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Relation-Theoretic Banach Contraction Principle in Topological Spaces with an Application

This paper extends the Banach contraction principle to topological spaces by introducing relation-preserving contraction mappings, validates the result with a MATLAB-visualized example, and demonstrates its application in solving fractional differential equations.

Original authors: Md Hasanuzzaman, Abhishikta Das, Sumit Som

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Md Hasanuzzaman, Abhishikta Das, Sumit Som

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to finding points that do not move when a rule is applied to them. Imagine a map of a city where every location is assigned a new location. A "fixed point" is a place on that map that, when you apply the rule, points back to itself. This concept is not just an abstract puzzle; it is a powerful tool used to solve real-world problems, from predicting the equilibrium of a market to understanding how a fluid settles. For decades, the most famous tool for finding these unmoving points has relied on a strict system of measuring distance, much like using a ruler to ensure that every step taken toward a solution brings you closer to the target. This method, known as the Banach contraction principle, has been a cornerstone of modern analysis, guaranteeing that a solution exists and is unique under very specific conditions.

However, the world is not always neatly measured by a single ruler. In many complex systems, the standard rules of distance break down, or the structure of the space is too irregular for traditional tools to work. Mathematicians have long sought ways to extend these powerful ideas into these more chaotic or abstract environments, such as general topological spaces where the usual concept of distance might not even exist. The challenge has been to find a way to guide a process toward a solution without a rigid measuring tape, relying instead on the relationships between the points themselves. This is where the work of Md Hasanuzzaman, Abhishikta Das, and Sumit Som comes in, offering a new way to navigate these uncharted territories.

The researchers set out to solve a specific problem: what happens when the standard methods for finding fixed points fail because the space is too strange or the rules are too loose? In their study, they introduced a new framework that combines the geometry of a space with a simple list of connections, or a "binary relation," between points. Think of this relation not as a distance, but as a set of allowed pathways. In their new system, a rule for moving from one point to another is only required to shrink the gap between points if those points are connected by this specific pathway. This is a significant departure from older methods, which demanded that the rule shrink the gap between every single pair of points in the entire space, regardless of whether they were related or not.

By introducing this relational structure, the authors were able to prove that a fixed point still exists even in situations where previous theorems could not guarantee one. They demonstrated that if a space is complete enough in a specific sense and the connections between points are preserved by the rule being applied, a solution is inevitable. To make this concrete, they constructed a specific example involving a two-dimensional space where the standard rules of distance failed to ensure a solution. In this scenario, the old methods would have left the problem unsolved, claiming that no guarantee of a fixed point could be made. Yet, by applying their new relational approach, they showed that a unique solution did indeed exist. They visualized this convergence using computer simulations, showing how a sequence of points, guided by the new rules, steadily and rapidly approached a single, unmoving destination.

The power of this new approach lies in its flexibility. It allows mathematicians to solve problems in spaces that are too irregular for traditional distance-based tools, provided there is a logical structure connecting the elements. The authors did not stop at theory; they applied their findings to a complex equation used to model economic growth. These equations, which describe systems with memory and non-local effects, are notoriously difficult to solve. By framing the problem within their new relational space, the team proved that a solution to this economic model exists and can be found through a specific iterative process. They ran a numerical simulation of this economic model, starting with a simple guess and repeatedly applying the rule. The results showed the error between steps dropping dramatically, confirming that the process was converging to a stable solution, just as their theory predicted.

This work represents a quiet but significant expansion of mathematical capability. It does not discard the old, reliable methods but rather builds a bridge to areas where those methods previously hit a wall. By showing that the strict requirement of shrinking every possible distance can be relaxed to only shrinking distances between related points, the authors have opened the door to solving a wider class of problems. Their findings suggest that in many complex systems, the key to finding a stable solution is not just measuring how far apart things are, but understanding how they are connected. This insight provides a robust new tool for researchers working in fields ranging from pure mathematics to applied economics, ensuring that even in the most irregular spaces, a path to a solution can be found.

Technical Summary: Relation-Theoretic Banach Contraction Principle in Topological Spaces with an Application

Problem Statement
Metric fixed point theory, anchored by the Banach Contraction Principle (BCP), is a fundamental tool in nonlinear analysis. However, many applications arise in settings where a metric structure is absent or insufficient, necessitating the study of mappings in general topological spaces. While previous work by Som et al. [9] extended the BCP to topological spaces using a continuous function $g: \Omega \times \Omega \to \mathbb{R}$ , certain mappings fail to satisfy the contraction conditions required by these existing theorems. Furthermore, classical results often require contraction conditions to hold for all pairs of points in the space, which can be overly restrictive. The authors address the need for a more flexible framework that can guarantee the existence and uniqueness of fixed points in topological spaces where standard metric-based or topological BCPs may fail.

Methodology
The paper introduces a framework that integrates three distinct structures:

Topological Space ( $\Omega$ ): The underlying space where no metric is assumed.
Continuous Function ( $g$ ): A function $g: \Omega \times \Omega \to \mathbb{R}$ satisfying specific properties (symmetry and triangle inequality) to define convergence and Cauchy sequences ( $g$ -convergence and $g$ -completeness). The condition $g(r,s)=0 \implies r=s$ is required for the main theorem but is not assumed to hold universally in all illustrative examples (e.g., Example 3.4 demonstrates a case where this condition is violated, necessitating the relational structure to recover the fixed point result).
Binary Relation ( $R$ ): A subset of $\Omega \times \Omega$ that imposes a relational structure on the space.

The authors define topologically $R$ -preserving contraction mappings. A mapping $S: \Omega \to \Omega$ is a topologically $R$ -preserving contraction if there exists $\alpha \in (0, 1)$ such that:
$|g(S\mu_1, S\mu_2)| \leq \alpha |g(\mu_1, \mu_2)|$
for all $\mu_1, \mu_2 \in \Omega$ such that $(\mu_1, \mu_2) \in R$ .

Key definitions introduced include:

$g$ - $R$ -continuity: Continuity of the mapping with respect to $R$ -preserving sequences.
$g$ - $R$ -completeness: Completeness of the space with respect to $R$ -preserving $g$ -Cauchy sequences.
$S$ -closedness: The property that if $(r, s) \in R$ , then $(Sr, Ts) \in R$ .
$g$ -self-closedness: A property of the relation $R$ regarding subsequences of convergent sequences.

Key Contributions and Results
The primary contribution is Theorem 3.2, which establishes the existence of a fixed point for a mapping $S$ on a topological space $\Omega$ under the following conditions:

$\Omega$ is $g$ - $R$ -complete.
$R$ is $S$ -closed.
The set $\Omega(S; R) = \{u \in \Omega : (u, Su) \in R\}$ is non-empty.
Either $S$ is $g$ - $R$ -continuous or $R$ is $g$ -self-closed.
$S$ is a topologically $R$ -preserving contraction with respect to $g$ .
The function $g$ satisfies $g(r, u) = 0 \implies r = u$ , symmetry, and the triangle inequality for relevant elements.

The theorem guarantees that the Picard sequence $\{S^n(r_0)\}$ converges to a fixed point for any $r_0 \in \Omega(S; R)$ . Theorem 3.3 further establishes the uniqueness of this fixed point if the image $S(\Omega)$ is $g$ - $R_s$ -connected.

The paper demonstrates the utility of this framework through:

Counterexamples: Examples (3.4 and 3.5) illustrate scenarios where the topological BCP by Som et al. [9] (Theorem 2.7) fails to guarantee a fixed point because the contraction condition does not hold for all pairs in $\Omega$ or because the function $g$ fails to satisfy the identity of indiscernibles ( $g(a,u)=0 \implies a=u$ ) globally. However, by restricting the contraction condition to pairs related by a specific binary relation $R$ (and ensuring $g$ satisfies the necessary properties within that relational context), the new theorem successfully guarantees a fixed point.
Visualization: MATLAB simulations are provided to visualize the geometric convergence of the $y$ -coordinate under the proposed mappings, validating the analytical results.
Generalization: The authors show that their results generalize existing literature. By choosing $R$ as the universal relation, the result reduces to Theorem 2.7 of Som et al. [9]. By choosing $g$ as a metric $d$ , it reduces to the relation-theoretic BCP of Alam and Imdad [2].

Application
The theoretical results are applied to the existence of solutions for a fractional differential equation modeling economic growth:
$D^\zeta(f(t)) = h(t, f(t)), \quad t \in [0, 1], \quad 1 < \zeta \leq 2$
subject to integral boundary conditions. The authors define the space $\Omega = C[0, 1]$ with the sup-metric topology and a binary relation $R$ based on pointwise ordering ( $p_1 \leq p_2$ ). They construct an integral operator $T$ and verify that it satisfies the conditions of Theorem 3.2. A numerical illustration using an iterative scheme ( $u_{n+1} = T u_n$ ) with $h(t, u) = \frac{1}{16}u + \sin(t)$ demonstrates rapid convergence to the unique solution, validating the theoretical findings.

Significance
The paper claims that its framework offers a more efficient and flexible approach to fixed point theory in topological spaces. By incorporating a binary relation, the contraction condition is relaxed to apply only to "related" elements rather than the entire space. This allows the authors to prove the existence of fixed points in cases where classical or previously generalized topological Banach contraction principles fail, particularly when the underlying function $g$ does not satisfy the identity of indiscernibles globally. The work bridges the gap between metric fixed point theory and topological spaces using relational structures, providing a robust tool for analyzing nonlinear problems such as fractional differential equations where metric structures may be insufficient or where order relations play a critical role.

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