IJPAM: Volume 56, No. 4 (2009)

ON THE SOLUTION OF NONLINEAR $\circledast^{k}$ OPERATOR

Wanchak Satsanit
Department of Mathematics
Regina Coeli College
166, Charoenprathet Road Changkran, Chiang Mai, 50100, THAILAND
e-mail: aunphue@live.com


Abstract.In this paper, we study the solution of nonlinear equation $\circledast^{k}u(x)=f(x,\triangle^{k-1}.L^{k}u(x))$, where $\circledast^{k}$ is the otimes operator iterated $k$ times and is defined by

\begin{eqnarray*}
\circledast^{k}&=&\left(\left(\sum^{p}_{i=1}\frac{\partial^2}...
..._{j=p+1}\frac{\partial^2}{\partial
x^2_j}\right)^{3}\right)^{k},
\end{eqnarray*}

where $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$, $u(x,t)$ is an unknown function for $x=(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n$, $f(x)$ is the given function, $k$ is a positive integer and $u(x)$ is an unknown function.

It is found that the existence of the solution $u(x)$ of such equation depends on the conditions of $f$ and $\triangle^{k-1}L^{k}u(x)$.

Received: September 2, 2009

AMS Subject Classification: 47F05

Key Words and Phrases: the hyperbolic kernel of Marcel Riesz, diamond operator, Schander's estimates

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 56
Issue: 4