An expansion of Basic Logic with fixed points

Abstract

We introduce an expansion of Basic Logic (BL) with new connectives which express fixed points of continuous formulas, i.e. formulas of BL whose connectives are among (Formula presented.). The algebraic semantics of this logic is studied together with some of its subclasses corresponding to extensions of the above-mentioned expansion. The axiomatic extensions are proved to be standard complete

Meir–Keeler Fixed-Point Theorems in Tripled Fuzzy Metric Spaces

“…Let (X, F, * ) be a G-fuzzy metric space and suppose that there exist k ∈ N and λ ∈ [0, 1) such that F f k x, f k y, f k z (t) ≥ λ for all x, y, z ∈ X and t > 0. A map f : X −→ X is said to be a generalized fuzzy Meir-Keeler-type contraction with respect to (k, λ, δ), where δ ∈ ∆, if the following condition holds: ∀ϵ ∈ (λ, 1), ϵ − δ(ϵ) < G x,y,z (t) ≤ ϵ ⇒ F f x, f y, f z (t) > ϵ, (19) where G x,y,z (t) = min{F x,y,z (t), F f x,x,x (t), F f y,y,y (t), F f z,z,z (t)} for all x, y, z ∈ X and t > 0.…”

“…Later, George and Veeramani [14] gave a necessary and sufficient condition for the completeness of fuzzy metric space. Since then, various fixed-point results for mappings satisfying different contractive conditions were established by many researchers [15][16][17][18][19][20][21]. Moreover, in 2019, Zheng and Wang [22] proposed the concept of fuzzy Meir-Keeler contractive mappings in fuzzy metric spaces, which covers fuzzy ψ-contractive mappings and fuzzy H-contractive mappings in [23,24] as special cases, and obtained some Meir-Keeler-type fixed-point theorems.…”