Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces

In this article, we prove that the omega-periodic discrete evolution family Gamma := {rho(n, k) : n, k is an element of Z(+), n >= k} of bounded linear operators is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions. More precisely, we prove that if for each real number. and each sequence (xi(n)) taken from some Banach space, the approximate solution of the nonautonomous omega-periodic discrete system theta(n+1) = Lambda(n)theta(n), n is an element of Z(+) is represented by phi(n+1) = Lambda(n)phi(n) + e(i gamma(n+1))xi(n + 1), n is an element of Z(+); phi(0) = theta(0), then the Hyers-Ulam stability of the nonautonomous omega-periodic discrete system theta(n+1) = Lambda(n)theta(n), n is an element of Z(+) is equivalent to its uniform exponential stability.