This article is part of the series Nonlocal Boundary Value Problems.

Open Access Research Article

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

Ruyun Ma* and Tianlan Chen

Author Affiliations

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

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Boundary Value Problems 2011, 2011:297578  doi:10.1155/2011/297578


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/297578


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

© 2011 Ruyun Ma and Tianlan Chen.

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.

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.

1. Introduction

The deformations of an elastic beam in an equilibrium state, whose both ends are simple supported, can be described by the fourth-order boundary value problem

(11)

(12)

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 [39] and the references therein. In [3], Ma was concerned with the existence of positive solutions of (1.1) and (1.2) under the assumptions:

(H1) is continuous and there exist constants , with such that

(13)

uniformly for , and

(14)

uniformly for , where ;

(H2) for and ;

(H3)there exist constants satisfying and

(15)

Ma proved the following.

Theorem A (see [3, Theorem ]).

Let (H1), (H2), and (H3) hold. Assume that either

(16)

or

(17)

where denotes the first generalized eigenvalue of the generalized eigenvalue problem

(18)

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

At the same time, we notice that a class of boundary value problems with integral boundary conditions appeared in heat conduction, chemical engineering underground water flow, thermoelasticity, and plasma physics. Such a kind of problems include two-point, three-point, multipoint and nonlocal boundary value problems as special cases and attracting the attention of a few readers; see [1013] and the references therein. For example, In particular, Zhang and Ge [10] used Guo-Krasnoselskii fixed-point theorem to study existence and nonexistence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions:

(P)

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

Motivated by [3, 10], in this paper, we consider the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions:

(19)

under the assumption

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

Theorem 1.1.

Let (H1), (H2), (H3), and (H4) hold. Assume that either

(110)

or

(111)

where

(112)

(113)

(114)

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.

Zhang and Ge [10] proved existence and nonexistence of positive solutions via Guo-Krasnoselskii fixed-point theorem under some conditions which do not involve the eigenvalues of (1.12)–(1.14). While our Theorem 1.1 is established under (1.10) or (1.11) which is related to the eigenvalues of (1.12)–(1.14). Moreover, (1.10) and (1.11) are optimal. Let us consider the problem

(115)

(116)

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!).

Suppose that is a real Banach space with norm . Let be a cone in . A nonlinear mapping is said to be 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

(117)

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.

Assume that

(i) has nonempty interior and ;

(ii) is -completely continuous and positive, for , for and

(118)

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

Then there exists an unbounded connected subset of

(119)

such that .

Moreover, if has a linear minorant and there exists a

(120)

such that and , then can be chosen in

(121)

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]).

Assume that (H4) holds. Then for any , the boundary value problem

(21)

has a unique solution which is given by

(22)

where

(23)

Lemma 2.2 (see [10]).

Assume that (H4) holds. Then for any , the boundary value problem

(24)

has a unique solution which is given by

(25)

where

(26)

Lemma 2.3 (see [10]).

Assume that (H4) holds. Then one has

(27)

Let

(H5) be two given constants with .

Definition 2.4.

One says thatis a generalized eigenvalue of linear problem

(28)

(29)

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

Let

(210)

Let with the norm . Let with the norm .

Let

(211)

For , from Lemma 2.1, it follows that

(212)

By simple calculations, we have

(213)

Combining this with (H4), we conclude that

(214)

This together with (2.12) and the fact that imply that

(215)

Since , we may define the norm of by

(216)

We claim that is a Banach space.

In fact, let be a Cauchy sequence, that is, as . From the definition of , it follows that

(217)

where is a normal in defined by . Thus,

(218)

By the completeness of , there exists , such that

(219)

From the fact that , we have that for arbitrary , there exists , such that

(220)

and subsequently,

(221)

Fixed and let , we get

(222)

This is,

(223)

Therefore, is a Banach space.

Let

(224)

Then the cone is normal and nonempty interior and .

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

(1)there exist real number , such that

(225)

(2), .

From and (H4), we obtain that for some . Moreover,

(226)

and subsequently,

(227)

for some . We may take satisfying

(228)

Now, let us define

(229)

Then , and

(230)

Thus, . Obviously, .

Lemma 2.5.

Assume that holds. Then for any , one has

(231)

Proof.

In fact, for , we have that

(232)

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

We have from the fact that , , that

(233)

which implies that , and consequently .

For , define a linear operator by

(234)

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.

If , then can be explicitly given by

(235)

and the corresponding eigenfunction .

Proof of Theorem 2.6.

From Lemma 2.2, it is easy to check that (2.8) and (2.9) is equivalent to the integral equation

(236)

We claim that .

In fact, for ,we have that

(237)

Since

(238)

and for some constant , it concludes that

(239)

Hence,

(240)

where , it follows that , and accordingly .

If , then on , and accordingly

(241)

Thus , and accordingly .

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

Next, we show that is positive.

For , if , from Lemma 2.3, we have

(242)

Combining this with (2.39), there exist such that

(243)

For , if , applying a similar proof process of (2.43), we have

(244)

Combining this with (2.39), there exist such that

(245)

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

Before proving Theorem 1.1, we denote by setting

(31)

where

(32)

It is easy to check that is compact.

Let be such that

(33)

Obviously,()impliesthat

(34)

(35)

Let

(36)

then is nondecreasing and

(37)

Let us consider

(38)

as a bifurcation problem from the trivial solution . It is to easy to check that (3.8) can be converted to the equivalent equation

(39)

From the proof process of Theorem 2.6, the operator ,

(310)

is compact and strongly positive. Define by

(311)

then we have from (3.4) and Lemma 2.5 that

(312)

locally uniformly in . Fromand Theorem 2.6 (with obvious changes), it follows that if is a nontrivial solution of (3.8) with , then . Combining this with Lemma 1.4, we conclude that there exists an unbounded connected subset of the set

(313)

such that .

Proof of Theorem 1.1.

It is clear that any solution of the form yields a solution of (1.9). We will show that crosses the hyperplane . To do this, it is enough to show that joins to . Let satisfy

(314)

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

Case 1 ().

In this case, we show that

(315)

We divide the proof into two steps.

Step 1.

We show that if there exists a constant number such that

(316)

then joins to .

From (3.16), we have that . We divide the equation

(317)

by and set . Since is bounded in , choosing a subsequence and relabeling if necessary, we see that for some with . Moreover, we have from (3.7) and Lemma 2.5 that

(318)

Since Thus,

(319)

where , again choosing a subsequence and relabeling if necessary. Thus,

(320)

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 .

By,there exist constants satisfying and

(321)

For , let

(322)

then is a linear minorant of . Moreover,

(323)

where

(324)

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

Case 2 ().

In this case, if such that

(325)

and then

(326)

and, moreover, .

Assume that there exists , such that for all ,

(327)

Applying a similar argument to that used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that

(328)

Again joins to and the result follows.

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.

References

  1. Gupta, CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis. 26(4), 289–304 (1988). Publisher Full Text OpenURL

  2. Gupta, CP: Existence and uniqueness results for the bending of an elastic beam equation at resonance. Journal of Mathematical Analysis and Applications. 135(1), 208–225 (1988). Publisher Full Text OpenURL

  3. Ma, R: Existence of positive solutions of a fourth-order boundary value problem. Applied Mathematics and Computation. 168(2), 1219–1231 (2005). Publisher Full Text OpenURL

  4. Ma, R, Xu, L: Existence of positive solutions of a nonlinear fourth-order boundary value problem. Applied Mathematics Letters. 23(5), 537–543 (2010). Publisher Full Text OpenURL

  5. Li, Y: Positive solutions of fourth-order boundary value problems with two parameters. Journal of Mathematical Analysis and Applications. 281(2), 477–484 (2003). Publisher Full Text OpenURL

  6. Ma, R, Wang, H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis. 59(1–4), 225–231 (1995)

  7. Bai, Z, Wang, H: On positive solutions of some nonlinear fourth-order beam equations. Journal of Mathematical Analysis and Applications. 270(2), 357–368 (2002). Publisher Full Text OpenURL

  8. Ma, R, Xu, J: Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. Nonlinear Analysis: Theory, Methods & Applications. 72(1), 113–122 (2010). PubMed Abstract | Publisher Full Text OpenURL

  9. Ma, Y: Existence of positive solutions of Lidstone boundary value problems. Journal of Mathematical Analysis and Applications. 314(1), 97–108 (2006)

  10. Zhang, X, Ge, W: Positive solutions for a class of boundary-value problems with integral boundary conditions. Computers & Mathematics with Applications. 58(2), 203–215 (2009). PubMed Abstract | Publisher Full Text OpenURL

  11. Gallardo, JM: Second-order differential operators with integral boundary conditions and generation of analytic semigroups. The Rocky Mountain Journal of Mathematics. 30(4), 1265–1291 (2000). Publisher Full Text OpenURL

  12. Karakostas, GL, Tsamatos, PCh: Multiple positive solutions of some integral equations arisen from nonlocal boundary-value problems. Electronic Journal of Differential Equations. 30, 1–17 (2002)

  13. Lomtatidze, A, Malaguti, L: On a nonlocal boundary value problem for second order nonlinear singular differential equations. Georgian Mathematical Journal. 7(1), 133–154 (2000)

  14. Dancer, EN: Global solution branches for positive mappings. Archive for Rational Mechanics and Analysis. 52, 181–192 (1973)

  15. Amann, H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review. 18(4), 620–709 (1976). Publisher Full Text OpenURL

  16. Zeidler, E: Nonlinear Functional Analysis and Its Applications. I. Fixed-point Theorems,p. xxi+897. Springer, New York, NY, USA (1986)

  17. Deimling, K: Nonlinear Functional Analysis,p. xiv+450. Springer, Berlin, Germany (1985)