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

Abstract

Hyers-Ulam stability of the difference equation $z_{n+1}=a_n{z}_n+b_n$ 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 ${\lim }_{n\to \infty }{\left({\prod }_{j=1}^n|a_j|\right)}^{\frac{1}{n}}$ is greater or less than one. These results contain Hyers-Ulam stability of the first order linear difference equation with periodic coefficients also.

36 pages