On precision of constants in some theorems on stability of difference equations. (English) Zbl 1153.39003

The authors study stability conditions (uniform and asymptotic stability) for the linear difference equation
\[ x(n+1)-x(n)=-a(n)\,x(n-h(n)), \quad n\in{\mathbb N}_0. \tag{\(*\)} \]
More precisely, they show that the conditions \[ a(n)\geq0 \quad\text{and}\quad \sup_{n\in{\mathbb N}_0}\,\sum_{i=n-h(n)}^n a(i)\leq\frac{3}{2} \] imply the uniform stability of equation (\(*\)). A large part of the paper is devoted to showing that the constant \(\frac{3}{2}\) above as well as other constants such as \(\frac{3}{2}+\frac{1}{2k+1}\) in similar results in the literature are sharp.


39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)