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Boundary Value Problems

, 2011:594128 | Cite as

Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations

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Research Article
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Abstract

This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps. Firstly, we establish a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from Open image in new window Our nonlinearity Open image in new window may be singular at Open image in new window and/or Open image in new window .

Keywords

Eigenvalue Problem Closed Subset Fixed Point Theorem Global Structure Point Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Singular differential equations arise in the fields of gas dynamics, Newtonian fluid mechanics, the theory of boundary layer, and so on. Therefore, singular boundary value problems have been investigated extensively in recent years (see [1, 2, 3, 4] and references therein).

This paper investigates the following fourth-order nonlinear singular eigenvalue problem:

where Open image in new window is a parameter and Open image in new window satisfies the following hypothesis:

Typical functions that satisfy the above sublinear hypothesis ( Open image in new window ) are those taking the form

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window . The hypothesis ( Open image in new window ) is similar to that in [5, 6].

Because of the extensive applications in mechanics and engineering, nonlinear fourth-order two-point boundary value problems have received wide attentions (see [7, 8, 9, 10, 11, 12] and references therein). In mechanics, the boundary value problem (1.1) (BVP (1.1) for short) describes the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. The term Open image in new window in Open image in new window represents bending effect which is useful for the stability analysis of the beam. BVP (1.1) has two special features. The first one is that the nonlinearity Open image in new window may depend on the first-order derivative of the unknown function Open image in new window , and the second one is that the nonlinearity Open image in new window may be singular at Open image in new window and/or Open image in new window .

In this paper, we study the existence of positive solutions and the structure of positive solution set for the BVP (1.1). Firstly, we construct a special cone and present a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from Open image in new window . Our analysis mainly relies on the fixed point theorem in a cone and the fixed point index theory.

By singularity of Open image in new window , we mean that the function Open image in new window in (1.1) is allowed to be unbounded at the points Open image in new window , Open image in new window , Open image in new window , and/or Open image in new window . A function Open image in new window is called a (positive) solution of the BVP (1.1) if it satisfies the BVP (1.1) ( Open image in new window for Open image in new window and Open image in new window for Open image in new window ). For some Open image in new window , if the Open image in new window (1.1) has a positive solution Open image in new window , then Open image in new window is called an eigenvalue and Open image in new window is called corresponding eigenfunction of the BVP (1.1).

The existence of positive solutions of BVPs has been studied by several authors in the literature; for example, see [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] and the references therein. Yao [15, 18] studied the following BVP:
where Open image in new window is a closed subset and Open image in new window , Open image in new window . In [15], he obtained a sufficient condition for the existence of positive solutions of Open image in new window (1.4) by using the monotonically iterative technique. In [13, 18], he applied Guo-Krasnosel'skii's fixed point theorem to obtain the existence and multiplicity of positive solutions of BVP (1.4) and the following BVP:

These differ from our problem because Open image in new window in (1.4) cannot be singular at Open image in new window , Open image in new window and the nonlinearity Open image in new window in (1.5) does not depend on the derivatives of the unknown functions.

In this paper, we first establish a necessary and sufficient condition for the existence of positive solutions of BVP (1.1) for any Open image in new window by using the following Lemma 1.1. Efforts to obtain necessary and sufficient conditions for the existence of positive solutions of BVPs by the lower and upper solution method can be found, for example, in [5, 6, 21, 22, 23]. In [5, 6, 22, 23] they considered the case that Open image in new window depends on even order derivatives of Open image in new window . Although the nonlinearity Open image in new window in [21] depends on the first-order derivative, where the nonlinearity Open image in new window is increasing with respect to the unknown function Open image in new window . Papers [24, 25] derived the existence of positive solutions of BVPs by the lower and upper solution method, but the nonlinearity Open image in new window does not depend on the derivatives of the unknown functions, and Open image in new window is decreasing with respect to Open image in new window .

Recently, the global structure of positive solutions of nonlinear boundary value problems has also been investigated (see [26, 27, 28] and references therein). Ma and An [26] and Ma and Xu [27] discussed the global structure of positive solutions for the nonlinear eigenvalue problems and obtained the existence of an unbounded connected branch of positive solution set by using global bifurcation theorems (see [29, 30]). The terms Open image in new window in [26] and Open image in new window in [27] are not singular at Open image in new window , Open image in new window , Open image in new window . Yao [14] obtained one or two positive solutions to a singular elastic beam equation rigidly fixed at both ends by using Guo-Krasnosel'skii's fixed point theorem, but the global structure of positive solutions was not considered. Since the nonlinearity Open image in new window in BVP (1.1) may be singular at Open image in new window and/or Open image in new window , the global bifurcation theorems in [29, 30] do not apply to our problem here. In Section 4, we also investigate the global structure of positive solutions for BVP (1.1) by applying the following Lemma 1.2.

The paper is organized as follows: in the rest of this section, two known results are stated. In Section 2, some lemmas are stated and proved. In Section 3, we establish a necessary and sufficient condition for the existence of positive solutions. In Section 4, we prove that the closure of positive solution set possesses an unbounded connected branch which comes from Open image in new window .

Finally we state the following results which will be used in Sections 3 and 4, respectively.

Lemma 1.1 (see [31]).

Let Open image in new window be a real Banach space, let Open image in new window be a cone in Open image in new window , and let Open image in new window , Open image in new window be bounded open sets of Open image in new window , Open image in new window . Suppose that Open image in new window is completely continuous such that one of the following two conditions is satisfied:

Open image in new window

Then, Open image in new window has a fixed point in Open image in new window .

Lemma 1.2 (see [32]).

Suppose also that Open image in new window is a family of connected subsets of Open image in new window , satisfying the following conditions:

Open image in new window and Open image in new window for each Open image in new window .

(2)For any two given numbers Open image in new window and Open image in new window with Open image in new window , Open image in new window is a relatively compact set of Open image in new window .

Then there exists a connected branch Open image in new window of Open image in new window such that

where Open image in new window there exists a sequence Open image in new window such that Open image in new window .

2. Some Preliminaries and Lemmas

It is easy to conclude that Open image in new window is a cone of Open image in new window . Denote
Then Open image in new window is the Green function of homogeneous boundary value problem

Lemma 2.1.

Open image in new window , Open image in new window , and Open image in new window have the following properties:

(1) Open image in new window , Open image in new window , Open image in new window , for all Open image in new window .

(2) Open image in new window , Open image in new window , Open image in new window (or Open image in new window ), for all Open image in new window .

(3) Open image in new window , Open image in new window , Open image in new window , for all Open image in new window .

(4) Open image in new window , Open image in new window , Open image in new window , for all Open image in new window .

Proof.

From (2.4), it is easy to obtain the property (2.18).

We now prove that property (2) is true. For Open image in new window , by (2.4), we have

Consequently, property (2) holds.

From property (2), it is easy to obtain property (3).

We next show that property (4) is true. From (2.4), we know that property (4) holds for Open image in new window .

Therefore, property (4) holds.

Lemma 2.2.

Proof.

Therefore, (2.9) holds. From (2.9), we get
By (2.9) and the definition of Open image in new window , we can obtain that

Thus, (2.10) holds.

Then, it is easy to know that

Lemma 2.3.

hold. Then Open image in new window .

Proof.

From ( Open image in new window ), for any Open image in new window , Open image in new window , Open image in new window , we easily obtain the following inequalities:
For every Open image in new window , Open image in new window , choose positive numbers Open image in new window min Open image in new window . It follows from ( Open image in new window ), (2.10), Lemma 2.1, and (2.17) that
Similar to (2.19), from ( Open image in new window ), (2.10), Lemma 2.1, and (2.17), for every Open image in new window , Open image in new window , we have

Thus, Open image in new window is well defined on Open image in new window .

From (2.4) and (2.14)–(2.16), it is easy to know that

Therefore, Open image in new window follows from (2.21).

Obviously, Open image in new window is a positive solution of BVP (1.1) if and only if Open image in new window is a positive fixed point of the integral operator Open image in new window in Open image in new window .

Lemma 2.4.

Suppose that ( Open image in new window ) and (2.17) hold. Then for any Open image in new window , Open image in new window is completely continuous.

Proof.

First of all, notice that Open image in new window maps Open image in new window into Open image in new window by Lemma 2.3.

Next, we show that Open image in new window is bounded. In fact, for any Open image in new window , by (2.10) we can get
Choose positive numbers Open image in new window , Open image in new window , Open image in new window . This, together with ( Open image in new window ), (2.22), (2.16), and Lemma 2.1 yields that

Thus, Open image in new window is bounded on Open image in new window .

Now we show that Open image in new window is a compact operator on Open image in new window . By (2.23) and Ascoli-Arzela theorem, it suffices to show that Open image in new window is equicontinuous for arbitrary bounded subset Open image in new window .

Since for each Open image in new window , (2.22) holds, we may choose still positive numbers Open image in new window , Open image in new window , Open image in new window . Then

From (2.25), (2.26), and the absolute continuity of integral function, it follows that Open image in new window is equicontinuous.

Therefore, Open image in new window is relatively compact, that is, Open image in new window is a compact operator on Open image in new window .

By (2.17), we know that Open image in new window is integrable on Open image in new window . Thus, from the Lebesgue dominated convergence theorem, it follows that

Thus, Open image in new window is continuous on Open image in new window . Therefore, Open image in new window is completely continuous.

3. A Necessary and Sufficient Condition for Existence of Positive Solutions

In this section, by using the fixed point theorem of cone, we establish the following necessary and sufficient condition for the existence of positive solutions for BVP (1.1).

Theorem 3.1.

Suppose ( Open image in new window ) holds, then BVP (1.1) has at least one positive solution for any Open image in new window if and only if the integral inequality (2.17) holds.

Proof.

Suppose first that Open image in new window be a positive solution of BVP (1.1) for any fixed Open image in new window . Then there exist constants Open image in new window ( Open image in new window ) with Open image in new window , Open image in new window such that
On the other hand,

Let Open image in new window let Open image in new window and let Open image in new window then (3.1) holds.

Choose positive numbers Open image in new window , Open image in new window , Open image in new window . This, together with ( Open image in new window ), (1.2), and (2.18) yields that
where Open image in new window . Hence, integrating (3.4) from Open image in new window to 1, we obtain
Notice that Open image in new window , integrating (3.7) from 0 to 1, we have
By an argument similar to the one used in deriving (3.5), we can obtain
Integrating (3.12) from 0 to 1, we have
That is,

This and (3.10) imply that (2.17) holds.

Now assume that (2.17) holds, we will show that BVP (1.1) has at least one positive solution for any Open image in new window . By (2.17), there exists a sufficient small Open image in new window such that
For any fixed Open image in new window , first of all, we prove

where Open image in new window .

From Lemma 2.1, (3.18), and ( Open image in new window ), we get

Thus, (3.17) holds.

Next, we claim that

where Open image in new window .

Therefore, by Lemma 2.1 and ( Open image in new window ), it follows that

This implies that (3.20) holds.

By Lemmas 1.1 and 2.4, (3.17), and (3.20), we obtain that Open image in new window has a fixed point in Open image in new window . Therefore, BVP (1.1) has a positive solution in Open image in new window for any Open image in new window .

4. Unbounded Connected Branch of Positive Solutions

In this section, we study the global continua results under the hypotheses ( Open image in new window ) and (2.17). Let

then, by Theorem 3.1, Open image in new window for any Open image in new window .

Theorem 4.1.

Suppose ( Open image in new window ) and (2.17) hold, then the closure Open image in new window of positive solution set possesses an unbounded connected branch Open image in new window which comes from Open image in new window such that

(i)for any Open image in new window , and

(ii) Open image in new window

Proof.

We now prove our conclusion by the following several steps.

First, we prove that for arbitrarily given Open image in new window is bounded. In fact, let
Therefore, by Lemma 2.1 and ( Open image in new window ), it follows that
Therefore, by Lemma 2.1 and ( Open image in new window ), it follows that

Therefore, Open image in new window has no positive solution in Open image in new window . As a consequence, Open image in new window is bounded.

By the complete continuity of Open image in new window , Open image in new window is compact.

Second, we choose sequences Open image in new window and Open image in new window satisfy
We are to prove that for any positive integer Open image in new window , there exists a connected branch Open image in new window of Open image in new window satisfying

If Open image in new window , then taking Open image in new window .

It is obvious that in Open image in new window , the family of Open image in new window makes up an open covering of Open image in new window . Since Open image in new window is a compact set, there exists a finite subfamily Open image in new window which also covers Open image in new window . Let Open image in new window , then
Hence, by the homotopy invariance of the fixed point index, we obtain
By the first step of this proof, the construction of Open image in new window , (4.4), and (4.7), it follows easily that there exist Open image in new window such that

However, by the excision property and additivity of the fixed point index, we have from (4.12) and (4.14) that Open image in new window , which contradicts (4.15). Hence, there exists some Open image in new window such that the connected branch Open image in new window of Open image in new window containing Open image in new window satisfies that Open image in new window . Let Open image in new window be the connected branch of Open image in new window including Open image in new window , then this Open image in new window satisfies (4.9).

By Lemma 1.2, there exists a connected branch Open image in new window of Open image in new window such that Open image in new window for any Open image in new window . Noticing Open image in new window , we have Open image in new window . Let Open image in new window be the connected branch of Open image in new window including Open image in new window , then Open image in new window for any Open image in new window . Similar to (4.4) and (4.7), for any Open image in new window , Open image in new window , we have, by ( Open image in new window ), (4.2), (4.3), (4.5), (4.6), and Lemma 2.1,
where Open image in new window is given by (3.16). Let Open image in new window in (4.16) and Open image in new window in (4.17), we have

Therefore, Theorem 4.1 holds and the proof is complete.

Notes

Acknowledgments

This work is carried out while the author is visiting the University of New England. The author thanks Professor Yihong Du for his valuable advices and the Department of Mathematics for providing research facilities. The author also thanks the anonymous referees for their carefully reading of the first draft of the manuscript and making many valuable suggestions. Research is supported by the NSFC (10871120) and HESTPSP (J09LA08).

References

  1. 1.
    Agarwal RP, O'Regan D: Nonlinear superlinear singular and nonsingular second order boundary value problems. Journal of Differential Equations 1998, 143(1):60-95. 10.1006/jdeq.1997.3353CrossRefMathSciNetGoogle Scholar
  2. 2.
    Liu L, Kang P, Wu Y, Wiwatanapataphee B: Positive solutions of singular boundary value problems for systems of nonlinear fourth order differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(3):485-498. 10.1016/j.na.2006.11.014CrossRefMathSciNetGoogle Scholar
  3. 3.
    O'Regan D: Theory of Singular Boundary Value Problems. World Scientific, River Edge, NJ, USA; 1994:xii+154.CrossRefGoogle Scholar
  4. 4.
    Zhang Y: Positive solutions of singular sublinear Emden-Fowler boundary value problems. Journal of Mathematical Analysis and Applications 1994, 185(1):215-222. 10.1006/jmaa.1994.1243CrossRefMathSciNetGoogle Scholar
  5. 5.
    Wei Z:Existence of positive solutions for Open image in new windowth-order singular sublinear boundary value problems. Journal of Mathematical Analysis and Applications 2005, 306(2):619-636. 10.1016/j.jmaa.2004.10.037CrossRefMathSciNetGoogle Scholar
  6. 6.
    Wei Z, Pang C:The method of lower and upper solutions for fourth order singular Open image in new window-point boundary value problems. Journal of Mathematical Analysis and Applications 2006, 322(2):675-692. 10.1016/j.jmaa.2005.09.064CrossRefMathSciNetGoogle Scholar
  7. 7.
    Aftabizadeh AR: Existence and uniqueness theorems for fourth-order boundary value problems. Journal of Mathematical Analysis and Applications 1986, 116(2):415-426. 10.1016/S0022-247X(86)80006-3CrossRefMathSciNetGoogle Scholar
  8. 8.
    Agarwal RP: On fourth order boundary value problems arising in beam analysis. Differential and Integral Equations 1989, 2(1):91-110.MathSciNetGoogle Scholar
  9. 9.
    Bai Z: The method of lower and upper solutions for a bending of an elastic beam equation. Journal of Mathematical Analysis and Applications 2000, 248(1):195-202. 10.1006/jmaa.2000.6887CrossRefMathSciNetGoogle Scholar
  10. 10.
    Franco D, O'Regan D, Perán J: Fourth-order problems with nonlinear boundary conditions. Journal of Computational and Applied Mathematics 2005, 174(2):315-327. 10.1016/j.cam.2004.04.013CrossRefMathSciNetGoogle Scholar
  11. 11.
    Gupta CP: Existence and uniqueness theorems for the bending of an elastic beam equation. Applicable Analysis 1988, 26(4):289-304. 10.1080/00036818808839715CrossRefMathSciNetGoogle Scholar
  12. 12.
    Li Y: On the existence of positive solutions for the bending elastic beam equations. Applied Mathematics and Computation 2007, 189(1):821-827. 10.1016/j.amc.2006.11.144CrossRefMathSciNetGoogle Scholar
  13. 13.
    Yao Q: Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(5-6):1570-1580. 10.1016/j.na.2007.07.002CrossRefGoogle Scholar
  14. 14.
    Yao Q: Existence and multiplicity of positive solutions to a singular elastic beam equation rigidly fixed at both ends. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(8):2683-2694. 10.1016/j.na.2007.08.043CrossRefMathSciNetGoogle Scholar
  15. 15.
    Yao Q: Monotonically iterative method of nonlinear cantilever beam equations. Applied Mathematics and Computation 2008, 205(1):432-437. 10.1016/j.amc.2008.08.044CrossRefMathSciNetGoogle Scholar
  16. 16.
    Yao Q: Solvability of singular cantilever beam equation. Annals of Differential Equations 2008, 24(1):93-99.MathSciNetGoogle Scholar
  17. 17.
    Yao QL: Positive solution to a singular equation for a beam which is simply supported at left and clamped at right by sliding clamps. Journal of Yunnan University. Natural Sciences 2009, 31(2):109-113.MathSciNetGoogle Scholar
  18. 18.
    Yao QL: Existence and multiplicity of positive solutions to a class of nonlinear cantilever beam equations. Journal of Systems Science & Mathematical Sciences 2009, 29(1):63-69.Google Scholar
  19. 19.
    Yao QL: Positive solutions to a class of singular elastic beam equations rigidly fixed at both ends. Journal of Wuhan University. Natural Science Edition 2009, 55(2):129-133.MathSciNetGoogle Scholar
  20. 20.
    Yao Q: Existence of solution to a singular beam equation fixed at left and clamped at right by sliding clamps. Journal of Natural Science. Nanjing Normal University 2007, 9(1):1-5.MathSciNetGoogle Scholar
  21. 21.
    Graef JR, Kong L: A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(11):2389-2412. 10.1016/j.na.2006.03.028CrossRefMathSciNetGoogle Scholar
  22. 22.
    Xu Y, Li L, Debnath L: A necessary and sufficient condition for the existence of positive solutions of singular boundary value problems. Applied Mathematics Letters 2005, 18(8):881-889. 10.1016/j.aml.2004.07.029CrossRefMathSciNetGoogle Scholar
  23. 23.
    Zhao J, Ge W: A necessary and sufficient condition for the existence of positive solutions to a kind of singular three-point boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):3973-3980. 10.1016/j.na.2009.02.067CrossRefMathSciNetGoogle Scholar
  24. 24.
    Zhao ZQ: Positive solutions of boundary value problems for nonlinear singular differential equations. Acta Mathematica Sinica 2000, 43(1):179-188.MathSciNetGoogle Scholar
  25. 25.
    Zhao Z:On the existence of positive solutions for Open image in new window-order singular boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(11):2553-2561. 10.1016/j.na.2005.09.003CrossRefMathSciNetGoogle Scholar
  26. 26.
    Ma R, An Y: Global structure of positive solutions for nonlocal boundary value problems involving integral conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(10):4364-4376. 10.1016/j.na.2009.02.113CrossRefMathSciNetGoogle Scholar
  27. 27.
    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.2009.06.061CrossRefMathSciNetGoogle Scholar
  28. 28.
    Ma RY, Thompson B: Nodal solutions for a nonlinear fourth-order eigenvalue problem. Acta Mathematica Sinica 2008, 24(1):27-34. 10.1007/s10114-007-1009-6CrossRefMathSciNetGoogle Scholar
  29. 29.
    Dancer E: Global solutions branches for positive maps. Archive for Rational Mechanics and Analysis 1974, 55: 207-213. 10.1007/BF00281748CrossRefMathSciNetGoogle Scholar
  30. 30.
    Rabinowitz PH: Some aspects of nonlinear eigenvalue problems. The Rocky Mountain Journal of Mathematics 1973, 3(2):161-202. 10.1216/RMJ-1973-3-2-161CrossRefMathSciNetGoogle Scholar
  31. 31.
    Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.Google Scholar
  32. 32.
    Sun JX: A theorem in point set topology. Journal of Systems Science & Mathematical Sciences 1987, 7(2):148-150.Google Scholar

Copyright information

© Huiqin Lu. 2011

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.

Authors and Affiliations

  1. 1.School of Mathematical SciencesShandong Normal UniversityJinan, ShandongChina

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