This study delves into the concept of -fuzzy mappings and their associated -fuzzy fixed points within the framework of Hausdorff intuitionistic fuzzy metric-like spaces. A general fixed point theorem for -fuzzy mappings is established in complete HIFMS and its subclasses. The results unify and generalize several classical and fuzzy fixed point theorems, extending them to more complex fuzzy structures. Additionally, recognizing the significant applications of differential inclusions as set-valued mappings, the research explores first-order nonlinear Cauchy differential inclusions within Hausdorff intuitionistic fuzzy metric spaces by leveraging the derived theoretical results. The findings demonstrate the robustness of fuzzy fixed point theory in modeling systems with uncertainty and imprecision, with potential applications in control theory and optimization. The gap between fuzzy metric theory and differential inclusions.
The study of fixed points is a dynamic and evolving area of mathematics, closely linked to functional analysis, topology, engineering, and science. This theory plays a pivotal role in the rapidly advancing domains of nonlinear analysis and nonlinear operators. Though relatively new, fixed point theory is experiencing rapid growth in both science and engineering. Historically, fixed point problems and their associated points have been essential mathematical tools across a wide range of disciplines, including topology, differential equations, economics, functional analysis, optimal control, dynamics, game theory, and various engineering fields. The development of precise computational methods for identifying fixed points has significantly enhanced the practical applications of this concept, making fixed point methods indispensable to scientists and engineers, as well as applied mathematicians.
Major areas of mathematics, such as algebraic topology, functional analysis, general topology, and set theory, offer the best environments for the application of fixed point problems. These problems are extensively employed throughout numerous fields, including theory of differential equations (Alam & Rohen, 2025; Alam et al., 2024), game theory, mathematical economics (Alam et al., 2024), approximation theory (Alam & Rohen, 2024; Alam et al., 2024), and the potential theory, to solve problems in these fields. For those passionate about systems of integral, differential, or functional equations (Alam et al., 2024), fixed point methodologies offer powerful tools for analyzing a wide range of problems in engineering and science. This approach is particularly valuable when addressing issues in control systems and elasticity, making it a crucial technique in both scientific and engineering contexts.
To solve mathematical models like variational inequalities, integral equations (Alam & Rohen, 2024, 2024?), and differential equations (Alam et al., 2023, 2024), which describe phenomena in areas such as fluid flow, neutron transport, epidemics, steady-state temperature distributions, chemical reactions, and economic theories, one of the most essential tools available are fixed point problems. These problems are also critical in assessing the complexity of selecting the appropriate center for these systems, providing deep insights into the underlying challenges. In 2021, Saleem et al. (2021), and Furqan et al. (2021) generalizs the concept of fuzzy metric space and defined fuzzy double and fuzzy triple controlled metric like spaces. Several authors proved some remarkable findings related to coincidence best proximity point along with numerous fixed point results metric, fuzzy metric and non-Archimedean fuzzy metric spaces, (for details see, Abbas et al., 2016; De la Sen et al., 2017; Saleem et al., 2019, 2020; Zhou et al., 2021).
The release of a landmark study by Zadeh Zadeh (1965) marked a significant pivotal moment in the development of the modern notion of unpredictability. Fuzzy sets are sets with uncertain bounds, according to a theory developed by Zadeh. Membership in a fuzzy set is a degree issue instead of acceptance or rejection. In order to solve problems across many disciplines, this notion is being employed and is found to be more effective.
Kramosil and Michalek (1975) were the ones who first presented the idea of fuzzy metric spaces (FMS). Later, in response to Kramosil et al., George and Veeramani (1994) extended the idea of FMS and investigated its Hausdorff topology. In Alam et al. (2024), Alam et al. generalized the groundbreaking notions of fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of FMS by introducing the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. Intuitionistic fuzzy metric space (IFMS) was a concept that Park (2004) first articulated in 2004. He demonstrated that for each space having an intuitionistic fuzzy metric (IFM), the topology produced by the IFM and the fuzzy metric are identical. The idea of intuitionistic fuzzy normed space (IFNS) was first suggested by Saadati and Park (2006) in 2006. Later in 2014, Shojaei (2014) generalized the theory by proposing Hausdorff Intuitionistic Fuzzy Metric Space (HIFMS).
The introduction of HIFMS allows for a refined measure of distance between fuzzy sets, which is essential in handling multivalued mappings in uncertain environments. Compared to other fuzzy set models like hesitant fuzzy sets, intuitionistic fuzzy metrics provide both membership and non-membership functions, capturing a richer form of uncertainty. The Hausdorff framework extends classical fuzzy metric spaces by incorporating set-based convergence and compactness, which are critical in proving fixed point theorems for fuzzy, fuzzy mapping (FM), and modeling differential inclusions. This dual capability is not easily addressed using standard fuzzy or hesitant fuzzy models.
This research paper focuses to bridge fuzzy fixed point theory and the analysis of differential inclusions within generalized metric frameworks (see Figure 1 for graphical representation). Introduction of FM in HIFMLS and HIFMS, drawing inspiration from the framework of Kelley (1959). We explore fuzzy fixed point (FFP) and fixed point conclusions, contributing to the evolving field of FM results in fuzzy type metric spaces. Further, we study first-order nonlinear differential inclusions in HIFMS. Our primary objective is to explore the conditions under which a solution to the differential inclusion corresponds to a continuous function so that . Under suitable fuzzy contractive conditions, FFPs exist in complete HIFMLS, and these results guarantee solutions to nonlinear Cauchy differential inclusions. Such investigation is particularly relevant to various applied mathematical problems, spanning control theory, economic systems, and Hamiltonian systems.
Graphical Representation of the Work.
Preliminaries
We begin by reviewing the fundamental definitions and key properties of a HIFMS.
Schweizer and Sklar (1960) A continuous norm is a commutative, associative, continuous binary operation so that and
Some of the easiest examples of norm are , and .
Schweizer and Sklar (1960) A continuous conorm is an commutative, associative, continuous binary operation so that and .
Some of the easiest examples of conorm are
Alaca et al. (2006) An IFMS is a five tuple , where is any set, are fuzzy sets on and are both respectively continuous norm and conorm, so that for and if
,
,
,
,
is left continuous,
,
,
,
,
is right continuous,
.
In this case, the pair is known as IFM on .
In the definition of IFMS, if we replace both side implication conditions (IFMS2) and (IFMS8) by right side implication only, then the IFMS will be called an intuitionistic fuzzy metric like space (IFMLS). In this regard, every IFMS is again an IFMLS.
If is an IFMS (respectively IFMLS), then is an FMS (respectively FMLS). Every FMS (respectively FMLS) is again an IFMS (respectively IFMLS) , if we define and .
Again, if all Cauchy sequences in an IFMS are convergent then, the IFMS is called complete and if all sequences in an IFMS have convergent subsequences then, the IFMS is called compact.
Drawing inspiration from the framework of Hausdorff metric (Kelley, 1959) and Rodriguez-Lopez and Romaguera (2004), in the set of all subsets of , for all and , we denote the following equality and inequality notations:
In case of MS ,
In case of IFMS ,
With the above inequality notations, we call an IFMS (respectively IFMLS) as a HIFMS (respectively HIFMLS). The Hausdorff distance measures in fuzzy metric settings provide a natural structure for extending fixed point concepts to set-valued and fuzzy mappings, facilitating applications in real-world modeling where outputs are not single-valued.
In a HIFMS , let , the collection of all non-empty closed and bounded subsets of . Then, for any there is a , so that
Zadeh (1965), we now recall some definitions related to the term fuzzy.
Allow to be any non-empty set. Any function having the domain and values between and is referred to as a fuzzy set in . The function-value is frequently referred to as the membership level of in if is a fuzzy set and is a member of . Unless otherwise specified, refers to the gathering of all fuzzy sets contained in . Assume and are two random non-empty sets. A fuzzy mapping is one that transforms from the set into . We denote
known as level set of and , where overline indicates closure of a set, when is induced with a topology. For a self fuzzy mapping from into , a point is refereed to FFP for some , if .
From the definition of and , we find a sequence in , so that
Again, applying (IFMS4) and (IFMS10), we get
Which gives
Being is compact and , the last implications gives , that is, is an FFP of .
The proposed method guarantees existence under minimal assumptions, where many classical methods fail without strong continuity or uniqueness conditions. However, computation of -level sets and verification of Hausdorff distances can be expensive in high dimensions. Numerical experiments in future work will further evaluate trade-offs. This paper focuses on theoretical existence results. While examples will be provided to illustrate applicability, numerical validation using benchmark systems (e.g., fuzzy control of uncertain dynamic systems) will be considered in future work. Comparisons with other fuzzy fixed point techniques, including hesitant fuzzy and Zadeh-type systems, are needed to empirically assess performance and robustness.
The following corollary is a direct consequence of Theorem 2 for set-valued mappings.
Let be a complete HIFMLS (or, HIFMS) and a set-valued mapping is so that
. Where is non-empty and compact in for each . Then, has a fixed point .
Obvious from Theorem 2.
Now we present two corollaries that will carry our results in HIFMS and HFMS.
Let be a complete HIFMS and a FM is so that
. Where so that is non-empty and compact in for each . Then, has an FFP .
The assertion that ’every IFMS is again an IFMLS’ leads to the proof at once.
Let be a complete HFMS and a FM is so that
. Where so that is non-empty and compact in for each . Then, has an FFP .
The proof adheres to Remark 1.
Now we state a corollary which will find FFP utilizing Corollary 4 in metric space.
Let be a complete MS and a FM is so that
. Where so that is non-empty and compact in for each . Then, has an FFP .
For any , let us define
Also, let the binary operations be defined as
Then, the five tuple will form a complete HIFMS, as the MS is complete.
Now,
and
This shows that the fuzzy mapping satisfies Corollary 4 and the outcome is as follows.
Notice that, from the procedure of the proof of Corollary 6, it is clear that every MS is again an IFMS.
In the Corollary 6, If we replace the binary operations given by
Then, the result will still hold.
Approximation and Numerical Computation of FFPs: While our primary focus is the existence of FFPs, practical implementations, especially in control systems and fuzzy optimization, require computational techniques for approximating these fixed points. Inspired by the structure of iterative approximation methods such as Picard, Mann, and Ishikawa iterations, one may construct a sequence in the Hausdorff intuitionistic fuzzy metric space as follows:
initialized with an arbitrary . This mimics the selection process in the proof of Theorem 2. Under the same contractive-type conditions used in the theorem, this sequence converges to an FFP. For numerical approximations, fuzzy similarity measures or centroid-based crisp representatives can be employed to approximate the level sets . Further, algorithms incorporating defuzzification and fuzzy arithmetic may aid in computing fixed points iteratively. These aspects will be explored in future computational studies.
The following theorem is an application to set-valued fixed point theory concluded using Corollary 6.
Let be a complete MS and a compact set valued mapping is so that
. Then, has a fixed point so that .
Let us define , for by
Then, and consequently, satisfy Corollary 6 so that there is , that is, has a fixed point.
For a pair , let us define
Then, is a complete MS.
For any , let us define
Also, let the binary operations be described as
Then, the five tuple will form a complete HIFMS, as the MS is complete.
Also,
Let us consider the fuzzy mapping defined as
For , we get
Consequently,
Then, for all and , satisfy Theorem 2 having an FFP .
Comparative Analysis with Classical Results: Theorem 2 and its corollaries generalize classical results such as Banach’s contraction (single-valued, complete metric spaces), Nadler’s fixed point theorem (multi-valued mappings), and Shojaei’s results in HIFMS.
Banach’s theorem is recovered when is single-valued, , and the fuzzy metric reduces to a classical metric.
Nadler’s theorem corresponds to the case with set-valued maps satisfying contraction in terms of the Hausdorff metric in classical metric spaces.
Our theorem extends Shojaei’s work by relaxing completeness to fuzzy structures and integrating fuzzy level sets, allowing a finer control over fixed point existence in uncertain or fuzzy environments.
These generalizations broaden applicability in settings involving partial membership and non-determinism.
To illustrate practical utility, FM can model decision-making scenarios where input parameters are imprecise. For instance, in a smart grid control system, -level fuzzy sets can represent acceptable voltage thresholds, and FFPs correspond to optimal safe states. While the primary contribution of this paper is theoretical, practical implementation involves approximating fuzzy mappings and computing Hausdorff distances over fuzzy sets. In high-dimensional settings, this can be computationally intensive. Future studies may consider extending the -fuzzy framework using hesitant fuzzy or intuitionistic hesitant fuzzy settings, as explored in recent work on decision-making under uncertainty (Gitinavard et al., 2015). This would potentially enhance flexibility in modeling ambiguity and confidence levels.
Application
A Cauchy differential inclusion generalizes the classical Cauchy problem by replacing the ordinary differential equation (ODE) with a differential inclusion. Unlike ODEs, where the rate of change is described by a unique function, differential inclusions define the rate of change through a set-valued map, allowing for multiple possible behaviors at any given point. These inclusions are particularly valuable for modeling systems with non-deterministic dynamics, uncertainty, or situations where multiple outcomes are possible. Applications span various fields, such as control theory, where they model systems influenced by a range of potential inputs or control actions; mechanics, for handling ambiguity in applied forces; and optimization and game theory, where strategies or decisions are shaped by competing factors.
Unlike classical fixed point approaches that assume precise data and deterministic systems, the proposed -fuzzy framework accommodates imprecision in both data and dynamic rules. This flexibility is especially advantageous in fields such as optimization under uncertainty or economic modeling, where multiple outcomes or strategies may be acceptable. The inclusion of fuzzy and multivalued operators leads to more robust modeling and broader applicability.
Consider a robotic path-planning system where sensor noise introduces ambiguity in control. The fuzzy differential inclusion model accommodates this uncertainty by permitting a set of possible accelerations. Our results ensure the existence of a feasible trajectory under these conditions. The Cauchy differential inclusion (Frigon, 1995) is defined by the following statement
where is a multivalued mapping from to with and is separable normed Banach space. Differential inclusion can be considered a broader concept than a differential equation. The key distinction lies in the reality that within a differential equation, the right-hand side, denoted as in the inclusion, simplifies to a function with a single point value, effectively transforming the inclusion into an equation sign.
Let be an complete HIFMS, where and
be defined as
and the binary operations specified as
Associated to , let us consider the multivalued Carathéodory mapping defined as
where be the Niemytzki operator associated with , that is
where is the Banach space of valued integrable mappings on .
We now have a useful lemma.
Frigon (1995) Let be a multivalued mapping from to so that
is continuous almost everywhere on ,
is measurable for all ,
so that for all
Then, is non-empty and compact.
Now we state our result.
Along with the circumstances of Lemma 8, for some , suppose that, satisfies
and
Then, the differential inclusion (7) has a solution.
Let be arbitrary. Then there exist so that
Now implies that
and
Again,
The other inequality corresponding to will come in a similar manner and by arbitrariness of
we see satisfies Corollary 3 for . Consequently, has a fixed point in , or we can confirm that the inclusion (7) has a solution.
Orientor equation concepts can be used to formulate each challenge of controlling first-order ordinary differential equations. This result was a crucial catalyst for the study of orientor differential equations, which led to the development of the still-relevant new concept of “differential inclusions.”
Consider a robotic system whose velocity is influenced by uncertain inputs (e.g., terrain resistance). Let , where and model two extreme behaviors (e.g., normal and maximum friction). We define the fuzzy metric space on the function space using exponential-type distance. With suitable bounds on , conditions (8) and (9) can be verified, ensuring the existence of a feasible path.
Further Directions with Neutrosophic Extensions: The FFP framework developed here may be enriched by recent advances in neutrosophic environments, which model indeterminacy explicitly. Works involving neutrosophic soft sets and novel norm structures for topological modeling suggest promising pathways to incorporate granularity and hesitation in decision spaces. These could potentially extend our results to intuitionistic or hesitant neutrosophic fuzzy metric-like settings, where level generalizations may reflect degrees of acceptance, rejection, and indeterminacy simultaneously. This remains an intriguing avenue for future investigation.
Conclusion
This study has developed a general theory of FFP in the setting of Hausdorff intuitionistic fuzzy metric spaces, culminating in existence results for nonlinear Cauchy differential inclusions. The work unifies and extends several known fixed point theorems while offering a flexible framework for modeling uncertain and multivalued systems. The findings of this investigation have wide-ranging practical applications in fields like mathematical economics, control theory, optimization, and various branches of engineering and science. We hope that these results will inspire continued exploration and innovation in this intriguing field, fostering advancements in both scientific and engineering disciplines. Future directions include the development of numerical schemes for computing FFPs, extending the theory to partial differential inclusions, and integrating this framework into real-world applications such as autonomous systems, game-theoretic models, and machine learning algorithms under uncertainty.
Footnotes
Acknowledgements
The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-87).
ORCID iD
Khairul Habib Alam
Naeem Saleem
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Authors’ Contributions
Conceptualization, K.H.A., N.S.; formal analysis, Y.R., N.S. and K.H.A.; investigation, Y.R., and N.S.; supervision, Y.R.; writing original draft preparation, K.H.A., N.S., A.A.; writing review and editing, K.H.A., M.A., and Y.R. All authors have read and agreed to the published version of the manuscript.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-87).
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Availability of data and materials
The data supporting the results of this study can be obtained from the corresponding author upon request.
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