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.
Keywords:
lattice; fixed point; integral boundary value problem; signchanging solution1 Introduction
In this paper, we consider the following secondorder integral boundary value problem:
where
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 [26] 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 [1121] and the references therein.
In [15], Zhang and Sun studied the following differential equation:
where
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 [2225]). 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 [15,16,2225], 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 [1121].
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 [2225]).
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
We call E a lattice under the partial ordering ≤ if
For
Let
Let
Let P be a cone of a Banach space E. x is said to be a positive fixed point of A if
LetPbe a normal cone ofE, and
(i) there exist a positive bounded linear operator
(ii) there exist a positive bounded linear operator
(iii)
(iv)
Then the operatorAhas 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
Then the operatorAhas at least three fixed points: one positive fixed point, one negative fixed point and one signchanging fixed point.
Let
For convenience, list the following condition.
(H_{0})
is the sequence of positive solutions of the equation
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)
(ii)
(iii)
(iv) the eigenvalues of the linear operatorBare
(v)
3 Main results
Let us list some conditions for convenience.
(H_{1}) There exists a constant
(H_{2}) There exists a constant
where
(H_{3})
(H_{4})
where
Theorem 3.1Suppose that (H_{0}), (H_{1}), (H_{2}), (H_{3}), (H_{4}) are satisfied andnis an odd number in (H_{4}). Then the boundary value problem (1.1) has at least a nontrivial solution.
Proof Choose
So, by (3.1) and (H_{1}), we know that
where
By (3.2) and (3.3), we have
where
By (H_{3}), we have
i.e.,
Theorem 3.2Suppose (H_{0}), (H_{2}), (H_{3}), (H_{4}) are satisfied andnis 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
Obviously, from (3.8) and (2.2), we easily get
From (2.1), we easily know that
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 [1121], and the results are different from [1121].
Example 3.1 We consider the following integral boundary value problem:
where
By simple calculations, we get that
Example 3.2 We consider the following integral boundary value problem:
where
By simple calculations, we get that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
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.
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