On the Bessel diamond and the nonlinear Bessel diamond operator related to the Bessel wave equation

In this article, we study the solution of the equation ♢Bku(x)=f(x) where u(x)u(x) is an unknown generalized function and ff is a generalized function, ♢Bk is the Bessel diamond operator iterated kk times and is defined by ♢Bk=[(Bx1+Bx2+⋯+Bxp)2−(Bxp+1+⋯+Bxp+q)2]k where p+q=n,Bxi=∂2∂xi2+2vixi∂∂xi, where 2vi=2αi+12vi=2αi+1, αi>−12 [B.M. Levitan, Expansion in Fourier series and integrals with Bessel functions, Uspekhi Matematicheskikh Nauk (N.S.) 6 2 (42) (1951) 102–143 (in Russian)], xi>0,i=1,2,…,n,kxi>0,i=1,2,…,n,k, is a nonnegative integer and nn is the dimension of the Rn+. Firstly, it found that the solution u(x)u(x) depends on the conditions of pp and qq and moreover such a solution is related to the solution of the Laplace Bessel equation and the Bessel wave equation. Finally, we study the solution of the nonlinear equation ♢Bku(x)=f(x,ΔBk−1□Bku(x)). It is found that the existence of the solution u(x)u(x) of such an equation depends on the condition of ff and ΔBk−1□Bku(x) and moreover such a solution u(x)u(x) related to the Bessel wave equation depends on the conditions of pp, qq and kk.