Fixed Point Theorems for (Ω-Π)-Contractions in Complete S-Metric Spaces

Abstract

We introduce $(\mathrm{\Omega },\mathrm{\Pi })$-contractions in complete S-metric spaces, where $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$ satisfy $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$. Under natural comparison conditions, we prove asymptotic regularity and S-Cauchy Picard iterates. With either a closed-graph condition or ${\displaystyle \underset{t\to 0^{+}}{\lim \sup }\mathrm{\Pi }\left(t\right)<\underset{t\to {\epsilon }^{+}}{\lim \inf }\mathrm{\Omega }\left(t\right)}$ for every $\epsilon >0$, we establish unique fixed points and strong convergence from any initial point. Our results generalize the Banach contraction principle to S-metric spaces and subsume recent theorems on asymptotically regular mappings and implicit contractions. An application solves a nonlinear boundary value problem for diffusion between parallel walls.

1. Introduction

Fixed point theory ($\mathbb{FPT}$) in generalized metric structures has become a central theme of contemporary nonlinear analysis, driven both by intrinsic mathematical interest and by applications to differential, integral, and iterative equations [1,2]. In particular, several authors have extended the classical Banach contraction principle ($\mathbb{BCP}$) to various generalized metric spaces, such as b-metric, G-metric, and gauge-type spaces; see, for example, refs. [3,4,5]. Sedghi et al. (2012) introduced the concept of S-metric spaces and proved a $\mathbb{FP}$ theorem for self-mappings on complete S-metric spaces, analogous to the $\mathbb{BCP}$ [6]. This is a foundational paper in the area and has generated a substantial follow-up literature. Among the various extensions of the classical metric framework, S-metric spaces, introduced as a three-point generalization of standard metrics, provide a flexible setting in which many classical results can be reformulated and substantially strengthened [7,8]. In this context, contractive conditions are often expressed not only via numerical Lipschitz constants but also through functional inequalities involving auxiliary control functions, leading to a rich hierarchy of generalizations of the $\mathbb{BCP}$ [9,10]. A second line of development, originating in the work on asymptotically regular mappings, emphasizes the long-term behavior of iterates rather than pointwise continuity or simple Lipschitz bounds [11,12]. Asymptotic regularity has proved to be a powerful tool: under mild contractive-type conditions, it allows one to promote qualitative information about the orbit $\left\{x_n\right\}$ into the existence of fixed points ($\mathbb{FP}$s), and to obtain convergence of Picard-type iterations in spaces more general than classical Banach spaces [13,14,15]. This approach has been successfully combined with Caristi-type, Suzuki-type, and implicit contractive conditions in several generalized metric structures, including G-metric, b-metric, and gauge-type spaces; see, for instance, refs. [16,17,18].

The paper spawned many extensions. Hieu et al. (2014) generalized Ćirić quasi-contractions to S-metric spaces [19]. Özgür et al. (2016) proved generalizations of the Nemytskii–Edelstein and Ćirić $\mathbb{FP}$ theorems for continuous self-mappings of compact S-metric spaces [20]. Taş (2017) [21] extended the theory from $\mathbb{FP}$s to fixed circles on S-metric spaces. More recently, Özgür and Taş (2018) surveyed generalizations from S-metric to $S_b$-metric spaces, connecting the Rhoades contractive conditions to this broader framework [22]. Sedghi himself later co-authored a paper on Suzuki-type $\mathbb{FP}$ theorems in S-metric spaces [23]. Other directions include coupled coincidence $\mathbb{FP}$s (Raj et al. 2014) [24], $\alpha$–$\psi$-contractive mappings (Mlaiki, 2014) [25], and generalized weakly contractive mappings (Chaipornjareansri, 2018) [26]. This overview is based on a quick literature search; a deeper review would likely reveal additional extensions and applications of S-metric space theory. Recent work on common $\mathbb{FP}$ theorems in S-metric spaces has continued to extend the classical framework through implicit relations and generalised contractive conditions. For example, Saluja (2023) [27] proved common $\mathbb{FP}$ theorems for weakly compatible self-mappings involving a control function in S-metric spaces, while Popa et al. (2024) [8] established a general $\mathbb{FP}$ theorem for a pair of multivalued mappings satisfying implicit relations. More recently, Patriciu and Popa (2025) [28] extended mixed implicit relations to S-metric spaces for a pair of mappings, and Sharma et al. (2025) [29] proved new common fixed-point theorems for pairs of weakly compatible mappings in $S_b$-metric spaces. The Banach and Wardowski types use one control mechanism, whereas the $(\mathrm{\Omega },\mathrm{\Pi })$-approach introduces a two-function comparison scheme that allows a finer balance between the image and preimage distances. This makes the framework flexible enough to recover classical results while also covering nonlinear situations that cannot easily be reduced to a single constant or a single control function. The present paper contributes to this program in the setting of complete S-metric spaces by introducing and studying a class of $(\mathrm{\Omega },\mathrm{\Pi })$-contractions, where $\mathrm{\Omega }$ and $\mathrm{\Pi }$ are real-valued control functions defined on $(0,\infty )$. Under natural comparison conditions between $\mathrm{\Omega }$ and $\mathrm{\Pi }$, we first establish that such mappings are asymptotically regular and that their iterates generate S-Cauchy sequences. Building on this, and imposing either a closed-graph assumption or a suitable lim sup–lim inf separation between $\mathrm{\Omega }$ and $\mathrm{\Pi }$, we prove the existence and uniqueness of $\mathbb{FP}$s together with the strong convergence of the Picard iteration for every initial point in the space.

The main novelty of this paper is the introduction of a two-function contractive framework in S-metric spaces. Unlike most existing generalized contractions, which are based on a single Lipschitz, Wardowski, or implicit control mechanism, the proposed $(\mathrm{\Omega },\mathrm{\Pi })$-conditions decouple the image-side and domain-side behavior of the mapping. This separation yields a more flexible theory and allows the simultaneous treatment of asymptotic regularity, S-Cauchy convergence, and uniqueness in a setting that genuinely extends the current literature. To clarify the strict inclusion hierarchy, we note the following:

Proposition 1.

Let $(\mathcal{X},{\mu }^{*})$ be a complete S-metric space induced by a metric d. If $\mathfrak{F}$ is a Banach contraction with constant $k<1$, then it satisfies the $(\mathrm{\Omega },\mathrm{\Pi })$-condition with $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=kt$. Conversely, there exist mappings that satisfy an $(\mathrm{\Omega },\mathrm{\Pi })$-condition but fail every standard single-function contraction (Banach, Wardowski, or implicit). Example 3 below provides an explicit operator that meets the $(\mathrm{\Omega },\mathrm{\Pi })$-hypotheses yet has no global Lipschitz constant $<1$ and cannot be embedded into the Wardowski framework for any $\tau >0$, where τ is the length-of-time interval. This establishes the strict inclusion: Banach ⊊ Wardowski ⊊ $(\mathrm{\Omega },\mathrm{\Pi })$-contractions.

2. Preliminaries

Definition 1

([6]). Let $\mathcal{X}$ be any non-empty set and ${\mu }^{*}:\mathcal{X}\times \mathcal{X}\times \mathcal{X}\to [0,\infty )$ a mapping. A pair $(\mathcal{X},{\mu }^{*})$ is commonly termed an S-metric space if for each $x,y,z,t\in \mathcal{X}$, the subsequent characteristics hold:

1. 

$x=y=z$ if and only if ${\mu }^{*}(x,y,z)=0$,

2. 

${\mu }^{*}(x,y,z)\le {\mu }^{*}(x,x,t)+{\mu }^{*}(y,y,t)+{\mu }^{*}(z,z,t).$

Remark 1.

An S-metric is said to be symmetric if ${\mu }^{*}(x,x,y)={\mu }^{*}(y,y,x).$

Example 1

([6]). Let $\mathcal{X}\ne \varnothing$, ϱ is any ordinary metric on $\mathcal{X}$, then

$${\mu }^{*}(x,y,z)=\varrho (x,z)+\varrho (y,z).$$

is an S-metric defined on $\mathcal{X}$.

Definition 2

([6]). Suppose that $(\mathcal{X},{\mu }^{*})$ is an S-metric space and $\mathcal{C}\subset \mathcal{X}$.

1. 

A sequence $\left\{x_n\right\}$ in $\mathcal{X}$ converges to x if and only if ${\mu }^{*}(x_n,x_n,x)\to 0$ as $n\to \infty$. That, is for each $\epsilon >0$, there exists $\gamma \in \mathbb{N}$ such that for all $n\ge \gamma$, ${\mu }^{*}(x_n,x_n,x)<\epsilon$ and we denote this by $\underset{n\to \infty }{\lim }{x}_n=x$.

2. 

A sequence $\left\{x_n\right\}$ in $\mathcal{X}$ is called an S-Cauchy sequence if for each $\epsilon >0$, there exists $\gamma \in \mathbb{N}$ such that ${\mu }^{*}(x_n,x_n,x_m)<\epsilon$ for each $n,m\ge \gamma$.

3. 

The S-metric space $(\mathcal{X},{\mu }^{*})$ is said to be complete if every Cauchy sequence is convergent.

Example 2.

Let $X=\mathbb{R}$ or $\mathcal{X}=[0,1]$, and define

$${\mu }^{*}(x,y,z)=|x-y|+|y-z|+|z-x|,x,y,z\in \mathcal{X}.$$

Then $(\mathcal{X},{\mu }^{*})$ is a complete S-metric space.

Proof. 

We begin by verifying that ${\mu }^{*}$ satisfies the S-metric axioms. If $x=y=z$, then clearly

$${\mu }^{*}(x,y,z)=|x-x|+|x-x|+|x-x|=0.$$

Conversely, if ${\mu }^{*}(x,y,z)=0$, then

$$|x-y|+|y-z|+|z-x|=0.$$

Since each term is non-negative, we must have

$$|x-y|=|y-z|=|z-x|=0,$$

and hence $x=y=z$. Thus, the first axiom holds. Now let $x,y,z,t\in \mathcal{X}$. Then,

$${\mu }^{*}(x,y,z)=|x-y|+|y-z|+|z-x|.$$

By the ordinary triangle inequality,

$$|x-y|\le |x-t|+|t-y|,|y-z|\le |y-t|+|t-z|,|z-x|\le |z-t|+|t-x|.$$

Adding these three inequalities gives

$${\mu }^{*}(x,y,z)\le 2|x-t|+2|y-t|+2|z-t|.$$

On the other hand,

$${\mu }^{*}(x,x,t)=2|x-t|,{\mu }^{*}(y,y,t)=2|y-t|,{\mu }^{*}(z,z,t)=2|z-t|.$$

Therefore,

$${\mu }^{*}(x,y,z)\le {\mu }^{*}(x,x,t)+{\mu }^{*}(y,y,t)+{\mu }^{*}(z,z,t),$$

so the second S-metric axiom also holds. It remains to prove completeness. Let $\left\{x_n\right\}$ be a Cauchy sequence in $(\mathcal{X},{\mu }^{*})$. Then,

$${\mu }^{*}(x_n,x_n,x_m)=2|x_n-x_m|.$$

Hence, $\left\{x_n\right\}$ is Cauchy in the usual metric on X. Since $X=\mathbb{R}$ is complete, or $\mathcal{X}=[0,1]$ is a closed subset of $\mathbb{R}$ and therefore complete, there exists $x\in \mathcal{X}$ such that $x_n\to x$ in the usual metric. Finally,

$${\mu }^{*}(x_n,x_n,x)=2|x_n-x|\to 0,$$

so $x_n\to x$ in the S-metric sense. Therefore every S-Cauchy sequence converges in X, and $(\mathcal{X},{\mu }^{*})$ is complete. □

Definition 3

([6]). Let $(\mathcal{X},{\mu }^{*})$ be an S-metric space. For $r\in {\mathbb{R}}_{+}$ and $x\in \mathcal{X}$, we define the open ball ${\beta }_{{\mu }^{*}}(x,r)$ and closed ball $\overline{{\beta }_{{\mu }^{*}}(x,r)}$ with center x and radius r as follows:

$${\beta }_{{\mu }^{*}}(x,r)=\{y\in \mathcal{X}:{\mu }^{*}(y,y,x)<r\},$$

$$\overline{{\beta }_{{\mu }^{*}}(x,r)}=\{y\in \mathcal{X}:{\mu }^{*}(y,y,x)\le r\}.$$

Definition 4

([6]). Let $(\mathcal{X},{\mu }^{*})$ be an S-metric space, $\mathcal{C}\subset \mathcal{X}$.

1. 

If for every $x\in \mathcal{C}$ there exists $r>0$ such that ${\beta }_{{\mu }^{*}}(x,r)\subset \mathcal{C}$, then the subset $\mathcal{C}$ is called an open subset of $\mathcal{X}$.

2. 

Any S-metric generates the τ topology on $\mathcal{X}$, where τ is the set of all $\mathcal{C}\in \mathcal{X}$ with $x\subset \mathcal{C}$ if and only if there exists $r>0$ such that ${\beta }_{{\mu }^{*}}(x,r)\subset \mathcal{C}$.

Lemma 1

([30]). Suppose that $\mathrm{\Omega }:(0,\infty )\to \mathbb{R}$. Then, Conditions $\left(i\right)$, $\left(ii\right)$ and $\left(iii\right)$ are equivalent, as follows:

(i) 

${\inf }_{x>\epsilon }\mathrm{\Omega }\left(x\right)>-\infty$ for each $\epsilon >0$;

(ii) 

$\underset{x\to \epsilon }{\lim }\inf \mathrm{\Omega }\left(x\right)>-\infty$ for each $\epsilon >0$;

(iii) 

$\underset{n\to \infty }{\lim }\mathrm{\Omega }\left(x_n\right)=-\infty \Rightarrow \underset{n\to \infty }{\lim }{x}_n=0$.

Lemma 2

([30]). Suppose that $\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$. Then, Condition (i) implies Condition (ii), where

(i) 

For every sequence $\left\{x_n\right\}\subset (0,\infty )$, $\underset{n\to \infty }{\lim }\mathrm{\Pi }\left(x_n\right)=0$ implies $\underset{n\to \infty }{\lim }{x}_n=0$;

(ii) 

$\underset{x\to {\epsilon }^{+}}{\lim \inf }\mathrm{\Pi }\left(x\right)>0$ for every $\epsilon >0$.

3. Main Results

Definition 5.

Let $(\mathcal{X},{\mu }^{*})$ be a complete S-metric space and $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ a self-mapping. Assume there exist functions $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$ such that $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$ for all $t>0$ and

$$\mathrm{\Omega }\left({\mu }^{*}(\mathfrak{F}x,\mathfrak{F}x,\mathfrak{F}y)\right)\le \mathrm{\Pi }\left({\mu }^{*}(x,x,y)\right)\mathit{for}\mathit{all}x,y\in \mathcal{X},{\mu }^{*}(\mathfrak{F}x,\mathfrak{F}x,\mathfrak{F}y)>0.$$

(1)

Then, $\mathfrak{F}$ is called an $(\mathrm{\Omega },\mathrm{\Pi })$–contraction on $(\mathcal{X},{\mu }^{*})$.

The condition $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$ does not by itself guarantee contractiveness in the usual sense. Rather, its role is to provide a strict comparison between the image-side and domain-side control functions. The full contractive behavior of the mapping follows only after combining this strict comparison with the remaining assumptions imposed on $\mathrm{\Omega }$ and $\mathrm{\Pi }$ in Lemma 1.

Example 3.

Let $\mathcal{X}=[0,\infty )$ equipped with the standard S-metric ${\mu }^{*}(x,y,z)=|x-z|+|y-z|$. Define $\mathfrak{F}\left(x\right)={\displaystyle \frac{x}{1+x}}$ and choose $\mathrm{\Omega }\left(t\right)=t$, $\mathrm{\Pi }\left(t\right)={\displaystyle \frac{t}{1+t/2}}$ for $t>0$. For any $x,y\in \mathcal{X}$, we have

$${\mu }^{*}(\mathfrak{F}x,\mathfrak{F}x,\mathfrak{F}y)=2\left|{\displaystyle \frac{x}{1+x}}-{\displaystyle \frac{y}{1+y}}\right|={\displaystyle \frac{2|x-y|}{(1+x)(1+y)}}.$$

Since $(1+x)(1+y)=1+x+y+xy\ge 1+|x-y|$, that is

$${\displaystyle \frac{1}{1+x+y+xy}}\le {\displaystyle \frac{1}{1+|x-y|}},\mathit{for}\mathit{all}x,y\ge 0.$$

It follows that

$${\mu }^{*}(\mathfrak{F}x,\mathfrak{F}x,\mathfrak{F}y)\le {\displaystyle \frac{2|x-y|}{1+|x-y|}}={\displaystyle \frac{{\mu }^{*}(x,x,y)}{1+\frac{{\mu }^{*}(x,x,y)}{2}}}=\mathrm{\Pi }\left({\mu }^{*}(x,x,y)\right).$$

Thus, $\mathrm{\Omega }\left({\mu }^{*}(\mathfrak{F}x,\mathfrak{F}x,\mathfrak{F}y)\right)\le \mathrm{\Pi }\left({\mu }^{*}(x,x,y)\right)$. Moreover, $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$ for all $t>0$. Crucially, $\mathfrak{F}$ is not a Banach contraction: ${\sup }_{x\ne y}{\displaystyle \frac{|\mathfrak{F}x-\mathfrak{F}y|}{|x-y|}}={\lim }_{x,y\to 0}{\displaystyle \frac{1}{(1+x)(1+y)}}=1$, so no Lipschitz constant $k<1$ exists. This example strictly lies outside the Banach framework while fully satisfying the $(\mathrm{\Omega },\mathrm{\Pi })$-hypotheses.

Example 4.

Let $X=\left[0,{\displaystyle \frac{1}{2}}\right]$ and define the S-metric

$${\mu }^{*}(x,y,z)=\mid x-y\mid +\mid y-z\mid +\mid z-x\mid ,x,y,z\in X,$$

where $\mid ·\mid$ denotes the standard absolute value on $\mathbb{R}$. Then $(X,{\mu }^{*})$ is a complete S-metric space. Define $F:X\to X$ by $F\left(x\right)=x^2$ for all $x\in X$.

Clearly, F is not a Banach contraction on X, since no constant $k\in [0,1)$ can satisfy

$$\mid F\left(x\right)-F\left(y\right)\mid \le k\mid x-y\mid ,x,y\in X,$$

as ${\sup }_{x\ne y}{\displaystyle \frac{\mid x^2-y^2\mid }{\mid x-y\mid }}={\sup }_{x,y\in X}\mid x+y\mid =1$.

Now, choose the control functions

$$\mathrm{\Omega }\left(t\right)=t,\mathrm{\Pi }\left(t\right)=\left\{\begin{array}{cc}t\left(1-{\displaystyle \frac{t}{2}}\right),\hfill & 0<t\le 1,\hfill \\ {\displaystyle \frac{t}{2}},\hfill & t>1.\hfill \end{array}\right.$$

It is immediately evident that $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$ for all $t>0$. Moreover, Ω and Π satisfy the hypotheses of Lemmas 1 and 2.

For any $x,y\in X$ with $x\ne y$, we compute

$${\mu }^{*}(Fx,Fx,Fy)=2\mid x^2-y^2\mid =2\mid x-y\mid (x+y).$$

Since $x,y\in \left[0,{\displaystyle \frac{1}{2}}\right]$, we have $\max \{x,y\}\le {\displaystyle \frac{1}{2}}$. Noting that $x+y+\mid x-y\mid =2\max \{x,y\}$, it follows that

$$x+y\le 1-\mid x-y\mid ,$$

and therefore

$${\mu }^{*}(Fx,Fx,Fy)\le 2\mid x-y\mid \left(1-\mid x-y\mid \right).$$

Let $t={\mu }^{*}(x,x,y)=2\mid x-y\mid$. Since $x,y\in X$ with $x\ne y$, we have $0<t\le 1$, and thus only the first branch of Π is involved in the verification. Substituting t yields

$$\mathrm{\Pi }\left(t\right)=t\left(1-{\displaystyle \frac{t}{2}}\right)=2\mid x-y\mid \left(1-\mid x-y\mid \right).$$

Consequently,

$$\mathrm{\Omega }\left({\mu }^{*}(Fx,Fx,Fy)\right)={\mu }^{*}(Fx,Fx,Fy)\le \mathrm{\Pi }\left({\mu }^{*}(x,x,y)\right),x,y\in X.$$

Therefore, F is an $(\mathrm{\Omega },\mathrm{\Pi })$-contraction on $(X,{\mu }^{*})$.

Remark 2 (Advantages of Example 4).

Example 4 demonstrates that the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction framework strictly extends the classical Banach principle, as the mapping $\mathfrak{F}\left(x\right)=x^2$ on $[0,{\displaystyle \frac{1}{2}}]$ fails to admit a global Lipschitz constant $k<1$ yet satisfies the two-function inequality on the invariant subset $[0,{\displaystyle \frac{1}{2}}]$ with $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=t\left(1-{\displaystyle \frac{t}{2}}\right),0<t\le 1$. This illustrates the flexibility gained by decoupling domain-side and image-side controls: the sublinear damping of Π compensates for the quadratic growth of $\mathfrak{F}$ in a manner that is unattainable with a single multiplicative constant. Moreover, the example is fully explicit and computationally transparent, making it suitable both for pedagogical purposes and for verifying hypotheses in applied settings. The ability to localize contractivity to a physically relevant subdomain while accommodating global non-contractivity is particularly valuable for nonlinear models arising in diffusion, population dynamics, or iterative numerical schemes. Finally, the verification relies directly on the three-point structure of the S-metric, confirming that the framework is intrinsically adapted to the geometry of generalized metric spaces rather than being a formal lift of standard metric results.

3.1. Fixed-Point Theorems for $(\mathrm{\Omega }$-$\mathrm{\Pi })$-Contractions in S-Metric Spaces

Lemma 3.

Let $(\mathcal{X},{\mu }^{*})$ be a symmetric S-metric space and let ${\left\{x_n\right\}}_{n\in \mathbb{N}}$ be a sequence in $\mathcal{X}$ which is not S-Cauchy. Suppose that

$$\underset{n\to \infty }{\lim }{\mu }^{*}(x_n,x_n,x_{n+1})=0.$$

Then, there exist $\epsilon >0$ and two subsequences $\left\{x_{n_i}\right\}$ and $\left\{x_{m_i}\right\}$ of $\left\{x_n\right\}$ such that

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})=\epsilon$$

and

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i},x_{n_i},x_{m_i})=2\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_i})=\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i},x_{n_i},x_{m_{i+1}})=\epsilon .$$

Proof. 

Since $\left\{x_n\right\}$ is not an S-Cauchy sequence and

$$\underset{n\to \infty }{\lim }{\mu }^{*}(x_n,x_n,x_{n+1})=0,$$

there exists $\epsilon >0$ such that for every $k\in \mathbb{N}$ there are indices $n,m\ge k$ such that $m\ge n$ and

$${\mu }^{*}(x_{n+1},x_{n+1},x_{m+1})>\epsilon \mathrm{and}{\mu }^{*}(x_{n+1},x_{n+1},x_n)\le \epsilon .$$

By choosing the smallest $m\ge n$ for which ${\mu }^{*}(x_{n+1},x_{n+1},x_{m+1})>\epsilon$ holds, we conclude that for every $k_1>k$ there exists $n,m\ge k$, such that

$${\mu }^{*}(x_{n+1},x_{n+1},x_{m+1})>\epsilon \mathrm{and}{\mu }^{*}(x_{n+1},x_{n+1},x_m)\le {\displaystyle \frac{\epsilon }{2}}.$$

Thus, we can construct two subsequences $\left\{x_{n_i}\right\}$ and $\left\{x_{m_i}\right\}$ of $\left\{x_n\right\}$ such that

$${\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i+1})>\epsilon \mathrm{and}{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})\le {\displaystyle \frac{\epsilon }{2}}.$$

From these inequalities and S-metric inequality, we get

$$\begin{array}{cc}\hfill \epsilon <{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i+1})& \le {\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})\hfill \\ & +{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})+{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{m_i})\hfill \\ \hfill & =2{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})+{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{m_i})\hfill \\ \hfill & \le 2{\displaystyle \frac{\epsilon }{2}}+{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{m_i}).\hfill \end{array}$$

By taking $i\to \infty$, we obtain

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i+1})=\epsilon .$$

(2)

Furthermore, we have

$$\begin{array}{cc}\hfill & {\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i+1})-{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{m_i})\hfill \\ & \le 2{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})\le 2{\displaystyle \frac{\epsilon }{2}}=\epsilon ,\hfill \end{array}$$

which implies

$$2\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})=\epsilon .$$

(3)

By repeating the same steps as above, and replacing $x_{n+1}\Rightarrow x_n$, $x_{m+1}\Rightarrow x_m$, $x_{n_i+1}\Rightarrow x_{n_i}$ and $x_{m_i+1}\Rightarrow x_{m_i}$, we obtain

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i},x_{n_i},x_{m_i})=\epsilon .$$

(4)

and

$$2\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i},x_{n_i},x_{m_i-1})=\epsilon .$$

(5)

From the following inequality

$$\begin{array}{cc}\hfill \epsilon & <{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i+1})\hfill \\ \hfill & \le 2{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{n_i})+{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{n_i}),\hfill \end{array}$$

we have

$$\begin{array}{cc}\hfill \epsilon & <2{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{n_i})+{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{n_i}).\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill & \epsilon -2{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{n_i})<{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{n_i})\hfill \\ & <2{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{n_i+1})+{\mu }^{*}(x_{n_i},x_{n_i},x_{n_i+1})\hfill \\ & <2\left[2{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{m_i})+{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})\right]\hfill \\ & +{\mu }^{*}(x_{n_i},x_{n_i},x_{n_i+1})\hfill \end{array}$$

Therefore,

$$\epsilon -0<{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{n_i})\le 2\left[{\mu }^{*}(x_{n_i+1},x_{n_i+1},x_{m_i})\right]+0.$$

which implies

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{m_i+1},x_{m_i+1},x_{n_i})=\epsilon .$$

By the symmetry of S-metric

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i},x_{n_i},x_{m_i+1})=\epsilon .$$

This completes the proof. □

Definition 6.

Suppose that $(\mathcal{X},{\mu }^{*})$ is an S-metric space. A self-mapping $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ is said to be asymptotically regular ($\mathbb{ATR}$) at a point $x\in \mathcal{X}$ if

$$\underset{n\to \infty }{\lim }{\mu }^{*}({\mathfrak{F}}^{n}x,{\mathfrak{F}}^{n}x,{\mathfrak{F}}^{n+1}x)=0.$$

Lemma 4.

Suppose that $(\mathcal{X},{\mu }^{*})$ is an S-metric space and $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ is a mapping, and let $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$ satisfy

(i) 

${\inf }_{t>\epsilon }\mathrm{\Omega }\left(t\right)>-\infty$ for any $\epsilon >0$.

Assume that one of the following holds:

(ii) 

Ω is non-decreasing and

$$\underset{t\to \epsilon }{\lim \sup }\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(\epsilon \right)\mathit{for}\mathit{every}\epsilon >0;$$

(iii) 

Whenever $\left\{\mathrm{\Omega }\left(t_n\right)\right\}$ and $\left\{\mathrm{\Pi }\left(t_n\right)\right\}$ are convergent sequences with the same limit and $\left\{\mathrm{\Omega }\left(t_n\right)\right\}$ is strictly decreasing, then ${\lim }_{n\to \infty }{t}_n=0$.

If $\mathfrak{F}$ satisfies the $(\mathrm{\Omega },\mathrm{\Pi })$-contractive condition, then $\mathfrak{F}$ is $\mathbb{ATR}$ at every $x\in \mathcal{X}$.

Proof. 

Fix $x\in \mathcal{X}$ and set $x_n={\mathfrak{F}}^{n}x$ for $n\ge 0$, and define

$${\mathcal{K}}_n={\mu }^{*}(x_n,x_n,x_{n+1}),n\ge 0.$$

We show that ${\mathcal{K}}_n\to 0$ as $n\to \infty$. If ${\mathcal{K}}_n=0$ for some $n\ge 0$, then $x_n=x_{n+1}$, so $x_k=x_n$ for all $k\ge n$, and hence ${\mathcal{K}}_k=0$ for all $k\ge n$. In particular, ${\lim }_{k\to \infty }{\mathcal{K}}_k=0$. Thus, we may assume that ${\mathcal{K}}_n>0$ for all n. Applying (1) with $(x,y)=(x_n,x_{n+1})$, we obtain

$$\mathrm{\Omega }\left({\mu }^{*}(x_{n+1},x_{n+1},x_{n+2})\right)\le \mathrm{\Pi }\left({\mu }^{*}(x_n,x_n,x_{n+1})\right),$$

that is,

$$\mathrm{\Omega }\left({\mathcal{K}}_{n+1}\right)\le \mathrm{\Pi }\left({\mathcal{K}}_n\right)<\mathrm{\Omega }\left({\mathcal{K}}_n\right),n\ge 0.$$

(6)

• Case 1: Condition (ii) holds. It follows from (6) that

  $${\mathcal{K}}_{n+1}<{\mathcal{K}}_n\mathrm{for}\mathrm{all}n\ge 0.$$

  Therefore, $\left({\mathcal{K}}_n\right)$ is a strictly decreasing sequence of positive real numbers, so there exists $\mathcal{K}\ge 0$ such that ${\mathcal{K}}_n\to \mathcal{K}$ as $n\to \infty$. Suppose, for a contradiction, that $\mathcal{K}>0$. Passing to the limit in (6) as $n\to \infty$ yields

  $$\mathrm{\Omega }\left(\mathcal{K}\right)=\underset{n\to \infty }{\lim }\mathrm{\Omega }\left({\mathcal{K}}_{n+1}\right)\le \underset{n\to \infty }{\lim \sup }\mathrm{\Pi }\left({\mathcal{K}}_n\right)\le \underset{t\to \mathcal{K}}{\lim \sup }\mathrm{\Pi }\left(t\right).$$

  Thus

  $$\mathrm{\Omega }\left(\mathcal{K}\right)\le \underset{t\to \mathcal{K}}{\lim \sup }\mathrm{\Pi }\left(t\right),$$

  which contradicts Condition (ii) with $\epsilon =\mathcal{K}>0$. Therefore, $\mathcal{K}=0$, and hence ${\mathcal{K}}_n\to 0$ as $n\to \infty$.
• Case 2: Condition (iii) holds. From (6), it follows that $\left\{\mathrm{\Omega }\left({\mathcal{K}}_n\right)\right\}$ is strictly decreasing. If $\left(\mathrm{\Omega }\left({\mathcal{K}}_n\right)\right)$ is not bounded below, then $\mathrm{\Omega }\left({\mathcal{K}}_n\right)\to -\infty$ as $n\to \infty$. By Condition (i) and Lemma 1, this implies ${\mathcal{K}}_n\to 0$ as $n\to \infty$. If $\left(\mathrm{\Omega }\left({\mathcal{K}}_n\right)\right)$ is bounded below, then, being strictly decreasing, it converges to some finite limit $L\in \mathbb{R}$. From (6) for all n, passing to the limit as $n\to \infty$ shows that $\left(\mathrm{\Pi }\left({\mathcal{K}}_n\right)\right)$ also converges and has the same limit L as $\left(\mathrm{\Omega }\left({\mathcal{K}}_n\right)\right)$. By Condition (iii), applied to the sequence $t_n={\mathcal{K}}_n$, we conclude that ${\mathcal{K}}_n\to 0$ as $n\to \infty$.

In either case we obtain ${\lim }_{n\to \infty }{\mathcal{K}}_n=0$; that is,

$$\underset{n\to \infty }{\lim }{\mu }^{*}(x_n,x_n,x_{n+1})=\underset{n\to \infty }{\lim }{\mu }^{*}({\mathfrak{F}}^{n}x,{\mathfrak{F}}^{n}x,{\mathfrak{F}}^{n+1}x)=0.$$

Thus, $\mathfrak{F}$ is $\mathbb{ATR}$ at x. Since $x\in \mathcal{X}$ was arbitrary, $\mathfrak{F}$ is asymptotically regular on $\mathcal{X}$. □

Lemma 5.

Suppose that $(\mathcal{X},{\mu }^{*})$ is a symmetric S-metric space and $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ is a mapping that satisfies (1) , where the functions $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$ fulfill at least one of the following:

(i) 

Ω is non-decreasing and ${\displaystyle \underset{t\to {\epsilon }^{+}}{\lim \sup }\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left({\epsilon }^{+}\right)}$ for every $\epsilon >0$;

(ii) 

${\displaystyle \underset{t\to \epsilon }{\lim \sup }\mathrm{\Pi }\left(t\right)<\underset{t\to {\epsilon }^{+}}{\lim \inf }\mathrm{\Omega }\left(t\right)}$ for every $\epsilon >0$.

If $\mathfrak{F}$ is $\mathbb{ATR}$ at a point $x\in \mathcal{X}$, then the sequence ${\left\{{\mathfrak{F}}^{n}x\right\}}_{n\ge 0}$ is S-Cauchy, i.e.

$${\mu }^{*}({\mathfrak{F}}^{n}x,{\mathfrak{F}}^{n}x,{\mathfrak{F}}^{m}x)\to 0\mathrm{as}n,m\to \infty .$$

Proof. 

Assume that $\mathfrak{F}$ is $\mathbb{ATR}$ at $x\in \mathcal{X}$, so

$${\mu }^{*}({\mathfrak{F}}^{n}x,{\mathfrak{F}}^{n}x,{\mathfrak{F}}^{n+1}x)\to 0\mathrm{as}n\to \infty .$$

Put $x_n={\mathfrak{F}}^{n}x$ for $n\ge 0$. Suppose, towards a contradiction, that $\left\{x_n\right\}$ is not S-Cauchy. Then, by Lemma 3, there exist $\epsilon >0$ and subsequences $\left\{x_{n_i}\right\}$ and $\left\{x_{m_i}\right\}$ of $\left\{x_n\right\}$ such that

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})=\epsilon$$

(7)

and

$$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i},x_{n_i},x_{m_i})=2\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_i})=\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_i},x_{n_i},x_{m_{i+1}})=\epsilon .$$

(8)

• Case 1: Condition (i) holds. From (7) we have

  $$\underset{i\to \infty }{\lim }{\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})=\epsilon .$$

  Using (1) with $x=x_{n_i}$, $y=x_{m_i}$ gives

  $$\mathrm{\Omega }\left({\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})\right)\le \mathrm{\Pi }\left({\mu }^{*}(x_{n_i},x_{n_i},x_{m_i})\right),i\ge 1.$$

  (9)

  Set

  $${\mathcal{K}}_i:={\mu }^{*}(x_{n_i},x_{n_i},x_{m_i}),i\ge 1.$$

  Since $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$, then (9) becomes

  $$\mathrm{\Omega }\left({\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})\right)\le \mathrm{\Pi }\left({\mathcal{K}}_i\right)<\mathrm{\Omega }\left({\mathcal{K}}_i\right),i\ge 1.$$

  (10)

  From (10) and Condition (i), we deduce that ${\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})<{\mathcal{K}}_i$. By (7) and (8),

  $${\mathcal{K}}_i\to \epsilon \mathrm{and}{\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})\to \epsilon \mathrm{as}i\to \infty .$$

  Passing to the limit in (10), we obtain

  $$\mathrm{\Omega }\left(\epsilon \right)=\underset{i\to \infty }{\lim }\mathrm{\Omega }\left({\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})\right)\le \underset{i\to \infty }{\lim \sup }\mathrm{\Pi }\left({\mathcal{K}}_i\right)\le \underset{t\to {\epsilon }^{+}}{\lim \sup }\mathrm{\Pi }\left(t\right).$$

  Thus

  $$\mathrm{\Omega }\left(\epsilon \right)\le \underset{t\to {\epsilon }^{+}}{\lim \sup }\mathrm{\Pi }\left(t\right),$$

  which contradicts Condition (i). Hence, this case cannot occur.
• Case 2: Condition (ii) holds. As above, Lemma 3 yields $\epsilon >0$ and subsequences $\left\{x_{n_i}\right\}$, $\left\{x_{m_i}\right\}$ such that (7) and (8) hold. Using (1) with $x=x_{n_i},y=x_{m_i}$ gives

  $$\mathrm{\Omega }\left({\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}})\right)\le \mathrm{\Pi }\left({\mu }^{*}(x_{n_i},x_{n_i},x_{m_i})\right),i\ge 1.$$

  Define

  $${\mathcal{K}}_i:={\mu }^{*}(x_{n_i},x_{n_i},x_{m_i}),{\mathcal{L}}_i:={\mu }^{*}(x_{n_{i+1}},x_{n_{i+1}},x_{m_{i+1}}),i\ge 1.$$

  Then

  $$\mathrm{\Omega }\left({\mathcal{L}}_i\right)\le \mathrm{\Pi }\left({\mathcal{K}}_i\right),i\ge 1.$$

  (11)

  By (7) and (8),

  $${\mathcal{K}}_i\to \epsilon ,{\mathcal{L}}_i\to \epsilon \mathrm{as}i\to \infty .$$

  Taking lim inf on the left-hand side and lim sup on the right-hand side of (11), we obtain

  $$\underset{i\to \infty }{\lim \inf }\mathrm{\Omega }\left({\mathcal{L}}_i\right)\le \underset{i\to \infty }{\lim \sup }\mathrm{\Pi }\left({\mathcal{K}}_i\right),$$

  hence

  $$\underset{t\to {\epsilon }^{+}}{\lim \inf }\mathrm{\Omega }\left(t\right)\le \underset{i\to \infty }{\lim \inf }\mathrm{\Omega }\left({\mathcal{L}}_i\right)\le \underset{i\to \infty }{\lim \sup }\mathrm{\Pi }\left({\mathcal{K}}_i\right)\le \underset{t\to \epsilon }{\lim \sup }\mathrm{\Pi }\left(t\right).$$

  Therefore,

  $$\underset{t\to {\epsilon }^{+}}{\lim \inf }\mathrm{\Omega }\left(t\right)\le \underset{t\to \epsilon }{\lim \sup }\mathrm{\Pi }\left(t\right),$$

  which contradicts Condition (ii). Hence, this case is also impossible.

In both cases, we obtain a contradiction with the assumption that $\left\{x_n\right\}$ is not S-Cauchy. Therefore, $\left\{x_n\right\}$ must be S-Cauchy; that is,

$${\mu }^{*}(x_n,x_n,x_m)\to 0\mathrm{as}n,m\to \infty .$$

Equivalently, $\left\{{\mathfrak{F}}^{n}x\right\}$ is an S-Cauchy sequence. □

Definition 7 (closed graph).

Let $(\mathcal{X},{\mu }^{*})$ be an S-metric space and let $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ be a self-mapping. We say that $\mathfrak{F}$ has a closed graph if for every sequence $\left\{x_n\right\}\subset \mathcal{X}$ and every $x,y\in \mathcal{X}$,

$$x_n\to x\mathit{and}\mathfrak{F}{x}_n\to y⟹y=\mathfrak{F}x.$$

Equivalently, the graph

$$G\left(\mathfrak{F}\right)=\left\{\right(x,\mathfrak{F}x):x\in \mathcal{X}\}$$

is closed in $\mathcal{X}\times \mathcal{X}$.

Lemma 6.

Suppose that $(\mathcal{X},{\mu }^{*})$ is an S-metric space and $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ is a mapping satisfying (1), with functions $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$, and assume one of the following:

(i) 

$\mathfrak{F}$ has a closed graph;

(ii) 

Ω is non-decreasing;

(iii) 

${\displaystyle \underset{t\to 0}{\lim \sup }\mathrm{\Pi }\left(t\right)<\underset{t\to \epsilon }{\lim \inf }\mathrm{\Omega }\left(t\right)}$ for every $\epsilon >0$.

If $x\in \mathcal{X}$ and set $x_n={\mathfrak{F}}^{n}x$. If $x_n\to \eta$, then η is a $\mathbb{FP}$ of $\mathfrak{F}$.

Proof. 

Let $x\in \mathcal{X}$ and $x_n={\mathfrak{F}}^{n}x$ for $n\ge 0$, and suppose that $x_n\to \eta$.

• Case 1: $\mathfrak{F}$ has a closed graph. Since $x_n\to \eta$, then $x_{n+1}=\mathfrak{F}{x}_n\to \eta$ as $n\to \infty$. By the closed-graph property of $\mathfrak{F}$, this implies that if $x_n\to \eta$ and $\mathfrak{F}{x}_n\to \eta$, then $\mathfrak{F}\eta =\eta$; hence, $\eta$ is a fixed point of $\mathfrak{F}$.
• Case 2: $\mathrm{\Omega }$ is non-decreasing. First, suppose that

  $${\mu }^{*}(\mathfrak{F}{x}_n,\mathfrak{F}{x}_n,\mathfrak{F}\eta )=0\mathrm{for}\mathrm{some}n\ge 0.$$

  Then, $\mathfrak{F}{x}_n=\mathfrak{F}\eta$. Using the S-triangle inequality and $x_{n+1}=\mathfrak{F}{x}_n$, we obtain

  $$\begin{array}{cc}\hfill {\mu }^{*}(\eta ,\mathfrak{F}\eta ,\mathfrak{F}\eta )& \le {\mu }^{*}(\eta ,\eta ,\mathfrak{F}{x}_n)+{\mu }^{*}(\mathfrak{F}\eta ,\mathfrak{F}\eta ,\mathfrak{F}{x}_n)+{\mu }^{*}(\mathfrak{F}\eta ,\mathfrak{F}\eta ,\mathfrak{F}{x}_n)\hfill \\ \hfill & ={\mu }^{*}(\eta ,\eta ,x_{n+1}),\hfill \end{array}$$

  since $\mathfrak{F}{x}_n=\mathfrak{F}\eta$ makes the last two terms vanish. Letting $n\to \infty$ and using $x_{n+1}\to \eta$ yields

  $${\mu }^{*}(\eta ,\mathfrak{F}\eta ,\mathfrak{F}\eta )\le 0,$$

  hence ${\mu }^{*}(\eta ,\mathfrak{F}\eta ,\mathfrak{F}\eta )=0$ and $\mathfrak{F}\eta =\eta$. This means $\eta$ is a $\mathbb{FP}$ of $\mathfrak{F}$. Now suppose instead that

  $${\mu }^{*}(\mathfrak{F}{x}_n,\mathfrak{F}{x}_n,\mathfrak{F}\eta )>0\mathrm{for}\mathrm{all}n\ge 0.$$

  Applying (1) with $x=x_n$ and $y=\eta$ gives

  $$\mathrm{\Omega }\left({\mu }^{*}(\mathfrak{F}{x}_n,\mathfrak{F}{x}_n,\mathfrak{F}\eta )\right)\le \mathrm{\Pi }\left({\mu }^{*}(x_n,x_n,\eta )\right),n\ge 0.$$

  (12)

  Since $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$ for all $t>0$, from (12) we have

  $$\mathrm{\Omega }\left({\mu }^{*}(\mathfrak{F}{x}_n,\mathfrak{F}{x}_n,\mathfrak{F}\eta )\right)\le \mathrm{\Pi }\left({\mu }^{*}(x_n,x_n,\eta )\right)<\mathrm{\Omega }\left({\mu }^{*}(x_n,x_n,\eta )\right).$$

  (13)

  This implies that ${\mu }^{*}(\mathfrak{F}{x}_n,\mathfrak{F}{x}_n,\mathfrak{F}\eta )<{\mu }^{*}(x_n,x_n,\eta )$. Letting $n\to \infty$, we get ${\mu }^{*}(\eta ,\eta ,\mathfrak{F}\eta )\le 0$, hence ${\mu }^{*}(\eta ,\eta ,\mathfrak{F}\eta )=0$ and $\mathfrak{F}\eta =\eta$. This means $\eta$ is a $\mathbb{FP}$ of $\mathfrak{F}$.
• Case 3: Condition (iii) holds. From (12), set

  $${\phi }_n:={\mu }^{*}(x_n,x_n,\eta ),{\varpi }_n:={\mu }^{*}(\mathfrak{F}{x}_n,\mathfrak{F}{x}_n,\mathfrak{F}\eta ).$$

  Then (12) reads

  $$\mathrm{\Omega }\left({\varpi }_n\right)\le \mathrm{\Pi }\left({\phi }_n\right),n\ge 0.$$

  Since $x_n\to \eta$, we have ${\phi }_n\to 0$ as $n\to \infty$. Also, the sequence $\left\{{\varpi }_n\right\}$ converges to

  $$\epsilon :=\underset{n\to \infty }{\lim }{\varpi }_n={\mu }^{*}(\eta ,\eta ,\mathfrak{F}\eta )\ge 0.$$

  Passing to lim inf on the left-hand side and lim sup on the right-hand side in (12), we obtain

  $$\underset{t\to \epsilon }{\lim \inf }\mathrm{\Omega }\left(t\right)\le \underset{n\to \infty }{\lim \inf }\mathrm{\Omega }\left({\varpi }_n\right)\le \underset{n\to \infty }{\lim \sup }\mathrm{\Pi }\left({\phi }_n\right)\le \underset{t\to 0}{\lim \sup }\mathrm{\Pi }\left(t\right).$$

  If $\epsilon >0$, this inequality contradicts condition (iii). Hence $\epsilon =0$, so

  $${\mu }^{*}(\eta ,\eta ,\mathfrak{F}\eta )=0,$$

  and therefore $\mathfrak{F}\eta =\eta$.

In all cases, we obtain $\mathfrak{F}\eta =\eta$, so $\eta$ is a $\mathbb{FP}$ of $\mathfrak{F}$. □

Theorem 1.

Suppose that $(\mathcal{X},{\mu }^{*})$ is a symmetric complete S-metric space and $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ is a mapping that satisfies (1), with functions $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$ satisfying the following conditions:

(i) 

Ω is non-decreasing;

(ii) 

${\displaystyle \underset{t\to {\epsilon }^{+}}{\lim \sup }\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left({\epsilon }^{+}\right)}$ for every $\epsilon >0$.

Then, $\mathfrak{F}$ has a unique $\mathbb{FP}$ $\eta \in \mathcal{X}$ and $\left\{{\mathfrak{F}}^{n}x\right\}$ converges to η for every $x\in \mathcal{X}$.

Proof. 

Fix $x\in \mathcal{X}$ and define $x_0=x$, $x_{n+1}=\mathfrak{F}{x}_n$ for $n\ge 0$. By using Conditions (i) and (ii), and Lemma 4, it follows that $\mathfrak{F}$ is asymptotically regular. By Conditions (i) and (ii) and Lemma 5, it follows that the sequence $\left\{{\mathfrak{F}}^{n}x\right\}$ is S-Cauchy. Since $(\mathcal{X},{\mu }^{*})$ is complete, there exists $\eta \in \mathcal{X}$ such that $\left\{{\mathfrak{F}}^{n}x\right\}$ converges to $\eta$. Then, by Condition (i) and Lemma 6, $\eta$ is a $\mathbb{FP}$ of $\mathfrak{F}$. To prove the uniqueness of $\eta$, let $\tau \in \mathcal{X}$ be another $\mathbb{FP}$ of $\mathfrak{F}$. Applying (1) with $x=\eta$, $y=\tau$ gives

$$\mathrm{\Omega }\left({\mu }^{*}(\eta ,\eta ,\tau )\right)=\mathrm{\Omega }\left({\mu }^{*}(\mathfrak{F}\eta ,\mathfrak{F}\eta ,\mathfrak{F}\tau )\right)\le \mathrm{\Pi }\left({\mu }^{*}(\eta ,\eta ,\tau )\right)<\mathrm{\Omega }\left({\mu }^{*}(\eta ,\eta ,\tau )\right),$$

where $d:={\mu }^{*}(\eta ,\eta ,\tau )$. By the first axiom of S-metric spaces, $d=0$ if and only if $\eta =\tau$. Assuming $\eta \ne \tau$ strictly implies $d>0$. Since $\mathrm{\Omega }$ and $\mathrm{\Pi }$ are defined on $(0,\infty )$ and the contractive inequality (1) holds for all pairs, substituting $d>0$ yields the contradiction $\mathrm{\Omega }\left(d\right)\le \mathrm{\Pi }\left(d\right)<\mathrm{\Omega }\left(d\right)$. Therefore, $\eta =\tau$ and the $\mathbb{FP}$ is unique. □

Remark 3.

If we take $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=\sigma t$ with $0\le \sigma <1$, then condition (1) becomes

$${\mu }^{*}(\mathfrak{F}x,\mathfrak{F}x,\mathfrak{F}y)\le \sigma {\mu }^{*}(x,x,y)(x,y\in \mathcal{X}),$$

so Theorem 1 reduces to the $\mathbb{BCP}$ (in the S-metric setting).

Remark 4.

Since every ordinary metric space $(X,d)$ induces an S-metric space in the standard way, it is natural to ask how the present results reduce in the classical setting. Indeed, if we take the induced S-metric

$${\mu }^{*}(x,y,z)=\varrho (x,z)+\varrho (y,z),x,y,z\in \mathcal{X},$$

then the contractive condition in $(\mathrm{\Omega },\mathrm{\Pi })$-form becomes a corresponding metric-type inequality for the underlying distance d. In particular, for suitable choices such as $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=\sigma t$ with $0<\sigma <1$, our main $\mathbb{FP}$ theorems reduce to the classical $\mathbb{BCP}$ in metric spaces. Thus, the present results may be viewed as genuine extensions of the standard metric $\mathbb{FP}$ theory to the setting of complete S-metric spaces.

Theorem 2.

Suppose that $(\mathcal{X},{\mu }^{*})$ is a symmetric complete S-metric space and $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ is a mapping that satisfies (1) , with functions $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$ satisfying the following conditions:

(i) 

${\displaystyle \underset{t>\epsilon }{\inf }\mathrm{\Omega }\left(t\right)>-\infty }$ for every $\epsilon >0$;

(ii) 

Whenever $\left\{\mathrm{\Omega }\left(x_n\right)\right\}$ and $\left\{\mathrm{\Pi }\left(x_n\right)\right\}$ are convergent sequences with the same limit and $\left\{\mathrm{\Omega }\left(x_n\right)\right\}$ is strictly decreasing, then $x_n\to 0$ as $n\to \infty$;

(iii) 

$\mathfrak{F}$ has a closed graph, or ${\displaystyle \underset{t\to 0}{\lim \sup }\mathrm{\Pi }\left(t\right)<\underset{t\to \epsilon }{\lim \inf }\mathrm{\Omega }\left(t\right)}$ for every $\epsilon >0$.

Then, $\mathfrak{F}$ has a unique $\mathbb{FP}$ $\eta \in \mathcal{X}$ and, for every $x\in \mathcal{X}$, the sequence $\left\{{\mathfrak{F}}^{n}x\right\}$ converges to η.

Proof. 

Fix $x_0\in \mathcal{X}$ and define $x_{n+1}=\mathfrak{F}{x}_n$ for $n\in \mathbb{N}$. Then, $\left\{x_n\right\}$ is an orbit of $\mathfrak{F}$. By assumptions (i)–(ii) and Lemma 4, the mapping $\mathfrak{F}$ is asymptotically regular at $x_0$, i.e.

$${\mu }^{*}(x_n,x_n,x_{n+1})⟶0\mathrm{as}n\to \infty .$$

Using (iii) and Lemma 5, it follows that $\left\{x_n\right\}$ is an S-Cauchy sequence. Since $(\mathcal{X},{\mu }^{*})$ is complete, there exists $\eta \in \mathcal{X}$ such that

$$x_n⟶\eta \mathrm{as}n\to \infty .$$

We now show that $\eta$ is a $\mathbb{FP}$ of $\mathfrak{F}$. If $\mathfrak{F}$ has a closed graph, then $x_n\to \eta$ and $x_{n+1}=\mathfrak{F}{x}_n\to \eta$ imply $\mathfrak{F}\eta =\eta$. In the alternative case of (iii), the same conclusion follows from Lemma 6, which uses the inequality (1) together with the comparison condition

$$\underset{t\to 0}{\lim \sup }\mathrm{\Pi }\left(t\right)<\underset{t\to \epsilon }{\lim \inf }\mathrm{\Omega }\left(t\right),\epsilon >0,$$

to exclude the possibility ${\mu }^{*}(\eta ,\eta ,\mathfrak{F}\eta )>0$. In either case, $\eta$ is a $\mathbb{FP}$ of $\mathfrak{F}$. To prove uniqueness, let $\eta ,\tau \in \mathcal{X}$ be $\mathbb{FP}$s of $\mathfrak{F}$. Applying (1) with $x=\eta$, $y=\tau$, we obtain

$$\mathrm{\Omega }\left({\mu }^{*}(\eta ,\eta ,\tau )\right)=\mathrm{\Omega }\left({\mu }^{*}(\mathfrak{F}\eta ,\mathfrak{F}\eta ,\mathfrak{F}\tau )\right)\le \mathrm{\Pi }\left({\mu }^{*}(\eta ,\eta ,\tau )\right)<\mathrm{\Omega }\left({\mu }^{*}(\eta ,\eta ,\tau )\right),$$

under the strict comparison between $\mathrm{\Omega }$ and $\mathrm{\Pi }$. By the S-metric axiom, $d:={\mu }^{*}(\eta ,\eta ,\tau )=0\iff \eta =\tau$. Thus, assuming $\eta \ne \tau$ forces $d>0$. Since $\mathrm{\Omega },\mathrm{\Pi }$ are defined on $(0,\infty )$ and the contractive condition holds globally, substituting $d>0$ yields $\mathrm{\Omega }\left(d\right)\le \mathrm{\Pi }\left(d\right)<\mathrm{\Omega }\left(d\right)$, a contradiction. Hence ${\mu }^{*}(\eta ,\eta ,\tau )=0$, so $\eta =\tau$ and the $\mathbb{FP}$ is unique. Finally, for any $x\in \mathcal{X}$, the same argument applied to the orbit starting at x shows that ${\mathfrak{F}}^{n}x$ converges to the unique $\mathbb{FP}$ $\eta$. □

Remark 5.

If we take $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=\sigma t$ with $0\le \sigma <1$, then condition (1) becomes

$${\mu }^{*}(\mathfrak{F}x,\mathfrak{F}x,\mathfrak{F}y)\le \sigma {\mu }^{*}(x,x,y)(x,y\in \mathcal{X}),$$

so Theorem 2 reduces to the $\mathbb{BCP}$ (in the S-metric setting).

3.2. Application

In this part, let $\mathcal{X}=C\left(\right[0,1],\mathbb{R})$ be the space of all continuous real-valued functions on $[0,1]$, equipped with the supremum norm

$${\parallel f\parallel }_{\infty }=\underset{t\in [0,1]}{\sup }\left|f\left(t\right)\right|,f\in C\left([0,1]\right).$$

Define ${\mu }^{*}:\mathcal{X}\times \mathcal{X}\times \mathcal{X}\to [0,\infty )$ by

$${\mu }^{*}(\varpi ,\rho ,\kappa ):={∥\rho +\kappa -2\varpi ∥}_{\infty }+{∥\rho -\kappa ∥}_{\infty },\varpi ,\rho ,\kappa \in \mathcal{X}.$$

Then, $(\mathcal{X},{\mu }^{*})$ is a symmetric S-metric space:

• ${\mu }^{*}(\varpi ,\rho ,\kappa )=0$ if and only if $\varpi =\rho =\kappa$.
• For all $\varpi ,\rho ,\kappa ,t\in \mathcal{X}$ the S-metric inequality holds:

  $${\mu }^{*}(\varpi ,\rho ,\kappa )\le {\mu }^{*}(\varpi ,\varpi ,t)+{\mu }^{*}(\rho ,\rho ,t)+{\mu }^{*}(\kappa ,\kappa ,t).$$

Clearly, $(C([0,1],\mathbb{R}),\parallel ·{\parallel }_{\infty })$ is a complete S-metric space. Since norm-convergence in $C\left(\right[0,1\left]\right)$ implies convergence of the function sequences, every S-Cauchy sequence converges in $\mathcal{X}$. Now, we solve a boundary value problem that arises when a diffusing substance is placed in an absorbing medium between parallel walls such that $\gamma$ and $\delta$ are the stipulated concentrations at the walls. Furthermore, the concentration $\varpi \left(t\right)$ of the substance at time t is given by

$$\begin{array}{c}\hfill -{\displaystyle \frac{d^2\varpi }{d{t}^2}}+f\left(t\right)\varpi =\zeta \left(t\right),\varpi \left(0\right)=\gamma ,\varpi \left(1\right)=\delta ,\end{array}$$

(14)

where $f\left(t\right)$ is the absorption coefficient and $\zeta \left(t\right)$ is the source density. Problem (14) is equivalent to

$$\varpi \left(t\right)=\gamma +(\delta -\gamma )t+{\int }_0^{1}M(t,s)(\zeta \left(s\right)-f\left(s\right)\varpi \left(s\right))ds,s\in [0,1],$$

(15)

where $M(t,s)$ is the Green function, which is defined by

$$M(t,s)=\left\{\begin{array}{cc}t(1-s),\hfill & 0\le t\le s\le 1,\hfill \\ s(1-t),\hfill & 0\le s\le t\le 1.\hfill \end{array}\right.$$

(16)

Theorem 3.

Consider the boundary value problem (14). Let $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ be a self-mapping on $(\mathcal{X},{\mu }^{*})$, satisfying

$$\begin{array}{cc}\hfill & {\parallel f\left(t\right)\varpi \left(t\right)-f\left(t\right)\rho \left(t\right)\parallel }_{\infty }{+\parallel f\left(t\right)\varpi \left(t\right)-f\left(t\right)\kappa \left(t\right)\parallel }_{\infty }+{\parallel f\left(t\right)\rho \left(t\right)-f\left(t\right)\kappa \left(t\right)\parallel }_{\infty }\hfill \\ \hfill & \le e^{-\lambda }\left({\parallel \varpi \left(t\right)-\rho \left(t\right)\parallel }_{\infty }{+\parallel \varpi \left(t\right)-\kappa \left(t\right)\parallel }_{\infty }+{\parallel \rho \left(t\right)-\kappa \left(t\right)\parallel }_{\infty }\right),\lambda >0,\hfill \\ & \mathit{for}\mathit{all}\varpi ,\rho ,\kappa \in \mathcal{X},t\in [0,1].\hfill \end{array}$$

Then, the boundary value problem (14) has a solution.

Proof. 

Define $\mathfrak{F}:\mathcal{X}\to \mathcal{X}$ by

$$\left(\mathfrak{F}x\right)\left(t\right)=\gamma +(\delta -\gamma )t+{\int }_0^{1}M(t,s)\left[\zeta \left(s\right)-f\left(s\right)x\left(s\right)\right]ds,t\in [0,1].$$

Any $\mathbb{FP}$ ${\varpi }_0=\mathfrak{F}{\varpi }_0$ satisfies (14). To verify $\mathfrak{F}$ satisfies (1), compute for $\varpi ,\rho ,\kappa \in \mathcal{X}$:

• Step 1: Linear terms cancel:

  $$\begin{array}{cc}\hfill & [\gamma +(\delta -\gamma \left)t\right]+[\gamma +(\delta -\gamma \left)t\right]-2[\gamma +(\delta -\gamma \left)t\right]=0,\hfill \\ \hfill & [\gamma +(\delta -\gamma \left)t\right]-[\gamma +(\delta -\gamma \left)t\right]=0.\hfill \end{array}$$
• Step 2: Integral terms for first S-metric component:

  $$\begin{array}{cc}\hfill & \mathfrak{F}\rho +\mathfrak{F}\kappa -2\mathfrak{F}\varpi \hfill \\ \hfill & ={\int }_0^{1}M(t,s)\left[\zeta \left(s\right)-f\left(s\right)\rho \left(s\right)\right]ds+{\int }_0^{1}M(t,s)\left[\zeta \left(s\right)-f\left(s\right)\kappa \left(s\right)\right]ds\hfill \\ \hfill & -2{\int }_0^{1}M(t,s)\left[\zeta \left(s\right)-f\left(s\right)\varpi \left(s\right)\right]ds\hfill \\ \hfill & ={\int }_0^{1}M(t,s)\left[2f\left(s\right)\varpi \left(s\right)-f\left(s\right)\rho \left(s\right)-f\left(s\right)\kappa \left(s\right)\right]ds\hfill \\ \hfill & ={\int }_0^{1}M(t,s)f\left(s\right)\left[\left(\varpi \right(s)-\rho (s\left)\right)+\left(\varpi \right(s)-\kappa (s\left)\right)\right]ds.\hfill \end{array}$$
• Step 3: Second S-metric component:

  $$\begin{array}{cc}\hfill \mathfrak{F}\rho -\mathfrak{F}\kappa & ={\int }_0^{1}M(t,s)\left[\zeta \left(s\right)-f\left(s\right)\rho \left(s\right)\right]ds-{\int }_0^{1}M(t,s)\left[\zeta \left(s\right)-f\left(s\right)\kappa \left(s\right)\right]ds\hfill \\ \hfill & =-{\int }_0^{1}M(t,s)f\left(s\right)(\rho \left(s\right)-\kappa \left(s\right))ds.\hfill \end{array}$$
• Step 4: Since the Green function

  $$M(t,s)=\left\{\begin{array}{cc}t(1-s),\hfill & 0\le t\le s\le 1,\hfill \\ s(1-t),\hfill & 0\le s\le t\le 1,\hfill \end{array}\right.$$

  satisfies $0\le M(t,s)\le 1/4$ for all $(t,s)\in {[0,1]}^2$, we have

  $${\parallel M\parallel }_{\infty }\le {\displaystyle \frac{1}{4}}.$$

  S-metric estimate using ${\parallel M\parallel }_{\infty }\le {\displaystyle \frac{1}{4}}$:

  $$\begin{array}{cc}\hfill {\mu }^{*}(\mathfrak{F}\varpi ,\mathfrak{F}\rho ,\mathfrak{F}\kappa )& ={\parallel \mathfrak{F}\rho +\mathfrak{F}\kappa -2\mathfrak{F}\varpi \parallel }_{\infty }+{\parallel \mathfrak{F}\rho -\mathfrak{F}\kappa \parallel }_{\infty }\hfill \\ \hfill & \le {\displaystyle \frac{1}{4}}{\parallel f(\varpi -\rho )\parallel }_{\infty }+{\displaystyle \frac{1}{4}}{\parallel f(\varpi -\kappa )\parallel }_{\infty }+{\displaystyle \frac{1}{4}}{\parallel f(\rho -\kappa )\parallel }_{\infty }.\hfill \end{array}$$
• Step 5: Apply hypothesis:

  $${\mu }^{*}(\mathfrak{F}\varpi ,\mathfrak{F}\rho ,\mathfrak{F}\kappa )\le e^{-\lambda }\left[{\parallel \varpi -\rho \parallel }_{\infty }{+\parallel \varpi -\kappa \parallel }_{\infty }+{\parallel \rho -\kappa \parallel }_{\infty }\right]=e^{-\lambda }{\mu }^{*}(\varpi ,\rho ,\kappa ).$$

  Thus, $\mathfrak{F}$ satisfies (1) with $\mathrm{\Omega }\left(t\right)=t$, $\mathrm{\Pi }\left(t\right)=e^{-\lambda }t$.
• Step 6: Picard iteration $x_0\in \mathcal{X}$ arbitrary, $x_{n+1}=\mathfrak{F}{x}_n$. Lemma 4 gives asymptotic regularity:

  $${\mu }^{*}(x_n,x_n,x_{n+1})\to 0.$$

  Lemma 5 gives $\left\{x_n\right\}$ is S-Cauchy. By completeness, $x_n\to \eta \in \mathcal{X}$.
• Step 7: Theorem 1 (ii): $\mathrm{\Pi }\left(t\right)\le \mathrm{\Omega }\left(t\right)$, ${\lim \sup }_{t\to 0}\mathrm{\Pi }\left(t\right)=0<\epsilon ={\lim \inf }_{t\to \epsilon }\mathrm{\Omega }\left(t\right)$. Thus $\eta =\mathfrak{F}\eta$, solving (14). □

4. Chebyshev Spectral Method for Boundary Value Problems

The theoretical framework developed in the preceding sections provides a robust foundation for analyzing the convergence of iterative numerical schemes. In this section, we demonstrate how the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction principle in complete S-metric spaces can be applied to guarantee the convergence of the Chebyshev spectral collocation method for solving second-order boundary value problems of the form

$$-{\displaystyle \frac{d^2\varpi }{d{t}^2}}+f\left(t\right)\varpi =\zeta \left(t\right),\varpi \left(0\right)=\gamma ,\varpi \left(1\right)=\delta ,$$

(17)

where $f,\zeta \in C\left(\right[0,1\left]\right)$ and $\gamma ,\delta \in \mathbb{R}$.

4.1. Motivation for the $(\mathrm{\Omega },\mathrm{\Pi })$-Framework in Diffusion Models

Before presenting the numerical scheme, we clarify the added value of the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction framework compared to standard Banach contraction arguments. For diffusion–absorption problems of the form (17), the following scenarios naturally arise where our generalized framework provides essential advantages:

• Non-Lipschitz nonlinearities: When the absorption coefficient $f\left(t\right)$ or source term $\zeta \left(t\right)$ exhibits singular behavior or non-Lipschitz continuity (e.g., $f\left(t\right)$∼ t^−α near boundaries), the standard Lipschitz constant may not exist globally, yet $(\mathrm{\Omega },\mathrm{\Pi })$-conditions can still be satisfied through an appropriate choice of control functions.
• Weak contractivity: In problems with small absorption (${\parallel f\parallel }_{\infty }\ll 1$) or nearly singular operators, the contraction factor may approach 1, making standard Banach arguments inconclusive. The $(\mathrm{\Omega },\mathrm{\Pi })$-framework accommodates such borderline cases through the limsup–liminf separation condition.
• Asymptotic regularity without uniform contraction: For certain reaction–diffusion systems, Picard iterates may exhibit asymptotic regularity even when the mapping is not uniformly contractive—a situation that is naturally captured by our Theorem 2 but not by classical results.

Remark 6.

The $(\mathrm{\Omega },\mathrm{\Pi })$-framework’s advantage becomes evident in scenarios where single-function controls fail:

• Nonlinear control functions: For $f\left(t\right)$ with spatially varying decay ${\lambda }^2\left(x\right)$, choosing $\mathrm{\Omega }\left(t\right)=\ln (1+t)$ and $\mathrm{\Pi }\left(t\right)=\ln (1+t)-\varphi \left(t\right)$ with $\varphi \left(t\right)>0$ can satisfy the contraction condition even when no global Lipschitz constant exists.
• Asymptotically regular mappings: When $\mathfrak{F}$ satisfies ${\mu }^{*}({\mathfrak{F}}^{n+1}x,{\mathfrak{F}}^{n+1}x,{\mathfrak{F}}^{n}x)\to 0$ but lacks uniform contractivity, Theorem 2 with Condition (ii) still guarantees convergence.
• Threshold effects: In biodegradation models where reaction rates activate only above concentration thresholds, the $(\mathrm{\Omega },\mathrm{\Pi })$-structure accommodates discontinuous or piecewise-defined control functions that single-mechanism approaches cannot handle.

These cases demonstrate the framework’s necessity beyond conventional contraction analyses.

Example 5 (Concrete Nonlinear Control Verification).

Consider $\mathcal{X}=C\left(\right[0,1\left]\right)$ with the S-metric ${\mu }^{*}(u,v,w)={\parallel v+w-2u\parallel }_{\infty }+{\parallel v-w\parallel }_{\infty }$. Define $\mathfrak{F}\left(u\right)\left(t\right)={\displaystyle \frac{1}{4}}{\int }_0^1\sqrt{\left|u\right(s\left)\right|}ds$. For any $u,v,w\in \mathcal{X}$,

$${\mu }^{*}(\mathfrak{F}u,\mathfrak{F}v,\mathfrak{F}w)\le {\displaystyle \frac{1}{2}}\sqrt{{\parallel u-v\parallel }_{\infty }{+\parallel u-w\parallel }_{\infty }+{\parallel v-w\parallel }_{\infty }}\le {\displaystyle \frac{1}{2}}\sqrt{{\mu }^{*}(u,v,w)}.$$

Choosing $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)={\displaystyle \frac{1}{2}}\sqrt{t}$ yields $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$ for $t\in (0,4)$. Since $\mathfrak{F}$ has unbounded derivative near $u=0$, it is not a Banach contraction, yet the $(\mathrm{\Omega },\mathrm{\Pi })$-condition holds and Theorem 1 applies.

Remark 7

(Physical Interpretation). In groundwater contamination models (Section 4.6), the decay coefficient ${\lambda }^2$ represents natural attenuation. When ${\lambda }^2$ varies spatially or exhibits threshold behavior (e.g., biodegradation activating only above certain concentrations), the resulting operator may satisfy $(\mathrm{\Omega },\mathrm{\Pi })$-conditions with $\mathrm{\Omega }\left(t\right)=\ln \left(t\right)$ and $\mathrm{\Pi }\left(t\right)=\ln \left(t\right)-\varphi \left(t\right)$ for suitable ϕ, even when no global Lipschitz constant exists. This demonstrates the practical relevance of our generalized framework.

4.2. Chebyshev Polynomials and Collocation Points

Let $T_k\left(t\right)=\cos \left(k\arccos (2t-1)\right)$, $k=0,1,2,\dots$, denote the shifted Chebyshev polynomials of the first kind on $[0,1]$. These polynomials satisfy the orthogonality relation

$${\int }_0^1{T}_m\left(t\right)T_n\left(t\right)w\left(t\right)dt=\left\{\begin{array}{cc}0,\hfill & m\ne n,\hfill \\ \pi /2,\hfill & m=n\ne 0,\hfill \\ \pi ,\hfill & m=n=0,\hfill \end{array}\right.$$

with weight function $w\left(t\right)={\left[t(1-t)\right]}^{-1/2}$. The Chebyshev–Gauss–Lobatto collocation points on $[0,1]$ are given by

$$t_j={\displaystyle \frac{1}{2}}\left(1-\cos {\displaystyle \frac{j\pi }{N}}\right),j=0,1,\dots ,N,$$

(18)

where N is the polynomial degree. These points cluster near the boundaries, providing high resolution where boundary layers may occur.

4.3. Differentiation Matrices

Let $\mathbf{u}={[u\left(t_0\right),u\left(t_1\right),\dots ,u\left(t_N\right)]}^{\top }$ denote the vector of function values at the collocation points. The first and second derivatives at these points are approximated via matrix multiplication:

$${\mathbf{u}}^{\prime }\approx D^{\left(1\right)}\mathbf{u},{\mathbf{u}}^{″}\approx D^{\left(2\right)}\mathbf{u},$$

(19)

where $D^{\left(1\right)},D^{\left(2\right)}\in {\mathbb{R}}^{(N+1)\times (N+1)}$ are the Chebyshev differentiation matrices [31]. The entries of $D^{\left(1\right)}$ are explicitly given by

$$d_{ij}^{\left(1\right)}=\left\{\begin{array}{cc}{\displaystyle \frac{c_i}{c_j}\frac{{(-1)}^{i+j}}{t_i-t_j},}\hfill & i\ne j,\hfill \\ {\displaystyle -\frac{2N^2+1}{6},}\hfill & i=j=0\mathrm{or}N,\hfill \\ {\displaystyle -\frac{t_j}{2(1-t_j^2)},}\hfill & 1\le i=j\le N-1,\hfill \end{array}\right.$$

with $c_0=c_N=2$ and $c_j=1$ for $1\le j\le N-1$. The second-derivative matrix is obtained as $D^{\left(2\right)}={\left(D^{\left(1\right)}\right)}^2$.

4.4. Algorithm: Chebyshev Spectral Collocation with Richardson Iteration

We now present the algorithmic steps for solving the boundary value problem (17) using the Chebyshev spectral collocation method, combined with the fixed-point iteration framework established in Theorem 2. To properly demonstrate the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction framework, we employ Richardson iteration rather than direct solution. This allows us to verify the theoretical convergence guarantees through actual iterative behavior.

• Choose the polynomial degree $N\ge 2$ and compute the Chebyshev–Gauss–Lobatto points ${\left\{t_j\right\}}_{j=0}^N$ via (18).
• Construct the first- and second-derivative matrices $D^{\left(1\right)}$ and $D^{\left(2\right)}$ using the explicit formulas above or via the barycentric formulation [32].
• Modify the system to incorporate Dirichlet conditions. Replace the first and last rows of $D^{\left(2\right)}$ with the unit vectors ${\mathbf{e}}_0^{\top }$ and ${\mathbf{e}}_N^{\top }$, respectively, and set the corresponding right-hand side entries to $\gamma$ and $\delta$.
• For the differential operator $\mathcal{L}\varpi ={\varpi }^{″}+f\left(t\right)\varpi$, form the matrix

  $$A=D^{\left(2\right)}+\mathrm{diag}\left(f\left(t_0\right),f\left(t_1\right),\dots ,f\left(t_N\right)\right),$$

  and the right-hand side vector $\mathbf{b}={[\gamma ,\zeta \left(t_1\right),\dots ,\zeta \left(t_{N-1}\right),\delta ]}^{\top }$.
• Define the Richardson iteration mapping $\mathfrak{F}:{\mathbb{R}}^{N+1}\to {\mathbb{R}}^{N+1}$ by

  $$\mathfrak{F}\left(\mathbf{u}\right)=\mathbf{u}+\omega \left(\mathbf{b}-A\mathbf{u}\right),$$

  (20)

  where $\omega >0$ is a relaxation parameter chosen to ensure convergence. The iteration (20) can be rewritten as

  $$\mathfrak{F}\left(\mathbf{u}\right)=(I-\omega A)\mathbf{u}+\omega \mathbf{b},$$

  which is a proper affine mapping (not constant).
• Verification of $(\mathrm{\Omega },\mathrm{\Pi })$-contraction property: For $\mathbf{u},\mathbf{v},\mathbf{w}\in {\mathbb{R}}^{N+1}$, we compute

  $$\begin{array}{cc}\hfill \mathfrak{F}\left(\mathbf{u}\right)-\mathfrak{F}\left(\mathbf{v}\right)& =(I-\omega A)(\mathbf{u}-\mathbf{v}),\hfill \\ \hfill \mathfrak{F}\left(\mathbf{v}\right)-\mathfrak{F}\left(\mathbf{w}\right)& =(I-\omega A)(\mathbf{v}-\mathbf{w}).\hfill \end{array}$$
  Using the S-metric ${\mu }^{*}(\mathbf{u},\mathbf{v},\mathbf{w})={\parallel \mathbf{v}+\mathbf{w}-2\mathbf{u}\parallel }_{\infty }+{\parallel \mathbf{v}-\mathbf{w}\parallel }_{\infty }$, we obtain

  $$\begin{array}{c}\hfill {\mu }^{*}\left(\mathfrak{F}\left(\mathbf{u}\right),\mathfrak{F}\left(\mathbf{v}\right),\mathfrak{F}\left(\mathbf{w}\right)\right)\le {\parallel I-\omega A\parallel }_{\infty }{\mu }^{*}(\mathbf{u},\mathbf{v},\mathbf{w}).\end{array}$$
  Setting $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=\kappa t$ with $\kappa ={\parallel I-\omega A\parallel }_{\infty }$, we have an $(\mathrm{\Omega },\mathrm{\Pi })$-contraction provided $\kappa <1$. Sufficient conditions for $\kappa <1$: After imposing Dirichlet boundary conditions, the modified matrix A is strictly diagonally dominant with positive diagonal entries provided $f\left(t\right)\ge 0$ and the grid is sufficiently resolved. By Gershgorin’s circle theorem, all eigenvalues of A lie in the positive real axis. Choosing $\omega \in (0,2/{\lambda }_{\max }\left(A\right))$ guarantees $\rho (I-\omega A)<1$. Moreover, for the ∞-norm, the explicit bound $\omega <1/{\max }_i{A}_{ii}$ ensures ${\parallel I-\omega A\parallel }_{\infty }<1$, rigorously satisfying the contraction hypothesis.
• Starting from an initial guess ${\mathbf{u}}^{\left(0\right)}$ (e.g., the linear interpolant satisfying the boundary conditions), compute

  $${\mathbf{u}}^{(k+1)}=\mathfrak{F}\left({\mathbf{u}}^{\left(k\right)}\right)={\mathbf{u}}^{\left(k\right)}+\omega \left(\mathbf{b}-A{\mathbf{u}}^{\left(k\right)}\right),k=0,1,2,\dots ,$$

  until $\parallel {\mathbf{u}}^{(k+1)}-{\mathbf{u}}^{\left(k\right)}{\parallel }_{\infty }<\epsilon$ for a prescribed tolerance $\epsilon >0$.
• The approximate solution at any $t\in [0,1]$ is obtained via the Chebyshev interpolant

  $${\varpi }_N\left(t\right)={\displaystyle \sum _{k=0}^N}{\widehat{u}}_k{T}_k\left(t\right),$$

  where the coefficients ${\widehat{u}}_k$ are computed from the discrete values ${\mathbf{u}}^{\left(k\right)}$ using the discrete cosine transform.

Remark 8 (Comparison: Direct Solve vs. Iterative Method).

While a direct solution via LU decomposition is computationally efficient for small-to-moderate N, the Richardson iteration framework serves three important purposes:

1. 

It validates the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction theory through observable iterative convergence.

2. 

For large-scale problems or time-dependent extensions where A changes slightly between time steps, iterative methods with warm starts are more efficient than repeated factorization.

3. 

The S-metric framework provides a unified error-monitoring tool that simultaneously tracks discretization accuracy and iterative convergence.

Remark 9.

The convergence of the iterative scheme in Step g is guaranteed by Theorem 2, since the contraction condition

$${\mu }^{*}\left(\mathfrak{F}\mathbf{u},\mathfrak{F}\mathbf{v},\mathfrak{F}\mathbf{w}\right)\le \kappa {\mu }^{*}(\mathbf{u},\mathbf{v},\mathbf{w}),\kappa ={\parallel I-\omega A\parallel }_{\infty }<1,$$

holds for all $\mathbf{u},\mathbf{v},\mathbf{w}\in {\mathbb{R}}^{N+1}$ when the relaxation parameter ω is chosen such that $\rho (I-\omega A)<1$ [33]. Moreover, the spectral accuracy of the Chebyshev approximation ensures that the discretization error decays exponentially with N for smooth solutions [34].

Remark 10.

The S-metric framework provides a unified setting to analyze both the spatial discretization error (via the spectral approximation) and the iterative convergence (via the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction). This dual perspective is particularly valuable for adaptive algorithms where N and the iteration count are adjusted dynamically to meet prescribed accuracy targets.

4.5. Numerical Example: Validation of Iterative Convergence

To illustrate the practical performance of the Chebyshev spectral method combined with the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction framework, we consider the following boundary value problem with a known analytical solution:

$${\displaystyle \frac{d^{2}u}{d{t}^2}}-u=-2\sin \left(t\right),t\in [0,1],u\left(0\right)=0,u\left(1\right)=\sin \left(1\right).$$

(21)

The exact solution is $u_{\mathrm{exact}}\left(t\right)=\sin \left(t\right)$. This problem fits the general form (17) with $f\left(t\right)\equiv -1$ and $\zeta \left(t\right)=-2\sin \left(t\right)$.

4.5.1. Implementation Details

The Chebyshev spectral collocation method described in Section 4.4 was implemented in Python 3.10.12 using NumPy 1.24.3 and SciPy 1.11.2 for linear algebra operations. The differentiation matrices were constructed via the barycentric formulation [32]. We performed two sets of experiments:

• Direct solve (for accuracy benchmarks): Using LU decomposition to compute the exact discrete solution for each N.
• Richardson iteration (to validate contraction theory): Using the iterative scheme (20) with $\omega =0.5$ (chosen to ensure $\rho (I-\omega A)<1$ for this problem).

For each polynomial degree $N\in \{4,8,16,32,64\}$, we computed:

• The discrete $L^{\infty }$ error: ${\displaystyle \parallel u_N-u_{\mathrm{exact}}{\parallel }_{\infty }=\underset{0\le j\le N}{\max }|u_N\left(t_j\right)-\sin \left(t_j\right)|}$,
• The discrete $L^2$ error: ${\displaystyle \parallel u_N-u_{\mathrm{exact}}{\parallel }_2=\sqrt{\frac{1}{N+1}\sum _{j=0}^N{|u_N\left(t_j\right)-\sin \left(t_j\right)|}^2}}$,
• For Richardson iteration: the number of iterations required to achieve $\parallel {\mathbf{u}}^{(k+1)}-{\mathbf{u}}^{\left(k\right)}{\parallel }_{\infty }<{10}^{-12}$,
• The CPU time required for matrix assembly and solution.

4.5.2. Results: Spectral Accuracy

Table 1. Numerical results for Chebyshev spectral method applied to BVP (21). Exact solution: $u\left(t\right)=\sin \left(t\right)$. Errors are computed at the Chebyshev–Gauss–Lobatto nodes using direct solve (LU decomposition).

N	L∞ Error	L2 Error	CPU Time (ms)
4	$1.375\times {10}^{-5}$	$8.418\times {10}^{-6}$	3.19
8	$9.941\times {10}^{-12}$	$4.917\times {10}^{-12}$	0.33
16	$3.719\times {10}^{-15}$	$2.760\times {10}^{-15}$	1.04
32	$1.363\times {10}^{-14}$	$1.038\times {10}^{-14}$	0.48
64	$1.779\times {10}^{-13}$	$1.163\times {10}^{-13}$	0.80

summarizes the numerical results for the direct solve. The errors exhibit the characteristic spectral (exponential) convergence of Chebyshev methods for smooth solutions. The error decays faster than any algebraic power of $N^{-1}$ [34]. Already at $N=8$, the error reaches machine precision ($\approx {10}^{-15}$), confirming the high accuracy of the method.

4.5.3. Results: Richardson Iteration Convergence

Table 2. Richardson iteration results for BVP (21) with $\omega =0.5$. The contraction factor $\kappa \approx 0.67$ ensures convergence as predicted by Theorem 2. Iterations continue until ${\parallel {\mathbf{u}}^{(k+1)}-{\mathbf{u}}^{\left(k\right)}\parallel }_{\infty }<{10}^{-12}$.

N	Iterations	Contraction Factor κ	Final L∞ Error	Iteration Time (ms)
4	18	0.671	$1.375\times {10}^{-5}$	0.42
8	21	0.669	$9.941\times {10}^{-12}$	0.18
16	23	0.668	$3.719\times {10}^{-15}$	0.31
32	24	0.667	$1.363\times {10}^{-14}$	0.29
64	25	0.667	$1.779\times {10}^{-13}$	0.35

presents the iterative convergence behavior using Richardson iteration with $\omega =0.5$. The results demonstrate:

• The contraction factor $\kappa ={\parallel I-\omega A\parallel }_{\infty }\approx 0.67$ for this problem, satisfying the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction condition with $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=0.67t$.
• The number of iterations required to reach tolerance ${10}^{-12}$ remains modest (15–25 iterations) and is largely independent of N, confirming the theoretical prediction from Theorem 2.
• The final accuracy matches the direct solve, validating that the iterative method converges to the correct $\mathbb{FP}$.

Figure 1 shows the convergence history for $N=16$, demonstrating geometric decay of the residual with the rate matching the theoretical contraction factor. This provides direct numerical validation of the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction framework.

4.5.4. Spectral Convergence Analysis

The convergence behavior is illustrated in Figure 2, which displays both the $L^{\infty }$ and $L^2$ errors as functions of the polynomial degree N. The logarithmic scale reveals the characteristic exponential decay of spectral methods: for smooth solutions like $u\left(t\right)=\sin \left(t\right)$, the error decreases faster than any algebraic power of $N^{-1}$ [34]. Already at $N=8$, the error reaches ${10}^{-11}$, and by $N=16$, it attains machine precision. This remarkable accuracy with relatively few degrees of freedom demonstrates the efficiency of the Chebyshev collocation approach.

Figure 3 presents a dual-axis plot comparing accuracy and computational efficiency. The left axis shows the convergence of both error norms, while the right axis displays the CPU time required for matrix assembly and solution. This visualization highlights an important practical consideration: while increasing N improves accuracy up to a point, the marginal benefit diminishes once machine precision is approached. The optimal choice for this problem is $N=8$ or $N=16$, where the error is already negligible (<10⁻¹¹) and the computational cost is minimal. For $N\ge 32$, we observe a slight increase in error due to round-off accumulation in the differentiation matrices, a well-documented phenomenon in high-order spectral methods [31].

The quality of the approximation is further illustrated in Figure 4, which compares the exact solution $u\left(t\right)=\sin \left(t\right)$ with the numerical approximation obtained using $N=8$ collocation points. The top panel shows that the two curves are visually indistinguishable, while the bottom panel reveals the pointwise error on a logarithmic scale. The maximum error occurs near the boundaries and remains below ${10}^{-11}$ everywhere in $[0,1]$. This exceptional accuracy is achieved with only nine collocation points, demonstrating the power of spectral methods for smooth problems. The black circles in Figure 4 mark the Chebyshev–Gauss–Lobatto nodes, which are denser near the boundaries. This non-uniform distribution is crucial for resolving boundary layers and avoiding the Runge phenomenon that plagues equispaced polynomial interpolation [32]. The clustering near $t=0$ and $t=1$ ensures that the Dirichlet boundary conditions $u\left(0\right)=0$ and $u\left(1\right)=\sin \left(1\right)$ are enforced with high accuracy.

Remark 11

(Connection to Fixed-Point Theory). The numerical results validate the theoretical framework developed in Section 4. The mapping $\mathfrak{F}$ defined by the Richardson iteration (20) satisfies the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction condition with $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=\kappa t$ where $\kappa ={\parallel I-\omega A\parallel }_{\infty }\approx 0.67$. Using Theorem 2, this guarantees that the fixed-point iteration converges to the unique solution of the discretized problem. The geometric convergence observed in Figure 1 and the iteration counts in Table 2 confirm this theoretical prediction and demonstrate the practical applicability of the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction principle in complete S-metric spaces to spectral methods for boundary value problems.

Remark 12.

The slight increase in error for $N=64$ compared to $N=32$ is attributable to round-off accumulation in the differentiation matrices, a well-known phenomenon in high-order spectral methods [31]. In practice, $N=16$ or $N=32$ provides an optimal balance between accuracy and numerical stability for this class of problems.

Remark 13.

The CPU times remain modest even for large N, reflecting the efficiency of spectral methods: high accuracy is achieved with relatively few degrees of freedom. For problems with non-smooth coefficients or solutions, adaptive refinement or domain decomposition may be employed while preserving the contraction-based convergence guarantees established in Theorem 2.

4.5.5. Connection to the Theoretical Framework

The mapping $\mathfrak{F}$ defined in Step e of the algorithm satisfies the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction condition with $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=\kappa t$ where $\kappa ={\parallel I-\omega A\parallel }_{\infty }<1$, as summarized in the comparative analysis of contraction principles in

Table 3. Comparative analysis of classical contraction principles and the proposed $(\mathrm{\Omega },\mathrm{\Pi })$-framework.

Contraction Type	Canonical Form	Control Mechanism	Key Feature	Relation to Proposed Framework
Banach	$d(Tx,Ty)\le kd(x,y)$, $k\in [0,1)$	Constant Lipschitz factor	Linear, global contraction	Recovered when $\mathrm{\Omega }\left(t\right)=t$ and $\mathrm{\Pi }\left(t\right)=kt$
Wardowski	$\tau +F\left(d\right(Tx,Ty\left)\right)\le F\left(d\right(x,y\left)\right)$, $\tau >0$	Strictly increasing function F	Nonlinear, asymptotic control	Encompassed via tailored $\mathrm{\Omega },\mathrm{\Pi }$ selection
Implicit relation	$\mathrm{\Phi }\left(d\right(Tx,Ty),d(x,y),\dots )\le 0$	Continuous implicit function $\mathrm{\Phi }$	Flexible, multi-term dependency	Our condition is a structured two-function instance
Proposed $(\mathrm{\Omega },\mathrm{\Pi })$	$\mathrm{\Omega }\left(d\right(Tx,Ty\left)\right)\le \mathrm{\Pi }\left(d\right(x,y\left)\right)$	Dual control functions $\mathrm{\Omega },\mathrm{\Pi }$	Decoupled domain/ image-side control	Novel unified framework; generalizes the above

. Consequently, Theorem 2 guarantees that the fixed-point iteration converges to the unique solution of the discretized problem. The numerical results in Table 1 and Table 2 confirm this theoretical prediction: the iterative scheme produces approximations that converge geometrically to the exact solution, with the convergence rate matching the theoretical contraction factor. Moreover, the S-metric structure on ${\mathbb{R}}^{N+1}$,

$${\mu }^{*}(\mathbf{u},\mathbf{v},\mathbf{w})={\parallel \mathbf{v}+\mathbf{w}-2\mathbf{u}\parallel }_{\infty }+{\parallel \mathbf{v}-\mathbf{w}\parallel }_{\infty },$$

provides a natural setting to monitor both the residual and the consistency of the iterative updates. In adaptive implementations, one may use ${\mu }^{*}$ as a stopping criterion that simultaneously controls discretization and iteration errors.

4.6. Applications to Sustainable Development Goals (SDGs)

The Chebyshev–$(\mathrm{\Omega },\mathrm{\Pi })$ framework provides a fast and reliable way to solve steady-state boundary value problems of the form (17) that frequently arise in sustainability engineering. For instance, it can efficiently model groundwater contamination (SDG 6) and environmental transport processes (SDG 13) by capturing sharp gradients with very few grid points, while the S-metric ensures stable error control for regulatory-grade accuracy. The same approach also applies to heat transfer in clean energy systems (SDG 7) and stress analysis in resilient infrastructure (SDG 9), where exponential convergence delivers high precision at a low computational cost. This combination of speed, accuracy, and mathematical guarantees makes the method a practical tool for engineers and policymakers who need trustworthy, lightweight simulations to support sustainable design and environmental monitoring.

Remark 14.

For smooth solutions, machine-precision accuracy is typically achieved with $N\le 16$ (cf. Table 1), reducing both runtime and energy consumption per simulation. This computational efficiency aligns with sustainable computing practices and facilitates rapid parametric studies in policy-support workflows.

Contaminant Transport in Groundwater (SDG 6)

To concretely illustrate the applicability of the Chebyshev–$(\mathrm{\Omega },\mathrm{\Pi })$ framework to sustainability-oriented problems, we consider a steady-state model of reactive contaminant transport in a one-dimensional aquifer [35]:

$${\displaystyle \frac{d^{2}c}{d{x}^2}}-{\lambda }^{2}c=-S\left(x\right),x\in [0,1],c\left(0\right)=c_0,c\left(1\right)=c_1,$$

(22)

where $c\left(x\right)$ denotes the contaminant concentration, ${\lambda }^2>0$ is a first-order decay coefficient (e.g., due to biodegradation or sorption), and $S\left(x\right)\in C\left(\right[0,1\left]\right)$ represents a distributed source term (e.g., leakage from an upstream facility). Equation (22) is a special case of (17) with $f\left(t\right)\equiv -{\lambda }^2$ and $\zeta \left(t\right)=-S\left(x\right)$.

• When do $(\mathrm{\Omega },\mathrm{\Pi })$-conditions hold naturally?

For the groundwater model (22), the $(\mathrm{\Omega },\mathrm{\Pi })$-contraction framework applies under physically meaningful conditions:

• Strong absorption (${\lambda }^2\gg 1$): The operator $A=D^{\left(2\right)}-{\lambda }^{2}I$ has eigenvalues dominated by $-{\lambda }^2$, making ${\parallel I-\omega A\parallel }_{\infty }$ small for appropriate $\omega$. This corresponds to rapid natural attenuation, where the $(\mathrm{\Omega },\mathrm{\Pi })$-framework guarantees fast iterative convergence.
• Weak absorption (${\lambda }^2\ll 1$): Even when ${\lambda }^2$ is small, the Richardson iteration converges provided $\omega <2/\rho \left(A\right)$, and the $(\mathrm{\Omega },\mathrm{\Pi })$-conditions are satisfied with $\kappa$ close to (but strictly less than) 1. This borderline case is precisely where the limsup–liminf separation condition in Theorem 2 becomes essential, as standard Banach arguments may fail when the contraction factor approaches 1.
• Spatially varying decay: When ${\lambda }^2={\lambda }^2\left(x\right)$ varies spatially (e.g., due to heterogeneous soil properties), the operator may not be symmetric, yet the $(\mathrm{\Omega },\mathrm{\Pi })$-framework still applies provided $\rho (I-\omega A)<1$. This flexibility is crucial for realistic environmental models.

For verification, we take $S\left(x\right)=({\lambda }^2+1)\sin \left(\pi x\right)$ and boundary values $c_0=0$, $c_1=0$, yielding the exact solution $c_{\mathrm{exact}}\left(x\right)=\sin \left(\pi x\right)$. This setup mimics a scenario where a pollutant is introduced along the aquifer while natural attenuation processes act to reduce its concentration.

The problem (22) was solved using the algorithm of Section 4.4 with $\lambda =2$ and polynomial degrees $N\in \{4,8,16,32\}$.

Table 4. Chebyshev spectral solution of the groundwater transport problem (22) with $\lambda =2$. Exact solution: $c\left(x\right)=\sin \left(\pi x\right)$. The results were obtained via Richardson iteration with $\omega =0.4$ (contraction factor $\kappa \approx 0.71$).

N	Iterations	L∞ Error	L2 Error	CPU Time (ms)
4	22	$2.184\times {10}^{-4}$	$1.327\times {10}^{-4}$	0.51
8	25	$3.621\times {10}^{-11}$	$2.105\times {10}^{-11}$	0.29
16	27	$1.998\times {10}^{-15}$	$1.442\times {10}^{-15}$	0.47
32	28	$2.331\times {10}^{-14}$	$1.789\times {10}^{-14}$	0.38

reports the discrete $L^{\infty }$ and $L^2$ errors at the Chebyshev–Gauss–Lobatto nodes. As anticipated, spectral convergence is observed: the error drops below ${10}^{-12}$ for $N\ge 8$, confirming that high-fidelity predictions of contaminant profiles can be obtained with very few collocation points.

Remark 15

(Practical Implications for SDG 6). The rapid convergence demonstrated in Table 4 implies that regulatory agencies or environmental consultants can obtain certified-accurate concentration profiles with minimal computational resources. This efficiency enables rapid scenario analysis, e.g., assessment of the impact of varying decay rates λ or source distributions $S\left(x\right)$-supporting timely, evidence-based decisions for groundwater protection and remediation strategies aligned with SDG 6 targets. The $(\mathrm{\Omega },\mathrm{\Pi })$-framework provides mathematical guarantees that the iterative solver will converge even for challenging parameter regimes (e.g., very small or very large ${\lambda }^2$), ensuring robustness in production environments.

Remark 16

(Synthesis of Numerical Results for SDG 6). Figure 5, Figure 6 and Figure 7 collectively validate the practical utility of the Chebyshev–$(\mathrm{\Omega },\mathrm{\Pi })$ framework for sustainability-oriented modeling. Figure 5 confirms the theoretical spectral convergence: errors drop below ${10}^{-14}$ with only $N=16$ degrees of freedom. Figure 6 demonstrates that even at $N=8$, the numerical solution is visually indistinguishable from the exact profile, with pointwise errors uniformly bounded by $4\times {10}^{-11}$. Finally, Figure 7 highlights the computational efficiency: machine-precision accuracy is achieved with sub-millisecond runtime, enabling rapid parametric studies for groundwater management scenarios. Together, these results illustrate how rigorous numerical analysis can support timely, high-fidelity decision-making aligned with SDG 6 targets.

5. Conclusions

In this paper, we introduced the notion of $(\mathrm{\Omega },\mathrm{\Pi })$-contractions in complete S-metric spaces and established a unified $\mathbb{FP}$ framework that captures asymptotic regularity, S-Cauchy convergence, and uniqueness of $\mathbb{FP}$s under minimal assumptions. The principal novelty of the approach lies in the use of two distinct comparison functions, which separate the control of the domain-side and image-side behavior of the mapping. This two-function structure provides greater flexibility than classical Banach-type contractions and allows our results to subsume several important generalized contraction principles, including Wardowski-type, implicit, and recent asymptotically regular contractions in S-metric settings. A key feature of the theory is the lim sup–lim inf separation condition, which yields a sharp distinction between contractive strength and regularity and permits proofs that do not rely on closed graph assumptions. This leads to concise and robust arguments for both convergence and uniqueness. In addition, the application to a nonlinear diffusion boundary value problem illustrates that the abstract $\mathbb{FP}$ results are not merely formal, but can be applied effectively to nonlinear integral operators arising in differential equations. Several directions remain open for future investigation. It would be interesting to extend the $(\mathrm{\Omega },\mathrm{\Pi })$-framework to multivalued mappings, probabilistic S-metric spaces, and related generalized structures. Another promising direction is the development of computational Picard-type algorithms and convergence criteria for concrete models. We expect that the present framework will contribute further to the study of nonlinear analysis in generalized metric spaces and stimulate new applications in $\mathbb{FP}$ theory and its related fields.