Remarks on Approximate Solutions to Difference Equations in Various Spaces

Abstract

Quite often (e.g., using numerical methods), we are only able to find approximate solutions of some equations, and it is necessary to know the size of the difference between such approximate solutions and the mappings that satisfy the equation exactly. This issue is the main subject of the theory of Ulam stability, and it is related to other areas of research such as, e.g., shadowing, optimization, and approximation theory. In this expository paper, we present several selected outcomes on Ulam stability of difference equations, show possible extensions of them and indicate further directions for research. We also present and discuss some simple methods that allow improvement of several already known results concerning Ulam stability of some difference equations in normed or metric spaces and extend them to b-metric and 2-normed spaces. Our results show that the noticeable symmetry exists between the outcomes of this type in normed and metric spaces and those obtained by us for other spaces. In particular, we extend the result of Pólya and Szegö concerning the stability of equation xn+m=xn+xm for m,n∈T, where T means either the set of integers Z or the set of positive integers N. We also consider the stability of equation xn+p+a1xn+p−1+…+apxn+bn=0 (with a fixed positive integer p) and of two more general difference equations.

Keywords

• Ulam stability
• difference equation
• 2-norm
• <i>b</i>-metric
• Banach space