Existence of Positive Solutions for Eigenvalue Problems of Fourth-order Elastic Beam Equations

LU Hai-xia

(School of Arts and Science,Suqian College,Suqian 223800,China)

Existence of Positive Solutions for Eigenvalue Problems of Fourth-order Elastic Beam Equations

LU Hai-xia

(School of Arts and Science,Suqian College,Suqian 223800,China)

In this paper,we investigate the positive solutions of fourth-order elastic beam equations with both end-points simply supported.By using the approximation theorem of completely continuous operators and the global bifurcation techniques,we obtain the existence of positive solutions of elastic beam equations under some conditions concerning the first eigenvalues corresponding to the relevant linear operators,when the nonlinear term is non-singular or singular,and allowed to change sign.

elastic beam equations;singular;positive solutions;global bifurcation

§1.Introduction and Preliminaries

In this paper,we consider the existence of positive solutions of the following fourth-order two-point boundary value problem

where λ is a positive parameter,f:[0,1]×R1→R1is continuous and may be singular at t=0,1.

Fourth-order two-point boundary value problems are useful for material mechanics because the problems usually characterize the deflection of an elastic beam.In mechanics,the problem (1.1)describes the deflection of an elastic beam with both end-points simply supported.The existence of positive solutions for the elastic beam equations has been studied extensively,see for example[19]and references therein,when the nonlinear term satisfies

And the major methods used are upper and lower solution method,contraction mapping and iterative technique,Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type, topological degree theory and global bifurcation technique.However,when(1.2)is not satisfied, which means the nonlinear term is allowed to change sign,there are only a few papers concerned with the fourth-order boundary value problems.Yao[10]considered the existence of positive solutions of elastic beam equations by constructing control functions and a special cone and using fixed point theorem of cone expansion-compression type.Lu[11]obtained the existence of positive solutions by the topological degree and fixed point theory of nonlinear operator on lattice.

In this paper,by using the approximation theorem of completely continuous operators and the global bifurcation techniques,we study the existence of positive solutions of the problem (1.1)under some conditions concerning the first eigenvalues corresponding to the relevant linear operators,when the nonlinear term f is non-singular or singular and in the case that(1.2)is not satisfied.The method and results in this paper improve those in[10-11].

For the remainder of this section,we present some definitions and lemmas which are used in Section 2 and Section 3.

Let E be a Banach space,P be a cone of E.

Definition 1.1[12-13]Let B:E→E be a linear operator.B is said to be a u0-bounded operator,if there exists u0∈P{θ},such that for any x∈P{θ},there exist a natural number n and real numbers ζ,η＞0,such that

Lemma 1.1[12-13]Let B be a completely continuous u0-bounded operator,λ1＞0 be the first eigenvalue of B,then B must have a positive eigenfunction∈P{θ},corresponding to λ1, and λ1is the unique positive eigenvalue of B corresponding to positive eigenfunction.

Lemma 1.2[14]Let B be a completely continuous u0-bounded operator,A:E→E be an operator(we don’t suppose A maps P to P).If there exist ?0∈P{θ}and λ＞0 such that A?0＞B?0,λA?0=?0,then λ＜λ1,where λ1＞0 is the first eigenvalue of B.

Let X be a Banach space and{Cn|n=1,2,···}be a family of connected subsets of X, we define

Lemma 1.3[14]Suppose that the following conditions are satisfied

(1)There exist zn∈Cn(n=1,2,···)and z?∈X,such that zn→z?;

(2)rn→+∞(n→∞),where rn=sup{‖x‖|x∈Cn};

Then there must exist an unbounded connected component C in D and z?∈C.

§2.Existence of Positive Solutions in the Case That f is Not Singular

In this section we consider the boundary value problem(1.1)in the case that f is not singular and f(t,u)=a(t)u+F(t,u).

We assume that

(H1)a∈C[0,1]with a(t)≥0 on[0,1]and a(t)/≡0 on any subinterval of[0,1];

(H3)There exists α∈(-∞,+∞),such that

where it is not supposed that f(t,u)≥0(u≥0).

Let

It is easy to verify that G(t,s)is nonnegative continuous and for t,s∈[0,1]×[0,1],

It is obvious that(1.1)can be converted to the following integral equation

It is easy to see that A:C[0,1]→C[0,1]is a completely continuous operator.Evidently,the fixed point of λA is the solution of(1.1).

We see from(H2)that the linearization of the boundary value problem(1.1)is

By Theorem 2.3 in Ma[15],we have

Lemma 2.1Suppose(H1),(H2)are satisfied.Then

(1)Problem(2.3)has an infinite sequence of positive eigenvalues

(2)To each eigenvalue λkthe algebraic is 1 and there corresponds an essential eigenfunction ?kwhich has exactly k-1 simple zero in(0,1)and is positive near 0.

Let

By Rabinowitz[16],Lemma 2 in Sun[17]and Lemma 2.1 we know that

Lemma 2.2Suppose(H1),(H2)are satisfied,then C+1is an unbounded connected component of((0,+∞)×S+1)∪{(λ1,θ)}in R1×C[0,1].

Define the linear operator

Lemma 2.3Operator B defined by(2.4)is a u0-bounded operator.

Let u0(t)=P(t),t∈[0,1].For any u∈P{θ},we have

which means the linear operator B is u0-bounded operator.

Let r(B)and λBdenote the spectral radius and the first eigenvalue of B respectively,then λB=(r(B))?1.

By(H3),there exists M0＞0 such that

Theorem 2.2Suppose that(H1)～(H3)hold and α≤0 in(H3),then for any λ∈(λ1,+∞),the boundary value problem(1.1)has at least a positive solution.

§3.Existence of Positive Solutions in the Case That f is Singular

In this section we consider the boundary value problem(1.1)in the case that f(t,u)= h(t)g(u)and h is allowed to be singular at t=0 or t=1.i.e.,

We assume that

Define nonlinear operator A and linear operator B

Then the fixed point of λA is the solution of(3.1)～(3.2).

For any natural number n(n≥2),we set

Then hn:[0,1]→[0,+∞)is continuous and hn(t)≤h(t),t∈(0,1).Let

then An,Bn:C[0,1]→C[0,1]are continuous.And the boundary value problem(3.5),(3.2)and (3.6),(3.2)can be converted into the following nonlinear integral equation and linear integral equation u(t)=λAnu(t)and u(t)=λBnu(t),respectively.

Then we have the following lemma.

Similarly,B:C[0,1]→C[0,1]is completely continuous.

Lemma 3.2Suppose that(H′3)is satisfied,then operators B and Bndefined by(3.3)and (3.7)are u0-bounded operators.

ProofBy(2.2)and(H′3)and by the same method as the proof of Lemma 2.3,we know that Lemma 3.2 holds.

Let λ1and λ1n(n=2,3,···)denote the first eigenvalue of u0-bounded linear operators B and Bnrespectively,then λ1＞0 and λ1n＞0 and λ1=(r(B))?1,λ1n=(r(Bn))?1(n= 2,3,···),where(r(B))?1and(r(Bn))?1denote the spectral radius of linear operators B and Bnrespectively.

(i)D is the subset of the solution of the boundary value problem(3.1),(3.2)and

For any(λ,u)∈D,it follows the definition of D that there exist the subsequence{nk}?{n} and(λnk,unk)∈C+1nk,such that λnk→λ,unk→u.Thus{λnk}and{unk}are bounded.And by the proof of Lemma 3.1 we know Anuniformly converges to A on a bounded set.So

which means u=λAu.So(i)holds.

If(3.8)does not hold,then for any δ1:0＜δ1＜δ,there exist λ＞λ1+ε0,u∈K and N2＞N1, such that u=λAN2u,0＜‖u‖＜δ1.By(3.9)we have

It follows from Lemma 3.2 τBN1,τBN2are u0-bounded linear operators.Let τ?1λ1N1and τ?1λ1N2be the first eigenvalue of τBN1and τBN2respectively.From(3.10)and Lemma 1.2 we know that λ＜τ?1λ1N2.Since hN2≥hN1,then τBN2≥τBN1and so τ?1λ1N2≤τ?1λ1N1. Then

which is a contradiction.Thus(3.8)holds.

By(3.8)and the definition of D,we know that(ii)holds.

It follows from Lemma 3.3 that(λ1n,θ)→(λ1,θ).Note that for any n≥2,C+1nis unbounded.Hence,by Lemma 1.3 there exists an unbounded connected component C in D, containing(λ1,θ).From the property(ii)of D we have

By(3.11)and the same method as the proof of Theorem 2.1,we have

which means Theorem 3.1 holds.

It follows from the same method as the proof of Theorem 2.2,we have

Theorem 3.2Suppose that(H′

1)～(H′3)are satis fied and β≤0 in(H′2),then for any λ∈(λ1,+∞),the boundary value problem(3.1)～(3.2)has at least a positive solution.

ExampleConsider the following fourth-order boundary value problem

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tion:34B16,34B18

:A

1002–0462(2017)01–0007–09

date:2016-05-13

Supported by the National Natural Science Foundation of China(11501260);Supported by the National Natural Science Foundation of Suqian City(Z201444)

Biography:LU Hai-xia(1976-),female,native of Jianhu,Jiangsu,an associate professor of Suqian College, M.S.D.,engages in nonlinear functional analysis.

CLC number:O175.8