- Open Access
Existence of solutions for integral boundary value problems of second-order ordinary differential equations
© Li and Sun; licensee Springer 2012
- Received: 24 June 2012
- Accepted: 26 November 2012
- Published: 17 December 2012
In this paper, we investigate the existence of solutions for some second-order 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.
- fixed point
- integral boundary value problem
- sign-changing solution
where , is nonnegative with .
The multi-point boundary value problems for ordinary differential equations have been well studied, especially on a compact interval. For example, the study of three-point boundary value problems for nonlinear second-order ordinary differential equations was initiated by Gupta (see ). Since then, the existence of solutions for nonlinear multi-point 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, thermo-elasticity and plasma physics (see [7, 8] and the references therein). Moreover, boundary value problems with Riemann-Stieltjes integral conditions constitute a very interesting and important class of problems. They include two, three, multi-point 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.
where , is nonnegative with . By fixed-point index theory, the existence and multiplicity of sign-changing 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 ).
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  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.
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 implies . P is called solid if it contains interior points, i.e., int .
We call E a lattice under the partial ordering ≤ if and exist for arbitrary .
and are called the positive part and the negative part of x, respectively, and obviously . Take , then . For convenience, we use the notations , .
Let be a bounded linear operator. B is said to be positive if .
Let P be a cone of a Banach space E. x is said to be a positive fixed point of A if is a fixed point of A; x is said to be a negative fixed point of A if is a fixed point of A; x is said to be a sign-changing fixed point of A if is a fixed point of A.
- (i)there exist a positive bounded linear operator , and such that
- (ii)there exist a positive bounded linear operator and such that
, , where is the spectral radius of ();
, the Fréchet derivative of A at θ exists, and 1 is not an eigenvalue of the linear operator , the sum μ of the algebraic multiplicities for all eigenvalues of lying in is an odd number.
Then the operator A has at least one nonzero fixed point.
Lemma 2.2 
Then the operator A has at least three fixed points: one positive fixed point, one negative fixed point and one sign-changing fixed point.
Let with the normal , then E is a Banach space. Let , 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.
is the sequence of positive solutions of the equation .
Lemma 2.3 
is a completely continuous linear operator;
is a completely continuous operator;
is quasi-additive on the lattice;
the eigenvalues of the linear operator B are and the algebraic multiplicity of is equal to 1, where is defined by (H0);
, where is the spectral radius of the operator B.
Let us list some conditions for convenience.
where is defined by (H0).
(H3) uniformly on .
where , , is defined by (H0).
Theorem 3.1 Suppose that (H0), (H1), (H2), (H3), (H4) are satisfied and n is an odd number in (H4). Then the boundary value problem (1.1) has at least a nontrivial solution.
where , , , , B is defined by (2.1). Obviously, , , is a positive completely continuous operator. By Lemma 2.3, we have , so .
i.e., . By Lemma 2.3, 1 is not an eigenvalue of the linear operator . Since , n is an odd number, the sum of the algebraic multiplicities for all eigenvalues of lying in 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. □
Then the boundary value problem (1.1) has at least three nontrivial solutions: one positive solution, one negative solution and one sign-changing solution.
By (3.1) and (3.8), (3.4) and (3.5) hold. From (H3), (3.6) holds, and by Lemma 2.3, 1 is not an eigenvalue of the linear operator . Since , n is an even number, the sum of the algebraic multiplicities for all eigenvalues of lying in is an even number.
From (2.1), we easily know that , .
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 sign-changing solution. □
By simple calculations, we get that , , , . So, . It is easy to know that the nonlinear term f satisfies (H1), (H2), (H3), (H4). Thus, the boundary value problem (3.10) has at least a nontrivial solution by Theorem 3.1.
By simple calculations, we get that , , , . So . It is easy to know that the nonlinear term f satisfies (H2), (H3), (H4) and , , . The boundary value problem (3.11) has at least three nontrivial solutions containing a positive solution, a negative solution and a sign-changing solution by Theorem 3.2.
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.
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