Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 932145, 10 pages

http://dx.doi.org/10.1155/2013/932145

Research Article

On Linear and Nonlinear Fourth-Order Eigenvalue Problems with Nonlocal Boundary Condition

Dongming Yan

School of Mathematics and Statistics, Zhejiang University of Finance & Economics, Hangzhou 310018, China

Received 18 December 2012; Revised 13 February 2013; Accepted 15 February 2013

Academic Editor: Feliz Minhós

Copyright © 2013 Dongming Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We determine the principal eigenvalue of the linear problem ,  , , where and . Moreover, we investigate the existence of positive solutions for the corresponding nonlinear problem. The proofs of our main results are based upon the Krein-Rutman theorem and fixed point index theory.

1. Introduction

The deformations of an elastic beam can be described by the boundary value problems of the fourth-order ordinary differential equations. For example, an elastic beam in an equilibrium state whose both ends are simply supported can be described by the fourth-order boundary value problem of the form

see Gupta [1, 2]. Owing to its significance in physics, it has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point index theory, and the method of lower and upper solutions; see, for example, [3–16].

Recently, Bai [3] investigated the existence of positive solutions for more general fourth-order nonlocal boundary value problem

By using the Krasnoselskii fixed point theorem, the sufficient conditions for the existence of positive solutions of (2) are obtained. We can find that, however, those conditions do not involve the eigenvalues with respect to the relevant linear operator, and those sufficient conditions are not optimal.

The likely reason is that the spectrum structure of the linear eigenvalue problem

is not clear.

It is the purpose of this paper to investigate the first eigenvalue of (3) by using the Krein-Rutman theorem [17, 18], and then we use this spectrum result to establish the existence of positive solutions of nonlinear nonlocal problem (2). The existence of positive solution is obtained by means of fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The obtained sufficient conditions in this paper are optimal. For the concepts and properties of fixed point index theory, we refer the reader to [19].

The rest of the paper is arranged as follows: in Section 2, we study the spectrum of (3). Finally, in Section 3, we give an application of our main result.

2. Spectrum of (3)

Let us assume that

(A1), , , , ,  ,  .(A2).

Definition 1. We say is an eigenvalue of linear problem (see (3)), if (3) has nontrivial solutions.

To study the spectrum of (3), we need several preliminary results.

Let

Lemma 2 (see [7]). Assume (A1) holds. Then one has

(i),  for ; ,  for ;(ii),  for ,

where , ; , .

Lemma 3 (see [3]). Assume (A1) holds. Then for any ,   solves the problem

if and only if

where

Let

Then, there exist such that

respectively.

In fact, from (ii) of Lemma 2, we can obtain

where we can take

and also we obtain

where

Let

for a certain , and let be the Banach space with the norm . For , we have

Combining (16) and (9) with the fact that , we conclude that

So, we may define the norm of by

This norm is so called -norm.

It is easy to get the following lemmas.

Lemma 4. is a Banach space.

Proof. Let be Cauchy sequence of . Then, we have

From (18) and (19), for all , there exists such that

Hence,

which implies that is a Cauchy sequence of . According to the completeness of , there exists such that

Let in (20). Then, we have

which implies that

Hence, for Cauchy sequence , there exists satisfying

Therefore, is a Banach space.

Lemma 5. is compactly embedded in .

Proof. Let be bounded. Then, according to the fact that is compactly embedded in , there exist and such that

From (26), for all , there exists such that

Since

combining this fact with (27) we have

which implies

Therefore, is compactly embedded in .

Let

Then, the cone is normal and has nonempty interior .

In the rest of this section, we will prove the existence of the first eigenvalue of (3). To wit, we get the following.

Theorem 6. Assume (A1) holds. Then (3) has an algebraically simple eigenvalue , with an eigenfunction , and there is no other eigenvalue with a positive eigenfunction.

Remark 7. If , then can be explicitly given by

and the corresponding eigenfunction , .

Proof of Theorem 6. For , define a linear operator by

Then, by the definition of , we have

Therefore, we can obtain

We claim that . In fact, for , let . Then, from (18) and (17) we have, for ,

On the one hand, from (36)-(37) and (10) we have

On the other hand, from (36)-(37) and (10) we have

Therefore, . If , then , on . According to (36) and the fact that

we have

Furthermore, according to (41), the definition of , and the fact that

we have

Then , and accordingly, .

Now, since and is compactly embedded in , we have that is compact.

Next, we show that is strongly positive.

For , it is easy to check that there exist , such that for

In fact, for every , it follows from Lemma 2 that

where

where

Therefore, it follows from (44) that .

Now, on the one hand, by the Krein-Rutman theorem ([17, Theorem ] and [18, Theorem 19.3]), has an algebraically simple eigenvalue with an eigenfunction . Moreover, there is no other eigenvalue with a positive eigenfunction. On the other hand, we have from the definition of and Lemma 3 that (3) is equivalent to the integral equation

Therefore, with a positive eigenfunction is a simple eigenvalue of (3). Moreover, for (3), there is no other eigenvalue with a positive eigenfunction.

3. An Application of Theorem 6

For convenience, we introduce the following notations:

Theorem 8. Assume that (A1) and (A2) hold, and . Then, for each satisfying

there exists at least one positive solution of (2).

Theorem 9. Assume that (A1) and (A2) hold, and . Then, for each satisfying

there exists at least one positive solution of (2).

Remark 10. Bai [3] proved existence of positive solutions via Guo-Krasnoselskii fixed point theorem under some conditions which do not involve the eigenvalue of (3). While our Theorems 8 and 9 are established under (51) or (52) which are related to the eigenvalue of (3). Our Theorems 8 and 9 cover an undefined case in [3]. Consider the following boundary value problem:

In this case, , , , and

According to Theorem 9, the above boundary value problem has at least one positive solution. For the above boundary value problem, however, we cannot obtain the above conclusion by [3, Theorem 3.1] since

Moreover, (51) and (52) are optimal. In order to illustrate this point, consider the problem

where

In this case, , , , and

However, (56) has no positive solution. In fact, suppose on the contrary that (56) has a positive solution . Multiplying the first equation of (56) with and integrating from 0 to 1, we get

which is a contradiction.

To prove Theorems 8 and 9, we need the following preliminary results.

Lemma 11. For every , there exist such that

Proof. By , there is , so

Hence

where .

By , there is a such that , and so, for

Hence , and accordingly,

We have from (18) that , which implies that

and consequently

where .

Let

It is easy to show that is a completely continuous operator. In addition, we can verify that the nonzero fixed points of the operator are positive solutions of the problem (2).

Lemma 12 (see [19]). Let be Banach space, a cone in , and a bounded open set in . Suppose that is a completely continuous operator. If there exists such that

then the fixed point index .

Lemma 13 (see [19]). Let be Banach space, a cone in , and a bounded open set in with . Suppose that is a completely continuous operator. If

then the fixed point index .

Proof of Theorem 8. It follows from the first inequality of (51) that there exists , such that

Let be the positive eigenfunction of (see (33)) corresponding to . Thus .

Let . Then for every , we have from Lemma 11 that

It follows from (70) that

We may suppose that has no fixed points on (otherwise, the proof is finished). Now we show that

Suppose the contrary, that there exists and such that . Hence and

Put

It is easy to see that and . We find from that

Therefore, by (72), we have

which contradicts the definition of . Hence (73) is true, and we have from Lemma 12 that

It follows from the second inequality of (51) that there exists and such that

Let , . Then is a bounded linear operator and .

Let

where . It is clear that .

Let

In the following, we prove that is bounded.

For any , set . Then,

Thus, , for every . Since is the first eigenvalue of and , the first eigenvalue of , . Therefore, the inverse operator exists and

It follows from that . So, we have , , and we conclude that is bounded.

Select . Then, from the invariance property of the fixed point index, we have

By (78) and (84), we have that

Then has at least one fixed point on . This means that the boundary value problem (2) has at least one positive solution.

Proof of Theorem 9. It follows from the second inequality of (52) that there exists , such that

Let . Then, for every , we have from Lemma 11 that

It follows from (86) that

Suppose there exists and satisfying . We may suppose that has no fixed points on (otherwise, the proof is finished). So . By (88), we have . By induction, we have , for all . Thus

By Gelfand’s formula, we have

which is a contradiction with . Hence

and we have from Lemma 13 that

It follows from the first inequality of (52) that there exist and such that

From (A2), we know that there exists such that

We can take large enough, such that

where is defined as in (46). Then by (94) and (33), for all ,  , , we have

Hence, for every ,  , we have

Let be the positive eigenfunction of corresponding to . Thus .

We may suppose that has no fixed points on (otherwise, the proof is finished). Now, we show that

Suppose the contrary, that there exists and such that

Hence and

Put

It is easy to see that and . We find from that

Therefore by (97), we have

which contradicts the definition of . Hence (98) is true, and we have from Lemma 12 that

By (92) and (104), we have that

Then, has at least one fixed point on . This means that the boundary value problem (2) has at least one positive solution.

Acknowledgment

The author would like to express his gratitude to anonymous referee for a number of valuable comments and suggestions.

References

C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,” Applicable Analysis, vol. 26, no. 4, pp. 289–304, 1988.

C. P. Gupta, “Existence and uniqueness results for the bending of an elastic beam equation at resonance,” Journal of Mathematical Analysis and Applications, vol. 135, no. 1, pp. 208–225, 1988.

Z. B. Bai, “Positive solutions of some nonlocal fourth-order boundary value problem,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4191–4197, 2010.

Z. B. Bai, “The upper and lower solution method for some fourth-order boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 6, pp. 1704–1709, 2007.

G. Q. Chai, “Existence of positive solutions for fourth-order boundary value problem with variable parameters,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 870–880, 2007.

H. Y. Feng, D. H. Ji, and W. G. Ge, “Existence and uniqueness of solutions for a fourth-order boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3561–3566, 2009.

Y. X. Li, “Positive solutions of fourth-order boundary value problems with two parameters,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 477–484, 2003.

Y. X. Li, “Positive solutions of fourth-order periodic boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 6, pp. 1069–1078, 2003.

X.-L. Liu and W.-T. Li, “Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters,” Mathematical and Computer Modelling, vol. 46, no. 3-4, pp. 525–534, 2007.

H. L. Ma, “Symmetric positive solutions for nonlocal boundary value problems of fourth order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 3, pp. 645–651, 2008.

R. Y. Ma, “Existence of positive solutions of a fourth-order boundary value problem,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1219–1231, 2005.

C. C. Pang, W. Dong, and Z. L. Wei, “Multiple solutions for fourth-order boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 464–476, 2006.

Z. L. Wei and C. C. Pang, “Positive solutions and multiplicity of fourth-order -point boundary value problems with two parameters,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1586–1598, 2007.

Y. Yang and J. H. Zhang, “Existence of solutions for some fourth-order boundary value problems with parameters,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1364–1375, 2008.

Q. L. Yao, “Local existence of multiple positive solutions to a singular cantilever beam equation,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 138–154, 2010.

J. F. Zhao and W. G. Ge, “Positive solutions for a higher-order four-point boundary value problem with a -Laplacian,” Computers & Mathematics with Applications, vol. 58, no. 6, pp. 1103–1112, 2009.

E. Zeidler, Nonlinear Functional Analysis and Its Applications. I Fixed-Point Theorems, Springer, New York, NY, USA, 1985.

K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988.