Bounded solutions of finite-difference equations (Q1820916)

Consider the equation (1): Dẋ\(+Bx=f\), where D, B denote operators of the form \((Dx)(t)=\sum^{n}_{m=1}a_ m(t)x(t+h_ m),\) \((Bx)(t)=\sum^{n}_{m=1}b_ m(t)x(t+h_ m).\) The author supposes that for every bounded function f (in C or \(L_ p\) metric) the equation (1) has a unique bounded solution. It is proved that this assumption yields the existence of an inverse operator \(D^{-1}\), hence the equation (1) can be written in the form \(\dot x+D^{-1}Bx=D^{-1}Bf.\) The main ideas of the proof are the following: All operators in this problem form a subalgebra of the algebra of all linear bounded operators. A special ideal I is selected in this subalgebra. Finally, the author shows that the invertibility of the operator D modulo I yields the invertibility of D in of usual sense.