The HK-Sobolev space and applications to one-dimensional boundary value problems

1 Introduction

By way of introduction, let us begin with the motivation for this work.

A classical variational problem. The Sobolev space is given by

Given $f\in L^2([0,1])$, find $u\in H_0^1([0,1])$such that

The space $H_0^1([0,1])$ has certain properties that are useful in order to find a solution to variational problems (2): it is a Hilbert, separable and reflexive space.

What happens if f is not of square Lebesgue integrable? In several physical phenomena, highly oscillating or singular functions appear (Condon et al., 2009; Hamed and Cummins, 1991; Hong and Xu, 2001; Samoilov et al., 2005). The Lebesgue integral is not enough for some highly oscillating functions leading to the possibility that the integral on the right side of the Eq. (2) does not exist for this type of functions and so the variational problem (2) would not be well defined. One way to solve this problem is to change the type of integral to be considered, in this work we will use the Henstock–Kurzweil integral. Different authors have studied differential equations involving Henstock–Kurzweil integrable functions. In León-Velasco et al. (2019) the authors use the Finite Element Method (FEM) for finding numerical solutions of elliptic problems with Henstock–Kurzweil integrable functions. They use open quadratures and Lobatto quadratures to approximate numerically the integrals that appear in the FEM. In Liu et al. (2018) are given conditions to establish the existence of a solution to nonlinear second-order differential equations of type $-D^{2}x=f(t,x)+g(t,x)\mathit{Du}$ subject to the boundary conditions $x(0)=\beta \mathit{Dx}(0),\mathit{Dx}(1)+\mathit{Dx}(\eta )=0$, where the derivatives are in the distributional sense, $x,u$ are regulated functions and g is of bounded variation. In that paper the Henstock–Kurzweil–Stieltjes integral is used to transform the distributional differential equation into an integral equation, then the Leray–Schauder nonlinear alternative theorem is applied for finding a solution. In Sánchez-Perales and Mendoza-Torres (2020) the existence and uniqueness of the Shrödinger equation, $-y^{″}+\mathit{qy}=f$ a.e. on $[a,b]$ subject to arbitrary boundary values, is guaranteed for functions $f,q$ Henstock–Kurzweil integrable. Properties of the inverse of the Shrödinger operator are established, then the authors give conditions so that the solution of the differential equation can be expressed as a Fourier type series.

Henstock–Kurzweil–Sobolev space. Around the 1960s, R. Henstock and J. Kurzweil, independently, define a Riemann-type integral, known as Henstock–Kurzweil integral, which is equivalent to Denjoy and Perron integrals. This integral is more general than the Lebesgue integral. In this work we introduce, using the Henstock–Kurzweil integral instead of the Lebesgue integral, a space analogous to $H^1([0,1])$, which we will call the Henstock–Kurzweil–Sobolev space and denote it by $W_{\mathit{HK}}$. Since the product of two Henstock–Kurzweil integrable functions is not necessarily an integrable function, the $W_{\mathit{HK}}$space is not provided with a natural internal product. Thus we cannot apply classical theorems, such as Lax–Milgram’s, to guarantee the existence and uniqueness of the solution to variational problems such as (2). In this paper, we will use Fredholm’s alternative for compact operators and the properties of the Henstock–Kurzweil–Sobolev space to solve such problems.

2 Preliminaries

The symbol $\mathbb{R}$ denotes the set of real numbers, $\mathbb{C}$ stands for the complex numbers and $[a,b]\subset \mathbb{R}$ is a closed finite interval. A tagged partition $\{([t_{i-1},t_i],{\xi }_i):i=1,\dots ,n\}$ of $[a,b]$ is a finite collection of non-overlapping intervals $[t_{i-1},t_i]$ such that $[a,b]={\cup }_{i=1}^n[t_{i-1},t_i]$, and ${\xi }_i\in [t_{i-1},t_i]$ for all $i=1,\dots ,n$. A function $\delta :[a,b]\to \mathbb{R}$ is a gauge on $[a,b]$ if $\delta (t)>0$ for every $t\in [a,b]$. Given a gauge $\delta$ on $[a,b]$ and a tagged partition $P=\{([t_{i-1},t_i],{\xi }_i):i=1,\dots ,n\}$ of $[a,b],P$ is $\delta$-fine if$$[t_{i-1},t_i]\subset ({\xi }_i-\delta ({\xi }_i),{\xi }_i+\delta ({\xi }_i)),i=1,\dots ,n\text{.}$$

A function $f:[a,b]\to \mathbb{C}$ is Henstock–Kurzweil integrable (HK-integrable) on $[a,b]$ if there exists a number I such that for every $∊>0$ there exists a gauge $\delta$ on $[a,b]$ such that for each $\delta$-fine tagged partition $\{([t_{i-1},t_i],{\xi }_i):i=1,\dots ,n\}$ of $[a,b]$,$$\left|{\displaystyle \sum _{i=1}^n}f({\xi }_i)(t_i-t_{i-1})-I\right|<∊\text{.}$$

The number I is called the integral of f over $[a,b]$ and it is denoted by ${\int }_a^{b}f$. The space of Henstock–Kurzweil integrable functions is denoted by $\mathit{HK}([a,b])$. The Alexiewicz semi-norm of a function $f\in \mathit{HK}([a,b])$ is defined by$$\Vert f{\Vert }_A=\underset{t\in [a,b]}{\sup }\left\{\left|{\int }_a^{t}f\right|\right\}\text{.}$$

A function $\phi :[a,b]\to \mathbb{C}$ is of bounded variation on $[a,b]$ ($\phi \in \mathit{BV}([a,b])$) if$$V_{[a,b]}\phi ≔\sup \left\{{\displaystyle \sum _{i=1}^n}|\phi (d_i)-\phi (c_i)|\right\}<\infty ,$$where the supremum is taken over all finite collections $\{[c_i,d_i]:i=1,\dots ,n\}$ of non-overlapping intervals of $[a,b]$.

The next Fubini’s Theorem is a direct consequence of Talvila (2002, Lemma 25).

A function $F:[a,b]\to \mathbb{C}$ is absolutely continuous (respectively, absolutely continuous in the restricted sense) on a set $E\subseteq [a,b]$, if for each $∊>0$ there exists $\delta >0$ such that ${\sum }_{i=1}^s|F(d_i)-F(c_i)|<∊$ (respectively, ${\sum }_{i=1}^s\sup \{|F(x)-F(y)|:x,y\in [c_i,d_i]\}<∊$) whenever ${\{[c_i,d_i]\}}_{i=1}^s$ is a collection of non-overlapping intervals with endpoints in E and such that ${\sum }_{i=1}^s(d_i-c_i)<\delta$. The space of absolutely continuous functions on E is denoted by $\mathit{AC}(E)$, and the space of absolutely continuous functions in the restricted sense on E is denoted by ${\mathit{AC}}_{\ast }(E)$.

The function F is generalized absolutely continuous in the restricted sense on $[a,b]$ ($F\in {\mathit{ACG}}_{\ast }([a,b])$), if F is continuous on $[a,b]$ and there exists a countable collection ${(E_n)}_{n=1}^{\infty }$ of subsets of $[a,b]$ such that $[a,b]={\cup }_{i=1}^{\infty }{E}_n$ and $F\in {\mathit{AC}}_{\ast }(E_n)$ for all $n\in \mathbb{N}$. This concept leads to a very strong version of the fundamental theorem of calculus:

3 The HK-Sobolev space

In what follows from this document, we have presented the results on the interval $[0,1]$ without loss of generality, since they can be generalized to any compact interval. Let $C_P^2([0,1])$ the space of all functions $\phi \in C([0,1])$ for which there exists ${\{[t_{i-1},t_i]\}}_{i=1}^n$ a partition of $[0,1]$ such that $\phi \in C^2((t_{i-1},t_i))$ for all $i=1,\cdots ,n$; and ${\phi }^{(k)}(t_0+),{\phi }^{(k)}(t_1-),{\phi }^{(k)}(t_1+),\cdots ,{\phi }^{(k)}(t_{n-1}-),{\phi }^{(k)}(t_{n-1}+),{\phi }^{(k)}(t_n-)$ exist for all $k=1,2$. We set$$V=\{\phi \in C_P^2([0,1]):\phi (0)=\phi (1)=0\}\text{.}$$

It is clear that if $\phi \in C_P^2([0,1])$, then $\phi$ and ${\phi }^{\prime }$ belong to $\mathit{AC}([0,1])(\subseteq \mathit{BV}([0,1]))$. The next theorem is proved in a similar way to Hestenes (1966, Lemma 15.2, p. 51) with some modifications.

For $u\in W_{\mathit{HK}}$ we define the weak derivative of u, denoted by $\dot{u}$, as$$\dot{u}=g,$$where g is the function given in Definition 1. Observe, by Corollary 3.2, $\dot{u}$ is well defined. Also, from Theorem 3.4 we have that for every $f\in \mathit{HK}([0,1])$, there exists a continuous function v defined on $[0,1]$ such that $\dot{v}=f$, that is, each HK-integrable function is the weak derivative of a some continuous function.

4 Existence and uniqueness of a solution of a boundary value problem

Define the spaces$$W_{\mathit{HK}}^2=\{u\in W_{\mathit{HK}}:\dot{u}\in W_{\mathit{HK}}\}$$and$$W_{{\mathit{HK}}_0}=\{u\in W_{\mathit{HK}}:u(0)=u(0+)=u(1)=u(1-)=0\}\text{.}$$

Let $f\in \mathit{HK}([0,1])$ and let $q,\rho$ be real valued functions such that $q\in L^2([0,1]),\rho \in {\mathit{ACG}}_{\ast }([0,1])\cap \mathit{BV}([0,1])$ and $|\rho (x)|⩾\alpha$ for all $x\in [0,1]$ and some $\alpha >0$. Consider the following problems:

• I. Find $u\in W_{\mathit{HK}}^2$ that satisfies

  (7)

  $$\begin{array}{c}-[\rho \dot{u}]̇+\mathit{qu}=f\mathrm{a}.\mathrm{e}.\mathrm{on}(0,1)\text{;}\hfill \\ u(0)=u(0+)=0,u(1)=u(1-)=0\text{.}\hfill \end{array}$$

• II. Find $u\in W_{{\mathit{HK}}_0}$ that satisfies

  (8)

  $${\int }_0^1\rho \dot{u}{\phi }^{\prime }+{\int }_0^1\mathit{qu}\phi ={\int }_0^{1}f\phi$$for all $\phi \in V$.

The boundary value problem (I) and the variational problem (II) are equivalent. Indeed, suppose that $u\in W_{\mathit{HK}}^2$ is a solution of the boundary value problem (I). Multiplying both sides of the differential equation in (7) by $\phi \in V$ and integrating it from 0 to 1, we obtain that$$-{\int }_0^1[\rho \dot{u}]̇\phi +{\int }_0^1\mathit{qu}\phi ={\int }_0^{1}f\phi \text{.}$$

Therefore$${\int }_0^1\rho \dot{u}{\phi }^{\prime }+{\int }_0^1\mathit{qu}\phi ={\int }_0^{1}f\phi$$for all $\phi \in V$. Conversely, suppose that $u\in W_{{\mathit{HK}}_0}$ satisfies (8) for all $\phi \in V$. Then,$${\int }_0^1\rho \dot{u}{\phi }^{\prime }=-{\int }_0^1(\mathit{qu}-f)\phi$$for all $\phi \in V$. Thus $\rho \dot{u}\in W_{\mathit{HK}}$. Since $\frac{1}{\rho }\in {\mathit{ACG}}_{\ast }([0,1])$, it follows by Proposition 3.5 and Corollary 3.7 that $\dot{u}\in W_{\mathit{HK}}$, consequently $u\in W_{\mathit{HK}}^2$. On the other hand, since $\rho \dot{u}\in W_{\mathit{HK}}$ we have that$${\int }_0^1\rho \dot{u}{\phi }^{\prime }=-{\int }_0^1(\rho \dot{u})̇\phi$$for all $\phi \in V$. Consequently, from (8),$${\int }_0^1[-(\rho \dot{u})̇+\mathit{qu}-f]\phi =0$$for all $\phi \in V$. Hence, by Corollary 3.2, $-(\rho \dot{u})̇+\mathit{qu}=f$ a.e. on $[0,1]$.

To find a solution to the boundary value problem (I), we demonstrate the existence and uniqueness of the variational problem (II). Define B on $W_{{\mathit{HK}}_0}\times V$ and $l_u,l_f$ on V by$$\begin{array}{cc}\hfill B(u,\phi )=& {\int }_0^1\rho \dot{u}{\phi }^{\prime },\hfill \\ \hfill l_u(\phi )=& {\int }_0^1\mathit{qu}\phi ,\hfill \\ \hfill l_f(\phi )=& {\int }_0^{1}f\phi \text{.}\hfill \end{array}$$

It is clear that $l_u$ and $l_f$ are linear operators and B is a bilinear operator. The variational problem (8) is equivalent to find $u\in W_{{\mathit{HK}}_0}$ such that

Affirmation 1. There exists an operator $A:W_{{\mathit{HK}}_0}\to W_{{\mathit{HK}}_0}$ such that $l_u(\phi )=B(A(u),\phi )$ for all $u\in W_{{\mathit{HK}}_0}$ and $\phi \in V$.

Affirmation 2. There exists a function $v_f\in W_{{\mathit{HK}}_0}$ for which $l_f(\phi )=B(v_f,\phi )$ for all $\phi \in V$.

Therefore, the problem (9) is equivalent to find $u\in W_{{\mathit{HK}}_0}$ such that$$B(u,\phi )+B(A(u),\phi )=B(v_f,\phi ),$$for all $\phi \in V$, or$$B(u+A(u)-v_f,\phi )=0,$$for all $\phi \in V$.

Remember that an operator $T:X\to Y$ between two normed spaces is compact if and only if for any bounded sequence $(x_n)$ in X, the sequence $({\mathit{Tx}}_n)$ contains a converging subsequence. The following is the Arzelá-Ascoli Theorem.

Affirmation 3. The operator $A:(W_{{\mathit{HK}}_0},\Vert ·\Vert )\to (W_{{\mathit{HK}}_0},\Vert ·\Vert )$ is compact, where $\Vert u\Vert =\Vert u{\Vert }_2+\Vert \dot{u}{\Vert }_A$.