Bielecki-Ulam stability of a hammerstein-type difference system

Abstract

In this study, we investigate the Bielecki-Ulam (B-U) stabilities of two forms of Hammerstein-type difference systems (HT-DS). Specifically, we consider the systems:

$$\left(0.1\right)\{\begin{array}{c}{x}_{m+1}-x_m={\overline{M}}_m{x}_m+\overline{F}(m,x_m,x_{hm})\left[{\sum }_{\left[j=0\right]}^{\left[m\right]}\overline{G}(m,j)\overline{H}(j,x_j,x_{hj})\right]\hfill \\ x_0=b_0\hfill \end{array},$$

and

$$\left(0.2\right)\{\begin{array}{c}{x}_{m+1}-x_m={\overline{M}}_m{x}_m+\overline{F}(m,x_m,\overline{L}{x}_m,\overline{J}{x}_m)\hfill \\ x_0=b_0\hfill \end{array},$$

by establishing conditions under which a unique solution exists. We derive sufficient conditions for the existence and uniqueness of solutions that satisfy B-U stability criteria. To demonstrate the theoretical findings, we provide an illustrative example that confirms the validity of our results.

• •Purpose: In this study, we examine the Bielecki-Ulam (B-U) stabilities of two forms of Hammerstein-type difference systems (HT-DS) to understand the conditions necessary for solution uniqueness and stability.
• •Methodology: We analyze two specific systems characterized by distinct recursive nonlinear structures and employ the Banach contraction principle under the Bielecki norm to establish stability results. The theoretical development involves verifying boundedness and Lipschitz continuity of the nonlinear terms and ensuring that the involved operators satisfy contractive conditions.
• •Findings: We derive sufficient conditions (outlined in Theorems 2 and 3) under which the systems possess unique solutions and are shown to be Bielecki-Ulam stable (Theorems 4 and 5). Specifically, these conditions include boundedness of system coefficients, Lipschitz continuity of nonlinear mappings, and the fulfillment of a contraction inequality using the Bielecki norm. Illustrative examples are provided to confirm the applicability of the results.

Graphical abstract

Specifications table

Subject area:	Mathematics and Statistics
More specific subject area:	Stability theory of dynamical systems
Name of your method:	Bielecki stability
Name and reference of original method:	N/A
Resource availability:	N/A

Background

Discrete models are widely used across various scientific disciplines such as biomathematics, physics, economics, statistics, mathematics, and engineering sciences to represent sample numbers rather than continuous ones. Many practical problems, including the national income models and cobweb models [1], are effectively described using difference equations. Similarly, models referenced in [2,3] are utilized to address real-world problems in fields like physics, mechanics, and biomathematics.

While differential equations are commonly employed to simulate most physical, technological, or biological processes such as the motion of celestial bodies, the structural development of buildings, and the interactions between neurons they often challenge exact solutions. Numerical methods become essential in these cases, offering approximations where analytical solutions are impractical. Applications of differential equations are vast, encompassing tasks such as calculating electricity flow [4], and elucidating thermodynamic principles [5]. Indeed, many foundational laws of chemistry and physics are articulated through differential equations, and they are also instrumental in modeling complex systems in biology [6] and economics [7]. The theory of stability has seen significant development over the past seven decades, with numerous new concepts emerging. Stability problems are critical for a variety of functional equations. In practice, many differential equations are too complex for symbolic solutions, making numerical approximations necessary. Although error analysis in these approximations is sometimes considered negligible, there are cases where it can pose significant challenges. Thus, understanding when and to what extent these errors can be disregarded is of great interest. Stability theory plays a crucial role in this analysis, particularly in determining whether the error remains bounded. A differential equation is generally considered stable if small perturbations lead to minor changes in its solution.

Asymptotic stability theory, a key area in mathematics, has many variants, including Ulam-Hyers, Bielecki-Ulam (B-U), Ulam-Hyers-Rassias, and their generalizations. The concept of Ulam-Hyers stability originates from a question posed by Ulam at a conference in 1940 [8], which was subsequently addressed in the context of Banach spaces by D. Hyers in 1941 [9]. This led to the development of Ulam-Hyers stability, a concept used to determine the stability of functional equations. Rassias further extended this idea in 1978, focusing on the case of linear mappings [13]. For further details, see [[10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]].

Bielecki-Ulam stability offers a more generalized approach to analyzing asymptotic behavior than Ulam-Hyers stability. In functional analysis, Bielecki-Hyers-Ulam (B-H-U) stability pertains to the stability of functional equations, refining the Hyers-Ulam stability by considering conditions under which a functional equation exhibits approximately additive or homogeneous solutions. B-H-U stability is vital for understanding stability across various mathematical contexts, including dynamical equations, Banach spaces, and other functional frameworks. For other recent works on Hyers stability in the field of difference equations we refere to [[22], [23], [24]].

The past few years have seen extreme growth in areas like biology and engineering applying difference systems of the Hammerstein type. Nonlinear discrete-time models are deployed to estimate the growth of species populations in population dynamics and ecology. Furthermore, the stability analysis guarantees that small measurement errors will not cause significant deviations in the results of the prediction [25,26]. Similar behavior is observed in economical modeling where Hammerstein-type systems can capture the underlying nonlinearities in cobweb models of supply and demand. In this case, Bielecki-Ulam (B-U) stability is robust to the small values of perturbations in market parameters [1,27]. In epidemiology, SIR models assume non-linear infection/recovery rates giving rise to discrete SIR models with Hammerstein-type difference systems structure. The stability of these models ensures that reliable predictions of the spread of disease can be made even with loosely bound initial data [28,29].

Additionally, Hammerstein-type frameworks arise organically in control systems and signal processing, where static nonlinear elements are combined with linear dynamic motion. In control systems, maintaining B-U stability is fundamental for passive systems with actuator nonlinearities or sensor errors, since it ensures that performance remains well-defined even when perturbed by small disturbances [30,31]. In the context of neural networks, particularly recurrent architectures, discrete-time dynamics with nonlinear activation functions tend to exhibit a Hammerstein-like structure which require some stabilization so as not to diverge when training or running the model [32,33].

Encouraged by the practical uses, in this work we aim to complete the Bielecki-Ulam stability gap for Hammerstein-type difference systems, which adds to the theoretical groundwork needed to guarantee dependable performance in such scenarios.

BASIC definitions AND preliminaries

This section, covers some fundamental ideas and terminologies from Functional Analysis pertaining to our main work of difference equations, B-H-U stability of delay difference system, uniqueness of solution of non-linear Hammerstein type difference system.

Throughout the paper, we will use the following notations: the vector norm will be denoted by $\parallel ·\parallel$ , the $n$-dimensional Euclidean space will be represented by the symbol ${\mathbb{R}}^n$. The collection of real, integer, and non-negative integer numbers are represented by the symbols $\mathbb{R},\mathbb{Z}\text{and}{\mathbb{Z}}_{+}$respectively. The space of all bounded sequences will be denoted by $B\left({\mathbb{Z}}_{+},X\right)$ $,$ with norm $\parallel x\parallel ={Sup}_{n\in {\mathbb{Z}}_{+}}\left|x_n\right|$ , the symbol ${\mathbb{M}}_{n\times n}$ presents the matrices of size $n\times n.$ The fixed point of a map $f$ is a point $x_0$ such that $f\left(x_o\right)=x_o$. A contraction is a map $f\left(x\right)$ such that $d\left(f\left(x\right),f\left(y\right)\right)\le Ld\left(x,y\right)$, for all $x,y$ in the metric space $X$, where $0\le L<1.$

Definition 1

A sequence $\left\{x_m\right\}$ is referred to be an exact solution of (0.1) if

$$\parallel x_{m+1}-x_m-{\overline{M}}_m{x}_m-\overline{F}(m,x_m,x_{hm})\left[\sum _{\left[j=0\right]}^{\left[m\right]}\overline{G}(m,j)\overline{H}(j,x_j,x_{hj})\right]\parallel =0$$

and same for (0.2), provided:

$$\parallel x_{m+1}-x_m-{\overline{M}}_m{x}_m-\overline{F}\left(m,x_m,\overline{L}{x}_m,\overline{J}{x}_m\right)\parallel =0$$

Definition 2

Let $\left\{{\phi }_m\right\}$ be a sequence. It is termed as $ϵ$-approximate solution of (0.1), if

$$\parallel {\phi }_{m+1}-{\phi }_m-{\overline{M}}_m{\phi }_m-\overline{F}\left(m,{\phi }_m,{\phi }_{hm}\right)\left[\sum _{\left[j=0\right]}^{\left[m\right]}\overline{G}\left(m,j\right)\overline{H}\left(j,{\phi }_j,x_{{\phi }_j}\right)\right]\parallel <ϵ$$

and same for (0.2), provided:

$$\parallel {\phi }_{m+1}-{\phi }_m-{\overline{M}}_m{\phi }_m-\overline{F}\left(m,{\phi }_m,\overline{L}{\phi }_m,\overline{J}{\phi }_m\right)\parallel <ϵ$$

Definition 3

The systems (0.1) and (0.2) become H-U stable provided we can find $\overline{L}>0$, with

$$\parallel {\phi }_m-x_m\parallel <\overline{L}ϵ,\forall m\in \mathbb{Z},$$

where $\left\{{\phi }_m\right\}$ is an exact and $\left\{x_m\right\}$ an $ϵ$-approximate solution respectively.

Definition 4

The systems (0.1) and (0.2) are considered Bielecki-Ulam (B-U) stable if there exists a constant $\overline{L}>0$ such that, for every $\left\{{\phi }_m\right\}$ and $\left\{x_m\right\}$, the following inequality holds for all $m\in N.$

$$\parallel {\phi }_m-x_m\parallel e^{-h\left(m\right)}\le \overline{L}ϵ$$

Definition 5

(0.1) and (0.2) are Bielecki-Ulam's Hyers-Rassias stable if there exists a constant $\overline{L}>0$ such that for every $\left\{{\phi }_m\right\}$ and $\left\{x_m\right\}$ of the system, the following condition is satisfied for all $m\in \mathbb{Z}$:

$$\parallel {\phi }_m-x_m\parallel e^{-h\left(m\right)}<\overline{L}{\phi }_m.$$

The next theorem is a The Banach contraction theorem which is a well-known result of Banach, will be used for proving the existence and uniqueness of the solutions of the proposed systems.

Theorem 1

[25].

Let $(\mathbb{Y},d)$ a complete metric space and $g:\mathbb{Y}\to \mathbb{Y}$ a contraction having constant $L_0$, then $g$ has a unique fixed point $v\in \mathbb{Y}$. Furthermore; for any $y_0\in \mathbb{Y}$ we have

$$\underset{m\to \infty }{\ell im}{g}^m\left(y_0\right)=v,\text{with}d(g^m\left(y_0\right),v)\le L_0^m/(1-L_0)d(y_0,g\left(y_0\right))$$

The system (0.1) and (0.2) have exact and approximate solutions in the following lemmas respectively.

Lemma 1

The system (0.1) and (0.2) have the exact solutions:

$$x_m=b_0+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j{x}_j\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\overline{F}(j,x_j,x_{h_j})\right]\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\overline{G}(j,k_0)\overline{H}(k_0,x_{k_0},x_{h_{k_0}})\right]$$

and

$$x_m=b_0+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j{x}_j\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\overline{F}(j,x_j,{\overline{L}}_{x_j},{\overline{J}}_{x_j})\right].$$

In the same manner.

Lemma 2

The system (0.1) and (0.2) have the approximate solutions:

$${\phi }_m=b_0+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j{\phi }_j\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\overline{F}(j,{\phi }_j,{\phi }_{h_j})\right]\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\overline{G}(m,k_0)\overline{H}(k_0,{\phi }_{k_0},{\phi }_{h_{k_0}})\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{f}_j\right],$$

and

$${\phi }_m=b_0+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j{\phi }_j\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\overline{F}(j,{\phi }_j,{\overline{L}}_{{\phi }_j},{\overline{J}}_{{\phi }_j})\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{f}_j\right].$$

The above solutions are easily determined for different values of $m$.

Method details

Existence and uniqueness of solutions

Let us note the following assumptions in order to determine existence and uniqueness of solutions of (0.1) and (0.2):

• ${\mathcal{B}}_1:$$\parallel {\overline{M}}_j\parallel \le m_j$

• ${\mathcal{B}}_2:$$\parallel \overline{G}(j,k_0)\parallel \le g_{j,k_0}$

• ${\mathcal{B}}_3:$$\parallel \overline{F}(j,x_j,x_{h_j})\parallel \le \overline{K}$

• ${\mathcal{B}}_4:$$\parallel \overline{H}(j,y_j,y_{h_j})\parallel \le \overline{N}$

• ${\mathcal{B}}_5:$$\parallel \overline{F}(j,x_j,x_{h_j})-\overline{F}(j,y_j,y_{h_j})\parallel \le {\overline{L}}_1\parallel x_j-y_j\parallel$

• ${\mathcal{B}}_6:$$\parallel \overline{H}(j,x_j,x_{h_j})-\overline{H}(j,y_j,y_{h_j})\parallel \le {\overline{L}}_2\parallel x_j-y_j\parallel$

• ${\mathcal{B}}_7:$${\sum }_{\left[j=0\right]}^{\left[\infty \right]}\left[m_j+(\overline{K}{\overline{L}}_2+\overline{N}{\overline{L}}_1)\left({\sum }_{\left[k_0=0\right]}^{\left[j\right]}{g}_{j,k}\right)\right]<1$.

The next two results are very important in which through the above and below assumptions we will prove the existence and uniqueness of the solutions.

Theorem 2

The system (0.1) contributes a unique solution upon the validity of assumptions ${\mathcal{B}}_1$ to ${\mathcal{B}}_7$.

Proof:

We can define a map as under:

$$\overline{T}{x}_m=b_0+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j{x}_j\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left(\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\overline{F}(j,x_j,x_{h_j})\overline{G}(j,k)\overline{H}(k_0,x_{k_0},x_{h_{k_0}})\right)\right].$$

Now consider,

$$\begin{array}{c}\parallel \overline{T}{x}_m-\overline{T}{y}_m\parallel =\parallel \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}({\overline{M}}_j{x}_j-{\overline{M}}_j{y}_j)\right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\overline{G}(j,k_0)\hfill \\ ((\overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,x_{k_0},x_{h_{k_0}})-\overline{F}(j,y_j,y_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0}))]\parallel \hfill \\ \le \underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\parallel {\overline{M}}_j\parallel \parallel x_j-y_j\parallel \right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\parallel \overline{G}(j,k_0)\parallel \hfill \\ [\parallel \overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,x_{k_0},x_{h_{k_0}})-\overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\hfill \\ +\overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})-\overline{F}(j,y_j,y_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\parallel ]\hfill \\ \le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\parallel {\overline{M}}_j\parallel \parallel x_j-y_j\parallel \right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\parallel \overline{G}(j,k_0)\parallel \hfill \\ \parallel \overline{F}(j,x_j,x_{h_j})\parallel \parallel \overline{H}(k_0,x_{k_0},x_{h_{k_0}})-\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\parallel ]\hfill \\ +\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\parallel \overline{G}(j,k_0)\parallel \parallel \overline{F}(j,x_j,x_{h_j})-\overline{F}(j,y_j,y_{h_j})\parallel \parallel \overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\parallel \right]\hfill \\ \le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j\parallel x_j-y_j\parallel \right]\hfill \\ +\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\overline{K}{\overline{L}}_2\parallel x_{k_0}-y_{k_0}\parallel +\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}{\overline{L}}_1\overline{N}\parallel x_j-y_j\parallel ]\hfill \end{array}$$

Taking supremum over $m$ on both sides we have

$$\parallel \overline{T}x-\overline{T}y\parallel \le \sum _{j=0}^{\infty }\left[{\overline{M}}_j\parallel x-y\parallel +\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\overline{K}{\overline{L}}_2\parallel x-y\parallel \right]+\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}{\overline{L}}_1\overline{N}\parallel x-y\parallel \right]\right].$$

From this we have

$$\parallel \overline{T}x-\overline{T}y\parallel \le \sum _{\left[j=0\right]}^{\left[\infty \right]}\left[{\overline{M}}_j+(\overline{K}{\overline{L}}_2+{\overline{L}}_1\overline{N})\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\right]\parallel x-y\parallel .$$

By remembering ${\mathcal{B}}_7$we see that $\overline{T}$ become contraction and thanks to Banach contraction principle, the unique fixed point of $\overline{T}$is in fact the unique solution to (0.1).We also require the following two presumptions for the next theorem:

• ${\mathcal{B}}_8:$$\parallel \overline{F}(j,x_j,{\overline{L}}_{x_j},{\overline{J}}_{x_j})-\overline{F}(j,y_j,{\overline{L}}_{y_j},{\overline{J}}_{y_j})\parallel \le {\overline{L}}_j\parallel x_j-y_j\parallel$,

• ${\mathcal{B}}_9:$${\sum }_{i=0}^{\infty }({\overline{M}}_j+{\overline{L}}_j)<1$.

Theorem 3

If the assumptions ${\mathcal{B}}_1$, ${\mathcal{B}}_8$ and ${\mathcal{B}}_9$ hold, then (0.2) takes a unique solution.

Proof:

Let us define a map as:

$$\overline{T}{x}_m=b_0+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j{x}_j\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\overline{F}(j,x_j,{\overline{L}}_{x_j},{\overline{J}}_{x_j})\right].$$

Now consider:

$$\begin{array}{c}\parallel \overline{T}{x}_m-\overline{T}{y}_m\parallel =\parallel \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}({\overline{M}}_j{x}_j-{\overline{M}}_j{y}_j)\right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\overline{G}(j,k_0)\hfill \\ ((\overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,x_{k_0},x_{h_{k_0}})-\overline{F}(j,y_j,y_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0}))]\parallel \hfill \\ \le \underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\parallel {\overline{M}}_j\parallel \parallel x_j-y_j\parallel \right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\parallel \overline{G}(j,k_0)\parallel \hfill \\ [\parallel \overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,x_{k_0},x_{h_{k_0}})-\overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\hfill \\ +\overline{F}(j,x_j,x_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})-\overline{F}(j,y_j,y_{h_j})\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\parallel ]\hfill \\ \le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\parallel {\overline{M}}_j\parallel \parallel x_j-y_j\parallel \right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\parallel \overline{G}(j,k_0)\parallel \hfill \\ \parallel \overline{F}(j,x_j,x_{h_j})\parallel \parallel \overline{H}(k_0,x_{k_0},x_{h_{k_0}})-\overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\parallel ]\hfill \\ +\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\parallel \overline{G}(j,k_0)\parallel \parallel \overline{F}(j,x_j,x_{h_j})-\overline{F}(j,y_j,y_{h_j})\parallel \parallel \overline{H}(k_0,{y_k}_0,y_{{h_k}_0})\parallel \right]\hfill \end{array}$$

$$\begin{array}{c}\le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j\parallel x_j-y_i\parallel \right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\overline{K}{\overline{L}}_2\parallel x_{k_0}-y_{k_0}\parallel \hfill \\ +\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}{\overline{L}}_1\overline{N}\parallel x_j-y_j\parallel ]\hfill \end{array}$$

Taking supremum over $m$ on both sides we have:

$$\parallel \overline{T}x-\overline{T}y\parallel \le \left[\sum _{\left[i=0\right]}^{\left[\infty \right]}{\overline{M}}_j\parallel x-y\parallel \right]+\left[\sum _{\left[j=0\right]}^{\left[\infty \right]}{\overline{L}}_j\parallel x-y\parallel \right].$$

From this we have:

$$\parallel \overline{T}x-\overline{T}y\parallel \le \left[\sum _{\left[j=0\right]}^{\left[\infty \right]}\left({\overline{M}}_j+{\overline{L}}_j\right)\right]\parallel x-y\parallel .$$

Thus the fixed point of $\overline{T}$ is the unique solution of (0.2), according to the Banach contraction principle, since we are employing the axiom ${\mathcal{B}}_9$ to make the operator $\overline{T}$ a contraction.

B-H-U stability OF ht-ds

In this part of manuscript, we require the following presumptions to examine B-H-U stability of (0.1):

• ${\mathcal{B}}_{10}:$$\parallel f_i\parallel \le ϵ$,

• ${\mathcal{B}}_{11}:$$\exists B\in {\mathbb{R}}_{+}$ With the property that:

  $$\frac{\left(m-1\right)e^{-rm}}{1-{{\sum }^{\left[m-1\right]}}_{\left[j=0\right]}\left[\left({\overline{M}}_j+(\overline{K}{\overline{L}}_2+\overline{N}{\overline{L}}_1\right)\left[{{\sum }^{\left[j\right]}}_{\left[k_0=0\right]}{g}_{j,k_0}\right]\right]}\le B.$$

The next two theorems are our main results, in which we prove the Bielecki-Ulam stabilities for both the system while using the above and below conditions.

Theorem 4

The validity of presumptions ${\mathcal{B}}_1$ to ${\mathcal{B}}_7$, ${\mathcal{B}}_{10}$, and ${\mathcal{B}}_{11}$ guarantee that (0.1) is B-U stable .

Proof:

For solutions $\left\{x_m\right\}$ and $\left\{{\phi }_m\right\}$ of (0.1) we consider:

$$\parallel x_m-{\phi }_m\parallel e^{-rn}\le \underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\parallel {\overline{M}}_j\parallel \parallel x_j-{\phi }_i\parallel e^{-rn}\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}$$

$$+\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\left[\parallel \overline{G}(j,k_0)\parallel \parallel \overline{F}(j,x_j,x_{h_j})\parallel \parallel \overline{H}(k_0,x_{k_0},x_{h_{k_0}})-\overline{H}(k_0,{\phi }_{k_0},{\phi }_{h_{k_0}})\parallel e^{-h\left(m\right)}\right]\right]$$

$$+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\left[\parallel \overline{G}(j,k_0)\parallel \parallel \overline{F}(j,x_j,x_{h_j})-\overline{F}(j,{\phi }_j,{\phi }_{h_j})\parallel \parallel \overline{H}(k_0,{\phi }_{k_0},{\phi }_{h_{k_0}})\parallel e^{-rm}\right]\right]$$

$$+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\parallel f_j\parallel e^{-rn}.$$

Now by using ${\mathcal{B}}_1$ to ${\mathcal{B}}_5$ and ${\mathcal{B}}_{10}$ we have:

$$\parallel x_m-{\phi }_m\parallel e^{-rn}\le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j\parallel x_j-{\phi }_j\parallel e^{-rm}\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\overline{K}{\overline{L}}_2\parallel x_{k_0}-{\phi }_{k_0}\parallel e^{-rm}\right]$$

$$+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\overline{N}{\overline{L}}_1\parallel x_j-{\phi }_j\parallel e^{-rm}\right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}ϵe^{-rm}.$$

Taking supremum over $\parallel x_m-{\phi }_m\parallel$ on both sides we have:

$$\parallel x-\phi \parallel e^{-rn}\le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j\parallel x-\phi \parallel e^{-rn}\right]+\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\overline{K}{\overline{L}}_2\parallel x-\phi \parallel e^{-rm}\right]$$

$$\sum _{\left[j=0\right]}^{\left[m-1\right]}\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\overline{N}{\overline{L}}_1\parallel x-\phi \parallel e^{-rm}+(m-1)ϵe^{-rm}\right],$$

From this we have:

$$\parallel x-\phi \parallel e^{-rm}\le \frac{ϵ(m-1)e^{-rm}}{\left[1-\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left[\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\left[{\overline{M}}_j+\left(g_{j,k_0}\overline{K}{\overline{L}}_2+g_{j,k_0}\overline{N}{\overline{L}}_1\right)\right]\right]\right]}.$$

$$\parallel x-\phi \parallel e^{-h\left(m\right)}\le \overline{B}ϵ$$

Hence the system (0.1) is B-H-U stable.

The following assumptions will be under consideration stability of the system (0.2):

${\mathcal{B}}_{12}:$ There exist a constant $\overline{C}$ such that $\frac{(m-1)e^{-rm}}{\left[1-{{\sum }^{\left[m-1\right]}}_{\left[j=0\right]}\left({\overline{M}}_j+{\overline{L}}_j\right)\right]}\le \overline{C},\forall m\in \mathbb{Z}.$

Theorem 5

If ${\mathcal{B}}_1$ to ${\mathcal{B}}_7$ and ${\mathcal{B}}_{12}$ hold, then (0.2) is B-U stable.

Proof:

For solutions $\left\{x_m\right\}$ and $\left\{{\phi }_m\right\}$ of (0.1) and (0.2) let us consider:

$$\begin{array}{c}\parallel x_m-{\phi }_m\parallel e^{-rn}=\parallel \underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}({\overline{M}}_j{x}_j-{\overline{M}}_j{\phi }_j)+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left(\overline{F}(j,x_j,{\overline{L}}_{x_j},{\overline{J}}_{x_j})-F(j,{\phi }_j,{\overline{L}}_{{\phi }_j},{\overline{J}}_{{\phi }_j}\right)\right]\hfill \\ -\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{f}_j\parallel e^{-rm}\hfill \end{array}$$

$$\le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\parallel {\overline{M}}_j\parallel \parallel x_j-{\phi }_j\parallel e^{-rm}\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\parallel \overline{F}(j,x_j,{\overline{L}}_{x_j},{\overline{J}}_{x_j})-\overline{F}(j,{\phi }_j,L_{{\phi }_j},{\overline{J}}_{{\phi }_j})\parallel e^{-rm}\right]$$

$$+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\parallel f_j\parallel e^{-rn}\right].$$

Now by using ${\mathcal{B}}_1$, ${\mathcal{B}}_8$ and ${\mathcal{B}}_{10}$, we have

$$\parallel x_m-{\phi }_m\parallel e^{-rn}\le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j\parallel x_j-{\phi }_j\parallel e^{-rm}\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{L}}_j\parallel x_m-{\phi }_m\parallel e^{-rm}\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}ϵe^{-rm}\right].$$

Taking supremum we achieve,

$$\parallel x-\phi \parallel e^{-rm}\le \left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{M}}_j\parallel x-\phi \parallel e^{-rm}\right]+\left[\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}{\overline{L}}_j\parallel x-\phi \parallel e^{-rm}+ϵ(m-1)e^{-rm}\right].$$

From this we have

$$\parallel x-\phi \parallel e^{-rm}\le \frac{ϵ(m-1)e^{-rm}}{\left[1-\underset{\left[j=0\right]}{\sum ^{\left[m-1\right]}}\left({\overline{M}}_j+{\overline{L}}_j\right)\right]}.$$

$$\parallel x-\phi \parallel e^{-h\left(m\right)}\le \overline{C}ϵ.$$

Hence (0.2) is B-H-U stable.

To verify the existence, uniqueness, and Bielecki-Ulam stability of solutions for Hammerstein-type difference systems, the following procedure can be followed:

• Step 1:
  Define the System

  • •
    The given difference system in the Hammerstein form, identifying the functions $M_m$, $F(m,x_m)$, $G(m,j,x_j)$ and $H(m,j,x_j)$ as appropriate.
• Step 2:
  Obtain the Exact Solution and Approximate Solution of the System

  • •
    By putting different values of $m$ in the system we obtain the exact $\left\{x_m\right\}$ and approximate solution $\left\{{\phi }_m\right\}$ in a recursive manner.
• Step 3:
  Verify the Boundedness Conditions

  • •
    Check that the sequences and nonlinear functions satisfy boundedness conditions (assumptions $B_1$ to $B_4$):

    • ○
      $\parallel M_m\parallel \le B$,
    • ○
      $\parallel G(m,j)\parallel \le g$,
    • ○
      $\parallel F(m,x_m)\parallel \le K$,
    • ○
      $\parallel H(m,x_m)\parallel \le N$ for all relevant indices.
• Step 4:
  Verify Lipschitz Conditions

  • •
    Confirm that the nonlinear mappings $F$ and $H$ are Lipschitz continuous with constants $L_1$ and $L_2$ (assumptions $B_5$ and$B_6$).
• Step 5:
  Check the Contraction Condition

  • •
    Verify that the contraction condition holds:

    $$\underset{m}{sup}\left(K{L}_1+N{L}_2+g\right)<1,$$

    ensuring that the mapping defined in the proof becomes a contraction.
• Step 6:
  Apply Banach Fixed Point Theorem

  • •
    Using the contraction property, conclude that a unique solution exists according to the Banach fixed point theorem.
• Step 7:
  Verify Bielecki-Ulam Stability

  • •
    For the approximate and exact solutions $\left\{{\phi }_m\right\}$ and $\left\{x_m\right\}$, check if the Bielecki-Ulam stability inequality holds:

    $$\parallel x_m-{\phi }_m\parallel \le C{e}^{-hm}ϵ,$$

    where $C$ is a stability constant derived from the boundedness and Lipschitz constants.

Bielecki stability is a specific concept used primarily in the context of differential and integral equations, especially in Banach spaces. It differs from other types of stability (like Lyapunov, asymptotic, exponential stability, Hyers-Ulam and many other stabilities) in how it measures the behavior of solutions over time, particularly with respect to an altered norm.

Key Differences of Bielecki Stability:

• 1. Norm Modification (Bielecki Norm): Bielecki stability is assessed using a modified norm or weighted norm, known as the Bielecki norm:

  $${\parallel x\parallel }_{\tau }=\underset{a\le t\le b}{Sup}{e}^{-\tau t}\parallel x\left(t\right)\parallel$$

  where $\tau >0$ is a weighting factor. This norm exponentially de-emphasizes the future behavior of the solution, allowing for better control of growth.

• 2. Focus on Integral and Operator Equations: Bielecki stability is mostly applied in the context of integral equations or functional differential equations, rather than classical dynamical systems.

• 3. Tool for Existence and Uniqueness: It’s often used to establish existence and uniqueness of solutions using fixed-point theorems, particularly Banach's Fixed Point Theorem.

• 4. Not Directly About System Behavior Over Time: Unlike Lyapunov or asymptotic stability, which focus on the long-term behavior of trajectories (e.g., whether they converge to equilibrium), Bielecki stability is more about the sensitivity of solutions to initial data or operators within the Bielecki norm.

Method validation

Examples

• Example 1.

We have the Hammerstein type difference system as follows:

$$\left(5.1\right)\{\begin{array}{c}{x}_{m+1}-x_m={\overline{M}}_m{x}_m+\overline{F}(m,x_m,x_{h_m})\left[\sum _{\left[k_0=0\right]}^{\left[m\right]}\overline{G}(m,k_0)\overline{H}(k_0,x_{k_0},x_{h_{k_0}})\right]\hfill \\ x_0=b_0,\hfill \end{array}$$

Where ${\overline{M}}_m=\frac{0.1}{3^m}$, $\overline{G}(j,k_0)=\frac{1}{2^{2j+k_0}}$, $\overline{F}(m,x_m,x_{h_m})=e^{-m}+\cos \left(m\right)$ and $\overline{H}(m,x_m,x_{h_m})=e^{-2m}+\sin \left(m\right)$.

Now since,

$$\left[\sum _{\left[j=0\right]}^{\left[\infty \right]}\overline{M_j}\right]=\left[\sum _{\left[j=0\right]}^{\left[\infty \right]}\frac{0.1}{3^j}\right]=0.2<1,$$

$$\parallel \overline{G}(j,k_0)\parallel \le \frac{1}{2},$$

$$\parallel \overline{F}(m,x_m,x_{h_m})\parallel =\parallel e^{-m}+\cos \left(m\right)\parallel \le 2,$$

$$\parallel \overline{H}(m,x_m,Z_{h_m})\parallel =\parallel e^{-2m}+\sin \left(m\right)\parallel \le 2,$$

$$\parallel \overline{F}(m,x_m,x_{h_m})-\overline{F}(m,y_m,Y_{h_m})\parallel \le \parallel x_m-y_m\parallel ,$$

$$\parallel \overline{H}(m,x_m,Z_{h_m})-\overline{H}(m,y_m,Y_{h_m})\parallel \le \parallel x_m-y_m\parallel ,$$

$$\left[\sum _{\left[j=0\right]}^{\left[\infty \right]}{\overline{M}}_j+(\overline{K}{\overline{L}}_2+\overline{N}{\overline{L}}_1)\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}{g}_{j,k_0}\right]=\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\left[\frac{0.1}{3^j}+\underset{\left[k_0=0\right]}{\sum ^{\left[j\right]}}\frac{4}{{16.2}^{\left(2j+k_0\right)}}\right]=0.82<1.$$

Also, since

$$\left[\frac{(m-1)e^{-rm}}{1-\left[{{\sum }^{\left[m-1\right]}}_{\left[j=0\right]}({\overline{M}}_j+(\overline{K}{\overline{L}}_2+\overline{N}{\overline{L}}_1)\right]\left[{{\sum }^{\left[j\right]}}_{\left[k_0=0\right]}{g}_{j,k_0}\right]}\right]=\left[\frac{(m-1)e^{-rm}}{\left[0.52+\frac{0.15}{3^m}+\frac{1}{2^m}-\frac{0.67}{4^m}\right]}\right]\stackrel{n\to \infty }{\to }0$$

Therefore, it needs to be bounded, meaning that there needs to be a positive number such that

$$\left[\frac{(m-1)e^{-rm}}{1-\left[{{\sum }^{\left[m-1\right]}}_{\left[j=0\right]}({\overline{M}}_j+(\overline{K}{\overline{L}}_2+\overline{N}{\overline{L}}_1)\right]\left[\underset{\left[k_0=0\right]}{{\sum }^{\left[j\right]}}{g}_{j,k_0}\right]}\right]\le \overline{B}.$$

Taking into account that ${\mathcal{B}}_1$ to ${\mathcal{B}}_9$ be true, (5.1) give unique solution bearing in mind Theorem. 2, and it is Bielecki-Ulam-Hyers stable according to Theorem. 4.

• Example 2.

Examine:

$$\left(5.2\right)\{\begin{array}{c}{x}_{m+1}-x_m={\overline{M}}_m{x}_m+\overline{F}(m,x_m,{\overline{L}}_{x_m},{\overline{J}}_{x_m})\hfill \\ x_0=b_0,\hfill \end{array}$$

where ${\overline{M}}_m=\frac{0.1}{3^m}$ and $\overline{F}(m,x_m,{\overline{L}}_{x_m},{\overline{J}}_{x_m})=e^{-3m}+\frac{1}{5^m}\cos \left(x_m\right)+\sin \left({\overline{L}}_{x_m}\right)$.

Clearly, all axioms of Theorem. 3 and Theorem. 5 are true. So (5.2) has unique solution and is B-H-U stable.

Conclusion

In this study, we investigated the Bielecki-Ulam (B-U) stability properties of two forms of nonlinear Hammerstein-type difference systems. The main objective was to identify sufficient conditions under which the proposed systems admit unique solutions that are stable in the Bielecki-Ulam sense. By employing fixed point theory particularly, the Banach contraction principle and leveraging weighted norms, we established a set of precise conditions (denoted $B_1$ to $B_{12}$) that guarantee the existence, uniqueness, and B-U stability of solutions.

These results contribute a refined framework for analyzing nonlinear discrete systems by incorporating exponential decay into the stability structure, which improves upon classical Ulam-type criteria. Illustrative examples were provided to demonstrate the validity and applicability of the main results. The findings confirm that the imposed assumptions are sufficient to ensure the qualitative behavior of the system's solutions, thereby fulfilling the research objective.

Limitations

Not applicable

Ethics statements

Hereby, I Muhammad Sarrwar, consciously assure that for the manuscript BIELECKI-ULAM STABILITY OF A HAMMERSTEIN-TYPE DIFFERENCE SYSTEM

Following are fulfilled:

• •This material is the authors' own original work, which has not been previously published elsewhere.
• •The paper is not currently being considered for publication elsewhere.
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CRediT authorship contribution statement

Gul Rahmat: Conceptualization, Writing – original draft, Writing – review & editing. Sohail Ahmad: Conceptualization, Writing – original draft, Writing – review & editing. Muhammad Sarwar: Conceptualization, Writing – original draft, Writing – review & editing, Methodology, Project administration, Supervision, Validation, Visualization. Kamaleldin Abodayeh: Data curation, Formal analysis, Funding acquisition, Methodology, Project administration, Supervision, Validation, Visualization. Saowaluck Chasreechai: Data curation, Formal analysis, Funding acquisition. Thanin Sitthiwirattham: Data curation, Formal analysis, Funding acquisition, Methodology, Project administration, Supervision, Validation, Visualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in the article.