On the Solution 𝑛-Dimensional of the Product ⊗^𝑘 Operator and Diamond Bessel Operator

Abstract

Firstly, we studied the solution of the equation ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥) where 𝑢(𝑥) is an unknown unknown function for 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛, 𝑓(𝑥) is the generalized function, 𝑘 is a positive integer. Finally, we have studied the solution of the nonlinear equation ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥,□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)). It was found that the existence of the solution 𝑢(𝑥) of such an equation depends on the condition of 𝑓 and □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥). Moreover such solution 𝑢(𝑥) is related to the inhomogeneous wave equation depending on the conditions of 𝑝, 𝑞, and 𝑘.

1. Introduction

The operator ◊^𝑘 has been first introduced by Kananthai (see [1]), is named as the Diamond operator iterated 𝑘−times, and is defined by◊^𝑘=((^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖)²−(^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑗)²)^𝑘,𝑝+𝑞=𝑛. (1.1)𝑛 is the dimension of the space ℝ^𝑛, for 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛 and 𝑘 is a nonnegative integer. The operator ◊^𝑘 can be expressed in the form ◊^𝑘=Δ^𝑘□^𝑘=□^𝑘Δ^𝑘, where Δ^𝑘 is the Laplacian operator itrerated 𝑘−times defined byΔ^𝑘𝜕=(²𝜕𝑥²₁+𝜕²𝜕𝑥²₂𝜕+⋯+²𝜕𝑥²_𝑛)^𝑘, (1.2) and □^𝑘 is the ultrahyperbolic operator iterated 𝑘−times defined by□^𝑘𝜕=(²𝜕𝑥²₁+𝜕²𝜕𝑥²₂𝜕+⋯+²𝜕𝑥²_𝑝−𝜕²𝜕𝑥²_𝑝+1−𝜕²𝜕𝑥²_𝑝+2𝜕−⋯−²𝜕𝑥²_𝑝+𝑞)^𝑘. (1.3)

Kananthai (see [1, Theorem  3.1, page 33]) has shown that the convolution (−1)^𝑘𝑅^𝑒_2k(𝑥)∗𝑅^𝐻_2𝑘(𝑥) is an elementary solution of the operator ◊^𝑘, that is, ◊^𝑘((−1)^𝑘𝑅^𝑒_2𝑘(𝑥)∗𝑅^𝐻_2𝑘(𝑥))=𝛿(𝑥). (1.4) Next, Kananthai (see [2]) has studied the linear equation◊^𝑘𝑢(𝑥)=𝑓(𝑥). (1.5) This equation is the generalization of the ultrahyperbolic equation and it can be applied to the wave equation. We obtain 𝑢(𝑥)=(−1)^𝑘𝑀_2𝑘,2𝑘(𝑥)∗𝑓(𝑥) as a solution of such an equation (1.5) where 𝑀_2𝑘,2𝑘=𝑅^𝐻_2𝑘(𝑥)∗𝑅^𝑒_2𝑘(𝑥). (1.6) The function 𝑅^𝐻_2𝑘(𝑥) is called the ultrahyperbolic kernel defined by (2.2) and 𝑅^𝑒_2𝑘(𝑥) is called the elliptic kernel defined by (2.8), with 𝛼=2𝑘.

Furthermore, Yıldırım et al. (see [3]) first introduced the ◊^𝑘_𝐵 operator that is named as Diamond Bessel operator, where ◊^𝑘_𝐵 is defined by◊^𝑘_𝐵=[(^𝑝_𝑖=1𝐵_𝑥𝑖)²−(^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑗)²]^𝑘, (1.7) and 𝐵_𝑥𝑖=𝜕²/𝜕𝑥²_𝑖+(2𝜐_𝑖/𝑥_𝑖)(𝜕/𝜕𝑥_𝑖),2𝜐_𝑖=2𝛼_𝑖+1,𝛼_𝑖>−1/2,𝑥_𝑖>0. The operator ◊^𝑘_𝐵 can be expressed by ◊^𝑘_𝐵=Δ^𝑘_𝐵□^𝑘_𝐵=□^𝑘_𝐵Δ^𝑘_𝐵, whereΔ^𝑘_𝐵=(^𝑛_𝑖=1𝐵_𝑥𝑖)^𝑘.□^𝑘_𝐵=(^𝑝_𝑖=1𝐵_𝑥𝑖−^𝑝+𝑞_𝑗=𝑝+1𝐵_𝑥𝑗)^𝑘. (1.8)(1.9) Next, W. Satsanit has first introduced ⊗^𝑘 operator and ⊗^𝑘 is defined by⊗^𝑘=[(^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖)³−(^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑗)³]^𝑘=(^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖−^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑗)^𝑘[(^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖)²+(^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖)⋅(^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑗)+(^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑗)²]^𝑘=□^𝑘(Δ²−14(Δ+□)(Δ−□))^𝑘3=(41◊Δ+4□³)^𝑘, (1.10) where ◊,Δ, and □ are defined by (1.1), (1.2), and (1.3) with 𝑘=1, respectively.

Now, firstly, the purpose of this work is to study the equation⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥), (1.11) where the operator ⊗^𝑘 is defined by (1.10) and ◊^𝑘_𝐵 defined by (1.7), 𝑓(𝑥) is a generalized function and 𝑢(𝑥) is an unknown function. Finally we study the equation⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥,□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)) (1.12) with 𝑓 having a continuous first derivative for all 𝑥∈Ω∪𝜕Ω, where Ω is an open subset of 𝑅^𝑛, and 𝜕Ω denotes the boundary of Ω, 𝑓 is bounded on Ω, that is, |𝑓|≤𝑁,𝑁 is constant, as well as □^𝑘−1,𝐿^𝑘,Δ^𝑘_𝐵, and □^𝑘_𝐵 are defined by (1.3), (2.46), (1.8) and (1.9), respectively.

We can find the solution 𝑢(𝑥) of (1.12) that is unique under the boundary condition □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=0 for 𝑥∈𝜕Ω. By [4, page 369] there exists a unique solution 𝑊(𝑥) of the equation □𝑊(𝑥)=𝑓(𝑥,𝑊(𝑥)) for all 𝑥∈Ω with the boundary condition 𝑊(𝑥)=0 for all 𝑥∈𝜕Ω where 𝑊(𝑥)=□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥).

Moreover, if we put 𝑝=𝑘=1 in □^𝑘□^𝑘_𝐵𝑀(𝑥)=𝑊(𝑥), then we found that 𝑀(𝑥)=𝐼^𝐻₂(𝑥)∗𝐼₂(𝑥)∗𝑊(𝑥) is a solution of the inhomogeneous equation where 𝐼^𝐻₂(𝑥) and 𝐼₂(𝑥) are defined by (2.6) and (2.20) with 𝛼=2,𝛾=2, respectively.

Before going into details, the following definitions and some important concepts are needed.

2. Preliminaries

Definition 2.1. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point of the 𝑛-dimensional Euclidean space ℝ^𝑛, denoted by 𝜐=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞. (2.1) The nondegenerated quadratic form 𝑝+𝑞=𝑛 is the dimension of the space ℝ^𝑛. Let Γ₊={𝑥∈ℝ^𝑛∶𝑥₁>0and𝑢>0} be the interior of forward cone and let Γ₊ denote its closure. For any complex number 𝛼, define the function 𝑅^𝐻_𝛼𝜐(𝜐)=𝐾^{(𝛼−𝑛)/2}_𝑛,(𝛼)for𝑥∈Γ₊,0,for𝑥∉Γ₊, (2.2) where the constant 𝐾_𝑛(𝛼) is given by the formula 𝐾_𝑛𝜋(𝛼)=^(𝑛−1)/2Γ((2+𝛼−𝑛)/2)Γ((1−𝛼)/2)Γ(𝛼).Γ((2+𝛼−𝑝)/2)Γ((𝑝−𝛼)/2) (2.3) The function 𝑅^𝐻_𝛼(𝜐) is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki (see [5]).

It is well known that 𝑅^𝐻_𝛼(𝜐) is an ordinary function if 𝑅𝑒(𝛼)≥𝑛 and is a distribution of 𝛼 if 𝑅𝑒(𝛼)<𝑛. Let supp 𝑅^𝐻_𝛼(𝜐) denote the support of 𝑅^𝐻_𝛼(𝜐) and suppose that supp 𝑅^𝐻_𝛼(𝜐)⊂Γ₊, that is, supp 𝑅^𝐻_𝛼(𝜐) is compact.

From Trione (see [6, page 11]), 𝑅^𝐻_2𝑘(𝜐) is an elementary solution of the operator □^𝑘, that is,□^𝑘𝑅^𝐻_2𝑘(𝜐)=𝛿(𝑥). (2.4)

By putting 𝑝=1 in 𝑅^𝐻_2𝑘(𝜐) and taking into account Legendre's duplication formula for Γ(𝑧), that is Γ(2𝑧)=2^2𝑧−1𝜋^−1/21Γ(𝑧)Γ(𝑧+2), (2.5) we obtain𝐼^𝐻_𝛼𝜐(𝜐)=^{(𝛼−𝑛)/2}𝐻_𝑛(𝛼), (2.6) and 𝜐=𝑥²₁−𝑥²₂−𝑥²₃⋯−𝑥²_𝑛 where 𝐻_𝑛(𝛼)=𝜋^(𝑛−2)/22^𝛼−1Γ(𝛼+2−𝑛2𝛼)Γ(2). (2.7)𝐼^𝐻_𝛼(𝜐) is the hyperbolic kernel of Marcel Riesz.

Definition 2.2. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) be a point of ℝ^𝑛 and 𝜔=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛, then the function 𝑅^𝑒_𝛼(𝜔) denoted the elliptic kernel of Marcel Riesz and is defined by 𝑅^𝑒_𝛼𝜔(𝜔)=^{(𝛼−𝑛)/2}𝑊_𝑛,(𝛼) (2.8) where 𝜔=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑛,𝑊_𝑛𝜋(𝛼)=^𝑛/22^𝛼Γ(𝛼/2)Γ((𝑛−𝛼)/2) (2.9)(2.10) where 𝛼 is a complex parameter and 𝑛 is the dimension of ℝ^𝑛.

It can be shown that 𝑅^𝑒_−2𝑘(𝑥)=(−1)^𝑘Δ^𝑘𝛿(𝑥) where Δ^𝑘 is defined by (1.2). It follows that 𝑅^𝑒₀(𝑥)=𝛿(𝑥), (see [7, page 118]).

Moreover, we obtain (−1)^𝑘𝑅^𝑒_2𝑘(𝑥) is an elementary solution of the operator Δ^𝑘 (see [8, Lemma  2.4, page 31]). That isΔ^𝑘((−1)^𝑘𝑅^𝑒_2𝑘(𝑥))=𝛿(𝑥). (2.11) By (2.2) and (2.3) with 𝑞=0, then 𝜐^{(𝛼−𝑛)/2} reduces to 𝜔_𝑝^{(𝛼−𝑝)/2} where 𝜔_𝑝=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝 and 𝐾_𝑛(𝛼) reduces to 𝐾_𝑝(𝛼)=(𝜋^(𝑝−1)/2Γ((1−𝛼)/2)Γ(𝛼))/Γ((𝑝−𝛼)/2). By using the formula Γ(2𝑧)=2^2𝑧−1𝜋^−1/21Γ(𝑧)Γ(𝑧+21),Γ(21+𝑧)Γ(2−𝑧)=𝜋sec(𝜋𝑧), (2.12) we obtain 𝐾_𝑝1(𝛼)=2sec(𝜋𝛼2)𝑊_𝑝(𝛼), (2.13) where 𝑊_𝑝(𝛼) is defined by (2.10) with 𝑛=𝑝. Thus, for 𝑞=0,𝑅^𝐻_𝛼𝜐(𝜐)=^{(𝛼−𝑝)/2}𝐾_𝑝(𝛼)=2cos(𝜋𝛼2)𝜐^{(𝛼−𝑝)/2}𝑊_𝑝(𝛼)=2cos(𝜋𝛼2)𝑅^𝑒_𝛼(𝜔_𝑝), (2.14) where 𝜔_𝑝=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝. Thus, if 𝛼=2𝑘, then 𝑅^𝐻_2𝑘(𝜔_𝑝)=2(−1)^𝑘𝑅^𝑒_2𝑘(𝜔_𝑝) (2.15) for 𝑞=0 and 𝜔_𝑝=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝.

Definition 2.3. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛),𝜈=(𝜈₁,𝜈₂,…,𝜈_𝑛)∈ℝ⁺_𝑛. For any complex number 𝛼, we define the function 𝑆_𝛼(𝑥) by 𝑆_𝛼2(𝑥)=^{𝑛+2|𝜈|−2𝛼}Γ((𝑛+2|𝜈|−𝛼)/2)|𝑥|^{𝛼−𝑛−2|𝜈|}∏^𝑛_𝑖=12^{𝜈𝑖−1/2}Γ(𝜈_𝑖.+1/2) (2.16)

Definition 2.4. Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛),𝜈=(𝜈₁,𝜈₂,…,𝜈_𝑛)∈ℝ⁺_𝑛, and 𝑉=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝−𝑥²_𝑝+1−𝑥²_𝑝+2−⋯−𝑥²_𝑝+𝑞 the nondegenerated quadratic form. Denote the interior of the forward cone by Γ₊={𝑥∈ℝ⁺_𝑛∶𝑥₁>0,𝑥₂>0,…,𝑥_𝑛>0,𝑉>0}. The function 𝑅_𝛾(𝑥) is defined by 𝑅_𝛾𝑉(𝑥)=^{(𝛾−𝑛−2|𝜈|)/2}𝐾_𝑛,(𝛾) (2.17) where 𝐾_𝑛𝜋(𝛾)=^{(𝑛+2|𝜈|−1)/2}Γ((2+𝛾−𝑛−2|𝜈|)/2)Γ((1−𝛾)/2)Γ(𝛾),Γ((2+𝛾−𝑝−2|𝜈|)/2)Γ((𝑝−2|𝜈|−𝛾)/2) (2.18) and 𝛾 is a complex number. By putting 𝑝=1 in 𝑅_𝛾(𝑥) and taking into account Legendre's duplication formula for Γ(𝑧), that is, Γ(2𝑧)=2^2𝑧−1𝜋^−1/21Γ(𝑧)Γ(𝑧+2) (2.19) we obtain 𝐼_𝛾𝑉(𝑥)=^{(𝛾−𝑛−2|𝜈|)/2}𝑁_𝑛(𝛾), (2.20) and 𝑉=𝑥²₁−𝑥²₂−𝑥²₃⋯−𝑥²_𝑛 where 𝑁_𝑛(𝛾)=𝜋^{(𝑛+2|𝜈|−1)/2}2^2𝑘−1Γ(2+𝛾−𝑛−2|𝜈|2𝛾)Γ(2). (2.21)

Lemma 2.5. Given the equation Δ^𝑘_𝐵𝑢(𝑥)=𝛿(𝑥) for 𝑥∈ℝ⁺_𝑛, where Δ^𝑘_𝐵 is defined by (1.8), then 𝑢(𝑥)=(−1)^𝑘𝑆_2𝑘(𝑥), (2.22) where 𝑆_2𝑘(𝑥) is defined by (2.16), with 𝛼=2𝑘.

Proof. (See [3, page 379] and [9]).

Lemma 2.6. Given the equation □^𝑘_𝐵𝑢(𝑥)=𝛿(𝑥) for 𝑥∈ℝ⁺_𝑛, where □^𝑘_𝐵 is defined by (1.9), then 𝑢(𝑥)=𝑅_2𝑘(𝑥), (2.23) where 𝑅_2𝑘(𝑥) is defined by (2.17), with 𝛾=2𝑘.

Proof. (See [3, page 379] and [9]).

Lemma 2.7. Given that 𝑃 is a hyperfunction, then 𝑃𝛿^𝑘(𝑝)+𝑘𝛿^(𝑘−1)(𝑝)=0, (2.24) where 𝛿^(𝑘) is the Dirac-delta distribution with 𝑘-derivatives.

Proof. (See [8, page 233]).

Lemma 2.8. Given the equation □^𝑘𝑢(𝑥)=0, (2.25) where □^𝑘 is defined by (1.3) and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛, then 𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚) is a solution of (2.25) with 𝑚=(𝑛−4)/2,𝑛≥4 and 𝑛 is even dimension. The function (𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚) is defined by (2.2) with 𝑚-derivatives, 𝛼=2(𝑘−1), and 𝜐 being defined by (2.1).

Proof. We first show the generalized function 𝛿^(𝑚)(𝑟²−𝑠²) where 𝑟²=𝑥²₁+𝑥²₂+⋯+𝑥²_𝑝 and 𝑠²=𝑥²_𝑝+1+𝑥²_𝑝+2+⋯+𝑥²_𝑝+𝑞,𝑝+𝑞=𝑛, is a solution of the equation □𝑢(𝑥)=0, (2.26) where □ is defined by (1.3) with 𝑘=1 and 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛, 𝜕𝜕𝑥_𝑖𝛿^(𝑚)(𝑟²−𝑠²)=2𝑥_𝑖𝛿^(𝑚+1)(𝑟²−𝑠²𝜕),₂𝜕𝑥²_𝑖𝛿^(𝑚)(𝑟²−𝑠²)=2𝛿^(𝑚+1)(𝑟²−𝑠²)+4𝑥²_𝑖𝛿^(𝑚+2)(𝑟²−𝑠²),□𝛿^(𝑚)(𝑟²−𝑠²)=^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖𝛿^(𝑚)(𝑟²−𝑠²)=2𝑝𝛿^(𝑚+1)(𝑟²−𝑠²)+4𝑟²𝛿^(𝑚+2)(𝑟²−𝑠²)=2𝑝𝛿^(𝑚+1)(𝑟²−𝑠²)+4(𝑟²−𝑠²)𝛿^(𝑚+2)(𝑟²−𝑠²)+4𝑠²𝛿^(𝑚+2)(𝑟²−𝑠²)=2𝑝𝛿^(𝑚+1)(𝑟²−𝑠²)−4(𝑚+2)𝛿^(𝑚+1)(𝑟²−𝑠²)+4𝑠²𝛿^(𝑚+2)(𝑟²−𝑠²)=(2𝑝−4(𝑚+2))𝛿^(𝑚+1)(𝑟²−𝑠²)+4𝑠²𝛿^(𝑚+2)(𝑟²−𝑠²). (2.27) By Lemma 2.5 with 𝑃=𝑟²−𝑠², similarly, ^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑗𝛿^(𝑚)(𝑟²−𝑠²)=(−2𝑞+4(𝑚+2))𝛿^(𝑚+1)(𝑟²−𝑠²)+4𝑟²𝛿^(𝑚+2)(𝑟²−𝑠²). (2.28) Thus □𝛿^(𝑚)(𝑟²−𝑠²)=^𝑝_𝑖=1𝜕²𝜕𝑥²_𝑖𝛿^(𝑚)(𝑟²−𝑠²)−^𝑝+𝑞_𝑗=𝑝+1𝜕²𝜕𝑥²_𝑖𝛿^(𝑚)(𝑟²−𝑠²)=(2(𝑝+𝑞)−8(𝑚+2))𝛿^(𝑚+1)(𝑟²−𝑠²)−4(𝑟²−𝑠²)𝛿^(𝑚+2)(𝑟²−𝑠²)=(2𝑛−8(𝑚+2))𝛿^(𝑚+1)(𝑟²−𝑠²)+4(𝑚+2)𝛿^(𝑚+1)(𝑟²−𝑠²)=(2𝑛−4(𝑚+2))𝛿^(𝑚+1)(𝑟²−𝑠²). (2.29) If 2𝑛−4(𝑚+2)=0, then we have □𝛿^(𝑚)(𝑟²−𝑠²)=0. That is, 𝑢(𝑥)=𝛿^(m)(𝑟²−𝑠²) is a solution of (2.26) with 𝑚=(𝑛−4)/2,𝑛≥4 and 𝑛 is even dimension. We write ^𝑘𝑢(𝑥)=□(^𝑘−1𝑢(𝑥))=0, (2.30) and from the above proof we have □^𝑘−1𝑢(𝑥)=𝛿^(𝑚)(𝑟²−𝑠²) with 𝑚=(𝑛−4)/2,𝑛≥4 and 𝑛 is even dimension. Convolving the above equation by 𝑅^𝐻_2(𝑘−1)(𝜐), we obtain 𝑅^𝐻_2(𝑘−1)(𝜐)∗□^𝑘−1𝑢(𝑥)=𝑅^𝐻_2(𝑘−1)(𝜐)∗𝛿^(𝑚)(𝑟²−𝑠²)□^𝑘−1(𝑅^𝐻_2(𝑘−1)(𝜐))∗𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚),where𝜐=(𝑟²−𝑠²)𝛿∗𝑢(𝑥)=𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚) (2.31) by (2.2), and 𝜐=𝑟²−𝑠² is defined by Definition (2.1).
Thus 𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚) is a solution of (2.25) with 𝑚=(𝑛−4)/2,𝑛≥4 and 𝑛 is even dimension.

Lemma 2.9. Given the equation ⊗^𝑘𝐺(𝑥)=𝛿(𝑥), (2.32) then 𝐺(𝑥)=(𝑅^𝐻_6𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥))∗(𝑂^∗𝑘(𝑥))^∗−1 (2.33) is an elementary solution for the ⊗^𝑘 operator iterated 𝑘−times where ⊗^𝑘 is defined by (1.10), and 3𝑂(𝑥)=4𝑅^𝐻₄1(𝑥)+4(−1)²𝑅^𝑒₄(𝑥) (2.34) where 𝑂^∗𝑘(𝑥) denotes the convolution of 𝑂(𝑥) itself 𝑘−times and (𝑂^∗𝑘(𝑥))^∗−1 denotes the inverse of 𝑂^∗𝑘(𝑥) in the convolution algebra. Moreover 𝐺(𝑥) is a tempered distribution.

Proof. From (3.1), we have ⊗^𝑘3𝐺(𝑥)=(41◊Δ+4³)^𝑘𝐺(𝑥)=𝛿(𝑥), (2.35) or we can write (341◊Δ+4□³3)(41◊Δ+4□³)^𝑘−1𝐺(𝑥)=𝛿(𝑥). (2.36) Convolving both sides of the above equation by 𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥), (341◊Δ+4□³)∗(𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄3(𝑥))(41◊Δ+4³)^𝑘−1𝐺(𝑥)=𝛿(𝑥)∗𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥), (2.37) or (34□(𝑅^𝐻₂(𝑥))∗Δ²(−1)²𝑅^𝑒₄(𝑥)∗𝑅^𝐻₄1(𝑥)+4□³𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄3(𝑥))∗(41◊Δ+4□³)^𝑘−1𝐺(𝑥)=𝛿(𝑥)∗𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥). (2.38) By (2.4) and (2.8), we obtain (34𝛿∗𝛿∗𝑅^𝐻₄1(𝑥)+4𝛿∗(−1)²𝑅^𝑒₄3(𝑥))∗(41◊Δ+4□³)^𝑘−1𝐺(𝑥)=𝛿(𝑥)∗𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥). (2.39) Thus (34𝑅^𝐻₄1(𝑥)+4(−1)²𝑅^𝑒₄3(𝑥))∗(41◊Δ+4³)^𝑘−1𝐺(𝑥)=𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥). (2.40) Keeping on convolving both sides of the above equation by 𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥) up to 𝑘−1 times, we obtain 𝑂^∗𝑘(𝑥)∗𝐺(𝑥)=(𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥))^∗𝑘 (2.41) where the symbol ∗𝑘 denotes the convolution of itself 𝑘−times. By properties of 𝑅_𝛼(𝑥), we have (𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥))^∗𝑘=𝑅^𝐻_6𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥). (2.42) Thus, 𝑂^∗𝑘(𝑥)∗𝐺(𝑥)=𝑅^𝐻_6𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥). (2.43)
Now, consider the function 𝑂^∗𝑘(𝑥), since 𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥) is a tempered distribution. Thus 𝑂(𝑥) defined by (2.34) is a tempered distribution, and we obtain that 𝑂^∗𝑘(𝑥) is a tempered distribution and 𝑅^𝐻_6𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥)∈𝒮^′ is the space of tempered distribution. Choose 𝒮^′⊂𝒟^′_ℛ where 𝒟^′_ℛ is the right-side distribution which is a subspace of 𝒟^′ of distribution.
Thus 𝑅^𝐻_6𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥)∈𝒟^′_ℛ. It follows that 𝑅^𝐻_6𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥) is an element of convolution algebra, since 𝒟^′_ℛ is a convolution algebra. Hence by the method of Zemanian (see [10]), (2.33) has a unique solution 𝐺(𝑥)=(𝑅^𝐻_6𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥))∗(𝑂^∗𝑘(𝑥))^∗−1, (2.44) where (𝑂^∗𝑘(𝑥))^∗−1 is an inverse of 𝑂^∗𝑘(𝑥) in the convolution algebra and 𝐺(𝑥) is called the Green function of the ⊗^𝑘 operator.

Lemma 2.10. Given the equation 𝐿^𝑘𝐾(𝑥)=𝛿(𝑥), (2.45) where 𝐿^𝑘 is the operator defined by 𝐿^𝑘3=(4Δ²+14□²)^𝑘 (2.46) and Δ and □ are defined by (1.2) and (1.3) with 𝑘=1, respectively, one obtains that 𝐾(𝑥) is an elementary solution of the 𝐿^𝑘 operator where 𝐾(𝑥)=(𝑅^𝐻_4𝑘(𝑥)∗(−1)^2𝑘𝑅^𝑒_4𝑘(𝑥))∗(𝑂^∗𝑘(𝑥))^∗−1,3𝑂(𝑥)=4𝑅^𝐻₄1(𝑥)+4(−1)²𝑅^𝑒₄(𝑥), (2.47) where 𝑂^∗𝑘(𝑥) denotes the convolution of 𝑂(𝑥) itself 𝑘−times and (𝑂^∗𝑘(𝑥))^∗−1 denotes the inverse of 𝑂^∗𝑘(𝑥) in the convolution algebra. Moreover 𝐾(𝑥) is a tempered distribution.

Proof. The proof of Lemma 2.10 is similar to the proof of Lemma 2.9.

Lemma 2.11. Given the equation □𝑢(𝑥)=𝑓(𝑥,𝑢(𝑥)), (2.48) where 𝑓 is defined and has continuous first derivatives for all 𝑥∈Ω∪𝜕Ω, where Ω is an open subset of 𝑅^𝑛 and 𝜕Ω is the boundary of Ω, assume that 𝑓 is bounded, that is, |𝑓(𝑥,𝑢(𝑥))|≤𝑁 for all 𝑥∈Ω. Then one obtains a continuous function 𝑢(𝑥) as unique solution of (2.48) with the boundary condition 𝑢(𝑥)=0 for 𝑥∈𝜕Ω.

Proof. We can prove the existence of the solution 𝑢(𝑥) of (2.48) by the method of iterations and Schuder's estimates. The details of the proof are given by Courant and Hilbert; (see [4, pages 369–372]).

Lemma 2.12. The function 𝑅^𝐻_−2𝑘(𝑥) and 𝑆_−2𝑘(𝑥) are the inverse of the convolution algebra of 𝑅^𝐻_2𝑘 and 𝑆_2𝑘, respectively, that is, 𝑅^𝐻_−2𝑘(𝑥)∗𝑅^𝐻_2𝑘(𝑥)=𝑅^𝐻_{−2𝑘+2𝑘}(𝑥)=𝑅^𝐻₀𝑆(𝑥)=𝛿,_−2𝑘(𝑥)∗𝑆_2𝑘(𝑥)=𝑆_{−2𝑘+2𝑘}(𝑥)=𝑆₀(𝑥)=𝛿. (2.49)

Proof. (See [7, page 158] and [11]).

3. Main Results

Theorem 3.1. Given the equation ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=0, (3.1) where ⊗^𝑘 is the Otimes operator iterated 𝑘−times and ◊^𝑘_𝐵 is Diamond Bessel operator iterated 𝑘− times defined by (1.10) and (1.7), respectively, and 𝑢(𝑥) is an unknown function, one obtains that 𝑢(𝑥) is a solution of (3.1) where 𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(−1)^𝑘−1(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚) (3.2) where 𝐾(𝑥) is defined by (2.47), as well as 𝑆_2𝑘(𝑥), 𝑅_2𝑘(𝑥), and (𝑅^𝐻_2(𝑘−1)(𝜐))^𝑚 are defined by (2.16),(2.17), and (2.2) with 𝛼=2𝑘, 𝛾=2𝑘 and 𝛼=2(𝑘−1), respectively.

Proof. Since ⊗^𝑘3=(41◊Δ+4□³)^𝑘,◊^𝑘_𝐵=Δ^𝑘_𝐵□^𝑘_𝐵. (3.3)
Consider the homogeneous equation⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=0. (3.4)
The above equation can be written as(341◊Δ+4□³)^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=0, (3.5) or □^𝑘(34Δ²+14□²)^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=0. (3.6) That is, □^𝑘𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=0, (3.7) where □^𝑘,𝐿^𝑘,Δ^𝑘_𝐵, and □^𝑘_𝐵 are defined by (1.3), (2.46), (1.8), and (1.9), respectively. By Lemma 2.8, we obtain 𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚). (3.8)
Since (−1)^𝑘𝑆_2𝑘(𝑥),𝑅_2𝑘(𝑥) are the elementary solution of the operators Δ^𝑘_𝐵 and □^𝑘_𝐵, respectively, and by Lemma 2.10, we have that 𝐾(𝑥) is an elementary of the operator 𝐿^𝑘 defined by (2.46), that is, Δ^𝑘_𝐵(−1)^𝑘𝑆_2𝑘(𝑥)=𝛿(𝑥),□^𝑘_𝐵𝑅_2𝑘𝐿(𝑥)=𝛿(𝑥),_𝑘𝐾(𝑥)=𝛿(𝑥). (3.9) Convolving both sides of (3.8) by 𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥), we obtain 𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗Ł^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚). (3.10) By properties of convolution Ł^𝑘𝐾(𝑥)∗Δ^𝑘_𝐵(−1)^𝑘𝑆_2𝑘(𝑥)∗□^𝑘_𝐵𝑅_2𝑘(𝑥)∗𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚). (3.11)
By Lemmas 2.10, 2.5, and 2.6, we obtain𝛿(𝑥)∗𝛿(𝑥)∗𝛿(𝑥)∗𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚). (3.12)
Thus𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚) (3.13) is the solution of (3.1).

Theorem 3.2. Given the equation ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥), (3.14) where ⊗^𝑘 is the Otimes operator iterated 𝑘−times defined by (1.10), and ◊^𝑘_𝐵 is the Diamond Bessel operator iterated 𝑘−times defined by (1.7), 𝑓(𝑥) is the generalized function, 𝑢(𝑥) is an unknown function, 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛 and 𝑛 is even,
One obtains that𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚)+𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗𝑓(𝑥) (3.15) is a general solution of (3.14) and 𝐺(𝑥) is defined by (2.33), 𝐾(𝑥) is defined by (2.47), as well as 𝑆_2𝑘(𝑥) and 𝑅_2𝑘(𝑥) are defined by (2.16) and (2.17) with 𝛼=2𝑘 and 𝛾=2𝑘, respectively.

Proof. Consider the equation ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥) (3.16) or ⊗^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥). (3.17)
Convolving both sides of (3.14) by 𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥), we obtain𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗⊗^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗𝑓(𝑥). (3.18) By properties of convolution, ⊗^𝑘𝐺(𝑥)∗Δ^𝑘_𝐵(−1)^𝑘𝑆_2𝑘(𝑥)∗□^𝑘_𝐵𝑅_2𝑘(𝑥)∗𝑢(𝑥)=𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗𝑓(𝑥). (3.19) By Lemmas 2.9, 2.5, and 2.6, we obtain 𝛿(𝑥)∗𝛿(𝑥)∗𝛿(𝑥)∗𝑢(𝑥)=𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗𝑓(𝑥). (3.20) Thus 𝑢(𝑥)=𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗𝑓(𝑥). (3.21) Consider the homogeneous equation ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=0. (3.22) By Theorem 3.1, we have a homogeneous solution 𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚). (3.23) Thus, the general solution of (3.14) is 𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−1)(𝜐))^(𝑚)+𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗𝑓(𝑥), (3.24) as required.

Theorem 3.3. Consider the nonlinear equation ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥,□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)) (3.25) where ⊗^𝑘,◊^𝑘_𝐵,□^𝑘−1,𝐿^𝑘,Δ^𝑘_𝐵, and □^𝑘_𝐵 are defined by (1.10), (1.7),(1.3),(2.44), and (1.9), respectively. Let 𝑓 be defined, and having continuous first derivative for all 𝑥∈Ω∪𝜕Ω,Ω is an open subset of 𝑅^𝑛 and 𝜕Ω denotes the boundary function, that is, |𝑓(𝑥,□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥))|≤𝑁 (3.26) for all 𝑥∈Ω and the boundary condition □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=0 (3.27) for all 𝑥∈𝜕Ω. Then one obtains 𝑢(𝑥)=𝑅^𝐻_2(𝑘−1)(𝑥)∗𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗𝑊(𝑥) (3.28) as a solution of (3.25) with the boundary condition 𝑢(𝑥)=(𝑅^𝐻_2(𝑘−2)(𝜐))^(𝑚)∗𝐺(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥) (3.29) for all 𝑥∈𝜕Ω,𝑚=(𝑛−4)/2, and 𝑊(𝑥) is a continuous function for 𝑥∈Ω∪𝜕Ω,while𝑅^𝐻_{2(𝑘−2)(𝜐)},𝑆_2𝑘(𝑥), and 𝑅_2𝑘(𝑥) are given by (2.2), (2.16), and (2.17) with 𝛼=2(𝑘−2),𝛼=2𝑘, and 𝛾=2𝑘, respectively. Moreover, for 𝑘=1 one obtains 𝑀(𝑥)=(𝑅^𝐻₋₄(𝑥)∗(−1)²𝑅^𝑒₋₄(𝑥))∗(𝑂^∗1(𝑥))∗(−1)^𝑘𝑆₋₂(𝑥)∗𝑢(𝑥) (3.30) as a solution of the inhomogeneous equation □□_𝐵𝑀(𝑥)=𝑊(𝑥), (3.31) where □ and □_𝐵 are defined by (1.3) and (1.9) with 𝑘=1, respectively, and 𝑢(𝑥) is obtained from (3.28). Furthermore, If one puts 𝑝=𝑘=1, then the operators □^𝑘 and □^𝑘_𝐵 reduce to 𝜕²𝜕𝑥²₁−𝜕²𝜕𝑥²₂−𝜕²𝜕𝑥²₃𝜕−⋯−²𝜕𝑥²_𝑛,𝐵_𝑥1−𝐵_𝑥2−𝐵_𝑥3−⋯−𝐵_𝑥𝑛, (3.32) respectively, and the solution 𝑀(𝑥)=𝐼^𝐻₂(𝑥)∗𝐼₂(𝑥)∗𝑊(𝑥) is the inhomogeneous wave equation (𝜕²𝜕𝑥²₁−𝜕²𝜕𝑥²₂−𝜕²𝜕𝑥²₃𝜕−⋯−²𝜕𝑥²_𝑛)⋅(𝐵_𝑥1−𝐵_𝑥2−𝐵_𝑥3−⋯−𝐵_𝑥𝑛)𝑀(𝑥)=𝑊(𝑥), (3.33) where 𝐼^𝐻₂(𝑥) is defined by (2.6) with 𝛼=2 and 𝐼₂(𝑥) is defined by (2.20) with 𝛾=2.

Proof. Since⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=□□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=𝑓(𝑥,□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)), (3.34)𝑢(𝑥) has continuous derivative up to order 6𝑘 for 𝑘=1,2,3,…, and □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥) exists as the generalized function. Thus we can assume that □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=𝑊(𝑥),∀𝑥∈Ω. (3.35) Then (3.34) can be written in the form ⊗^𝑘◊^𝑘_𝐵𝑢(𝑥)=□𝑊(𝑥)=𝑓(𝑥,𝑊(𝑥)). (3.36) By (3.26) |𝑓(𝑥,𝑊(𝑥))|≤𝑁,𝑥∈Ω, (3.37) and by (3.27) 𝑊(𝑥)=0,𝑥∈𝜕Ω, or □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=0,∀𝑥∈𝜕Ω. (3.38) We obtain a unique solution of (3.28) which satisfies (3.27) by Lemma 2.8.
Since 𝑅^𝐻_2(𝑘−1)(𝑥),(−1)^𝑘𝑆_2𝑘(𝑥), and 𝑅_2𝑘(𝑥) are the elementary solution of the operators □^𝑘−1,Δ^𝑘_𝐵, and □^𝑘_𝐵, respectively, and by Lemma 2.10, we have that 𝐾(𝑥) is an elementary of the operator 𝐿^𝑘 where 𝐿^𝑘=((3/4)Δ²+(1/4)□²)^𝑘, that is, □^𝑘−1𝑅^𝐻_2(𝑘−1)(𝑥)=𝛿,Δ^𝑘_𝐵(−1)^𝑘𝑆_2𝑘(𝑥)=𝛿,_𝑘𝐵𝑅_2𝑘(𝑥)=𝛿,𝐿^𝑘𝐾(𝑥)=𝛿. (3.39) From (3.35), we have □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=𝑊(𝑥). (3.40) Convolving the above equation by 𝑅^𝐻_2(𝑘−1)(𝑥)∗𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥), (3.41) we obtain (𝑅^𝐻_2(𝑘−1)(𝑥)∗𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))∗(□^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥))=(𝑅^𝐻_2(𝑘−1)(𝑥)∗𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))∗𝑊(𝑥). (3.42) By properties of convolution, we obtain (□^𝑘−1𝑅^𝐻_2(𝑘−1)(𝑥))∗(Ł^𝑘𝐾(𝑥))∗(Δ^𝑘_𝐵∗(−1)^𝑘𝑆_2𝑘)∗(□^𝑘_𝐵𝑅_2𝑘)∗𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝑥)∗𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))∗𝑊(𝑥). (3.43) By (3.39) we obtain 𝛿∗𝛿∗𝛿∗𝛿∗𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝑥)∗𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))∗𝑊(𝑥). (3.44) Thus 𝑢(𝑥)=(𝑅^𝐻_2(𝑘−1)(𝑥)∗𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))∗𝑊(𝑥), (3.45) as a solution of (3.25).
Next, consider the boundary condition (3.38). From □^𝑘−1𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=0, (3.46) by Lemma 2.8, we have 𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵𝑢(𝑥)=(𝑅^𝐻_2(𝑘−2)(𝜐))^(𝑚), (3.47) where 𝑚=(𝑛−4)/2,𝑛≥4 and 𝑛 is even. Convolving both sides of (3.47) by 𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥), (3.48) we obtain (𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))∗(𝐿^𝑘Δ^𝑘_𝐵□^𝑘_𝐵)∗𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−2)(𝜐))^(𝑚). (3.49) By the properties of convolution, we obtain (𝐿^𝑘𝐾(𝑥))∗(Δ^𝑘_𝐵(−1)^𝑘𝑆_2𝑘)∗(□^𝑘_𝐵𝑅_2𝑘)∗𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−2)(𝜐))^(𝑚). (3.50) By (3.39), we obtain 𝛿∗𝛿∗𝛿∗𝑢(𝑥)=(𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥))∗(𝑅^𝐻_2(𝑘−2)(𝜐))^(𝑚). (3.51) Thus, for 𝑥∈𝜕Ω and 𝑘=2,3,4,5,….,𝑢(𝑥)=𝐾(𝑥)∗(−1)^𝑘𝑆_2𝑘(𝑥)∗𝑅_2𝑘(𝑥)∗(𝑅^𝐻_2(𝑘−2)(𝜐))^(𝑚), (3.52) as required.
Now, for 𝑘=1 in (3.28), we have 𝑢(𝑥)=𝛿(𝑥)∗𝐺(𝑥)∗(−1)𝑆₂(𝑥)∗𝑅₂(𝑥)∗𝑊(𝑥). (3.53) By (2.47), we have 𝐺(𝑥)=(𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥))∗(𝑂^∗1(𝑥))^∗−1. (3.54) Taking into account (3.53), we obtain 𝑢(𝑥)=(𝑅^𝐻₆(𝑥)∗(−1)²𝑅^𝑒₄(𝑥))∗(𝑂^∗1(𝑥))^∗−1∗(−1)¹𝑆₂(𝑥)∗𝑅₂(𝑥)∗𝑊(𝑥) (3.55) as a solution of (3.25) for 𝑘=1.
Convolving both sides of (3.55) by (𝑅^𝐻₋₄(𝑥)∗(−1)²𝑅^𝑒₋₄(𝑥))∗(𝑂^∗1(𝑥))∗(−1)𝑆₋₂(𝑥), (3.56) by Lemma 2.12, we obtain (𝑅^𝐻₋₄(𝑥)∗(−1)²𝑅^𝑒₋₄(𝑥))∗(𝑂^∗1(𝑥))∗(−1)𝑆₋₂(𝑥)∗𝑢(𝑥)=𝑅^𝐻₂(𝑥)∗𝑅₂(𝑥)∗𝑊(𝑥). (3.57) By Lemma 2.6, we obtain 𝑀(𝑥)=(𝑅^𝐻₋₄(𝑥)∗(−1)²𝑅^𝑒₋₄(𝑥))∗(𝑂^∗1(𝑥))∗(−1)𝑆₋₂(𝑥)∗𝑢(𝑥) (3.58) as a solution of the inhomogeneous equation □□_𝐵𝑀(𝑥)=𝑊(𝑥). (3.59) Now, consider the boundary condition for 𝑘=1 in (3.27); we have 𝐿Δ_𝐵□_𝐵𝑢(𝑥)=0,or□_𝐵𝐿Δ_𝐵𝑢(𝑥)=0 (3.60) for 𝑥∈𝜕Ω. Thus by Lemma 2.8, for 𝑘=1, we have 𝐿Δ_𝐵𝑢(𝑥)=𝛿^(𝑚)(𝜐)for𝑥∈𝜕Ω, (3.61) where 𝛿^(𝑚)(𝑥)=𝑅^𝐻₀(𝑥). Convolving the above equation by 𝐾(𝑥)∗(−1)𝑆₂(𝑥) where 𝐾(𝑥) is defined by (2.47) with 𝑘=1 and 𝑆₂(𝑥) is defined by (2.16) with 𝛼=2, we obtain 𝐾(𝑥)∗(−1)𝑆₂(𝑥)∗(𝐿Δ_𝐵𝑢(𝑥))=𝛿^(𝑚)(𝜐)∗𝐾(𝑥)∗(−1)𝑆₂(𝑥). (3.62) By properties of convolution, 𝐿𝐾(𝑥)∗Δ_𝐵(−1)𝑆₂(𝑥)∗𝑢(𝑥)=𝛿^(𝑚)(𝜐)∗𝐾(𝑥)∗(−1)𝑆₂(𝑥). (3.63) By Lemmas 2.10 and 2.5, we obtain 𝛿(𝑥)∗𝛿(𝑥)∗𝑢(𝑥)=𝛿^(𝑚)(𝜐)∗𝐾(𝑥)∗(−1)𝑆₂(𝑥). (3.64) It follows that 𝑢(𝑥)=𝛿^(𝑚)(𝜐)∗𝐾(𝑥)∗(−1)𝑆₂(𝑥). (3.65) By (2.47) with 𝑘=1, we have 𝐾(𝑥)=(𝑅^𝐻₄(𝑥)∗(−1)²𝑅^𝑒₄(𝑥))∗(𝑂^∗1(𝑥))^∗−1. (3.66) Taking into account (3.65), we obtain 𝑢(𝑥)=𝛿^(𝑚)(𝜐)∗(𝑅^𝐻₄(𝑥)∗(−1)²𝑅^𝑒₄(𝑥))∗(𝑂^∗1(𝑥))^∗−1∗(−1)𝑆₂(𝑥)for𝑥∈𝜕Ω. (3.67)
Now consider the case 𝑘=1,𝑝=1, and 𝑞=𝑛−1, that is, from (3.59), 𝑅^𝐻₂(𝑥) reduced to 𝐼^𝐻₂(𝑥) where 𝐼^𝐻₂(𝑥) is defined by (2.2) with 𝛼=2 and 𝑅₂(𝑥) reduced to 𝐼₂(𝑥) where 𝐼₂(𝑥) is defined by (2.17) with 𝛾=2, and then the operator □ defined by (1.3) reduces to the wave operator □^∗=𝜕²𝜕𝑥²₁−𝜕²𝜕𝑥²₂−𝜕²𝜕𝑥²₃𝜕−⋯−²𝜕𝑥²_𝑛, (3.68)□_𝐵 defined by (1.9) reduces to the Bessel wave operator □^∗_𝐵=𝐵_𝑥1−𝐵_𝑥2−𝐵_𝑥3−⋯−𝐵_𝑥𝑛, (3.69) and then the solution 𝑀(𝑥) reduced to 𝑀(𝑥)=𝐼^𝐻₂(𝑥)∗𝐼₂(𝑥)∗𝑊(𝑥), (3.70) which is the solution of inhomogeneous wave equation □^∗□^∗_𝐵𝑀(𝑥)=𝑊(𝑥), (3.71) or (𝜕²𝜕𝑥²₁−𝜕²𝜕𝑥²₂−𝜕²𝜕𝑥²₃𝜕−⋯−²𝜕𝑥²_𝑛)⋅(𝐵_𝑥1−𝐵_𝑥2−𝐵_𝑥3−⋯−𝐵_𝑥𝑛)𝑀(𝑥)=𝑊(𝑥). (3.72) With the boundary condition for 𝑥∈𝜕Ω, 𝐿^∗□^∗_𝐵Δ_𝐵𝑢(𝑥)=0, (3.73) where 𝐿^∗=(3/4)Δ²+(1/4)(□^∗)² and □^∗ is defined by (3.68), or for 𝑥∈𝜕Ω and by (3.65), we obtain 𝑢(𝑥)=𝛿^(𝑚)(𝑠)∗(𝐼^𝐻₄(𝑥)∗(−1)²𝑅^𝑒₄(𝑥))∗(𝐷^∗1(𝑥))^∗−1∗(−1)𝑆₂(𝑥), (3.74) where 𝐼₄(𝑥) is defined by (2.20) with 𝛾=4, 𝑠=𝑥²₁−𝑥²₂−𝑥²₃−⋯−𝑥²_𝑛, and 𝐷(𝑥) reduced from 𝑂(𝑥) where it is defined by (2.34), that is, 𝐷(𝑥)=(3/4)𝐼^𝐻₄(𝑥)+(1/2)(−1)²𝑅^𝑒₄(𝑥).

Acknowledgment

The authors would like to thank The Thailand Research Fund and Graduate School, Chiang Mai University, Thailand, for financial support.