R E S E A R C H
Open Access
New Riesz representations of linear maps
associated with certain boundary value
problems and their applications
Wei Yang
1, Jiannan Duan
1, Wenmin Hu
2and Jing Zhang
3**Correspondence:
3Faculty of Science and Technology,
Tapee College, Surathani, 84000, Thailand
Full list of author information is available at the end of the article
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.
Keywords: Riesz decomposition method; representation; vector-valued map
1 Introduction
Physicists have long been using so-called singular functions such as the Dirac delta func-tionδ, although these cannot be properly defined within the framework of classical func-tion theory. The Dirac delta funcfunc-tionδ(x–ξ) is equal to zero everywhere except atξ, 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 δ when calculating the transition rates of certain particle
interac-tions []. In [], a definition of a product of distribuinterac-tions was given using delta sequences. However,δ as a product ofδ with itself, was shown not to exist. In [], Bremermann used the Cauchy representations of distributions with compact support to define√δ+and
logδ+. Unfortunately, his definition did not carry over to √
δand logδ. In , Gel’fand and Shilov [] definedδ[(k)](P) for an infinitely differentiable functionP(x,x, . . . ,xn) such
that theP= hypersurface has no singular points, where
P=P(x,x, . . . ,xn) =x+x+· · ·+xp–xp+–· · ·–xp+q, (.)
p+q=nis the dimension of the Euclidean spaceRn, theP= hypersurface is a
hyper-cone with a singular point (the vertex) at the origin. Then they also defined the generalized functionsδ(k)(P) andδ(k)(P) as in the casesp,q> andp,q= , respectively. To establish the numerous properties ofPdefined by (.) Bliedtner and Hansen first showed that it was a quotient of the larger Feller compactification in []. 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 onP. They called them continuous lin-ear maps. In developing their properties, Ikegami gave several equivalent conditions that
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force them to have an integral representation even with respect to minimal representing measures onthe boundary ofPin []. Several examples given by the Laplace equation and the heat equation showed thatPwas in general different from the Martin compactifica-tion; 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 []. Based on earlier work of the authors in [], and [] concerning the boundary behavior of continuous linear maps, the second author and Weizsäcker had shown that a required condition was naturally satisfied when the under-lying measure space was second countable. Samuelsson [] studied the residue of the gen-eralized functionGλ, whereλwas a complex number. This generalized functionGλhave
been used for various purposes by several authors; notably for instance the explicit proof of the duality theorem for a complete intersection in [], explicit versions of the fundamen-tal principle in [], sharp approximation by polynomials [], and estimates of solutions to the Bezout equation in []; for further examples in [] and the references therein. One can also use such generalized functions to obtain sharp estimates at the boundary, such asHp-estimates, of explicit solutions to division problems in []. In , Buriol and
Fer-reira [] 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 [], where the authors applied a linear boundary control force in position and velocity at the free end of the beam. Recently, Yan [] studied the generalization of distributional product of Dirac’s delta in a hypercone, whose results are a generalization of formulas that appear in []. Furthermore, he also used a much simpler method of deriving the product
f(r)·δ(k)(r– ) for all non-negative integerkandr= (x
+x+· · ·+xn)/, and then studied
a more general productf(H)·δ(k)(H), whereHis a regular hypersurface. And they found
the productPn·δ(k)(P) as well as a general productf(P)·δ(k)(P), wheref is aC∞-function
onR. Another study of the products of particular distributions and the development of other work can be found in [, ].
By using augmented Riesz decomposition methods developed by Wang, Huang and Yamini [], the purpose of this paper is to study the productGl·δ(k)(G) and then study
a more general product off(G)·δ(k)(G), wheref is aC∞-function onRandδ(k)(G) is the
Dirac delta function withk-derivatives. Meanwhile, we shall show that we can control the
L∞norm by theHnorm and a stronger norm with a logarithmic growth or double
log-arithmic growth. The inequality is sharp for the double loglog-arithmic 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 follow-ing definitions and concepts are required.
2 Preliminaries
Definition . Letx= (x,x, . . . ,xn) be a point ofn-dimensional Euclidean spaceRnand
mbe a positive integer. The hypersurfaceG=G(m,x) is defined by
G=G(m,x) =
p
i=
xi
m
–
p+q
j=p+
xj
m
, (.)
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where p+q=nis the dimension ofRn. The hypersurfaceGis due to Berndtsson and
Passare []. We observe that puttingm= in (.), we obtain
G=G(,x) =
p
i=
xi –
p+q
j=p+
xj =P(x) =P, (.)
where the quadratic formPis due to Gel’fand and Shilov [] and is given by (.). The hypersurfaceG= is a generalization of a hyperconeP= with a singular point (the vertex) at the origin.
Definition . Let gradG= , which means there is no singular point onG= . Then we
define
δ(k)(G),φ=
δ(k)(G)φ(x)dx, (.)
whereδ(k) is the Dirac delta function withk-derivatives,φis any testing function in the
Schwartz space S, x= (x,x, . . . ,xn)∈Rn and dx=dxdxdxn. In a sufficiently small
neighborhoodUof any point (x,x, . . . ,xn) of the hypersurfaceG= , we can introduce a
new coordinate system such thatG= becomes one of the coordinate hypersurface. For this purpose, we writeG=uand choose the remaininguicoordinates (withi= , , . . . ,n)
for which the Jacobian
D
x u > ,
where
D
x
u =
∂(x,x, . . . ,xn) ∂(G,u, . . .un)
.
Thus (.) can be written as
δ(k)(G),φ= (–)k ∂
k
∂Gk
φD
x
u G=dudu· · ·dun. (.)
The proof of the following lemma is given in [].
Lemma . Given the hypersurface
G=
p
i=
xi
m
–
p+q
j=p+
xj
m
,
where p+q=n is the dimension ofRn,and m is a positive integer.If we transform to bipolar
coordinates defined by
x=rω, . . . , xp=rωp, xp+=sωp+, . . . , xp+q=sωp+q,
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where
p
i= ωi =
and
p+q
j=p+ ωj= .
Then the hypersurface G can be written by
G=rm–sm,
and we obtain
δ(k)(G),φ=
∞
msm–
∂ ∂s
k
sq–mψ(r,s)
m
s=r
rp–dr (.)
or
δ(k)(G),φ= (–)k
∞
mrm–
∂ ∂r
k
rp–mψ(r,s)
m
r=s
sq–ds, (.)
where
ψ(r,s) =
φd (p)d (q),
and d (p) and d (q) are the elements of surface area on the unit sphere inRp andRq,
respectively.
Now, we assume thatφvanishes in the neighborhood of the origin, so that these integrals will converge for anyk. Now for
(p– ) + (q– m)≥mk
or
k<
m(p+q– m),
the integrals in (.) converge for anyφ(x)∈S. Similarly, for
(q– ) + (p– m)≥mk
or
k<
m(p+q– m),
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the integrals in (.) also converge for anyφ(x)∈S. Thus we take (.) and (.) to be the defining equation forδ(k)(G). On the other hand, if
k≥
m(p+q– m),
then we shall defineδ∗(G),φ andδ∗(G),φ as the regularization of (.) and (.), re-spectively. Forp> andq> , the generalized functionδ∗(k)(G) andδ∗(k)(G) are defined by
δ∗(k)(G),φ=
∞
msm–
∂ ∂s
k
sq–mψ(r,s)
m
s=r
rp–dr
for all
k≥
m(p+q– m),
we have
δ∗(k)(G),φ= (–)k
∞
mrm–
∂ ∂r
k
rp–mψ(r,s)
m
r=s
sq–ds (.)
for
k≥
m(p+q– m).
In particular, form= ,δ∗(k)(G) is reduced toδ(k)(G), andδ∗(k)(G) is reduced toδ(k)(G) (see [, p.]).
3 Main results
Assume that bothp> andq> . Let
G(x) =G(x,x, . . . ,xn) =
x+x+· · ·+xpm–xp++· · ·+xp+qm,
withp+q=n, then theG= hypersurface is a hypercone with a singular point (the vertex) at the origin.
We start by assuming thatφ(x) vanishes in a neighborhood of the origin. The distribu-tionδ(k)(G) is defined by
δ(k)(G),φ= (–)k ∂
k
∂Gk
m
rm–G
q
m–φ
G=
rp–dr d (p)d (q), (.)
which is convergent.
Furthermore, if we transform fromGto
s=rm–G
m,
then we note that
∂ ∂G= –
msm––∂
∂s.
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We may write this in the form
δ(k)(G),φ= msm–
∂ ∂s
k
sq–m φ
m
s=r
rp–dr d (p)d (q). (.)
Let us now define
ψ(r,s) =
φd (p)d (q).
Hence
δ(k)(G),φ=
∞
msm–
∂ ∂s
k
sq–mψ(r,s)
m
s=r
rp–dr. (.)
See Lemma . for more details.
Theorem . The product of Glandδ(k)(G)exists and
Gl·δ(k)(G) =
⎧ ⎨ ⎩
(–)l k! k–lδ
k–l(G) if k≥l,
if k≥l. (.)
Proof From (.), we start with
Gl·δ(k)(G),φ= (–)k ∂
k
∂Gk
Gl
m
rm–G
q
m–φ
G=
rp–dr d (p)d (q)
=
∞
msm–
∂ ∂s
k
rm–smlsq–mψ(r,s)
m
s=r
rp–dr.
Making the substitutionsu=rm,v=smand puttingψ(r,s) =ψ
(u,v), we have
Gl·δ(k)(G),φ= m
∞ ∂ ∂v k
(u–v)lvqm–ψ (u,v)
u=v
upm–du.
Clearly,
∂k ∂vk
(u–v)lvqm–ψ
(u,v)u–v= k i= k i D i
v(u–v)lDkv–i
vqm–ψ
(u,v)u–v
=
i<l
k
i D
i
v(u–v)lDkv–i
vqm–ψ
(u,v)u–v
+
k
l D
i
v(u–v)lDkv–i
vqm–ψ(u,v) u–v
+
i>l
k i D
i
v(u–v)lDkv–i
vqm–ψ(u,v) u–v
=I+I+I,
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where
Div=∂/∂vi. It follows that
I=I=
sincei=l. As forI, we obtain
I= ⎧ ⎨ ⎩
(–)l k! k–lDkv–l{v
q
m–ψ(u,v)} ifk≥l,
ifk≥l. SubstitutingIback and using (.), we obtain
Gl·δ(k)(G) =
⎧ ⎨ ⎩
(–)l k!
k–lδk–l(G) ifk≥l,
ifk≥l,
which completes the proof of theorem.
Example . By lettingm=n=p= in (.) andl=k= in (.), we have
x·δx= –δx.
Obviously, we can extend Theorem . to a more general product as follows.
Theorem . Let f be aC∞-function onR.Then the product of f(G)andδ(k)(G)exists and
f(G)δ(k)(G) =
k
i=
k i = (–)
if(i)()δ(k–i)(G).
Proof LetGl=f(G) and use Theorem .. Moreover, note that
∂k ∂vk
f(u–v)vqm–ψ
(u,v)u–v= k
i=
k
i D
i
vf(u–v)Dkv–i
vqm–ψ
(u,v)u–v
=
k
i=
k i (–)
if(i)()Dk–i v
vqm–ψ
(u,v)u–v.
In particular, we have
sinG·δ(k)(G) =
k
i=
k i (–)
isiniπ
δ
(k–i)(G) (.)
and
eG·δ(k)(G) =
k
i=
k i (–)
iδ(k–i)(G). (.)
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Example . By lettingm=n=p= in (.) andk= in (.), we have
sinx·δx= –δx+δx.
Similarly, by lettingm=n=p= in (.) andk= in (.), we have
ex·δ()x=δ()x– δx+ δx– δx+δx.
4 Numerical simulations
In this section, we give the bifurcation diagrams, phase portraits of model (.) 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 (.) we chooseμ= .,N= .,β= .,γ = .,h∈[, .], and the initial value (S,I) = (., .). We see that model (.) has only one positive equilibriumE.
By calculation we have
E
S∗,I∗=E(., .), α= –., α= ., h=
– √, , and
(μ,N,β,h,γ)∈M,
which shows the correctness of Theorem .. From Theorem ., we see that the equilib-riumE(., .) is stable for
h< –
√
, ,
and loses its stability whenh=–
√ , . If
– √,
<h< .,
then there exist orbits. Moreover, orbits, orbits and period- orbits appear in the rangh∈[., .). At last, the nperiod orbits disappear and the dynamical behaviors are from non-period orbits to the chaotic set with the increasing ofh. We also can find that the rangehis decreasing with the doubled increasing of the period orbits which indicates the Feigenbaum constantδ. The dynamical behavior pro-cesses 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 [].
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,
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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. Fi-nally, 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.
Acknowledgements
The authors thank the referees for their valuable comments, which greatly improved their paper. This work was supported by the National Natural Science Foundation of China (Grant No. 41171176) and Postgraduate Technology Innovation Project of Hunan Province (Grant No. CX2015B243).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JZ drafted the manuscript. WY helped to prepare the revised manuscript and JD carried out the transformation process according to the referee reports. WH corrected typos and grammatical errors throughout the manuscript, making it more readable. All authors read and approved the final manuscript.
Author details
1College of Resources and Environment, Hunan Agricultural University, Changsha, 410128, China.2College of Forestry,
Central South University of Forestry and Technology, Changsha, 410004, China.3Faculty of Science and Technology,
Tapee College, Surathani, 84000, Thailand.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 26 May 2017 Accepted: 27 October 2017
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