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Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 614376, 7 pages
Global Structure of Positive Solutions for a Singular Fourth-Order Integral Boundary Value Problem
Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou 730050, China
Received 11 November 2013; Revised 14 December 2013; Accepted 16 December 2013; Published 8 January 2014
Academic Editor: Gabriele Bonanno
Copyright © 2014 Wenguo Shen and Tao He. 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 consider fourth-order boundary value problems , where is a Stieltjes integral with being nondecreasing and being not a constant on may be singular at and , with on any subinterval of ; and for all , and We investigate the global structure of positive solutions by using global bifurcation techniques.
Recently, fourth-order boundary value problem has been investigated by the fixed point theory in cones, see [1–4] (). By applying bifurcation techniques, see Rynne  (), Korman  (), Xu and Han  (), Shen [8, 9] (), and references therein. However, these papers only studied the nonsingular boundary value problems.
In 2008, Webb et al.  studied the existence of multiple positive solutions of nonlinear nonlocal boundary value problems (BVPs) for equations of the form where are continuous and nonnegative functions and is a function of bounded variation. They treat many boundary conditions appearing in the literature in a unified way. The main tool they used is the fixed point index theory in cones. In 2009, Ma and An  studied the global structure for second-order nonlocal boundary value problem involving Stieltjes integral conditions by applying bifurcation techniques.
Motivated by above papers, in this paper, we will use global bifurcation techniques to study the global structure of positive solutions of the singular problem where may be singular at and , and is a parameter.
In order to prove our main result, let us make the assumptions as follows:(A1) is nondecreasing and is not a constant on , for , and with ;(A2) with on any subinterval of , and ;(A3) satisfies for all ;(A4);(A5).
Remark 1. For other results on the existence and multiplicity of positive solutions and nodal solutions for the boundary value problems of fourth-order ordinary differential equations based on bifurcation techniques, see Ma et al. [12–15] and Bai and Wang  and their references.
The rest of the paper is arranged as follows: In Section 2, we state some properties of superior limit of certain infinity collection of connected sets. In Section 3, we will give some preliminary results. In Section 4, we state and prove our main results.
2. Superior Limit and Component
In order to treat the case , we will need the following definition and lemmas.
Definition 2 (see ). Let be a Banach space and let be a family of subsets of . Then the superior limit of is defined by
Lemma 3 (see ). Each connected subset of metric space is contained in a component, and each connected component of is closed.
Lemma 4 (see ). Let be a Banach space and let be a family of closed connected subsets of . Assume that(i)there exist , and , such that ;(ii); (iii)for all , is a relative compact set of , where Then there exists an unbounded connected component in and .
We consider the problem as follows:
Lemma 5. For any , the problem (6) has a unique solution where .
Let be the Banach space with the norm .
Let with the norm
Let and for , let .
In order to use bifurcation technique to study the problem (3), we consider the linear eigenvalue problem
By , it is easy to show the following lemma.
Lemma 9. Assume that (A1)–(A3) hold the following.
is a completely continuous linear operator and , and the fixed points of the operator in are the positive solutions of the BVP (23).
is a completely continuous operator and , and the fixed points of the operator in are the positive solutions of the BVP (3).
By virtue of Krein-Rutman theorem (Theorem 2.5 in ), one has (see  or ) the following lemma.
Lemma 10. Suppose that is a completely continuous linear operator and . If there exist and a constant such that , then the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue , that is, .
Lemma 11. Suppose (A1) and (A2) are satisfied, then for the operator defined by (24), the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .
Proof. It is easy to see that there is such that . Thus there exists such that and . Take such that , and . Then for , So there exists a constant such that . From Lemma 10, we know that the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .
Lemma 12. Let (A1)–(A3) hold. The solution of the problem (3) satisfies
Proof. From , there exists , such that . Using a similar proof of (10) in [21, page 212], it is easy to show that Furthermore, we obtain
Lemma 13. Let (A1)–(A3) hold. Assume that is a sequence of positive solutions of (3). Assume that for some constant , and Then
Proof. Assume on the contrary that
for some constant .
Since is a solution of the problem (3), we have Thus, where , (see ).
Furthermore, it follows that where , together with , which implies that is bounded whenever is bounded.
Together with Lemma 12, we obtain for some constant . This is a contradiction.
4. Main Results
Let be the closure of the set of positive solutions of (3) in . The main results of this paper are the following.
Theorem 14. Let (A1)–(A5) hold, then (3) has at least one solution for any .
Let be an operator defined by with Then is a closed operator and is completely continuous.
For each , define by Then with By , it follows that To apply the global bifurcation theorem, one extends to an odd function by Similarly one may extend to an odd function for each .
Now let one consider the auxiliary family of the equations Let be such that Then Let one consider as a bifurcation problem from the trivial solution .
From Lemma 5, (46) can be converted to the equivalent equation Further one has that
Indeed, (8) implies that, for all , where , , (see ).
So, the compactness of together with (45) and (A2) imply that and consequently
Let , , then . Let .
By Lemma 11 and the fact , the global bifurcation result (see Rabinowitz ) for (46) can be stated as follows: there exists a continuum of positive solutions of (46) joining to infinity in . Moreover, is the only positive bifurcation point of (46) lying on trivial solutions line . Moreover, .
Since condition (i) in Lemma 4 is satisfied with . Obviously and accordingly (ii) in Lemma 4 holds. (iii) in Lemma 4 can be deduced directly from the Arzela-Ascoli theorem and the definition of . Therefore, the superior limit of , , contained an unbounded connected component with . Since , one concludes . Moreover, by (3).
Proof of Theorem 14. We firstly prove
Assume on the contrary that Then there exists a sequence such that for some positive constant with doing not depend on . From Lemma 13, we have
This together with the fact imply that for arbitrary Since , we have that
Set . Then
Now, choosing a subsequence and relabeling if necessary, it follows that there exists with such that
By (A3), let Then is nondecreasing and Further it follows from (66) that
Thus, Notice that (62) is equivalent to
Furthermore, by (59), (68), and (69), together with the Lebesgue dominated convergence theorem, it follows that It follows that
This contradicts (63). Therefore
Noticing that is the only solution of the problem (3), thus
Furthermore, it follows the proof of Theorem 14.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Thanks are given to the anonymous referee for his/her valuable suggestions. The authors were supported by the NSFC (no. 11261052) and the Scientific Research Foundation of the Education department of Gansu Province (no. 1114-04).
- R. P. Agarwal and Y. M. Chow, “Iterative methods for a fourth order boundary value problem,” Journal of Computational and Applied Mathematics, vol. 10, no. 2, pp. 203–217, 1984.
- R. Ma and H. P. Wu, “Positive solutions of a fourth-order two-point boundary value problem,” Acta Mathematica Scientia A, vol. 22, no. 2, pp. 244–249, 2002.
- Q. Yao, “Multiple positive solutions to a singular beam equation fixed at both ends,” Acta Mathematica Scientia A, vol. 28, no. 4, pp. 768–778, 2008.
- Q. Yao, “Positive solutions for eigenvalue problems of fourth-order elastic beam equations,” Applied Mathematics Letters, vol. 17, no. 2, pp. 237–243, 2004.
- B. P. Rynne, “Infinitely many solutions of superlinear fourth order boundary value problems,” Topological Methods in Nonlinear Analysis, vol. 19, no. 2, pp. 303–312, 2002.
- P. Korman, “Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems,” Proceedings of the Royal Society of Edinburgh A, vol. 134, no. 1, pp. 179–190, 2004.
- J. Xu and X. Han, “Nodal solutions for a fourth-order two-Ooint boundary value problem,” Boundary Value Problem, vol. 2010, Article ID 570932, 11 pages, 2010.
- W. G. Shen, “Existence of nodal solutions of a nonlinear fourth-order two-point boundary value problem,” Boundary Value Problems, vol. 2012, p. 31, 2012.
- W. G. Shen, “Global structure of nodal solutions for a fourth-order two-point boundary value problem,” Applied Mathematics and Computation, vol. 219, no. 1, pp. 88–98, 2012.
- J. R. L. Webb, G. Infante, and D. Franco, “Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions,” Proceedings of the Royal Society of Edinburgh Section A, vol. 138, no. 2, pp. 427–446, 2008.
- R. Ma and Y. An, “Global structure of positive solutions for superlinear second order m-point boundary value problems,” Topological Methods in Nonlinear Analysis, vol. 34, no. 2, pp. 279–290, 2009.
- R. Ma, “Nodal solutions for a fourth-order two-point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 314, no. 1, pp. 254–265, 2006.
- R. Ma, “Nodal solutions of boundary value problems of fourth-order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 424–434, 2006.
- R. Ma and J. Xu, “Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 1, pp. 113–122, 2010.
- R. Ma, “Existence of positive solutions of a fourth-order boundary value problem,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1219–1231, 2005.
- Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations,” Journal of Mathematical Analysis and Applications, vol. 270, no. 2, pp. 347–368, 2002.
- G. T. Whyburn, Topological Analysis, Princeton Mathematical Series. No. 23, Princeton University Press, Princeton, NJ, USA, 1958.
- G. W. Zhang and J. X. Sun, “Positive solutions of m-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 406–418, 2004.
- M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Limited, Groningen, The Netherlands, 1964.
- D. J. Guo and J. X. Sun, Nonlinear Integral Equations, Shandong Science and Technology Press, Jinan, China, 1987.
- Y. J. Liu and W. G. Ge, “Positive solutions of two-point boundary value problems for 2n-order differential equations dependent on all derivatives,” Applied Mathematics Letters, vol. 18, no. 2, pp. 209–218, 2005.
- P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems,” Journal of Functional Analysis, vol. 7, no. 3, pp. 487–513, 1971.