Tensor product solutions of certain Abstract Cauchy problems of Euler type

Abstract

Our concern in this paper, is to find exact solutions, using tensor product techniques, of two types of second order homogeneous Abstract Cauchy problems of the following forms $u^{″}\left(t\right)+C{u}^{\prime }\left(t\right)+Fu\left(t\right)=0,$ and $t^2{u}^{\prime \prime }\left(t\right)+2tC{u}^{\prime }\left(t\right)+Fu\left(t\right)=0.$ The initial conditions to be used are $u\left(t_0\right)=u_0,u^{\prime }\left(t_0\right)=u_0^{\prime },$ where $C$ and $F$ are closed linear operators that are densely defined on a Banach space $Y$, $u_0,u_0^{\prime }\in Y$ and $u\in C^2(I,Y).$