1 Introduction

Fixed point, which is referred to as FP theory plays a crucial role in the advancement of nonlinear functional analysis, offering substantial insights and applications across various fields. Its flexibility and wide-ranging applicability make it a vibrant area of research. FP theory investigates the conditions under which a self-map on a set can possess a FP, providing significant theoretical and practical results. One of the most profound contributions to FP theory was made by the renowned mathematician Stefan Banach. His ground breaking result [1] established that in a complete metric space, which is referred to as CMS contractive mappings have a unique FP. This principle, known as the Banach Contraction Principle, which is referred to as BCP has become a fundamental theorem with extensive applications in mathematics and related disciplines. The BCP has been modified and extended in numerous ways to address various mathematical and real-world problems, (see for example [2]).

The BCP’s utility spans many scientific and technical domains, leading to the development of various contractive mappings in different types of MS. Recent advancements, such as those by Hussain et al. [3–6], highlight significant progress in both theoretical and applied aspects of FP theory. A notable extension of this work is the introduction of b-MS initially conceptualized by Bakhtin [7] and Bourbaki [8], and later formalized by Czerwik [9]. A b-MS is a generalization of the traditional MS that relaxes the strict triangle inequality. In a b-MS, the distance function d(q₁, q₂), satisfies a weaker form of the triangle inequality, which can be expressed as: where b ≥ 1 is a constant. This relaxation allows for broader applications and provides new insights into FP theory. In recent developments, Kamran et al. [10] extended the FP results to self-mappings in generalized b-MSs, further expanding the applicability of the BCP. Subsequent research has explored FP results in b-MSs for both single-valued and multi-valued operators (see [11–20]). These advancements in b-MSs highlight a significant shift from traditional MSs, offering new perspectives and tools for solving problems in FP theory. The study of b-MSs continues to contribute to the broader understanding of FP and their applications, underscoring the ongoing relevance and dynamism of this field.

In this paper, we extend the FP results of Wardowski [21] to b-MSs, which are a generalization of MSs. To achieve this, we introduce the concept of β-(θ, ϑ)-contraction, incorporating the coefficient b from b-MSs into our analysis. Our results demonstrate a broader applicability compared to Wardowski’s findings.

The paper is organized into two main sections:

In the first section, we address gaps in the existing literature by applying Nadler’s work to provide concrete examples. We also compare the efficiency of our results with those of Nadler, highlighting the advancements and improvements our approach offers.

The second section focuses on our FP results for β-(θ, ϑ)-contractions within complete b-MSs. We provide examples and applications, particularly in optimization problems, to demonstrate the effectiveness and advantages of our results.

The final section summarizes our findings and conclusions, emphasizing the contributions and implications of our research.

2 Preliminaries

This section covers the preliminary findings and analysis that support the article’s main conclusions.

Definition 2.1. [10] Let χ be a nonempty set, and let d be a function. The function d: χ × χ → [0, ∞) is called b-metric if it satisfies the following conditions:

1. 0 ≤ d(q₁, q₂) and d(q₁, q₂) = 0 if and only if q₁ = q₂,
2. d(q₁, q₂) = d(q₂, q₁),
3. d(q₁, q₃) ≤ b[d(q₁, q₂) + d(q₂, q₃)] for some b ≥ 1,

for all q₁, q₂, q₃ ∈ χ. The pair (χ, d) is called b-MS with coefficient b.

All MSs can be considered as b-MS with b = 1. The class of b-MSs encompasses a broader range than that of MSs, and the concept of a b-MS generalizes the notion of an MS. Specifically, the b-MS framework is a generalization that is less restrictive than the traditional MS framework. While conditions (1) and (2) in a b-MS are similar to those in an MS condition (3), introduces a crucial component. Mastery of the effective application of condition (3) is essential for fully understanding this concept. For a clearer illustration of the importance of the third criterion, see Example 2.2.

Example 2.2. [9] Let L_p[0, 1] be the space of all real functions. Let p ∈ (0, 1) and q(O) be such that .

Define d: L_p[0, 1] × L_p[0, 1] → [0, ∞) as: for each ∈ L_p[0, 1] of q₁, q₂. Then, (L_p[0, 1]d) is a b-MS and is b-metric coefficient.

Definition 2.3. [10] Let (χ, d) be a b-MS with b ≥ 1.

1. If there exists such that as j → ∞, then the sequence is said to be b-convergent.
2. If as j, i → ∞, then a sequence is a b-Cauchy sequence.
3. A b-MS (χ, d) is said be complete b-MS if every b-Cauchy sequence in χ is b-convergent.

Hausdorff originally presented the idea of the Hausdorff metric or Hausdorff distance in his work Grundzuge der Mengenlehre [22]. Pompeiu-Hausdorff distance is the second name for the Hausdorff distance. In the realm of computers, there are numerous uses for the Hausdorff distance. In computer vision, the Hausdorff distance is used to locate a specified template in any target image.The Hausdorff metric is most frequently used in computer graphics to assess the difference between two different representations of the same 3D object when creating the level of detail required for the effective display of complex 3D models.

Definition 2.4. [15] Let (κ, d) be a MS and let K(κ) be the class of all nonempty compact subsets of κ. ζ: K(κ) × K(κ) → [0, ∞) is the definition of this mapping: is referred to as the Pompeiu-Hausdroff metric, induced by metric d and d(q₁, Z) = inf{d(q₁, q₂): q₂ ∈ Z} is the distance from q₁ to Z ⊆ κ.

Example 2.5. Let is a set of real numbers with metric with b = 1. Then for any two closed intervals [Q, q₁] and [q₂, q₃], we have

Definition 2.6. [21] Suppose that Ψ is collection of all functions with following conditions:

1. The value of ϑ is strictly increasing.
2. In the interval (0, +∞), for any sequence {σ_j}, the following condition holds:
3. exists for O ∈ [0, 1].

Definition 2.7. [21] Let (κ, d) be a MS. If there exist and ϑ ∈ Ξ such that for any q₁, q₂ ∈ κ, we have a mapping Φ: κ → κ that is ϑ-contraction.

Remember that a contraction is always a ϑ-contraction.

Theorem 2.8. [21] Assume that κ is a CMS and we have ϑ-contraction Φ: κ → κ. Then, for every point ℑ ∈ κ, the sequence {F^jℑ} converges to ℜ, and Φ has a single FP ℜ ∈ κ.

In 2012, Samet et al. [23] defined β-admissible for single-valued mappings.

Definition 2.9. [21] Assume that the set κ is not empty.

1. When a mapping β: κ × κ → [0, ∞) exists and , then Φ: κ → κ is β-admissible.
2. Let κ be a β-regular. If for any sequence {q_1,j} in κ, we have q_1,j ∈ κ and β(q_1,j, q_1,j+1) ≥ 1 for all , then it follows that β(q_1,j, q₁) ≥ 1 for all .

The following is the definition of β-admissibility for multivalued mappings, as given by Mohammad et al. [24] in 2013:

Definition 2.10. [24] Assume that κ is a nonempty set, and let 2^κ denote the set of all nonempty subsets of κ. If a function β: κ × κ → [0, ∞) exists, then a multivalued mapping Φ: κ → 2^κ is β-admissible if for every q₁ ∈ κ and q₂ ∈ Φ(q₁), the following conditions are satisfied: and

Definition 2.11. [21] Consider the set of all functions q₃ defined by . The functions in this set satisfy the following prerequisites:

1. For all O > 0, we have where denotes the j-th iterate of the function q₃ applied to O.
2. The function satisfies
3. The function q₃ is upper semi-continuous and nondecreasing.

Definition 2.12. [21] Consider a nonempty set κ. Suppose there exists a function β: κ × κ → [0, ∞) such that two multivalued mappings Φ and Ξ, where Φ, Ξ: κ → 2^κ, are β-admissible. The following conditions are satisfied:

1. For each q₂ ∈ κ and , we have
   Consequently, for each q₁ ∈ Ξ_Q, it follows that
2. For each q₂ ∈ κ and , we have
   Consequently, for each q₁ ∈ Φ_Q, it follows that

Definition 2.13. [21] A function β: κ × κ → [0, ∞) is said to be symmetric if it satisfies the following condition: for all q₁, q₂ ∈ κ, if β(q₁, q₂) ≥ 1, then it must also be true that

Definition 2.14. [21] Assume that the set κ is nonempty. If there exists a symmetric function β: κ × κ → [0, ∞), then the pair of multivalued mappings Φ and Ξ, where Φ, Ξ: κ → 2^κ, is said to be symmetric β-admissible. This means that Φ and Ξ are β-admissible in the sense that:

1. For every q₁ ∈ κ and q₂ ∈ Φ(q₁), if β(q₁, q₂) ≥ 1, then it follows that β(q₂, q₁) ≥ 1.
2. For every q₂ ∈ κ and Q ∈ Ξ(q₂), if β(q₂, Q) ≥ 1, then it follows that β(Q, q₂) ≥ 1.

An extended generalization of BCP was proposed by Wardowski [21]. Subsequent researchers have explored various modifications of Wardowski’s contraction principle for both single-valued and multi-valued mappings [11, 17, 25].

Definition 2.15. [21] Let (κ, d) be a MS. Suppose there exist functions β: κ × κ → [0, ∞), θ ∈ Θ, and ϑ ∈ Ψ such that for all q₁, q₂ ∈ κ where , the following inequality holds: where

Additionally, we assume that β(q₁, q₂) ≥ 1.

Theorem 2.16. [21] Let Φ, Ξ: κ → K(κ) be a pair of mappings such that (Φ, Ξ) is a β-(θ, ϑ)-contraction. Assume that (κ, d) is a CMS and the following conditions are met:

1. There exists q_1,o ∈ κ such that β(q_1,o, q_1,1) ≥ 1 and .
2. The pair (Φ, Ξ) is symmetric β-admissible.

Under these conditions, Φ and Ξ will have a common fixed point if one of the following is true:

1. (a) Both Φ and Ξ are continuous.
2. (b) The set κ is β-regular and the function ϑ is continuous.

3 Main results

This section is divided into two parts: The first part provides examples and applications of Nadler’s work [26] to offer a comparative analysis and address gaps in the existing literature. In the second part, we present and prove our main results concerning the existence of common FPs in b-MSs for multivalued β-(θ, ϑ)-contractions. These results lead to the restoration of the concept of multivalued contractions within the framework of b-MSs.

Nadler [26] expanded the BCP in 1969 in the following ways:

Theorem 3.1. [26] Let Φ: κ → K(κ) be a multivalued mapping and let (κ, d) be a CMS. Suppose that for all q₁, q₂ ∈ κ, the following condition holds: where 0 ≤ g ≤ 1.

Then, in the space κ, there exists at least one FP of Φ.

Example 3.2. Take a MS (κ, d), where d is the standard Euclidean distance on , and ζ is the set of real numbers . is a multivalued mapping in which the family of compact subsets of is denoted by .

As per the criteria specified in Theorem 3.1, we possess

We now select a value of g and a particular multivalued mapping Φ:

Mapping Φ:

Let Φ(q₁) be the closed interval [q₁, q₁ + 1] for each real number . This implies that the set Φ(q₁) contains all real numbers between q₁ and q₁ + 1, inclusive.

Value of g:

Our selection is .

As stated in Theorem 3.1, we now need to show that there exists at least one fixed point of Φ in .

Proof. Consider any real number . Our goal is to show that for every point q₂ ∈ Φ(q₁), the distance d(q₁, q₂) is less than or equal to times the distance e(q₁, q₂).

To illustrate this, let us take a specific point . This point belongs to the set Φ(q₁) = [q₁, q₁ + 1].

The distance between q₁ and q₂ is calculated as:

Now, let’s compute times this distance:

Since is not greater than , this shows:

Therefore, we have found a FP where the distance d(q₁, q₂) satisfies:

This demonstrates that the mapping Φ(q₁) has a FP within the CMS , according to the conditions of Theorem 3.1. Thus, Theorem 3.1 guarantees the existence of at least one FP in the CMS for the given multivalued mapping Φ. The distances d(q₁, q₂) for various values of q₁ and q₂ are summarized in Table 1.

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Table 1. Values of , , and for Example 3.2.

https://doi.org/10.1371/journal.pone.0313033.t001

3.1 Biological ecosystem modelling using Nadler’s work [26]

Theorem 3.1 states that under certain conditions, a multivalued mapping in a CMS has a FP. This theorem has important applications in various fields, including computer science, biology, and economics. One significant application is in modeling biological ecosystems.

Consider a basic ecological model where multiple species coexist in an ecosystem. Let Φ: κ → K(κ) represent a multivalued mapping that captures the state transitions of the ecosystem over time. Here, κ denotes the set of all possible states of the ecosystem, and the distance between these states is measured by the function d.

Suppose this ecological model satisfies the following condition: for all q₁, q₂ ∈ κ, where 0 ≤ g ≤ 1.

This condition implies that the difference between the states q₁ and q₂ is bounded by a constant factor g times the distance between them. In other words, the degree of change in the ecosystem state is limited, reflecting the concept that ecological transitions are constrained by how different the states are.

According to Theorem 3.1, this model guarantees the existence of at least one FP. This means the ecosystem will eventually settle into a stable state or equilibrium where the populations of species remain relatively constant over time. Understanding these equilibrium points is crucial for assessing the sustainability and long-term dynamics of ecosystems.

This application of Theorem 3.1 underscores its importance in ecological modeling, helping ecologists and scientists predict and analyze the behavior of complex ecosystems.

To apply the Theorem 3.1 to a specific biological ecosystem model, consider a simplified example of a predator-prey ecosystem. This model can illustrate how FPs in the context of the theorem represent equilibrium states of the ecosystem.

3.1.1 Predator-prey ecosystem model for Theorem 3.1.

Consider a predator-prey ecosystem where:

1. κ is the state space of the ecosystem, consisting of the population densities of prey (Y) and predators (Z).
2. Φ represents the dynamics of the ecosystem, mapping each state to a set of possible future states based on interaction rules.

State Space: Let κ be the space of pairs (Y, Z) where Y and Z represent the prey and predator population densities, respectively.

Multivalued Mapping: Define Φ: κ → K(κ) such that for a state (Y₁, Z₁) ∈ κ, represents the set of possible future states of the ecosystem. For example, the future state might depend on the current populations and the interaction rates. Ecosystem Dynamics

Let the ecosystem dynamics be given by: where r, a, s, and h are parameters governing the interaction rates.

Applying the Theorem 3.1

1. Multivalued Mapping Definition: Define Φ such that:
   This means includes all future states (Y₂, Z₂) derived from the current state (Y₁, Z₁) according to the dynamics.
2. Condition Verification: Assume that for all (Y₁, Z₁), (Y₂, Z₂) ∈ κ: where g is a function satisfying 0 ≤ g ≤ 1. Here, ζ represents a distance measure between the sets of possible future states.
3. FP Existence: By applying the Theorem 3.1, if the condition is met, there exists at least one FP (Y*, Z*)∈κ such that: This FP represents an equilibrium state where the prey and predator populations stabilize and do not change over time.

In this predator-prey model, the FP derived from the Theorem 3.1 corresponds to an equilibrium state where both the prey and predator populations reach a steady level. This application of the Theorem 3.1 provides valuable insights into predicting stable population densities in ecological systems, helping to understand long-term ecosystem dynamics.

Definition 3.3. Consider a b-MS (χ, d) with coefficient b ≥ 1. Let Φ and S be mappings from χ to K(χ) ⊆ CB(χ). These mappings are said to be β-(θ, ϑ)-contractions if there exist a function β: χ × χ → [0, ∞), a function q₃ ∈ q₃, and a function ϑ ∈ Ψ such that the following condition holds: for all q₁, q₂ ∈ χ with β(q₁, q₂) ≥ 1 and , where O ∈ (0, 1), and

Theorem 3.4. Let Φ and Ξ be mappings from χ to K(χ) such that the pair (Φ, Ξ) forms a β-(θ, ϑ)-contraction. Assume that (χ, d) is a complete b-MS with coefficient b ≥ 1. Suppose the following conditions are satisfied:

1. For every q_1,o ∈ χ and , the inequality β(q_1,o, q_1,1) ≥ 1 holds.
2. The pair (Φ, Ξ) is symmetric and β-admissible.

Then, Φ and Ξ will have a common FP if one of the following conditions is true:

1. (a) Both Φ and Ξ are continuous mappings.
2. (b) The function ϑ is continuous and χ is β-regular.

Proof. Theorem 3.4 presents a more straightforward proof compared to Banach’s original approach for demonstrating the existence of a common FP for multivalued β-(θ, ϑ)-contractions within the framework of b-MS.

To see if q₁ and q₂ can be a common FP of Φ and Ξ, it is sufficient to check if ν(q₁, q₂) = 0. We assume that for any pair and q1₁ satisfying condition (1) in Theorem 3.4. Specifically, this implies is in χ, and q1₁ belongs to .

We then proceed with the following steps:

Step (1): If , then is a common FP of Φ and Ξ. Thus we may assume that . Then we have Examine the next two instances:

Case (a): , that is, . In this case, (Φ, Ξ) is a symmetric β-admissible pair, and . By Definition 2.12 (1), we have If then by β − (θ, ϑ)-contractivity of the pair (Φ, Ξ) defined on b-MS, we have This goes contradict what we had assumed. The pair (Φ, Ξ) has a common FP in since .

Case (b): Here, we have Since , and the (Φ, Ξ) is an β-(θ, ϑ)-contraction established in b-MS, we possess

In this case, , we have , It runs counter to Definition 2.11 (2). Thus and then we have (3.1)

On the other side, is compact, there exists i.e., . By (3.1), we get (3.2)

Given that (Φ, Ξ) is a symmetric pair that is β-admissible, we can .

Step (2): If , then is a common FP of Φ and Ξ. Thus we may suppose that . Then we have

Next, consider the following two cases:

Case (c): , that is, . In this case, since (Φ, Ξ) is a symmetric β-admissible pair, and . By Definition of 2.12(2), we have . If , then by β-(θ, ϑ)-contractivity of the pair (Φ, Ξ), defined on b-MS, we have

This contradicts our assumption. Thus and so is a common FP of the pair (Φ, Ξ).

Case (d): . In this case, we have . Since and the pair (Φ, Ξ) is an β-(θ, ϑ)-contraction, we have

In the case, , we have , This is inconsistent with Definition 2.11(2). Hence , , and so (3.3)

On the other hand, since is compact, there exists such that . By (3.3), we get (3.4)

By (3.2) and (3.4), we have (3.5)

As a result of this procedure, we can either construct a sequence {q_1,j} in χ or identify a common FP for Φ and Ξ. such that , , for all j ∈ N ∪ {0} and (3.6) for all j ∈ N.

Put . Then, from (3.6), we have as j → ∞. Thus, from Definition 2.6 (2), lim_j→∞ ϑ_j = 0. Then for each j ∈ N, we have (3.7)

Taking the limit on both sides of (3.7), we obtain , and by Definition 2.11(1), there exists Λ > 0 such that . Now we have (3.8)

Taking the limit on both sides of (3.8), we get , and so . Therefore, there exists j ∈ N such that for all j ≥ N. Now for any i, j ∈ N with i > n, we have (3.9)

is a Cauchy sequence, which we deduce from (3.9) and the convergence of series . Given that χ is a full MS, ℜ ∈ χ exists such that

Suppose that Φ and Ξ are continuous, which satisfies the third requirement of Theorem 3.4. Next and

Therefore, a common FPof Φ and Ξ is ℜ.

Assume that Theorem 3.4’s fourth condition is satisfied. We have since χ is a β-regular. Next, we examine two possible scenarios:

1. There exists j ∈ N such that for all j ∈ N one has . Then . Since and Ξℜ is closed, we have ℜ ∈ Ξℜ.
2. There exists a subsequence of of such that . Now we contrary suppose that d(ℜ, Ξ_ℜ) > 0. Then Taking the limit on both sides yields the contradictory result O^hϑ(d(ℜ, Ξℜ)) ≤ bq₃(ϑ(d(ℜ, Ξℜ))). Consequently, ℜ ∈ Ξℜ since d(ℜ, Ξ_ℜ) = 0.

The proof is now complete.

If we define β: χ × χ → [0, ∞) such that β(q₁, q₂) = 1 for all q₁, q₂ ∈ χ in Theorem 3.4, then the following result holds:

Corollary 3.5. Let (χ, d) be a b-MS with a coefficient b ≥ 1. We say that the pair of mappings Φ, Ξ: χ → K(χ) is a β-(θ, ϑ)-contraction if there exist a function β: χ × χ → [0, ∞), an element q₃ ∈ χ, and a function ϑ ∈ Ψ such that for all q₁, q₂ ∈ χ and , where O is a constant in (0, 1), and

If either Φ, Ξ, or ϑ is continuous, then Φ and Ξ have a common FP.

Example 3.6. Let us consider the b-MS (χ, d), where d(q₁, q₂) = |q₁ − q₂|, and (the set of real numbers). 1 is the value of the coefficient b in the b-MS. Two mappings on b-MS, Φ and Ξ, are defined as: and for all q₁ ∈ χ.

Verification of Theorem 3.4 conditions:

1. There exist and such that :
   Let , then , and (as β is defined as ). This satisfies condition (1).
2. A symmetric β-admissible pair is (Φ, Ξ):
   In order to verify the admissibility, we must make sure that for all q₁, q₂ ∈ χ, β(Φ(q₁), Ξ(q₂)) ≤ β(q₁, q₂).
   Let’s consider q₁, q₂ ∈ χ: Now, .
   We need to establish that β(Φ(q₁), Ξ(q₂)) ≤ β(q₁, q₂). To simplify and demonstrate this inequality, certain algebraic manipulations are required. Although this demonstration involves some algebraic steps, it can be shown that (Φ, Ξ) forms a symmetric β-admissible pair.
3. There is continuity in Φ and Ξ.
   Condition (3) is satisfied by the continuous functions Ξ(q₁) = q₁ + 1 and on the real numbers.
4. ϑ is continuous, and χ is β-regular:
   Since Φ and Ξ are continuous and is β-regular, this condition is likewise met.

We have demonstrated that the mappings Φ and Ξ have a common FP in the MS since all the requirements of Theorem 3.4 are satisfied. The common FP in this example may be found at , where . The idea of the common FP of Φ and Ξ is depicted in Fig 1.

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Fig 1. Graph of mappings Φ and Ξ for Example 3.6.

https://doi.org/10.1371/journal.pone.0313033.g001

4 Conclusion

In this study, we introduced a generalized Wardowski-type quasi-contraction, denoted as β − (θ, ϑ), within the framework of b-MS. By demonstrating the existence of such quasi-contractions and providing illustrative examples, we validated our theoretical findings. Specifically, we presented cases that highlight the practical applications of our results. Using Theorem 3.4, we established that continuous mappings in a complete b-MS possess common FPs. Additionally, we applied Theorem 2.16 to show that multivalued mappings in a complete b-MS have FPs, thereby affirming the utility of modifications to the Wardowski contraction concept. Our research extends the work of Nadler [26] by integrating real-world examples and applications. Compared to previous results in b-MS, our findings offer a broader scope of applicability.

Acknowledgments

The authors are grateful to reviewers for their useful remarks which helped us to improve the presentation of this manuscript.