0ue problems for the0system of ordinary di ff erential equations have been studied extensively.The main tools are fi xed point theorems in cones for completely continuous operators,and the readers may refer to[1–5].Lu¨ et al.[5]considered the following problem:

+

By the f i xed point theorem of cone expansion and compression,they obtained the existence of solutions to the system(1.2).

With regard to integral boundary value problems,many authors have studied the existence of solutions,and the interested readers may refer to[6–8].Furthermore,a large amount of literature has been devoted to the study of the existence of positive solutions to boundary value problems in which the nonlinear functions are allowed to change sign.Ji et al.[9]obtained the existence of solutions for boundary value problem with sign changing nonlinearity by the f i xed point theorem.

However,it seems that there are not much study of singular fourth order systems of ordinary dif f erential equations with sign change on nonlinear terms.Motivated by the above works,we discuss the existence of positive solutions to the problem(1.1).

The main features of this paper are as follows:Firstly,the system(1.1)consists of two fourth order ordinary dif f erential equations with integral boundary conditions.Secondly, the functions f and g depend on u,v,u′′,v′′,and the functions ω1,ω2are allowed to be singular at t=0 or t=1.Moreover,the nonlinear terms f,g are allowed to change sign. By making a priori estimates and calculating accurately we overcome some difficulties and apply Guo-Krasnoselskii f i xed point theorem to prove the existence of solutions by choosing suitable function class to which the solutions belong.

The outline of this paper is as follows:In Section 2,we give some properties of the Green's function associated with the problem(1.1)and some necessary preliminaries.Section 3 is devoted to the proof of the existence of solutions to the problem(1.1).In Section 4,we give some examples to illustrate how the main results can be used in practice.

2 Preliminaries

In this section,we give some preliminaries and lemmas.

where

and

Set

where δ will be given later.Def i ne K=K×K.It is easily seen that K is a cone in E, and

Firstly,we consider the following boundary value problem:

where ωi,gi(s)and hi(s)satisfy the conditions(H2),(H3),i=1,2,and f1,f2satisfy

Evidently,(u,v)∈C4(0,1)×C4(0,1)is a solution of the problem(2.1)if and only if (u,v)∈C2[0,1]×C2[0,1]is a solution of the following nonlinear integral system:

where

andμi,νi,i=1,2,are the same as in the condition(H3).

By the expression of G(t,s),Gi(t,s)and Ki(t,s),i=1,2,we can prove the following proposition.

Proposition 2.1Under the condition(H3),for all t,s∈[0,1],we have

where

Proof.The proof is similar to the proof of Proposition 2.4 in[10],and we omit the details here.

Def i ne two integral operators

and an integral operator T0:E→E by

Then(u,v)is a f i xed point of the integral operator T0if and only if(u,v)is a solution of the problem(2.2).

To obtain the existence of a positive solution of the problem(1.1),we also need the following lemmas.

Lemma 2.1Under the conditions(H11),(H2),(H3),if

then T0(u,v)∈K for(u,v)∈K.

Proof.By the properties of G(t,s),Ki(t,s),Gi(t,s)and the continuity of fi,ωi,i=1,2, if(u,v)∈E,then T0(u,v)∈E.We can easily derive that

By Proposition 2.1,we f i nd that∫∫

where

satisf i es 0＜δ＜1,i=1,2.Therefore,T0(u,v)∈K and

This completes the proof of Lemma 2.1.

Lemma 2.2Under the conditions(H11),(H2),(H3),T0:K→K is completely continuous.

3 Main Results

In this section,we show the existence of positive solutions to the problem(1.1).

Before stating our main results,we give some notations as follows:

Theorem 3.1Suppose that the conditions(H1)–(H3)hold,and there exists an r＞0 such that for t∈[0,1],and|u|+|v|+|u′′|+|v′′|∈[δr,+∞].

Then the problem(1.1)has at least one positive solution(u(t),v(t)),if fδr,r,gδr,r≤a,and f,g=∞,where

Proof.Def i ne

Consider the following boundary value problem:

Def i ne two integral operators

by

and an integral operator T:E→E by

where K1,K2,G1,G2are the same as in Section 2.We claim that T is a completely continuous operator.The proof is similar to the proof of Lemmas 2.1 and 2.2.

Set

Then for all(u,v)∈K∩?Ω1,t∈[0,1],from f*0,r,g*0,r≤a it follows that

with

Now,from(3.3)–(3.4)and Proposition 2.1,we have

Applying(A1)of Lemma 1.1,we obtain that T has a f i xed pointand.Then,the problem(3.1)has at least one positive solution(u,v)and (u,v)is also a positive solution of the problem(1.1).

Theorem 3.2Suppose that the conditions(H1)–(H3)hold,and there exists an R＞0 such that

Then the problem(1.1)has at least one positive solution(u(t),v(t)),if fδR,R,gδR,R≥,where }

Proof.Consider the following boundary value problem:

where

and

We def i ne the operators T,A,B as in Theorem 3.1.Then it is easy to see that T is completely continuous.Fromwe have

Let

Then,similar to the proof of(3.5)-(3.6),for all(u,v)∈K∩?Ω1,t∈[0,1],we conclude that

with

By(3.6)–(3.7)and Proposition 2.2,we f i nd that

Hence,

Applying(A2)of Lemma 1.1,we obtain that T has a f i xed point(u,v)R≤Thus,the problem(3.5)has at least one positive solution(u,v)which implies that it is also the solution of the problem(1.1).

Theorem 3.3Suppose that the conditions(H1)–(H3)hold,and there exists an r＞0 such that

Then the problem(1.1)has at least two positive solutions,ifand there exists an r1＞r＞0 such that

where a is as given in Theorem 3.1,and η＞0 satisf i es

Proof.Def i ne two functions f*,g*and three operators T,A,B which are the same as in Theorem 3.1.It is easily seen that,and

Let

Then,following the lines of the proof of(3.5)-(3.6),we know that for all(u,v)∈K∩?Ω3,

Applying Theorems 3.1 and 3.2,we have

Choosing R2so large that R2＞r1＞r in Theorem 3.2,using(A1)and(A2)of Lemma 1.1, we obtain that T has two f i xed points

Therefore,the problem(1.1)has at least two positive solutions(u1,v1)and(u2,v2)

Theorem 3.4Suppose that the conditions(H1)–(H3)hold,and there exists an R＞0 such thatfor

Then the problem(1.1)has at least two positive solutions,if fδR,R,gδR,R＞b,f∞,g∞=∞, and there exists an r1＞R＞0 such that

where b is as given in Theorem 3.2,and λ＞0 satisf i es

Proof.The proof is similar to that of Theorem 3.3,and we omit the details here.

4 Some Examples

Example 4.1Set

Example 4.2Set

and

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tion:47H10,34B15

A

1674-5647(2013)02-0167-12

Received date:Dec.18,2011.

The under-graduation base items(J0630104,J0730104,J1030101)of School of Mathematics,Jilin University,and the 985 program of Jilin University.

*Corresponding author.

E-mail address:641829005@qq.com(Xia T),caocl@jlu.edu.cn(Cao C L).