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

Abstract

In this paper, we study the solution of nonlinearequation$\otimes^{k}\diamondsuit^{k}_{c_{1}}u(x)=f(x,\Box^{k-1}L^{k}\diamondsuit^{k}_{c_{1}}u(x))$where $\otimes^{k}\diamondsuit^{k}_{c_{1}}$ is the product of theOtimes operator and Diamond operator and $\otimes^{k}$defined by\begin{eqnarray*} \otimes^{k}&=&\left(\left(\sum^{p}_{i=1}\frac{\partial^2}{\partialx^2_i}\right)^{3}-\left(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partialx^2_j}\right)^{3}\right)^{k}\end{eqnarray*}and $\diamondsuit^{k}_{c_{1}}$ defined by \begin{eqnarray*} \diamondsuit^{k}_{c_{1}}&=&\left(\frac{1}{c^{4}_{1}}\left(\sum^{p}_{i=1}\frac{\partial^2}{\partialx^2_i}\right)^{2}-\left(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partialx^2_j}\right)^{2}\right)^{k}\\\end{eqnarray*} where $c_{1}$ is positive constants, $k$ is a positive integer, $p+q=n$, $n$ is the dimension of the Euclidean space$\mathbb{R}^n$, for $x=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$,$u(x)$is an unknown function and $f(x,\Box^{k-1}.L^{k}\diamondsuit^{k}_{c_{1}}u(x))$ 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 $\Box^{k-1}L^{k}\diamondsuit^{k}_{c_{1}}u(x).$