Let X be a Banach space and T be a bounded linear operator defined on X. In this work we study the existence of solutions of a class of nonlinear difference equation of fractional order in the form (Formula presented.) where (Formula presented.) corresponds to the fractional difference operator of order (Formula presented.) in sense of Riemann–Liouville and (Formula presented.) is a function satisfying suitable conditions. More specifically, by using operator-theoretical methods and fixed point theory, we show the existence of solutions of such class of equations on the vector-valued weighted space of sequences (Formula presented.).
