On k-fuzzy metric spaces with applications - Patel, Uma Devi; Chandra, Vineeta; Savatović, Milica; Radenović, Stojan (Nonlinear Analysis: Modelling and Control, 2025)
With application point of view, Gopal et al. [D. Gopal, W. Sintunavarat, A.S. Ranadive, S. Shukla, The investigation of k-fuzzy metric spaces with the first contraction principle in such spaces, Soft Comput., 27:11081–11089, 2023] generalized the conceptions of a fuzzy metric space and introduced the definition of k-fuzzy metric space. Here a fuzzy set defined in k-fuzzy metric space is a membership function FY : X × X × (0, +∞)k -> [0; 1], that is, the fuzzy distance between two points of the set depends on more than one parameter, and then also introduced first contraction principle in this space. In this sequel, we extend the work on k-fuzzy metric spaces by generalizing Banach contraction principle by introducing various type of inequalities. Here we introduce Tirado-type k-fuzzy contraction condition and prove fixed point theorem for Tirado-type contractive mapping. We also discuss the k-fuzzy ψ-contractive mapping, where ψ ∈ Ψ, and Ψ is a class of mappings defined from ψ : [0; 1] -> [0; 1] that has certain properties, and also obtained fixed point for such class of mappings. Later, we define Ćirić-type contraction inequalities to prove fixed point results by restricting ourselves on l-natural property of the fuzzy space to ensure the existence of fixed point. Between all results, a set of supportive examples are also produced to validate the results. In application section, we discuss the solutions of Volterra-type integral equations and second-order nonlinear ordinary differential equation.
Keywords: fuzzy metric space; k-fuzzy metric space; k-fuzzy Tirado-type contractive mapping; k-fuzzy psi-contractive mapping; fixed point