Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

Round 1

Reviewer 1 Report

Referee's report on paper titled "Hyers-Ulam stability for linear difference with time dependent and periodic coefficients: The case when the monodromy matrix has simple eigenvalue"} by C.Buse, D.O'Regan and O. Saierli

In this paper the authors have proved  Hayers-Ulam stability of linear difference equation of third order with dependent and periodic coefficients. All the proofs are correct and clear. \\

Example 5.1 page 6, Equation (5.1) on the left hand side instead of $x=1$ it should be $x_{n+1}.$\\

 I recommend this paper for publication.

Author Response

Is done. Thank you.

Author Response File: Author Response.pdf

Reviewer 2 Report

The authors consider Hyers-Ulam stability for linear difference equations with variable but periodic coefficients. The topic is of interest to researchers working in the field of difference equations and the result is solid, though it could be extended to more general cases. In lattice equations, a Darboux transformation iteratively generates different solutions (see, e.g., Anal. Math. Phys., 9(2019), https://doi.org/10.1007/s13324-018-0267-z), yielding a difference equation with a fixed number of steps of applications. An interesting question could be as follows: are such resulting equations, either for Darboux transformations or solutions, are Hyers-Ulam stabile? If so, Darboux transformations are stable in the Lyanopov sense and could present approximate solutions to integrable lattices. This is an optional question for the authors to think of possible applications to other research fields. Also, In (5.1), “n=1” should be “n+1”, and it would be better to repeat (0.1) and (0.2) before Theorem 3.1. I recommend publication of a slightly revised manuscript in the journal since the manuscript present an interesting and significant result.  

Author Response

The problem raised by you is very interesting, but we are still far

from being able to solve it; at least at this time. We read the reference

[15] and wrote a remark suggested by the content of that article. The

other requirements have been met in full.

Author Response File: Author Response.pdf