Search

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this article, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a) for non-autonomous equations with finite delays and uniformly bounded compact coefficient operators in Banach spaces and (b) for Volterra difference equations with infinite delay in finite dimensional spaces.

In this work, we consider the first-order difference equation with general argument

\[ \varDelta x(n)+p(n)x\left( \tau (n)\right) =0,\quad n\geq 0, \]

where $(p(n))$

is a sequence of non-negative real numbers, $(\tau (n))$

is a sequence of integers such that $\tau (n)< n$

for$\ n\in \mathbf {N},\,$

and$\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$

Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.

Search Results

Refine search

Refine search

Actions for selected content:

2 results

Shadowing and hyperbolicity for linear delay difference equations

Upper and lower limit oscillation conditions for first-order difference equations

Search Results

Refine search

Refine search

Actions for selected content:

Save Search

2 results

Shadowing and hyperbolicity for linear delay difference equations

Upper and lower limit oscillation conditions for first-order difference equations