The following notations will be used throughout this article. ${R}^{+}$ denotes the set of all real non-negative numbers; ${R}^{n}$ denotes the *n*-dimensional space with the scalar product $\u3008\cdot ,\cdot \u3009$ and the vector norm $\parallel \cdot \parallel $; ${R}^{n\times r}$ denotes the space of all matrices of $(n\times r)$-dimension. ${A}^{T}$ denotes the transpose of *A*; a matrix *A* is symmetric if $A={A}^{T}$, a matrix *I* is the identity matrix of appropriate dimension.

Matrix *A* is semi-positive definite ($A\ge 0$) if $\u3008Ax,x\u3009\ge 0$, for all $x\in {R}^{n}$; *A* is positive definite ($A>0$) if $\u3008Ax,x\u3009>0$ for all $x\ne 0$; $A\ge B$ means $A-B\ge 0$.

Consider a nonlinear uncertain delay-difference systems with polytopic uncertainties of the form where

$x(k)\in {R}^{n}$ is the state, the system matrices are subjected to uncertainties and belong to the polytope Ω given by

$\mathrm{\Omega}=\{[A,D](\xi ):=\sum _{i=1}^{p}{\xi}_{i}[{A}_{i},{D}_{i}],\sum _{i=1}^{p}{\xi}_{i}=1,{\xi}_{i}\ge 0\},$

(2.1)

where

${A}_{i}$,

${D}_{i}$,

$i=1,2,\dots ,p$, are given constant matrices with appropriate dimensions.

The nonlinear perturbations

$f(k,x(k-h(k)))$ satisfies the following condition

${f}^{T}(k,x(k-h(k)))f(k,x(k-h(k)))\le {\beta}^{2}{x}^{T}(k-h(k))x(k-h(k)),$

(2.2)

where *β* is positive constants. For simplicity, we denote $f(k,x(k-h(k)))$ by *f*, respectively.

The time-varying uncertain matrices

$\mathrm{\Delta}A(k)$ and

$\mathrm{\Delta}D(k)$ are defined by

$\mathrm{\Delta}A(k)={E}_{a}{F}_{a}(k){H}_{a},\phantom{\rule{2em}{0ex}}\mathrm{\Delta}D(k)={E}_{d}{F}_{d}(k){H}_{d},$

(2.3)

where

${E}_{a}$,

${E}_{d}$,

${H}_{a}$,

${H}_{d}$ are known constant real matrices with appropriate dimensions.

${F}_{a}(k)$,

${F}_{d}(k)$ are unknown uncertain matrices satisfying

${F}_{a}^{T}(k){F}_{a}(k)\le I,\phantom{\rule{2em}{0ex}}{F}_{d}^{T}(k){F}_{d}(k)\le I,\phantom{\rule{1em}{0ex}}k=0,1,2,\dots ,$

(2.4)

where *I* is the identity matrix of appropriate dimension.

The time-varying function

$h(k)$ satisfies the condition:

$0<{h}_{1}\le h(k)\le {h}_{2},\phantom{\rule{1em}{0ex}}\mathrm{\forall}k=0,1,2,\dots .$

**Remark 2.1** It is worth noting that the time delay is a time-varying function belonging to a given interval, which allows the time-delay to be a fast time-varying function and the lower bound is not restricted to being zero as considered in [2, 6, 9–11].

**Definition 2.1** The nonlinear uncertain system (${\mathrm{\Sigma}}_{\xi}$) is robustly stable if the zero solution of the system is asymptotically stable for all uncertainties which satisfy (2.1), (2.3), and (2.4).

**Proposition 2.1** *For real numbers*${\xi}_{i}\ge 0$,

$i=1,2,\dots ,p$,

${\sum}_{i=1}^{p}{\xi}_{i}=1$,

*the following inequality hold*$(p-1)\sum _{i=1}^{p}{\xi}_{i}^{2}-2\sum _{i=1}^{p-1}\sum _{j=i+1}^{p}{\xi}_{i}{\xi}_{j}\ge 0.$

*Proof* The proof is followed from the completing the square:

$(p-1)\sum _{i=1}^{p}{\xi}_{i}^{2}-2\sum _{i=1}^{p-1}\sum _{j=i+1}^{p}{\xi}_{i}{\xi}_{j}=\sum _{i=1}^{p-1}\sum _{j=i+1}^{p}{({\xi}_{i}-{\xi}_{j})}^{2}\ge 0.$

□

**Proposition 2.2** (Cauchy inequality)

*For any symmetric positive definite matrix*
$N\in {M}^{n\times n}$
*and*
$a,b\in {R}^{n}$
*we have*
$\underline{+}{a}^{T}b\le {a}^{T}Na+{b}^{T}{N}^{-1}b.$

**Proposition 2.3** ([1])

*Let* *E*,

*H* *and* *F* *be any constant matrices of appropriate dimensions and*${F}^{T}F\le I$.

*For any*$\u03f5>0$,

*we have*$EFH+{H}^{T}{F}^{T}{E}^{T}\le \u03f5E{E}^{T}+{\u03f5}^{-1}{H}^{T}H.$