Implicit relations framework for fixed-point theorems in probabilistic metric spaces

Introduction

Fixed point theory is an important area of nonlinear analysis and has wide applications in differential equations, optimization, game theory, and dynamic programming. One of the most fundamental results in this area is the Banach contraction principle, which guarantees the existence and uniqueness of a fixed point for a contraction mapping defined on a complete metric space. In order to deal with situations involving uncertainty in measuring distances, probabilistic metric spaces were introduced by Menger. In these spaces, the distance between two points is described by a probability distribution function instead of a real number. This concept generalizes classical metric spaces and has been widely studied by many researchers. Several extensions of fixed-point results in probabilistic metric spaces have been obtained by imposing various conditions such as commuting mappings, weak compatibility, and occasionally weak compatibility. Researchers such as Jungck, Rhoades, Popa, and Dhage developed important fixed-point theorems under different contractive conditions. Recently, implicit relation techniques have been used to obtain more generalized fixed-point results without imposing strong compatibility conditions. Motivated by these developments, in this paper we establish a common fixed-point theorem in probabilistic metric spaces using implicit relations. The obtained results generalize several existing results in the literature.

Preliminaries

The fundamental definitions and ideas required to derive fixed-point findings for probabilistic metric spaces in terms of implicit relations are provided in this section. These preliminaries lay the groundwork for our understanding of the critical aspects of self-maps, contractive conditions, and convergence criteria probabilistically.

Definition 2.1 (Probabilistic Metric Space)

Let 𝒀𝒀 be a non-empty set. A probabilistic metric space is a triple (𝑌𝑌, 𝜎𝜎, * ) where

* is a continuous triangular norm (t-norm), satisfying the following conditions for all 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 ∈ 𝑌𝑌and 𝑡𝑡, 𝑠𝑠 > 0: 1. 𝜎𝜎 𝑥𝑥𝑥𝑥 (0) = 0 2. 𝜎𝜎 𝑥𝑥𝑥𝑥 (𝑡𝑡) = 1for all 𝑡𝑡 > 0if and only if 𝑥𝑥 = 𝑦𝑦 3. 𝜎𝜎 𝑥𝑥𝑥𝑥 (𝑡𝑡) = 𝜎𝜎 𝑥𝑥𝑥𝑥 (𝑡𝑡) 4. 𝜎𝜎 𝑥𝑥𝑥𝑥 (𝑡𝑡 + 𝑠𝑠) ≥ * (𝜎𝜎 𝑥𝑥𝑥𝑥 (𝑡𝑡), 𝜎𝜎 𝑥𝑥𝑥𝑥 (𝑠𝑠))

Definition 2.2 Distribution Function

Definition 2.5 Cauchy Sequence

A sequence 𝑥𝑥 𝑛𝑛 is Cauchy if for every 𝑡𝑡 > 0

Definition 2.6 Implicit Relations

An implicit relation is a function

satisfying suitable contractive conditions involving the probabilistic metric values 𝜎𝜎 𝑥𝑥𝑥𝑥 (𝑡𝑡). Implicit relations allow the formulation of fixed-point results without requiring explicit contraction conditions.

Definition 2.7 (Completeness)

A probabilistic metric space (𝑌𝑌, 𝜎𝜎, * )is said to be complete if every Cauchy sequence in 𝑌𝑌converges to a point in 𝑌𝑌.. The background required to contextualize the subsequent theoretical findings for fixed points in probabilistic metric spaces will be established by these preambles. Specifically, we provide a few expansions of conventional fixed-point theorems that guarantee fixed points' existence and uniqueness. Under broader circumstances via implications and contractive mappings. The property of completeness gives us convergence of iterative sequences, forming a solid basis for the later study of probabilistic metric spaces.

Function G Satisfying Conditions

The following examples illustrate functions 𝐺𝐺 that satisfy the conditions 𝐺𝐺 1 , 𝐺𝐺 2 , 𝐺𝐺 3 , as detailed in [15]. These functions serve as a basis for testing implicit relations in the probabilistic metric space context. where 𝜆𝜆 > 0, 𝜇𝜇, 𝜈𝜈, 𝜌𝜌 ≥ 0, 𝜆𝜆 + 𝜇𝜇 < 1, and 𝜆𝜆 + 𝜈𝜈 + 𝜌𝜌 < 1.

Example 2.4

Define 𝐺𝐺: 𝑅𝑅 + 6 → 𝑅𝑅 as:

Example 2.5

Define 𝐺𝐺: 𝑅𝑅 + 6 → 𝑅𝑅 as:

where 𝜆𝜆 > 0, 𝜇𝜇 ≥ 0, and 𝜆𝜆 + 𝜇𝜇 < 1.

Implicit Relation and Convergence

The implicit relation introduced in our modified theorem is given by:

This connection ensures that the iteration sequence will converge, regardless of whether or not the mappings and commute, weakly commute, or are compatible between the mappings 𝑈𝑈 and 𝑉𝑉. Using these generalized conditions, the theorem extends classical results and also is helpful for providing a flexible route towards fixed-point theory in probabilistic metric spaces.

Main Theorem

Let A and B be self-maps on Y, and let (Y,σ,*) be a complete probabilistic metric space. Assume that for every u,v in Y, the following condition is true due to an implicit relation Ψ:

where 𝑒𝑒 is the probabilistic metric induced by 𝜎𝜎. Furthermore, assume the following conditions: a) Contractive Condition: 𝑒𝑒(𝐴𝐴𝐴𝐴, 𝐴𝐴𝐵𝐵) ≤ 𝜎𝜎(𝑒𝑒(𝐴𝐴, 𝐵𝐵)) for all 𝐴𝐴, 𝐵𝐵 ∈ 𝑌𝑌, where 𝜎𝜎 is a non-decreasing contractive function satisfying 𝜎𝜎(𝑡𝑡) < 𝑡𝑡 for all 𝑡𝑡 > 0. b) Continuity: 𝐴𝐴 and 𝐵𝐵 are continuous. c) Convergence Condition: For any sequence �𝐴𝐴 𝑝𝑝 � in 𝑌𝑌, if 𝑒𝑒�𝐴𝐴 𝑝𝑝 , 𝐴𝐴𝐴𝐴 𝑝𝑝 � → 0 as 𝑛𝑛 → ∞, then 𝐴𝐴𝐴𝐴 𝑝𝑝 converges to a point 𝑞𝑞 ∈ 𝑌𝑌.

Under these assumptions, because the mappings A and B share a single fixed point in Y, there is a unique 𝑞𝑞 ∈ 𝑌𝑌 such that: 𝐴𝐴𝑞𝑞 = 𝐵𝐵𝑞𝑞 = 𝑞𝑞 Proof Let 𝐴𝐴 0 ∈ 𝑌𝑌 be an arbitrary point. Define the sequence �𝐴𝐴 𝑝𝑝 � by:

Applying the implicit relation, for all 𝐴𝐴, 𝐵𝐵 ∈ 𝑌𝑌, we have:

Applying this to 𝐴𝐴 𝑛𝑛 and 𝐴𝐴 𝑛𝑛+1 , we obtain:

Since 𝐴𝐴 𝑝𝑝+1 = 𝐴𝐴𝐴𝐴 𝑝𝑝 , this simplifies to:

By the properties of the probabilistic metric 𝑒𝑒 𝑟𝑟 we assume: ), which is non-decreasing and satisfies 𝜎𝜎(𝑡𝑡) < 𝑡𝑡 for all 𝑡𝑡 > 0. The next step is to demonstrate that �𝐴𝐴 𝑝𝑝 � is a Cauchy sequence. We suppose that the sequence �𝐴𝐴 𝑝𝑝 � is not a Cauchy sequence for the sake of contradiction. By definition, a sequence {𝐴𝐴 𝑛𝑛 } is Cauchy if:

Since we are assuming that {𝐴𝐴 𝑛𝑛 } is not a Cauchy sequence, this means there exists some 𝜖𝜖 > 0 such that:

This means that no matter how far we go in the sequence, there are two terms that remain at least 𝜖𝜖 apart. We are given that 𝐴𝐴 is a contractive mapping with a contraction constant 𝜎𝜎, where: 𝑒𝑒(𝐴𝐴𝑥𝑥, 𝐴𝐴𝑦𝑦) ≤ 𝜎𝜎𝑒𝑒(𝑥𝑥, 𝑦𝑦) for all 𝑥𝑥, 𝑦𝑦 This property tells us that applying 𝐴𝐴 to two elements 𝑥𝑥, 𝑦𝑦 brings them closer together by a factor of at most 𝜎𝜎. We have a sequence {𝐴𝐴 𝑛𝑛 } defined by an iterative process:

We want to analyze the difference between successive terms in this sequence, which means we consider 𝑒𝑒�𝐴𝐴 𝑝𝑝+1 , 𝐴𝐴 𝑝𝑝 �. Using the definition of the sequence:

Now, applying the contraction property 𝑒𝑒(𝐴𝐴𝑥𝑥, 𝐴𝐴𝑦𝑦) ≤ 𝜎𝜎𝑒𝑒(𝑥𝑥, 𝑦𝑦) with 𝑥𝑥 = 𝐴𝐴 𝑛𝑛 and 𝑦𝑦 = 𝐴𝐴 𝑛𝑛-1 , we obtain:

We apply this inequality recursively:

Observing this pattern, we see that each step reduces the distance by another factor of 𝜎𝜎.

Generalization by Induction

𝑒𝑒�𝐴𝐴 𝑝𝑝+𝑘𝑘 , 𝐴𝐴 𝑝𝑝+𝑘𝑘-1 � ≤ 𝜎𝜎 𝑘𝑘 𝑒𝑒�𝐴𝐴 𝑝𝑝 , 𝐴𝐴 𝑝𝑝-1 �

We now prove by induction that for any ≥ 1 :

This is already given by the contraction property:

We need to prove the formula holds for 𝑘𝑘 + 1, i.e., 𝑒𝑒�𝐴𝐴 𝑝𝑝+𝑘𝑘+1 , 𝐴𝐴 𝑝𝑝+𝑘𝑘 � ≤ 𝜎𝜎 𝑘𝑘+1 𝑒𝑒�𝐴𝐴 𝑝𝑝 , 𝐴𝐴 𝑝𝑝-1 � Using the contraction property:

Now, using the inductive hypothesis:

Thus, by mathematical induction, we conclude:

Applying for 𝑛𝑛 = 1 Finally, setting 𝑛𝑛 = 1, we obtain: 𝑒𝑒(𝐴𝐴 1+𝑘𝑘 , 𝐴𝐴 1+𝑘𝑘-1 ) ≤ 𝜎𝜎 𝑘𝑘 𝑒𝑒(𝐴𝐴 1 , 𝐴𝐴 0 ) Since 0 ≤ 𝜎𝜎 < 1, we know that 𝜎𝜎 is a number strictly less than 1 . Now, consider the sequence 𝜎𝜎 𝑘𝑘 , which is the repeated multiplication of 𝜎𝜎 with itself:

Since 𝜎𝜎 is a fraction (i.e., 0 ≤ 𝜎𝜎 < 1 ), multiplying it by itself repeatedly makes it smaller and smaller:

This follows from a well-known mathematical property of sequences:

Since 𝜎𝜎 satisfies this condition, we conclude:

Since we already showed that 𝜎𝜎 𝑘𝑘 → 0 as 𝑘𝑘 → ∞, we multiply this by 𝑒𝑒(𝐴𝐴 1 , 𝐴𝐴 0 ), which is a fixed number (let's call it 𝐶𝐶 ):

𝜎𝜎 𝑘𝑘 𝐶𝐶

Since 𝜎𝜎 𝑘𝑘 → 0 and 𝐶𝐶 is a constant, we apply the standard limit property:

Since 𝑒𝑒�𝐴𝐴 𝑝𝑝+𝑘𝑘 , 𝐴𝐴 𝑝𝑝+𝑘𝑘-1 � ≤ 𝜎𝜎 𝑘𝑘 𝑒𝑒(𝐴𝐴 1 , 𝐴𝐴 0 ), and we just proved that:

This means that for any given 𝜖𝜖 > 0, we can find a sufficiently large 𝑘𝑘 such that:

This contradicts our assumption that {𝐴𝐴 𝑛𝑛 } is not a Cauchy sequence. Hence, {𝐴𝐴 𝑛𝑛 } must be Cauchy. Y is complete, hence there is a q in Y such that:

Since 𝐴𝐴 is continuous, taking limits on both sides gives:

From the sequence definition, we have 𝐴𝐴𝐴𝐴 𝑝𝑝 = 𝐴𝐴 𝑝𝑝+1 , so taking the limit:

Similarly, since 𝐵𝐵 satisfies the implicit relation, we also have:

Consequently, q is a fixed point that both A and B share. To prove uniqueness, suppose that there exist two fixed points 𝑞𝑞, 𝑟𝑟 ∈ 𝑌𝑌 such that:

Applying the implicit relation, we get: Ψ(𝐴𝐴𝑞𝑞, 𝐵𝐵𝑟𝑟, 𝑒𝑒(𝑞𝑞, 𝑟𝑟)) = Ψ(𝑞𝑞, 𝑟𝑟, 𝑒𝑒(𝐴𝐴𝑞𝑞, 𝐵𝐵𝑟𝑟))

Since 𝐴𝐴𝑞𝑞 = 𝑞𝑞 and 𝐵𝐵𝑟𝑟 = 𝑟𝑟, this simplifies to:

This equation holds trivially, but to prove uniqueness, we must show that 𝑒𝑒(𝑞𝑞, 𝑟𝑟) = 0. Using the properties of Ψ and: a) Non-negativity of: In probabilistic metric spaces, 𝑒𝑒(𝑞𝑞, 𝑟𝑟) ≥ 0. b) Strict Contractiveness of Ψ: If 𝑒𝑒(𝑞𝑞, 𝑟𝑟) > 0, then Ψ(𝑞𝑞, 𝑟𝑟, 𝑒𝑒(𝑞𝑞, 𝑟𝑟)) should yield a strictly decreasing value, which contradicts the equality.

Thus, the only possible solution is:

This entirely confirms the uniqueness of the fixed point. This shows that the fixed point is unique, which indicates that no other fixed points can exist. This guarantees that the solution set is stable and consistent, further solidifying the robustness of the theorem.

The presence of a single shared fixed-point results from the self-maps A and B on Y satisfying the contractive and implicit relation criteria under the specified circumstances. The Cauchy property and completeness of 𝑌𝑌 ensure convergence, and the implicit relation guarantees uniqueness. Hence, the theorem is proven.

Example 3.1

Let 𝑌𝑌 = [0,1]and consider the probabilistic metric space (𝑌𝑌, 𝜎𝜎, * ), and let 𝐴𝐴 and 𝐵𝐵 be operators defined as follows:

Here, we have two nonlinear operators, and we need to verify the conditions of the modified theorem and prove convergence and uniqueness of the fixed point. We apply the implicit relation for 𝑦𝑦 𝑛𝑛 and 𝑦𝑦 𝑛𝑛+1 in the sequence. The implicit relation is:

Rewriting in terms of distance:

This shows that the contraction depends nonlinearly on both the difference 𝑒𝑒(𝑦𝑦 𝑛𝑛 , 𝑦𝑦 𝑛𝑛+1 ) and the sum 𝑦𝑦 𝑛𝑛 + 𝑦𝑦 𝑛𝑛+1 . For (𝑦𝑦) :

This shows that 𝐵𝐵 has a strict contraction rate of 1/3. Thus, 𝐵𝐵 is a linear contraction, while 𝐴𝐴 is a nonlinear contraction requiring further verification. Nevertheless, both exhibit distance reduction as 𝑛𝑛 increases.

Applying the contractive condition iteratively: For 𝐵𝐵 :

For 𝐴𝐴, the bound is more complex:

While this does not simplify to a geometric progression like 𝐵𝐵, it still shows that successive distances decrease. Since both operators are contractions (though 𝐴𝐴 is more complex), the sequence {𝑦𝑦 𝑛𝑛 } will converge to a fixed point. Let 𝑞𝑞 be the limit of {𝑦𝑦 𝑛𝑛 }. By the continuity of 𝐴𝐴 and 𝐵𝐵 𝑡𝑡 we have:

Thus, 𝑞𝑞 is a fixed point of both operators.

for all 𝑦𝑦 ∈ 𝑌𝑌. If A and B If both A and B meet the continuity criteria and the contractive requirement, they will have a single fixed point in common. This example uses certain contraction conditions and implicit relations to demonstrate that there is a single shared fixed point in a probabilistic metric space. Because of the iterative contraction and the non-linear structure of A, all of the sequences converge to the same limit. The fixed-point theorem's universal application under implicit relations is further supported by this conclusion.

Conclusion

We have proved here the generalization of some well-known common fixed-point theorem in probabilistic metric spaces. The major improvement of this work is to eliminate the usual assumptions, e.g. commutativity, weak commutativity, compatibility conditions, required for the existence and uniqueness of fixed points. Using implicit relations and contractive conditions of integral type, we generalised classical results and opened them up to more flexibility and applicability in a more general setting. We extend previous work (in particular, S. R. Dhage work on contractive mappings in probabilistic metric spaces) that requested compatibility conditions on mappings for getting fixed points. Removing this constraint from the approach discussed in the work aforementioned improves the general applicability of the theorem. Besides that, our study investigated fixed point theorems under contractive circumstances in probabilistic and fuzzy fields. Their results are based on particular compatibility conditions; on the other hand, our scheme does not make such assumptions, which generalizes the applicability of fixed-point theory.

Implications and Future Work

Such results are important for problems related to probabilistic metric spaces, and thus the consequences of this paper are considered substantial for the fixed-point theory. Our approach gives us the freedom to extend even further to fuzzy metric spaces and beyond, allowing for new generalisations which may unlock future avenues within applied mathematics. Future research directions might be: a) Dealing with most general classes of spaces: fuzzy metric spaces, non-Archimedean metric spaces, etc. b) Examining conditions for the stability and convergence of iterated sequences based on contractive integral types. c) Developing its applications in real life, particularly in economics, biology, and decision theory, as there is uncertainty present in these subjects.

In conclusion, this work expands the generality and usefulness of fixed-point theorems, and presents a very powerful and flexible ground for future mathematical and interdisciplinary innovations.

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