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The present paper deals with Hyers–Ulam stability of the first-order linear difference equation Δ h x ( t ) − a x ( t ) = f ( t ) on h Z , where Δ h x ( t ) = ( x ( t + h ) − x ( t ) ) / h and h Z = { h k | k ∈ Z } for the constant stepsize h  > 0; a is a real number; f ( t ) is a real-valued function on h Z . The main purpose of this paper is to find the best HUS constant on h Z . Several relationships between solutions of two different perturbed difference equations are also given. [ABSTRACT FROM AUTHOR]

We show that the following system of difference equations x n = x n − 1 y n − 2 a y n − 2 + b y n − 1 , y n = y n − 1 x n − 2 c x n − 2 + d x n − 1 , n ∈ N 0 , where parameters a, b, c, d , as well as initial values x − 2 , x − 1 , y − 2 , y − 1 , are real numbers, is solvable in closed form and by using obtained formulas we give some results on the long term behavior of their solutions. We also give natural explanations and considerably extend some recent results in the literature. [ABSTRACT FROM AUTHOR]