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

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 \(p+q = n\) 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].

Now, we assume that ϕ vanishes in the neighborhood of the origin, so that these integrals will converge for any k. Now for or the integrals in (2.5) converge for any \(\phi(x)\in S\). Similarly, for or 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 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 for all we have for

Assume that both \(p > 1\) and \(q > 1\). Let 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 which is convergent.

Furthermore, if we transform from G to then we note that

We may write this in the form

Let us now define

Hence See Lemma 2.1 for more details.

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