- Open Access
Nontrivial solutions for a boundary value problem with integral boundary conditions
© Liu et al.; licensee Springer. 2014
- Received: 27 July 2013
- Accepted: 13 November 2013
- Published: 13 January 2014
This paper concerns the existence of nontrivial solutions for a boundary value problem with integral boundary conditions by topological degree theory. Here the nonlinear term is a sign-changing continuous function and may be unbounded from below.
- Nonlinear Term
- Nontrivial Solution
- Continuous Operator
- Real Banach Space
- Positive Linear Operator
where , , , , , α and β are right continuous on , left continuous at and nondecreasing on with ; , and denote the Riemann-Stieltjes integral of u with respect to α and β, respectively. Here the nonlinear term is a continuous sign-changing function and f may be unbounded from below, with is continuous and is allowed to be singular at .
Problems with integral boundary conditions arise naturally in thermal conduction problems , semiconductor problems , hydrodynamic problems . Integral BCs (BCs denotes boundary conditions) cover multi-point BCs and nonlocal BCs as special cases and have attracted great attention, see [4–14] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu , Agarwal and O’Regan . Yang , Boucherif , Chamberlain et al. , Feng , Jiang et al.  focused on the existence of positive solutions for the cases in which the nonlinear term is nonnegative. Although many papers investigated two-point and multi-point boundary value problems with sign-changing nonlinear terms, for example, [15–20], results for boundary value problems with integral boundary conditions when the nonlinear term is sign-changing are rarely seen except for a few special cases [7, 12, 13].
Inspired by the above papers, the aim of this paper is to establish the existence of nontrivial solutions to BVP (1.1) under weaker conditions. Our findings presented in this paper have the following new features. Firstly, the nonlinear term f of BVP (1.1) is allowed to be sign-changing and unbounded from below. Secondly, the boundary conditions in BVP (1.1) are the Riemann-Stieltjes integral, which includes multi-point boundary conditions in BVPs as special cases. Finally, the main technique used here is the topological degree theory, the first eigenvalue and its positive eigenfunction corresponding to a linear operator. This paper employs different conditions and different methods to solve the same BVP (1.1) as ; meanwhile, this paper generalizes the result in  to boundary value problems with integral boundary conditions. What we obtain here is different from [6–20].
We assume that the following condition holds throughout this paper.
Let , , , .
(H2) , , .
Lemma 2.1 ()
where is the Green function for (1.1).
It is easy to show that is a completely continuous nonlinear operator, and if is a fixed point of A, then u is a solution of BVP (1.1) by Lemma 2.1.
Lemma 2.2 ()
If (H1) holds, then there is such that is a subcone of P and .
Lemma 2.3 ()
Let E be a real Banach space and be a bounded open set with . Suppose that is a completely continuous operator. (1) If there is with such that for all and , then . (2) If for all and , then . Here deg stands for the Leray-Schauder topological degree in E.
Lemma 2.4 Assume that (H1), (H2) and the following assumptions are satisfied:
(C1) There exist , and such that (2.3), (2.4) hold and K maps P into .
(C3) There exist a bounded continuous operator and such that for all .
(C4) There exist and such that for all .
provided that R is sufficiently large.
where is a constant.
where is a constant.
Theorem 3.1 Assume that (H1), (H2) hold and the following conditions are satisfied:
(A1) There exist two nonnegative functions with and one continuous even function such that for all . Moreover, B is nondecreasing on and satisfies .
(A2) is continuous.
(A3) uniformly on .
(A4) uniformly on .
Here is the first eigenvalue of the operator K defined by (2.2).
Then BVP (1.1) has at least one nontrivial solution.
that is, . Take , for any , where . Obviously, holds. Therefore H satisfies condition (C2) in Lemma 2.4.
which shows that condition (C3) in Lemma 2.4 holds.
So condition (C4) in Lemma 2.4 is satisfied.
If there exist and such that . Let . Then and by (3.4), . The n th iteration of this inequality shows that (), so , that is, . This yields , which is a contradictory inequality. Hence, (3.5) holds.
So A has at least one fixed point on , namely, BVP (1.1) has at least one nontrivial solution. □
Corollary 3.1 Using () instead of (A1), the conclusion of Theorem 3.1 remains true.
The first two authors were supported financially by the National Natural Science Foundation of China (11201473, 11271364) and the Fundamental Research Funds for the Central Universities (2013QNA35, 2010LKSX09, 2010QNA42). The third author was supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
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