Abstract
This paper investigates the existence of concave symmetric positive solutions and establishes corresponding iterative schemes for a secondorder boundary value problem with integral boundary conditions. The main tool is a monotone iterative technique. Meanwhile, an example is worked out to demonstrate the main results.
Keywords:
integral boundary conditions; iterative; monotone positive solution; symmetric; completely continuous1 Introduction
The theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can be reduced to the nonlinear problems with integral boundary conditions; we refer readers to [13] for examples and references.
At the same time, boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary value problems as special cases.
Hence, increasing attention is paid to boundary value problems with integral boundary conditions [48]. Generally, the GuoKrasnosel’ skii fixed point theorem in a cone, the LeggettWilliams fixed point theorem, the method of upper and lower solutions and the monotone iterative technique play extremely important roles in proving the existence of solutions to boundary value problems. In particular, we would like to mention some excellent results.
In [4], Ma studied the following problem:
where
In 2010, Wang et al.[7] considered the secondorder boundary value problem with the integral boundary conditions
where ϕ, f,
Different from [7], [9] is not based on the assumption that the upper and lower solutions to the boundary value problem should exist, but constructs the specific form of the symmetric upper and lower solutions. The author in [9] investigated a secondorder SturmLiouville boundary value problem
And by applying monotone iterative techniques, author proved the existence of n symmetric positive solutions.
To the best of our knowledge, no contribution exists concerning the existence of solutions for a boundary value problem with integral boundary conditions by applying monotone iterative techniques. Inspired by the work mentioned above, we concentrate on the following problem:
where
The difficulty of this paper is that the nonlinear term f depends on
2 Preliminaries
Definition 2.1 Let E be a Banach space, a nonempty convex closed set
(i) if
(ii) if
Every cone
Definition 2.2 Let
Definition 2.3 Let
for any
We consider the Banach space
In this paper, we always suppose that the following assumptions hold:
(H_{1})
(H_{2})
(H_{3})
Denote
It is easy to see that P is a cone in E.
For any
Then we can easily get the solution:
where
and
During the process of getting the above solution, we can also know
for
Lemma 2.1If (H_{3}) is satisfied, the following results are true:
1.
2.
For any
Lemma 2.2If (H_{3}) is satisfied,
Proof For any
Obviously, Ty is concave. From the expression of Ty, combining with Lemma 2.1, we know that Ty is nonnegative on
For
Similarly, we have
So,
For any
Therefore, from (2.3), we have
So,
Moreover,
And the similar results can be obtained for
The ArzelàAscoli theorem guarantees that TΩ is relatively compact, which means T is compact.
Finally, we show that Ty is nondecreasing about y.
For any
Hence, for
Furthermore, we have
In order to prove
A similar result can be obtained for
3 Existence and iterative of solutions for BVP (1.1) and (1.2)
Theorem 3.1Assume that (H_{1})(H_{3}) hold. If there exist two positive numbers
whereaand
then problem (1.1) and (1.2) has a concave symmetric positive solution
Proof We denote
Let
By assumption (H_{2}) and (3.1), for
For any
and
Hence,
Since
On the other hand, we notice that
So,
Let
Similarly to
Remark The existence of a solution under the assumptions of Theorem 3.1 is just a consequence of Schauder’s fixed point theorem. The monotone iterative technique adds the information about the approximation sequences.
Example Consider the following secondorder boundary value problem with integral boundary conditions:
And we have
It is easy to check that the assumptions (H_{1})(H_{3}) hold and
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Chinese Universities Scientific Fund (Project No. 2013QJ004).
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