Let X denote a locally compact metric space and φ : X → X be a continuous map. In the 1970s L. Zadeh presented an extension principle, helping us to fuzzify the dynamical system (X, φ), i.e., to obtain a map Φ for the space of fuzzy sets on X. We extend an idea mentioned in [P. Diamond, A. Pokrovskii, Chaos, entropy and a generalized extension principle, Fuzzy Sets and Systems 61 (1994)] and we...

In this paper, the unit interval 0, 1 will be denoted by I. LetX be a nonempty set. A member of I is called a fuzzy subset of X 1 . Throughout this paper, for A,B ∈ I we write A ≤ B if and only if A x ≤ B x for all x ∈ X. By A B, we mean that A ≤ B and B ≤ A, that is, A x B x for all x ∈ X. Also we write A < B if and only if A ≤ B and A/ B. If {Aj : j ∈ J} is a collection of fuzzy sets in X, th...

We explore a notion of fuzzy T -neighbourhood spaces, for any continuous triangular norm T , and we present on this notion a uni6ed treatment. Our theory, on one hand, generalizes the theory of Lowen (Fuzzy Sets and Systems 7 (1982) 65) from T =Min to arbitrary T , which has been the progenitor of this work, and on the other hand it is strongly related to the theory of L-neighbourhoods of H; oh...

The main goal of this work is to study the classical models of Partial Differential Equations (PDE) as the heat, wave and Poisson with uncertain parameters, considering the parameter as a fuzzy number. The fuzzy solution is built from the fuzzification of the deterministic solution. The continuity of the Zadeh extension is used to obtain qualitative properties upon the regular α − cuts of the f...

In this paper, we consider the problem of solving systems of fuzzy relation equations in a space with fuzzy preorder. Two types of these systems with different compositions are considered. New solvability criteria are proposed for systems of both types. The new criteria are weaker than all the known ones that are based on the assumption that fuzzy sets on the left-hand side of a system establis...

The overlap function, a particular kind of binary aggregate has been extensively utilized in decision-making, image manipulation, classification, and other fields. With regard to function theory, many scholars have also obtained achievements, such as pseudo-overlap quasi-overlap semi-overlap etc. above generalized functions contain commutativity continuity, which makes them some limitations pra...

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rn , let K(Y ) be the family of all fuzzy sets of Rn , which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in Y. We show that the space K(Y ) with the topology of endograph metric is homeomorphic to the Hilbert cube Q = [−1, 1] iff Y is compact; and the space K(Y ) is homeomorphic to {(...

In this paper, the well-known Egoroff’s theorem in classical measure theory is established on monotone non-additive measure spaces. Taylor’s theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff’s theorem i...