EXISTENCE AND UNIQUENESS OF SOLUTIONS TO DISTRIBUTIONAL DIFFERENTIAL EQUATIONS INVOLVING HENSTOCK–KURZWEIL–STIELTJES INTEGRALS

This paper is concerned with the existence and uniqueness of solutions to the second order distributional differential equation with Neumann boundary value problem via Henstock–Kurzweil–Stieltjes integrals.The existence of solutions is derived from Schauder’s fixed point theorem, and the uniqueness of solutions is established by Banach’s contraction principle. Finally, two examples are given to demonstrate the main results.


Introduction
In this paper we apply the Henstock-Kurzweil-Stieltjes integral to establish the existence and uniqueness of solutions to the second order distributional differential equation (in short DDE) − D 2 x = f (t, x) + g(t, x)Du, t ∈ [0, 1], (1.1) subject to the Neumann boundary value condition Dx(0) = Dx(1) = 0, (1.2) where D, D 2 stand for the first order and the second order distributional derivative, respectively, x and u are regulated functions (see Definition 2.2) such that both are left-continuous on (a, b] and right-continuous at a, the function f (·, x(·)) is Henstock-Kurzweil integrable, and g(·, x(·)) is a function of bounded variation. We know that regulated functions contain continuous functions and functions of bounded variation as special cases. Especially, when u is an absolutely continuous function, its distributional derivative is the usual derivative and we obtain the ordinary differential equation (in short ODE); when u is a function of bounded variation, Du can be identified with a Stieltjes measure, and (1.1) is called measure differential equation (in short MDE). Results concerning existence, uniqueness and stability of solutions for MDEs have been extensively studied in many papers, see [2,11]. For example, in [2], the authors establish the existence of solutions for the first order MDE Dx = f (t, x) + G(t, x)Du. Recently, in [8], authors obtained the existence result for the first order DDE of the form Dx = f (t, x)Du. As far as we know, few papers are concerned with the second order DDE. The purpose of this paper is to fill this gap, considering the second order DDE with the Neumann boundary value problem.
Neumann boundary value problems have played an important role in mathematical physics, and hence have attracted the attention of many researchers. Under suitable conditions, some existence and uniqueness results and the multiplicity of positive solutions for the Neumann boundary value problem have been established. For example, in [1], a Lyapunov-type inequality and Schauder's fixed point theorem were used to obtain existence and uniqueness results for the Neumann boundary value problem as In [13], by using a fixed point theorem in a cone, the authors established the existence and multiplicity of the positive solutions to the second order Neumann boundary value problem (1.3) with parameter. Particularly, problem (1.3) is a special case of problem (1.1)-(1.2). In order to deal with the problem (1.1)-(1.2), we apply the Henstock-Kurzweil-Stieltjes integral, which is a powerful tool for the study of DDEs and contains the Henstock-Kurzweil integral ( [6,12,9,10]), the regulated primitive integral ( [14]), the Lebesgue-Stieltjes integral, and the Riemann-Stieltjes integral. With the help of Henstock-Kurzweil-Stieltjes integrals, we establish the existence and uniqueness of solutions for the second order DDE with Neumann boundary value problem. Therefore, our results can be seen as a generalization of the results in [1,13].
This paper is organized as follows. In Section 2, we recall the definition and some basic properties of Henstock-Kurzweil-Stieltjes integrals and regulated functions. In Section 3, two results of the existence and uniqueness of solutions are established by using Schauder's fixed point theorem and Banach's contraction principle. In Section 4, we give two examples to illustrate Theorem 3.4 and Theorem 3.6.

Regulated functions.
In this subsection, we recall the definition and some basic properties of regulated functions. First, we introduce functions of bounded variation, which are a special case of regulated functions.
We consider a nondegenerate compact interval [a, b] ⊂ R, and denote by D a,b the set of all divisions of the form (2.1) The set of functions of bounded variation on [a, b] is denoted by We now list some basic properties of the above spaces which we will need later.

The Henstock-Kurzweil-Stieltjes integral.
In this subsection, we recall some basic properties of the Henstock-Kurzweil-Stieltjes integral. We cite most of the results without proof, and interested readers can find more information in [6,12,7,9,10]. We consider a division d = {t 0 , . . . , t m } ∈ D a,b from (2.1), and define a partition D as (2. 2) The basic concept in the Henstock-Kurzweil-Stieltjes integration theory is a δ-fine partition.
We define the set An element δ ∈ Γ(a, b) is called a gauge.
For given functions f, g : [a, b] → R and a partition D of the form (2.2), we define the Henstock-Kurzweil-Stieltjes integral sum K D (f, g) by the formula Definition 2.7 ([7]). Let f, g : [a, b] → R be given. We say that J ∈ R is the Henstock-Kurzweil-Stieltjes integral over [a, b] of f with respect to g and is denoted by Particularly, if g(t) ≡ t, then Definition 2.7 is the definition of Henstock-Kurzweil integrals (HK-integrals for short).
Remark 2.8. According to [7], we know that the Henstock-Kurzweil-Stieltjes integral is a generalization of the Lebesgue-Stieltjes integral and the Riemann-Stieltjes integral. For example, when g is a regulated function, the function f is not Lebesgue-Stieltjes integrable with respect to the function g. Also, the Riemann-Stieltjes integral will not be appropriate when the function f and the function g have a common point of discontinuity.
The proof of the following proposition is similar to that for the Young integral in [5,Proposition 3.12]; however, in order to make the paper self-contained we give the proof here.
∞, such that f − w < ε, and there exists N > 0 for which f − f n < ε and g − g n < ε when n > N . Then we obtain by Lemma 2.3 and Lemma 2. Hence, This completes the proof.
For the HK-integral, we have the following controlled convergence theorem, which will be used later. (
The following statement is the well-known Schauder's fixed point theorem.

K(t, s)g(s, x(s)) du(s). (3.15)
Now we prove this theorem in three steps.
Step 1. There exists r > 0 such that For all x ∈ B r , by (3.14) and (3.15), we have
As a consequence of Steps 1 and 2 together with Lemma 2.5, we can conclude that the set A(B r ) is relatively compact in G − ([0, 1]; R).
Step 3. A is continuous on B r .
According to (C 3 ) and the controlled convergence theorem of Lemma 2.11, we have lim Therefore, lim n→∞ Ax n (·) = Ax(·). This shows that A is continuous. Thus, A satisfies the hypotheses of Lemma 3.3, and the application of Schauder's fixed point theorem shows that A has at least a fixed point which is a solution of the problem (1.1)-(1.2). This completes the proof.
Then we have the following uniqueness result.

Examples
In this section, we give two examples to illustrate Theorem 3.4 and Theorem 3.6. is a highly oscillating function, which is not Lebesgue integrable. The function g(t, x) = u(t) = H(t− 2 3 ) is of bounded variation but not continuous; g(t, x) and u(t) have a common point of discontinuity at t = 2 3 , so g(t, x) is not Riemann-Stieltjes integrable with respect to u(t). However, in the two examples, the Henstock-Kurzweil-Stieltjes integral is valid. This implies that our results are more extensive.