Existence and Uniqueness Results for ψ-α-θ Fuzzy Quasi-Controlled Hybrid Contractions with Applications to Integral, Fractional, and Directed Systems
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Abstract
In this paper, we introduce a new class of ψ–α–θ fuzzy quasi-controlled hybrid contractions in the framework of controlled fuzzy quasimetric spaces. The proposed model integrates fuzziness, asymmetry, and control mechanisms into a unified structure, thereby extending classical Banach and Kannan contraction principles as well as several recent developments in fuzzy fixed point theory. A fixed point theorem guaranteeing existence, uniqueness, and convergence of Picard iterations is established under admissibility and continuity assumptions. Unlike classical contraction frameworks, the proposed hybrid condition does not require a global Lipschitz constant less than one, but instead employs a combination of simulation, control, and hybrid aggregation functions to regulate nonlinear behavior. The effectiveness of the framework is illustrated through nontrivial examples and counterexamples, including cases where the mappings are neither Banach nor Kannan contractions and where classical fixed point theorems fail. These results demonstrate that the proposed approach constitutes a strict and nontrivial generalization of existing contraction principles. Applications to nonlinear integral equations, fractional differential equations, and directed systems are presented. In particular, we establish existence and uniqueness of solutions in settings where classical contraction conditions are violated, including non-expansive operators and cases with effective Lipschitz constants exceeding unity. The results provide a flexible and robust analytical framework for studying nonlinear problems in fuzzy and asymmetric environments, with potential applications in applied mathematics, engineering, and computational sciences.
Keywords
- Fixed point theorem
- Fractional Differential Equations
- Fuzzy quasi-controlled metric space
- Generalized contraction
- Nonlinear integral equation
- Picard iteration
References
[1] S. Banach, “Sur les operations dans les ensembles abstraits et leurs applications aux équations intégrales”, Fundamenta Mathematicae, Vol.3, (1922), pp.133–181.
[2] I.A. Bakhtin, “The contraction mapping principle in almost metric space”, Functional Analysis and Its Applications, (1989), pp.26–37.
[3] S. Czerwik, “Contraction mappings in b-metric spaces”, Acta Mathematica et Informatica Universitatis Ostraviensis, Vol.1, (1993), pp.5–11.
[4] N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, “Controlled metric type spaces and the related contraction principle”, Mathematics, Vol.6, No.10, (2018), p.194.
[5] L.A. Zadeh, “Fuzzy sets”, Information and Control, Vol.8, (1965), pp.338–353.
View more references (25)
[6] I. Kramosil, J. Michalek, “Fuzzy metric and statistical metric spaces”, Kybernetika, Vol.11, (1975), pp.336–344.
[7] A. George, P. Veeramani, “On some results in fuzzy metric spaces”, Fuzzy Sets and Systems, Vol.64, No.3, (1994), pp.395–399.
[8] D. Mihet, “Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces”, Fuzzy Sets and Systems, Vol.159, (2008), pp.739–744.
[9] S. Heilpern, “Fuzzy mappings and fixed point theorem”, Journal of Mathematical Analysis and Applications, Vol.83, (1981), pp.566–569.
[10] H. Qawaqneh, M. S. M. Noorani and H. Aydi, “Some new characterizations and results for fuzzy contractions in fuzzy b-metric spaces and applications”, AIMS Mathematics, Vol.8, (2023), pp.6682–6696.
[11] A. Moussaoui, F. I. A. Amir and M. Elomari, “Fixed-point results connected to fuzzy F-contraction and simulation functions with application”, Fixed Point Theory and Algorithms for Sciences and Engineering, (2025), Article 14, pp.1–15.
[12] A. Moussaoui, S. Radenovic, O. Taghipour Birgani and S. Melliani, “A New Contraction Principle via Fuzzy L-Simulation Functions”, Boletim da Sociedade Paranaense de Matematica, Vol.43, (2025), pp.1–9.
[13] A. Moussaoui and S. Radenovic, “Fuzzy Metric Spaces: A Survey on Fixed Point Results, Contraction Principles and Simulation Functions”, Fixed Point Theory and Algorithms for Sciences and Engineering, (2025).
[14] J.C. Kelly, “Bitopological spaces”, Proceedings of the London Mathematical Society, Vol.13, No.3, (1963), pp.71–89.
[15] V. Gregori, S. Romaguera, “Fuzzy quasi-metric spaces”, Applied General Topology, Vol.5, No.1, (2004), pp.129–136.
[16] Y. Zhao, F. He and X. Liu, “Fuzzy H-Quasi-Contraction and Fixed Point Theorems in Tripled Fuzzy Metric Spaces”, Axioms, Vol.13, (2024), Article 536.
[17] A. Bataihah and A. A. Hazaymeh, “Quasi Contractions and Fixed Point Theorems in the Context of Neutrosophic Fuzzy Metric Spaces”, European Journal of Pure and Applied Mathematics, Vol.18, (2025).
[18] N. Souayah, M. Mrad, “On fixed point results in controlled partial metric type spaces with a graph”, Mathematics, (2019).
[19] J. Pamba, H. Chirwa, Y. Sakala, A.S. Mungo, “Fixed point theory in graph-structured controlled partial metric spaces with applications to integral equations”, Journal of Advanced Applied Computational Mathematics, Vol.12, (2025), pp.121–142.
[20] J. Pamba, T. Nazir, “On some fixed point results of Banach type contraction in controlled G-metric spaces”, Nonlinear Analysis and Computational Techniques (ICNACT 2024), Springer Proceedings in Mathematics & Statistics, Vol.501, (2025).
[21] J. Pamba, I.D. Tembo, “On fixed point results of Suzuki-type contractions on controlled metric spaces with applications”, International Journal of Science and Technology Society, Vol.13, No.5, (2025).
[22] A. Hussain, U. Ishtiaq and H. Al Sulami, “Fixed Point Results in Fuzzy Strong Controlled Metric Spaces with an Application to the Domain Words”, Advances in Mathematical Physics, (2023), Article ID 4350504.
[23] F.M. Azmi, “Fixed-point results for (α–ψ)-fuzzy contractive mappings on fuzzy double-controlled metric spaces”, Symmetry, Vol.15, (2023), p.716.
[24] Z. Bibi, U. Ishtiaq, T. Kamran, M. Akram, I.-L. Popa, “On new controlled fuzzy metric space and related fixed point results”, Journal of Inequalities and Applications, Vol.2025, (2025), p.138.
[25] S. Nadaban, “Fuzzy b-metric spaces”, International Journal of Computers Communications & Control, Vol.11, No.2, (2016), pp.273–281.
[26] F. Mehmood, R. Ali, C. Ionescu, “Extended fuzzy b-metric spaces”, Journal of Mathematical Analysis, Vol.8, (2017), pp.124–131.
[27] M.S. Sezen, “Controlled fuzzy metric spaces and some related fixed point results”, Numerical Methods for Partial Differential Equations, Vol.37, (2020), pp.583–593.
[28] N. Saleem, H. Isik, S. Furqan, C. Park, “Fuzzy double controlled metric spaces and related results”, Journal of Intelligent & Fuzzy Systems, Vol.40, (2021), pp.9977–9985.
[29] J. Pamba, H. Sabao, I.D. Tembo, “Uniform continuity in fuzzy quasi-metric spaces: foundations, extensions, and structural properties”, Bulletin da Sociedade Paranaense de Matematica, (2025).
[30] A. Moussaoui, C. Park and S. Melliani, “New Best Proximity Point Results via Simulation Functions in Fuzzy Metric Spaces”, Boletim da Sociedade Paranaense de Matematica, Vol.43, (2025), pp.1–12.