Solvability conditions for the difference equations with an initial condition in a subspace (Q630282)

The author establishes necessary and suffcient conditions for the existence of a unique solution \(x\in\ell_p(\mathbb{Z}_+,X)\) of the following difference equation \[ x_{n+1}=Bx_n+f_n,\quad f\in\ell_p(\mathbb{Z}_+,X),\;p\geq1,\;n\in\mathbb{Z}_+, \] such that \(x_0\in E\), where \(E\) is a closed subspace of a Banach space \(X\) and \(B\) belongs to the Banach algebra \(LB(X)\) of bounded linear operators in \(X\). In addition, he also discusses some relations between the considered difference equation and the differential equation \[ x'(t)=Ax(t)+f(t),\quad f\in L_p(\mathbb{R}_+,X),\;t\geq0, \] where \(A\) denotes an infinitesimal operator of a strongly continuous semigroup of operators \(T:[0,\infty)\to LB(X)\) of class \(C_0\). Finally, an estimate of solutions of the difference equation is given.