Abstract

In this paper, we study the solution of nonlinear
equation
$\otimes^{k}\diamondsuit^{k}_{c_{1}}u(x)=f(x,\Box^{k-1}L^{k}\diamondsuit^{k}_{c_{1}}u(x))$
where $\otimes^{k}\diamondsuit^{k}_{c_{1}}$ is the product of the
Otimes operator and Diamond operator and $\otimes^{k}$defined by
\begin{eqnarray*}
  \otimes^{k}&=&\left(\left(\sum^{p}_{i=1}\frac{\partial^2}{\partial
x^2_i}\right)^{3}
-\left(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial
x^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}{\partial
x^2_i}\right)^{2}
-\left(\sum^{p+q}_{j=p+1}\frac{\partial^2}{\partial
x^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
 $\Box^{k-1}L^{k}\diamondsuit^{k}_{c_{1}}u(x).$ \\

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