On the Green Function of the Operator ((⊕)_B + m~6)~k Related to the Bessel Helmholtz Operator and the Bessel Klein-Gordon Operator

摘要

In this article, we study the Green function of the form ((⊕)_b + m~6)~k which is iterated k-times and is defined byrn((⊕)_B+m~6)~k= [Σ_(i=1)~p B_(x_i))~3 + (Σ_(i=p+1)~(p+q) B_(x_i))~3 + m~6 ]~krnwhere p + q = n is the dimension of R_n~+, B_(x_i) = (e~2)/(ex_i~2) + (2v_i)/(x_i) e-(e_(x_i)), 2v_i = 2α_i + 1,rnα_i > - ?, X_i > 0, i = 1,2,..., n, m is a positive real number and k is a positive integer. At first, we study the Green function or elementary solution of the operator ((⊕)_B + m~6)~k. Moreover, the operator ((⊕)_B + m~6)~k can be related to the Bessel Helmholtz operator (△_B + m~2)~k and the Bessel ultra-hyperbolic Klein-Gordon operator (⊕_B + m~2)~k. After that, we apply such a Green function to solve the solution of the equation ((⊕)_B+m~6)~k u(x) = f(x) where / is a generalized function and u(x) is an unknown function for x ∈ R~n.