In this paper, we study the uniqueness and existence of the solutions of four types of non‐singular delay difference equations by using the Banach contraction principles, fixed point theory, and Gronwall’s inequality. Furthermore, we discussed the Hyers–Ulam stability of all the given systems over bounded and unbounded discrete intervals. The exponential stability and controllability of some of the given systems are also characterized in terms of spectrum of a matrix concerning the system. The spectrum of a matrix can be easily obtained and can help us to characterize different types of stabilities of the given systems. At the end, few examples are provided to illustrate the theoretical results.

In this article, we study the approximate fixed point sequence of an evolution family. A family of a bounded nonlinear operator acting on a metric space is said to be an evolution family if and for all . We prove that the common approximate fixed point sequence is equal to the intersection of the approximate fixed point sequence of two operators from the family. Furthermore, we apply the Ishikawa iteration process to construct an approximate fixed point sequence of an evolution family of nonlinear mapping.

Let be a strongly continuous and -periodic evolution family acting on a complex Banach space . We prove that if and then the growth bound of the family is less than or equal to .

We prove that a discrete semigroup????={T(n):n∈ℤ+}of bounded linear operators acting on a complex Banach spaceXis uniformly exponentially stable if and only if, for eachx∈AP0(ℤ+,X), the sequencen↦∑k=0n‍T(n-k)x(k):ℤ+→Xbelongs toAP0(ℤ+,X). Similar results for periodic discrete evolution families are also stated.

Let X be a Banach space, be an operator-valued sequence and let be the discrete evolution family associated to In this paper we prove that the family is non-uniformly strongly stable (i.e. for every nonnegative integer m and every if and only if it is -approximative admissible, i.e. for every sequence in and every positive number there exists the sequence in satisfying such that the solution of the discrete Cauchy Problem belongs to Other types of asymptotic behavior of the family are also analyzed.

In this study, we establish some results for strong convergence of a sequence to a common fixed point of a subfamily of a nonexpansive and periodic evolution family of bounded linear operators acting on a closed and bounded subset of a strictly convex Banach space . In fact, we generalized the results from semigroups of the operator to an evolution family of operators.