Existence of Solutions of Second Order Boundary Value Problems with Integral Boundary Conditions and Singularities
© G. Ye and X. Li. 2010
Received: 28 December 2009
Accepted: 20 March 2010
Published: 12 May 2010
By the notation and monotone convergence theorem of Henstock-Kurzweil integral, we investigate the existence of continuous solutions for the second order boundary value problems with integral boundary conditions in which the nonlinearities are allowed to have the singularities in t and are not Lebesgue integrable.
In , Taliaferro showed that problem (1.1) has a solution, where , and with and .
Since then, there are many improvements of this result in literatures for more general case.
In  and other literatures, the authors studied (1.1) in the case where , is continuous, and with or in the case where is continuous and satisfies with and . We note that admit a time singularity at and/or and space singularity at .
In , the authors considered (1.1) when , , is continuous, and a.e. (in particular, is allowed to have a finite number of singularities).
In , Agarwal and O'Regan studied (1.1) when , and satisfies the following caratheodory conditions.
In , the authors studied (1.1) with and supposed that , is continuous, and .
Henstock-Kurzweil integral encompasses the Newton, Riemann and Lebesgue integrals. A particular feature of this integral is that integrals of highly oscillating function which occur in quantum theory and nonlinear analysis such as , where on and , can be defined.
This paper is organized as follows. In Section 2, we make some preliminaries in Henstock-Kurzweil integral; in Section 3, we will prove the equivalence of problem (1.3) and an integral equation as well as existence and uniqueness of solution for the linear problem which associate with (1.3); in Section 4, we are devoted to the existence results for the singular problem (1.3). An example will be given in Section 5.
In this section we introduce the basic facts on Henstock-Kurzweil integrability, a concept that extends the classical Lebesgue integrability on the real line. All notations and properties can be found in the references (see, e.g., [13, 14]).
One says that is a tagged partition of if is a finite family of closed subintervals of which are nonoverlapping, that is, their interiors are pairwise disjoint, and whose union is , and if . Given a function (called a gauge of ), one says that a tagged partition is -fine if for every .
Denote by the continuous functions space on , by the absolutely continuous functions space on , by the generalized absolutely continuous functions space on , and by the space of -integrable functions from to . Assume that the space is equipped with pointwise ordering and normed by the maximum norm, and that the space is equipped with a.e. pointwise ordering and normed by the Alexiewicz norm.
3. Linear Problem
It is easy to prove the following lemma.
4. The Nonlinear Problems
To prove our results, we need the following fixed point theorem for mappings of which is proved in .
Let be an increasing mapping which maps every monotone sequence of to a sequence which converges pointwise to a function of . If , , , and , then has in an order interval of least and greatest fixed points and they are increasing in .
We prove an existence result for solutions of (4.1).
Thus, by Lemma . We know that has in the order interval of least fixed point and greatest fixed point . The functions and are least and greatest solutions of (4.1) in . The hypothesis implies also that if , then . Thus all the solutions of (4.1) belong to the order interval , whence and are least and greatest of all solutions in of (4.1).
The proof is completed.
5. An Example
Since function is not Lebesgue integrable, the results in literature do not hold for (5.1). Let , , then and is -integrable for every continuous since is Lebesgue integrable for every continuous and HK-integrable.
Hence, the existence of continuous solution of problem (5.1) is guaranteed by Theorem 4.2.
This work was supported by NNSF of China (10871059).
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