Existence of Solutions of Second Order Boundary Value Problems with Integral Boundary Conditions and Singularities

Abstract

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.

1. Introduction

The singular boundary value problems

(1.1)

where may be singular at and , have been studied extensively; see, for example, [1–8], and the references contained therein.

In [7], 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 [5] 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 [4], the authors considered (1.1) when , , is continuous, and a.e. (in particular, is allowed to have a finite number of singularities).

In [1], Agarwal and O'Regan studied (1.1) when , and satisfies the following caratheodory conditions.

The map is continuous for .

The map is measurable for all .

There exists with such that for and .

In [8], the authors studied (1.1) with and supposed that , is continuous, and .

It is noticed that the case

(1.2)

with being continuous is not included in all those papers abovementioned.

In this paper, motivated by this case, relying on theory of Henstock-Kurzweil integral, we investigate the following second order boundary value problems with integral boundary conditions

(1.3)

where are nonnegative constants and are not certainly -integrable.

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.

For the literatures in which the theory of Henstock-Kurzweil integral to study differential equations is used we refer to [9–14] and so on.

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.

2. Preliminaries

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]).

Let be the real unit interval provided with the -algebra of Lebesgue measurable sets with the Lebesgue measure .

Definition 2.1 (see [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 .

Definition 2.2 (see [13, 14]).

A function is said to be Henstock-Kurzweil (shortly ) integrable if there exists a real satisfying that, for every , there is a gauge such that

(2.1)

for every -finite partition . One says that

(2.2)

is a Henstock-Kurzweil (shortly ) integral of over .

A function is absolutely continuous (or ) on if for each there exists such that whenever is a finite collection of nonoverlapping intervals that have endpoints in and satisfy while denotes the oscillation of over ; that is,

(2.3)

A function is generalized absolutely continuous (or ) on if is continuous on and if can be expressed as a countable union of sets on each of which is absolutely continuous (or ).

For the Lebesgue integral of function , we denote that .

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.

The following Lemma 2.3–Lemma 2.7 are from [13, 14].

Lemma 2.3.

The Henstock-Kurzweil integral is linear, and additive over nonoverlapping intervals of .

Lemma 2.4.

Let be -integrable and let be bounded variation. Then is -integrable, and for every

(2.4)

Lemma 2.5.

Let be -integrable. If for almost every , and if , then

(2.5)

Lemma 2.6.

Let be -integrable. Then the relation

(2.6)

defined a function , which is continuous and belongs to , a.e. derivable and a.e. on .

is called a primitive of .

Lemma 2.7.

Assume that functions and are -integrable, that the sequence is increasing (respectively decreasing) for almost every , and that

(2.7)

for all and a.e. . Then there exists such an -integrable function , that for a.e. , and that

(2.8)

3. Linear Problem

We know that the homogeneous problem

(3.1)

has only the trivial solution and Green's function is

(3.2)

It is easy to prove the following lemma.

Lemma 3.1.

For every , functions and are derivable on and and their derivations are absolutely continuous.

Lemma 3.2.

Let be an -integrable function, then

()for every , and are -integrable in ;

()the function, where ,

(3.3)

is derivable a.e. on and

(3.4)

() satisfies the following conditions:

(3.5)

() is derivable a.e. on and

(3.6)

Proof.

() From Lemma 3.1, since we know that and are absolutely continuous respect to , and , the conclusions are in as follows.

() Since

(3.7)

it follows from Lemma 2.6 that, for a.e. ,

(3.8)

() Since

(3.9)

we claim that

(3.10)

In fact, by Lemma 2.4,

(3.11)

Denote that ; then and . There exists such that

(3.12)

Therefore,

(3.13)

Thus, we have

(3.14)

The proof of another condition is similar.

() Since

(3.15)

for a.e. , there exists a subset of with such that on . Relying on Lemma 2.6, is derivable a.e. on and, therefore, a.e. on , and

(3.16)

Theorem 3.3.

Given functions . Then the following nonhomogeneous linear problem

(3.17)

has a unique solution and

(3.18)

where

(3.19)

Proof.

We notice that and

(3.20)

The facts associated with Lemma 3.2 deduce that the function satisfies , is derivable a.e. on , and

(3.21)

and verifies the boundary conditions. The uniqueness of solution of (3.17) follows from Lemma 3.1.

4. The Nonlinear Problems

In this section we consider the following nonlinear problems:

(4.1)

We impose the following hypotheses on the functions and .

and are -integrable whenever .

and are increasing in for almost every .

There exist -integrable functions and such that

(4.2)

a.e. hold on for all .

To prove our results, we need the following fixed point theorem for mappings of which is proved in [10].

Lemma 4.1.

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).

Theorem 4.2.

Assume that the hypotheses ()–() are satisfied, then (4.1) has least and greatest solutions in .

Proof.

We know from Theorem 3.3 that the solutions of (4.1) are the solutions of following operator equation:

(4.3)

where

(4.4)

The hypothesis and Lemma 2.5 imply that if and , then

(4.5)

That is, is increasing in .

Let be an increasing sequence in , then the hypothess ()–() imply that the functions sequences are increasing in and belong to , and

(4.6)

Thus, by Lemma 2.7, there exist -integrable functions such that

(4.7)

Denote that

(4.8)

Then we can easily get that for every and

(4.9)

which implies also that for every . Therefore we obtain

(4.10)

Denoting that

(4.11)

then, by Lemma 2.6, . In addition, the hypothesis implies that

(4.12)

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).

On the other hand, if is a solution of (4.1), then, from Lemma 2.6,

(4.13)

The proof is completed.

5. An Example

Consider the following problem:

(5.1)

where

(5.2)

and satisfies the following caratheodory conditions:

the map is continuous for ,

the map is measurable for all ,

there exists with such that for and ,

is increasing in for .

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.