Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral ()
1. Introduction
This paper is devoted to the study of the existence of solutions of the second order periodic boundary value problem (PBVP for brevity)
(1.1)
where 
and 
are the first and second order distributional derivatives of 
respectively, 
and 
is a distribution (generalized function).
If the distributional derivative in the system (1.1) is replaced by the ordinary derivative and
, then (1) converts into
(1.2)
here
, and 
and 
denote the first and second ordinary derivatives of
. The existence of solutions of (1.2) have been extensively studied by many authors [1,2]. It is well-known, the notion of a distributional derivative is a general concept, including ordinary derivatives and approximate derivatives. As far as we know, few papers have applied distributional derivatives to study PBVP. In this paper, we have come up with a new way, instead of the ordinary derivative, using the distributional derivative to study the PBVP and obtain some results of the existence of solutions.
This paper is organized as follows. In Section 2, we introduce fundamental concepts and basic results of the distributional Henstock-Kurzweil integral or briefly the 
-integral. A distribution 
is 
-integrable on 
if there is a continuous function F on 
with 
whose distributional derivative equals
. From the definition of the 
-integral, it includes the Riemann integral, Lebesgue integral, HK-integral and wide Denjoy integral (for details, see [3-5]). Furthermore, the space of 
-integrable distributions is a Banach space and has many good properties, see [6-8].
In Section 3, with the 
-integral and the distributional derivative, we generalize the PBVP (1.2) to (1.1). By using the method of upper and lower solutions and a fixed point theorem, we achieve some interesting results which are the generalizations of some corresponding results in the references.
2. The Distributional Henstock-Kurzweil Integral
In this section, we present the definition and some basic properties of the distributional Henstock-Kurzweil integral.
Define the space
where the support of a function 
is the closure of the set on which 
does not vanish, denote by
. A sequence 
converges to 
if there is a compact set 
such that all 
have support in 
and for every 
the sequence of derivatives 
converges to 
uniformly on
. Denote 
endowed with this convergence property by
. Where 
is called test function if
. The distributions are defined as continuous linear functionals on
. The space of distributions is denoted by
, which is the dual space of
. That is, if 
then
, and we write
, for
.
For all
, we define the distributional derivative 
of 
to be a distribution satisfying 
, where 
is a test function.
Let 
be an open interval in
, we define
the dual space of 
is denoted by
.
Remark 2.1. 
and 
are 
and 
respectively if
,
.
Let 
be the space of continuous functions on
, and
Note that 
is a Banach space with the uniform norm
.
Now we are able to introduce the definition of the 
-integral.
Definition 2.1. A distribution 
is distributionally Henstock-Kurzweil integrable or briefly 
-integrable on 
if 
is the distributional derivative of a continuous function
.
The 
-integral of 
on 
is denoted by
where 
is called the primitive of
and “
” denotes the 
-integral. Analogously, we denote 
-integral and Lebesgue integral.
The space of 
-integrable distributions is defined by
With this definition, if 
then we have for all
.
(2.1)
With the definition above, we know that the concept of the 
-integral leads to its good properties. We firstly mention the relation between the 
-integral and the 
-integral.
Recall that 
is Henstock-Kurzweil integrable on 
if and only if there exists a continuous function 
which is 
(generalized absolutely continuous, see [4]) on 
such that 
almost everywhere. P. Y. Lee pointed out that if 
is a continuous function and pointwise differentiable nearly everywhere on
, then 
is
. Furthermore, if 
is a continuous function which is differentiable nowhere on
, then 
is not
. Therefore, if 
but differentiable nowhere on
, then 
exists and is 
-integrable but not 
- integrable. Conversely, if 
and it also belongs to
. Then 
is not only 
-integrable but also 
-integrable. Here 
denotes the ordinary derivative of
. Obviously, the 
-integral includes the 
-integral.
Now we shall give some corresponding results of the distributional Henstock-Kurzweil integral.
Lemma 2.1. ([3, Theorem 4], Fundamental Theorem of Calculus).
1) Let
, define
. Then 
and
.
2) Let
. Then 
for all 
For
, we define the 
norm by
The following result has been proved.
Lemma 2.2. ([3, Thoerem 2]). With the 
norm, 
is a Banach space.
We now impose a partial ordering on
: for
, we say that 
(or
) if and only if 
is a measure on 
(see details in [9]). By this definition, if 
then
(2.2)
whenever
,
. We also have other usual relations between the 
-integral and the ordering, for instance, the following result.
Lemma 2.3. ([9, Corollary 1]). If
, 
and if 
and 
are 
-integrable, then 
is also 
-integrable.
We say a sequence 
converges strongly to 
if 
as
. It is also shown that the following two convergence theorems hold.
Lemma 2.4. ([9, Corollary 4], Monotone convergence theorem for the 
-integral). Let 
be a sequence in 
such that 
and that 
as
. Then 
in
and
.
Lemma 2.5. ([7, Lemma 2.3], Dominated convergence theorem for the 
-integral). Let 
be a sequence in 
such that 
in
. Suppose there exist 
satisfying
.
Then 
and
.
We now give another result about the distributional derivative.
Lemma 2.6. Let 
be the distributional derivative of
, where
. Then
(23)
Proof. It follows from the definition of the distributional derivative and (3.1) that
Consequently, the result holds.
If
, its variation is
where the supremum is taken over every sequence 
of disjoint intervals in
, then 
is called a function with bounded variation. The set of functions with bounded variation is denoted
. It is known that the dual space of 
is 
(see details in [3]), and the following statement holds.
Lemma 2.7. ([3, Definition 6], Integration by parts). Let
, and
. Define
, where 
. Then 
and
3. Periodic Boundary Value Problems
Consider the second order periodic boundary value problem (1.1)
where 
and 
denote the first and second order distributional derivatives of
, respectively, 
and 
is a distribution (generalized function).
The distributional derivative subsumes the ordinary derivative. And if the first ordinary derivative of 
exists, the first ordinary derivative and first order distributional derivative of 
are equivalent. For
, then the distributional derivative 
and
, hence 
.
Recall that we say 
if and only if 
and 
for all
.
We impose the following hypotheses on the functions 
and
.
(D0) There exist 
with 
such that
and
, 
, with 
and
, 
such that
(D1) 
is Lesbesgue integrable on 
when
, 
, and 
is 
-integrable on
(D2) 
is nonincreasing with respect to 
for all
.
We say that 
is a solution of PBVP (1) if 
and satisfies (1). Before giving our main results in this paper, we first apply Lemma 2.1 to convert the PBVP (1) into an integral equation.
Lemma 3.1. Let 
be a distribution and
, a function
is a solution of the PBVP (1.1) on 
if and only if 
and 
satisfy for any
, 
on
, with 
and
, the integral equation
(3.1)
where
(3.2)
and
(3.3)
Proof. Let
, then the function 
with 
is continuous on
, so 
is 
-integrable. Let
, then by (1.1) we have
, or equivalently,
(3.4)
Integrating (3.4) we have
This implies
. We can prove that 
by the same way. Thus 
and 
satisfy the operator equation (3.1).
Conversely, assume that 
satisfy (3.1). In view of (2) we then have for each 
(3.5)
Noticing that
, then (3.5) implies by differentiation that
(3.6)
It follows from (3.1) and (3.3) that for each
,
(3.7)
Applying Lemma 2.6 to (3.7), we obtain for all 
which together with (3.6) implies that
It follows from (5) that
, and from (7) that
, so that 
is a solution of the PBVP (1.1). □
Let 
be an ordered Banach space, 
a nonempty subset of
. The mapping 
is increasing if and only if
, whenever 
and
.
An important tool which will be used latter concerns a fixed point theorem for an increasing mapping and is stated next.
Lemma 3.2. ([10, Theorem 3.1.3]) Let 
with
, and 
be an increasing mapping satisfying
. If 
is relatively compact, then 
has a maximal fixed point 
and a minimal fixed point 
in
. Moreover,
(3.8)
where 
and
,
(3.9)
Lemma 3.3. Let conditions (d0)-(d2) be satisfied. Denoting
(3.10)
then 
and 
Proof. The hypotheses (d0) and (d2) imply that for all 
in
, satisfying
,
(3.11)
This and (d1) ensure that 
and 
in (3.2) and (3.3) are defined for
. Condition (d0) implies that for each 
It follows from (3.7), (3.10) and (d0) that for each 
Thus, 
and
, whence
. The proof that 
is similar.
Lemma 3.4. Assume that conditions (D0)-(D2) hold. Denoting
then the equations (1)-(3) define a nondecreasing mapping
.
Proof. Let
be given. The hypotheses (D0)-(D2) imply that for each 
and
Thus 
This and Lemma 3.3 imply the assertion.
With the preparation above , we will prove our main result on the existence of the extremal solutions of the periodic boundary value problem (1.1).
Theorem 3.1. Assume that conditions (D0)-(D2) are satisfied. Then the PBVP (1.1) has such solutions 
and 
in 
that 
and 
for each solution 
of (1.1) in 
such that 
.
Proof. In view of Lemma 3.4 the equations (3.1)-(3.3) define a nondecreasing mapping
. For any 
, we have
Since 
and
, there exists constant 
such that, for each
,
(3.12)
which implies 
is uniformly bounded on 
.
Let
. Then by (3.2) and (3.3), for each 
(3.13)
(3.14)
Since
, 
,
is continuous and so is uniformly continuous on
, i.e., for all
, there exists 
such that
It is easy to see that 
(so is
) on
. Hence, there exists 
such that
The result 
on 
implies by Lemma 2.6 that 
and
are
-integrable on
, because 
and 
are 
-integrable for all
. This result and the monotonicity of 
and
imply
and
Then by (3.12)-(3.14), there exists 
such that
(3.15)
and
(3.16)
Since 
and 
are 
-integrable on
, the primitives of 
and 
are continuous and so are uniformly continuous on
. Similarly, the primitives of 
and 
are uniformly continuous on
. Therefore, by inequalities (15) and (16), 
and 
are equiuniformly continuous on 
for all
. So 
is equiuniformly continuous on 
for all
.
In view of the Ascoli-Arzelàtheorem, 
is relatively compact. This result implies that 
satisfies the hypotheses of Lemma 3.2, whence 
has the minimal fixed point 
and the maximal fixed point
. It follows from Lemma 3.1 that 
are solutions of PBVP (1), and that 
and
.
Let
, and
, 
, then (3.8) and (3.9) hold. If 
with 
is a solution of (1), it follows from Lemma 3.1 that 
is a fixed point of
. It follows from the extremality of 
and 
that
, i.e., 
and 
.
As a consequence of Theorem 3.1 we have Corollary 3.1. Given the functions
, assume that conditions (D0) and (D1) hold for the function
If 
is nonincreasing in 
for all
, and if 
is nonincreasing in 
for all
, then the PBVP (1.1) has the extremal solutions in
.
NOTES