(Springer Plus, 2016-09-02) Saleem, Naeem; De la Sen, Manuel; Abbas, Mujahid
This paper discusses some convergence properties in fuzzy ordered proximal approaches defined by {(gn, Tn)}—sequences of pairs, where g : A → A is a surjective self-mapping and T : A → B, where Aand Bare nonempty subsets of and abstract nonempty set X and (X, M, ∗, ≺) is a partially ordered non-Archimedean fuzzy metric space which is endowed with a fuzzy metric M, a triangular norm * and an ordering ≺. The fuzzy set M takes values in a sequence or set {Mσn } where the elements of the socalled switching rule {σn} ⊂ Z+ are defined from X × X × Z0+ to a subset of Z+. Such a switching rule selects a particular realization of M at the nth iteration and it is parameterized by a growth evolution sequence {αn} and a sequence or set {ψσn } which belongs to the so-called Ψ (σ, α)-lower-bounding mappings which are defined from [0, 1] to [0, 1]. Some application examples concerning discrete systems under switching rules and best approximation solvability of algebraic equations are discussed.