Discrete maps with distributed memory fading parameter

Discrete maps with uniformly distributed memory fading parameter are proposed. These maps are derived from equations with fractional derivatives and fractional integrals of uniformly distributed orders and periodic sequence of kicks that are described by Dirac delta-functions. The fractional differential and fractional integral equations, for which the discrete maps are derived, are nonlinear equations with fractional operators proposed by Nakhushev and Pskhu. The solutions of these fractional differential and integral equations of distributed orders are derived. It is emphasized that the fractional integrals and derivatives with a uniformly distributed order cannot be considered as mutually inverse operators for which the fundamental theorems hold. Fundamental theorems of fractional calculus for fractional operators with uniformly distributed order, which are proved by Pshu, are used to obtain discrete maps with uniformly distributed memory fading parameter. In this paper, we also derive discrete maps with memory from equations with FDs and FIs that are inverse operators to FDs and FIs of a distributed order. The all suggested discrete maps are derived from fractional differential and integral equations without an approximation.