This article concerns sufficient conditions for the solvability of implicit difference equations. These are cast in a very general framework that relies on the α-covering and Lipschitz properties of the implicit recursive map with respect relatively to the first and second arguments. Difference equations have a prominent role in many scientific and engineering fields and may arise due to a wide variety of reasons. These may range from the discretization of time and other variables in dynamic systems given by either ordinary or partial differential equations required by computational procedures, or from the intrinsically discrete nature of many systems. While in some of these, discrete state transitions are caused by discrete events, in others, their decentralized nature requires iterative procedures built in the control synthesis in order to achieve some kind of global consensus. © 2015 Springer International Publishing.