General theorems for uniform asymptotic stability and boundedness in finitely delayed difference systems

Abstract

The paper deals with boundedness of solutions and uniform asymptotic stability of the zero solution. In our current undertaking, we aim to solve two open problems that were proposed by the author in his book Qualitative theory of Volterra difference equations (2018, Springer, Cham). Our approach centers on finding the appropriate Lyapunov functional that satisfies specific conditions, incorporating the concept of wedges.

1 Introduction

Let $\mathbb {Z},\; \mathbb {Z}^+,\;\mathbb {R},\;\mathbb {R}^+,$ and $\mathbb {R}^d$ be the sets of integers, nonnegative integers, real numbers, nonnegative real numbers, and d-dimensional real space, respectively. This paper is concerned with the uniform asymptotic stability of the zero solution of the finite delay difference equation

(1.1) $$ \begin{align} \displaystyle x(n+1) = f(n,x_n), \end{align} $$

where $x_n$ is the segment of $x(s)$ for $n-h \leq s\leq n,\ h $ is a nonzero positive integer. Here, the function f is continuous in x with $\displaystyle f: \mathbb {Z}\times C \rightarrow \mathbb {R}^d$ , where C is the set of functions $ \displaystyle \phi :\{n_0 -h, n_0 -h +1, \ldots , n_0-1,n_0\} \rightarrow \mathbb {R}^d,\; h> 0$ and integer, and $n_0 \geq 0 $ is the initial time. Let

$$ \begin{align*}C(n) = \{\phi : \{n-h, n-h +1, \ldots, n-1,n\}\rightarrow \mathbb{R}^d\}.\end{align*} $$

It is to be understood that $C(n)$ is C when $n=0.$ Also, $\phi _n$ denotes $\phi \in C(n)$ and $ \displaystyle ||\phi _n|| = \max _{n-h \leq s \leq n} |\phi (n\textit {})|,$ where $|\cdot |$ is a convenient norm on $\mathbb {R}^d.$ For $n=0$ ,

$$ \begin{align*}C(0) = \{\phi : \{-h, -h +1, \ldots, -1,0\}\rightarrow \mathbb{R}^d\}.\end{align*} $$

We say that $x(n)\equiv x(n,n_0,\varphi )$ is a solution of (1.1) if $x(n)$ satisfies (1.1) for $n= n_0 +1, n_0+ 2,\ldots $ and $x(n) = \varphi (n),\; n = n_0 -h, n_0 -h+1,\ldots , n_0 -1, n_0$ , where $\varphi $ is a given initial sequence such that $\varphi :\{n_0-h, n_0-h +1, \ldots , n_0-1,n_0\} \rightarrow \mathbb {R}^d.$ If $x(n)$ is any solution of system (1.1), then the variation of the function $V,$ where

$$ \begin{align*}V:\mathbb{Z}^+\times C \rightarrow \mathbb{R^{+}},\end{align*} $$

is defined as

$$ \begin{align*}\triangle V(x(n)) = V(f(n,x(n))) - V(x(n)) = V(x(n+1)) - V(x(n)).\end{align*} $$

Throughout this paper, we assume that $f(n,0) = 0, \text {for all}\; n \in \mathbb {Z}, $ when we are considering the stability of the zero solution. In this paper, we will prove two general theorems regarding the uniform asymptotic stability of the zero solution and the uniform boundedness of all solutions in terms of wedges by assuming the existence of a Lyapunov functional.

Delay discrete systems play a pivotal role in mathematical models that describe how a system changes over time, but with the added complexity of incorporating delays in the process. These delays represent the time it takes for a system to respond to inputs or changes in its environment. Studying the boundedness of solutions and stability of the zero solution is important since many real-world systems exhibit delays in their responses, such as control systems, biological processes, communication networks, and economic systems. Understanding and analyzing delay discrete systems help in designing and optimizing such systems for better performance. It is important to study the boundedness of solutions in delay discrete systems to ensure that the system’s behavior remains manageable and does not diverge or go to infinity, which could lead to instability or unpredictable outcomes. Additionally, the analysis of the stability of the zero solution is as important as boundedness. The zero solution represents the equilibrium state where the system remains unchanged over time. Stability analysis of the zero solution helps in understanding whether small perturbations or disturbances in the system will die out over time (stable), grow indefinitely (unstable), or remain at a constant level (marginally stable). This information is crucial for ensuring the reliability and robustness of the system. For more reading, we refer the interested reader to [1, 4, 6].

The use of Lyapunov functionals in the context of delay difference equations is rooted in stability analysis and control theory. Lyapunov methods provide a powerful tool for studying the stability and convergence properties of dynamic systems, including those described by delay difference equations. Aleksandr Lyapunov introduced the concept of Lyapunov functions in the late nineteenth century. His work laid the foundation for stability analysis in differential equations. Lyapunov methods were later extended to difference equations, which describe systems evolving in discrete time steps. As the study of systems with time delays gained prominence, researchers began applying Lyapunov methods to analyze stability in the presence of delays. The advantages and effectiveness of the use of Lyapunov functional can be seen in many areas including but not limited to stability analysis, and control design.

Lyapunov functionals provide a systematic way to analyze the stability of solutions to delay difference equations. They provide a mathematically rigorous framework for stability analysis. They allow researchers to derive explicit stability criteria and prove the convergence properties of systems described by delay difference equations. In summary, the use of Lyapunov functionals in delay difference equations has a rich history and remains a powerful and widely adopted methodology.

We assume that the readers are familiar with the calculus of difference equations, and for a comprehensive study of the calculus of difference equations, we refer to the books [3, 7], and for an excellent reference to the use of Lyapunov functionals in discrete systems, we refer to the book [17].

In this paper, we refer to wedges as $W_{i}:[0,\infty ) \rightarrow [0,\infty )$ that are continuous with $W_{i}(0)=0$ , $W_{i}(r)$ strictly increasing, and $W_{i}(r)\rightarrow \infty $ as $r\rightarrow \infty , i=1,2,3,4.$

Using Lyapunov functionals, in [14], the author proved general theorems regarding the stability of the zero solution and the boundedness of all solutions of functional systems of difference equations of the form

(1.2) $$ \begin{align} x(n+1) = G(n,x(s);\ 0\leq s \leq n) \overset{def}{=} G(n,x(\cdot)), \end{align} $$

where $G: \mathbb {Z^{+}}\times \mathbb {R}^{k}\rightarrow \mathbb {R}^{k}$ is continuous in $x.$ During our analysis of (1.2), we encountered endless difficulties due to finding a suitable Lyapunov functional that satisfied the pair of inequalities

(1.3) $$ \begin{align} W_{1}(|x(n)|)\leq V(n,x(\cdot))\leq W_{2}(|x(n)|) \end{align} $$

and

(1.4) $$ \begin{align} \triangle V(n,x(\cdot))\leq - \rho W_{3}(|x(n)|)+K \end{align} $$

for some positive constant $\rho $ and nonnegative constant $K.$

In this paper, we try to close the gap and prove parallel theorems regarding the stability of the zero solution and the boundedness of all solutions of the finitely delayed functional difference equation (1.1). Those parallel theorems that we attempt to prove were proposed as open problems in the book [17].

This paper is organized as follows: In the introduction, we define our functional delay problem and provide relevant stability definitions, along with an overview of previous research conducted in this context. Section 2 focuses on stating and proving a comprehensive theorem concerning the uniform asymptotic stability of the zero solution for equation (1.1). The theorem necessitates the presence of a Lyapunov functional meeting specific conditions, which involve wedges. Similarly, Section 3 addresses another pivotal theorem, emphasizing the uniform boundedness of all solutions to equation (1.1). Once again, this theorem calls for a Lyapunov functional that satisfies certain conditions related to wedges. Finally, in Section 4, we present practical applications in the form of examples derived from our findings.

The next definition and theorem can be found in [18].

Definition 1 [18]

Let $x(t)= 0$ be a solution of (1.1).

(a) The zero solution of (1.1) is stable if for each $\varepsilon>0$ and $t_1 \geq t_0$ there exists $\delta>0$ such that $[\phi \in C(t_1), \:\|\phi \|< \delta , t \geq t_1 ]$ imply that $|x(t,t_1,\phi )|< \varepsilon .$

(b) The zero solution of (1.1) is uniformly stable if it is stable and if $\delta $ is independent of $t_1 \geq t_0.$

(c) The zero solution of (1.1) is asymptotically stable if it is stable and if for each $t_1\geq t_0$ there is an $\eta>0$ such that $[\phi \in C(t_1), \:\|\phi \|< \eta ]$ imply that $|x(t,t_1,\phi )|\rightarrow 0$ as $t\rightarrow \infty .$ Note that if this is true for every $\eta>0$ , then $x=0$ is asymptotically stable in the large or globally asymptotically stable.

(d) The zero solution of (1.1) is uniformly asymptotically stable (UAS) if it is uniformly stable and if there is an $\eta>0$ such that for each $\gamma>0$ there exist a $T>0$ such that $[t_1\geq t_0, \phi \in C(t_1), \:\|\phi \|< \eta , t \geq t_1 + T]$ imply that $|x(t,t_1,\phi )|\rightarrow 0$ as $t\rightarrow \infty .$ We note also that if this is true for every $\eta>0$ , then $x=0$ is UAS in the large.

In [18], this author proved a general theorem of three parts in which stability, uniform stability, and asymptotic stability were proved concerning the zero solution of (1.1). However, the result concerning the uniform asymptotic stability was left open, which we solve, in this paper. The results of [18] are summarized in Theorem 1.1.

Theorem 1.1 [18]

Let $D> 0$ and there is a scalar functional $\displaystyle V(t, \psi _t)$ that is continuous in $\psi $ and locally Lipschitz in $\psi _t$ when $t \ge t_0$ and $\psi _t \in C(t)$ with $||\psi _t|| <D.$ Suppose also that $V(t,0) = 0$ and

(1.5) $$ \begin{align} W_{1}(|\psi(t)|)\leq V(t,\psi_t). \end{align} $$

1. (a) If

   (1.6) $$ \begin{align} \triangle V(t,\psi_t)\leq 0\;\mbox{for}\;t_0 \leq t < \infty\; \mbox{and}\; ||\psi_t|| \leq D,\qquad\qquad \end{align} $$
   then the zero solution of (1.1) is stable.

2. (b) If, in addition to (a),

   (1.7) $$ \begin{align} V(t,\psi_t)\leq W_{2}(||\psi_t||),\qquad\qquad \end{align} $$
   then the zero solution of (1.1) is uniformly stable.

3. (c) If there is an $\displaystyle M>0$ with $\displaystyle |F(t, \psi _t)| \leq M\; \mbox {for} \;t_0 \leq t < \infty \; \mbox {and}\;||\psi _t|| \leq D,$ and if

   (1.8) $$ \begin{align} \triangle V(t,\psi_t)\leq - W_{2}(|\psi(t)|),\qquad\qquad \end{align} $$

then the zero solution of (1.1) is asymptotically stable.

2 General theorem on UAS

In this section, we state and solve the open problem that was posed in [17, 18] regarding the UAS of the zero solution of (1.1).

Theorem 2.1 [17]

Let $D> 0$ and there is a scalar functional $\displaystyle V(n, \psi _n)$ that is continuous in $\psi $ and locally Lipschitz in $\psi _n$ when $n \ge n_0$ and $\psi _n \in C(n)$ with $||\psi _n|| <D.$ In addition we assume that if $\displaystyle x:[n_0 - h, \infty ) \rightarrow \mathbb {R}^d $ is a bounded sequence, then $F(n, x_n)$ is bounded on $[n_0, \infty ).$ Suppose that there is a function V such that $V(n,0) = 0,$

$$ \begin{align*} W_{1}(|\psi(n)|)\leq V(n,\psi_n)\leq W_{2}(||\psi_n||), \end{align*} $$

and

$$ \begin{align*} \triangle V(n,\psi_n)\leq - W_{3}(|\psi(n)|), \end{align*} $$

then the zero solution of (1.1) is UAS.

Proof Find $\delta $ of the uniform stability from part (b) of Theorem 1.1 for the given $\varepsilon>0$ where $\varepsilon = \min [1, \frac {D}{2}].$ For a given $\gamma> 0,$ we need to find an integer $T> 0$ such that $[n_1\geq n_0, \phi \in C(n_1), \:\|\phi \|< \eta , n \geq n_1 + T]$ imply that $|x(n,n_1,\phi )| < \gamma .$ We determine a $\delta $ of uniform stability for this same $\gamma $ , so that $[n_2\geq n_0, ||\phi _{n_2}|| < \delta , n \geq n_2]$ imply that $|x(n,n_1,\phi )| < \gamma .$ In order to distinguish this new $\delta $ from the prior $\delta $ , denote it by $\mu .$ Summing $\triangle V(n,\psi _n)\leq 0,$ from $s= n_1 $ to $n-1$ , leads to

(2.1) $$ \begin{align}V(n, x_n) \leq V(n_1, \phi_{n_1}) \leq W_{2}(||\phi_n||) < W_{2}(\eta).\end{align} $$

For $n \in [n_2, n_3 -1],$ suppose that $|x(n)|> \frac {\mu }{2}.$ Then we have

$$ \begin{align*}\triangle V(n,x_n)\leq -W_3(\mu/2).\end{align*} $$

Consequently, summing $\triangle V(n,x_n)\leq - W_{3}(|x(n)|)$ will end up with

$$ \begin{align*} 0 \leq V(n_3, x_{n_3}) &\leq V(n_2, x_{n_2}) - \sum^{n_3 -1}_{s= n_2}W_3(\mu/2)\\ &\leq W_2(\eta) - (n_3-n_2)W_3(\mu/2). \end{align*} $$

This implies that

(2.2) $$ \begin{align} n_3-n_2 < \Big \lfloor \frac{W_2(\eta)}{W_3(\mu/2)} \Big \rfloor.\end{align} $$

Here, the notation $\lfloor z \rfloor = \max \{m \in \mathbb {Z}^+: m \leq z\}. $ Moreover, it was previously mentioned that if $|x(n)| < \mu $ holds within the interval $[n_4, n_5-1]$ , satisfying $n_5 - n_4 \geq h$ , then it follows that

(2.3) $$ \begin{align} |x(n)|< \gamma\quad \text{for}\;\; n \geq n_5 - 1.\end{align} $$

A final and crucial fact to know is that when $|x(n_6)| \leq \frac {\mu }{2}$ and $|x(n_7)| \geq \mu $ with $n_6 < n_7$ , and given that $f(n, x_n)$ remains bounded in n, there exists a positive constant S such that

(2.4) $$ \begin{align} n_7 - n_6> S.\end{align} $$

Thus, by summing $\triangle V(n,x_n)\leq - W_{3}(|x(n)|),$ on the interval $[n_6, n_7 -1]$ , we get

$$ \begin{align*} 0 \leq V(n_7, x_{n_7}) -V(n_6, x_{n_6}) &\leq - \sum^{n_7 -1}_{s= n_6}W_3(|x|)\\ &\leq -(n_7-n_6)W_3(\mu/2) \\ &\leq - S W_3(\mu/2), \end{align*} $$

since $W_3$ is continuous and increasing. Thus, $V(n, x_n) $ decreases by the value $T W_3(\mu /2)$ on the interval $[n_6, n_7 -1].$ Consequently, we are now able to find an integer N with

(2.5) $$ \begin{align} N S W_3(\mu/2)> W_2(\eta). \end{align} $$

The information provided in equation (2.2) indicates the existence of an integer $e_i$ within each interval of size $\Big \lceil \frac {W_2(\eta )}{W_3(\mu /2)}\Big \rceil $ , where $|x(e_i)|\leq \frac {\mu }{2}.$ Here, the notation $\lceil z \rceil = \max \{m \in \mathbb {Z}^+: m \geq z\}. $ The assertion (2.3) implies the existence of an integer point $E_i$ within each interval of length r for every n, such that $|x(E_i)| \geq \mu $ . Otherwise, the magnitude of $|x(n)|$ will stay below $\gamma $ . On the other hand, as a consequence of statement (2.4), we see that T time units pass between $e_i$ and $E_i.$ Let

$$ \begin{align*}K = r + \Big \lfloor \frac{W_2(\eta)}{W_3(\mu/2)} \Big \rfloor.\end{align*} $$

Then, on each interval of length K, we have that $V(n, x_n)$ decreases $S W_3(\mu /2)$ units. Consequently, the value $T = N K$ suffices, and as a result, we have

$$ \begin{align*}|x(n, n_1, \phi)| < \gamma \;\text{for}\; n> n_1 + T.\end{align*} $$

This completes the proof.

For more reading on the notion of stability by different approaches, we refer to [5, 10, 11, 18, 19, 21]. For more reading on the concepts of boundedness of solutions and stability of the zero solutions we refer to [8, 13, 15, 16].

3 General theorem on uniform boundedness

Now we shift our focus to the study of boundedness of solutions of system (1.1). When Lyapunov functionals are used to study the behavior of solutions of functional difference equations with finite delays of the form of (1.1), we are likely to encounter a pair of inequalities of the form

(3.1) $$ \begin{align} V(n,x(\cdot)) = W_{1}(x(n))+ \sum^{n-1}_{s=n-r}C(n,s)W_{2}(x(s)), \end{align} $$

(3.2) $$ \begin{align} \triangle V(n,x(\cdot)) \leq - W_{3}(x(n)) + F(n), \end{align} $$

where V is a Lyapunov functional bounded below, x is the known solution of the functional difference equation, and K, F, and $W_{i}, i=1,2,3,$ are scalar positive functions.

Inequalities (3.1) and (3.2) are full of information that is not visible to the naked eye. Our job now is to prove a general theorem and try to extract boundedness of the solutions. The next theorem was proposed as an open problem in [17] so that equations of the form

(3.3) $$ \begin{align} x(n+1) = a(n)x(n) + \sum^{n-1}_{s= n-r}C(n,s) g(x(s)) + p(n) \end{align} $$

can be handled, where the function g is continuous. Before we state and prove the open problem, we state a definition regarding uniform boundedness of solutions of (1.1). For more on inequalities (3.1) and (3.2), we refer to [12].

Definition 2 Solutions of (1.1) are uniformly bounded (UB) if for each $B_{1}>0$ there is $B_{2}>0$ such that $\big [n_{0}\geq 0, \phi \in C, ||\phi ||< B_{1},\; n \geq n_0\big ]$ implies $|x(n,n_{0},\phi )|< B_{2}.$

Theorem 3.1 Suppose that there is a scalar and differentiable functional $V(n,x_n)$ that is defined for $n \in \mathbb {Z}.$ Assume that the delay in (1.1) is r instead of $h.$ Let ${\phi :[n_0-r, \infty ) \rightarrow \mathbb {R}^d.}$ Suppose that every solution $\phi (n)$ of (1.1) satisfies

(3.4) $$ \begin{align} W_{4}(|\phi(n)|)\leq V(n,\phi_n)\leq W_{1}(|\phi(n)|)+W_2\Big(\sum^{n-1}_{s=n-r}W_{3}(|\phi(s)|)\Big) \end{align} $$

and

(3.5) $$ \begin{align} \triangle V(n,\phi_n)\leq -W_{3}(|\phi(n)|)+M \end{align} $$

for some positive constant M. Then solutions of (1.1) are UB.

Proof For $n_1 \geq n_0,$ and $ \phi \in C(n_1),$ we let $||\phi || \leq B_1, $ for positive constant $B_1.$ Let $x(n) = x(n, n_1,\phi ).$ A summation of the inequality in (3.5) from $s = n-r$ to $s= n-1$ with $n-1 \geq n_1 - r$ gives

$$ \begin{align*}V(n, x_n) - V(n-r, x_{n-r}) \leq -\sum^{n-1}_{s= n-r} W_3(|x(s)|) + M r.\end{align*} $$

This gives us the relation

(3.6) $$ \begin{align} \sum^{n-1}_{s= n-r} W_3(|x(s)|) \leq V(n-r, x_{n-r}) - V(n, x_n) + M r.\end{align} $$

Set $V(s)=V(s,x_s)$ on an arbitrary interval $[n_1, L]$ for any $L> n_1 + r.$ Since V is continuous in x, it has a maximum. Hence, let $V(n^{*})=\max _{n_1\leq n^{*} \leq L}V(n).$ Suppose $n^{*} \leq n_1 + r.$ Then, summing (3.5) from $n_1$ to $n^{*}-1$ followed by the use of (3.4) gives

$$ \begin{align*} V(n) &\leq V(n^{*}) \leq V(n_1) -\sum_{s= n_1}^{n^{*}-1} W_3(|x(s)|) + (n^{*} - n_1)M\\ &\leq V(n_1) + (n^{*} - n_1)M\\ & \leq W_{1}(B_1)+W_{2}(r W_3(B_1)) + Mr. \end{align*} $$

From the left side of (3.4), we have that $W_4(|x(n)|)\leq V(n),$ and hence the above inequality gives

$$ \begin{align*}|x(n)|\leq W^{-1}_4\big[W_{1}(B_1)+W_{2}(r W_3(B_1)) + Mr \big].\end{align*} $$

On the other hand, if $n^{*} \in [n_1 + r, L],$ then $V(n^*-r, x_{n^*-r}) - V(n^*, x_n^*) \leq 0,$ and hence from (3.6), we have that

$$ \begin{align*}\sum^{n^*-1}_{s= n^*-r} W_3(|x(s)|) \leq M r.\end{align*} $$

We observe that for such $n^*,\; \triangle V(n^*) \geq 0,$ and hence from (3.5), we have that $ 0 \leq -W_{3}(|\phi (n)|)+M.$ This gives

$$ \begin{align*}|x(n^*)| \leq W_3^{-1}(M).\end{align*} $$

Thus, for $n \in [n_1, L]$ , we have from (3.4) that

$$ \begin{align*}W_4(|x(n)|) \leq V(n) \leq V(n^{*}) \leq W_1\big(W_3^{-1}(M)\big) + W_2(M r).\end{align*} $$

This yields the bound

$$ \begin{align*}|x(n)| \leq W^{-1}_4\big[W_1\big(W_3^{-1}(M)\big) + W_2(M r)\big].\end{align*} $$

The proof is concluded since L is arbitrary and by taking

$$ \begin{align*}B_2 = \max\Big\{W^{-1}_4\big[W_{1}(B_1)+W_{2}(r W_3(B_1)) + Mr \big],\;W^{-1}_4\big[W_1\big(W_3^{-1}(M)\big) + W_2(M r)\big]\Big\}.\end{align*} $$

For more reading on the notion of boundedness, we refer to [2, 5, 6, 12, 20].

4 Applications

This section is devoted to applications of Theorems 2.1 and 3.1. Our applications will be presented in the forms of examples. We begin with the following example.

Example 4.1 We consider the scalar nonlinear finitely delayed difference equation

(4.1) $$ \begin{align} x(n+1) = b(n)h(x(n)) + a(n)g(x(n- r)) + c(n), \end{align} $$

where $a, b, c:\mathbb {Z}^+\rightarrow \mathbb {R}$ and r is a positive integer. The functions g and h are considered to be continuous in $x.$ Suppose that there are three positive constants $ \zeta _1, \zeta _2,$ and $\zeta _3$ such that $|h(x)| \leq \zeta _1 |x|, |g(x)| \leq \zeta _2 |x|,$ and $|c(n)| \leq \zeta _3.$ Additionally, we assume that

(4.2) $$ \begin{align} \lim_{n \rightarrow \infty} \zeta_1 |b(n)| \ne 1, \end{align} $$

(4.3) $$ \begin{align} \zeta_1 |b(n)| + \zeta_2 |a(n+r)| - 1 \leq -\zeta_2 |a(n+r)|,\end{align} $$

and

(4.4) $$ \begin{align} \sum^{\infty}_{n= 0} |a(n)| <\infty.\end{align} $$

Then all solutions of (4.1) are UB.

Proof We consider the Lyapunov functional $V(n) := V(n,x(n)),$

$$ \begin{align*}V(n) = |x(n)| + \sum^{n-1}_{s=n-r}|a(s+r)||g(x(s))|.\end{align*} $$

Then, along the solutions of (4.1), we have

$$ \begin{align*} \triangle V(n)&= |x(n+1)|-|x(n)| + |a(n+r)| |g(x(n))| - |a(n)||g(x(n-r))|\\ &\leq |b(n)| |h(x(n))| + |a(n)||g(x(n-r))| + \zeta_3 - |x(n)| \\ &+ |a(n+r)| |g(x(n))| - |a(n)||g(x(n-r))|\\ &= \Big(\zeta_1 |b(n)| + \zeta_2 |a(n+r)| - 1\Big)|x(n)| + \zeta_3. \end{align*} $$

Now we make sure that the requirements of Theorem 3.1 are met. We may take $W(|x(n)|) = W_1(|x(n)|) = |x(n)|.$ From the definition of $V(n)$ and due to condition (4.4), we have that

$$ \begin{align*}V(n) = |x(n)| + \sum^{n-1}_{s=n-r}|a(s+r)||g(x(s))| \leq |x(n)| +\zeta_2 \sum^{n-1}_{s=n-r}|a(s+r)||x(s)|.\end{align*} $$

Thus, we take $W_3(|x(s)|) = \zeta _2 |a(s+r)||x(s)|.$ Consequently,

$$ \begin{align*} \triangle V(n) &\leq \Big(\zeta_1 |b(n)| + \zeta_2 |a(n+r)| - 1\Big)|x(n)| + \zeta_3\\ &\leq - \zeta_2 |a(n+r)||x(n)| + \zeta_3\\ &= -W_3(|x(n)|) + \zeta_3. \end{align*} $$

Thus, all the requirements of Theorem 3.1 are satisfied and all the solutions of (4.1) are UB.

For example, the nonlinear delay equation

$$ \begin{align*}x(n+1) = \frac{1}{2}\frac{n}{n+1}x(n) + \frac{1}{6}\frac{1}{n^2+1} \frac{x(n-r)}{x^2(n) +1} + \sin(n),\quad n \geq 0,\end{align*} $$

satisfies the conditions of Theorems 2.1 and 3.1 with

$$ \begin{align*}\zeta_1= \zeta_2 =\zeta_3= 1,\; |b(n)| \leq \frac{1}{2},\; |a(n)| \leq \frac{1}{6},\;\text{and}\; \sum^{\infty}_{n=0} |a(n)| < \infty.\end{align*} $$

Remark 1 In Example 4.1, the boundedness of solutions did not depend on the size of the delay.

In the next example, we use a Lyapunov functional and show that all the solutions of equations of the form of (3.3) are UB.

Example 4.2 Assume that $D(n,s) \ne 0$ for all $-r\leq s \leq n,$ and there is a positive constant M such that $\displaystyle |p(n)| \leq M$ for all $n = 0, 1,2,\ldots $ . Then the solutions of the scalar finitely delayed Volterra difference equation

(4.5) $$ \begin{align} x(n+1) = a(n)x(n) + \sum ^{n-1}_{s=n-r} D(n,s) x(s) + p(n) \end{align} $$

are UB provided that

(4.6) $$ \begin{align} \lim_{n \rightarrow \infty} (-1 + |a(n)|) \ne 0, \end{align} $$

(4.7) $$ \begin{align} -1 + |a(n)| +\sum^\infty_{u =n+1} |D(u,n)| \leq -\sum^\infty_{u =n} |D(u,n)|, \end{align} $$

and

(4.8) $$ \begin{align} \sum ^{n-1}_{s=n-r}\sum ^{\infty}_{u=n}|D(u,s)| \leq L,\; \text{for a positive constant}\; L. \end{align} $$

Proof Consider the Lyapunov functional

(4.9) $$ \begin{align} V(n,x_n) = |x(n)| + \sum ^{n-1}_{s=n-r}\sum ^{\infty}_{u=n}|D(u,s)| |x(s)|. \end{align} $$

Then, along the solutions of (4.5), we have

(4.10) $$ \begin{align} \triangle V(n,x_n) &\leq \Big(|a(n)| -1 \Big)|x(n)| + \sum ^{n-1}_{s=n-r}|D(n,s)| |x(s)|+M\nonumber \\ &+\sum ^{n}_{s=n-r+1}\sum ^{\infty}_{u=n+1}|D(u,s)| |x(s)| -\sum ^{n-1}_{s=n-r}\sum ^{\infty}_{u=n}|D(u,s)| |x(s)|\nonumber \\ &\leq \Big(|a(n)| + \sum^\infty_{u =n+1} |D(u,n)| -1 \Big)|x(n)| + \sum ^{n-1}_{s=n-r}|D(u,s)| |x(s)|+M\nonumber \\ &+ \sum ^{n-1}_{s=n-r+1}\sum ^{\infty}_{u=n+1}|D(u,s)| |x(s)| -\sum ^{n-1}_{s=n-r}\sum ^{\infty}_{u=n}|D(u,s)| |x(s)|. \end{align} $$

We observe that

$$ \begin{align*} \sum ^{n-1}_{s=n-r}\sum ^{\infty}_{u=n}|D(u,s)| |x(s)| &= \sum ^{n-1}_{s=n-r}\Big[|D(n,s)| |x(s)|+ \sum ^{\infty}_{u=n+1}|D(u,s)| |x(s)|\Big]\\ &= \sum ^{n-1}_{s=n-r}|D(n,s)| |x(s)| +\sum ^{n-1}_{s=n-r}\sum ^{\infty}_{u=n+1}|D(u,s)| |x(s)|\\ &= \sum ^{n-1}_{s=n-r}|D(n,s)| |x(s)| +\sum ^{\infty}_{u=n+1}|D(u,n-r)| |x(n-r)|\Big]\\ &+ \sum ^{n-1}_{s=n-r+1}\sum ^{\infty}_{u=n+1}|D(u,s)| |x(s)|. \end{align*} $$

Substituting back into (4.10), we arrive at the inequality

$$ \begin{align*} \triangle V(n,x_n) &\leq \Big(|a(n)| + \sum^\infty_{u =n+1} |D(u,n)| -1 \Big)|x(n)| - \sum ^{\infty}_{u=n+1}|D(u,n-r)| |x(n-r)| + M\\ &\leq \Big(|a(n)| + \sum^\infty_{u =n+1} |D(u,n)| -1 \Big)|x(n)| + M. \end{align*} $$

It is clear from the definition of $V(n)$ that $W_4(|x|) = W_1(|x|) = |x|.$ As a consequence of (4.8), we take $W_3(|x(s)|) = \sum ^{\infty }_{u=n}|D(u,s)| |x(s)|. $ Then, from $\triangle V$ and (4.7), we have

$$ \begin{align*} \triangle V(n,x_n) &\leq \Big(|a(n)| + \sum^\infty_{u =n+1} |D(u,n)| -1 \Big)|x(n)| + M\\ &< -\sum^\infty_{u =n} |D(u,n)| |x(n)| + M\\ &=- W_3(|x(n)|) + M. \end{align*} $$

Since all the conditions of Theorem 3.1 are met, we conclude that all the solutions of (4.5) are UB.

We end this study with an example showing that the zero solution of a totally nonlinear difference equation is UAS.

Example 4.3 Consider the higher-order highly nonlinear and finitely delayed difference equation

(4.11) $$ \begin{align} x(n+1) = b(n)x^3(n-r) + c(n)x^3(n), \end{align} $$

where $b, c:\mathbb {Z}^+\rightarrow \mathbb {R}$ , and r is a positive integer. Let

$$ \begin{align*}\displaystyle Q = \{\psi \in C(n): ||\psi_n|| = \max_{n-r \leq s \leq n} |\psi(n\textit{})| < 1\}.\end{align*} $$

Assume that

(4.12) $$ \begin{align} \lim_{n \rightarrow \infty} (-1 + c^2(n)) \ne 0. \end{align} $$

If there is an $\alpha \in (0,1)$ such that

(4.13) $$ \begin{align} c^2(n) + \beta - 1 \leq -\alpha, \end{align} $$

and for positive constant $\gamma ,$

(4.14) $$ \begin{align} \alpha - \frac{b^2(n)c^2(n)}{\beta - b^2(n)}> \gamma, \end{align} $$

with $\beta> b^2(n),$ and $\beta $ is to be defined shortly. Additionally, if

(4.15) $$ \begin{align} \lim_{n \rightarrow \infty} \left(\alpha - \frac{b^2(n)c^2(n)}{\beta - b^2(n)}\right) \ne 0, \end{align} $$

then the zero solution of (4.11) is asymptotically stable.

Proof Let $x(n)$ be a solution of (4.11) with $x \in Q$ and consider the Lyapunov functional $V(n) := V(n,x(n)),$

$$ \begin{align*}V(n) = x^2(n) +\beta \sum^{n-1}_{s=n-r}x^6(s).\end{align*} $$

Then, along the solutions of (4.11), we have

$$ \begin{align*} \triangle V(n)&= x^2(n+1)-x^2(n) + \beta x^6(n) - \beta x^6(n-r)\\ &= b^2(n) x^6(n-r) + c^2(n) x^6(n) + 2b(n)c(n)x^3(n)x^3(n-r)\\ &+\beta x^6(n) - \beta x^6(n-r) - x^2(n). \end{align*} $$

Since $x(n) \in Q,$ we have that $x^2(n)>x^6(n),$ and hence

$$ \begin{align*} \triangle V(n)&\leq \Big[c^2(n) + \beta(n) -1\Big]x^6(n) +\Big(b^2(n) - \beta\Big)x^6(n-r)\\ &+ 2b(n)c(n)x^3(n)x^3(n-r)\\ &\leq -\alpha x^6(n) +\Big(b^2(n) - \beta\Big)x^6(n-r)\\ &+ 2b(n)c(n)x^3(n)x^3(n-r)\\&= -\Big[\alpha - \frac{b^2(n)c^2(n)}{\beta - b^2(n)}\Big]x^6(n)\\ &-\Big[\frac{b(n)c(n)}{\sqrt{\beta - b^2(n)}}x^3(n) - \sqrt{\beta - b^2(n)} x^3(n-r)\Big]^2\\&\leq -\gamma x^6(n). \end{align*} $$

Next, we verify that all the conditions of Theorem 2.1 are met. Let $W_1(|x(n)|)= x^2(n)$ and $W_3(|x(n)|)= \gamma x^6(n).$ Since $x(n) \in Q,$ we obtain from $V(n)$ that

$$ \begin{align*} x^2(n) +\beta \sum^{n-1}_{s=n-r}x^6(s) &\leq |x(n)| +\beta \sum^{n-1}_{s=n-r}|x(s)|\\ &\leq \Big(1+r\beta\Big) ||x_n||. \end{align*} $$

Thus, $W_2(||x_n||) =\Big (1+ r \beta \Big ) ||x_n||,$ and the zero solution of (4.11) is UAS, by Theorem 2.1. For example, if we let

$$ \begin{align*}c^2 = \frac{1}{10}, b^2 = \frac{2}{10}, \beta = \frac{1}{4},\end{align*} $$

then all the conditions of Example 4.3 are satisfied with $\alpha = \frac {13}{20}.$

We end this paper by comparing our results with those of [9]. In [9], the authors prove the discrete analogue of continuous Halanay inequality and apply it to derive sufficient conditions for the global asymptotic stability of the equilibrium of certain generalized difference equations. However, their results regarding stability will not work for equations like (4.5) when $p(n) = 0,$ for all $n \in \mathbb {Z}^+.$ This is due to the fact that our kernel, $D(n,s),$ is not constant. Additionally, the right side of (6) of [9] must have the linear term $ax(n)$ for constant a in order to invert and conclude the results. Of course, our theorems do not ask for such a requirement.