Existence and Uniqueness Results for ψ-α-θ Fuzzy Quasi-Controlled Hybrid Contractions with Applications to Integral, Fractional, and Directed Systems

Source 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.