^{1}

^{*}

^{1}

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.

This paper is devoted to the study of the existence of solutions of the second order periodic boundary value problem (PBVP for brevity)

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

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.

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.

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 [

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 [

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

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 [

Lemma 2.7. ([3, Definition 6], Integration by parts). Let, and. Define, where . Then and

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

where

and

Proof. Let, then the function with is continuous on, so is -integrable. Let, then by (1.1) we have, or equivalently,

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

Noticing that, then (3.5) implies by differentiation that

It follows from (3.1) and (3.3) that for each,

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,

where and,

Lemma 3.3. Let conditions (d0)-(d2) be satisfied. Denoting

then and

Proof. The hypotheses (d0) and (d2) imply that for all in, satisfying

,

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,

which implies is uniformly bounded on .

Let. Then by (3.2) and (3.3), for each

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

and

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.