A new contraction principle in menger spaces

Some results on best proximity points of cyclic alpha-psi contractions in Menger probabilistic metric spaces

“…Note that Theorem 2.1 generalizes some results on fixed points given in [7,8,15,16,28] for either non-cyclic selfmappings or cyclic self-mappings on union of sets which intersect to quasi-best proximity points and best proximity points in the case that such sets do not intersect. On the other hand, a direct consequence of Theorem 2.1 is the following corollary for the case that u 2 U D .…”

“…On the other hand, a direct consequence of Theorem 2.1 is the following corollary for the case that u 2 U D . The results are based on the fact that u D À ð Þ ¼ D and u t ð Þ ¼ 0 if t 2 0; D ½ and u 2 U D while it generalizes results on fixed points for the cases of either non-cyclic self-mappings or cyclic self-mappings with nonempty intersections of the involved subsets obtained in [7,8,15,16,28]: Corollary 2.1 Let X; F; D ð Þ be a G-complete Menger PM-space and T : S i2 p A i ! S i2 p A i be a p-cyclic a-wtype contraction satisfying the following conditions:…”

“…Note also that the particular set of functions U 0 coincides with the set of functions U of [15,16] which have continuity at cero. Definition 1.2 will be used in the following to establish the class of contractions under investigation using functions in the sets U D and U DD , where D is the distance in-between adjacent subsets of the cyclic disposal in X.…”

“…See, for instance, [2][3][4][7][8][9][10][11][12][13]. In addition, Menger probabilistic metric spaces are a special case of the wide class of probabilistic metric spaces which are endowed with a triangular norm [2,3,7,9,11,15,16,30]. In probabilistic metric spaces, the deterministic notion of distance is considered to be probabilistic in the sense that, given any two points x and y of a metric space, a measure of the distance between them is a probabilistic metric F x;y t ð Þ, rather than the deterministic distance d x; y ð Þ, which is interpreted as the probability of the distance between x and y being less than t t [ 0 ð Þ [3].…”

“…Further extensions to contractive mappings in complete fuzzy metric spaces using generalized distance distribution functions have been studied in [8,9] and references therein. The concept of altering distances was exploited in a very general context to derive fixed point results in [14], and extended later on in [15] to Menger probabilistic metric spaces. On the other hand, general fixed point theorems have been very recently obtained in [16] for two new classes of contractive mappings in Menger probabilistic metric spaces.…”