Abstract
AbstractThis paper investigates some properties of cyclic fuzzy maps in metric spaces. The convergence of distances as well as that of sequences being generated as iterates defined by a class of contractive cyclic fuzzy mapping to fuzzy best proximity points of (non-necessarily intersecting adjacent subsets) of the cyclic disposal is studied. An extension is given for the case when the images of the points of a class of contractive cyclic fuzzy mappings restricted to a particular subset of the cyclic disposal are allowed to lie either in the same subset or in its next adjacent one.
Highlights
Fixed point theory has received much attention in the last decades with a rapidly increasing number of related theorems on nonexpansive and on contractive mappings, variational inequalities, optimization etc
Variants and extensions have been considered in the frameworks of probabilistic metric spaces and multivalued mappings
Fixed point theory has proved to be an important tool for the study of relevant problems in science and engineering concerning local and global stability, asymptotic stability, stabilization, convergence of trajectories and sequences to equilibrium points [ – ], dynamics switching in dynamic systems and in differential/difference equations, etc
Summary
Fixed point theory has received much attention in the last decades with a rapidly increasing number of related theorems on nonexpansive and on contractive mappings, variational inequalities, optimization etc. If ( α∈( , ][Tzi]α) = ∅ and [Tzi] (= ∅) ⊂ clXi+ and Di = Di , Di = d(zi, [Tzi] ) = d(zi, zi+ ) for some zi+ ∈ [Tzi] ⊂ Xi+ which is a -fuzzy fixed point of Xi+ through T and a best proximity point of Xi+ to Xi. Definition Let p(≥ ) ∈ Z+ and let Xi be nonempty subsets of a nonempty abstract set X, ∀i ∈ p , and let d be a metric on X. The following assumption establishes that if T(= gf ) : i∈p Xi → F( i∈p Xi) is a decomposable p-cyclic fuzzy mapping, the -level set of Tx intersects with fx for any x ∈ i∈p Xi. Equivalently, there is at least a point y ∈ fx such that the grade of membership of y in Tx is unity, i.e. X is an -fuzzy best proximity point of Xi through T
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