Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Shift operators and stability in delayed dynamic equations   In this paper, we use what we call the shift operator so that general delay
dynamic equations of the form \[
x^{\Delta}(t)=a(t)x(t)+b(t)x(\delta_{-}(h,t))\delta_{-}^{\Delta}% (h,t),\ \ \
t\in\lbrack t_{0},\infty)_{\mathbb{T}}% \] can be analyzed with respect to
stability and existence of solutions. By means of the shift operators we define
a general delay function opening an avenue for the construction of Lyapunov
functional on time scales. Thus, we use the Lyapunov's direct method to obtain
inequalities that lead to stability and instability. Therefore, we extend and
unify stability analysis of delay differential, delay difference, delay
$h-$difference, and delay $q-$difference equations which are the most important
particular cases of our delay dynamic equation. \textbf{Keywords}: Delay
dynamic equation, instability, shift operators, stability, time scales.

20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org