On the Product of the Non-linear of Diamond Operator and ⊗ k Operator

In this paper, we study the solution of nonlinear equation ⊗♦c1 u(x) = f (x, L♦c1 u(x)) where ⊗♦c1 is the product of the Otimes operator and Diamond operator where c1 is positive constants, k is a positive integer, p + q = n, n is the dimension of the Euclidean space Rn, for x = (x1, x2, . . . , xn) ∈ Rn,u(x) is an unknown function and f (x, .L♦c1 u(x)) is a given function. It was found that the existence of the solution u(x) of such equation depending on the conditions of f and L♦c1 u(x).


Introduction
The operator ♦ k has been first by A. Kananthai (1997) and is named as the Diamond operator iterated k times and is defined by p + q = n, n is the dimension of the space R n , for x = (x 1 , x 2 , . . ., x n ) ∈ R n and k is a nonnegative integer.The operator ♦ k can be expressed in the form where k is the Laplacian operator iterated k times defined by and 2 k is the ultra-hyperbolic operator iterated k times defined by (3) Next, W. Satsanit has been first introduced ⊗ k operator and ⊗ k is defined by where ♦, and 2 are defined by ( 1), ( 2) and (3) with k = 1 respectively.
Consider the nonlinear equation where ⊗ k is the operator iterated k times is defined (4), and L k is the operator iterated k times is defined by and 2 k−1 is the ultra-hyperbolic operator iterated k − 1 times defined by (3).Let f defined and have continuous first derivatives for all x ∈ Ω ∪ ∂Ω, where Ω is an open subset of R n and ∂Ω denotes the boundary of Ω and f be a bounded function, that is with the boundary condition Then, we obtain as a solution of (5) with the boundary condition  15) with α = 2(k − 2) and , R e 4k (x) is given by ( 20) with γ = 4k.The purpose o this work to extend the ♦ k operator defined by (1) to be Where c 1 is positive constant and k is a non-negative integer Now, we study the nonlinear equation of the form with f defined and having continuous first derivative for all x ∈ Ω ∪ ∂Ω where Ω is an open subset of R n and ∂Ω denoted the boundary of 11) and 2 k defined by (3) and L k defined by (6).
We can find the solution u(x) of (12) which unique under the boundary condition Courant, 1996 p.369) there exists a unique solution W(x) of the equation 2W(x) = f (x, W(x)) for all x ∈ Ω with the boundary condition W(x) = 0 for all x ∈ ∂Ω where W is solution of the inhomogeneous wave equation where I H 2 (x) and N H 2 (x) are defined by ( 18) and ( 19) with α = β = 2 respectively.Before going that points , the following definitions and some concepts are needed.

Preliminaries Definition
where p + q = n.The interior of forward cone defined by For any complex number α, define the function and where the constant K n (α), K n (β) is given by the formula The function R H α (u), S H β (w) is called the ultra-hyperbolic kernel of Marcel Riesz and was introduced by Y. Nozaki (1964, p.72).It is well known that R H α (u) and S H β (w) is an ordinary function if Re(α) ≥ n and Re(β) ≥ n and is a distribution if Re(α) < n and Re(β) < n.By putting p = 1 in ( 13), ( 14) and ( 17) using the Legendre's duplication of then ( 15) and ( 16) reduce to and respectively, where and N H β (x) are precisely called the Hyperbolic kernel of Marcel Riesz.Definition 2.2 Let x = (x 1 , x 2 , ..., x n ) be a point of R n and the function R e γ (x) and L e ρ (x) is defined by and where γ is a complex parameter and n is the dimension of R n .
Definition 2.3 Let c 1 be positive number, p + q = n and k is a nonnegative integer.The ultra-hyperbolic operators iterated k times 2 k and 2 k c 1 are defined by The Laplacian operators iterated k times k and k c 1 are defined by Lemma 1 Given the equation Where 2 k and 2 k c 1 defined by ( 3) and ( 13) respectively, x ∈ R n and δ is the Dirac-delta distribution.Then we obtain are an elementary solution of ( 15) and ( 16) respectively where R H 2k (x) and S H 2k (x) are defined by ( 15) and ( 16) with α = β = 2k.
Lemma 3 Let S α (x)and R β (x) be the function defined by ( 13) and ( 14) respectively.Then where α and β are a positive even number.
Lemma 5 Given P is a hyper-function then where δ (k) is the Dirac-delta distribution with k derivatives.
Proof.We first to show that the generalized function δ (m) (r 2 − s 2 ) where Where 2 is defined by (3) with k = 1 and x = (x 1 , x 2 , . . ., By Lemma 1 with P = r 2 − s 2 .Similarly, Thus ) is a solution of ( 29) with m = n−4 2 , n ≥ 4 and n is even dimension.We write From the above proof we have ) is a solution of ( 29) with m = n−4 2 , n ≥ 4 and n is even dimension.
Lemma 7 Given the equation where 2 k c 1 is defined by ( 23).Then we obtain u(x as a solution of ( 31) where S H 2(k−1) (x) is defined by ( 16) with m derivative and β = 2(k − 1), m = n−4 2 , n ≥ 4 and n is even dimension.
Proof.The proof of Lemma 7 is similar to the proof of Lemma 6.
Proof.We can prove the existence of the solution u(x) of ( 35) by the method of iterations and the Schuder's estimates.The details of the proof are given by Courant and Hilbert, (R.Courant, 1966, pp.369-372).

Theorem
Consider the nonlinear equation Where 2 k , ♦ k c 1 are defined by ( 3) , ( 11) respectively and the operator L k is defined by ( 6).Let f be defined and having continuous first derivatives for all x ∈ Ω ∪ ∂Ω , Ω is an open subset of R n and ∂Ω denotes the boundary of Ω and n is even with n ≥ 4. Suppose f is bounded function, that is for all x ∈ Ω and the boundary condition for all x ∈ ∂Ω.Then we obtain as a solution of (36) with the boundary condition for all x ∈ ∂Ω , m = (n − 4)/2, W(x) is a continuous function for x ∈ Ω ∪ ∂Ω, and G(x) defined by ( 33).The function L e 2k (x), S H 2k (x) are defined by ( 22), ( 16) with ρ = 2k, β = 2k respectively and (R H 2(k−2) (υ)) (m) is defined by ( 15) with α = 2(k − 2).Moreover, for k = 1 we obtain as a solution of the inhomogeneous equation Where 2 and 2 c 1 are defined by (3), ( 23) with k = 1 respectively and u(x) is obtained from (48).Furthermore, if we put p = k = 1 then the operator 2 k and 2 k c 1 reduces to n respectively and the solution M(x) = I H 2 (x) * N H 2 (x) * W(x) which is the inhomogeneous wave equation Where I H 2 (x) is defined by ( 18) with α = 2 and N H 2 (x) is defined by ( 19) with β = 2. Proof.We have Since u(x) has continuous derivative up to order 6k for k = 1, 2, 3, . . .and 2 k−1 L k ♦ k c 1 u(x) exists as the generalized function.Thus we can assume Then (41) can be written in the form and by (38), W(x) = 0, x ∈ ∂Ω or We obtain a unique solution of (43) which satisfies (37) by Lemma 9. Since and n is even dimension for k = 2, 3, 4, 5, ..... and W(x) is a continuous function for x ∈ Ω∪∂Ω.