Hyers-Ulam stability of the difference equation $ z_{ n+1} = a_nz_n + b_n $ is investigated. If $ \prod_{ j=1}^{ n}|a_j| $ has subexponential growth rate, then difference equation generated by linear maps has no Hyers-Ulam stability. Other complementary results are also found where $ \lim_{ n \rightarrow \infty} \left(\prod_{ j=1}^{ n}|a_j| \right)^{ \frac{ 1}{ n}} $ is greater or less than one. These results contain Hyers-Ulam stability of the first order linear difference equation with periodic coefficients also.
Categories: math.DS