Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral *

We apply the distributional derivative to study the existence of solutions of the second order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The distributional Henstock-Kurzweil integral is a general intergral, which contains the Lebesgue and Henstock-Kurzweil integrals. And the distributional derivative includes ordinary derivatives and approximate derivatives. By using the method of upper and lower solutions and a fixed point theorem, we achieve some results which are the generalizations of some previous results in the literatures.


Introduction
where and are the first and second order distributional derivatives of  and f 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 here  : 0, g T       , and x and x 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 whose distributional derivative equals f .From the definition of the HK -integral, it includes the Riemann integral, Lebesgue integral, HK-integral and wide Denjoy integral (for details, see [3][4][5]).Furthermore, the space of D HK -integrable distributions is a Banach space and has many good properties, see [6][7][8].

D
In Section 3, with the HK -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.

The Distributional Henstock-Kurzweil Integral
In this section, we present the definition and some basic properties of the distributional Henstock . 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 all , we define the distributional derivative of , where  is a test function.

Let
be an open interval in , we define and are and respectively if , .
Note that is a Banach space with the uniform norm With the definition above, we know that the concept of the HK -integral leads to its good properties.We firstly mention the relation between the , we define the Alexiewicz norm by The following result has been proved.

Lemma 2.2. ([3, Thoerem 2]). With the Alexiewicz norm, HK
We now impose a partial ordering on D is a Banach space.
e details in [9]).By this definition, if , (se . We also have other usual relations between the HK -integral and the ordering, for instance, the following result.
, and if and are HK D -integrable, then g is also HK We say a sequence It is also shown that the following two convergence theorems hold.
Lemma 2.4.([9, Corollary 4], Monotone convergence theorem for the HK -integral).Let   be a se- quence in We now give another result about the distributional derivative.
Lemma 2.6.Let f g be the distributional derivative of , F G , where where the supremum is taken over every sequence   where and denote the first and second order distributional derivatives of 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 Recall we say   if and only if for all We impose the following hypotheses on the functions and on 0, x is a solution of PBVP (1) if We say that 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.

 
: 0, f T   be a distribution and Lemma 3.1.
where     : , is relatively compact, then has a maximal fixed point x and a minimal fixed point * .
then the Equations (1)-( 3) define a nondecreasing mapping , , , , , ,  ,  x y x y x y x y   For any    , we have   This paper is devoted to the study of the existence of solutions of the second order periodic boundary value problem (PBVP for brevity)

HKD
-integral.A distribution f is HK -integrable on D     , a b   if there is a continuous function F on , . Analogously, we denote HK -integral and Lebesgue integral.The space of HK   D -integrable distributions is defined by It follows from the definition of the distributional derivative and (3.1) that

 3 .
Periodic Boundary Value ProblemsConsider the second order periodic boundary value problem(1.1)

6 )
y by the same way.Thus x and = It follows from (3.1) and (3.3) that for each

3 . 1 .
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 Assume that conditions (D0)-(D2) are satisfied.Then the PBVP (1.1) has such solutions x and x in   , v u that x x x   and Dx Dx Dx   for each solution x of (1.1) in   , In view of Lemma 3.4 the Equations (3.1)-(3.3)define a nondecreasing mapping    : , , G     .
on   0,T for all     of (1), it follows from Lemma 3.1 that is a fixed point of .It follows from the extremality of x  and x  that x z x     v x Dv Dx Du , i.e., and u     .As a consequence of Theorem 3.1 we have then g is called a function with bounded variation.The set of functions with bounded variation is denoted .It is known that the dual space of D  Lemma 2.7.([3, Definition 6], Integration by parts). , 1 , , d