Communications on Applied Nonlinear Analysis 

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Probabilistic Perspective of Jungck Contractive Situations in Machine 

Learning 

 

Ashutosh Mishra1, Piyush Kumar Tripathi2, Mukesh Kumar Singh3, Manisha Gupta4, Alok 

Kumar Agrawal5 

1,2,5Department of Mathematics, ASAS, Amity University Uttar Pradesh (Lucknow Campus), India 
3Armapore P. G. College, Kanpur, India 

4University of Technology and Applied Science, Muscat 

Email: 1amishra94154@gmail.com, 1ashutosh.mishra8@s.amity.edu; 2pktripathi@amity.edu; 3mksinghuptti@gmail.com; 
4manisha.gupta@utas.edu.om; 5akagrawal@lko.amity.edu 

 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: 

In the realm of machine learning (ML), the underlying mathematical frameworks significantly 

influence the development and efficacy of algorithms. One such framework is the 

probabilistic metric space, which provides a robust foundation for handling uncertainty and 

variability inherent in real-world problems and data. This work explores the applicability of 

probabilistic metric spaces in machine learning, with a particular focus on the Jungck 

contractive notion. The Jungck contraction theorem extends the classical Banach contraction 

principle to the setting of probabilistic metric spaces. This theorem is pivotal for proving the 

existence and uniqueness of fixed points in probabilistic settings, which is crucial for the 

convergence analysis of ML algorithms. This generalises invariant point propositions to 

probabilistic settings, offers powerful tools for the convergence analysis of iterative ML 

algorithms. 

We delve into the theoretical underpinnings of probabilistic metric spaces, explain the Jungck 

contractive notion, and bespeak its relevance and application in ML algorithms. 

Keywords: Probabilistic space, Contractive conditions, Machine learning, Convergence 

analysis. 

1. Introduction 

Machine learning is an underlying collection as a subset of AI (artificial intelligence) that concentrates 

the advancement of algorithms, as well as statistical model which enables computers to execute certain 

tasks without distinct instructions, depending on patterns and inference instead. In other words, it is 

about creating systems that can learn from data. 

The crux behind such learning is to empower computer related programmes to train automatically from 

given data and to improve its effectiveness over time, without being distinctively programmed for 

every task. This is brought about through various learning techniques viz. supervised, unsupervised, 

reinforced, and deep. 

Supervised Learning 

In this approach, the algorithm retains information from some specific labelled data, meaning there by 

that it is given input-output pairs and learns to map inputs to outputs. For example, let us have a given 

set of labelled impressions (images) of cats and dogs, a supervised learning algorithm would learn to 

classify new impressions (images) as either cats or dogs [8]. 

mailto:amishra94154@gmail.com
mailto:1ashutosh.mishra8@s.amity.edu
mailto:2pktripathi@amity.edu
mailto:mksinghuptti@gmail.com3
mailto:manisha.gupta@utas.edu.om


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Unsupervised Learning 

In this stage, the algorithm is given input data without explicit labels, and it tries to find patterns or 

structure in the data. Clustering is a common task in unsupervised learning, where the algorithm groups 

similar data points together. An example could be grouping similar customer purchase behaviour for 

market segmentation [8]. 

Reinforced Learning 

This is decision making training algorithms that gets learning from feedback. The algorithm trains 

itself to achieve a outcome or goal in an uncertain, potentially complex environment by taking actions 

and receiving rewards or penalties. It is often used in areas like gamming, robotics, and auto driving 

[10]. 

Deep Learning 

It is an underlying subset of machine learning which uses artificial neural networks with many blankets 

of layers, hence proposed as “deep”, to model and process complex patterns in large amounts of data. 

It's particularly effective for tasks such as image and speech recognition, natural language processing, 

and playing strategic games like Go [9]. 

Overall, machine learning has numerous applications across various industries, including finance, e-

commerce, healthcare, cyber security, and many more. Its ability to find patterns in data and make 

predictions or decisions based on those patterns has made it a powerful tool for solving a wide range 

of real-world problems. 

Machine learning algorithms often rely on the convergence properties of iterative processes. Classical 

metric spaces and their associated invariant point theorems have provided substantial insights into 

these processes. However, real-world data is frequently stochastic in nature, demanding a probabilistic 

approach to better model the uncertainty and variability. Probabilistic metric spaces, introduced by 

Menger[2]in the year 1940, extend the concept of metric spaces by incorporating probability 

distributions instead of deterministic distances. Since then, a variety of propositions were introduced 

using Menger theory. Some implications of this notion are illustrated in population dynamics of cells 

[11, 14]. This probabilistic framework is particularly useful in ML, where data and models are 

inherently noisy and probabilistic. 

The Jungck theorem on contraction mapping[4], a generalization of Banach's fixed point result to 

probabilistic metric spaces, offers a robust mechanism to analyze the convergence of sequences in 

these spaces[1]. This theorem is instrumental in ensuring the stability and reliability of iterative ML 

algorithms. By leveraging the Jungck contraction theorem, we can develop and analyze ML algorithms 

that are more resilient to the uncertainties of real-world data. 

The K-means clustering algorithm is a fundamental technique in machine learning for partitioning data 

into clusters based on similarity[7]. In traditional K-means, the distance between points and centroids 

is deterministic. However, real-world data often contains uncertainty and noise, which can lead to 

instability in the clustering results. By incorporating a probabilistic approach into K-means, we can 

account for this uncertainty, leading to more robust and reliable clustering. Before delving into the 

probabilistic K-means algorithm, it's essential to understand probabilistic metric spaces (PMS). 

 

 



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2. Objectives 

By employing the Jungck contractive statement, we ensure that the probabilistic gradient descent 

algorithm converges to an optimal solution, providing robustness against the uncertainties in the 

training data. 

3. Methods and Preliminaries 

A. Probabilistic Space: A probabilistic distance (or metric) space (PMS) is a generalization of a metric 

space where the distance between any two points is not a single real number but a distribution function. 

Formally, a PMS is a triplet (𝑃, 𝐹, 𝑇), where 𝑃 is a set, 𝐹 is the set of distribution functions on [0, ∞), 

and 𝑇: 𝑃 × 𝑃 → 𝐹is a mapping that satisfies certain axioms analogous to those of a metric space. 

Definition 3.1: [11] A distribution function 𝐹 on [0, ∞) is a non-decreasing, left-continuous function, 

such that, 𝐹(0) = 0, and 𝑙𝑖𝑚
𝑛→∞

𝐹(𝑥) = 1. 

The function 𝑇(𝑥, 𝑦) gives the probability that the distance between 𝑥 and 𝑦is less than or equal to a 

given value. This probabilistic approach is particularly useful in modelling uncertainties in data points. 

Definition 3.2: [11] A triangular norm (t-norm) can be represented as a mapping 𝑡: [0,1] × [0,1] →

[0,1] satisfying: 

i. t(𝑎, 1) = 𝑎 for every 𝑎 ∈ [0,1] 
ii. t(𝑎, 𝑏) = t(𝑏, 𝑎) for every 𝑎, 𝑏 ∈ [0,1] 

iii. t(𝑎, 𝑐) = t(𝑏, 𝑑) if 𝑎 ≥ 𝑏, 𝑐 ≥ 𝑑 

iv. t(𝑎, t(𝑏, 𝑐)) = t(t(𝑎, 𝑏), 𝑐); 𝑎, 𝑏,𝑐 ∈ [0,1]. 

B. Axioms of PMS: 

i. Non-negativity: 𝑇(𝑥, 𝑦)  =  𝜖0 (where 𝜖0 is the degenerate distribution at 0), iff, 𝑥 = 𝑦. 

ii. Symmetry: 𝑇(𝑥, 𝑦) =  𝑇(𝑦, 𝑥), 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑃𝑎𝑛𝑑 𝑦 ∈ 𝑃. 

iii. Triangle inequality: For all 𝑥, 𝑦, 𝑧 in 𝑃 and for all 𝑡 ≥ 0, 

𝑇(𝑥, 𝑧)(𝑡) ≥  sup𝑢+𝑣=𝑡 𝑚𝑖𝑛 {𝑇(𝑥, 𝑦)(𝑢), 𝑇(𝑦, 𝑧)(𝑣)}. 

These axioms ensure that the probabilistic metric retains the essential properties of a classical metric, 

allowing us to extend various analytical techniques to the probabilistic domain. 

C.  Jungck Contractive Condition: The Jungck contraction theorem extends the classical Banach 

contraction principle to the setting of probabilistic metric spaces. This theorem is pivotal for proving 

the existence and uniqueness of fixed points in probabilistic settings, which is crucial for the 

convergence analysis of ML algorithms. 

Theorem 3.1 (Jungck Contraction Theorem): [4] Let (𝑃, 𝐹, 𝑇) be a complete probabilistic metric space. 

Suppose 𝑓, 𝑔: 𝑃 → 𝑃are two mappings satisfying: 

𝑇(𝑓(𝑥), 𝑓(𝑦))(𝑡) ≥  𝛽 𝑇(𝑔(𝑥), 𝑔(𝑦))(𝑡), 

for all 𝑥, 𝑦 ∈ 𝑃 and for some constant 𝛽 ∈ (0,1).Then,𝑓 and 𝑔 have a unique common invariant point 

in 𝑃. 

This result ensures that under a contraction condition, iterative applications of the mappings 𝑓 and 𝑔 

will converge to a common invariant point. This result is useful in analyzing machine learning based 

algorithms where such mappings often represent iterative update steps, for example, in analysing the 

convergence of probabilistic K-means clustering. 



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Application in Machine Learning: Many algorithms, in machine learning, can be viewed as iterative 

processes that seek to minimize an objective function or optimize a model. The convergence of these 

processes is crucial for the reliability and stability of the algorithms. By framing these iterative 

processes within the context of probabilistic metric spaces, we can leverage the Jungck contractive 

condition to assure convergence even in the presence of uncertainty. 

K-means Clustering in Probabilistic Space-A Case Study: The K-means clustering algorithm is a 

widely used method for partitioning data into clusters. However, in the presence of uncertainty and 

noise, the classical K-means algorithm may struggle to find stable clusters. By adopting a probabilistic 

approach, we can enhance the robustness of K-means clustering. 

Probabilistic Distance Metric: Instead of using a deterministic distance, we define a probabilistic 

distance between points. This distance is modelled as a distribution function capturing the uncertainty 

in the data. 

Probabilistic Centroid Update: The update step for centroids in the classical K-means is replaced by 

a probabilistic update. Given a cluster, the new centroid is chosen such that it minimizes the expected 

probabilistic distance to the points in the cluster. 

Convergence Analysis: Using the Jungck contraction theorem, we can analyse the convergence of the 

probabilistic K-means algorithm. By ensuring that the probabilistic distance satisfies the contraction 

condition, we can guarantee that the iterative updates of centroids will converge to a stable 

configuration. 

K-means Clustering Algorithm in Probabilistic Space: The probabilistic K-means algorithm [7] 
extends the classical K-means by incorporating probabilistic distances, which are represented by 

distribution functions. This approach provides a more robust clustering method in the presence of 

uncertainty and noise. 

Algorithm of K-means in probabilistic space is as follows: 

1. Initialization: Randomly initialize 𝑘 probabilistic centroids, where each centroid is represented by a 

distribution function. 

2. Assignment Step: Here, we assign each data point to the cluster whose centroid minimizes the 

probabilistic distance. 

3. Update Step: For each and every cluster, update the centroid to minimize the expected probabilistic 

distance to the points in the cluster. 

4. Convergence Check: Repeat the assignment and update steps until the centroids converge. 

Step-by-Step Explanation: 

1. Initialization: 

Randomly select 𝑘 points from the dataset as initial centroids. 

Represent each centroid as a distribution function capturing the uncertainty. 

2. Assignment Step: 



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For each data point 𝑥𝑖, compute the probabilistic distance to each centroid 𝐶𝑗 using the distribution 

function 𝑇(𝑥𝑖 , 𝐶𝑗). 

Assign 𝑥𝑖to the cluster with the centroid that has the smallest expected probabilistic distance. 

3. Update Step: 

For each cluster, 𝐶𝑗, update the centroid to the point that minimizes the expected probabilistic distance 

to all points in the cluster. 

This involves finding a new distribution function that best represents the central tendency of the points 

in the cluster. 

4. Convergence Check: 

We check, if the centroids have stabilized (i.e., there is little to no change in the distribution functions 

of the centroids). 

If not so, we repeat the assignment and update steps until the convergence is obtained, e.g. 

Consider a simple 2D dataset with three clusters. The classical K-means algorithm may struggle to 

identify the clusters accurately due to noise and uncertainty in the data points. We apply the 

probabilistic K-means algorithm to handle this uncertainty. 

Dataset: 

Points: [(1.0, 2.0), (1.5, 1.8), (5.0, 8.0), (8.0, 8.0), (1.0, 0.6), (9.0, 11.0)] 

Initialization: 

Randomly initialize 𝑘 = 2 probabilistic centroids. Assume initial centroids are (1.0, 2.0) and 

(5.0, 8.0), represented by distribution functions with small variances. 

Iteration 1: Assignment Step 

Compute probabilistic distances between each point and the centroids using a Gaussian distribution 

function. 

Assign points to the nearest centroid based on expected probabilistic distance. 

Iteration 1: Update Step 

Update centroids by recalculating the distribution functions that minimize the expected probabilistic 

distance within each cluster. 

Iteration 1: Results 

Cluster 1: [(1.0, 2.0), (1.5, 1.8), (1.0, 0.6)] 

Cluster 2: [(5.0, 8.0), (8.0, 8.0), (9.0, 11.0)] 

New Centroids (distribution functions): 

Cluster 1: Mean = (1.17, 1.47), Variance = small 



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Cluster 2: Mean = (7.33, 9.0), Variance = small 

Iteration 2: Assignment Step 

Recompute probabilistic distances using updated centroids. 

Reassign points to the nearest centroid. 

Iteration 2: Update Step 

Recalculate centroids for the new clusters. 

Iteration 2: Results 

Cluster 1: [(1.0, 2.0), (1.5, 1.8), (1.0, 0.6)] 

Cluster 2: [(5.0, 8.0), (8.0, 8.0), (9.0, 11.0)] 

New Centroids (distribution functions): 

Cluster 1: Mean = (1.17, 1.47), Variance = very small 

Cluster 2: Mean = (7.33, 9.0), Variance = very small 

Convergence Check: 

Centroids have stabilized (no significant change in mean and variance of distribution functions). 

Algorithm converges. 

Final Clusters: 

Cluster 1: [(1.0, 2.0), (1.5, 1.8), (1.0, 0.6)] 

Cluster 2: [(5.0, 8.0), (8.0, 8.0), (9.0, 11.0)] 

By applying the Jungck contraction theorem, we ensure that the probabilistic K-means algorithm 

converges to a stable clustering configuration, providing robustness against the inherent uncertainties 

in the data. 

Probabilistic Neural Networks (PNN)-A Case Study: Neural networks (NNs) are a cornerstone of 

modern ML. Training a NN involves iteratively updating weights to minimize a loss function. In the 

presence of noisy data, a probabilistic framework can enhance the training process[13].PNNs and 

related probabilistic models are being applied to various domains such as healthcare (medical diagnosis 

and prognosis) and finance (risk assessment and fraud detection). These applications benefit from the 

ability of PNNs to handle uncertainty and complex patterns in data[13]. 

1. Probabilistic Weight Updates: The weight updates are modelled probabilistically, capturing the 

uncertainty in the gradient estimates due to noisy data. 

2. Probabilistic Loss Function: The loss function is defined in terms of probabilistic distances, 

providing a more robust objective for optimization [16]. 



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3. Convergence Analysis: Using the Jungck contraction theorem, we analyse the convergence of the 

probabilistic weight updates [16]. By ensuring that the probabilistic distance satisfies the contraction 

condition, we can assure convergence to an optimal set of weights. 

Probabilistic Gradient Descent Algorithm: 

1. Initialization: Randomly initialize weights probabilistically. 

2. Forward Pass: Compute the output of the NN using the current probabilistic weights. 

3. Loss Computation: Compute the probabilistic loss based on the output and the true labels. 

4. Backward Pass: Compute the probabilistic gradient of the loss with respect to the weights. 

5. Weight Update: Update the weights probabilistically based on the computed gradient. 

6. Convergence Check: We repeat steps 2-5, until the loss converges. 

By employing the Jungck contractive statement and commutativity [15], we ensure that the 

probabilistic gradient descent algorithm converges to an optimal solution, providing robustness against 

the uncertainties in the training data. 

4. Main Results & Statements 

We use the following as a result in harnessing the situation as stated above. 

Lemma 4.1: [6] If (𝑃, 𝐹, 𝑇) be a Menger generalized PM-space with a Cauchy sequence < 𝛼𝑛 > in 

𝑃along with some 𝛽 ∈ (0,1) such that 𝐹𝛼𝑛,𝛼𝑛+1
(𝛽𝑥) ≥ 𝐹𝛼𝑛−1,𝛼𝑛

(𝑥) for all positive values of 𝑥. 

Also, if for 𝑟 > 1 and 𝑖 = 𝑛 𝑡𝑜 ∞, we have 

lim
𝑛→∞

𝑇𝑖 (𝐹𝛼0,𝛼1
(𝑟𝑖)) = 1 

Therefore, we say the sequence < 𝛼𝑛 > is 𝐹-Cauchy. 

To get the common coincidence point for mappings under generalized Menger space for producing 

some machine learning results, we can apply the following statements with some specific constraints. 

Theorem 4.1: With (𝑃, 𝐹, 𝑇)a generalized PM-space and a continuous 𝑡-norm𝑇 in (𝑎, 1) for all 0 <

𝑎 < 1; 0 < 𝜃 < 1;  𝛿 ∈ ∆ and 𝑓, 𝑔: 𝑄 → 𝑃, we have 

(i). 𝛿(𝐹𝑓𝛼,𝑓𝛽(𝜃𝑥), 𝐹𝑔𝛼,𝑔𝛽(𝑥), 𝐹𝑓𝛼,𝑔𝛼(𝑥), 𝐹𝑓𝛽,𝑔𝛽(𝜃𝑥)) ≥ 0 for every 𝛼, 𝛽 ∈ 𝑄 and positive values of 𝑥 

(ii). 𝑓(𝑄) ⊂ 𝑔(𝑄) 

(iii).𝛼0, 𝛼1 ∈ 𝑄 so that 𝑓𝛼0 = 𝑔𝛼1 and lim
𝑛→∞

𝑇𝑖 (𝐹𝑓𝛼0,𝑓𝛼1
(𝑟𝑖)) = 1for 𝑟 > 1 and 𝑖 = 𝑛 𝑡𝑜 ∞ 

Then, we can say to have common coincidence value for𝑓and 𝑔. 

Proof: Suppose, 𝛼0, 𝛼1 ∈ 𝑄 so that 𝑓𝛼0 = 𝑔𝛼1 and lim
𝑛→∞

𝑇𝑖 (𝐹𝑓𝛼0,𝑓𝛼1
(𝑟𝑖)) = 1     (1) 

As 𝑓(𝑄) ⊂ 𝑔(𝑄) and 𝑓𝛼0 = 𝑔𝛼1, so a sequence < 𝛼𝑛 > can be constructed to have 𝑓𝛼𝑛 = 𝑔𝛼𝑛+1 

Consider,𝑧𝑛 = 𝑓𝛼𝑛 



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Then, (1) gives, lim
𝑛→∞

𝑇𝑖 (𝐹𝑧0,𝑧1
(𝑟𝑖)) = 1 ; 𝑖 = 𝑛 𝑡𝑜 ∞. Substituting 𝛼 = 𝛼𝑛 and 𝛽 = 𝛽𝑛+1 in (𝑖), we 

get 

𝛿(𝐹𝑓𝛼𝑛,𝑓𝛼𝑛+1
(𝜃𝑥), 𝐹𝑔𝛼𝑛,𝑔𝛼𝑛+1

(𝑥), 𝐹𝑓𝛼𝑛,𝑔𝛼𝑛
(𝑥), 𝐹𝑓𝛼𝑛+1,𝑔𝛼𝑛+1

(𝜃𝑥)) ≥ 0, 

𝑖. 𝑒.  𝛿(𝐹𝑧𝑛,𝑧𝑛+1
(𝜃𝑥), 𝐹𝑧𝑛−1,𝑧𝑛

(𝑥), 𝐹𝑧𝑛−1,𝑧𝑛
(𝑥), 𝐹𝑧𝑛,𝑧𝑛+1

(𝜃𝑥)) ≥ 0 

and using virtue of 𝛿, for all 𝑛, we obtain 

𝐹𝑧𝑛,𝑧𝑛+1
(𝜃𝑥) ≥ 𝐹𝑧𝑛−1,𝑧𝑛

(𝑥) 

So, by above Lemma, the sequence < 𝑧𝑛 >= < 𝑓𝛼𝑛 > is Cauchy. 

If 𝑔(𝑄) is 𝐹- complete, then there must be 𝛼 ∈ 𝑔(𝑄), so that,< 𝑧𝑛 > → 𝛼, and for any 𝑧 ∈ 𝑄 so that 

𝑔𝑧 = 𝛼. Substituting 𝛼 = 𝛼𝑛 and 𝑦 = 𝑧, in (𝑖), we have 

𝛿(𝐹𝑓𝛼𝑛,𝑓𝑧(𝜃𝑥), 𝐹𝑔𝛼𝑛,𝑔𝑧(𝑥), 𝐹𝑓𝛼𝑛,𝑔𝛼𝑛
(𝑥), 𝐹𝑓𝑧,𝑔𝑧(𝜃𝑥)) ≥ 0 

𝑖. 𝑒. 𝛿(𝐹𝑧𝑛,𝑓𝑧(𝜃𝑥), 𝐹𝑧𝑛−1,𝑔𝑧(𝑥), 𝐹𝑧𝑛−1,𝑓𝑧(𝑥), 𝐹𝑔𝑧,𝑓𝑧(𝜃𝑥)) ≥ 0 

As 𝑛 → ∞, we get, 

𝛿(𝐹𝑔𝑧,𝑓𝑧(𝜃𝑥), 𝐹𝑔𝑧,𝑔𝑧(𝑥), 𝐹𝑔𝑧,𝑓𝑧(𝑥), 𝐹𝑓𝑧,𝑔𝑧(𝜃𝑥)) ≥ 0 

𝑖. 𝑒. 𝛿(𝐹𝑔𝑧,𝑓𝑧(𝜃𝑥), 1, 𝐹𝑔𝑧,𝑓𝑧(𝑥), 𝐹𝑓𝑧,𝑔𝑧(𝜃𝑥)) ≥ 0 

Again, as 𝐹𝑔𝑧,𝑓𝑧(𝜃𝑥) ≥ 1, so 𝑓𝑧 = 𝑔𝑧 = 𝛼 

Taking limit as n→ , we have, 

Hence, 𝑓 and 𝑔 have𝑧 as a coincidence point and thus, for 𝑓(𝑄) to be complete, we have  < 𝑧𝑛 > →

𝛼 ∈ 𝑓(𝑄) ⊂ 𝑔(𝑄) so, 𝑧 is a common coincidence point. 

Theorem 4.2: With (𝑃, 𝐹, 𝑇)a generalized PM-space and a continuous 𝑡-norm𝑇 in (𝑎, 1) for all 0 <

𝑎 < 1; 0 < 𝜃 < 1;  𝛿 ∈ ∆ and 𝑓, 𝑔: 𝑄 → 𝑃, we have 

(i). 𝛿(𝐹𝑓𝛼,𝑓𝛽(𝜃𝑥), 𝐹𝑔𝛼,𝑔𝛽(𝑥), 𝐹𝑓𝛼,𝑔𝛼(𝑥), 𝐹𝑓𝛽,𝑔𝛽(𝜃𝑥)) ≥ 0 for every 𝛼, 𝛽 ∈ 𝑄 and positive values of 𝑥 

(ii). 𝑓(𝑃) ⊂ 𝑔(𝑃) 

(iii). ∃ 𝛼0, 𝛼1 so that 𝑓𝛼0 = 𝑔𝛼1 and lim
𝑛→∞

𝑇𝑖 (𝐹𝑓𝛼0,𝑓𝛼1
(𝑟𝑖)) = 1for 𝑟 > 1 and 𝑖 = 𝑛 𝑡𝑜 ∞ 

(iv). Either 𝑓(𝑃) or 𝑔(𝑃) is 𝐹- complete 

(v).  𝑓 and 𝑔 are commuting functions that commute at their coincidence point. 

Then, 𝑓 and 𝑔 have a common invariant point that is unique in, 𝑃. 

Proof: Let us choose 𝑃 = 𝑄 in previous theorem, then we obtain 𝑧𝑛 = 𝑓𝛼𝑛 and sequence < 𝑧𝑛 > is 

Cauchy. If 𝑔(𝑃) is 𝐹- complete, then 𝑧𝑛 → 𝛼 ∈ 𝑔(𝑃), and we have 𝑧 ∈ 𝑃 so that 𝑔(𝑧) = 𝛼. Replacing 

𝛼 by 𝛼𝑛 and 𝛽 by 𝑧 in inequality(𝑖) we have; 𝑓𝑧 = 𝑔𝑧 = 𝛼. As 𝑓 and 𝑔 are commuting, therefore, we 

must have, 𝑓𝑔𝑧 = 𝑔𝑓𝑧, so that,𝑓𝛼 = 𝑔𝛼. 

Again, using 𝛼 = 𝑧 and 𝛽 = 𝑓𝑧 in inequality (𝑖), we get 

𝛿(𝐹𝑓𝑧,𝑓𝑓𝑧(𝜃𝑥), 𝐹𝑔𝑧,𝑔𝑓𝑧(𝑥), 𝐹𝑔𝑧,𝑓𝑧(𝑥), 𝐹𝑓𝑓𝑧,𝑔𝑓𝑧(𝜃𝑥)) ≥ 0 



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𝑖. 𝑒. 𝛿(𝐹𝛼,𝑓𝛼(𝜃𝑥), 𝐹𝛼,𝑓𝛼(𝑥), 1, 𝐹𝑓𝛼,𝑔𝛼(𝜃𝑥)) ≥ 0 

Using the properties of distance metric(𝛿) for all positive,𝑥 we obtain, 𝐹𝛼,𝑓𝛼(𝜃𝑥) ≥ 𝐹𝛼,𝑓𝛼(𝑥) which is 

true only if 𝛼 = 𝑓𝛼, otherwise we reach to a contradiction with respect to Lemma 4.1. This implies 

that, 𝛼 is a common invariant point of 𝑓 and 𝑔. 

To prove the uniqueness, we suppose 𝛼 and 𝛽 are two common invariant points of 𝑓 and 𝑔. 

Using (𝑖), we get 

𝛿(𝐹𝛼,𝛽(𝜃𝑥), 𝐹𝛼,𝛽(𝑥), 𝐹𝛼,𝛼(𝑥), 𝐹𝛽,𝛽(𝜃𝑥)) ≥ 0⇒ 𝐹𝛼,𝛽(𝜃𝑥) ≥ 1 ⇒ 𝛼 = 𝛽. 

This shows the uniqueness of the result. 

Definition 4.1: [6] Aforementioned t–norm T is said Hadžić type or 𝐻- type, denoted as 𝑇 ∈ 𝐻, if the 

group {𝑇𝑛}𝑛∈𝑁 of it iterates is inductively depicted, ∀𝑥 ∈ [0,1] by, 

𝑇0(𝑥) = 1, 𝑇𝑛+1(𝑥) = 𝑇(𝑇𝑛(𝑥), 𝑥), for all non-negative 𝑥 and as 𝑇𝑛 is defined, it is equi-continuous 

at 𝑥 = 1. This states the following, 

휀 ∈ (0,1)∃ a 𝜗 ∈ (0,1), so that, 𝑥 > 1 − 𝜗⇒ 𝑇𝑛(𝑥) > 1 − 휀, for all 𝑛 ≥ 1. 

The statement is followed by Hadžić: 

Statement 4.1: [6] 

(i). If we take 𝑇 ≥ 𝑇𝐿, then for 𝑖 = 𝑛 𝑡𝑜 ∞, 

lim
𝑛→∞

𝑇𝑖𝑥𝑛+1 = 1, both sides imply ∑ 1 − 𝑥𝑛 < ∞ and 𝑇 = 𝑇µ
𝑆𝑊 

(By 𝑇 ≥ 𝑇𝐿 𝑖. 𝑒. 𝑎 ≥ 𝑐, 𝑏 ≥ 𝑑⇒ 𝑇(𝑎, 𝑏) = 𝑇(𝑐, 𝑑)). 

(ii). When, 𝑇 ∈ 𝐻, then for each (𝑥𝑛)𝑛∈𝑁 in [0,1], lim
𝑛→∞

𝑥𝑛 = 1⇒ lim
𝑛→∞

𝑇𝑖𝑥𝑛+𝑖 = 1,for 𝑖 = 𝑛 𝑡𝑜 ∞. 

(iii). When, 𝑇 ∈ 𝑇𝜇
𝐷, 𝑇𝜇

𝐴𝐴𝑇, then for 𝑖 = 𝑛 𝑡𝑜 ∞,  lim
𝑛→∞

𝑇𝑖𝑥𝑛+𝑖 = 1, both sides imply ∑(1 − 𝑥𝑛)𝜇 <

∞. 

We wish to repeat the notion of distribution functions and PMS for the interval[0, ] .  

Definition 4.2: [6] Consider a class of distribution map𝐿+as 𝐹: [0, ∞] → [0, ∞] with, 

(i). 𝐹 has zero value at 𝑥 = 0 

(ii). 𝐹 is a non-decreasing map and 

(iii). 𝐹 is left continuous for 0 < 𝑥 < ∞. 

Let 𝐷+ ⊂ 𝐿+ contains𝐹, so that, lim
𝑛→∞

𝐹(𝑥) = 1. Then, the distribution function, 휀0 is defined as 휀0 

휀0 = {
0, 𝑥 = 0
1, 𝑥 > 0

 lies in 𝐷+. 

Statement 4.2: Consider (𝑃, 𝐹, 𝑇) asa generalized PM-space and a continuous 𝑡-norm𝑇 ∈ 𝐻 with 0 <

𝜃 < 1;  𝛿 ∈ ∆ and 𝑓, 𝑔: 𝑃 → 𝑃, so that, 

(i). 𝛿(𝐹𝑓𝛼,𝑓𝛽(𝜃𝑥), 𝐹𝑔𝛼,𝑔𝛽(𝑥), 𝐹𝑓𝛼,𝑔𝛼(𝑥), 𝐹𝑓𝛽,𝑔𝛽(𝜃𝑥)) ≥ 0 for every 𝛼, 𝛽 ∈ 𝑃 and positive values of 𝑥 

(ii).  𝑓(𝑃) ⊂ 𝑔(𝑃) 

(iii). ∃ 𝛼0, 𝛼1 so that 𝑓𝛼0 = 𝑔𝛼1 for which 𝐹𝑓𝛼0,𝑓𝛼1
∈ 𝐷+ 

(iv). Either 𝑓(𝑃) or 𝑔(𝑃) is 𝐹- complete 



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(vi). 𝑓 and 𝑔 are commuting functions that commute at their coincidence point. 

Then, 𝑓 and 𝑔 have a unique common invariant point in 𝑃. 

Proof: In Theorem 4.1, if we consider a 𝜇 > 1 with a sequence < 𝑧𝑛 > along with a mapping, 𝑓𝑧0,𝑧1
∈

𝐷+, so that, lim
𝑛→∞

𝐹𝑧0,𝑧1
(𝜇𝑛) = 1. Therefore, using (𝑖𝑖) of Statement 4.1, we get 

lim
𝑛→∞

𝑇𝑖 (𝐹𝑧0,𝑧1
(𝜇𝑖)) = 1; 𝑖 = 𝑛 𝑡𝑜 ∞. 

Hence, Theorem 4.2 follows the required results. 

5. Discussion 

Probabilistic spaces offer a powerful framework for handling the uncertainties and variability inherent 

in real-world data [12]. By incorporating probability distributions into the metric space, we can model 

the stochastic nature of data more effectively. The Jungck contraction theorem provides a robust tool 

for analysing the convergence of iterative processes in probabilistic metric spaces, which is crucial for 

the stability and reliability of ML algorithms. This work has demonstrated the application of 

probabilistic metric spaces and the Jungck contraction theorem in enhancing the robustness and 

convergence properties of ML algorithms. Through case studies on probabilistic K-means clustering 

and probabilistic neural networks, we have illustrated the practical benefits of this approach. Future 

research can extend these concepts to other ML algorithms and explore the integration of probabilistic 

metric spaces with other probabilistic modelling techniques. 

In a nutshell, the probabilistic metric space approach, underpinned by the Jungck contraction theorem, 

offers a promising direction for developing more robust and reliable ML algorithms in the face of 

uncertainty and variability. 

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