Kannan-type cyclic contraction results in 2-Menger space

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:…”

“…On the other hand, 2-cyclic u-contractions on intersecting subsets of complete Menger spaces were discussed in [7] for contractions based on control u-functions. See also [8]. It was found that fixed points are unique.…”

“…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 parallel, mixed monotone operators in such Banach spaces were discussed while the existence of coupled minimal and maximal fixed points for these operators was analyzed and discussed in detail. 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.…”