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

TL;DR

This paper investigates the Hyers-Ulam stability of first-order linear difference equations with variable coefficients, establishing conditions related to the growth rate of the product of coefficients for stability or instability.

Contribution

It provides new criteria based on the growth rate of coefficient products for the stability analysis of linear difference equations, including those with periodic coefficients.

Findings

If the product of coefficients has subexponential growth, the equation lacks Hyers-Ulam stability.

When the limit of the nth root of the product exceeds one, stability conditions are characterized.

Results extend to equations with periodic coefficients, broadening the stability analysis scope.

Abstract

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.