Multiple monotone positive solutions for higher order differential equations with integral boundary conditions
© Hao and Liu; licensee Springer. 2014
Received: 17 January 2014
Accepted: 19 March 2014
Published: 28 March 2014
This paper investigates the higher order differential equations with nonlocal boundary conditions
The existence results of multiple monotone positive solutions are obtained by means of fixed point index theory for operators in a cone.
where , is continuous in which , A and B are right continuous on , left continuous at , and nondecreasing on , with ; and denote the Riemann-Stieltjes integrals of v with respect to A and B, respectively.
where , (), , . The authors obtained the existence, nonexistence and multiplicity of positive solutions by using the Krasnosel’skii-Guo fixed point theorem, the upper-lower solutions method and topological degree theory.
The arguments are based upon a specially constructed cone and fixed point theory in a cone for strict set contraction operators.
Motivated by the works mentioned above, in this paper, we consider the existence of multiple monotone positive solutions for BVP (1.1) and (1.2). In comparison with previous works, our paper has several new features. Firstly, we consider higher order boundary value problems, and we allow the nonlinearity f to contain derivatives of the unknown function up to order. Secondly, we discuss the boundary value problem with integral boundary conditions, i.e., BVP (1.1) and (1.2), which includes two-point, three-point, multi-point and nonlocal boundary value problems as special cases. Thirdly, we consider the existence of multiple monotone positive solutions. To our knowledge, few papers have considered the monotone positive solutions for a higher order differential equation with integral boundary conditions. We shall emphasize here that with these new features our work improves and generalizes the results of  and some other known results to some degree. In this work we shall also utilize the following fixed point theorem in cones.
If , , then .
If there exists such that for all and , then .
Let U be open in E such that . If and , then A has a fixed point in . The same result holds if and .
2 Preliminary lemmas
Let , then E is a Banach space with the norm for each .
We make the following assumptions:
() is continuous.
Hence, (2.2) follows from (2.6) and (2.7).
Hence (2.6) follows from (2.2), (2.9) and (2.10), and thus , . This completes the proof. □
i.e., , therefore , and so .
From (2.14) and , we obtain (2.13). This completes the proof of Lemma 2.3. □
Remark 2.2 From Lemma 2.3, if u is a positive solution of BVP (1.1) and (1.2), then u is nondecreasing on , i.e., u is a monotone positive solution of BVP (1.1) and (1.2).
Then u is a solution of BVP (1.1) and (1.2) if and only if u solves the operator equation .
Lemma 2.4 Suppose that () and () hold, then is completely continuous.
Hence, and .
Next by standard methods and the Ascoli-Arzela theorem, one can prove that is completely continuous. So this is omitted. □
Proceeding as for the proof of Lemma 2.5 in , we have the following.
is open relative to K;
if and only if ;
if , then for .
To prove our main results, we need the following lemmas.
This implies that for . By the point (1) in Lemma 1.1, we have . □
This implies that , and so by the point (c) in Lemma 2.5, this is a contradiction. It follows from the point (2) of Lemma 1.1 that . □
3 Main results
In the following, we shall give the main results on the existence of multiple positive solutions of BVP (1.1) and (1.2).
Theorem 3.1 Suppose that () and () are satisfied. In addition, assume that one of the following conditions holds.
Then BVP (1.1) and (1.2) has two nondecreasing positive solutions , in K. Moreover, if in () is replaced by , then BVP (1.1) and (1.2) has a third nondecreasing positive solution .
Proof Assume that () holds. We show that either T has a fixed point or in . If for , by Lemmas 2.6 and 2.7, we have , , and . By Lemma 2.5(b), we have since . It follows from Lemma 1.1(3) that T has a fixed point . Similarly, T has a fixed point . The proof is similar when () holds. □
Corollary 3.1 If there exists such that one of the following conditions holds:
() , , for , ,
() , , for , ,
then BVP (1.1) and (1.2) has at least two nondecreasing positive solutions in K.
This implies that and () holds. Similarly, () implies (). This completes the proof. □
Remark 3.1 We establish the multiplicity of monotone positive solutions for a higher order differential equation with integral boundary conditions, and we allow the nonlinearity f to contain derivatives of the unknown function up to order, so our work improves and generalizes the results of  to some degree.
Research supported by the National Natural Science Foundation of China (11371221, 11201260), the Specialized Research Fund for the Doctoral Program of Higher Education of China (20123705120004, 20123705110001), a Project of Shandong Province Higher Educational Science and Technology Program (J11LA06) and Foundation of Qufu Normal University (BSQD20100103).
- Eloe PW, Ahmad B: Positive solutions of a nonlinear n th order boundary value problem with nonlocal conditions. Appl. Math. Lett. 2005, 18: 521-527. 10.1016/j.aml.2004.05.009MathSciNetView ArticleGoogle Scholar
- Graef JR, Yang B: Positive solutions to a multi-point higher order boundary value problem. J. Math. Anal. Appl. 2006, 316: 409-421. 10.1016/j.jmaa.2005.04.049MathSciNetView ArticleGoogle Scholar
- Graef JR, Henderson J, Wong PJY, Yang B: Three solutions of an n th order three-point focal type boundary value problem. Nonlinear Anal. 2008, 69: 3386-3404. 10.1016/j.na.2007.09.024MathSciNetView ArticleGoogle Scholar
- Hao X, Liu L, Wu Y: Positive solutions for second order differential systems with nonlocal conditions. Fixed Point Theory 2012, 13: 507-516.MathSciNetGoogle Scholar
- Hao X, Liu L, Wu Y: On positive solutions of m -point nonhomogeneous singular boundary value problem. Nonlinear Anal. 2010, 73: 2532-2540. 10.1016/j.na.2010.06.028MathSciNetView ArticleGoogle Scholar
- Henderson J, Luca R: On a system of second-order multi-point boundary value problems. Appl. Math. Lett. 2012, 25: 2089-2094. 10.1016/j.aml.2012.05.005MathSciNetView ArticleGoogle Scholar
- Karaca IY: Positive solutions of an n th order multi-point boundary value problem. J. Comput. Anal. Appl. 2012, 14: 181-193.MathSciNetGoogle Scholar
- Ma R: Existence of positive solutions for superlinear semipositone m -point boundary-value problems. Proc. Edinb. Math. Soc. 2003, 46: 279-292. 10.1017/S0013091502000391MathSciNetView ArticleGoogle Scholar
- Hao X, Xu N, Liu L: Existence and uniqueness of positive solutions for fourth-order m -point boundary value problems with two parameters. Rocky Mt. J. Math. 2013, 43: 1161-1180. 10.1216/RMJ-2013-43-4-1161MathSciNetView ArticleGoogle Scholar
- Zhang X: Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives. Appl. Math. Comput. 2008, 201: 361-370. 10.1016/j.amc.2007.12.031MathSciNetView ArticleGoogle Scholar
- Zhang X, Liu L: A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions. Appl. Math. Comput. 2010, 215: 3501-3508. 10.1016/j.amc.2009.10.044MathSciNetView ArticleGoogle Scholar
- Zhang X, Liu L: Positive solutions of four-order multi-point boundary value problems with bending term. Appl. Math. Comput. 2007, 194: 321-332. 10.1016/j.amc.2007.04.028MathSciNetView ArticleGoogle Scholar
- Gallardo JM: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 2000, 30: 1265-1292. 10.1216/rmjm/1021477351MathSciNetView ArticleGoogle Scholar
- Karakostas GL, Tsamatos PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002, 2002: 1-17.Google Scholar
- Lomtatidze A, Malaguti L: On an nonlocal boundary-value problems for second order nonlinear singular differential equations. Georgian Math. J. 2000, 7: 133-154.MathSciNetGoogle Scholar
- Boucherif A: Second-order boundary value problems with integral boundary conditions. Nonlinear Anal. 2009, 70: 364-371. 10.1016/j.na.2007.12.007MathSciNetView ArticleGoogle Scholar
- Feng M, Ji D, Ge W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 2008, 222: 351-363. 10.1016/j.cam.2007.11.003MathSciNetView ArticleGoogle Scholar
- Hao X, Liu L, Wu Y, Sun Q: Positive solutions for nonlinear n th-order singular eigenvalue problem with nonlocal conditions. Nonlinear Anal. 2010, 73: 1653-1662. 10.1016/j.na.2010.04.074MathSciNetView ArticleGoogle Scholar
- Hao X, Liu L, Wu Y, Xu N: Multiple positive solutions for singular n th-order nonlocal boundary value problem in Banach spaces. Comput. Math. Appl. 2011, 61: 1880-1890. 10.1016/j.camwa.2011.02.017MathSciNetView ArticleGoogle Scholar
- Infante G, Webb JRL: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 2006, 49: 637-656. 10.1017/S0013091505000532MathSciNetView ArticleGoogle Scholar
- Jiang J, Liu L, Wu Y: Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. Appl. Math. Comput. 2009, 215: 1573-1582. 10.1016/j.amc.2009.07.024MathSciNetView ArticleGoogle Scholar
- Kang P, Wei Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. Nonlinear Anal. 2008, 70: 444-451.MathSciNetView ArticleGoogle Scholar
- Kong L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 2010, 72: 2628-2638. 10.1016/j.na.2009.11.010MathSciNetView ArticleGoogle Scholar
- Ma H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Anal. 2008, 68: 645-651. 10.1016/j.na.2006.11.026MathSciNetView ArticleGoogle Scholar
- Webb JRL: Nonlocal conjugate type boundary value problems of higher order. Nonlinear Anal. 2009, 71: 1933-1940. 10.1016/j.na.2009.01.033MathSciNetView ArticleGoogle Scholar
- Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 2006, 74: 673-693. 10.1112/S0024610706023179MathSciNetView ArticleGoogle Scholar
- Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differ. Equ. Appl. 2008, 15: 45-67. 10.1007/s00030-007-4067-7MathSciNetView ArticleGoogle Scholar
- Liu L, Hao X, Wu Y: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model. 2013, 57: 836-847. 10.1016/j.mcm.2012.09.012MathSciNetView ArticleGoogle Scholar
- Yang Z: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Anal. 2006, 65: 1489-1511. 10.1016/j.na.2005.10.025MathSciNetView ArticleGoogle Scholar
- Yang Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions. Nonlinear Anal. 2008, 68: 216-225. 10.1016/j.na.2006.10.044MathSciNetView ArticleGoogle Scholar
- Zhang X, Feng M, Ge W: Symmetric positive solutions for p -Laplacian fourth-order differential equations with integral boundary conditions. J. Comput. Appl. Math. 2008, 222: 561-573. 10.1016/j.cam.2007.12.002MathSciNetView ArticleGoogle Scholar
- Zhang X, Han Y: Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. Appl. Math. Lett. 2012, 25: 555-560. 10.1016/j.aml.2011.09.058MathSciNetView ArticleGoogle Scholar
- Lan KQ: Multiple positive solutions of semilinear differential equations with singularities. J. Lond. Math. Soc. 2001, 63: 690-704. 10.1112/S002461070100206XView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.