1. Introduction

Zadeh [1] established the foundation for fuzzy mathematics in 1965. Banach [2] established fixed point theory in 1922 with his famous Banach contraction principle. Mathematicians have generalized it in numerous ways and used it to address issues in a variety of domains [3–5]. Kramosil and Michalek [6] initially introduced the idea of fuzzy metric space and then refitted it by George and Veeramani [7]. Grabiec [8] pioneered the investigation of fixed-point theorems in fuzzy metric space. In 2012, Wardowski [9] proposed an F-contraction as a new kind of contraction in complete metric spaces. Then, other mathematicians [10–16] then extended F-contraction to various spaces; see also [17]. Huang et al. [18] recently published fuzzy F-contraction, an extension of F-contraction in fuzzy metric spaces with simplified conditions, and established various fixed-point theorems for fuzzy F-contraction. Gopal and Vetro [19] extended the notion through the development of α-admissible and β-admissible fuzzy metric spaces. Park [20] proposed the idea of intuitionistic fuzzy metric space using continuous t-norms and continuous t-conorms. In 1998, Smarandache [21] developed the ideas of neutrosophic logic and neutrosophic set. The idea of neutrosophic metric space was established by Kirisci and Simsek [22], which dealt with membership, nonmembership, and neutralness. While fuzzy metric spaces represent uncertainty using a single membership degree, neutrosophic metric spaces (NMSs) expand this framework by incorporating three independent degrees: membership, indeterminacy, and nonmembership. This allows for a more flexible and nuanced representation of uncertainty, making NMSs more suitable for problems involving incomplete or contradictory information. These spaces generalize fuzzy metric spaces, as every fuzzy metric space can be viewed as a special case of an NMS when indeterminacy and nonmembership are appropriately constrained. This richer structure provides a theoretical foundation for addressing complex problems in mathematics and applied sciences. Jeyaraman and Sowndrarajan [23] and Jeyaraman, Hassen, and Manuel [24] demonstrated the applicability of NMSs by establishing fixed-point results for contraction theorems within these spaces.

We have provided generalized neutrosophic $$ contraction with admissible properties in NMSs, inspired by the work of Gopal and Vetro [19], Koon et al. [25], and Huang et al. [18]. By establishing fixed-point theorems for these contractive maps, the theoretical framework is extended, and their practical applications are demonstrated. Our contributions include expanding the understanding of neutrosophic generalized $$ contraction and offering tools for solving integral equations. Additionally, the results have implications for optimization, dynamical systems, and decision-making models under uncertainty.

2. Preliminaries

The neutrosophic set ℵ can also be expressed as an ordered triple ℵ = ($$ in Ξ. In this context, ($$ are subsets of ] −0, +1[ without further constraints.

If the triangular inequalities are replaced in the definition above, with the condition

N5^∗. $$, $$, $$, $$, $$, $$, $$, $$, and $$, $$, $$, $$ for all $$ and $$ then $$ is known as a strong NMS (SNMS).

For the remaining part of this paper, the class of all mappings $$ such that for every $$ is indicated by ξ.

3. Main Results

$$ as $$, then $$ for every $$ Then, $$ has an invariant point.

As $$ is a neutrosophic generalized β-$$-contraction, there exists 0 < ι < 1 fulfilling Equations (1)–(3). It follows that $$ for all $$ and $$ Substituting $$ and $$ we obtain $$), for all $$ where $$ ($$ = min{$$), $$), $$)} = min {$$), $$), $$)} = min{$$)}.

If max {$$)} = $$), then

$$), for all $$ where $$ ($$ = max{$$), $$), $$)} = max {$$), $$), $$)} = max{$$)}.

Since $$ is a strictly increasing function, we get $$, $$, and $$ for all $$ and $$.

Hence, for all $$, the sequence {$$ is bounded above and strictly monotonic increasing in the interval [0,1], the sequence { $$ is strictly monotonic decreasing in the interval [0,1] and bounded below and the sequence {$$ is strictly monotonic decreasing in the interval [0,1] and bounded below.

Therefore, {$$, {$$, and {$$ are convergent, that is, for any $$, there exists $$, $$ such that $$, $$, and $$

We assert that $$ and $$ for all $$

Assume that $$ for some $$. Applying the limit on both sides of Equation (5), we get $$ which is a contradiction.

Suppose on the contrary, $$ is not Cauchy.

By condition (3) there exist $$ two subsequences $$, $$ of $$ and $$ such that $$ for all $$ with ν ≥ ν₀ and $$ and β ($$

Let $$ be the least integer that fulfills the above inequality.

Then, $$, $$, and $$ Since $$ is a SNMS, by (N5^∗), (N10^∗) and (N15^∗), we get $$, $$, and $$.

Letting the limit as ν → ∞, by Equations (8)–(10), (T1) and (TC1), we get

$$1∗(1 − ε) = 1 − ε, $$ 0 ⋄ε = ε and $$ 0 ⋄ε = ε.

This implies that $$, $$, and $$. Again, by (N5^∗), (N10^∗), and (N15^∗), we get $$)$$$$)$$, and $$)$$.

Letting ν → ∞, by Equations (8) and (9), (T1) and (TC1), we have $$ = 1 ∗ (1 − ε) ∗ 1 = 1 − ε, $$ = 0⋄ε⋄0 = ε and $$ = 0 ⋄ε⋄0 = ε.

Therefore, letting ν → ∞ in both sides of Equations (11)–(13), we obtain $$ ($$ and $$ ($$.

This implies that $$ and $$, this contradicts the fact that $$.

Thus $$ is a Cauchy sequence. Since $$ is complete, there exists $$ such that

$$ = 1, $$ = 0, and $$ = 0 for all $$, or other words, $$. Since β ($$ for all $$and $$ by condition (4), we have $$ for all $$ Now, we show that $$ is an invariant point of $$

If $$, that is, $$ and $$.

Then, by Equations (5)–(7), we obtain $$)>ιβ ($$) $$ for all $$, where $$ min {$$}.

$$)<ιβ ($$) $$ for all $$, where $$ max {$$}.

$$)<ιβ ($$) $$ for all $$, where $$ max {$$}.

Letting $$ in the above inequalities, we have $$)>ιβ ($$)$${1,1,$$), $$)<ιβ ($$)$${0,0,$$) and $$)<ιβ ($$)$${0,0,$$) this is not possible. Hence, $$ that is, $$ is a fixed point of $$

if $$ as $$, then $$ for every $$ Then, $$ has an invariant point.

$$, $$, and $$ for all $$ and $$.

Then, $$ is a complete but not a SNMS. For instance, take $$ and $$ Then, we have $$ and $$ and $$ and $$ and $$ and $$

Checking with Conditions F5 ^∗, F10 ^∗, and F15 ^∗, it is obvious that on the right side, we obtain $$ = min{$$, $$ = max {$$ and $$ = max {$$.

However, 0.287≱0.375. Therefore, $$ is not SNMS.

Define the mapping $$ by $$ for $$. Let $$ for $$, and let $$

Also, define β : Ξ × Ξ × (0, ∞) → (0, ∞) as $$ for every $$ and $$. It is straightforward to demonstrate that $$ is a neutrosophic generalized $$-contraction and Conditions 1, 2, 3, and 4 of Theorem 1 are satisfied. Note, however, that the mapping $$ does not have any fixed point.

$$ ($$, $$ 1–$$, and $$, for $$, $$ and $$

We can easily verify that $$ is a complete SNMS. Let $$ for all $$, and let $$ for 0 $$ and $$ Also, define $$ for all $$

First, we establish that $$ is a β-admissible mapping. Let $$, $$ be such that $$

It follows that $$. Thus $$ is a β-admissible mapping.

Next, we check whether the contractive condition of Theorem 1 is fulfilled or not. Let $$, $$

We consider the following cases:

This is seen in Figures 1–3, which depict the behavior of $$ contraction mapping.

Hence, all the conditions in Theorem 1 are fulfilled. Therefore, $$ has a fixed point.

$$ as →∞, then $$ for every $$ Then, $$ has a fixed point.

By the hypothesis there exists ι ∈ (0, 1) satisfying Equations (14)–(16).

It is clear that $$ for all $$ and $$

$$), $$),$$}

 = min {$$), $$), $$),$$), $$)}

Since the t-norm is minimum, t-conorm is maximum, by (N5 ^∗), (N10 ^∗), and (N15 ^∗), we get $$ = min {$$, $$.

So, we have $$ ($$), $$ for all $$ and $$

$$ = max {$$, $$.

So, we have $$ ($$), $$ for all $$ and $$

$$ = max {$$, $$.

So we have $$ ($$), $$ for all $$and $$

If max {$$)} = $$), then, $$)$$)$$)), which is a contradiction.

If max {$$)} = $$), then $$)$$)$$)), which is a contradiction.

Since $$ is a strictly increasing function, we get $$, $$, and $$ for all $$ and $$ the sequence {$$ is bounded above and strictly increasing in the interval [0,1], the sequence {$$ is strictly monotonic decreasing in the interval [0,1] and bounded below and the sequence {$$ is strictly monotonic decreasing in the interval [0,1] and bounded below.

Therefore, {$$, {$$, and {$$ are convergent, that is, for any $$, there exists $$, $$ such that $$, $$, and $$.

We claim that $$, $$, and $$ for all $$

Suppose $$ is not a Cauchy sequence.

By condition (3) there exist $$, ε > 0, two sub sequences $$, $$ of $$ and $$ such that $$ for all $$ with ν ≥ ν₀ and $$ ($$

Let $$ be the least integer that fulfills the above inequality.

Then, $$, $$, and $$

Using (N5^∗), (N10^∗), and (N15^∗) minimum t-norm and maximum t-conorm,

$$} and $$} Taking the limit as ν → ∞, by Equations (20)–(22), we get $$1, (1 − ε)} = 1 − ε, $$0 , ε} = ε and $$ 0 , ε} = ε.

This implies that $$, $$, and $$ Again, by (N5^∗), (N10^∗), and (N15^∗) minimum t-norm and maximum t-norm, we get $$)$$}, $$)$$, and $$)$$

Letting ν → ∞, by Equations (20)–(22), we have $$ = min {1, (1 − ε), 1} = 1 − ε,

$$ = max {0, ε, 0} = ε and $$ = max {0 , ε, 0} = ε.

Since $$, $$ and $$.

Therefore, letting ν → ∞ in both sides of Equations (23)–(25), we get $$ ($$ and $$ ($$.

This implies that $$ and $$, this is a contradiction to the fact that $$. Thus, $$ is a Cauchy sequence.

Since $$ is complete, there exists $$ such that $$ = 1, $$ = 0 and $$ = 0 for all $$, or other words, $$.

Since β ($$ for all $$ and $$ by condition (4), we have $$ for every $$ We shall now prove that $$ is a fixed point of $$

If $$, i.e., $$, and $$

To determine the uniqueness of the fixed point, apply the following condition:

(I) For any pair of fixed points $$, β ($$ for all $$

It follows that $$, $$, and $$ this is a contradiction. Hence, we conclude that $$.

By condition (I), we have

β ($$ for all $$ By Equations (14)–(16), we obtain $$ ($$ for all $$ where $$ min{$$ }, $$ ($$ for all $$ where $$ max{$$ } and $$ ($$ for all $$ where $$ max{$$ }.

It follows that $$, $$, and $$ this is impossible. Hence, we conclude that $$ Therefore, $$ has a unique fixed point.

4. Applications

5. Conclusion

In this article, we have introduced the concept of neutrosophic generalized $$-contraction, expanding the framework of neutrosophic $$ -contraction in the context of admissible mappings. The sufficient conditions are established for the existence and uniqueness of fixed points in complete SNMSs, advancing the theoretical understanding of this field. The practical application provided demonstrates the potential impact of neutrosophic generalized $$-contraction in solving integral equations. However, there are several limitations to this work. First, the application of neutrosophic generalized $$-contraction is primarily focused on integral equations, and future research could expand its use to other domains such as fuzzy control systems or image processing [26]. Since our study is based on NMSs, extending these results to neutrosophic fuzzy metric spaces or Banach spaces could open new avenues for exploration. Furthermore, the theoretical results presented here could be complemented with computational methods to make the theory more accessible for practical applications, similar to the approaches employed in solving nonlinear differential equations [27] or in iterative processes for fractals and dynamical systems [28]. Future work could also explore interdisciplinary applications, such as using neutrosophic logic in medical decision-making and conducting a comparative analysis with other related approaches, such as fuzzy and intuitionistic fuzzy metric spaces, to better understand the unique advantages of NMSs.

Conflicts of Interest

The authors declare no conflicts of interest.

Author Contributions

M. Pandiselvi, M. Jeyaraman, and Rahul Shukla: investigation, methodology, writing-original draft. Rahul Shukla: supervision. M. Jeyaraman: writing–review and editing. All authors contributed to various stages of the study’s design and execution and have reviewed and approved the final manuscript for submission.

Funding

No funding was received. Open access funding was enabled and organized by SANLiC Gold.