RETRACTED ARTICLE: New Riesz representations of linear maps associated with certain boundary value problems and their applications

Abstract

In this paper, we obtain new Riesz representations of continuous linear maps associated with certain boundary value problems in the set of all closed bounded convex non-empty subsets of any Banach space. As applications, the Riesz integral representation results are also given.

1 Introduction

Physicists have long been using so-called singular functions such as the Dirac delta function δ, although these cannot be properly defined within the framework of classical function theory. The Dirac delta function $\delta(x-\xi)$ is equal to zero everywhere except at ξ, where it is infinite, and its integral is one. According to the classical definition of a function and an integral these conditions are inconsistent. In elementary particle physics, one found the need to evaluate $\delta^{2}$ when calculating the transition rates of certain particle interactions [1]. In [2], a definition of a product of distributions was given using delta sequences. However, $\delta^{2}$ as a product of δ with itself, was shown not to exist. In [3], Bremermann used the Cauchy representations of distributions with compact support to define $\sqrt{\delta_{+}}$ and log $\delta_{+}$. Unfortunately, his definition did not carry over to $\sqrt{\delta}$ and log δ. In 1964, Gel’fand and Shilov [4] defined $\delta^{[(k)]}(P)$ for an infinitely differentiable function $P(x_{1},x_{2},\ldots,x_{n})$ such that the $P =0$ hypersurface has no singular points, where

$$ P = P(x_{1},x_{2},\ldots,x_{n}) = x_{1}^{2}+x_{2}^{2}+\cdots+x_{p}^{2}-x_{p+1}^{2}- \cdots -x_{p+q}^{2}, $$

(1.1)

$p+q = n$ is the dimension of the Euclidean space $\mathbb{R}^{n}$, the $P = 0$ hypersurface is a hypercone with a singular point (the vertex) at the origin. Then they also defined the generalized functions $\delta_{1}^{(k)}(P)$ and $\delta_{2}^{(k)}(P)$ as in the cases $p, q > 1$ and $p, q = 1$, respectively. To establish the numerous properties of P defined by (1.1) Bliedtner and Hansen first showed that it was a quotient of the larger Feller compactification in [5]. It then turned out that functions that were exactly the uniform limits on compact sets of sequences of bounded harmonic functions allowed a nice integral representation on P. They called them continuous linear maps. In developing their properties, Ikegami gave several equivalent conditions that force them to have an integral representation even with respect to minimal representing measures onthe boundary of P in [6]. Several examples given by the Laplace equation and the heat equation showed that P was in general different from the Martin compactification; It was, however, the same for ordinary harmonic functions on Lipschitz domains. Conditions were also presented that force all positive harmonic functions to be sturdy, extending the results first presented in [5]. Based on earlier work of the authors in [7], and [8] concerning the boundary behavior of continuous linear maps, the second author and Weizsäcker had shown that a required condition was naturally satisfied when the underlying measure space was second countable. Samuelsson [9] studied the residue of the generalized function $G^{\lambda}$, where λ was a complex number. This generalized function $G^{\lambda}$ have been used for various purposes by several authors; notably for instance the explicit proof of the duality theorem for a complete intersection in [10], explicit versions of the fundamental principle in [11], sharp approximation by polynomials [9], and estimates of solutions to the Bezout equation in [12]; for further examples in [13] and the references therein. One can also use such generalized functions to obtain sharp estimates at the boundary, such as $H^{p}$-estimates, of explicit solutions to division problems in [12]. In 2015, Buriol and Ferreira [14] studied the asymptotic behavior in time of the solutions of a coupled system of linear Maxwell equations with thermal effects. The Riesz basis property and the stability of a damped Euler-Bernoulli beam with nonuniform thickness or density have been studied in [15], where the authors applied a linear boundary control force in position and velocity at the free end of the beam. Recently, Yan [16] studied the generalization of distributional product of Dirac’s delta in a hypercone, whose results are a generalization of formulas that appear in [3]. Furthermore, he also used a much simpler method of deriving the product $f(r)\cdot\delta^{(k)}(r-1)$ for all non-negative integer k and $r =(x_{1}^{2}+ x_{2}^{2}+ \cdots+ x_{n}^{2})^{1/2}$, and then studied a more general product $f(H)\cdot\delta^{(k)}(H)$, where H is a regular hypersurface. And they found the product $P^{n}\cdot\delta ^{(k)}(P)$ as well as a general product $f(P)\cdot\delta^{(k)}(P)$, where f is a $C^{\infty}$-function on $\mathbb{R}$. Another study of the products of particular distributions and the development of other work can be found in [8, 17].

By using augmented Riesz decomposition methods developed by Wang, Huang and Yamini [17], the purpose of this paper is to study the product $G^{l}\cdot\delta ^{(k)}(G)$ and then study a more general product of $f(G)\cdot\delta^{(k)}(G)$, where f is a $C^{\infty}$-function on $\mathbb{R}$ and $\delta^{(k)}(G)$ is the Dirac delta function with k-derivatives. Meanwhile, we shall show that we can control the $L^{\infty}$ norm by the $H^{1}$ norm and a stronger norm with a logarithmic growth or double logarithmic growth. The inequality is sharp for the double logarithmic growth. The result there is used earlier in our paper to obtain a boundary limit theorem for sturdy harmonic functions and continuous linear maps. Before proceeding to our main results, the following definitions and concepts are required.

2 Preliminaries

Definition 2.1

Let $x = (x_{1}, x_{2}, \ldots, x_{n})$ be a point of n-dimensional Euclidean space $\mathbb{R}^{n}$ and m be a positive integer. The hypersurface $G = G(m,x)$ is defined by

$$ G = G(m,x) = \Biggl(\sum_{i=1}^{p} x_{i}^{2} \Biggr)^{m}- \Biggl(\sum _{j=p+1}^{p+q} x_{j}^{2} \Biggr)^{m}, $$

(2.1)

where $p+q=n$ is the dimension of $\mathbb{R}^{n}$. The hypersurface G is due to Berndtsson and Passare [11]. We observe that putting $m =1$ in (2.1), we obtain

$$ G= G(1,x) =\sum_{i=1}^{p} x_{i}^{2}-\sum_{j=p+1}^{p+q} x_{j}^{2}= P(x) = P, $$

(2.2)

where the quadratic form P is due to Gel’fand and Shilov [4] and is given by (1.1). The hypersurface $G = 0$ is a generalization of a hypercone $P = 0$ with a singular point (the vertex) at the origin.

Definition 2.2

Let grad $G \neq0$, which means there is no singular point on $G = 0$. Then we define

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = \int\delta ^{(k)}(G)\phi(x) \,dx, $$

(2.3)

where $\delta^{(k)}$ is the Dirac delta function with k-derivatives, ϕ is any testing function in the Schwartz space S, $x = (x_{1},x_{2}, \ldots,x_{n})\in\mathbb{R}^{n}$ and $dx = dx_{1}\,dx_{2}\,dx_{n}$. In a sufficiently small neighborhood U of any point $(x_{1},x_{2},\ldots ,x_{n})$ of the hypersurface $G = 0$, we can introduce a new coordinate system such that $G = 0$ becomes one of the coordinate hypersurface. For this purpose, we write $G = u_{1}$ and choose the remaining $u_{i}$ coordinates (with $i = 2,3,\ldots,n$) for which the Jacobian

$$D\binom{x}{u}> 0, $$

where

$$D\binom{x}{u}=\frac{\partial(x_{1},x_{2},\ldots, x_{n})}{\partial (G,u_{2},\ldots u_{n})}. $$

Thus (2.3) can be written as

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = (-1)^{k} \int \biggl[\frac {\partial^{k}}{\partial G^{k}} \biggl\{ \phi D\binom{x}{u} \biggr\} \biggr]_{G=0}\,du_{2}\,du_{3} \cdots \,du_{n}. $$

(2.4)

The proof of the following lemma is given in [17].

Lemma 2.1

Given the hypersurface

$$G = \Biggl(\sum_{i=1}^{p} x_{i}^{2} \Biggr)^{m}- \Biggl(\sum _{j=p+1}^{p+q} x_{j}^{2} \Biggr)^{m}, $$

where $p + q = n$ is the dimension of $\mathbb{R}^{n}$, and m is a positive integer. If we transform to bipolar coordinates defined by

$$x_{1} = r\omega_{1},\qquad \ldots,\qquad x_{p} = r \omega_{p},\qquad x_{p+1} = s\omega_{p+1},\qquad \ldots ,\qquad x_{p+q} = s\omega_{p+q}, $$

where

$$\sum_{i=1}^{p} \omega_{i}^{2}=1 $$

and

$$\sum_{j=p+1}^{p+q} \omega_{j}^{2}=1. $$

Then the hypersurface G can be written by

$$G =r^{2m}-s^{2m}, $$

and we obtain

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2ms^{2m-1}} \frac{\partial}{\partial s} \biggr)^{k} \biggl\{ s^{q-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{s=r}r^{p-1}\,dr $$

(2.5)

or

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle =(-1)^{k} \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2mr^{2m-1}} \frac{\partial}{\partial r} \biggr)^{k} \biggl\{ r^{p-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{r=s}s^{q-1}\,ds, $$

(2.6)

where

$$\psi(r,s) = \int\phi \, d\Omega^{(p)}\, d\Omega^{(q)}, $$

and $d\Omega^{(p)}$ and $d\Omega^{(q)}$ are the elements of surface area on the unit sphere in $\mathbb{R}^{p}$ and $\mathbb{R}^{q}$, respectively.

Now, we assume that ϕ vanishes in the neighborhood of the origin, so that these integrals will converge for any k. Now for

$$(p-1)+(q-2m) \ge2mk $$

or

$$k< \frac{1}{2m}(p+q-2m), $$

the integrals in (2.5) converge for any $\phi(x)\in S$. Similarly, for

$$(q-1)+(p-2m) \ge2mk $$

or

$$k< \frac{1}{2m}(p+q-2m), $$

the integrals in (2.6) also converge for any $\phi(x)\in S$. Thus we take (2.5) and (2.6) to be the defining equation for $\delta^{(k)}(G)$. On the other hand, if

$$k\ge\frac{1}{2m}(p+q-2m), $$

then we shall define $\langle\delta_{1}^{*}(G),\phi \rangle$ and$\langle\delta_{2}^{*}(G),\phi \rangle$ as the regularization of (2.5) and (2.6), respectively. For $p>1$ and $q>1$, the generalized function $\delta_{1}^{*(k)}(G)$ and $\delta_{2}^{*(k)}(G)$ are defined by

$$\bigl\langle \delta_{1}^{*(k)}(G),\phi \bigr\rangle = \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2ms^{2m-1}} \frac{\partial}{\partial s} \biggr)^{k} \biggl\{ s^{q-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{s=r}r^{p-1}\,dr $$

for all

$$k\ge\frac{1}{2m}(p+q-2m), $$

we have

$$ \bigl\langle \delta_{2}^{*(k)}(G),\phi \bigr\rangle =(-1)^{k} \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2mr^{2m-1}} \frac{\partial}{\partial r} \biggr)^{k} \biggl\{ r^{p-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{r=s}s^{q-1}\,ds $$

(2.7)

for

$$k\ge\frac{1}{2m}(p+q-2m). $$

In particular, for $m = 1$, $\delta_{1}^{*(k)}(G)$ is reduced to $\delta _{1}^{(k)}(G)$, and $\delta_{2}^{*(k)}(G)$ is reduced to $\delta _{2}^{(k)}(G)$ (see [4, p.250]).

3 Main results

Assume that both $p > 1$ and $q > 1$. Let

$$G(x) = G(x_{1},x_{2},\ldots,x_{n}) = \bigl(x_{1}^{2}+x_{2}^{2}+\cdots +x_{p}^{2}\bigr)^{m}-\bigl(x_{p+1}^{2}+ \cdots+x_{p+q}^{2}\bigr)^{m}, $$

with $p+q=n$, then the $G=0$ hypersurface is a hypercone with a singular point (the vertex) at the origin.

We start by assuming that $\phi(x)$ vanishes in a neighborhood of the origin. The distribution $\delta^{(k)}(G)$ is defined by

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle =(-1)^{k} \int \biggl[ \frac {\partial^{k}}{\partial G^{k}} \biggl\{ \frac{1}{2m} \bigl(r^{2m}-G\bigr)^{\frac {q}{2m}-1}\phi \biggr\} \biggr]_{G=0}r^{p-1} \, dr\, d\Omega^{(p)}\, d\Omega^{(q)}, $$

(3.1)

which is convergent.

Furthermore, if we transform from G to

$$s=\bigl(r^{2m}-G\bigr)^{\frac{1}{2m}}, $$

then we note that

$$\frac{\partial}{\partial G}= -\bigl(2ms^{2m-1}\bigr)^{-1} \frac{\partial}{\partial s}. $$

We may write this in the form

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = \int \biggl[ \biggl( \frac{1}{2ms^{2m-1}}\frac{\partial}{\partial s} \biggr)^{k} \biggl\{ s^{q-2m}\frac{\phi}{2m} \biggr\} \biggr]_{s=r}r^{p-1}\, dr\, d\Omega ^{(p)}\, d \Omega^{(q)}. $$

(3.2)

Let us now define

$$\psi(r,s)= \int\phi \, d\Omega^{(p)}\, d\Omega^{(q)}. $$

Hence

$$ \bigl\langle \delta^{(k)}(G),\phi \bigr\rangle = \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2ms^{2m-1}} \frac{\partial}{\partial s} \biggr)^{k} \biggl\{ s^{q-2m} \frac{\psi(r,s)}{2m} \biggr\} \biggr]_{s=r}r^{p-1}\,dr. $$

(3.3)

See Lemma 2.1 for more details.

Theorem 3.1

The product of $G^{l}$ and $\delta^{(k)}(G)$ exists and

$$ G^{l}\cdot\delta^{(k)}(G)= \textstyle\begin{cases} (-1)^{l}\frac{k!}{k-l}\delta^{k-l}(G)&\textit{if } k \ge l, \\ 0&\textit{if } k\ge l. \end{cases} $$

(3.4)

Proof

From (3.1), we start with

$$\begin{aligned} \bigl\langle G^{l}\cdot\delta^{(k)}(G),\phi \bigr\rangle =&(-1)^{k} \int \biggl[ \frac{\partial^{k}}{\partial G^{k}} \biggl\{ G^{l}\frac{1}{2m} \bigl(r^{2m}-G\bigr)^{\frac{q}{2m}-1}\phi \biggr\} \biggr]_{G=0}r^{p-1} \, dr\, d\Omega ^{(p)}\, d\Omega^{(q)} \\ =& \int_{0}^{\infty}\biggl[ \biggl( \frac{1}{2ms^{2m-1}} \frac{\partial }{\partial s} \biggr)^{k} \biggl\{ \bigl(r^{2m}-s^{2m} \bigr)^{l} s^{q-2m}\frac{\psi (r,s)}{2m} \biggr\} \biggr]_{s=r}r^{p-1}\,dr. \end{aligned}$$

Making the substitutions $u=r^{2m}$, $v=s^{2m}$ and putting $\psi (r,s)=\psi_{1}(u,v)$, we have

$$\bigl\langle G^{l}\cdot\delta^{(k)}(G),\phi \bigr\rangle = \frac{1}{4m^{2}} \int_{0}^{\infty}\biggl[ \biggl( \frac{\partial}{\partial v} \biggr)^{k} \bigl\{ (u-v)^{l} v^{\frac{q}{2m}-1} \psi_{1}(u,v) \bigr\} \biggr]_{u=v}u^{\frac{p}{2m}-1}\,du. $$

Clearly,

$$\begin{aligned} \frac{\partial^{k}}{\partial v^{k}} \bigl\{ (u-v)^{l} v^{\frac{q}{2m}-1}\psi _{1}(u,v) \bigr\} \big|_{u-v} =&\sum_{i=0}^{k} \binom{k}{i}D_{v}^{i}(u-v)^{l}D_{v}^{k-i} \bigl\{ v^{\frac {q}{2m}-1}\psi_{1}(u,v) \bigr\} \big|_{u-v} \\ =&\sum^{i< l}\binom{k}{i}D_{v}^{i}(u-v)^{l}D_{v}^{k-i} \bigl\{ v^{\frac {q}{2m}-1}\psi_{1}(u,v) \bigr\} \big|_{u-v} \\ &{}+\binom{k}{l}D_{v}^{i}(u-v)^{l}D_{v}^{k-i} \bigl\{ v^{\frac{q}{2m}-1}\psi _{1}(u,v) \bigr\} \big|_{u-v} \\ &{}+\sum^{i>l}\binom{k}{i}D_{v}^{i}(u-v)^{l}D_{v}^{k-i} \bigl\{ v^{\frac {q}{2m}-1}\psi_{1}(u,v) \bigr\} \big|_{u-v} \\ =&I_{1}+I_{2}+I_{3}, \end{aligned}$$

where

$$D_{v}^{i}= \partial/\partial v^{i}. $$

It follows that

$$I_{1}=I_{3}=0 $$

since $i\neq l$. As for $I_{2}$, we obtain

$$I_{2}= \textstyle\begin{cases} (-1)^{l}\frac{k!}{k-l}D_{v}^{k-l} \{v^{\frac{q}{2m}-1}\psi_{1}(u,v) \}&\mbox{if } k \ge l, \\ 0&\mbox{if } k\ge l. \end{cases} $$

Substituting $I_{2}$ back and using (3.1), we obtain

$$G^{l}\cdot\delta^{(k)}(G)= \textstyle\begin{cases} (-1)^{l}\frac{k!}{k-l}\delta^{k-l}(G)&\mbox{if } k \ge l, \\ 0&\mbox{if } k\ge l, \end{cases} $$

which completes the proof of theorem. □

Example 3.1

By letting $m = n = p = 1$ in (2.1) and $l = k = 3$ in (3.4), we have

$$x^{6}\cdot\delta^{\prime\prime\prime}\bigl(x^{2}\bigr)=-6\delta \bigl(x^{2}\bigr). $$

Obviously, we can extend Theorem 3.1 to a more general product as follows.

Theorem 3.2

Let f be a $\mathcal{C}^{\infty}$-function on $\mathbb{R}$. Then the product of $f(G)$ and $\delta^{(k)}(G)$ exists and

$$f(G)\delta^{(k)}(G)=\sum_{i=0}^{k} \binom{k}{i}=(-1)^{i} f^{(i)}(0)\delta ^{(k-i)}(G). $$

Proof

Let $G^{l} = f(G)$ and use Theorem 3.1. Moreover, note that

$$\begin{aligned} \frac{\partial^{k}}{\partial v^{k}} \bigl\{ f(u-v) v^{\frac{q}{2m}-1}\psi _{1}(u,v) \bigr\} \big|_{u-v} =&\sum_{i=0}^{k} \binom{k}{i}D_{v}^{i} f(u-v)D_{v}^{k-i} \bigl\{ v^{\frac {q}{2m}-1}\psi_{1}(u,v) \bigr\} \big|_{u-v} \\ =&\sum_{i=0}^{k}\binom{k}{i}(-1)^{i} f^{(i)}(0)D_{v}^{k-i} \bigl\{ v^{\frac {q}{2m}-1} \psi_{1}(u,v) \bigr\} \big|_{u-v}. \end{aligned}$$

In particular, we have

$$ \sin G\cdot\delta^{(k)}(G)=\sum_{i=0}^{k} \binom{k}{i}(-1)^{i}\sin\frac {i\pi}{2}\delta^{(k-i)}(G) $$

(3.5)

and

$$ e^{G}\cdot\delta^{(k)}(G)=\sum_{i=0}^{k} \binom{k}{i}(-1)^{i}\delta ^{(k-i)}(G). $$

(3.6)

 □

Example 3.2

By letting $m=n= p=1$ in (2.1) and $k=3$ in (3.5), we have

$$\sin x^{2}\cdot\delta^{\prime\prime\prime}\bigl(x^{2}\bigr)=-3 \delta^{\prime\prime }\bigl(x^{2}\bigr)+\delta\bigl(x^{2} \bigr). $$

Similarly, by letting $m=n=p=1$ in (2.1) and $k=4$ in (3.6), we have

$$e^{x^{2}}\cdot\delta^{(4)}\bigl(x^{2}\bigr)= \delta^{(4)}\bigl(x^{2}\bigr)-4\delta^{\prime\prime \prime} \bigl(x^{2}\bigr)+6\delta^{\prime\prime}\bigl(x^{2}\bigr)-4 \delta^{\prime}\bigl(x^{2}\bigr)+\delta\bigl(x^{2} \bigr). $$

4 Numerical simulations

In this section, we give the bifurcation diagrams, phase portraits of model (2.1) to confirm the above theoretic analysis and show the new interesting complex dynamical behaviors by using numerical simulations. The bifurcation parameters are considered in the following two cases:

In model (2.1) we choose $\mu=0.3$, $N=0.7$, $\beta=1.9$, $\gamma=0.1$, $h\in [1,2.9]$, and the initial value $(S_{0},I_{0})=(0.01,0.01)$. We see that model (2.1) has only one positive equilibrium $E_{2}$. By calculation we have

$$\begin{aligned}& E_{2}\bigl(S^{*},I^{*}\bigr)=E_{2}(0.1474,0.4145), \qquad \alpha_{1}=-0.9524, \\& \alpha _{2}=0.8811, \qquad h= \frac{570-4\sqrt{2\text{,}306}}{180}, \end{aligned}$$

and

$$(\mu,N,\beta,h,\gamma)\in M_{1}, $$

which shows the correctness of Theorem 3.1. From Theorem 3.2, we see that the equilibrium $E_{2}(0.1474,0.4145)$ is stable for

$$h< \frac{570-4\sqrt{2\text{,}306}}{180}, $$

and loses its stability when $h=\frac{570-4\sqrt{2\text{,}306}}{180}$. If

$$\frac{570-4\sqrt{2\text{,}306}}{180}< h< 2.64, $$

then there exist period-2 orbits. Moreover, period-4 orbits, period-8 orbits and period-16 orbits appear in the rang $h\in[2.65,2.85)$. At last, the $2^{n}$ period orbits disappear and the dynamical behaviors are from non-period orbits to the chaotic set with the increasing of h. We also can find that the range h is decreasing with the doubled increasing of the period orbits which indicates the Feigenbaum constant δ. The dynamical behavior processes from period-one orbit to chaos sets show self-similar characteristics. Further, the period-doubling transition leads to the chaos sets as May and Odter obtained in [3].

5 Conclusions

In this paper, we firstly obtained the representation of continuous linear maps in the set of all closed bounded convex non-empty subsets of any Banach space. As applications, we secondly deduced the Riesz integral representation results for set-valued maps, for vector-valued maps of Diestel-Uhl and for scalar-valued maps of Dunford-Schwartz. Finally, we gave the bifurcation diagrams, phase portraits of related models to confirm the above theoretic analysis and showed the new interesting complex dynamical behaviors by using numerical simulations.