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

This paper investigates properties of convergence of distances of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-cyclic α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}-type contractions on the union of the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} subsets of a space X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X$$\end{document} defining probabilistic metric spaces and Menger spaces. The paper also investigates the characterization of both Cauchy and G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G$$\end{document}-Cauchy sequences which are convergent, in particular, to best proximity points. On the other hand, the existence and uniqueness of fixed points and best proximity points of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-cyclic α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}-ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi$$\end{document}-type contractions are also investigated. The fixed points of the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-composite self-mappings, which are obtained from the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-cyclic self-mapping restricted to each of the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document} subsets in the cyclic disposal, are also investigated while a generalization and some illustrative examples are also given.


Introduction
Fixed point theory in the framework of probabilistic metric spaces [1][2][3][4] is receiving important research attention. 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]. Fixed point theorems in complete Menger spaces for probabilistic concepts of B and C-contractions can be found in [2] together with a new notion of contraction, referred to as W; C ð Þ-contraction. Such a contraction was proved to be useful for multivalued mappings while it generalizes the previous concept of C-contraction. 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. In addition, u-contractions in complete probabilistic Menger spaces have been also studied in [11] through the use of altering distances. See also [14,26]. On the other hand, probabilistic Banach spaces versus Fixed Point Theory were discussed in [10]. The concept of probabilistic complete metric space was adapted to the formalism of Banach spaces defined with norms being defined by triangular functions and under a suitable ordering in the considered space. 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. 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. The results have been established for aÀw contractive mappings and for a generalized b-type one. It has also to be pointed out that the parallel background literature related to results on best proximity points and fixed points in cyclic mappings in metric and Banach spaces as well as topics related to common fixed points is exhaustive including studies of fixed point theory and applications in the fuzzy framework. See, for instance, [5,6,13,[17][18][19][20][21][22][23][24][25][26][27][31][32][33][34][35][36][37] as well as references therein.
This paper investigates properties of convergence of distances of p-cyclic contractions on the union of the p subsets of the abstract set X defining the probabilistic metric spaces and the Menger spaces as well as the characterization of Cauchy and G-Cauchy sequences which converge to best proximity points of p-cyclic a-w-type contractions. The existence and uniqueness of fixed points and best proximity points of p-cyclic a-w-type contractions. The fixed points of the p-composite self-mappings, which are obtained from the cyclic self-mapping restricted to each of the p subsets in the cyclic disposal, are also investigated while illustrative examples and a further generalization are also given. Denote . . .; ng, and denote also by L, the set of distance distribution functions H : R ! 0; 1 ½ , [1], which are nondecreasing and left continuous such that H 0 ð Þ ¼ 0 and sup t2R H t ð Þ ¼ 1. Let X be a nonempty set and let the probabilistic metric (or distance) F : X Â X ! L a symmetric mapping from X Â X, where X is an abstract set, to the set of distance distribution functions L of the form H : R ! 0; 1 ½ which are functions of elements F x;y for every x; y ð Þ 2 X Â X. Then, the ordered pair X; F ð Þ is a probabilistic metric space (PM) [2,3,29] . F x;y t ð Þ ¼ F y;x t ð Þ; 8x; y 2 X, 8t 2 R 3. 8x; y; z 2 X; 8t 1 ; t 2 2 R þ F x;y t 1 ð Þ À À A particular distance distribution function F x;y 2 L is a probabilistic metric (or distance) which takes values F x;y t ð Þ identified with a probability distance density function H : R ! 0; 1 ½ in the set of all the distance distribution functions L.
A Menger PM-space is a triplet X; F; D ð Þ, where X; F ð Þ is a PM-space which satisfies: is a t-norm (or triangular norm) belonging to the set T of t-norms which satisfies the properties: A property which follows from the above ones is D a; 0 ð Þ ¼ 0 for a 2 0; 1 ½ . Typical continuous t-norms are the minimum t-norm defined by D M a; b ð Þ ¼ min a; b ð Þ, the product t-norm defined by D P a; b ð Þ ¼ a:b and the Lukasiewicz (or nilpotent-minimum) t-norm defined by The (probabilistic) diameter of a subset A of X is a function from R 0þ to 0; 1 2]. The diameter of a subset A & X in the PM-space X; F ð Þ, induced by a metric space X; d ð Þ, refers to maximum real interval measure, where the argument of the probabilistic metric is unity, that is, with some given positive real functions subject to b x; y being, in particular, infinity if a ¼ b (i.e., the probability one is reached as a limit as t ! 1) or if sup x;y2A d x; y ð Þ is arbitrarily large (i.e., if A is unbounded as a subset of the metric space The (probabilistic) distance in-between the subsets A and B of X defines the argument interval length of zero probability distance in-between points of two subsets A and B of X and it is defined as: A sequence x n f g in X is said to be convergent to x in X if, for every e; k 2 R þ , there exists n 0 ¼ n 0 e; k ð Þ 2 Z 0þ such that F x n ; x e ð Þ [ 1 À k, whenever n ! n 0 . 2. A sequence x n f g in X is said to be a Cauchy sequence if, for every e; k 2 R þ , there exists n 0 ¼ n 0 e; k ð Þ 2 Z 0þ such that F x n ; x m e ð Þ [ 1 À k, whenever n; m ! n 0 . 3. X; F; D ð Þis complete if every Cauchy sequence in X is convergent to a point in X.

4.
A sequence x n f g is said to be G-Cauchy if, for every e 2 R þ , lim The following properties hold: 1. x Proof Proof of (i) Step 1 We first prove that x n f g convergent ) x n f g is Cauchy. Since x n f g is convergent then for every e; k 2 0; 1 ð Þ 2 R þ , there exists n 0 ¼ n 0 e; k ð Þ 2 Z 0þ such that F x m ; x e=2 ð Þ[ 1 À k=2, 8n; m ! n 0 ð Þ2Z 0þ . Then, since F : R ! 0; 1 ½ is non-decreasing, one gets: and then x n f g is a Cauchy. Since the above inequalities hold for any n; m ! n 0 ð Þ2Z 0þ , it turns out that lim inf Proof of (i) Step 2 We next prove by contradiction that x nþm e 1 ð Þ\1 À 2k, for some m 2 Z 0þ , some e 1 2 R þ and some given k ¼ k e 1 ð Þ 2 0; 1=2 ð Þ. Since F x; y t ð Þ is non-decreasing in t for all x; y 2 X, so that F x; y e 1 ð Þ!F x; y e 1 =2 ð Þ, then: ð1:7Þ 8m 2 Z þ : But then k\k=2. Then, x n f g is G-Cauchy. Proof of (ii): Let X; F; D ð Þbe G-complete and let x n f g & X be any given Cauchy sequence. Then x n f g is G-Cauchy, from property (1), and convergent to some x 2 X since X; F; D ð Þis G-complete. Since x n f g is an arbitrary Cauchy sequence convergent in X, it turns out that X; F; D ð Þis complete. h The e; k ð Þ-topology in a Menger in a PM-space Þis a Hausdorff topology introduced by the family of neighborhoods N x of a point x 2 x given by In this topology, a function is continuous at x 0 2 X if and only if f x n ð Þ f g! f x 0 ð Þ for every convergent sequence x n f g ! x 0 . See [1,16] for more details.
h We next denote by u z þ ð Þ and u z À ð Þ, respectively, the right and left limits of u t ð Þ as t ! z.
Definition 1.2 A function u : R ! R 0þ is said to be a U xy -function if, for given real constants x; y 2 R 0þ , with y ! x, it satisfies the following conditions: The set of functions U xx is simply denoted by U x . If 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. Definition 1.3 [16] A function w : R 0þ ! R 0þ is said to be a W-function if it continuous with w 0 ð Þ ¼ 0 and w n a n ð Þ ! 0 when a n ! 0 as n ! 1.
Main results on best proximity points for p-cyclic a-w-type contractions The definition of a p ! 2 ð Þ-cyclic a-w-type contraction follows: Þis the common distance in-between adjacent subsets, 8i 2 p.
ð Þ, for any given u 2 U DD and any given realD ! D.
Þ-cyclic and if x 2 A i for some i 2 p then for any j 2 p and n 2 Z 0þ , T npþiþj x 2 A npþiþj ¼ A k for some k 2 p since if n; m 2 Z þ and n m mod p ð Þ then A m ¼ A n . In particular, p, 8n 2 Z 0þ . It can be pointed out that pcyclic contractions include the case of cyclic self-mappings T on X such that X ¼ S i2 p A i . In this case, A i f g i2 p is said to be a cyclic representation of X; T ð Þ. On the other hand, note that is an a-w-type contraction if (2.1) holds with D ¼ 0 for t 2 R þ [16]. The distances in-between adjacent subsets are assumed to be identical just to facilitate the exposition by simplifying the contractive condition to the form (2.1) so as to make less involved their associate calculations. Note that the distances inbetween adjacent subsets in non-expansive cyclic selfmappings are identical in uniformly convex Banach spaces [27].
An equivalent constraint to (2.1) is now discussed: Þ is a strictly increasing, bijective and bicontinuous function of (then continuous) inverse Thus, a p-cyclic B-contraction is a particular type of p-cyclic a-w-type contraction. See [2] for the case D ¼ 0. Some basic properties of a p-cyclic a-w-type contraction are now given: Þ being the distance in-between adjacent subsets, 8i 2 p. Then, the following properties hold provided that u 2 U DD for any givenD ! D: Proof Since A i and A iþ1 are bounded then the maximum distance in-between any two points of adjacent subsets is not larger than D þ 2 D. Then, lim ½ is non-decreasing and left-continuous, and Now, proceed by complete induction by assuming that is non-decreasing and left continuous and w 0 ð Þ ¼ 0, one has for Hence, the proofs of Properties (1) and (2) follow by complete induction.
Properties (3)-(5) follow directly from the definitions of the sets U DD for s ¼ u t ð Þ, being equivalent to It turns out that Proposition 2.2 holds, in particular, for u 2 U D . The a-admissibility of a-w-type contractions is defined to state layer on the main result: ð Þ a PM-space. Then: 1. an a-w-type contraction T : X ! X is a-admissible for a given function a : p A i be a a À w-type pcyclic mapping, u 2 U DD and w 2 W. Then: 1. the pair x; Tx ð Þ2 clA i ; clA iþ1 ð Þ , where cl : ð Þ stands for the closure of the : ð Þ-set, for any given i 2 p is a pair of quasi-best proximity points if F x;TxD þ e À Á [ 1 À k for any given e 2 R þ and F x;Tx D ð Þ ¼ F x;Tx D À ð Þ ¼ 0.
Each of them is a quasi-best proximity point in the corresponding subset A i .

A quasi-best proximity point is a best proximity point
if F x;Tx D þ e ð Þ[ 1 À k for any given e 2 R þ and If [ 1 À k for arbitrarily small positive real constants e and k so that F x;TxD Þis a pair of best proximity points if u is not continuous at D. h The most important of the main results of this paper follows below: ð Þ!D satisfying the following conditions: Þ ! x is a Picard iteration generated as Then, the following properties hold: Furthermore, T n x 0 f g& S i2 p A i and T npþj x 0 f g& A iþj are both Cauchy and G-Cauchy convergent sequences to a limit point x 2 T i2 p cl A i . If the subsets A i are closed for i 2 and it is also a quasibest proximity point (in particular, a best proximity for any given e 2 R þ , k 2 0; 1 ð Þ and for some n 0 ¼ n 0 e; k ð Þ, 8n ! n 0 ð Þ2Z 0þ : Proof Let x 0 2 S i2 p A i such that the condition (3) holds.
It has been proved by complete induction that a x n ; Tx n ; t ð Þ!1, 8n 2 Z 0þ , 8t 2 R þ provided that a x 0 ; Tx 0 ; t ð Þ!1, 8t 2 R þ .Then a x n ; Tx n ; t ð Þ2 1; þ1 ½ Þ and a À1 x n ; Tx n ; t On the other hand, since u 2 U DD and since T : Since the distance distribution function is non-decreasing and left-continuous and K À1 [ 1, then and replacing in (2.5) x 0 ! Tx 0 , Tx 0 ! T 2 x 0 with the use of (2.6) leads to: and proceeding recursively in the same way: ð2:8aÞ equivalently, Since K À1 [ 1 and u 2 U DD then lim n!þ1 We first prove that lim sup The above condition is identical to lim sup

Þ, one concludes for Cases a and b that lim
Then, for any given real e 2 R þ and k 2 0; 1 ð Þ, there is n 0 ¼ n 0 e; k ð Þ 2 Z 0þ such that since the distance distribution function is non-decreasing and left-continuous. On the other hand,  (1) has been proved. Property (2) relies on the case when u 2 U 0 and T i2 p A i 6 ¼ ; ð2:11Þ and lim x jþk for some j; k 2 p. Since X; F; D M ð Þis a Menger PM-space then from (2.10) with D ¼D ¼ u 0 ð Þ ¼ 0: for ' 2 k À 1 [ 0 f g, any e 2 R þ and any real k 2 0; 1 ð Þwhat implies that lim n!1 F T npþj x 0 ; T npþjþk x 0 t ð Þ ¼ 1, 8t 2 R þ and, from the first property of (1.1), x j ¼ x jþk , a contradiction, and then So, the p sequences T npþi x 0 f g, 8i 2 p have a unique limit point in T i2 p A i provided that such a set is nonempty and from Assertion 1.1 they are Cauchy and G-Cauchy sequences and then convergent since X; F; D ð Þis Gcomplete. In addition, for any e 2 R þ and any real k 2 0; 1 ð Þ there is n 01 ¼ n 01 e; k ð Þ 2 Z 0þ such that from (2.2) for D ¼ 0 since T n x 0 f gis Cauchy, then G-Cauchy, and convergent to x, then lim so that for some n 0 ¼ n 0 e; k ð Þ !n 01 ð Þ2Z 0þ and for any arbitrary e 2 R þ and k 2 0; 1 ð Þ using the third and fourth properties of the triangular norms, one gets: Thus, F x;Tx 0 þ ð Þ ¼ 1 so that x ¼ Tx from the property 1 of (1.1). By replacing x ! Tx and Tx ! T 2 x in (2.13), we prove x ¼ T 2 x. Proceeding in the same way, it is proved that T i x ¼ x, 8i 2 p. So, x is a limit point of T npþi x 0 f g, 8i 2 p and T n x 0 f g which is also a fixed point of T : if the subsets A i are closed 8i 2 p. Hence, Property (2) has been proved.
On the other hand, it follows from (1.2) for Menger PMspaces and the properties of (1.3) for triangular norms for the general case thatD Then, one gets for any given t; k \1 ð Þ 2 R þ , some n 0 ¼ n 0 t; k ð Þ and 8n ! n 0 .
ð2:17:aÞ x j is a fixed point of the composite self-mapping To prove that the fixed points of the composite self-mapping are quasi-best proximity points of the p-cyclic a-w-type contraction T : S i2 p A i ! S i2 p A i , we proceed by contradiction. Assume that this is not the case, so that there is a pair x i ; T x i ð Þ for some i 2 p such that there exist e 2 R þ ; k 2 0; 1 ð Þ and a sequence n k f g & Z 0þ for some n 0 ¼ n 0 e; k ð Þ 2 Z 0þ such that, for any j 2 p, one gets that lim k!1 F T n k p Þis a Menger space and, since any triangular norm is associative and commutative and since D M x; x ð Þ!x, one gets the contradiction: Note that if the image of a x; y; t ð Þis extended to be in clR 0þ (i.e., a x; y; t ð Þcan take also values at þ1 f g), then the proof of Property (1) of Theorem 2.1 is valid with a slight extension by considering also the case that a À1 T n 1 x 0 ; T n 1 þ1 x 0 ; K À1 t À D ð Þ þ D ð Þ ¼ 0 for some finite n 1 2 Z 0þ and 8t [ D ð Þ2R þ . Since w 2 W, it is continuous, and then w n 1 F À1 for any finite n 1 2 Z þ so that Then, we could use a similar recursive procedure as that used for the case up till the n 1 À 1 ð Þ-iteration, since Proposition 2.1 remains valid, see Eqs. (2.2) and (2.3), so that: 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 . 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]: 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: Þ ! x is a Picard iteration generated as x nþ1 ¼ Tx n , 8n 2 Z 0þ with x 0 2 S i2 p A i , such that a x n ; Tx n ; t ð Þ!1, 8n 2 Z 0þ ,8t 2 R þ then a x n ; x; t ð Þ ! 1, 8n 2 Z 0þ , 8t 2 R þ .
Then, the following properties hold: 1. If u 2 U 0 and T i2 p A i 6 ¼ ; then T n x 0 f g& S i2 p A i and T npþj x 0 f g& A iþj are both Cauchy and G-Cauchy convergent sequences to a limit point x 2 T i2 p cl A i , with cl A i being the closure of A i . If A i are closed for i 2 p then x ¼ Tx ¼ T p x, that is, it is a fixed point of the self-mappings T : S i2 p A i ! S i2 p A i and and being also a best proximity point of for any given e 2 R þ , k 2 0; 1 ð Þ\R and some n 0 ¼ n 0 e; k ð Þ, 8n ! n 0 ð Þ2Z 0þ . h The following result is an extended version of a parallel result given in [16] for uniqueness of fixed points of (noncyclic) self-mappings of a p-cyclic a-w-type contraction on X. The result is concerned with (a) the uniqueness of best proximity points of p-cyclic a-w-type contractions, being corresponding fixed points of the composite self-mappings restricted to each subset of the cyclic disposal, and (b) their confluence to a unique fixed point of the p-cyclic a-w-type contraction if the subsets intersect.
Theorem 2.2 Assume all the hypotheses of Corollary 2.1 and the additional one which follow: p, which is the unique best proximity point of T : S i2 p A i ! S i2 p A i and the unique fixed point of Proof Let u 2 clA i , v 2 clA j be best proximity points for any given i; j 2 k such that u ¼ Thus, there exist integers m ¼ m i; u; z ð Þ2p À 1 [ 0 f g and ' ¼ ' j; v; z ð Þ2p À 1 [ 0 f g, such that T pnþm z 2 A iþ1 and T pnþ' z 2 A jþ1 , and since T is a-admissible and min a x; z; t ð Þ; a y; z; t ð Þ ð Þ ! 1, 8t [ D ð Þ2R þ , one gets: ð2:22Þ Note that w npÀ1 a npÀ1 t; m; u; z ð Þ À Á ! 0 as a npÀ1 t; m; ð u; zÞ ¼ Assume that u 6 ¼ v are best proximity points in clA i , then Tu; Tv 2 A iþ1 are corresponding adjacent best proximity points and T pnþm z f g! Tu and T pnþm z f g! Tv. Thus, Tu ¼ Tv p is arbitrary then the set of adjacent best proximity points is unique.
In the particular case that The sequence x n f g & A [ B is generated by 2-cyclic self-mapping T on A [ B for some real constant K 2 0; 1 ½ Þ and any given initial with the extended definition sgn x ð Þ ¼ 0 if x ¼ 0 for the case that D ¼ 0. The above sequence has the two following subsequences, in A and B if x 0 2 A, respectively, in B and A if x 0 2 B: ð2:27aÞ ð2:28Þ and it turns out that x 2n f g ! D=2, x 2nþ1 f g! ÀD=2 if x 0 2 A and x 2n f g ! ÀD=2, x 2nþ1 f g! D=2 if x 0 2 B and x ¼ AED=2 are the unique best proximity points in A and B, respectively, and unique fixed points of T 2 : A ! A and, respectively, of T 2 : B ! B, which are confluent at x ¼ 0 if D ¼ 0 and then a unique fixed point of T : D be the set of all generalized distance distribution functions of elements defined by • a probability mapping F : for some given real constants k 2 0; 1 ð and d ! 1.
Note that R; F; D M ð Þ is a G-complete generalized Menger PM-space since the distance distribution function is a generalized one, [12]. It follows from Proposition 2.1 that since a À1 x n ; Tx n ; Then, note that T is a-admissible and since then, from Theorem 2.1, there exists the following limit so that x ¼ AED=2 are also best proximity points in the probabilistic sense. In addition, Main results on best proximity points for generalized p-cyclic a-w-type contractions We generalize the concept of p ! 2 ð Þ-cyclic a-w-type contractions as follows: diam A i ð Þ being the distance in-between adjacent subsets, 8i 2 p and u 2 U DD , then: Þso that it follows in a similar way as in the proof of Proposition 2.2 that w 0 x; y; Tx; Ty; u K À1 t À D ð Þ þ D À Á À Á ¼ w 0 x; y; Tx; Ty; 0 ð Þ ¼ 0 ð3:6Þ for t 2 D þ 2 D; 1 ð Þsince K 2 0; 1 ð Þ, the distance distribution function F : R ! 0; 1 ½ is non-decreasing and left-continuous, for any given e 2 R þ , k 2 0; 1 ð Þ and some n 0 ¼ n 0 e; k ð Þ, 8n ! n 0 ð Þ2Z 0þ . h Proof Since T : S i2 p A i ! S i2 p A i is a p-cyclic a-w-type generalized contraction, x 0 2 S i2 p A i , a x 0 ; Tx 0 ; t ð Þ!1, it is proved by complete induction as in Theorem 2.1 that a x n ; Tx n ; t ð Þ!1, 8n 2 Z 0þ , 8t 2 R þ since u 2 U DD , u is strictly increasing in D; 1 ð Þwith u D ð Þ ¼D ! D [ 0 and then there exists such that min F x 0 ;Tx 0 u t ð Þ ð Þ; F Tx 0 ; Since the distance distribution function is non-decreasing and left-continuous and K À1 [ 1, Þ2R þ and taking inverses in (3.10): The cases a-c of the proof of Theorem 2.1 are re-addressed via the changes: ð3:12Þ w 0 T nÀ1 x 0 ; T n x 0 ; T n x 0 ; T nþ1 x 0 ; u t ð Þ À Á since F x;x t ð Þ ¼ 1, 8t 2 R þ , since X; F ð Þ is a probabilistic metric space, and u : R ! R 0þ is nonzero, 8t 2 R þ for all x 2 X can be removed from the evaluation of the maximum in (3.13). Equations (2.8a)-(2.8b) are changed to: a À1 u; T npþk z; t À Á Â Ã w npÀ1 0 u; T k z; Tu; T kþ1 v; À where t 0 n ¼ t 0 n t ð Þ ¼ K Ànp t À D ð ÞþD since  since w npÀ1 a npÀ1 t; k; u; z