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Nontrivial solutions for a boundary value problem with integral boundary conditions
Boundary Value Problemsvolume 2014, Article number: 15 (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.
Consider the following Sturm-Liouville problem with integral boundary conditions
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].
2 Preliminaries and lemmas
Let be a Banach space with the maximum norm for . Define and . Then P is a total cone in E, that is, . denotes the dual cone of P, namely, . Let denote the dual space of E, then by Riesz representation theorem, is given by
We assume that the following condition holds throughout this paper.
(H1) is the unique solution of the linear boundary value problem
Let solve the following inhomogeneous boundary value problems, respectively:
Let , , , .
(H2) , , .
Lemma 2.1 ()
If (H1) and (H2) hold, then BVP (1.1) is equivalent to
where is the Green function for (1.1).
Define an operator as follows:
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.
For any , define a linear operator as follows:
It is easy to show that is a completely continuous nonlinear operator and holds. By , the spectral radius of K is positive. The Krein-Rutman theorem  asserts that there are and corresponding to the first eigenvalue of K such that
Here is the dual operator of K given by:
The representation of , the continuity of G and the integrability of h imply that . Let . Then , and (2.4) can be rewritten equivalently as
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 .
(C2) There exists a continuous operator such that
(C3) There exist a bounded continuous operator and such that for all .
(C4) There exist and such that for all .
Let , then there exists such that
Proof Choose a constant . From (C2), for , there exists such that implies
Now we shall show
provided that R is sufficiently large.
In fact, if (2.7) is not true, then there exist and satisfying
Since , , . Multiply (2.8) by on both sides and integrate on . Then, by (C4), (2.5), we get
By (2.9), holds. Then (2.3), (2.6) and (2.10) imply
where is a constant.
(C3) shows and (C1) implies . Then (C1), (2.8) and Lemma 2.2 tell us that
The definition of yields
It follows from (2.6), (2.11) and (2.12) that
where is a constant.
Since , then (2.13) deduces that (2.7) holds provided that R is sufficiently large such that . By (2.13) and Lemma 2.3, we have
3 Main results
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.
Proof We first show that all the conditions in Lemma 2.4 are satisfied. By Lemma 2.2, condition (C1) of Lemma 2.4 is satisfied. Obviously, is a continuous operator. By (A1), for any , there is such that when , holds. Thus, for with , holds. The fact that B is nondecreasing on yields for any , . Since is an even function, for any and , holds, which implies for . Therefore,
that is, . Take , for any , where . Obviously, holds. Therefore H satisfies condition (C2) in Lemma 2.4.
Take and for , , then it follows from (A1) that
which shows that condition (C3) in Lemma 2.4 holds.
By (A3), there exist and a sufficiently large number such that
Combining (3.1) with (A1), there exists such that
Since K is a positive linear operator, from (3.2) we have
So condition (C4) in Lemma 2.4 is satisfied.
According to Lemma 2.4, we derive that there exists a sufficiently large number such that
From (A4) it follows that there exist and such that
Next we will prove that
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.
It follows from (3.5) and Lemma 2.3 that
By (3.3), (3.6) and the additivity of Leray-Schauder degree, we obtain
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.
() There exist three constants , and such that
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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.
The authors declare that no conflict of interest exists.
All authors participated in drafting, revising and commenting on the manuscript. All authors read and approved the final manuscript.