We consider linear and weakly nonlinear di erence equations with a linear operator coe cient jump in the in nite dimensional complex Banach space X. Such equations often arise in theoretical and applied problems of physics, mechanics, mathematical physics, biology and mathematical economics. The results of the research describe existence conditions of the unique solutions for some types of di erence equations. In particular we study summable solution existence and uniqueness for linear di erence equation in the space lp(Z;X). Also we address the issue of the bounded solution existence and uniqueness for linear di erence equation in the space X and the explicit form of such a solution is determined. Furthermore we formulate su cient conditions of existence and uniqueness of the unique bounded solution of the weakly nonlinear di erence equation in the in nite dimensional complex Banach space X.
