A Fixed Point Framework for Nonlinear Fractional Systems with Memory Effects

Abstract

In this work, we develop two new classes of rational contraction mappings along with the corresponding fixed point theorems for these types of contractions in suprametric spaces. Furthermore, we use the obtained results to investigate two nonlinear systems, namely, a fractional chaotic financial system and a nonlinear fractional differential equation under integral boundary conditions. Both these nonlinear problems are transformed into fixed point problems in appropriate suprametric spaces, thereby demonstrating the applicability of the developed rational contraction results to nonlinear systems with memory effects.

1. Introduction

Fixed-point theorems play an important role in mathematical analysis. In fact, the famous Banach fixed-point theorem presents a simple scheme that ensures both the existence and uniqueness of the solution to a broad class of problems: any contraction map on a complete metric space admits a unique fixed point (“$FP$” refers to “fixed point” from here on) that can be obtained by iteration [1]. Applications of this result can be found in differential and integral equations [2].

In order to deal with increasingly sophisticated nonlinear scenarios, several types of contraction mappings have been defined. Among these generalized contractions, the most significant one is the rational contraction mapping, as it can model nonlinearity through ratios and function mappings [3]. Rational contraction mappings generalize the classical definition of contraction and provide stronger tools for working with operators that are not necessarily subject to linear Lipschitz-like constraints. The current state of research demonstrates that generalized contraction approaches provide an important extension of the scope of $FP$ techniques in nonlinear analysis [4,5].

Along with the above-listed concepts, suprametric spaces have been studied. A suprametric space provides a relaxed version of a metric and b-metric space, as it does not require adherence to the usual triangle inequality constraint. Suprametric spaces thus allow us to work with nonlinear operators having a weak contractive property or weaker continuity, making them an ideal framework for $FP$ research on generalized contraction mappings, for example, see [6].

Fractional-order differential equations have become essential for modeling physical phenomena, especially those involving memory and inheritance characteristics. Unlike classical models, fractional order dynamical systems take into account non-locality and dependence at distant points, thus becoming increasingly relevant in physics, engineering, biology, and finance [7]. Such systems can model anomalies and dynamical features not covered by classical models [8].

There are close connections between $FP$ theory and fractional calculus. Theorems concerning the existence and uniqueness of the solution of fractional differential equations depend on the use of $FP$ methods like the Banach contraction mapping principle, the Schauder $FP$ theorem, and its extension [9,10]. In the first step, one transforms the differential equation into an integral equation, and then using that equation, one uses the $FP$ techniques to solve the differential equation. Given certain assumptions, the $FP$ techniques can be used in the existence and uniqueness of solutions to nonlinear fractional differential equations [11].

Chaotic behavior in a specific system is an intrinsic randomness, not induced by external perturbation. The chaotic behavior of a new fractional order dynamical system was studied through the construction of the model and circuit realization by Hammouch and Mekkaoui [12]. Chaos in the fractional-order model was analyzed using bifurcation and Lyapunov exponents by Sene [13]. Recently, Atangana proposed the fractal-fractional derivative, a new fractional differential operator with two parameters: fractal dimension and fractional order [14]. For more related work, one can refer to [15,16].

The financial chaotic model is another name for chaotic finance theory, which originated during the late 20th century and explains the nature of financial markets that are unpredictable and highly volatile. It states that the financial market is a chaotic system, which shows non-linear dynamics and extreme dependence on initial conditions. For further analysis on this topic, one can refer to [17]. The roots of the financial chaotic model can be traced back to the work of mathematicians and physicists who were studying chaos theory in other fields. In the 1960s and 1970s, researchers like Edward Lorenz and Benoit Mandelbrot discovered that seemingly random and unpredictable behavior could arise from simple mathematical equations [18,19]. The idea became better known after the global financial crisis of 2008, as many classic economic theories could not predict nor interpret the severity of the crisis. It became clear that interconnections between various sectors can increase vulnerability in an economy. The present study concentrates on the chaotic financial model under fractional calculus due to the fact that the concept of memory plays an important role within the dynamical system, especially in the case of economics and finance-related models.

Fractional calculus extends the classical notions of integer-order differentiation and integration. One standard way to introduce fractional derivatives is through fractional integrals, as recalled below.

Definition 1

([7,20]). Let $\mathfrak{p}:[0,+\infty )\to \mathbb{R}$. The fractional integral of order $\alpha >0$ in the Riemann–Liouville sense is given by

$$I_t^{\alpha }\mathfrak{p}\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_a^t{(t-s)}^{\alpha -1}\mathfrak{p}\left(s\right)ds,t\in [a,b],$$

(1)

where $\mathsf{\Gamma }(·)$ is the Euler-gamma function.

Correspondingly, the Riemann–Liouville fractional derivative is defined as follows:

Definition 2

([7,20]). Let $\mathfrak{p}:[0,+\infty )\to \mathbb{R}$ be a sufficiently smooth function. The fractional derivative of order $\alpha \in (0,1)$ in the sense of Riemann-Liouville is defined by

$$D_t^{\alpha }\mathfrak{p}\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\displaystyle \frac{d}{dt}}{\int }_a^t\mathfrak{p}\left(s\right){(t-s)}^{-\alpha }ds,t\in [a,b],$$

(2)

where $\mathsf{\Gamma }(·)$ denotes the Euler-gamma function, and the classical derivative is given by

$${\displaystyle \frac{d\mathfrak{p}\left(t\right)}{dt}}=\underset{h\to 0}{lim}{\displaystyle \frac{\mathfrak{p}(t+h)-\mathfrak{p}\left(t\right)}{h}},t>0.$$

(3)

Although the Riemann–Liouville approach is classical and mathematically well developed, its direct use may be inconvenient in initial value problems involving classical initial conditions. For this reason, the Caputo derivative is often preferred in applications.

Definition 3

([20]). Consider a continuous map $g:[0,\infty )$ and $\alpha >0$. The Caputo fractional derivative of order α, denoted by ${}^{\mathrm{C}}D^{\alpha }g\left(t\right)$, is defined as

$${}^{\mathrm{C}}D^{\alpha }g\left(t\right)={\int }_a^x{K}_{\alpha }(t-s)g^{\left(m\right)}\left(s\right)ds,$$

(4)

with

$$K_{\alpha }(t-s)={\displaystyle \frac{{(t-s)}^{m-\alpha -1}}{\mathsf{\Gamma }(m-\alpha )}}$$

where $\alpha \in [m-1,m)$, $m=\lfloor \alpha \rfloor +1\in \mathbb{N}$, and $\lfloor \alpha \rfloor$ denotes the integer part of $\alpha$.

$g^{\left(m\right)}$ represents the commonly used m-order derivative, having special physical significance. From Equation (4), we notice that the $\alpha$-order fractional derivative at time t is not defined locally. It requires the sum total of the effect of the commonly used m-order integer derivative in the interval $[a,t]$. Thus, it can be employed to analyze the dynamics of systems in which the rate of change at any instant is dependent on its history; thus, it is termed “memory effect” in a graphic sense.

The fractional-order financial system exhibits several dynamic behaviors such as $FP$s, periodic orbits, and chaotic orbits. Recently in [21], the authors extended the concept of financial chaos in fractal-fractional form and established the existence of the solution of such model over a Banach space under the metric defined by norm. Additionally, in [22], the authors introduced the concept of minimum interest rate “d” in order to establish a more realistic and better financial model. The derivatives are defined using Caputo-fractional-order and Atangana-Baleanu derivatives, and their solution’s existence and uniqueness are analyzed using the $FP$ theorem.

Inspired by the recent advances in fractional calculus and $FP$ theory mentioned above, the goal of this paper is to study rational contraction mappings in suprametric spaces and employ $FP$ results to fractional differential equations and the fractional chaotic financial model. Although both fixed point theory and fractional differential equations have received extensive research attention, rational-type contractions in the setting of complete suprametric spaces for fractional nonlinear systems are still in their infancy. In addition, there is not enough literature regarding the existence and uniqueness of solutions of fractional chaotic financial models in the context of these generalized contractive assumptions. The aim of the present paper is to provide a unified and robust framework for analyzing complex nonlinear systems with memory effects and contribute to the growing intersection of $FP$ theory, fractional calculus, and financial mathematics.

2. Preliminaries

Definition 4

([23]). Let $\mathcal{X}$ be a nonempty set and let $\delta :\mathcal{X}\times \mathcal{X}\to [0,\infty )$ satisfy

(ds1) 

$\delta (\mathfrak{p},\nu )=0\iff \mathfrak{p}=\nu$ (identity);

(ds2) 

$\delta (\mathfrak{p},\nu )=\delta (\nu ,\mathfrak{p})$ (symmetry);

(ds3) 

$\delta (\mathfrak{p},\nu )\le \delta (\mathfrak{p},\xi )+\delta (\xi ,\nu )+\rho \delta (\mathfrak{p},\xi )\delta (\xi ,\nu ),$ for some constant $\rho \in (0,\infty )$ and for all $\mathfrak{p},\nu ,\xi \in \mathcal{X}$.

Then $(\mathcal{X},\delta )$ is said to be a suprametric space and δ is a suprametric on $\mathcal{X}$.

Example 1

([23]). Let $(\mathcal{X},\delta )$ be a metric space and let $\alpha ,\beta >0.$ Then,

(i) 

$${\delta }_1(\mathfrak{p},\nu )=\delta (\mathfrak{p},\nu )\left(\delta (\mathfrak{p},\nu )+\alpha \right)$$

is a suprametric with constant $\rho ={\displaystyle \frac{2}{\alpha }}$.

(ii) 

$${\delta }_2(\mathfrak{p},\nu )=\beta \left(e^{\delta (\mathfrak{p},\nu )}-1\right)$$

is a suprametric with $\rho ={\displaystyle \frac{1}{\beta }}$.

However, ${\delta }_1$ and ${\delta }_2$ are not usual metrics. For example, if $\delta (\mathfrak{p},\nu )=|\mathfrak{p}-\nu |$ on $\mathbb{R},$ then the triangle inequality does not hold for ${\delta }_1$ and ${\delta }_2.$ Particularly when $\mathfrak{p}=0,\nu =1,\xi =2.$

$${\delta }_1(0,1)+{\delta }_1(1,2)<{\delta }_1(0,2).$$

$${\delta }_2(0,1)+{\delta }_2(1,2)<{\delta }_2(0,2).$$

Every suprametric with a constant $\rho$ is also a suprametric with a constant ${\rho }^{\prime }>\rho$; however, the reverse is not always true.

Remark 1

([23]). If $\delta ={\delta }_1$ (or $\delta ={\delta }_2)$ with $\delta (\mathfrak{p},\nu )=|\mathfrak{p}-\nu |$, then ${\delta }_1$ with constant $\rho ={\displaystyle \frac{1}{3}}$ is not a suprametric, since

$$\delta (0,1)>\delta \left(0,{\displaystyle \frac{1}{2}}\right)+\delta \left({\displaystyle \frac{1}{2}},1\right)+{\displaystyle \frac{1}{3}}\delta (0,1)\delta \left({\displaystyle \frac{1}{2}},1\right).$$

Definition 5

([23]). Let $(\mathcal{X},\delta )$ be a suprametric space. The set

$$\beta ({\mathfrak{p}}_0,r)=\{\mathfrak{p}\in \mathcal{X}:\delta ({\mathfrak{p}}_0,\mathfrak{p})<r\},$$

where $r>0$ and ${\mathfrak{p}}_0\in \mathcal{X}$, is said to be an open ball. A subset $\mathcal{Y}\subseteq \mathcal{X}$ is called open if for every $\nu \in \mathcal{Y}$ there is an $r>0$ such that $\beta (\nu ,r)$ is a subset of $\mathcal{Y}$.

Definition 6

([23]). A sequence $\left\{{\mathfrak{p}}_n\right\}$ in a suprametric space $\mathcal{X}$ is said to be a Cauchy sequence if for every $\epsilon >0$ there is an $N\in \mathbb{N}$ such that

$$\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_m)<\epsilon \mathrm{for}\mathrm{all}n,m\ge N.$$

• A suprametric space is said to be complete if every Cauchy sequence converges.

Proposition 1

([23]). If $(\mathcal{X},\delta )$ is a complete metric space and ${\delta }_s$ is defined as in Example 1, then $(\mathcal{X},{\delta }_s)$ is a complete suprametric space.

Lemma 1

([23]). Every suprametric is continuous.

Example 2

([24]). Assume that $\rho =2$ and $\delta (x,y)={(x-y)}^2$, where $\mathcal{X}$ is a set of integers. Then $(\mathcal{X},\delta )$ is a suprametric space.

Example 3.

Assume that $\rho \ge 2$ is an integer and $\delta (\mathfrak{p},\nu )={|\mathfrak{p}-\nu |}^p$, where $\mathcal{X}$ is a set of integers. Then $(\mathcal{X},\delta )$ is a suprametric space for $\rho ={\sum }_{n=1}^{p-1}{}^{p}C_n$.

• Jleli and Samet [25] introduced the following class $\mathsf{\Theta }$ of auxiliary functions.

Definition 7.

Let Θ be the set of $\theta :(0,\infty )\to (1,\infty )$ satisfying $\left({\theta }_1\right)$–$\left({\theta }_3\right),$ where

(${\theta }_1$)

$\theta \left(\mathfrak{p}\right)\le \theta \left(\nu \right),\forall \mathfrak{p},\nu \in (0,\infty )$ and $\mathfrak{p}\le \nu$.

(${\theta }_2$)

For any sequence $\left({\lambda }_n\right)\subset (0,\infty )$,

$$\underset{n\to \infty }{lim}\theta \left({\lambda }_n\right)=1\iff \underset{n\to \infty }{lim}{\lambda }_n=0.$$

(${\theta }_3$)

There exist $l^{\prime }\in (0,\infty ]$ and $\gamma \in (0,1)$ such that

$$\underset{\lambda \to 0^{+}}{lim}{\displaystyle \frac{\theta \left(\lambda \right)-1}{{\lambda }^{\gamma }}}=l^{\prime }.$$

Jleli and Samet [25] studied the existence and uniqueness of $FP$s for $\theta -$ contractions in generalized metric spaces [26]. Moreover, Jleli and Samet [25] exhibited that every $\theta$-contraction possesses at most one $FP$ within the setting of complete generalized metric spaces.

Example 4.

The function $\theta :(0,\infty )\to (1,\infty )$ defined by $\theta \left(t\right)=e^{\sqrt{t}}$ satisfies the conditions $\left({\theta }_1\right),\left({\theta }_2\right)$ and $\left({\theta }_3\right)$ for $\gamma ={\displaystyle \frac{1}{2}}$.

Example 5.

The function $\theta :(0,\infty )\to (1,\infty )$ defined by $\theta \left(t\right)=1+t^m(1+\left[t\right])$ for any $p>1$ and $m\in (0,{\displaystyle \frac{1}{p}}),$ where $\left[t\right]$ denotes the integer part of $t,$ satisfies the conditions $\left({\theta }_1\right),\left({\theta }_2\right)$ and $\left({\theta }_3\right)$ for any $\gamma \in ({\displaystyle \frac{1}{p}},1)$.

Materials and Methods

The main setting of the paper is a complete suprametric space $(\mathcal{X},\delta )$. In this framework, the distance function satisfies the identity and symmetry properties together with a weakened triangle-type inequality involving a positive constant $\rho$. This setting is broader than the usual metric framework and is suitable for studying nonlinear mappings that may not satisfy a classical Lipschitz condition.

We also use the class $\mathsf{\Theta }$ of functions $\theta :(0,\infty )\to (1,\infty )$. These functions are non-decreasing and satisfy the asymptotic properties stated above. They allow us to formulate rational-type contractive conditions and to study the convergence of Picard iterations.

The methodology used in the paper consists of the following steps:

• We recall the Riemann–Liouville fractional integral, the Riemann–Liouville fractional derivative, and the Caputo fractional derivative. The Caputo derivative is used in the applications because it is compatible with classical initial conditions.
• We introduce two rational-type contractions, denoted by ${\mathcal{RT}}_1$ and ${\mathcal{RT}}_2$, in the setting of complete suprametric spaces.
• We prove fixed point theorems for these contractions by constructing Picard sequences, establishing their Cauchy property, and then proving that their limits are fixed points.
• We apply the results to a fractional chaotic financial model. Since the nonlinear financial vector field is not globally Lipschitz, the analysis is carried out on a closed bounded subset where explicit local Lipschitz constants are available.
• We also apply the fixed point framework to a nonlinear fractional differential equation with integral boundary conditions by rewriting the problem as an equivalent integral equation.

3. Fixed-Point Results

We begin this section by introducing rational-type contraction $1,$ denoted by ${\mathcal{RT}}_1,$ and rational-type contraction 2 denoted by ${\mathcal{RT}}_2.$

Definition 8.

Let $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ be a self-mapping on a suprametric space $(\mathcal{X},\delta )$. Suppose that there is a non-decreasing function $\theta :(0,\infty )\to (1,\infty )$ with $\theta \in \mathsf{\Theta }$ and a constant ${\kappa }^{\prime }\in (0,1)$ satisfying

$$\mathcal{T}\mathfrak{p}\ne \mathcal{T}\nu \Rightarrow \theta \left(\delta (\mathcal{T}\mathfrak{p},\mathcal{T}\nu )\right)\le {\left(\theta \left({\mathcal{RT}}_1(\mathfrak{p},\nu )\right)\right)}^{{\kappa }^{\prime }},$$

(5)

where

$${\mathcal{RT}}_1(\mathfrak{p},\nu )=max\left\{\begin{array}{cc}& \delta (\mathfrak{p},\nu ),\hfill \\ & {\displaystyle \frac{\delta (\mathfrak{p},\mathcal{T}\mathfrak{p})[1+\delta (\nu ,\mathcal{T}\nu \left)\right]}{1+\delta (\mathfrak{p},\nu )}},\hfill \\ & {\displaystyle \frac{\delta (\nu ,\mathcal{T}\nu )[1+\delta (\mathfrak{p},\mathcal{T}\mathfrak{p}\left)\right]}{1+\delta (\mathfrak{p},\nu )}},\hfill \\ & {\displaystyle \frac{\delta (\mathfrak{p},\mathcal{T}\mathfrak{p})\delta (\nu ,\mathcal{T}\nu )}{1+\delta (\mathfrak{p},\nu )}}\hfill \end{array}\right\}.$$

Then $\mathcal{T}$ is called a rational-type contraction 1.

Theorem 1.

Let $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ be a rational-type contraction 1 on a complete suprametric space $(\mathcal{X},\delta )$. Assume, in addition, that θ is continuous on $(0,\infty )$. Then $\mathcal{T}$ has a unique $FP$ in $\mathcal{X}$.

Proof. 

Let ${\mathfrak{p}}_0\in \mathcal{X}$ be arbitrary, and define the Picard sequence $\left({\mathfrak{p}}_n\right)$ by

$${\mathfrak{p}}_n=\mathcal{T}{\mathfrak{p}}_{n-1},n\in \mathbb{N}.$$

If ${\mathfrak{p}}_n={\mathfrak{p}}_{n-1}$ for some n, then ${\mathfrak{p}}_{n-1}$ is an $FP$ of $\mathcal{T}$, and there is nothing more to prove. Hence, we may assume that

$${\mathfrak{p}}_n\ne {\mathfrak{p}}_{n-1},n\in \mathbb{N}.$$

Set

$$a_n=\delta ({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n),n\in \mathbb{N}.$$

Then $a_n>0$ for all $n\in \mathbb{N}$. Since ${\mathfrak{p}}_n=\mathcal{T}{\mathfrak{p}}_{n-1}$ and ${\mathfrak{p}}_{n+1}=\mathcal{T}{\mathfrak{p}}_n$, the contractive condition (5) gives

$$\theta \left(a_{n+1}\right)=\theta \left(\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\right)\le {\left(\theta \left({\mathcal{RT}}_1({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)\right)\right)}^{{\kappa }^{\prime }}.$$

(6)

Moreover,

$${\mathcal{RT}}_1({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=max\left\{a_n,{\displaystyle \frac{a_n(1+a_{n+1})}{1+a_n}},a_{n+1},{\displaystyle \frac{a_n{a}_{n+1}}{1+a_n}}\right\}.$$

We first show that $a_{n+1}<a_n$ for every $n\in \mathbb{N}$. Suppose, to the contrary, that $a_{n+1}\ge a_n$ for some n. Then

$${\displaystyle \frac{a_n(1+a_{n+1})}{1+a_n}}\le a_{n+1},{\displaystyle \frac{a_n{a}_{n+1}}{1+a_n}}<a_{n+1}.$$

Thus,

$${\mathcal{RT}}_1({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=a_{n+1}.$$

From (6), we obtain

$$\theta \left(a_{n+1}\right)\le {\left(\theta \left(a_{n+1}\right)\right)}^{{\kappa }^{\prime }}.$$

Since $a_{n+1}>0$, we have $\theta \left(a_{n+1}\right)>1$. As $0<{\kappa }^{\prime }<1$, this implies

$${\left(\theta \left(a_{n+1}\right)\right)}^{{\kappa }^{\prime }}<\theta \left(a_{n+1}\right),$$

which is impossible. Therefore,

$$a_{n+1}<a_n,n\in \mathbb{N}.$$

It follows that

$${\mathcal{RT}}_1({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=a_n.$$

Hence, by (6),

$$\theta \left(a_{n+1}\right)\le {\left(\theta \left(a_n\right)\right)}^{{\kappa }^{\prime }},n\in \mathbb{N}.$$

(7)

Iterating (7), we obtain

$$1<\theta \left(a_{n+1}\right)\le {\left(\theta \left(a_1\right)\right)}^{{\left({\kappa }^{\prime }\right)}^n}.$$

Letting $n\to \infty$, we obtain

$$\underset{n\to \infty }{lim}\theta \left(a_n\right)=1.$$

By property $\left(\theta 2\right)$ of the class $\mathsf{\Theta }$, this gives

$$\underset{n\to \infty }{lim}{a}_n=\underset{n\to \infty }{lim}\delta ({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=0.$$

(8)

We now prove that $\left({\mathfrak{p}}_n\right)$ is a Cauchy sequence. Suppose, on the contrary, that $\left({\mathfrak{p}}_n\right)$ is not Cauchy. Then there is an $\epsilon >0$ and a strictly increasing sequence of positive integers $\left(n_i\right)$ such that, for each i, the index

$$m_i=min\left\{m>n_i:\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_m)\ge \epsilon \right\}$$

is well defined. By minimality of $m_i$, we have

$$\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})\ge \epsilon \mathrm{and}\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})<\epsilon .$$

Let $\rho >0$ be the constant appearing in the suprametric inequality. By the suprametric inequality, we have

$$\begin{array}{cc}\hfill \epsilon \le \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})& \le \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})+\delta ({\mathfrak{p}}_{m_i-1},{\mathfrak{p}}_{m_i})\hfill \\ & +\rho \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})\delta ({\mathfrak{p}}_{m_i-1},{\mathfrak{p}}_{m_i})\hfill \\ & <\epsilon +a_{m_i}+\rho \epsilon a_{m_i}.\hfill \end{array}$$

Using (8), we obtain

$$\underset{i\to \infty }{lim}\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})=\epsilon .$$

(9)

Next, let

$$q_i=\delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1}).$$

Again, by the suprametric inequality,

$$\begin{array}{cc}\hfill q_i& \le \delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{n_i})+\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})+\rho \delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{n_i})\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})\hfill \\ & <a_{n_i}+\epsilon +\rho \epsilon a_{n_i}.\hfill \end{array}$$

Therefore,

$$\underset{i\to \infty }{lim\; sup}{q}_i\le \epsilon .$$

On the other hand, applying the suprametric inequality twice gives

$$\begin{array}{cc}\hfill \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})& \le a_{n_i}+q_i+a_{m_i}+\rho q_i{a}_{m_i}\hfill \\ & +\rho a_{n_i}\left(q_i+a_{m_i}+\rho q_i{a}_{m_i}\right).\hfill \end{array}$$

Since $\left(q_i\right)$ is bounded by the preceding estimate and $a_{n_i},a_{m_i}\to 0$, (9) implies

$$\epsilon \le \underset{i\to \infty }{lim\; inf}{q}_i.$$

Consequently,

$$\underset{i\to \infty }{lim}\delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1})=\underset{i\to \infty }{lim}{q}_i=\epsilon .$$

(10)

Now,

$${\mathcal{RT}}_1({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1})=max\left\{\begin{array}{cc}& q_i,\hfill \\ & {\displaystyle \frac{a_{n_i}(1+a_{m_i})}{1+q_i}},\hfill \\ & {\displaystyle \frac{a_{m_i}(1+a_{n_i})}{1+q_i}},\hfill \\ & {\displaystyle \frac{a_{n_i}{a}_{m_i}}{1+q_i}}\hfill \end{array}\right\}.$$

By (8) and (10), we have

$$q_i\to \epsilon >0,$$

while

$${\displaystyle \frac{a_{n_i}(1+a_{m_i})}{1+q_i}}\to 0,{\displaystyle \frac{a_{m_i}(1+a_{n_i})}{1+q_i}}\to 0,{\displaystyle \frac{a_{n_i}{a}_{m_i}}{1+q_i}}\to 0.$$

Hence, there is an $i_0\in \mathbb{N}$ such that, for all $i\ge i_0$,

$${\mathcal{RT}}_1({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1})=q_i=\delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1}).$$

This conclusion follows from the above estimates; it is not assumed without loss of generality.

Since

$${\mathfrak{p}}_{n_i}=\mathcal{T}{\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i}=\mathcal{T}{\mathfrak{p}}_{m_i-1},$$

and $\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})\ge \epsilon >0$, we may apply (5). Thus, for all $i\ge i_0$,

$$\theta \left(\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})\right)\le {\left(\theta \left(q_i\right)\right)}^{{\kappa }^{\prime }}.$$

Letting $i\to \infty$ and using the continuity of $\theta$, together with (9) and (10), we obtain

$$\theta \left(\epsilon \right)\le {\left(\theta \left(\epsilon \right)\right)}^{{\kappa }^{\prime }}.$$

This is impossible because $\theta \left(\epsilon \right)>1$ and $0<{\kappa }^{\prime }<1$. Therefore, $\left({\mathfrak{p}}_n\right)$ is a Cauchy sequence.

Since $(\mathcal{X},\delta )$ is complete, there is a ${\mathfrak{p}}^{\ast }\in \mathcal{X}$ such that

$$\underset{n\to \infty }{lim}{\mathfrak{p}}_n={\mathfrak{p}}^{\ast }.$$

(11)

We now show that ${\mathfrak{p}}^{\ast }$ is an $FP$ of $\mathcal{T}$. Suppose, to the contrary, that

$$\mathcal{T}{\mathfrak{p}}^{\ast }\ne {\mathfrak{p}}^{\ast }.$$

Let

$$d=\delta ({\mathfrak{p}}^{\ast },\mathcal{T}{\mathfrak{p}}^{\ast })>0.$$

If $\mathcal{T}{\mathfrak{p}}^{\ast }=\mathcal{T}{\mathfrak{p}}_n$ for infinitely many n, then, along a subsequence,

$$\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}_{n+1}\to {\mathfrak{p}}^{\ast },$$

which gives $\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}^{\ast }$, a contradiction. Hence, for all sufficiently large n,

$$\mathcal{T}{\mathfrak{p}}^{\ast }\ne \mathcal{T}{\mathfrak{p}}_n.$$

For such n, by (5),

$$\theta \left(\delta (\mathcal{T}{\mathfrak{p}}^{\ast },{\mathfrak{p}}_{n+1})\right)=\theta \left(\delta (\mathcal{T}{\mathfrak{p}}^{\ast },\mathcal{T}{\mathfrak{p}}_n)\right)\le {\left(\theta \left({\mathcal{RT}}_1({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)\right)\right)}^{{\kappa }^{\prime }}.$$

(12)

Now

$${\mathcal{RT}}_1({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)=max\left\{\begin{array}{cc}& \delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n),\hfill \\ & {\displaystyle \frac{d[1+\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})]}{1+\delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)}},\hfill \\ & {\displaystyle \frac{\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})(1+d)}{1+\delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)}},\hfill \\ & {\displaystyle \frac{d\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})}{1+\delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)}}\hfill \end{array}\right\}.$$

By (8) and (11),

$$\delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)\to 0,\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\to 0.$$

Therefore,

$$\underset{n\to \infty }{lim}{\mathcal{RT}}_1({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)=d.$$

(13)

Additionally, since every suprametric is continuous and ${\mathfrak{p}}_{n+1}\to {\mathfrak{p}}^{\ast }$, we have

$$\underset{n\to \infty }{lim}\delta (\mathcal{T}{\mathfrak{p}}^{\ast },{\mathfrak{p}}_{n+1})=\delta (\mathcal{T}{\mathfrak{p}}^{\ast },{\mathfrak{p}}^{\ast })=d.$$

(14)

Passing to the limit in (12), and using the continuity of $\theta$ together with (13) and (14), we obtain

$$\theta \left(d\right)\le {\left(\theta \left(d\right)\right)}^{{\kappa }^{\prime }}.$$

This is impossible, since $d>0$, $\theta \left(d\right)>1$, and $0<{\kappa }^{\prime }<1$. Hence,

$$\delta ({\mathfrak{p}}^{\ast },\mathcal{T}{\mathfrak{p}}^{\ast })=0.$$

By the identity property of $\delta$, we conclude that

$$\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}^{\ast }.$$

Thus, ${\mathfrak{p}}^{\ast }$ is an $FP$ of $\mathcal{T}$.

Uniqueness still needs to be proven. Suppose that $\lambda \in \mathcal{X}$ is another $FP$ of $\mathcal{T}$ with $\lambda \ne {\mathfrak{p}}^{\ast }$. Then

$$\mathcal{T}\lambda =\lambda ,\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}^{\ast },$$

and therefore $\mathcal{T}\lambda \ne \mathcal{T}{\mathfrak{p}}^{\ast }$. By (5),

$$\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)=\theta \left(\delta (\mathcal{T}{\mathfrak{p}}^{\ast },\mathcal{T}\lambda )\right)\le {\left(\theta \left({\mathcal{RT}}_1({\mathfrak{p}}^{\ast },\lambda )\right)\right)}^{{\kappa }^{\prime }}.$$

Since both ${\mathfrak{p}}^{\ast }$ and $\lambda$ are $FP$s of $\mathcal{T}$, we have

$${\mathcal{RT}}_1({\mathfrak{p}}^{\ast },\lambda )=\delta ({\mathfrak{p}}^{\ast },\lambda ).$$

Consequently,

$$\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)\le {\left(\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)\right)}^{{\kappa }^{\prime }}.$$

This is impossible because $\delta ({\mathfrak{p}}^{\ast },\lambda )>0$, $\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)>1$, and $0<{\kappa }^{\prime }<1$. Hence $\lambda ={\mathfrak{p}}^{\ast }$, and the $FP$ is unique. □

Example 6.

Let $\mathcal{X}=\left\{0,1,2\right\}$ be a complete suprametric space with suprametric

$$\delta (\mathfrak{p},\nu )={|\mathfrak{p}-\nu |}^2,\rho =2.$$

Define the mapping $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ by

$$\mathcal{T}\left(\mathfrak{p}\right)=\left\{\begin{array}{cc}2,\hfill & \mathfrak{p}=0,\hfill \\ 1,\hfill & \mathfrak{p}=1,\hfill \\ 1,\hfill & \mathfrak{p}=2.\hfill \end{array}\right.$$

Then 1 is the only fixed point of $\mathcal{T}$.

We now verify that this self-mapping satisfies the assumptions of Theorem 1. The example is intended only to illustrate the theorem; the validity of the theorem itself follows from the proof given above.

Now, for the rational expression ${\mathcal{RT}}_1$, we compute

$${\mathcal{RT}}_1(0,1)=2,{\mathcal{RT}}_1(1,0)=2,$$

and

$${\mathcal{RT}}_1(0,2)=4,{\mathcal{RT}}_1(2,0)=4.$$

Moreover, for the pairs $(1,2)$ and $(2,1)$, we have

$$\mathcal{T}1=\mathcal{T}2=1.$$

Hence, the contractive condition is not required for these pairs.

Define $\theta :(0,\infty )\to (1,\infty )$ by

$$\theta \left(t\right)=e^{\sqrt{t}}.$$

Clearly, θ is non-decreasing, continuous on $(0,\infty )$, and $\theta \in \mathsf{\Theta }$.

The contractive condition only for the pairs

$$(0,1),(1,0),(0,2),(2,0)$$

still needs to be verified, because these are precisely the pairs for which

$$\mathcal{T}\mathfrak{p}\ne \mathcal{T}\nu .$$

For $(\mathfrak{p},\nu )=(0,1)$, we have

$$\delta (\mathcal{T}0,\mathcal{T}1)=\delta (2,1)=1$$

and

$${\mathcal{RT}}_1(0,1)=2.$$

Thus, for ${\kappa }^{\prime }={\displaystyle \frac{3}{4}}$,

$$\theta \left(\delta (\mathcal{T}0,\mathcal{T}1)\right)=e\le {\left(e^{\sqrt{2}}\right)}^{3/4}={\left(\theta \left({\mathcal{RT}}_1(0,1)\right)\right)}^{3/4}.$$

For $(\mathfrak{p},\nu )=(1,0)$, we have

$$\delta (\mathcal{T}1,\mathcal{T}0)=\delta (1,2)=1$$

and

$${\mathcal{RT}}_1(1,0)=2.$$

Thus,

$$\theta \left(\delta (\mathcal{T}1,\mathcal{T}0)\right)=e\le {\left(e^{\sqrt{2}}\right)}^{3/4}={\left(\theta \left({\mathcal{RT}}_1(1,0)\right)\right)}^{3/4}.$$

For $(\mathfrak{p},\nu )=(0,2)$, we have

$$\delta (\mathcal{T}0,\mathcal{T}2)=\delta (2,1)=1$$

and

$${\mathcal{RT}}_1(0,2)=4.$$

Therefore,

$$\theta \left(\delta (\mathcal{T}0,\mathcal{T}2)\right)=e\le {\left(e^2\right)}^{3/4}={\left(\theta \left({\mathcal{RT}}_1(0,2)\right)\right)}^{3/4}.$$

For $(\mathfrak{p},\nu )=(2,0)$, we have

$$\delta (\mathcal{T}2,\mathcal{T}0)=\delta (1,2)=1$$

and

$${\mathcal{RT}}_1(2,0)=4.$$

Therefore,

$$\theta \left(\delta (\mathcal{T}2,\mathcal{T}0)\right)=e\le {\left(e^2\right)}^{3/4}={\left(\theta \left({\mathcal{RT}}_1(2,0)\right)\right)}^{3/4}.$$

Hence, for ${\kappa }^{\prime }={\displaystyle \frac{3}{4}}\in (0,1)$, we have

$$\mathcal{T}\mathfrak{p}\ne \mathcal{T}\nu \Rightarrow \theta \left(\delta (\mathcal{T}\mathfrak{p},\mathcal{T}\nu )\right)\le {\left(\theta \left({\mathcal{RT}}_1(\mathfrak{p},\nu )\right)\right)}^{{\kappa }^{\prime }}.$$

Thus, all the conditions of Theorem 1 are satisfied. Therefore, 1 is the unique fixed point of the mapping $\mathcal{T}$.

This confirms that the example is a concrete illustration of the hypotheses and conclusion of Theorem 1.

Definition 9.

Let $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ be a self-mapping on $(\mathcal{X},\delta )$ such that there is a non-decreasing function $\theta :(0,\infty )\to (1,\infty )$ in Θ and ${\kappa }^{\prime }\in (0,1)$ satisfying

$$\mathcal{T}\mathfrak{p}\ne \mathcal{T}\nu \Rightarrow \theta \left(\delta (\mathcal{T}\mathfrak{p},\mathcal{T}\nu )\right)\le {\left(\theta \left({\mathcal{RT}}_2(\mathfrak{p},\nu )\right)\right)}^{{\kappa }^{\prime }},$$

(15)

where

$${\mathcal{RT}}_2(\mathfrak{p},\nu )=max\left\{\begin{array}{cc}& \delta (\mathfrak{p},\nu ),\hfill \\ & \delta (\mathfrak{p},\mathcal{T}\mathfrak{p}),\hfill \\ & \delta (\nu ,\mathcal{T}\nu ),\hfill \\ & {\displaystyle \frac{\delta (\mathfrak{p},\mathcal{T}\mathfrak{p})\delta (\nu ,\mathcal{T}\nu )}{1+\delta (\mathfrak{p},\nu )}}\hfill \end{array}\right\}.$$

Then $\mathcal{T}$ is called a rational-type contraction 2.

Theorem 2.

Let $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ be a rational-type contraction 2 on a complete suprametric space $(\mathcal{X},\delta )$. Assume, in addition, that θ is continuous on $(0,\infty )$. Then $\mathcal{T}$ has a unique $FP$ in $\mathcal{X}$.

Proof. 

Let ${\mathfrak{p}}_0\in \mathcal{X}$ be arbitrary and define a sequence $\left({\mathfrak{p}}_n\right)$ by

$${\mathfrak{p}}_n=\mathcal{T}{\mathfrak{p}}_{n-1},n\in \mathbb{N}.$$

If ${\mathfrak{p}}_n={\mathfrak{p}}_{n-1}$ for some n, then ${\mathfrak{p}}_{n-1}$ is an $FP$ of $\mathcal{T}$. Otherwise, assume that

$${\mathfrak{p}}_n\ne {\mathfrak{p}}_{n-1},n\in \mathbb{N}.$$

Set

$$a_n=\delta ({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n),b_n=\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}).$$

Then $a_n>0$ and $b_n>0$ for all $n\in \mathbb{N}$. Since

$${\mathfrak{p}}_n=\mathcal{T}{\mathfrak{p}}_{n-1},{\mathfrak{p}}_{n+1}=\mathcal{T}{\mathfrak{p}}_n,$$

the contractive condition (15) gives

$$\theta \left(b_n\right)=\theta \left(\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\right)\le {\left(\theta \left({\mathcal{RT}}_2({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)\right)\right)}^{{\kappa }^{\prime }}.$$

(16)

Moreover, by the definition of ${\mathcal{RT}}_2$,

$${\mathcal{RT}}_2({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=max\left\{a_n,a_n,b_n,{\displaystyle \frac{a_n{b}_n}{1+a_n}}\right\}.$$

We first prove that $b_n<a_n$ for every $n\in \mathbb{N}$. Suppose, on the contrary, that $b_n\ge a_n$ for some n. Since

$${\displaystyle \frac{a_n{b}_n}{1+a_n}}<b_n,$$

we have

$${\mathcal{RT}}_2({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=b_n.$$

Hence, from (16),

$$\theta \left(b_n\right)\le {\left(\theta \left(b_n\right)\right)}^{{\kappa }^{\prime }}.$$

This is impossible, because $b_n>0$, $\theta \left(b_n\right)>1$, and $0<{\kappa }^{\prime }<1$. Therefore,

$$b_n<a_n,n\in \mathbb{N}.$$

Consequently,

$${\displaystyle \frac{a_n{b}_n}{1+a_n}}<b_n<a_n,$$

and therefore

$${\mathcal{RT}}_2({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=a_n.$$

It follows from (16) that

$$\theta \left(a_{n+1}\right)=\theta \left(b_n\right)\le {\left(\theta \left(a_n\right)\right)}^{{\kappa }^{\prime }},n\in \mathbb{N}.$$

(17)

Iterating (17), we obtain

$$1<\theta \left(a_{n+1}\right)\le {\left(\theta \left(a_1\right)\right)}^{{\left({\kappa }^{\prime }\right)}^n}.$$

Letting $n\to \infty$, we obtain

$$\underset{n\to \infty }{lim}\theta \left(a_n\right)=1.$$

By property $\left(\theta 2\right)$ of the class $\mathsf{\Theta }$, we obtain

$$\underset{n\to \infty }{lim}{a}_n=\underset{n\to \infty }{lim}\delta ({\mathfrak{p}}_{n-1},{\mathfrak{p}}_n)=0.$$

(18)

We now show that $\left({\mathfrak{p}}_n\right)$ is a Cauchy sequence. Suppose, on the contrary, that $\left({\mathfrak{p}}_n\right)$ is not Cauchy. Then there is an $\epsilon >0$ and a strictly increasing sequence of positive integers $\left(n_i\right)$ such that, for each i, the index

$$m_i=min\left\{m>n_i:\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_m)\ge \epsilon \right\}$$

is well defined. By minimality of $m_i$, we have

$$\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})\ge \epsilon \mathrm{and}\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})<\epsilon .$$

Let $\rho >0$ be the constant appearing in the suprametric inequality. Then

$$\begin{array}{cc}\hfill \epsilon \le \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})& \le \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})+\delta ({\mathfrak{p}}_{m_i-1},{\mathfrak{p}}_{m_i})\hfill \\ & +\rho \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})\delta ({\mathfrak{p}}_{m_i-1},{\mathfrak{p}}_{m_i})\hfill \\ & <\epsilon +a_{m_i}+\rho \epsilon a_{m_i}.\hfill \end{array}$$

Using (18), we obtain

$$\underset{i\to \infty }{lim}\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})=\epsilon .$$

(19)

Let

$$q_i=\delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1}).$$

Again, by the suprametric inequality,

$$\begin{array}{cc}\hfill q_i& \le \delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{n_i})+\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})\hfill \\ & +\rho \delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{n_i})\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i-1})\hfill \\ & <a_{n_i}+\epsilon +\rho \epsilon a_{n_i}.\hfill \end{array}$$

Thus,

$$\underset{i\to \infty }{lim\; sup}{q}_i\le \epsilon .$$

On the other hand, applying the suprametric inequality twice, we obtain

$$\begin{array}{cc}\hfill \delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})& \le a_{n_i}+q_i+a_{m_i}+\rho q_i{a}_{m_i}\hfill \\ & +\rho a_{n_i}\left(q_i+a_{m_i}+\rho q_i{a}_{m_i}\right).\hfill \end{array}$$

Since $\left(q_i\right)$ is bounded by the preceding estimate and $a_{n_i},a_{m_i}\to 0$, (19) gives

$$\epsilon \le \underset{i\to \infty }{lim\; inf}{q}_i.$$

Consequently,

$$\underset{i\to \infty }{lim}\delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1})=\underset{i\to \infty }{lim}{q}_i=\epsilon .$$

(20)

Now,

$${\mathcal{RT}}_2({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1})=max\left\{q_i,a_{n_i},a_{m_i},{\displaystyle \frac{a_{n_i}{a}_{m_i}}{1+q_i}}\right\}.$$

By (18) and (20),

$$q_i\to \epsilon >0,$$

whereas

$$a_{n_i}\to 0,a_{m_i}\to 0,{\displaystyle \frac{a_{n_i}{a}_{m_i}}{1+q_i}}\to 0.$$

Hence, there is an $i_0\in \mathbb{N}$ such that, for all $i\ge i_0$,

$${\mathcal{RT}}_2({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1})=q_i=\delta ({\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i-1}).$$

Thus, the equality above follows from the estimates; it is not assumed.

Since

$${\mathfrak{p}}_{n_i}=\mathcal{T}{\mathfrak{p}}_{n_i-1},{\mathfrak{p}}_{m_i}=\mathcal{T}{\mathfrak{p}}_{m_i-1},$$

and $\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})\ge \epsilon >0$, we may apply (15). Therefore, for all $i\ge i_0$,

$$\theta \left(\delta ({\mathfrak{p}}_{n_i},{\mathfrak{p}}_{m_i})\right)\le {\left(\theta \left(q_i\right)\right)}^{{\kappa }^{\prime }}.$$

Letting $i\to \infty$ and using the continuity of $\theta$, together with (19) and (20), we obtain

$$\theta \left(\epsilon \right)\le {\left(\theta \left(\epsilon \right)\right)}^{{\kappa }^{\prime }}.$$

This is a contradiction, because $\theta \left(\epsilon \right)>1$ and $0<{\kappa }^{\prime }<1$. Hence, $\left({\mathfrak{p}}_n\right)$ is a Cauchy sequence.

Since $(\mathcal{X},\delta )$ is complete, there is a ${\mathfrak{p}}^{\ast }\in \mathcal{X}$ such that

$$\underset{n\to \infty }{lim}{\mathfrak{p}}_n={\mathfrak{p}}^{\ast }.$$

(21)

We now prove that ${\mathfrak{p}}^{\ast }$ is an $FP$ of $\mathcal{T}$. Suppose, to the contrary, that

$$\mathcal{T}{\mathfrak{p}}^{\ast }\ne {\mathfrak{p}}^{\ast }.$$

Let

$$d=\delta ({\mathfrak{p}}^{\ast },\mathcal{T}{\mathfrak{p}}^{\ast })>0.$$

If $\mathcal{T}{\mathfrak{p}}^{\ast }=\mathcal{T}{\mathfrak{p}}_n$ for infinitely many n, then, along a subsequence,

$$\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}_{n+1}\to {\mathfrak{p}}^{\ast },$$

which implies $\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}^{\ast }$. This contradicts $d>0$. Therefore, for all sufficiently large n,

$$\mathcal{T}{\mathfrak{p}}^{\ast }\ne \mathcal{T}{\mathfrak{p}}_n.$$

For such n, by (15),

$$\theta \left(\delta (\mathcal{T}{\mathfrak{p}}^{\ast },{\mathfrak{p}}_{n+1})\right)=\theta \left(\delta (\mathcal{T}{\mathfrak{p}}^{\ast },\mathcal{T}{\mathfrak{p}}_n)\right)\le {\left(\theta \left({\mathcal{RT}}_2({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)\right)\right)}^{{\kappa }^{\prime }}.$$

(22)

By the definition of ${\mathcal{RT}}_2$, we have

$${\mathcal{RT}}_2({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)=max\left\{\begin{array}{cc}& \delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n),\hfill \\ & d,\hfill \\ & \delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}),\hfill \\ & {\displaystyle \frac{d\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})}{1+\delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)}}\hfill \end{array}\right\}.$$

No averaged term occurs in this expression; it is obtained directly from the stated definition of ${\mathcal{RT}}_2$.

By (18) and (21),

$$\delta ({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)\to 0,\delta ({\mathfrak{p}}_n,{\mathfrak{p}}_{n+1})\to 0.$$

Therefore,

$$\underset{n\to \infty }{lim}{\mathcal{RT}}_2({\mathfrak{p}}^{\ast },{\mathfrak{p}}_n)=d.$$

(23)

Additionally, since every suprametric is continuous and ${\mathfrak{p}}_{n+1}\to {\mathfrak{p}}^{\ast }$, we have

$$\underset{n\to \infty }{lim}\delta (\mathcal{T}{\mathfrak{p}}^{\ast },{\mathfrak{p}}_{n+1})=\delta (\mathcal{T}{\mathfrak{p}}^{\ast },{\mathfrak{p}}^{\ast })=d.$$

(24)

Passing to the limit in (22), and using the continuity of $\theta$ together with (23) and (24), we obtain

$$\theta \left(d\right)\le {\left(\theta \left(d\right)\right)}^{{\kappa }^{\prime }}.$$

This is impossible, since $d>0$, $\theta \left(d\right)>1$, and $0<{\kappa }^{\prime }<1$. Hence,

$$\delta ({\mathfrak{p}}^{\ast },\mathcal{T}{\mathfrak{p}}^{\ast })=0.$$

By the identity property of $\delta$, we conclude that

$$\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}^{\ast }.$$

Thus, ${\mathfrak{p}}^{\ast }$ is an $FP$ of $\mathcal{T}$.

Uniqueness still needs to be proved. Suppose that $\lambda \in \mathcal{X}$ is another $FP$ of $\mathcal{T}$ and $\lambda \ne {\mathfrak{p}}^{\ast }$. Then

$$\mathcal{T}\lambda =\lambda ,\mathcal{T}{\mathfrak{p}}^{\ast }={\mathfrak{p}}^{\ast }.$$

Therefore, $\mathcal{T}\lambda \ne \mathcal{T}{\mathfrak{p}}^{\ast }$. By (15),

$$\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)=\theta \left(\delta (\mathcal{T}{\mathfrak{p}}^{\ast },\mathcal{T}\lambda )\right)\le {\left(\theta \left({\mathcal{RT}}_2({\mathfrak{p}}^{\ast },\lambda )\right)\right)}^{{\kappa }^{\prime }}.$$

Since both ${\mathfrak{p}}^{\ast }$ and $\lambda$ are $FP$s of $\mathcal{T}$, we have

$${\mathcal{RT}}_2({\mathfrak{p}}^{\ast },\lambda )=\delta ({\mathfrak{p}}^{\ast },\lambda ).$$

Thus,

$$\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)\le {\left(\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)\right)}^{{\kappa }^{\prime }}.$$

This is impossible because $\delta ({\mathfrak{p}}^{\ast },\lambda )>0$, $\theta \left(\delta ({\mathfrak{p}}^{\ast },\lambda )\right)>1$, and $0<{\kappa }^{\prime }<1$. Hence, $\lambda ={\mathfrak{p}}^{\ast }$, and the $FP$ is unique. □

Example 7.

Let $\mathcal{X}=\left\{0,1,2\right\}$ with suprametric

$$\delta (\mathfrak{p},\nu )={|\mathfrak{p}-\nu |}^3,\rho =6,$$

be a complete suprametric space.

Define $\mathcal{T}:\mathcal{X}\to \mathcal{X}$ by

$$\mathcal{T}\mathfrak{p}=\left\{\begin{array}{cc}1,\hfill & \mathfrak{p}=0,\hfill \\ 1,\hfill & \mathfrak{p}=1,\hfill \\ 0,\hfill & \mathfrak{p}=2.\hfill \end{array}\right.$$

Then 1 is the only fixed point of $\mathcal{T}$.

We now verify that this self-mapping satisfies the assumptions of Theorem 2. This example illustrates the theorem by checking its hypotheses in a concrete finite suprametric space.

For the rational expression ${\mathcal{RT}}_2$, we compute

$${\mathcal{RT}}_2(0,2)=8,{\mathcal{RT}}_2(2,0)=8,$$

and

$${\mathcal{RT}}_2(1,2)=8,{\mathcal{RT}}_2(2,1)=8.$$

Moreover, for the pairs $(0,1)$ and $(1,0)$, we have

$$\mathcal{T}0=\mathcal{T}1=1.$$

Define $\theta :(0,\infty )\to (1,\infty )$ by

$$\theta \left(t\right)=e^{\sqrt{t}}.$$

Clearly, θ is non-decreasing, continuous on $(0,\infty )$, and $\theta \in \mathsf{\Theta }$.

The contractive condition only for the pairs

$$(0,2),(2,0),(1,2),(2,1)$$

still needs to be verified, because these are precisely the pairs for which

$$\mathcal{T}\mathfrak{p}\ne \mathcal{T}\nu .$$

For $(\mathfrak{p},\nu )=(0,2)$, we have

$$\delta (\mathcal{T}0,\mathcal{T}2)=\delta (1,0)=1$$

and

$${\mathcal{RT}}_2(0,2)=8.$$

Thus, for ${\kappa }^{\prime }={\displaystyle \frac{1}{2}}$,

$$\theta \left(\delta (\mathcal{T}0,\mathcal{T}2)\right)=e\le {\left(e^{\sqrt{8}}\right)}^{1/2}={\left(\theta \left({\mathcal{RT}}_2(0,2)\right)\right)}^{1/2}.$$

For $(\mathfrak{p},\nu )=(2,0)$, we have

$$\delta (\mathcal{T}2,\mathcal{T}0)=\delta (0,1)=1$$

and

$${\mathcal{RT}}_2(2,0)=8.$$

Thus,

$$\theta \left(\delta (\mathcal{T}2,\mathcal{T}0)\right)=e\le {\left(e^{\sqrt{8}}\right)}^{1/2}={\left(\theta \left({\mathcal{RT}}_2(2,0)\right)\right)}^{1/2}.$$

For $(\mathfrak{p},\nu )=(1,2)$, we have

$$\delta (\mathcal{T}1,\mathcal{T}2)=\delta (1,0)=1$$

and

$${\mathcal{RT}}_2(1,2)=8.$$

Thus,

$$\theta \left(\delta (\mathcal{T}1,\mathcal{T}2)\right)=e\le {\left(e^{\sqrt{8}}\right)}^{1/2}={\left(\theta \left({\mathcal{RT}}_2(1,2)\right)\right)}^{1/2}.$$

For $(\mathfrak{p},\nu )=(2,1)$, we have

$$\delta (\mathcal{T}2,\mathcal{T}1)=\delta (0,1)=1$$

and

$${\mathcal{RT}}_2(2,1)=8.$$

Thus,

$$\theta \left(\delta (\mathcal{T}2,\mathcal{T}1)\right)=e\le {\left(e^{\sqrt{8}}\right)}^{1/2}={\left(\theta \left({\mathcal{RT}}_2(2,1)\right)\right)}^{1/2}.$$

Hence, for ${\kappa }^{\prime }={\displaystyle \frac{1}{2}}\in (0,1)$, we have

$$\mathcal{T}\mathfrak{p}\ne \mathcal{T}\nu \Rightarrow \theta \left(\delta (\mathcal{T}\mathfrak{p},\mathcal{T}\nu )\right)\le {\left(\theta \left({\mathcal{RT}}_2(\mathfrak{p},\nu )\right)\right)}^{{\kappa }^{\prime }}.$$

Thus, all the conditions of Theorem 2 are satisfied. Therefore, 1 is the unique fixed point of $\mathcal{T}$.

This confirms that the example is a concrete illustration of the hypotheses and conclusion of Theorem 2.

4. Applications

4.1. Fractional Chaotic Financial Model

We now apply the fixed point results to a fractional chaotic financial model formulated in the Caputo sense. The goal of this section is to establish the local existence and uniqueness of solutions to the fractional system (27), not to analyze its chaotic dynamics. The term fractional chaotic financial model is the standard name for System (25) in the literature [21,22], reflecting that the classical integer-order system is known to exhibit chaotic orbits for certain parameter ranges; the present analysis addresses only the well-posedness of the fractional counterpart.

The interest rate $\mathfrak{p}$, investment demand $\nu$, and price index $\xi$ are the three state variables included in the model. Changes in $\mathfrak{p}$ are mainly influenced by structural changes in product prices and imbalances in the investment market. Similarly, the interest rate and investment cost affect the rate of change of $\nu$. The price index $\xi$ is influenced by commercial supply and demand, together with the inflation rate. Assuming that commercial supply and demand are proportional to prices and remain constant over time, changes in the difference between nominal and real interest rates describe variations in the inflation rate. The corresponding financial chaotic model is given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\mathfrak{p}}{dt}}& =\xi +\mathfrak{p}\nu -A\mathfrak{p},\hfill \\ \hfill {\displaystyle \frac{d\nu }{dt}}& =1-B\nu -{\mathfrak{p}}^2,\hfill \\ \hfill {\displaystyle \frac{d\xi }{dt}}& =-\mathfrak{p}-C\xi ,\hfill \end{array}$$

(25)

subject to the initial conditions

$$\mathfrak{p}\left(0\right)={\mathfrak{p}}_0,\nu \left(0\right)={\nu }_0,\xi \left(0\right)={\xi }_0.$$

(26)

The parameter A represents the savings rate, B represents the cost per investment, and C represents the elasticity of demand. Since chaotic systems are sensitive to initial conditions, fractional derivatives are introduced to include memory effects and hereditary properties. Replacing the classical derivatives in System (25) with Caputo fractional derivatives, we obtain

$$\begin{array}{cc}\hfill {}^{\mathrm{C}}D_t^{\alpha }\mathfrak{p}\left(t\right)& =\xi \left(t\right)+\mathfrak{p}\left(t\right)\nu \left(t\right)-A\mathfrak{p}\left(t\right),\hfill \\ \hfill {}^{\mathrm{C}}D_t^{\alpha }\nu \left(t\right)& =1-B\nu \left(t\right)-{\mathfrak{p}}^2\left(t\right),\hfill \\ \hfill {}^{\mathrm{C}}D_t^{\alpha }\xi \left(t\right)& =-\mathfrak{p}\left(t\right)-C\xi \left(t\right),\hfill \end{array}$$

(27)

where ${}^{\mathrm{C}}D_t^{\alpha }$ denotes the Caputo fractional derivative of order $\alpha \in (0,1)$ and $t\in I=[0,t_{max}]$, $t_{max}>0$. The corresponding initial conditions are

$$\mathfrak{p}\left(0\right)={\mathfrak{p}}_0>0,\nu \left(0\right)={\nu }_0>0,\xi \left(0\right)={\xi }_0>0.$$

(28)

The use of fractional derivatives allows the model to incorporate memory effects, providing a more flexible description of the evolution of the financial variables $\mathfrak{p}$, $\nu$, and $\xi$.

We now formulate the problem in a vector-valued functional space. Let

$$\mathcal{W}=X^3,X=C(I,\mathbb{R}),$$

equipped with the norm

$${\parallel \mathcal{H}\parallel }_{\mathcal{W}}=\underset{t\in I}{max}\left\{\left|\mathfrak{p}\right(t\left)\right|+\left|\nu \right(t\left)\right|+\left|\xi \right(t\left)\right|\right\},$$

where

$$\mathcal{H}\left(t\right)={\{\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right)\}}^T.$$

For brevity, denote the right-hand sides of (27) by ${\mathsf{\Pi }}_1,{\mathsf{\Pi }}_2,{\mathsf{\Pi }}_3$ as follows:

$$\begin{array}{cc}\hfill {\mathsf{\Pi }}_1(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right))& =\xi \left(t\right)+\mathfrak{p}\left(t\right)\nu \left(t\right)-A\mathfrak{p}\left(t\right),\hfill \\ \hfill {\mathsf{\Pi }}_2(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right))& =1-B\nu \left(t\right)-{\mathfrak{p}}^2\left(t\right),\hfill \\ \hfill {\mathsf{\Pi }}_3(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right))& =-\mathfrak{p}\left(t\right)-C\xi \left(t\right).\hfill \end{array}$$

(29)

Hence,

$$\begin{array}{cc}\hfill {}^{\mathrm{C}}D_t^{\alpha }\mathfrak{p}\left(t\right)& ={\mathsf{\Pi }}_1(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right)),\hfill \\ \hfill {}^{\mathrm{C}}D_t^{\alpha }\nu \left(t\right)& ={\mathsf{\Pi }}_2(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right)),\hfill \\ \hfill {}^{\mathrm{C}}D_t^{\alpha }\xi \left(t\right)& ={\mathsf{\Pi }}_3(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right)).\hfill \end{array}$$

(30)

In vector form, (30) becomes

$$\left\{\begin{array}{c}{}^{\mathrm{C}}D_t^{\alpha }\mathcal{H}\left(t\right)=\mathsf{\Pi }(t,\mathcal{H}\left(t\right)),\hfill \\ \mathcal{H}\left(0\right)={\mathcal{H}}_0,\hfill \end{array}\right.$$

(31)

where

$${\mathcal{H}}_0={\{{\mathfrak{p}}_0,{\nu }_0,{\xi }_0\}}^T$$

and

$$\mathsf{\Pi }(t,\mathcal{H}\left(t\right))={\{{\mathsf{\Pi }}_1(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right)),{\mathsf{\Pi }}_2(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right)),{\mathsf{\Pi }}_3(t,\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right))\}}^T.$$

The equivalent integral form of (31) is

$$\mathcal{H}\left(t\right)={\mathcal{H}}_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\mathsf{\Pi }(s,\mathcal{H}\left(s\right))ds.$$

Equivalently,

$$\begin{array}{cc}\hfill \mathfrak{p}\left(t\right)& ={\mathfrak{p}}_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{\mathsf{\Pi }}_1(s,\mathfrak{p}\left(s\right),\nu \left(s\right),\xi \left(s\right))ds,\hfill \\ \hfill \nu \left(t\right)& ={\nu }_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{\mathsf{\Pi }}_2(s,\mathfrak{p}\left(s\right),\nu \left(s\right),\xi \left(s\right))ds,\hfill \\ \hfill \xi \left(t\right)& ={\xi }_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{\mathsf{\Pi }}_3(s,\mathfrak{p}\left(s\right),\nu \left(s\right),\xi \left(s\right))ds.\hfill \end{array}$$

(32)

The Picard sequence corresponding to (32) is therefore given by

$$\begin{array}{cc}\hfill {\mathfrak{p}}_n\left(t\right)& ={\mathfrak{p}}_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{\mathsf{\Pi }}_1(s,{\mathfrak{p}}_{n-1}\left(s\right),{\nu }_{n-1}\left(s\right),{\xi }_{n-1}\left(s\right))ds,\hfill \\ \hfill {\nu }_n\left(t\right)& ={\nu }_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{\mathsf{\Pi }}_2(s,{\mathfrak{p}}_{n-1}\left(s\right),{\nu }_{n-1}\left(s\right),{\xi }_{n-1}\left(s\right))ds,\hfill \\ \hfill {\xi }_n\left(t\right)& ={\xi }_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{\mathsf{\Pi }}_3(s,{\mathfrak{p}}_{n-1}\left(s\right),{\nu }_{n-1}\left(s\right),{\xi }_{n-1}\left(s\right))ds.\hfill \end{array}$$

(33)

By Example 1, the space $\mathcal{W}=X^3$ becomes a suprametric space with

$$\delta (\mathcal{H},\mathcal{K})={\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}}\left({\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}}+\beta \right),\beta \in (0,1),$$

where $\rho =2/\beta$.

Since the nonlinear terms $\mathfrak{p}\nu$ and ${\mathfrak{p}}^2$ are not globally Lipschitz on $\mathcal{W}$, we work on a closed bounded subset of $\mathcal{W}$. Fix $R>0$ and define

$${\overline{B}}_R=\left\{\mathcal{H}\in \mathcal{W}:\parallel \mathcal{H}-{\mathcal{H}}_0{\parallel }_{\mathcal{W}}\le R\right\}.$$

This set is closed in $\mathcal{W}$, so it is complete with respect to the induced suprametric. Let

$$M_R={\parallel {\mathcal{H}}_0\parallel }_{\mathcal{W}}+R.$$

Then, for every $\mathcal{H}\in {\overline{B}}_R$ and every $t\in I$,

$$\left|\mathfrak{p}\left(t\right)\right|\le M_R,\left|\nu \left(t\right)\right|\le M_R,\left|\xi \left(t\right)\right|\le M_R.$$

For $\mathcal{H},\mathcal{K}\in {\overline{B}}_R$, where

$$\mathcal{H}\left(t\right)={\{\mathfrak{p}\left(t\right),\nu \left(t\right),\xi \left(t\right)\}}^T,\mathcal{K}\left(t\right)={\{\overline{\mathfrak{p}}\left(t\right),\overline{\nu }\left(t\right),\overline{\xi }\left(t\right)\}}^T,$$

we have

$$\begin{array}{cc}\hfill |{\mathsf{\Pi }}_1(t,\mathcal{H}\left(t\right))-{\mathsf{\Pi }}_1(t,\mathcal{K}\left(t\right))|& \le (A+M_R)|\mathfrak{p}\left(t\right)-\overline{\mathfrak{p}}\left(t\right)|+M_R|\nu \left(t\right)-\overline{\nu }\left(t\right)|\hfill \\ & +|\xi \left(t\right)-\overline{\xi }\left(t\right)|,\hfill \\ \hfill |{\mathsf{\Pi }}_2(t,\mathcal{H}\left(t\right))-{\mathsf{\Pi }}_2(t,\mathcal{K}\left(t\right))|& \le 2M_R|\mathfrak{p}\left(t\right)-\overline{\mathfrak{p}}\left(t\right)|+B|\nu \left(t\right)-\overline{\nu }\left(t\right)|,\hfill \\ \hfill |{\mathsf{\Pi }}_3(t,\mathcal{H}\left(t\right))-{\mathsf{\Pi }}_3(t,\mathcal{K}\left(t\right))|& \le |\mathfrak{p}\left(t\right)-\overline{\mathfrak{p}}\left(t\right)|+C|\xi \left(t\right)-\overline{\xi }\left(t\right)|.\hfill \end{array}$$

Consequently,

$$\parallel \mathsf{\Pi }(t,\mathcal{H}\left(t\right))-\mathsf{\Pi }(t,\mathcal{K}\left(t\right))\parallel \le L_R{\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}},$$

where one may take

$$L_R=max\left\{A+3M_R+1,B+M_R,C+1\right\}.$$

Thus, the vector field is Lipschitz on the bounded set ${\overline{B}}_R$ with the explicit local Lipschitz constant $L_R$.

Define the operator $\mathcal{T}:{\overline{B}}_R\to \mathcal{W}$ by

$$\left(\mathcal{T}\mathcal{H}\right)\left(t\right)={\mathcal{H}}_0+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}\mathsf{\Pi }(s,\mathcal{H}\left(s\right))ds.$$

(34)

We assume that R and $t_{max}$ are chosen so that

$${\displaystyle \frac{t_{max}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}{P}_R\le R,$$

(35)

where

$$P_R=1+2M_R^2+(A+B+C+2)M_R.$$

Indeed, for $\mathcal{H}\in {\overline{B}}_R$,

$$\parallel \mathsf{\Pi }(t,\mathcal{H}\left(t\right))\parallel \le P_R,t\in I,$$

and therefore

$$\parallel \mathcal{T}\mathcal{H}-{\mathcal{H}}_0{\parallel }_{\mathcal{W}}\le {\displaystyle \frac{t_{max}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}{P}_R\le R.$$

Thus, $\mathcal{T}\left({\overline{B}}_R\right)\subseteq {\overline{B}}_R$.

Next, assume that

$$q_R={\displaystyle \frac{L_R{t}_{max}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}<1.$$

(36)

Then, for all $\mathcal{H},\mathcal{K}\in {\overline{B}}_R$,

$$\begin{array}{cc}\hfill {\parallel \mathcal{T}\mathcal{H}-\mathcal{T}\mathcal{K}\parallel }_{\mathcal{W}}& \le {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}}\underset{t\in I}{max}{\int }_0^t{(t-s)}^{\alpha -1}\parallel \mathsf{\Pi }(s,\mathcal{H}\left(s\right))-\mathsf{\Pi }(s,\mathcal{K}\left(s\right))\parallel ds\hfill \\ & \le {\displaystyle \frac{L_R}{\mathsf{\Gamma }\left(\alpha \right)}}\underset{t\in I}{max}{\int }_0^t{(t-s)}^{\alpha -1}{\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}}ds\hfill \\ & \le {\displaystyle \frac{L_R{t}_{max}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}}{\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}}\hfill \\ & =q_R{\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}}.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill \delta (\mathcal{T}\mathcal{H},\mathcal{T}\mathcal{K})& ={\parallel \mathcal{T}\mathcal{H}-\mathcal{T}\mathcal{K}\parallel }_{\mathcal{W}}\left({\parallel \mathcal{T}\mathcal{H}-\mathcal{T}\mathcal{K}\parallel }_{\mathcal{W}}+\beta \right)\hfill \\ & \le q_R{\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}}\left({\parallel \mathcal{H}-\mathcal{K}\parallel }_{\mathcal{W}}+\beta \right)\hfill \\ & =q_R\delta (\mathcal{H},\mathcal{K}).\hfill \end{array}$$

Let

$$\theta \left(t\right)=e^{\sqrt{t}},t>0.$$

Then $\theta \in \mathsf{\Theta }$ and $\theta$ is continuous on $(0,\infty )$. Moreover,

$$\theta \left(\delta (\mathcal{T}\mathcal{H},\mathcal{T}\mathcal{K})\right)\le e^{\sqrt{q_R\delta (\mathcal{H},\mathcal{K})}}={\left(\theta \left(\delta \right(\mathcal{H},\mathcal{K}\left)\right)\right)}^{\sqrt{q_R}}.$$

Since

$$\delta (\mathcal{H},\mathcal{K})\le {\mathcal{RT}}_1(\mathcal{H},\mathcal{K}),$$

we obtain

$$\theta \left(\delta (\mathcal{T}\mathcal{H},\mathcal{T}\mathcal{K})\right)\le {\left(\theta \left({\mathcal{RT}}_1(\mathcal{H},\mathcal{K})\right)\right)}^{\sqrt{q_R}},$$

where $\sqrt{q_R}\in (0,1)$.

Thus, $\mathcal{T}$ satisfies the rational-type contraction condition of Theorem 1 on the complete suprametric space ${\overline{B}}_R$. Therefore, $\mathcal{T}$ has a unique fixed point in ${\overline{B}}_R$. Consequently, the fractional chaotic financial system (27) has a unique local solution in the closed bounded set ${\overline{B}}_R$.

4.2. Existence of Solutions for a Nonlinear Fractional Differential Equation

In this section, we establish an existence and uniqueness result for a nonlinear fractional differential equation.

Let $\mathcal{X}=C\left(\right[0,1],\mathbb{R})$ be the Banach space of all continuous functions on $[0,1].$ Let $(\mathcal{X},\delta )$ be the required suprametric space as defined in Example 1, where

$$\delta (x,y)=\parallel x-y\parallel \left(\parallel x-y\parallel +\beta \right),x,y\in \mathcal{X},\beta \in (0,1).$$

Note that $(\mathcal{X},\delta )$ is a suprametric space with $\rho ={\displaystyle \frac{2}{\beta }}.$

Consider the nonlinear fractional differential equation

$${}^{\mathrm{C}}D^{\varsigma }u\left(x\right)=f(x,u\left(x\right)),x\in [0,1],$$

(37)

subject to the integral boundary conditions

$$u\left(0\right)=0,u\left(1\right)={\int }_0^{\eta }u\left(\tau \right)d\tau ,$$

(38)

where $\eta \in (0,1)$, $\varsigma \in (1,2]$, and $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous.

It is well known that $u\in \mathcal{X}$ is a solution of (37) if and only if it satisfies the equivalent integral equation

$$\begin{array}{cc}\hfill u\left(x\right)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^x{(x-\tau )}^{\varsigma -1}f(\tau ,u\left(\tau \right))d\tau \hfill \\ \hfill & -{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^1{(1-\tau )}^{\varsigma -1}f(\tau ,u\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^{\eta }\left({\int }_0^{\tau }{(\tau -r)}^{\varsigma -1}f(r,u\left(r\right))dr\right)d\tau .\hfill \end{array}$$

(39)

Define the operator $\mathcal{K}:\mathcal{X}\to \mathcal{X}$ by

$$\begin{array}{cc}\hfill \left(\mathcal{K}u\right)\left(x\right)& ={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^x{(x-\tau )}^{\varsigma -1}f(\tau ,u\left(\tau \right))d\tau \hfill \\ \hfill & -{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^1{(1-\tau )}^{\varsigma -1}f(\tau ,u\left(\tau \right))d\tau \hfill \\ \hfill & +{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^{\eta }\left({\int }_0^{\tau }{(\tau -r)}^{\varsigma -1}f(r,u\left(r\right))dr\right)d\tau .\hfill \end{array}$$

(40)

Theorem 3.

The nonlinear fractional differential Equation (37) admits a unique solution in $\mathcal{X}$ if there is a constant $L>0$ such that

$$\left|f\right(x,u)-f(x,v\left)\right|\le L|u-v|,x\in [0,1],u,v\in \mathbb{R},$$

(41)

where

$$L\le {\displaystyle \frac{\mathsf{\Gamma }(\varsigma +1)}{4M_{\eta ,\varsigma }}},$$

and

$$M_{\eta ,\varsigma }=\underset{x\in [0,1]}{sup}\left(x^{\varsigma }+{\displaystyle \frac{2x}{2-{\eta }^2}}+{\displaystyle \frac{2x{\eta }^{\varsigma +1}}{(2-{\eta }^2)(\varsigma +1)}}\right).$$

Proof. 

Let $u,v\in \mathcal{X}$. For $x\in [0,1]$, using the definition of the operator $\mathcal{K}$, we have

$$\begin{array}{cc}\hfill \left|\right(\mathcal{K}u\left)\right(x)-(\mathcal{K}v\left)\right(x\left)\right|& =|{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^x{(x-\tau )}^{\varsigma -1}\left[f(\tau ,u(\tau \left)\right)-f(\tau ,v(\tau \left)\right)\right]d\tau \hfill \\ \hfill & -{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^1{(1-\tau )}^{\varsigma -1}\hfill \\ \hfill & \left[f(\tau ,u(\tau \left)\right)-f(\tau ,v(\tau \left)\right)\right]d\tau \hfill \\ \hfill & +{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^{\eta }{\int }_0^{\tau }{(\tau -r)}^{\varsigma -1}\hfill \\ \hfill & \left[f(r,u(r\left)\right)-f(r,v(r\left)\right)\right]drd\tau |.\hfill \end{array}$$

(42)

Applying the triangle inequality, we obtain

$$\begin{array}{cc}\hfill \left|\right(\mathcal{K}u\left)\right(x)-(\mathcal{K}v\left)\right(x\left)\right|& \le {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^x{(x-\tau )}^{\varsigma -1}|f(\tau ,u\left(\tau \right))-f(\tau ,v\left(\tau \right))|d\tau \hfill \\ \hfill & +{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^1{(1-\tau )}^{\varsigma -1}\hfill \\ \hfill & \left|f\right(\tau ,u\left(\tau \right))-f(\tau ,v\left(\tau \right)\left)\right|d\tau \hfill \\ \hfill & +{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^{\eta }{\int }_0^{\tau }{(\tau -r)}^{\varsigma -1}\hfill \\ \hfill & \left|f\right(r,u\left(r\right))-f(r,v\left(r\right)\left)\right|drd\tau .\hfill \end{array}$$

(43)

Using Condition (41), we obtain

$$\begin{array}{cc}\hfill \left|\right(\mathcal{K}u\left)\right(x)-(\mathcal{K}v\left)\right(x\left)\right|& \le {\displaystyle \frac{L}{\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^x{(x-\tau )}^{\varsigma -1}|u\left(\tau \right)-v\left(\tau \right)|d\tau \hfill \\ \hfill & +{\displaystyle \frac{2xL}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^1{(1-\tau )}^{\varsigma -1}\hfill \\ \hfill & \left|u\right(\tau )-v(\tau \left)\right|d\tau \hfill \\ \hfill & +{\displaystyle \frac{2xL}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^{\eta }{\int }_0^{\tau }{(\tau -r)}^{\varsigma -1}\hfill \\ \hfill & \left|u\right(r)-v(r\left)\right|drd\tau .\hfill \end{array}$$

(44)

Since

$$\parallel u-v\parallel =\underset{x\in [0,1]}{sup}|u\left(x\right)-v\left(x\right)|,$$

we obtain

$$\begin{array}{cc}\hfill \left|\right(\mathcal{K}u\left)\right(x)-(\mathcal{K}v\left)\right(x\left)\right|& \le L\parallel u-v\parallel [{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^x{(x-\tau )}^{\varsigma -1}d\tau \hfill \\ \hfill & +{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^1{(1-\tau )}^{\varsigma -1}d\tau \hfill \\ \hfill & +{\displaystyle \frac{2x}{(2-{\eta }^2)\mathsf{\Gamma }\left(\varsigma \right)}}{\int }_0^{\eta }{\int }_0^{\tau }{(\tau -r)}^{\varsigma -1}drd\tau ].\hfill \end{array}$$

(45)

Now,

$${\int }_0^x{(x-\tau )}^{\varsigma -1}d\tau ={\displaystyle \frac{x^{\varsigma }}{\varsigma }},{\int }_0^1{(1-\tau )}^{\varsigma -1}d\tau ={\displaystyle \frac{1}{\varsigma }},$$

$${\int }_0^{\eta }{\int }_0^{\tau }{(\tau -r)}^{\varsigma -1}drd\tau ={\displaystyle \frac{{\eta }^{\varsigma +1}}{\varsigma (\varsigma +1)}}.$$

Therefore,

$$\begin{array}{cc}\hfill \left|\right(\mathcal{K}u\left)\right(x)-(\mathcal{K}v\left)\right(x\left)\right|& \le {\displaystyle \frac{L}{\mathsf{\Gamma }(\varsigma +1)}}\parallel u-v\parallel \left(x^{\varsigma }+{\displaystyle \frac{2x}{2-{\eta }^2}}+{\displaystyle \frac{2x{\eta }^{\varsigma +1}}{(2-{\eta }^2)(\varsigma +1)}}\right).\hfill \end{array}$$

(46)

Taking the supremum over $x\in [0,1]$, we obtain

$$\parallel \mathcal{K}u-\mathcal{K}v\parallel \le {\displaystyle \frac{L{M}_{\eta ,\varsigma }}{\mathsf{\Gamma }(\varsigma +1)}}\parallel u-v\parallel .$$

By the assumption on L, this gives

$$\parallel \mathcal{K}u-\mathcal{K}v\parallel \le {\displaystyle \frac{1}{4}}\parallel u-v\parallel .$$

(47)

We now verify that $\mathcal{K}$ satisfies the rational-type contraction condition of Definition 9 directly in terms of ${\mathcal{RT}}_2$, without passing through the auxiliary inequality $\delta (u,v)\le {\mathcal{RT}}_2(u,v)$. For $\beta \in (0,1)$, define

$$\delta (u,v)=\parallel u-v\parallel \left(\parallel u-v\parallel +\beta \right),u,v\in \mathcal{X}.$$

Set

$$\mathcal{M}(u,v)=max\left\{\parallel u-v\parallel ,\parallel u-\mathcal{K}u\parallel ,\parallel v-\mathcal{K}v\parallel \right\},$$

where $\parallel ·\parallel$ denotes the supremum norm on $\mathcal{X}$. Since $\parallel \mathcal{K}u-\mathcal{K}v\parallel \le \frac{1}{4}\parallel u-v\parallel \le \frac{1}{4}\mathcal{M}(u,v)$ by (47), and the map $t\mapsto t(t+\beta )$ is increasing on $[0,\infty )$, we obtain

$$\begin{array}{cc}\hfill \delta (\mathcal{K}u,\mathcal{K}v)& =\parallel \mathcal{K}u-\mathcal{K}v\parallel \left(\parallel \mathcal{K}u-\mathcal{K}v\parallel +\beta \right)\hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{M}(u,v)}{4}}\left({\displaystyle \frac{\mathcal{M}(u,v)}{4}}+\beta \right)\hfill \\ \hfill & \le {\displaystyle \frac{\mathcal{M}(u,v)}{4}}\left(\mathcal{M}(u,v)+\beta \right),\hfill \end{array}$$

(48)

where the second inequality uses $\frac{\mathcal{M}(u,v)}{4}+\beta \le \mathcal{M}(u,v)+\beta$. We now identify the right-hand side of (48) with ${\mathcal{RT}}_2(u,v)$. Since $t\mapsto t(t+\beta )$ is increasing, we have

$$\begin{array}{cc}\hfill \mathcal{M}(u,v)\left(\mathcal{M}(u,v)+\beta \right)& =max\{\parallel u-v\parallel \left(\parallel u-v\parallel +\beta \right),\hfill \\ \hfill & \parallel u-\mathcal{K}u\parallel \left(\parallel u-\mathcal{K}u\parallel +\beta \right),\hfill \\ \hfill & \parallel v-\mathcal{K}v\parallel \left(\parallel v-\mathcal{K}v\parallel +\beta \right)\}\hfill \\ \hfill & =max\left\{\delta (u,v),\delta (u,\mathcal{K}u),\delta (v,\mathcal{K}v)\right\}.\hfill \end{array}$$

(49)

Since the fourth term in ${\mathcal{RT}}_2(u,v)$ is non-negative, namely,

$${\displaystyle \frac{\delta (u,\mathcal{K}u)\delta (v,\mathcal{K}v)}{1+\delta (u,v)}}\ge 0,$$

it follows from (49) and the definition of ${\mathcal{RT}}_2$ that

$$\mathcal{M}(u,v)\left(\mathcal{M}(u,v)+\beta \right)\le {\mathcal{RT}}_2(u,v).$$

(50)

Combining (48) and (50), we arrive at

$$\delta (\mathcal{K}u,\mathcal{K}v)\le {\displaystyle \frac{1}{4}}{\mathcal{RT}}_2(u,v).$$

(51)

This is the central inequality of the proof. The self-distance terms $\delta (u,\mathcal{K}u)$ and $\delta (v,\mathcal{K}v)$ appear in the bound through (49), so the full structure of ${\mathcal{RT}}_2$ is genuinely used, not merely invoked via a weaker inequality. Now let $\theta \left(t\right)=e^{\sqrt{t}}$, $t>0$. By Example 4, $\theta \in \mathsf{\Theta }$ and $\theta$ is continuous on $(0,\infty )$. Suppose $\mathcal{K}u\ne \mathcal{K}v$, so that $\delta (\mathcal{K}u,\mathcal{K}v)>0$. Taking square roots of (51) gives

$$\sqrt{\delta (\mathcal{K}u,\mathcal{K}v)}\le {\displaystyle \frac{1}{2}}\sqrt{{\mathcal{RT}}_2(u,v)},$$

(52)

and since $t\mapsto e^t$ is increasing,

$$\theta \left(\delta (\mathcal{K}u,\mathcal{K}v)\right)=e^{\sqrt{\delta (\mathcal{K}u,\mathcal{K}v)}}\le e^{{\displaystyle \frac{1}{2}}\sqrt{{\mathcal{RT}}_2(u,v)}}={\left[\theta \left({\mathcal{RT}}_2(u,v)\right)\right]}^{{\displaystyle \frac{1}{2}}}.$$

(53)

Hence,

$$\mathcal{K}u\ne \mathcal{K}v\Rightarrow \theta \left(\delta (\mathcal{K}u,\mathcal{K}v)\right)\le {\left[\theta \left({\mathcal{RT}}_2(u,v)\right)\right]}^{{\displaystyle \frac{1}{2}}},$$

which is precisely condition (15) of Definition 9 with ${\kappa }^{\prime }=\frac{1}{2}$. Since $(\mathcal{X},\delta )$ is a complete suprametric space by Proposition 1, all hypotheses of Theorem 2 are satisfied. Therefore, $\mathcal{K}$ has a unique fixed point in $\mathcal{X}$, which is the unique solution of the fractional differential Equation (37). □

Remark 2.

The proof of Theorem 3 verifies the rational-type contraction condition of Definition 9 directly in terms of ${\mathcal{RT}}_2(u,v)$, rather than by way of the weaker inequality $\delta (u,v)\le {\mathcal{RT}}_2(u,v)$. The key step is the identity (49): because $t\mapsto t(t+\beta )$ is increasing, the maximum of the three suprametric distances $\delta (u,v)$, $\delta (u,\mathcal{K}u)$, and $\delta (v,\mathcal{K}v)$ equals $\mathcal{M}(u,v)\left(\mathcal{M}(u,v)+\beta \right)$, and this quantity is bounded above by ${\mathcal{RT}}_2(u,v)$ since the rational term in ${\mathcal{RT}}_2$ is non-negative. Consequently, the self-distance terms $\delta (u,\mathcal{K}u)$ and $\delta (v,\mathcal{K}v)$ play an active role in the estimate (51): whenever one of $\parallel u-\mathcal{K}u\parallel$ or $\parallel v-\mathcal{K}v\parallel$ exceeds $\parallel u-v\parallel$, the bound is dominated by a self-distance term rather than by $\delta (u,v)$ alone. This is precisely the mechanism that distinguishes a rational-type contraction from a classical Banach-type one, and it shows that Theorem 2 is applied here in a non-trivial way.

5. Discussion

The results obtained in this paper provide a fixed point framework for studying nonlinear problems in complete suprametric spaces through rational-type contractions. The main theoretical contribution is given by Theorems 1 and 2, where two rational contractive conditions, denoted by ${\mathcal{RT}}_1$ and ${\mathcal{RT}}_2$, are introduced and used to establish the existence and uniqueness of fixed points. These results extend the classical Banach contraction principle in two directions. First, the underlying space is not required to be a usual metric space; instead, the arguments are carried out in a suprametric space, where the triangle inequality is replaced by a weaker nonlinear inequality. Second, the contraction conditions are not expressed only in terms of $\delta (\mathfrak{p},\nu )$, but also involve the distances between points and their images under the mapping. This makes the contractive framework more flexible than the standard Banach-type condition, while still retaining a constructive Picard iteration scheme.

The present results are related to earlier developments in generalized metric and fixed point theory. Banach’s theorem [1] gives a fundamental existence and uniqueness principle under a strict Lipschitz contraction in a complete metric space. Later, Jleli and Samet [25] introduced $\theta$-contractions as a useful generalization of the Banach contraction principle, while Berzig [23] developed the basic structure of suprametric spaces and showed that this setting can accommodate mappings that are not naturally handled by ordinary metric spaces. The theorems proved here combine these two lines of development: the $\theta$-function approach is used together with rational expressions in the suprametric distance. In comparison with the classical metric setting, the present framework allows the contractive behavior to depend not only on the distance between two points, but also on the interaction between the points and their images. This is the main mathematical role of the rational terms appearing in ${\mathcal{RT}}_1$ and ${\mathcal{RT}}_2$.

The examples following Theorems 1 and 2 illustrate how the abstract assumptions can be verified in finite suprametric spaces. These examples are not intended to replace the general proofs of the theorems; rather, they show that the hypotheses are nonempty and can be checked directly for concrete self-maps. In particular, the examples demonstrate that the rational-type contractive conditions can hold even when the mapping is defined on a space equipped with a nonstandard distance such as $\delta (\mathfrak{p},\nu )={|\mathfrak{p}-\nu |}^2$ or $\delta (\mathfrak{p},\nu )={|\mathfrak{p}-\nu |}^3$. Such examples are useful because they clarify the practical meaning of the abstract inequalities and show how the maximum terms in ${\mathcal{RT}}_1$ and ${\mathcal{RT}}_2$ are evaluated.

The application to the fractional chaotic financial model shows how the fixed point framework can be used in a system with memory effects. Fractional derivatives are well suited to such models because present states may depend on the accumulated history of the system, a feature that is important in financial dynamics. Earlier studies, such as those dealing with fractional financial systems and chaotic behavior [21,22], usually formulate the problem in a Banach space and apply fixed point tools under suitable Lipschitz-type assumptions. In the present work, the financial model is first written in vector form on $\mathcal{W}=C{(I,\mathbb{R})}^3$, and the Caputo problem is then transformed into an equivalent integral equation. Since the nonlinear terms $\mathfrak{p}\nu$ and ${\mathfrak{p}}^2$ are not globally Lipschitz on the whole space, the analysis is carried out on a closed bounded subset. On this set, an explicit local Lipschitz constant can be obtained, and the integral operator becomes a self-map under suitable restrictions on the radius of the ball and the length of the time interval. This gives a mathematically consistent local existence and uniqueness result for the fractional financial system.

The nonlinear fractional differential equation with integral boundary conditions provides a second application of the theory. Similar types of boundary value problems have been studied through Banach, Schauder, and related fixed point methods; see, for example, [9,11]. In Theorem 3, the rational-type contraction condition of Definition 9 is verified directly in terms of ${\mathcal{RT}}_2(u,v)$. The key step is the identity

$$\mathcal{M}(u,v)\left(\mathcal{M}(u,v)+\beta \right)=max\left\{\delta (u,v),\delta (u,\mathcal{K}u),\delta (v,\mathcal{K}v)\right\}\le {\mathcal{RT}}_2(u,v),$$

where $\mathcal{M}(u,v)=max\{\parallel u-v\parallel ,\parallel u-\mathcal{K}u\parallel ,\parallel v-\mathcal{K}v\parallel \}$, which shows that the self-distance terms $\delta (u,\mathcal{K}u)$ and $\delta (v,\mathcal{K}v)$ play an active role in controlling $\delta (\mathcal{K}u,\mathcal{K}v)$. When either self-distance exceeds $\delta (u,v)$, it is that term, rather than $\delta (u,v)$ alone, that governs the contraction estimate. This distinguishes the present argument from a classical Banach-type one, and confirms that Theorem 2 is applied here in a non-trivial way. The point is not that the boundary value problem cannot be treated by classical methods, but that the rational contraction framework developed in the paper captures a strictly richer contractive structure than the Banach contraction principle alone.

6. Conclusions

A generalized approach based on $FP$ theorems in suprametric spaces and rational contraction mappings was developed. Theorems 1 and 2 extend the classical contraction principle by establishing existence and uniqueness of $FP$s for nonlinear operators satisfying rational-type contractive conditions in complete suprametric spaces. In the applications, the developed rational contractive framework was used to obtain local existence and uniqueness for the fractional chaotic financial model and existence and uniqueness for a nonlinear fractional differential equation with integral boundary conditions.

The significance of the results is illustrated by applying the framework to a problem related to the fractional chaotic financial model. Fractional derivatives allowed modeling memory and hereditary processes specific to the financial markets, and the introduced $FP$ theory ensured existence and uniqueness of the solution. The study demonstrated the power of employing the framework of suprametric spaces along with rational contractions and fractional derivatives for solving nonlinear equations.

Future Scope

The results of this study suggest several directions for future research. The concept of rational contraction in suprametric spaces may be extended to other generalized spaces, such as partial, fuzzy, and cone metric spaces, enhancing its applicability. In addition, while this work establishes existence and uniqueness, further studies on stability, convergence, and efficient numerical methods for fractional differential equations are needed. More specifically, we provide the following open problems:

1.

Can the rational-type contraction conditions introduced in this paper be extended to orthogonal suprametric spaces [27] and used to obtain coupled fixed point results?

2.

It will be interesting to establish $\alpha$-$ϵ$-Suzuki-type rational contraction principles as enunciated in [28] in complete suprametric spaces.

3.

Can the proposed results be generalized to covariant and contravariant mappings as studied in [29] in suprametric spaces?

4.

Based on [30], one can consider developing the rational-type contraction theorems proposed in this paper for the case of fractional differential equations that are modeled using the $(k,\psi )$-generalized Laplace transform. Here, it would be important to explore the conditions for parameters k and $\psi$ along with properties of the nonlinear operator that would guarantee a unique fixed point of the problem in the suprametric space.

5.

Prompted by [31], another open problem is related to designing numerical algorithms for the approximate computation of fixed points obtained with rational-type contraction theorems proposed here. This would be particularly relevant for L-fractional Abel problems.