# On The Solutionn-Dimensional of the Composite ♢k Operator and ♢Bk Operator

Authors: Wanchak Satsanit
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

Firstly, we study the solution of the equation♢k♢Bku(x)=f(x), where ♢k♢Bk is the composite of the diamond operator and Bessel diamond  operator. Finally, we study of the nonlinear equation ♢k♢Bku(x)=f(x,∆k-1□k♢Bk). It was found that the existence of the solution u(x) of such an equation depends on the condition of f and ∆k-1□k♢Bku(x). Moreover, such equation u(x) is related to the elastic wave equation.


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## Body

## 1. Introduction

Let□k be ultrahyperbolic operator iterated  k-times defined by
(1.1)□k=(∂2∂x12+∂2∂x22+⋯+∂2∂xp2-∂2∂xp+12-∂2∂xp+22-⋯-∂2∂xp+q2)k,
where p+q=n,n is the dimension of space ℝn and k is a nonnegative integer.Consider the linear differential equation of the form(1.2)□ku(x)=f(x),
where u(x) and f(x) are generalized function and  x=(x1,x2,…,xn)∈ℝn.Gel’fand and Shilov [1, pages 279–282] first introduced the fundamental solution of (1.2) which is complicated form. Later Trione [2] has shown that the generalized function R2k(x) which is defined by (2.2) is the unique fundamental solution of (1.2) and Aguirre Tellez [3] also proved that R2k(x) exists only in case p is odd and n is odd or even and p+q=n.In 1996, Kananthai [4] has been the first to introduce  the operator ♢k which is named as the diamond operator iterated k-times and is defined by
(1.3)♢k=((∑i=1p‍∂2∂xi2)2-(∑j=p+1p+q‍∂2∂xj2)2)k,
where p+q=n is the dimension of the space ℝn, for x=(x1,x2,…,xn)∈ℝn and k is a nonnegative integer. The operator ♢k can be expressed in the form
(1.4)♢k=∆k□k=□k∆k,
where ∆k is the Laplace operator defined by
(1.5)∆k=(∂2∂x12+∂2∂x22+⋯+∂2∂xn2)k,
and □k is the ultrahyperbolic operator iterated k-times and is defined by (1.1). Tellez and Kananthai [5, lemma 3.1, page 46] have shown that the convolution (-1)kR2ke(x)*R2kH(x) is a fundamental solution of the operator ♢k, where R2ke(x) and R2kH(x) are defined by (2.8) and (2.2), respectively. That is,
(1.6)♢k{(-1)kR2ke(x)*R2kH(x)}=δ(x).Furthermore, Yildirim et al. [6] first introduced the Bessel diamond operator ♢Bk iterated k-times defined by
(1.7)♢Bk=[(∑i=1p‍Bxi)2-(∑j=p+1p+q‍Bxj)2]k,
where Bxi=∂2/∂xi2+(2υi/xi)(∂/∂xi), 2υi=2αi+1,αi>-1/2,xi>0. The operator ♢Bk can be expressed by ♢Bk=∆Bk□Bk=□Bk∆Bk, where
(1.8)∆Bk=(∑i=1p‍Bxi)k,(1.9)□Bk=[∑i=1p‍Bxi-∑j=p+1p+q‍Bxj]k.Yildirim et al. [6] have shown that the solution of the convolution form u(x)=(-1)kS2k(x)*R2k(x) is a unique fundamental solution of the operator ♢Bk, that is,
(1.10)♢Bk((-1)kS2k(x)*R2k(x))=δ,
where S2k(x) and R2k(x) are defined by (2.11) and (2.15) with α=γ=2k, respectively.Now, firstly the purpose of this paper is to study the following equation:(1.11)♢k♢Bku(x)=f(x),
where the operator ♢k defined by (1.3) and ♢Bk defined by (1.7) with f(x) is a generalized function and u(x) is an unknown function.Finally, we will study the nonlinear of the form(1.12)♢k♢Bku(x)=f(x,∆k-1□k♢Bku(x)),
with f defined and having continuous first derivative for all x∈Ω∪∂Ω, where Ω is an open subset of ℝn and ∂Ω denotes the boundary of Ω, and f is bounded on Ω, that is, |f|≤N,N is constant. We can find the solution u(x) of (1.12) which is unique under the boundary condition ∆k-1□k♢Bku(x)=0 for x∈∂Ω, and we obtain the solution related to the elastic wave equation.Before going to that point, the following definitions and some concepts are needed.


## 2. Preliminaries

Definition 2.1.
Letx=(x1,x2,…,xn) be a point of the n-dimensional Euclidean space ℝn. Denote by
(2.1)υ=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2
the nondegenerated quadratic form and p+q=n is the dimension of the space ℝn. Let Γ+={x∈ℝn:x1>0 and υ>0} and Γ¯+ denote its closure. For any complex number α, define the function
(2.2)RαH(υ)={υ(α-n)/2Kn(α),forx∈Γ+,0,forx∉Γ+,
where the constant Kn(α) is given by the formula
(2.3)Kn(α)=π(n-1)/2Γ((2+α-n)/2)Γ((1-α)/2)Γ(α)Γ((2+α-p)/2)Γ((p-α)/2).
The function RαH(υ) is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki [7].
It is well known thatRαH(υ) is a function of Re(α)≥n and is a distribution of α if Re(α)<n. Let supp RαH(υ) denote the support of RαH(υ) and suppose supp RαH(υ)⊂Γ-+, that is, supp RαH(υ), is compact.
From Trione [2, page 11], R2kH(υ) is a fundamental solution of the operator □k, that is,
(2.4)□kR2kH(υ)=δ(x).
By puttingp=1 in R2k(υ) and taking into account Legendre's duplication formula for Γ(z)(2.5)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
then the formula (2.1) reduces to
(2.6)MαH(u)={u(α-n)/2Hn(α),forx∈Γ+,0,forx∉Γ+,
and u=x12-x22-⋯-xn2, where
(2.7)Hn(α)=π(n-2)/22α-1Γ(α-n+2|ν|2)Γ(α2).
M
α
(
u
) is the hyperbolic kernel of Riesz [8, page 31].Definition 2.2.
Letx=(x1,x2,…,xn) be a point of ℝn and |x|=x12+x22+⋯+xn2 the function Rαe(x) denoted by the elliptic kernel of Marcel Riesz which is defined by
(2.8)Rαe(x)=|x|(α-n)/2Wn(α),
where
(2.9)Wn(α)=πn/22αΓ(α/2)Γ((n-α)/2).α is a complex parameter and n is the dimension of the space ℝn.
Letα and β be complex numbers such that α+β≠n+2r,r=0.1,2,… The function Rαe(x) has the following properties [9]:
(2.10)R0(x)=δ(x),R-2k(x)=(-1)k∆kδ(x),∆k{Rα(x)}=(-1)kRα-2k(x),Rα(x)*Rβ(x)=Rα+β(x).Definition 2.3.
Letx=(x1,x2,…,xn),ν=(ν1,ν2,…,νn)∈ℝn+. For any complex number α, we define the distribution family Sα(x) by
(2.11)Sα(x)=|x|α-n-2|ν|wn(α),
where |x|=x12+x22+⋯+xn2,|ν|=ν1+ν2+⋯+νn and
(2.12)wn(α)=∏i=1n2νi-1/2Γ(νi+1/2)2n+2|ν|-2αΓ((n+2|ν|-α)/2).Definition 2.4.
Letx=(x1,x2,…,xn),ν=(ν1,ν2,…,νn)∈ℝn+, and denote by
(2.13)V=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2
the nondegenerated quadratic form. Denote the interior of the forward cone by
(2.14)Γ+={x∈ℝn+:x1>0,x2>0,…,xn>0,V>0},
and Γ¯+ denotes its closure. For any complex number γ the distribution family Rγ(x) is defined by
(2.15)Rγ(x)={V(γ-n-2|ν|)/2Kn(γ),forx∈Γ+,0,forx∉Γ+,
where
(2.16)Kn(γ)=π(n+2|ν|-1)/2Γ((2+γ-n-2|ν|)/2)Γ((1-γ)/2)Γ(γ)Γ((2+γ-p-2|ν|)/2)Γ((p-γ)/2),
where γ is a complex number.
By puttingp=1 in R2k(x) and taking into account Legendre's duplication formula for Γ(z)(2.17)Γ(2z)=22z-1π-1/2Γ(z)Γ(z+12),
we obtain
(2.18)Nγ(x)=u(γ-n-2|ν|)/2En(γ),
and u=x12-x22-⋯-⋯-xn2, where
(2.19)En(γ)=π(n+2|ν|-1)/22γ-1Γ(2+γ-n-2|ν|2)Γ(γ2).Lemma 2.5.
Given the equation∆Bku(x)=δ(x) for x∈ℝn+, where ∆Bk is defined by (1.8), then
(2.20)u(x)=(-1)kS2k(x),
where S2k(x) is defined by (2.11), with α=2k.Proof.
See [6, page 379].Lemma 2.6.
Given the equation□Bku(x)=δ(x) for x∈ℝn+, where □Bk is defined by (1.9). Then
(2.21)u(x)=R2k(x),
where R2k(x) is defined by (2.15), with γ=2k.Proof.
See [6, page 379].Lemma 2.7.
LetSα(x)and Rβ(x) be the function defined by (2.11) and (2.15), respectively. Then
(2.22)Sα(x)*Sβ(x)=Sα+β(x),(-1)kS-2k(x)*(-1)kS2k(x)=(-1)2kS-2k+2k(x)=S0(x)=δ(x),
where α and β are a positive even number.Proof.
See [10, pages 171–190].Lemma 2.8.
The functionR-2k(x) and (-1)kS-2k(x) are the inverse in the convolution algebra of R2k(x) and (-1)kS2k(x), respectively, that is,
(2.23)R-2k(x)*R2k(x)=R-2k+2k(x)=R0(x)=δ(x),Rβ(x)*Rα(x)=Rβ+α(x).Proof.
See [6].Lemma 2.9.
GivenP is a hyper-function, then
(2.24)Pδk(P)+kδ(k-1)(P)=0,
where δ(k) is the Dirac-delta distribution with k derivatives and
(2.25)P=P(x)=x12+x22+⋯+xp2-xp+12-xp+22-⋯-xp+q2.Proof.
See [1, page 233].Lemma 2.10.
Given the following equation:(2.26)∆ku(x)=0,
where ∆k is defined by (1.5) and x=(x1,x2,…,xn)∈ℝn, then
(2.27)u(x)=δ(m)(r2)
or
(2.28)u(x)=(-1)2m-n/2+1πn/2Γ(m-n/2+2)4m-n/2+1R-2(m-n/2+1)e(x)
is a homogeneous solution of (2.26) with m=n/2-k-1 for k=1,2,3,…. The function R-2(m-n/2+1)e  is defined by (2.8) and α=-2(m-n/2+1).Proof.
We first need to show that the generalized functionu(x)=δ(m)(r2), where r2=|x|2=x12+x22+⋯+xn2, and
(2.29)∆u(x)=0,
where ∆=(∂2/∂x12+∂2/∂x22+⋯+∂2/∂xn2) is a Laplace operator. In fact,
(2.30)∂∂xiδ(m)(r2)=2xiδ(m+1)(r2),∂2∂xi2δ(m)(r2)=2δ(m+1)r2+4xi2δ(m+2)(r2).
Thus
(2.31)∆δ(m)(r2)=∑i=1n‍∂2∂xi2δ(m)(r2)=2nδ(m+1)(r2)+4r2δ(m+2)(r2)=2nδ(m+1)(r2)-4(m+2)δ(m+1)(r2)=(2n-4(m+2))δ(m+1)(r2),∆2δ(m)(r2)=(2n-4(m+2))∆δ(m+1)(r2)=(2n-4(m+2))(2n-4(m+3))δ(m+2)(r2),∆3δ(m)(r2)=(2n-4(m+2))(2n-4(m+3))(2n-4(m+4))δ(m+3)(r2),∆kδ(m)(r2)=(2n-4(m+2))(2n-4(m+3))(2n-4(m+4))⋯(2n-4(m+k+1))δ(m+k)(r2)=22(n2-(m+2))22(n2-(m+3))22(n2-(m+4))⋯22(n2-(m+k+1))δ(m+k)(r2).
Thus
(2.32)∆kδ(m)(r2)=22k[(n2-m)-2][(n2-m)-3][(n2-m)-4]⋯[(n2-m)-(k+1)]δ(m+k)(r2).
Using the following formula:
(2.33)(z-1)(z-2)⋯(z-k)=(-1)kΓ(-z+k+1)Γ(m+1),
the above expression can be written in the following form:
(2.34)∆kδ(m)(r2)=22k(-1)kΓ(n/2-m-2+1)Γ(n/2-m-2-k+1)δ(m+k)(r2)=22k(-1)kΓ(n/2-m-1)Γ(n/2-m-k-1)δ(m+k)(r2).
If we put m=n/2-k-1 for k=1,2,3,… in (2.34), we obtain
(2.35)∆kδ(m)(r2)=0δ(m+k)(r2)=0.
It follows that
(2.36)u(x)=δ(m)(r2)
is homogeneous solution of the equation ∆ku(x)=0. On the other hand, by Aguirre Tellez [11], we have
(2.37)δ(m)(r2)=(-1)mπn/2Γ(m-n/2+1)4m-n/2+1∆m-n/2+1δ(x)=(-1)mπn2(-1)m-n2+1Γ(m-n2+1)4m-n2+1R-2(m-n2+1)e(x).
If we put m=n/2-k-1 in (2.37), we obtain
(2.38)δ(n/2-k-1)(r2)=(-1)n/2-k-1πn/2(-1)n/2-k-1-n/2+1Γ(n/2-k-1-n/2+1)4n/2-k-1-n/2+1R-2(n/2-k-1-n/2+1)e(x)=(-1)n/2-k-1πn/2(-1)-kΓ(-k)4-kR2ke(x)=0.
By (2.36) and (2.37), we conclude
(2.39)u(x)=δ(m)(r2)
or
(2.40)u(x)=(-1)2m-n/2+1πn/2Γ(m-n/2+2)4m-n/2+1R-2(m-n/2+1)e(x)
is a homogeneous solution of the equation ∆ku(x)=0. This completes the proof.Lemma 2.11.
Given the following equation:(2.41)♢k♢Bku(x)=0,
where ♢k and ♢Bk are diamond operator and Bessel diamond operator iterated k-times defined by (1.3) and (1.7), respectively, u(x) is an unknown function, we obtain
(2.42)u(x)=R2kH(u)*(-1)kS2k(x)*R2k(υ)*δ(m)(r2)
or
(2.43)u(x)=R2kH(u)*(-1)kS2k(x)*R2k(υ)*(-1)2m-n/2+1πn/2Γ(m-n/2+2)4m-n/2+1R-2(m-n/2+1)e(x)
with m=n/2-k-1 as a homogeneous solution of (2.41).Proof.
Since(2.44)♢k=∆k□k,♢Bk=∆Bk□Bk.
Consider the following homogeneous equation:
(2.45)♢k♢Bku(x)=0.
The above equation can be written as(2.46)∆k□k∆Bk□Bku(x)=0.
By Lemma2.10, we have
(2.47)□k∆Bk□Bku(x)=δ(m)(r2).
Convolving both sides byR2kH(u)*(-1)kS2k(x)*T2k(x), we obtain
(2.48)R2kH(υ)*(-1)kS2k(x)*R2k(x)*□k∆Bk□Bku(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*δ(m)(r2).
By properties of convolution, we have
(2.49)□R2kH(υ)*∆Bk(-1)kS2k(x)*□BkR2k(x)*u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*δ(m)(r2).
By (2.4), Lemmas 2.5, and 2.6, we obtain
(2.50)δ(x)*δ(x)*δ(x)*u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*δ(m)(r2).
Thus
(2.51)u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*δ(m)(r2)
or
(2.52)u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*(-1)2m-n/2+1πn/2Γ(m-n/2+2)4m-n/2+1R-2(m-n/2+1)e(x)
is a homogeneous solution of (2.41).Lemma 2.12.
Consider the following:(2.53)∆u(x)=f(x,u(x)),
where f is defined and has continuous first derivatives for all x∈Ω∪∂Ω, Ω is an open subset of Rn, and ∂Ω is the boundary of Ω. Assume that f is bounded, that is, |f(x,u)|≤N, and the boundary condition u(x)=0 for x∈∂Ω. Then we obtain u(x) as a unique solution of (2.53).Proof.
We can prove the existence of the solutionu(x) of (2.53) by the method of iterations and the Schuder's estimates. The details of the proof are given by Courant and Hilbert, [12, pages 369–372].


## 3. Main Results

Theorem 3.1.
Given the following equation:(3.1)♢k♢Bku(x)=f(x),
where ♢k and ♢Bk are defined by (1.3) and (1.7), respectively, f(x) is the generalized function, u(x) is an unknown function x=(x1,x2,…,xn)∈ℝn, and m=n/2-k-1. We obtain
(3.2)u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*δ(m)(r2)+R2kH(υ)*(-1)kR2ke(x)*(-1)kS2k(x)*R2k(x)*f(x)
or
(3.3)u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*(-1)2m-n/2+1πn/2Γ(m-n/2+2)4m-n/2+1R-2(m-n/2+1)e(x)+R2kH(υ)*(-1)kR2ke(x)*(-1)kS2k(x)*R2k(x)*f(x)
as a general solution of (3.1).Proof.
Consider the following equation:(3.4)♢k♢Bku(x)=f(x),
or
(3.5)□k∆k∆Bk♢Bku(x)=f(x).
Convolving both sides of (3.1) by R2kH(υ)*(-1)kR2ke(x)*(-1)kS2k(x)*R2k(x), we obtain
(3.6)R2kH(υ)*(-1)kR2ke(x)*R2k(x)*(-1)kS2k(x)*□k∆k□Bk∆Bku(x)=R2kH(υ)*(-1)kR2ke(x)*R2k(x)*(-1)kS2k(x)*f(x).
By properties of convolution, we have
(3.7)□kR2kH(υ)*∆k(-1)kR2ke(x)*∆Bk(-1)kS2k(x)*□BkR2k(x)*u(x)=R2kH(υ)*(-1)kR2ke(x)*R2k(x)*(-1)kS2k(x)*f(x).
By (2.4), Lemmas 2.5, and 2.6, we obtain
(3.8)δ(x)*δ(x)*δ(x)*δ(x)*u(x)=R2kH(υ)*(-1)kR2ke(x)*(-1)kS2k(x)*R2k(x)*f(x).
Thus(3.9)u(x)=R2kH(υ)*(-1)kR2ke(x)*(-1)kS2k(x)*R2k(x)*f(x).
Consider the following homogeneous equation:
(3.10)♢k♢Bku(x)=0.
By Lemma 2.10, we have a homogeneous solution as
(3.11)u(x)=R2kH(u)*(-1)kS2k(x)*R2k(x)*δ(m)(r2).
Thus, the general solution of (3.1) is
(3.12)u(x)=R2kH(x)*(-1)kS2k(x)*R2k(x)*δ(m)(r2)+R2kH(υ)*(-1)kR2ke(x)*(-1)kS2k(x)*R2k(x)*f(x),
or
(3.13)u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*(-1)2m-n/2+1πn/2Γ(m-n/2+2)4m-n/2+1R-2(m-n/2+1)e(x)+R2kH(υ)*(-1)kR2ke(x)*(-1)kS2k(x)*R2k(x)*f(x).
The proof is complete.Theorem 3.2.
Consider the following nonlinear equation:(3.14)♢k♢Bku(x)=f(x,∆k-1□k♢Bku(x)),
where ♢k,♢Bk,∆k-1, and □k are defined by (1.3), (1.7), (1.5), and (1.1), respectively. Let f be defined and having continuous first derivatives for all x∈Ω∪∂Ω,Ω is an open subset of ℝn and ∂Ω denotes the boundary of Ω and n is even with n≥4. Suppose f is bounded, that is,
(3.15)|f(x),∆k-1□k♢Bku(x)|≤N,
and, the boundary condition for all x∈∂Ω let be
(3.16)∆k-1□k♢Bku(x)=0.
We can assume ∆k-1□k♢Bku(x)=U(x) and U(x) is a continuous function for x∈∂Ω, then we obtain
(3.17)u(x)=(-1)k-1R2(k-1)e(x)*R2kH(υ)*(-1)kS2k(x)*R2k(x)*U(x)
as a solution of (3.14) with the boundary condition as
(3.18)u(x)=δ(m)(r2)*R2kH(υ)*(-1)kS2k(x),
for all x∈∂Ω and m=n/2-k. The function S2k(x),R2k(x),R2(k-2)e(x), and R2kH(υ) are given by (2.11), (2.15), (2.8), and (2.2), respectively. Moreover,
(3.19)W(x)=(-1)k-1R2(1-k)e(x)*(-1)kS-2k(x)*U(x)
is a solution of the following equation:
(3.20)□k□BkW(x)=U(x),
where □k,□Bk are defined by (1.1), (1.9), respectively, and U(x) is obtained from (3.11). Furthermore, if we put p=k=1, then W(x) is reduced to
(3.21)W(x)=M2H(u)*N2H(u)*U(x),
which is a solution of the following inhomogeneous elastic wave equation:
(3.22)(∂2∂x12-∂2∂x22-∂2∂x32-⋯-∂2∂xn2)·(Bx1-Bx2-Bx3-⋯-Bxn)W(x)=U(x).Proof.
We have(3.23)♢k♢Bku(x)=∆∆k-1□k♢Bku(x)=f(x,∆k-1□k♢Bku(x)).
Since u(x) has continuous derivative up to order 4p for k=1,2,3,…. thus we can assume
(3.24)∆k-1□k♢Bku(x)=U(x),∀x∈Ω.
Then (3.17) can be written in the following form:
(3.25)♢k♢Bku(x)=∆U(x)=f(x,U(x)).
By (3.2), we have(3.26)|f(x,U(x))|≤N,∀x∈Ω,
For U(x)=0 or
(3.27)∆k-1□k♢Bku(x)=0∀x∈∂Ω,
Convolving both sides of (3.24) by
(3.28)(-1)kR2(k-1)e(x)*R2kH(x)*(-1)kS2k(x)*R2k(x),
we obtain
(3.29)((-1)kR2(k-1)e(x)*R2kH(υ)*(-1)kS2k(x)*R2k(x))∆k-1□k♢Bku(x)=((-1)kR2(k-1)e(x)*R2kH(x)*(-1)kS2k(x)*R2k(x))*U(x).
By properties of convolution, we have
(3.30)(∆k-1(-1)kR2(k-1)e(x))*(□kR2kH(υ))*(♢Bk(-1)kS2k(x)*R2k(x))*u(x)=((-1)kR2(k-1)e(x)*R2kH(υ)*(-1)kS2k(x)*R2k(x))*U(x).
By Lemma 2.8, we obtain
(3.31)δ*δ*δ*u(x)=((-1)kR2(k-1)e(x)*R2kH(υ)*(-1)kS2k(x)*R2k(x))*U(x).
Thus
(3.32)u(x)=((-1)kR2(k-1)e(x)*R2kH(υ)*(-1)kS2k(x)*R2k(x))*U(x),
as a solution (3.14).
Now, considering the boundary condition we have(3.33)∆k-1□k♢Bku(x)=0.
By Lemma 2.10, we obtain
(3.34)□k♢Bku(x)=δ(m)(r2),
with m=n/2-k. Convolving both sides of (3.34) by R2kH(υ)*(-1)kS2k(x)*R2k(x), we obtain
(3.35)R2kH(υ)*(-1)kS2k(x)*R2k(x)□k♢ku(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*δ(m)(r2)
or
(3.36)u(x)=R2kH(υ)*(-1)kS2k(x)*R2k(x)*(-1)m+k-2(-1)mπn/2Γ(m-n/2+1)4m-n/2+1R-2(m-n/2+1)e(x),
for x∈∂Ω.
Lastly, convolving both sides of (3.36) by (-1)k-1R2(1-k)e(x)*(-1)kS-2k(x), we obtain
(3.37)(-1)k-1R2(1-k)e(x)*(-1)kS-2k(x)*u(x)=R2kH(x)*R2k(x)*U(x).
Setting
(3.38)W(x)=(-1)k-1R2(1-k)e(x)*(-1)kS-2k(x)*u(x).
By Lemmas 2.8 and 2.5, we obtain W(x) as a solution of the following equation:
(3.39)□k□BkW(x)=U(x).
If we put p=1, then R2kH(υ) and R2k(x) are reduced to M2kH(u) and N2k(u) and are defined by (2.6) and (2.18), respectively. Moreover, if we put p=k=1, then the operator □k and □Bk is reduced to
(3.40)∂2∂x12-∂2∂x22-∂2∂x32-⋯-∂2∂xn2,Bx1-Bx2-Bx3-⋯-Bxn,
respectively, and the solution W(x) is reduced to
(3.41)W(x)=M2H(u)*N2H(u)*U(x),
which is solution of the following inhomogeneous elastic wave equation:
(3.42)(∂2∂x12-∂2∂x22-∂2∂x32-⋯-∂2∂xn2)·(Bx1-Bx2-Bx3-⋯-Bxn)W(x)=U(x).
The proof is complete.



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*Source: 952191-2012-10-17.xml*