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SUMMARYIn this paper, we consider the problem of existence of certain global solutions for general discrete-time backward nonlinear equations, defined on infinite dimensional ordered Banach spaces. This class of nonlinear equations includes as special cases many of the discrete-time Riccati equations arising both in deterministic and stochastic optimal control problems. On the basis of a linear matrix inequalities approach, we give necessary and sufficient conditions for the existence of maximal, stabilizing, and minimal solutions of the considered discrete-time backward nonlinear equations. As an application, we discuss the existence of stabilizing solutions for discrete-time Riccati equations of stochastic control and filtering on Hilbert spaces. The tools provided by this paper show that a wide class of nonlinear equations can be treated in a uniform manner.