 Research
 Open Access
 Published:
Existence of solutions for integral boundary value problems of secondorder ordinary differential equations
Boundary Value Problems volume 2012, Article number: 147 (2012)
Abstract
In this paper, we investigate the existence of solutions for some secondorder integral boundary value problems, by applying new fixed point theorems in Banach spaces with the lattice structure derived by Sun and Liu.
MSC:34B15, 34B18, 47H11.
1 Introduction
In this paper, we consider the following secondorder integral boundary value problem:
where f\in C([0,1]\times R,R), a\in L[0,1] is nonnegative with {\int}_{0}^{1}{a}^{2}(s)<1.
The multipoint boundary value problems for ordinary differential equations have been well studied, especially on a compact interval. For example, the study of threepoint boundary value problems for nonlinear secondorder ordinary differential equations was initiated by Gupta (see [1]). Since then, the existence of solutions for nonlinear multipoint boundary value problems has received much attention from some authors; see [2–6] for reference.
The integral boundary value problems of ordinary differential equations arise in different areas of applied mathematics and physics such as heat conduction, underground water flow, thermoelasticity and plasma physics (see [7, 8] and the references therein). Moreover, boundary value problems with RiemannStieltjes integral conditions constitute a very interesting and important class of problems. They include two, three, multipoint and integral boundary value problems as special cases (see [9, 10]). For boundary value problems with other integral boundary conditions, we refer the reader to the papers [11–21] and the references therein.
In [15], Zhang and Sun studied the following differential equation:
where f\in C([0,1]\times R,R), a\in L[0,1] is nonnegative with {\int}_{0}^{1}{a}^{2}(s)<1. By fixedpoint index theory, the existence and multiplicity of signchanging solutions was discussed.
As we know, nearly all the methods computing the topological degree depend on cone mappings. Recently, Sun and Liu introduced some new computation of topological degree when the concerned operators are not cone mappings in ordered Banach spaces with the lattice structure (for details, see [22–25]). To the best of our knowledge, there is only one paper to use this new computation of topological degree to study an integral boundary value problem with the asymptotically nonlinear term (see [16]).
Motivated by [15, 16, 22–25], this paper is concerned with the boundary value problem (1.1) under sublinear conditions. The method we use is based on some recent fixed point theorems derived by Sun and Liu [22, 23], which are different from [16] and the results we obtain are different from [11–21].
This paper is organized as follows. In Section 2, we recall some properties of the lattice, new fixed point theorems and some lemmas that will be used to prove the main results. In Section 3, we prove the main results and, finally, we give concrete examples to illustrate the applicability of our theory.
2 Preliminaries
We first give some properties of the lattice and give new fixed point theorems with the lattice structure (see [22–25]).
Let E be a Banach space with a cone P. Then E becomes an ordered Banach space under the partial ordering ≤ which is induced by P. P is said to be normal if there exists a positive constant N such that \theta \le x\le y implies \parallel x\parallel \le N\parallel y\parallel. P is called solid if it contains interior points, i.e., int P\ne \theta.
We call E a lattice under the partial ordering ≤ if sup\{x,y\} and inf\{x,y\} exist for arbitrary x,y\in E.
For x\in E, let
{x}^{+} and {x}^{} are called the positive part and the negative part of x, respectively, and obviously x={x}^{+}{x}^{}. Take x={x}^{+}+{x}^{}, then x\in P. For convenience, we use the notations {x}_{+}={x}^{+}, {x}_{}={x}^{}.
Let B:E\to E be a bounded linear operator. B is said to be positive if B(P)\subset P.
Let D\subset E and A:D\to E be a nonlinear operator. A is said to be quasiadditive on lattice if there exists {v}^{\ast}\in E such that
Let P be a cone of a Banach space E. x is said to be a positive fixed point of A if x\in (P\mathrm{\setminus}\{\theta \}) is a fixed point of A; x is said to be a negative fixed point of A if x\in ((P)\mathrm{\setminus}\{\theta \}) is a fixed point of A; x is said to be a signchanging fixed point of A if x\notin (P\cup (P)) is a fixed point of A.
Let P be a normal cone of E, and A:E\to E be completely continuous and quasiadditive on lattice. Suppose that

(i)
there exist a positive bounded linear operator {B}_{1}, {u}^{\ast}\in P and {u}_{1}\in P such that
{u}^{\ast}\le Ax\le {B}_{1}x+{u}_{1},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in P; 
(ii)
there exist a positive bounded linear operator {B}_{2} and {u}_{2}\in P such that
Ax\ge {B}_{2}x{u}_{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in (P); 
(iii)
r({B}_{1})<1, r({B}_{2})<1, where r({B}_{i}) is the spectral radius of {B}_{i} (i=1,2);

(iv)
A\theta =\theta, the Fréchet derivative {A}_{\theta}^{\prime} of A at θ exists, and 1 is not an eigenvalue of the linear operator {A}_{\theta}^{\prime}, the sum μ of the algebraic multiplicities for all eigenvalues of {A}_{\theta}^{\prime} lying in (1,+\mathrm{\infty}) is an odd number.
Then the operator A has at least one nonzero fixed point.
Lemma 2.2 [22]
Let the conditions (i), (ii) and (iii) of Lemma 2.1 be satisfied. Suppose, in addition, that A\theta =\theta; the Fréchet derivative {A}_{\theta}^{\prime} of A at θ exists; 1 is not an eigenvalue of the linear operator {A}_{\theta}^{\prime}; the sum μ of the algebraic multiplicities for all eigenvalues of {A}_{\theta}^{\prime} lying in (1,+\mathrm{\infty}) is an even number and
Then the operator A has at least three fixed points: one positive fixed point, one negative fixed point and one signchanging fixed point.
Let E=C[0,1] with the normal \parallel x(t)\parallel ={max}_{t\in [0,1]}x(t), then E is a Banach space. Let P=\{x\in E:x(t)\ge 0,t\in [0,1]\}, then P is a cone of E. It is easy to see that E is a lattice under the partial ordering ≤ that is induced by P.
For convenience, list the following condition.
(H_{0})
is the sequence of positive solutions of the equation sin\sqrt{u}={\int}_{0}^{1}a(s)sin(\sqrt{u}s)\phantom{\rule{0.2em}{0ex}}ds.
Define the operators A, B and F:
where
It is obvious that the fixed points of the operator A defined by (2.3) are the solutions of the boundary value problem (1.1) (see [15, 16]).
Lemma 2.3 [16]

(i)
B:E\to E is a completely continuous linear operator;

(ii)
A:E\to E is a completely continuous operator;

(iii)
A=BF is quasiadditive on the lattice;

(iv)
the eigenvalues of the linear operator B are \{\frac{1}{{\lambda}_{n}},n=1,2,\dots \} and the algebraic multiplicity of \frac{1}{{\lambda}_{n}} is equal to 1, where {\lambda}_{n} is defined by (H_{0});

(v)
r(B)=\frac{1}{{\lambda}_{1}}, where r(B) is the spectral radius of the operator B.
3 Main results
Let us list some conditions for convenience.
(H_{1}) There exists a constant b>0 such that
(H_{2}) There exists a constant \u03f5>0 such that
where {\lambda}_{1} is defined by (H_{0}).
(H_{3}) f(t,0)=0 uniformly on t\in [0,1].
(H_{4})
where {\lambda}_{n}<\lambda <{\lambda}_{n+1}, {\lambda}_{n}, {\lambda}_{n+1} is defined by (H_{0}).
Theorem 3.1 Suppose that (H_{0}), (H_{1}), (H_{2}), (H_{3}), (H_{4}) are satisfied and n is an odd number in (H_{4}). Then the boundary value problem (1.1) has at least a nontrivial solution.
Proof Choose 0<\delta <\u03f5, then h={\lambda}_{1}\u03f5+\delta <{\lambda}_{1}. By (H_{2}), there exists a constant M>0 such that
So, by (3.1) and (H_{1}), we know that
where C={sup}_{0\le t\le 1,x<M}f(t,x).
By (3.2) and (3.3), we have
where {C}_{1}=C{\int}_{0}^{1}G(t,s)\phantom{\rule{0.2em}{0ex}}ds, {C}^{\ast}=b{\int}_{0}^{1}G(t,s)\phantom{\rule{0.2em}{0ex}}ds, t\in [0,1], \overline{B}=hB, B is defined by (2.1). Obviously, {C}_{1}\in P, {C}^{\ast}\in P, \overline{B}:P\to P is a positive completely continuous operator. By Lemma 2.3, we have r(B)=\frac{1}{{\lambda}_{1}}, so r(\overline{B})=hr(B)<{\lambda}_{1}r(B)=1.
By (H_{3}), we have A\theta =\theta, and
i.e., {A}_{\theta}^{\prime}=\lambda B. By Lemma 2.3, 1 is not an eigenvalue of the linear operator {A}_{\theta}^{\prime}. Since {\lambda}_{n}<\lambda <{\lambda}_{n+1}, n is an odd number, the sum of the algebraic multiplicities for all eigenvalues of {A}_{\theta}^{\prime} lying in (1,+\mathrm{\infty}) is an odd number. By Lemma 2.1, the operator A has at least one nonzero fixed point. So, the boundary value problem (1.1) has at least one nontrivial solution. □
Theorem 3.2 Suppose (H_{0}), (H_{2}), (H_{3}), (H_{4}) are satisfied and n is an even number in (H_{4}). In addition, assume that
Then the boundary value problem (1.1) has at least three nontrivial solutions: one positive solution, one negative solution and one signchanging solution.
Proof By (3.7), we have
By (3.1) and (3.8), (3.4) and (3.5) hold. From (H_{3}), (3.6) holds, and by Lemma 2.3, 1 is not an eigenvalue of the linear operator {A}_{\theta}^{\prime}. Since {\lambda}_{n}<\lambda <{\lambda}_{n+1}, n is an even number, the sum of the algebraic multiplicities for all eigenvalues of {A}_{\theta}^{\prime} lying in (1,+\mathrm{\infty}) is an even number.
Obviously, from (3.8) and (2.2), we easily get
From (2.1), we easily know that B(P\mathrm{\setminus}\{\theta \})\subset P\mathrm{\setminus}\{\theta \}, B((P)\mathrm{\setminus}\{\theta \})\subset (P)\mathrm{\setminus}\{\theta \}.
So, by (3.9), we have
By Lemma 2.2, the boundary value problem (1.1) has at least three nontrivial solutions containing a positive solution, a negative solution and a signchanging solution. □
Remark By Theorem 3.1 and Theorem 3.2, we can see that the methods used in this paper are different from [11–21], and the results are different from [11–21].
Example 3.1 We consider the following integral boundary value problem:
where
By simple calculations, we get that {\lambda}_{1}\approx 7.53, {\lambda}_{2}\approx 37.41, {\lambda}_{3}\approx 86.80, \lambda =10. So, {\lambda}_{1}<\lambda <{\lambda}_{2}. It is easy to know that the nonlinear term f satisfies (H_{1}), (H_{2}), (H_{3}), (H_{4}). Thus, the boundary value problem (3.10) has at least a nontrivial solution by Theorem 3.1.
Example 3.2 We consider the following integral boundary value problem:
where
By simple calculations, we get that {\lambda}_{1}\approx 7.53, {\lambda}_{2}\approx 37.41, {\lambda}_{3}\approx 86.80, \lambda =40. So {\lambda}_{2}<\lambda <{\lambda}_{3}. It is easy to know that the nonlinear term f satisfies (H_{2}), (H_{3}), (H_{4}) and f(t,x)x>0, \mathrm{\forall}t\in [0,1], x\ne 0. The boundary value problem (3.11) has at least three nontrivial solutions containing a positive solution, a negative solution and a signchanging solution by Theorem 3.2.
References
Gupta CP: Solvability of a threepoint nonlinear boundary value problem for a second order ordinary differential equations. J. Math. Anal. Appl. 1992, 168: 540551. 10.1016/0022247X(92)90179H
Zhang GW, Sun JX: Multiple positive solutions of singular second order threepoint boundary value problems. J. Math. Anal. Appl. 2006, 317: 442447. 10.1016/j.jmaa.2005.08.020
Zhang GW, Sun JX: Positive solutions of m point boundary value problems. J. Math. Anal. Appl. 2004, 291: 406418. 10.1016/j.jmaa.2003.11.034
Xu X, Sun JX: On signchanging solution for some threepoint boundary value problems. Nonlinear Anal. 2004, 59: 491505.
Ma RY: Nodal solutions for a secondorder m point boundary value problem. Czechoslov. Math. J. 2006, 56: 12431263. 10.1007/s1058700600927
Zhang KM, Xie XJ: Existence of signchanging solutions for some asymptotically linear threepoint boundary value problems. Nonlinear Anal. 2009, 70: 27962805. 10.1016/j.na.2008.04.004
Gallardo JM: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 2000, 30: 12651292. 10.1216/rmjm/1021477351
Timoshenko S: Theory of Elastice Stability. McGrawHill, New York; 1961.
Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 2006, 74: 673693. 10.1112/S0024610706023179
Karakostas GL, Tsamatos PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundaryvalue problems. Electron. J. Differ. Equ. 2002, 30: 117.
Yang ZL: Existence of nontrivial solutions for a nonlinear SturmLiouville problem with integral boundary value conditions. Nonlinear Anal. 2008, 68: 216225. 10.1016/j.na.2006.10.044
Jankowski T: Differential equations with integral boundary conditions. J. Comput. Appl. Math. 2002, 147: 18. 10.1016/S03770427(02)003710
Yang ZL: Positive solutions of a second order integral boundary value problem. J. Math. Anal. Appl. 2006, 321: 751765. 10.1016/j.jmaa.2005.09.002
Li Y, Li F: Signchanging solutions for secondorder integral boundary value problems. Nonlinear Anal. 2008, 69: 11791187. 10.1016/j.na.2007.06.024
Zhang XQ, Sun JX: On multiple signchanging solutions for some secondorder integral boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2010, 44: 115.
Li HT, Liu YS: On signchanging solutions for a secondorder integral boundary value problem. Comput. Math. Appl. 2011, 62: 651656. 10.1016/j.camwa.2011.05.046
Zhang XM, Feng MQ, Ge WG: Existence of solutions of boundary value problems with integral boundary conditions for secondorder impulsive integrodifferential equations in Banach spaces. J. Comput. Appl. Math. 2010, 233: 19151926. 10.1016/j.cam.2009.07.060
Feng MQ, Zhang XM, Ge WG: New existence results for higherorder nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 720702
Feng MQ, Ji DH, Ge WG: Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 2008, 222: 351363. 10.1016/j.cam.2007.11.003
Feng MQ: Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions. Appl. Math. Lett. 2011, 24: 14191427. 10.1016/j.aml.2011.03.023
Feng MQ, Pang HH: A class of threepoint boundaryvalue problems for secondorder impulsive integrodifferential equations in Banach spaces. Nonlinear Anal. 2009, 70: 6482. 10.1016/j.na.2007.11.033
Sun JX: Nonlinear Functional Analysis and Applications. Science Press, Beijing; 2008.
Sun JX, Liu XY: Computation of topological degree for nonlinear operators and applications. Nonlinear Anal. 2008, 69: 41214130. 10.1016/j.na.2007.10.042
Sun JX, Liu XY: Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations. J. Math. Anal. Appl. 2008, 348: 927937. 10.1016/j.jmaa.2008.05.023
Liu XY, Sun JX: Computation of topological degree of unilaterally asymptotically linear operators and its applications. Nonlinear Anal. 2009, 71: 96106. 10.1016/j.na.2008.10.032
Acknowledgements
The authors would like to thank the reviewers for carefully reading this article and making valuable comments and suggestions. The project is supported by the National Natural Science Foundation of P.R. China (10971179), Research Award Fund for Outstanding Young Scientists of Shandong Province (BS2012SF022, BS2010SF023), Natural Science Foundation of Shandong Province (ZR2010AM035) and SDUST CISE Research Fund.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, H., Sun, F. Existence of solutions for integral boundary value problems of secondorder ordinary differential equations. Bound Value Probl 2012, 147 (2012). https://doi.org/10.1186/168727702012147
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/168727702012147
Keywords
 lattice
 fixed point
 integral boundary value problem
 signchanging solution