- Research Article
- Open Access

# Existence of Positive Solutions of Fourth-Order Problems with Integral Boundary Conditions

- Ruyun Ma
^{1}Email author and - Tianlan Chen
^{1}

**Received:**5 May 2010**Accepted:**7 July 2010**Published:**3 August 2010

## Abstract

We study the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions, , , , , , , where is continuous, are nonnegative. The proof of our main result is based upon the Krein-Rutman theorem and the global bifurcation techniques.

## Keywords

- Linear Operator
- Nontrivial Solution
- Spectral Radius
- Cauchy Sequence
- Real Banach Space

## 1. Introduction

where is continuous; see Gupta [1, 2]. In the past twenty more years, the existence of solutions and positive solutions of these kinds of problems and the Lidstone problem has been extensively studied; see [3–9] and the references therein. In [3], Ma was concerned with the existence of positive solutions of (1.1) and (1.2) under the assumptions:

uniformly for , where ;

(H2) for and ;

Ma proved the following.

Theorem A (see [3, Theorem ]).

Then (1.1) and (1.2) have at least one positive solution.

where may be singular at and (or) ; is continuous, and are nonnegative.

under the assumption

(H4) are nonnegative, and . The main result of this paper is the following.

Theorem 1.1.

Then (1.9) has at least one positive solution.

Remark 1.2.

Theorem 1.1 generalizes [3, Theorem ] where the special case and was treated.

Remark 1.3.

In this case, and the corresponding eigenfunction is . However, (1.15) and (1.16) has no positive solution. (In fact, suppose on the contrary that (1.15) and (1.16) has a positive solution . Multiplying (1.15) with and integrating from to , we get a desired contradiction!).

*positive*if . It is said to be

*-completely continuous*if is continuous and maps bounded subsets of to precompact subset of . Finally, a positive linear operator on is said to be a

*linear minorant*for if for . If is a continuous linear operator on , denote the spectral radius of . Define

The following lemma will play a very important role in the proof of our main results, which is essentially a consequence of Dancer [14, Theorem ].

Lemma 1.4.

- (i)
has nonempty interior and ;

- (ii)

where is a strongly positive linear compact operator on with the spectral radius , satisfies as locally uniformly in .

such that .

Proof.

Since is a strongly positive compact endomorphism of and has nonempty interior, we have from Amann [15, Theorem ] that the set in [14, Theorem ] reduces to a single point . Now the desired result is a consequence of Dancer [14, Theorem ].

The rest of the paper is arranged as follows. In Section 2, we state and prove some preliminary results about the spectrum of (1.12)–(1.14). Finally, in Section 3, we proved our main result.

## 2. Generalized Eigenvalues

Lemma 2.1 (see [10]).

Lemma 2.2 (see [10]).

Lemma 2.3 (see [10]).

Let

(H5) be two given constants with .

Definition 2.4.

if (2.8) and (2.9) have nontrivial solutions.

Let with the norm . Let with the norm .

We claim that is a Banach space.

Therefore, is a Banach space.

Then the cone is normal and nonempty interior and .

In fact, for any , it follows from the definition of that

(2) , .

Thus, . Obviously, .

Lemma 2.5.

Proof.

From , we have that , and so , and accordingly .

which implies that , and consequently .

Theorem 2.6.

Assume that (H4) and (H5) hold. Let be the spectral radius of . Then (2.8) and (2.9) has an algebraically simple eigenvalue, , with a positive eigenfunction . Moreover, there is no other eigenvalue with a positive eigenfunction.

Remark 2.7.

and the corresponding eigenfunction .

Proof of Theorem 2.6.

We claim that .

where , it follows that , and accordingly .

Thus , and accordingly .

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

Next, we show that is positive.

This together with (2.9) and (H4) imply on .

Therefore, it follows from (2.43) and (2.45) that .

Now, by the Krein-Rutman theorem ([16, Theorem C]; [17, Theorem ]), has an algebraically simple eigenvalue with an eigenfunction . Moreover, there is no other eigenvalue with a positive eigenfunction. Correspondingly, with a positive eigenfunction of , is a simple eigenvalue of (2.8) and (2.9). Moreover, for (2.8) and (2.9), there is no other eigenvalue with a positive eigenfunction.

## 3. The Proof of the Main Result

It is easy to check that is compact.

such that .

Proof of Theorem 1.1.

we note that for all since is the only solution of (3.8) for and .

Case 1 ( ).

We divide the proof into two steps.

Step 1.

then joins to .

This together with Theorem 2.6 imply that . Therefore, joins to .

Step 2.

We show that there exists a constant be such that for all .

By Lemma 1.4, we only need to show that has a linear minorant and there exists a such that and .

Combining this with (2.39), we conclude that , here, . Therefore, we have that from Lemma 1.4 that .

Case 2 ( ).

and, moreover, .

Again joins to and the result follows.

## Declarations

### Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This paper was supported by the NSFC (no. 11061030), the Fundamental Research Funds for the Gansu Universities.

## Authors’ Affiliations

## References

- Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation.
*Applicable Analysis*1988, 26(4):289-304. 10.1080/00036818808839715View ArticleMathSciNetMATHGoogle Scholar - Gupta CP: Existence and uniqueness results for the bending of an elastic beam equation at resonance.
*Journal of Mathematical Analysis and Applications*1988, 135(1):208-225. 10.1016/0022-247X(88)90149-7View ArticleMathSciNetMATHGoogle Scholar - Ma R: Existence of positive solutions of a fourth-order boundary value problem.
*Applied Mathematics and Computation*2005, 168(2):1219-1231. 10.1016/j.amc.2004.10.014View ArticleMathSciNetMATHGoogle Scholar - Ma R, Xu L: Existence of positive solutions of a nonlinear fourth-order boundary value problem.
*Applied Mathematics Letters*2010, 23(5):537-543. 10.1016/j.aml.2010.01.007View ArticleMathSciNetMATHGoogle Scholar - Li Y: Positive solutions of fourth-order boundary value problems with two parameters.
*Journal of Mathematical Analysis and Applications*2003, 281(2):477-484. 10.1016/S0022-247X(03)00131-8View ArticleMathSciNetMATHGoogle Scholar - Ma R, Wang H: On the existence of positive solutions of fourth-order ordinary differential equations.
*Applicable Analysis*1995, 59(1–4):225-231.MathSciNetMATHGoogle Scholar - Bai Z, Wang H: On positive solutions of some nonlinear fourth-order beam equations.
*Journal of Mathematical Analysis and Applications*2002, 270(2):357-368. 10.1016/S0022-247X(02)00071-9View ArticleMathSciNetMATHGoogle Scholar - Ma R, Xu J: Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem.
*Nonlinear Analysis: Theory, Methods & Applications*2010, 72(1):113-122. 10.1016/j.na.2009.06.061View ArticleMathSciNetMATHGoogle Scholar - Ma Y: Existence of positive solutions of Lidstone boundary value problems.
*Journal of Mathematical Analysis and Applications*2006, 314(1):97-108.View ArticleMathSciNetMATHGoogle Scholar - Zhang X, Ge W: Positive solutions for a class of boundary-value problems with integral boundary conditions.
*Computers & Mathematics with Applications*2009, 58(2):203-215. 10.1016/j.camwa.2009.04.002View ArticleMathSciNetMATHGoogle Scholar - Gallardo JM: Second-order differential operators with integral boundary conditions and generation of analytic semigroups.
*The Rocky Mountain Journal of Mathematics*2000, 30(4):1265-1291. 10.1216/rmjm/1021477351View ArticleMathSciNetMATHGoogle Scholar - Karakostas GL, Tsamatos PCh: Multiple positive solutions of some integral equations arisen from nonlocal boundary-value problems.
*Electronic Journal of Differential Equations*2002, 30: 1-17.Google Scholar - Lomtatidze A, Malaguti L: On a nonlocal boundary value problem for second order nonlinear singular differential equations.
*Georgian Mathematical Journal*2000, 7(1):133-154.MathSciNetMATHGoogle Scholar - Dancer EN: Global solution branches for positive mappings.
*Archive for Rational Mechanics and Analysis*1973, 52: 181-192.View ArticleMathSciNetMATHGoogle Scholar - Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces.
*SIAM Review*1976, 18(4):620-709. 10.1137/1018114View ArticleMathSciNetMATHGoogle Scholar - Zeidler E:
*Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems*. Springer, New York, NY, USA; 1986:xxi+897.View ArticleMATHGoogle Scholar - Deimling K:
*Nonlinear Functional Analysis*. Springer, Berlin, Germany; 1985:xiv+450.View ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.