#### Abstract

Firstly, we studied the solution of the equation where is an unknown unknown function for , is the generalized function, is a positive integer. Finally, we have studied the solution of the nonlinear equation . It was found that the existence of the solution of such an equation depends on the condition of and . Moreover such solution is related to the inhomogeneous wave equation depending on the conditions of , , and .

#### 1. Introduction

The operator has been first introduced by Kananthai (see ), is named as the Diamond operator iterated times, and is defined by is the dimension of the space for and is a nonnegative integer. The operator can be expressed in the form where is the Laplacian operator itrerated times defined by and is the ultrahyperbolic operator iterated times defined by

Kananthai (see [1, Theorem  3.1, page 33]) has shown that the convolution is an elementary solution of the operator that is, Next, Kananthai (see ) has studied the linear equation This equation is the generalization of the ultrahyperbolic equation and it can be applied to the wave equation. We obtain as a solution of such an equation (1.5) where The function is called the ultrahyperbolic kernel defined by (2.2) and is called the elliptic kernel defined by (2.8), with .

Furthermore, Yıldırım et al. (see ) first introduced the operator that is named as Diamond Bessel operator, where is defined by and The operator can be expressed by where Next, W. Satsanit has first introduced operator and is defined by where , and are defined by (1.1), (1.2), and (1.3) with , respectively.

Now, firstly, the purpose of this work is to study the equation where the operator is defined by (1.10) and defined by (1.7), is a generalized function and is an unknown function. Finally we study the equation with having a continuous first derivative for all where is an open subset of , and denotes the boundary of , is bounded on , that is, is constant, as well as , and are defined by (1.3), (2.46), (1.8) and (1.9), respectively.

We can find the solution of (1.12) that is unique under the boundary condition for . By [4, page 369] there exists a unique solution of the equation for all with the boundary condition for all where .

Moreover, if we put in , then we found that is a solution of the inhomogeneous equation where and are defined by (2.6) and (2.20) with respectively.

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

#### 2. Preliminaries

Definition 2.1. Let be a point of the -dimensional Euclidean space denoted by The nondegenerated quadratic form is the dimension of the space Let be the interior of forward cone and let denote its closure. For any complex number define the function where the constant is given by the formula The function is called the ultrahyperbolic kernel of Marcel Riesz and was introduced by Nozaki (see ).

It is well known that is an ordinary function if and is a distribution of if . Let supp denote the support of and suppose that supp that is, supp is compact.

From Trione (see [6, page 11]), is an elementary solution of the operator that is,

By putting in and taking into account Legendre's duplication formula for , that is we obtain and where is the hyperbolic kernel of Marcel Riesz.

Definition 2.2. Let be a point of and then the function denoted the elliptic kernel of Marcel Riesz and is defined by where where is a complex parameter and is the dimension of

It can be shown that where is defined by (1.2). It follows that , (see [7, page 118]).

Moreover, we obtain is an elementary solution of the operator (see [8, Lemma  2.4, page 31]). That is By (2.2) and (2.3) with , then reduces to where and reduces to . By using the formula we obtain where is defined by (2.10) with Thus, for where Thus, if , then for and

Definition 2.3. Let . For any complex number , we define the function by

Definition 2.4. Let and the nondegenerated quadratic form. Denote the interior of the forward cone by The function is defined by where and is a complex number. By putting in and taking into account Legendre's duplication formula for , that is, we obtain and where

Lemma 2.5. Given the equation for where is defined by (1.8), then where is defined by (2.16), with

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

Lemma 2.6. Given the equation for , where is defined by (1.9), then where is defined by (2.17), with

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

Lemma 2.7. Given that is a hyperfunction, then where is the Dirac-delta distribution with -derivatives.

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

Lemma 2.8. Given the equation where is defined by (1.3) and , then is a solution of (2.25) with and is even dimension. The function is defined by (2.2) with -derivatives, , and being defined by (2.1).

Proof. We first show the generalized function where and , is a solution of the equation where is defined by (1.3) with and , By Lemma 2.5 with similarly, Thus If , then we have . That is, is a solution of (2.26) with and is even dimension. We write and from the above proof we have with and is even dimension. Convolving the above equation by , we obtain by (2.2), and is defined by Definition (2.1).
Thus is a solution of (2.25) with and is even dimension.

Lemma 2.9. Given the equation then is an elementary solution for the operator iterated times where is defined by (1.10), and where denotes the convolution of itself times and denotes the inverse of in the convolution algebra. Moreover is a tempered distribution.

Proof. From (3.1), we have or we can write Convolving both sides of the above equation by , or By (2.4) and (2.8), we obtain Thus Keeping on convolving both sides of the above equation by up to times, we obtain where the symbol denotes the convolution of itself times. By properties of , we have Thus,
Now, consider the function , since is a tempered distribution. Thus defined by (2.34) is a tempered distribution, and we obtain that is a tempered distribution and is the space of tempered distribution. Choose where is the right-side distribution which is a subspace of of distribution.
Thus It follows that is an element of convolution algebra, since is a convolution algebra. Hence by the method of Zemanian (see ), (2.33) has a unique solution where is an inverse of in the convolution algebra and is called the Green function of the operator.

Lemma 2.10. Given the equation where is the operator defined by and and are defined by (1.2) and (1.3) with respectively, one obtains that is an elementary solution of the operator where where denotes the convolution of itself times and denotes the inverse of in the convolution algebra. Moreover 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 where is defined and has continuous first derivatives for all where is an open subset of and is the boundary of , assume that is bounded, that is, for all Then one obtains a continuous function as unique solution of (2.48) with the boundary condition for

Proof. We can prove the existence of the solution 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 and are the inverse of the convolution algebra of and respectively, that is,

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

#### 3. Main Results

Theorem 3.1. Given the equation where is the Otimes operator iterated times and is Diamond Bessel operator iterated times defined by (1.10) and (1.7), respectively, and is an unknown function, one obtains that is a solution of (3.1) where where is defined by (2.47), as well as , , and are defined by (2.16),(2.17), and (2.2) with , and respectively.

Proof. Since
Consider the homogeneous equation
The above equation can be written as or That is, where , and are defined by (1.3), (2.46), (1.8), and (1.9), respectively. By Lemma 2.8, we obtain
Since are the elementary solution of the operators and respectively, and by Lemma 2.10, we have that is an elementary of the operator defined by (2.46), that is, Convolving both sides of (3.8) by , we obtain By properties of convolution
By Lemmas 2.10, 2.5, and 2.6, we obtain
Thus is the solution of (3.1).

Theorem 3.2. Given the equation where is the Otimes operator iterated times defined by (1.10), and is the Diamond Bessel operator iterated times defined by (1.7), is the generalized function, is an unknown function, and is even,
One obtains that is a general solution of (3.14) and is defined by (2.33), is defined by (2.47), as well as and are defined by (2.16) and (2.17) with and respectively.

Proof. Consider the equation or
Convolving both sides of (3.14) by , we obtain By properties of convolution, By Lemmas 2.9, 2.5, and 2.6, we obtain Thus Consider the homogeneous equation By Theorem 3.1, we have a homogeneous solution Thus, the general solution of (3.14) is as required.

Theorem 3.3. Consider the nonlinear equation where , and are defined by (1.10), (1.7),(1.3),(2.44), and (1.9), respectively. Let be defined, and having continuous first derivative for all is an open subset of and denotes the boundary function, that is, for all and the boundary condition for all . Then one obtains as a solution of (3.25) with the boundary condition for all , and is a continuous function for , and are given by (2.2), (2.16), and (2.17) with , and respectively. Moreover, for one obtains as a solution of the inhomogeneous equation where and are defined by (1.3) and (1.9) with respectively, and is obtained from (3.28). Furthermore, If one puts , then the operators and reduce to respectively, and the solution is the inhomogeneous wave equation where is defined by (2.6) with and is defined by (2.20) with .

Proof. Since has continuous derivative up to order for , and exists as the generalized function. Thus we can assume that Then (3.34) can be written in the form By (3.26) and by (3.27) , or We obtain a unique solution of (3.28) which satisfies (3.27) by Lemma 2.8.
Since , and are the elementary solution of the operators , and respectively, and by Lemma 2.10, we have that is an elementary of the operator where , that is, From (3.35), we have Convolving the above equation by we obtain By properties of convolution, we obtain By (3.39) we obtain Thus as a solution of (3.25).
Next, consider the boundary condition (3.38). From by Lemma 2.8, we have where and is even. Convolving both sides of (3.47) by we obtain