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On first-order periodic boundary value problems and distributional Henstock-Kurzweil integrals
Boundary Value Problems volume 2014, Article number: 54 (2014)
This paper is devoted to the study of existence and dependence of solutions of the first-order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The methods used are mainly the method of upper and lower solutions and a fixed point theorem.
In this paper we study the existence of extremal solutions of the first-order periodic boundary value problem (PBVP for short)
where Dx stands for the distributional derivative of , , and f is a distribution (generalized function). More precisely, we study the dependence of the extremal solutions of PBVP (1.1) on f and h.
It is known that the notion of a distributional derivative is very general, including, for example, ordinary derivatives and approximate derivatives. The first-order PBVP of the form
with ordinary derivatives and has been studied extensively in recent years; see, for example, [1–4]. In , by using the distributional derivative, PBVP (1.2) has been generalized to (1.1) in the case when h is continuous with respect to x. Results about the existence of solutions and the topological structure of the solution set are given. In this paper, we study PBVP (1.1) in the case when h is monotone with respect to x and obtain some extended results.
The outline of this paper is as follows. In Section 2 we define the distributional Henstock-Kurzweil integral or briefly the -integral. We say that a distribution f is -integrable on if there is a unique continuous function F on with whose distributional derivative is f. From the definition of the -integral, we know that the -integral includes the Riemann, Lebesgue, Henstock-Kurzweil (briefly HK), and wide Denjoy integrals (for details, see [6–11]). Furthermore, the space of such integrable distributions is a Banach space and has many basic properties; see [5, 12].
In Section 3, by using the -integral and the distributional derivative, we generalize PBVP (1.2) to (1.1). Our main tools are the method of upper and lower solutions and a fixed point theorem. The main result is Theorem 3.1, which extends some corresponding results in [1, 2]. This section also contains an illustration of the results.
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 ϕ is the closure of the set on which ϕ does not vanish, denoted by . A sequence converges to if there is a compact set K such that all have supports in K and for every the sequence of m th derivatives converges to uniformly on K. Let be endowed with this convergence property and denote it by . Also, ϕ is called test function if . Distributions are defined to be continuous linear functionals on . The space of distributions is denoted by , which is the dual space of . That is, if then is a linear functional and we write for .
For all , we recall that the distributional derivative Df of f is a distribution satisfying , where ϕ is a test function and is the ordinary derivative of ϕ. With this definition, all distributions have derivatives of all orders and each derivative is a distribution.
Let be an open interval in ℝ. We define
Of course, is endowed with the above convergence property. The dual space of is denoted by . In this case, if , , then and .
Define to be the space of continuous functions on , and
Note that is a Banach space with the uniform norm .
We are now able to present the definition of the -integral.
Definition 2.1 ([, Definition 2.1])
A distribution f is distributionally Henstock-Kurzweil integrable or briefly -integrable on if f is the distributional derivative of a continuous function .
The space of -integrable distributions is defined by
With this definition, if then, for all ,
The second equality holds because F and ϕ are continuous on and has compact support in . The integral in the last equality exists as a Riemann integral for the same reason.
For convenience, we write , where F is called the primitive of f and ‘’ denotes the -integral. As usual, if and , then the function F is a primitive of f. Notice that if then f has many primitives in , all differing by a constant, but f has exactly one primitive in .
Remark 2.1 Integrals defined in the same way have also been proposed in other papers. For example, Ang et al.  defined it in the plane and called it the G-integral, and Talvila  defined the -integral on the extended real line. In that case of integration over one-dimensional interval, these two integrals coincide.
The following result is known as the fundamental theorem of calculus.
Lemma 2.1 ([, Theorem 4])
Let and . Then and .
Let . Then for all .
For , we say that if and only if for all . Similarly, for , we say that if and only if
The following lemma will be needed later.
Lemma 2.2 ([, Corollary 1])
If , , and if and are -integrable, then is also -integrable.
Let , with . Then, under the Alexiewicz norm
is a Banach space (see [, Theorem 2]).
We say that a sequence converges strongly to (or in ) if as . The following two convergence theorems hold.
Lemma 2.3 ([, Corollary 4, monotone convergence theorem])
Let be a sequence in such that , and that . Then in and .
Lemma 2.4 ([, Corollary 5, dominated convergence theorem])
Let be a sequence in such that in . Suppose that there exist satisfying , . Then and .
If , its variation is , where the supremum is taken over every sequence of disjoint intervals in . If then g is called a function with bounded variation. Denote the set of functions with bounded variation by . As it is known that the dual space of is (see details in ), we have the next result.
Lemma 2.5 ([, Definition 6, integration by parts])
Let and . Define , where . Then and
Denote by L the space of Lebesgue integrable functions and by the norm on L.
Definition 2.2 Let and . Let such that . Define fg as the unique element in such that .
This definition, which is modified from [, Definition 5], makes sense since is dense in L. Moreover, the next statement holds.
Lemma 2.6 ([, Lemma 2.7])
Let f, g be the distributional derivatives of F, G, respectively, where . Then
3 Periodic boundary value problems
In this section, we shall study the first-order periodic boundary value problem
where Dx denotes the distributional derivative of , and f is a distribution on . Throughout this section, we denote by (respectively, HK, L) the space of (respectively, HK, Lebesgue)-integrable functions and by ‘’ the ∗-integral.
For convenience, let us list the following assumptions on the functions f and h.
(D1) There exist , such that and
and there is such that on , holds for the function
and the inequalities
are true for .
(D2) is Lebesgue integrable for every fixed , and the distribution f is -integrable on .
(D3) is nondecreasing in for all .
We recall that .
Before coming to the main results in this paper, we give a result following from Lemma 2.1, that is, PBVP (1.1) can be converted to an integral equation.
Lemma 3.1 Let f and h satisfy (D2). A function is a solution of PBVP (1.1) on if and only if the equality
is true for any such that on and
Remark 3.1 In view of Lemma 2.5, the result on implies that is -integrable on , because is -integrable on for all .
As a matter of fact, the proof of Lemma 3.1 follows exactly the lines of [, Lemma 3.1], so we omit it here.
In what follows we recall a fixed point theorem for increasing mappings, which is an important tool for proving the existence theorem.
Let E be an ordered Banach space, K be a nonempty subset of E. The mapping is increasing if and only if , whenever and .
Lemma 3.2 ([, 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 (),
We are now ready to give the main results.
Theorem 3.1 Let the functions f and h in (1.1) satisfy the assumptions (D1)-(D3). Then the extremal solutions of PBVP (1.1) exist in the ordering interval .
where is given in (D1). Then the hypotheses (D1)-(D3) imply, for all , that is Lebesgue integrable and
Define a mapping on by
It follows from (3.7) that, for each ,
Let . Then, by (D1), (3.5) and (3.8), the inequalities
hold. It follows from (3.9) that
Inequalities (3.10) and (3.11) imply, for each , that
This result implies
In view of (3.11) and (3.12), we then have , i.e., . We can similarly verify that .
It follows from (D3) and (3.5) that is nondecreasing on for all . Moreover, for each , whence (3.7) defines a nondecreasing mapping .
We now only need to prove that is relatively compact.
The properties of imply that
Since are bounded, there exists such that
This implies that is uniformly bounded.
Let . Then, by (3.7), for each ,
Since and on , is continuous and so it is uniformly continuous on . Then, for any , there exists such that
Moreover, the continuity of on implies that there exists such that
On the other hand, the monotonicity of on for all and (2.2) implies that
This result, (3.13), (3.14), (3.15) and (3.16) imply that
Since and are -integrable on , the primitives of them are continuous and so they are uniformly continuous on . Hence, by inequality (3.17), is equiuniformly continuous on for all . In view of the Ascoli-Arzelà theorem, is relatively compact. Thus, satisfies the hypotheses of Lemma 3.2, whence has the minimal fixed point and the maximal fixed point . From the definitions of g and and Lemma 3.1 it follows that and are also solutions of PBVP (1.1) in . Moreover, (3.3) and (3.4) hold with , replaced by y, z.
If x is any solution of PBVP (1.1) in , then it is, by Lemma 3.1, a fixed point of , whence . Thus and are the minimal and maximal solutions of PBVP (1.1) in , respectively. □
Remark 3.2 Let , and , where M is a positive constant, and . Then (D1) is reduced to
() There exist , , such that , on , and .
If in (D1), then we obtain
() There exist , , such that and on , and .
Further, the -integral includes the Lebesgue integral, and the distributional derivative contains the ordinary derivative. Thus, we can see that Theorem 3.1 is a proper generalization of [, Theorem 2.1] if on .
Corollary 3.1 Given a function g, assume that is nondecreasing on for all , is Lebesgue integrable on for every fixed , and there exist such that, for all ,
If on , with nonzero at and f is a distribution on , then the PBVP
has the extremal solutions.
Proof Let y be a solution of the PBVP
and z a solution of the PBVP
According to Lemma 3.1, it is easy to see that y, z are unique.
By choosing , , , we have that the following two inequalities
hold if y, z are given as above. Thus, the conditions (D1)-(D3) are satisfied and, by Theorem 3.1, PBVP (3.19) has the extremal solutions and in . Moreover, if x is any solution of PBVP (3.19), it follows from Lemma 3.1 that for each ,
This result, (3.18), (3.20) and (3.21) imply that each solution of PBVP (3.19) belongs to , whence and are the minimal and maximal solutions of PBVP (3.19). □
We now give an example to illustrate the main results.
Example 3.1 Consider the PBVP given by
where Dr is the distributional derivative of
Then PBVP (3.22) has extremal solutions.
Proof PBVP (3.22) can be regard as a PBVP of the form (3.19), where
It is easy to see that and . Choose , then and (3.18) holds. This result and is nondecreasing imply that the hypotheses of Corollary 3.1 are satisfied. The proof is therefore completed. □
It is well known that the function in (3.23) given by Riemann and proved by Hardy  is continuous but pointwise differentiable nowhere on ℝ. Then, by [, Example 2.4], the distributional derivative Dr is neither HK-integrable nor Lebesgue integrable. Hence, [, Theorem 2.1] is not applicable in this case, which implies that the main results in this paper are more general.
The following result shows the dependence of the extremal solutions of PBVP (1.1) on f and h.
Proposition 3.1 If the hypotheses of Theorem 3.1 hold, then the extremal solutions of PBVP (1.1) in are nondecreasing with respect to f and h.
Proof Let f, , h and satisfy, for all ,
Assume that the hypotheses of Theorem 3.1 hold for f, h and , with the same fixed functions y, z, , and p. Let be the minimal solution of PBVP (1.1) in , the minimal solution of the PBVP
in . Let be defined by (3.7), where g is given by (3.5). By Lemma 3.1, one has
It follows from (3.5), (3.24) and (3.27) that for all ,
whence (3.7) and (3.26) imply that , yet Lemma 3.2 holds with , . Indeed, by (3.3), is also the minimal fixed point of in . This and (3.4) imply that .
Similarly, it is easy to verify that , where denotes the maximal solution of PBVP (1.1) in and is the maximal solution of PBVP (3.25) in . □
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The authors would like to express their heartfelt appreciations to the expert referees for their helpful comments and suggestions.
The authors declare that they have no competing interests.
WL wrote the first draft and TA and GY corrected and improved the final version. All authors read and approved the final draft.
About this article
- periodic boundary value problem
- distributional Henstock-Kurzweil integral
- distributional derivative
- extremal solutions
- upper and lower solutions
- fixed point