Hyers-Ulam stability of the first order difference equation generated by linear maps

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} \prod_{j=1}^{n}|a_j|^{\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.Comment: 32 page