🔢 mathematics

Path Averaged Polynomial Contractions: A New Generalization of Polynomial Contractions, Path-Averaged Contractions, and Banach Contractions

This paper introduces the new concept of path-averaged polynomial contractions as a generalization of Banach, polynomial, and path-averaged contractions, establishing a corresponding fixed point theorem in metric spaces and providing an example to demonstrate that this new class properly extends Banach contractions.

Original authors: Clement Boateng Ampadu, Nicola Fabiano

Published 2026-07-17

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Clement Boateng Ampadu, Nicola Fabiano

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

Imagine you are trying to find a specific spot on a map where, no matter how many times you take a step, you always end up landing right back on that same spot. In the world of mathematics, this is called a "fixed point." It's a concept that shows up everywhere, from predicting how a ball bounces to understanding how computer algorithms settle down. For decades, mathematicians have been hunting for rules that guarantee such a spot exists. The most famous rule is the "Banach Contraction," which is like a strict bouncer at a club: if you and a friend start apart, the bouncer forces you to get closer together every time you move, until you are practically hugging. But life (and math) isn't always that simple. Sometimes, the rules are a bit more flexible, or the way you get closer is a bit more complicated. This is where the story gets interesting.

In this new paper, two researchers, Clement Boateng Ampadu and Nicola Fabiano, decide to mix two different, slightly more complex rules into one super-rule. They take the idea of a "polynomial contraction" (where the distance between points shrinks based on a fancy formula involving powers, like squaring or cubing the distance) and combine it with "path-averaged contractions" (where you don't just look at the immediate next step, but you look at the average distance covered over a whole journey of steps). By blending these, they create something they call a "Path-Averaged Polynomial Contraction." Think of it as upgrading a simple "get closer" rule into a smart navigation system that looks at your entire route history and uses a complex formula to ensure you eventually stop moving.

The paper's main finding is a proof that if you have a complete map (a mathematical space where every path leads somewhere) and you use this new, super-flexible rule, you are guaranteed to find a unique fixed point. The authors prove that if you start anywhere and keep following the rules, your path will eventually settle down on one specific spot and stay there. They also show that this new rule is powerful enough to handle situations where the old, strict "Banach" rule fails. In fact, they provide a specific example of a tiny world with just three points where the new rule works perfectly, but the old Banach rule would say "no fixed point here" because the points don't shrink fast enough in a single step.

The researchers are very sure about their results; they haven't just guessed or simulated this on a computer. They have provided a rigorous mathematical proof that works for any complete metric space, provided the mapping (the rule for moving) is continuous and meets a few specific conditions about how the distances are measured. They explicitly rule out the idea that this new rule is just a fancy version of the old Banach contraction; they prove it is a broader, more general category that includes the old rules but also catches cases the old rules miss. So, while the math might look like a wall of symbols, the story is simple: they found a new, more versatile way to guarantee that a journey will eventually come to a stop.

Technical Summary: Path Averaged Polynomial Contractions

Problem Statement and Context
This paper addresses the generalization of fixed point theory within the framework of metric spaces. It seeks to unify and extend two distinct classes of contractive mappings: polynomial contractions and path-averaged (PA) contractions.

Polynomial contractions, introduced in [2], involve inequalities where a weighted sum of powers of distances between iterates is bounded by a similar sum of the original distances, scaled by a constant $\lambda \in [0, 1)$ .
Path-averaged contractions, introduced in [3] for metric spaces (and extended to $b$ -metric and suprametric spaces in [4–6]), involve inequalities where the average (or sum) of distances over a sequence of iterates is bounded by the average of the initial distances, scaled by $\alpha \in (0, 1)$ .

The authors identify a gap in the literature where these two concepts have remained separate. The paper aims to introduce a new class of mappings that encompasses both, as well as the classical Banach contractions, to establish a more robust fixed point theorem.

Methodology
The authors define a new class of mappings termed Path-Averaged Polynomial (PA-Polynomial) Contractions.

Definition: A mapping $T: X \to X$ on a metric space $(X, d)$ is a PA-polynomial contraction if there exist $\alpha \in (0, 1)$ and $N \in \mathbb{N}$ such that for all $x, y \in X$ and all $n \ge N$ :
$\sum_{k=0}^{n-1} \sum_{i=0}^{r} a_i(T^{k+1}x, T^{k+1}y) d^i(T^{k+1}x, T^{k+1}y) \le \alpha \sum_{k=0}^{n-1} \sum_{i=0}^{r} a_i(T^k x, T^k y) d^i(T^k x, T^k y)$
Here, $a_i: X \times X \to [0, \infty)$ are auxiliary mappings, and $d^i$ represents the $i$ -th power of the metric.
Proof Strategy: The authors prove a fixed point theorem for these contractions in complete metric spaces under two conditions:
1. Continuity of $T$ .
2. A lower bound condition on at least one coefficient function $a_j$ (specifically $a_j(x, y) \ge A_j > 0$ for some $j \in \{1, \dots, r\}$ ).
The proof proceeds by constructing a Picard sequence $\{z_n\}$ where $z_{n+1} = T z_n$ . By defining a partial sum $P_n$ representing the polynomial evaluation along the sequence, the authors demonstrate that the series of these sums converges. This convergence implies that the distance between consecutive terms $d(z_k, z_{k+1})$ approaches zero. Under the assumption that the index $j$ corresponds to the linear term ( $j=1$ ), the triangle inequality ensures the sequence is Cauchy. The completeness of the space guarantees the existence of a limit $z^*$ , and the continuity of $T$ confirms $z^*$ is a fixed point. Uniqueness is established by contradiction, showing that assuming two distinct fixed points leads to a violation of the contraction constant $\alpha < 1$ .

Key Contributions and Results

New Generalization: The paper introduces the PA-polynomial contraction, which strictly generalizes:
- Polynomial contractions (by setting $n=1$ ).
- Path-averaged contractions (by setting $r=1, a_0 \equiv 0, a_1 \equiv 1$ ).
- Banach contractions (by setting $n=1, r=1, a_0 \equiv 0, a_1 \equiv 1$ ).
Fixed Point Theorem: Theorem 2.3 establishes the existence and uniqueness of a fixed point for PA-polynomial contractions in complete metric spaces, provided the mapping is continuous and satisfies the coefficient lower bound condition.
Strict Generalization Example: The paper provides Example 3.2 to demonstrate that the new class is strictly broader than Banach contractions. It constructs a mapping on a discrete metric space $X=\{0, 1, 2\}$ $X = {0, 1, 2}$ that:
- Fails to be a Banach contraction (as $d(Tx, Ty) \le \alpha d(x, y)$ cannot hold for any $\alpha < 1$ for all pairs).
- Satisfies the PA-polynomial contraction condition with specific parameters ( $\alpha = 1/2, N=2, r=2$ ).

Significance and Claims
The paper claims that the introduced notion of PA-polynomial contractions serves as a unifying framework for existing contraction types in metric fixed point theory. By combining the structural flexibility of polynomial contractions with the iterative averaging of path-averaged contractions, the authors provide a broader sufficient condition for the existence of unique fixed points. The significance lies in the ability to guarantee fixed points for mappings that do not satisfy the stricter requirements of Banach or standard polynomial contractions, thereby expanding the scope of applicable mappings in metric spaces. The authors do not propose experimental applications or future extensions beyond the theoretical generalization presented.

More like this