This paper investigates properties of convergence of distances of p-cyclic contractions on the union of the p subsets of an abstract set X defining probabilistic metric spaces and Menger probabilistic metric spaces as well as the characterization of Cauchy sequences which converge to the best proximity points. The existence and uniqueness of fixed points and best proximity points of p-cyclic contractions defined in induced complete Menger spaces are also discussed in the case when the associate complete metric space is a uniformly convex Banach space. On the other hand, the existence and the uniqueness of fixed points of the p-composite mappings restricted to each of the p subsets in the cyclic disposal are also investigated and some illustrative examples are given.

1. Introduction

Fixed point theory in the framework of probabilistic metric spaces [1–4] is receiving important research attention. See, for instance, [2–11]. On the other hand, 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, 5, 7, 9, 12, 13]. 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 Fx,yt, rather than the deterministic distance dx,y, which is interpreted as the probability of the distance between x and y being less than a real value tt>0, [3].

Fixed point theorems in complete Menger probabilistic metric spaces for probabilistic concepts of B and C-contractions can be found in [2] together with a new notion of contraction, referred to as a Ψ,C-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 ϕ-contractions on intersecting subsets of complete Menger spaces were discussed in [5] for contractions based on control ϕ-functions. See also [6]. It was found that fixed points are unique. Also, ϕ-contractions in complete probabilistic Menger spaces have been also studied in [9] through the use of altering distances. On the other hand, probabilistic Banach spaces versus fixed point theory were discussed in [8]. 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 by using generalized distribution functions have been studied in [6, 7] and some 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 [12] to Menger probabilistic metric spaces. On the other hand, general fixed point theorems have been very recently obtained in [13] for two new classes of contractive mappings in Menger probabilistic metric spaces. The results have been established for α-ψ contractive mappings and for a generalized β-type one. It has also to be pointed out that the parallel background literature related to best proximity points and fixed points in cyclic mappings in metric and Banach spaces is exhaustive. See, for instance, [15–28] and references therein. Fixed point theory has also been widely applied to stability and equilibrium problems since, even based on intuition, the convergence of trajectory-solutions of differential or difference equations or dynamic systems to a point can be typically associated to the convergence of sequences to fixed points; see, for instance, [27, 29, 30] and references therein, and to ergodic processes [31].

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 probabilistic metric spaces as well as the characterization of Cauchy sequences which converge to best proximity points. The existence and the uniqueness of fixed points and best proximity points of p-cyclic contractions is also discussed in induced complete Menger probabilistic metric spaces in the case that the associate complete metric space is a uniformly convex Banach space. The fixed points of the p-composite mappings restricted to each of the p subsets in the cyclic framework disposal are also investigated. Finally, some examples are discussed.

2. Main Results

Denote R+=R0+∪0=z∈R:z>0, R0+=z∈R:z≥0, Z+=Z0+∪0=z∈Z:z>0, Z0+=z∈Z:z≥0, and n-=1,2,…,n. Denote by L the set of distribution functions H:R→0,1, which are nondecreasing and left continuous, such that F0=0 and supt∈RFt=1. Let X be a nonempty abstract set of elements and let F:X×X→L be a mapping from X×X into the set of distribution functions L, which are symmetric functions of elements Fx,y for every x,y∈X×X. Then, the ordered pair X,F is a probabilistic metric space (PM), [1–4], if

∀x,y∈XFx,yt=1;∀t∈R+⇔x=y,

Fx,yt=Fy,xt; ∀x,y∈X, ∀t∈R,

∀x,y,z∈X;∀t1,t2∈R+,Fx,yt1=Fy,zt2=1⇒Fx,zt1+t2=1.

Note that an interpretation of the PM-space is that F:X×X→L is a set of distribution functions. A particular distribution function Fx,y is a probabilistic metric (or distance) which takes values Fx,yt, identified with the values of a mapping H:R→0,1 in the set of all the distribution functions L, so that two points are (probabilistically) identical if the probabilistic metric in-between them gives probabilistic certainty (namely, probability equal to one) for a mutual distance being smaller than any given positive real number.

A Menger PM-space is a triplet X,F,Δ, where X,F is a PM-space which satisfies(1)Fx,yt1+t2≥ΔFx,zt1,Fz,yt2;∀x,y,z∈X,∀t1,t2∈R0+under Δ:0,1×0,1→0,1 which is a t-norm (or triangular norm) belonging to the set T of t-norms which satisfy the properties(2)1Δa,1=a2Δa,b=Δb,a3Δc,d≥Δa,bifc≥a,d≥b4ΔΔa,b,c=Δa,Δb,c.A property which is a consequence of the above ones is Δa,0=0. The probabilistic diameter of a subset A of X is a function from R0+ to 0,1 defined by DAp0z=supt<zinfx,y∈AFx,yt and A which is probabilistically bounded if DAp=supz∈R+DAp0z=1 [2]. Note that the diameter of a set refers to the real interval length where the argument of the probabilistic metric is nonzero while the probabilistic diameter is a measure of boundedness or unboundedness of such a set. 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(3)D=dA,B=infz∈R0+:supx∈A,y∈BFx,yt=0;∀t∈0,z.The concept of p-cyclic contraction in a PM-space is recalled below (see, e.g., [1–6]).

Definition 1.

A mapping T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction in a X,F PM-space, where Ai are nonempty subsets of X with D being the distance between the adjacent subsets Ai and Ai+1, ∀i∈p-; if TAi⊆Ai+1, ∀i∈p- and there exists a real constant K∈0,1 such that, for each pair x,y∈Ai×Ai+1, ∀i∈p-, the following constraint holds:(4)FTx,TyKt+1-KD≥Fx,yt;∀t>D∈R+,equivalently, FTx,Tyt≥Fx,yK-1t-D+D; ∀t>D∈R+, and (5)FTx,Tyt-D≥Fx,yK-1t-D;∀t>D∈R+.Note that, if T:⋃i∈p-Ai→⋃i∈p-Ai is p-cyclic contraction in a metric space X,d, then there is K∈0,1 such that, for each x,y∈Ai×Ai+1, ∀i∈p-,(6)dTx,Ty≤Kdx,y+1-KD.The space X,F is the induced PM-space of the metric space X,d if Fx,yt=Ht-dx,y; ∀x,y∈X; ∀t∈R+.

The definitions of convergent sequence and Cauchy sequence in a PM-space follows.

Definition 2.

A sequence xn⊂X converges to x∈X if, for any given real constants ε,λ∈R+, with λ∈0,1, there is N0=N0ε,λ such that Fxn,xε>1-λ for n≥N0.

Definition 3.

A sequence xn is a Cauchy sequence in X if for any given real constants ε,λ∈R+, with λ∈0,1, there is N0=N0ε,λ such that Fxn,xmε>1-λ for n,m≥N0.

Note that if xn⊂X converges to some x∈X then limn→∞Fxn,xt=1, ∀t∈R+, and, if xn is a Cauchy sequence in X, then limn,m→∞Fxn,xmt=1, ∀t∈R+. A Menger PM-space X,F,Δ is complete if every Cauchy sequence is convergent in X.

Definition 4.

x∈clAi and y∈clAi+1 are adjacent best proximity points so that the pair x,y∈clAi×clAi+1 is an adjacent pair of best proximity points for any given i∈p- if Fx,yt=0; t∈-∞,D and Fx,yD+=1, where D=dAi,Ai+1, ∀i∈p-.

Since F:R→0,1 is nondecreasing and left continuous with F0=F0-=0 then if x,y are adjacent best proximity points Fx,yt=Fx,yD+=1 for t>D∈R+. If D=0, that is, all subsets Ai⊂X, ∀i∈p-, pair-wise intersect, so that ⋂i∈p-Ai≠Ø, then all the best proximity points are coincident at a unique fixed point in ⋂i∈p-clAi and Fx,Txt=Fx,y0+=1 for t∈R+ which implies x=Tx=y from the first property of the probabilistic space. The subsequent result addresses the fact that the sequences built by iterations through p-cyclic contractions T:⋃i∈p-Ai→⋃i∈p-Ai in complete Menger PM-spaces lead to convergent Cauchy sequences on each of the subsets Ai⊂X, ∀i∈p-, if such subsets are closed, the limits of the subsequences being in the closures of the subsets of the cyclic disposal in the general case. The convergence points of such subsequences are fixed points of the composite mappings Tp restricted to each one of the subsets Ai, ∀i∈p-, and best proximity points located at each pair of the closures of the adjacent subsets Ai and Ai+1, ∀i∈p-, and at the sets themselves if the subsets are closed.

Theorem 5.

Assume that T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction in a Menger PM-space X,F,Δ endowed with a t-norm Δ:0,1×0,1→0,1, where Ai are nonempty subsets of X such that D=dAi,Ai+1, ∀i∈p-, under a probability density function F:R→0,1 such that Fx,yt=0 for t∈-∞,D and Fx,yt=1, ∀t>D∈R+, for any x,y∈Ai×Ai+1, ∀i∈p-.

Then, the following property holds:(7)(i)limt→∞FTnpx,Tnp+1xt=limn→∞limt→D+FTnx,Tn+1xKnt+1-KnD=1;aaaaaa∀x,y∈Ai×Ai+1,∀i∈p-,∀t>D,limt→D-FTnx,TnyKnt+1-KnD=FTnx,Tnyt=0;∀x,y∈Ai×Ai+1,∀i∈p-,∀n∈Z0+,t∈-∞,D.Assume, furthermore, that the Menger PM-space X,F,Δ is complete and that Δ:0,1×0,1→0,1 is continuous and Δt,t≥t for each t∈R+.Then, the subsequent additional property holds:(8)iilimn→∞FTnpx,Tn+mpxt=limn→∞FTnpx,Tn+mpyt=1;∀x∈⋃i∈p-Ai,∀t>D∈R+,with x,y∈Aj for some j∈p-, and, for any given ε∈R+ and λ∈0,1, there is n0=n0ε,λ such that minFTnpx,Tn+mpxt,FTnpx,Tnpyt>1-λ, ∀n≥n0∈Z0+, so that the sequences Tnpx⊆Ai, Tnp+1x⊆Ai+1 are Cauchy sequences; then convergent to adjacent best proximity points, xi∗∈clAi(xi∗∈Ai if Ai is closed) and xi+1∗∈clAi+1(xi+1∗∈Ai+1 if Ai+1 is closed), ∀x∈Ai, ∀i∈p-, which are fixed points of the composite mapping Tp on Ai and Ai+1, respectively.

Proof.

Note that Tnpx⊂Ai and Tnpy⊂Ai+1, ∀x,y∈Ai×Ai+1, ∀i∈p-. Thus, one gets from (5) that (9)FTnpx,Tnpyt-D≥FTnp-1x,Tnp-1yK-1t-D≥⋯≥Fx,yK-npt-D;∀t>D∈R+;limn→∞FTnpx,Tnpyt=Fx,y∞=1, ∀x,y∈Ai×Ai+1, ∀t>D∈R+. It turns out that we can fix the argument of the first left-hand side to Kt+1-KD and, correspondingly, its first right-hand side to t. In the same way, we can use for both arguments with the corresponding pairs t+K-1-1D,K-1t and t,K-1t-D+D. Furthermore, it is obvious that(10)limn→∞limt→t0-FTnpx,Tnpyt=Fx,yD=Ft0=0;∀x,y∈Ai×Ai+1,t0∈-∞,Dand then, for any given ε∈R+ and λ∈0,1, there is N=Nε,λ∈Z0+ such that FTnpx,TnpyD+ε>1-λ for n>N∈Z0+, ∀x,y∈Ai×Ai+1, ∀t>D∈R+. Hence, property (i) follows. On the other hand, note from property (i) for y=Tx and any x∈⋃i∈p-Ai that(11)limt→∞FTnpx,Tnp+1xt=1;∀t>D∈R+and that, for any given ε∈R+ and λ∈0,1, there is N1=N1ε,λ∈Z0+ such that FTnpx,Tnp+1xD+ε>1-λ for n>N1∈Z0+. Then, one gets from (1), (5), the second and third conditions of (2), and the condition Δx,x≥x for each x∈0,1 and n,m∈Z+:(12)FTnpx,Tn+mpxt≥ΔFTnpx,Tnp+1x1-Kt,FTnp+1x,Tn+mpxKt≥ΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,ssssss1-Kmp1-KΔFTnp+1x,Tnp+2xK1-Kt,FTnp+2x,Tn+mpxK2t≥ΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,sssssssΔFx,Tx1-Kmp1-KK-np-1K1-Kt-D+D,sssssssssss1-Kmp1-KFTnp+2x,Tn+mpxK2t=ΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,ssssssssΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,sssssssssss1-Kmp1-KFTnp+2x,Tn+mpxK2t⋮≥ΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,sssssssssΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,sssssssssssss1-Kmp1-KFTn+mp-1x,Tn+mpxKmp-1t≥ΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,ssssssssΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,ssssssssssss1-Kmp1-KFx,TxK-n+mp+1Kmp-1t-D+D=ΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,sssssssΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,ssssssssssFx,Tx1-Kmp1-KK-np1-Kt-D+D≥ΔFx,Tx1-Kmp1-KK-np1-Kt-D+D,sssssssFx,Tx1-Kmp1-KK-np1-Kt-D+D≥Fx,Tx1-Kmp1-KK-np1-Kt-D+D;sssssssssssssssssssSSSSSSsssssssssssssss∀t∈R+.Then, limn→∞FTnpx,Tn+mpxt=Fx,Tx∞=1, ∀x∈⋃i∈p-Ai, ∀t∈R+ since Fx,Txt is nondecreasing with supremum over t∈R+ equalizing unity. Then, for any given ε∈R+ and λ∈0,1, there is n0=n0ε,λ such that FTnpx,Tn+mpxt>1-λ, ∀n≥n0∈Z0+ so that Tnpx⊂Aj is a Cauchy sequence which is convergent to a point in clAj, since X,F,Δ is complete, if x∈Aj for any j∈p-. Now, if we take x,y≠x∈Ai for some arbitrary given i∈p-, one obtains in a similar way that (13)FTnpx,Tnpyt≥Fx,yK-np1-Kt-D+Dand then limn→∞FTnpx,Tnpyt=1, ∀t∈R+. Then, for any given ε∈R+ and λ∈0,1, there is n1=n1ε,λ such that FTnpx,Tn+mpyt>1-λ, ∀n≥n01∈Z0+ and so that Tnpx⊂Ai for any x∈Ai is a Cauchy sequence, and a subsequence of Tnx⊂⋃i∈p-Ai, which is convergent to a point in clAi, since X,F,Δ is complete. Since Tnpx and Tnp+1x are convergent to xi∗∈clAi and xi+1∗∈clAi+1 for any x∈Ai, any i∈p-, then one gets by taking into account property (i) that xi∗ and xi+1∗ are adjacent best proximity points; that is, xi+1∗=Txi∗. Assume the contrary, so that Tnpx→xi∗, Tnp+1x→xi+1∗, and xi+1∗≠Txi∗ for some x∈Ai and some i∈p- so that:(14)Fxi∗,xi+1∗t≥ΔFTnpx,xi∗1-Kt,FTnpx,xi+1∗Kt=ΔFTnpx,xi∗1-Kt,Kt2ssssssΔFTnp+1x,TnpxKt2,FTnp+1x,xi+1∗Kt2≥ΔFTnpx,xi∗D+,ΔFx,TxD+,FTx,xi+1∗D+;ssssssssssssssssssssssssssssssssssssssssss∀t>D∈R+and then Fxi∗,xi+1∗t=1 thus xi+1∗=Txi∗, ∀x∈⋃i∈p-Ai, ∀i∈p-. It also turns out that those points are also fixed points of the restricted composite self-map Tp to Ai and, respectively, to Ai+1 since, otherwise, the respective sequences Tnpx and Tnp+1x would not be convergent. Property (ii) has been proved.

Note that Theorem 5 does not address the uniqueness of the best proximity points in the subsets Ai⊂X, ∀i∈p-. Their existence is guaranteed in the closures of the subsets Ai since all sequences of the form Tnpx,Tnp+1x are Cauchy convergent sequences to two respective limits at distance D with probability one allocated in adjacent sets clAi and clAi+1 for any given initial point x∈Ai for any i∈p-. The subsequent result addresses the existence and the uniqueness of best proximity points if X, is a uniformly convex Banach space and X,F,Δ is a complete Menger PM-space provided that the subsets Ai⊂X, ∀i∈p-, are closed and convex.

The following Corollary to Theorem 5 follows for the case when Ai=A⊂X, ∀i∈p- by making D=0.

Corollary 6.

Assume that T:A→A is a contraction in a Menger PM-space X,F,Δ endowed with a t-norm Δ:0,1×0,1→0,1, where A is a nonempty subset of X under a probability density function F:R→0,1 such that Fx,yt=0 for t∈-∞,0 and Fx,yt=1 for t>0 for any given x,y∈A.

Then, the following property holds:(15)ilimt→∞FTnx,Tn+1xt=limn→∞FTnx,Tn+1x0+=1,∀x,y∈A,∀i∈p-,∀t>0,limt→0-FTnx,TnyKnt=FTnx,Tnyt=0,∀x,y∈Ai×Ai+1,∀i∈p-,∀n∈Z0+,t∈-∞,0.If, furthermore, the Menger PM-space X,F,Δ is complete and that Δ:0,1×0,1→0,1 is continuous and Δt,t≥t for each t∈R+.Then, the subsequent additional property holds:(16)iilimn→∞FTnx,Tn+mxt=limn→∞FTnx,Tn+myt=1,∀x,y∈A,∀t∈R+,with x,y∈A, and, for any given ε∈R+ and λ∈0,1, there is n0=n0ε,λ∈Z0+ such that minFTnx,Tn+mxt,FTnx,Tnyt>1-λ, ∀n≥n0∈Z0+, so that the sequence Tnx⊆A is a Cauchy sequence, then convergent to a fixed point x∗∈clA(x∗∈A if A is closed) which is also a fixed point of the composite mapping Tp on A.

A second main result of the paper and a corresponding corollary for the case when all the subsets of the cyclic disposal coincide follow below.

Theorem 7.

Assume that T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction in a PM-space X,F, where Ai are nonempty subsets of X such that D=dAi,Ai+1, ∀i∈p-, endowed with a probability density function F:R→0,1 such that Fx,yt=0 for t∈-∞,D and Fx,yt=1 for t>D for any x,y∈Ai×Ai+1, ∀i∈p-. Then, the following properties hold:(17)(i)limn→∞limt→D+FTnx,TnyKnt+1-KnD=1;∀x,y∈Ai×Ai+1,∀i∈p-,∀t>D,limt→D-FTnx,TnyKnt+1-KnD=FTnx,Tnyt=0;∀x,y∈Ai×Ai+1,∀i∈p-,∀n∈Z0+,t∈-∞,D.Assume, furthermore, that X,d≡X, is a uniformly convex Banach space under a complete norm-induced metric d:X×X→R0+, so that X,F,Δ is a complete Menger PM-space under the distribution function Fx,yt=Ht-dx,y, ∀x,y∈X, for any t∈R subject to(18)Fx,yt=0ift≤D,Fx,yt=1ift>D;∀x,y∈Ai×Ai+1,∀i∈p-.Assume also that the subsets Ai of X are closed and convex, ∀i∈p-. Then, the following two further properties hold.

Consider sequences Tnx,Tnz,Tny⊂⋃i∈p-Ai for any x,z∈Ai and y∈Ai+1 for some arbitrarily given i∈p- such that

FTnpx,TnpzD+→1 as n→∞,

FTmpx,TnpyD+ε>1-λ for all m>n≥N1 and each given ε,λ<1∈R+ and some N1=N1ε,λ.

Then, for each given ε,λ<1∈R+, there exists N∈Z0+ such that FTmpx,Tnpzε>1-λ for all m>n≥N; then limn,m>n→∞limε→0+FTmpx,Tnpzε=1.

Tnpx⊂Ai, Tnpz⊂Ai, and Tnpy⊂Ai+1 are Cauchy sequences for any x,z∈Ai, y∈Ai+1, and any given i∈p- and then bounded, and convergent to unique fixed points yi=Tpyi and xi=zi=Tpzi of the composite mapping Tp restricted to each Ai into itself, ∀i∈p-. Also, zi and Tzi are the unique best proximity points in the adjacent subsets Ai and Ai+1 of X, ∀i∈p-.

Proof.

Since T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction in a PM-space X,F, one gets from (4) for any x,y∈Ai×Ai+1, ∀i∈p-, that(19)FTnx,TnyKnt+1-KnD≥⋯≥FT2x,T2yK2t+K1-KD+1-KD=FT2x,T2yK2t+1-K2D≥FTx,TyKt+1-KD≥Fx,yt;t∈R.Thus, since F:R→0,1 is nondecreasing and left-continuous,(20)limn→∞limt→D+FTnx,TnyKnt+1-KnD=limn→∞FTnx,TnyD+=1;∀x,y∈Ai×Ai+1,∀i∈p-,∀t>Dlimn→∞limt→D-FTnx,TnyKnt+1-KnD=limn→∞FTnx,TnyD+=FTnx,Tnyt=0;∀x,y∈Ai×Ai+1,∀i∈p-,∀n∈Z0+,t∈-∞,D.Hence, property (i) has been proved. To prove property (ii), first note that Fx,yt=Ht-dx,y=0, ∀x,y∈Ai×Ai+1, for any t≤D∈R since t-dx,y≤0 and then Fx,yt=H0=0 and also Fx,yt=Ht-dx,y=0 for t>D.

We now proceed by contradiction. Assume the contrary. Then, ∃ε0,λ0≤1∈R+ such that, for each k∈Z0+, there are mk>nk≥k for which FTmkpx,Tnkpzε0≤1-λ0. Choose 0<γ<1 such that ε0/γ>D and choose 0<ε<minε0/γ-D,Dδγ/(1-δγ). For such an ε∈R+, there is λ<1∈R+ and N1,N2∈Z+ such that, since FTnkpz,Tnkpyt+ε→1 as n→∞, ∀t≥D, then FTnkpz,Tnkpyt+ε>1-λ for mk>nk≥N2. Since, furthermore, FTmkpx,Tnkpyt+ε>1-λ for mk>nk≥N1, ∀t>D, and some N1,N2∈Z0+ since X,F,Δ is a Menger PM-space. Then, if N=maxN1,N2, since Ai are closed and convex, ∀i∈p-, and X, is a uniformly convex (and hence reflexive) Banach space with modulus of convexity δε>0 for ε>0 and δ· is strictly increasing, one has that, for any x,y,p∈X, R∈R+ and r∈0,2R,(21)x-p≤R∧y-p≤R∧x-y≥r⟹x+y2-p≤1-δrRRand then, for any x,z∈Ai, y∈Ai+1 and any arbitrary given i∈p-,(22)1>1-λ0≥FTmkpx+Tnkpz/2,Tnkpyt+ε=Ht+ε-Tmkpx+Tnkpz2-Tnkpy≥Ht+ε-1-δε0D+εD+ε;t>Dand there is t1>D such that the following contradiction is got if Tnpz is not a Cauchy sequence in the Menger PM-space X,F,Δ so that FTmkpx,Tnkpzε0≤1-λ0 for mk>nk≥k and some ε0,λ0≤1∈R+:(23)1≥Ht+ε-1-δε0D+εD+ε=1;t≥t1.Hence, property (ii) follows. To prove property (iii), note from property (ii) that, for any i∈p- and initial points x=z∈Ai, one has that FTmpz,Tnpzε>1-λ for m>N≥N0 and some N0=N0ε,λ and any ε,λ<1∈R+, then limm>nn→∞FTmpz,Tnpz0+=1 so that Tnpx⊂Ai is a Cauchy sequence for any x∈Ai and any i∈p- and then bounded, which is convergent to some zi∈Ai since the Menger PM-space X,F,Δ is complete. For the other two Cauchy sequences, the same proof applies. Since the sequence Tnpx⊂Ai is convergent to zi∈Ai (since Ai is closed) then limn→∞FTnpz,zi0+=1. Assume that zi≠Tpzi for some i∈p-. Then, since Tn+1pz→zi does not converge to Tpzi≠zi then 1>limn→∞FTn+1pz,Tpzi0+=limn→∞FTpTnpz,Tpzi0+=1 so that Tn+1pz→Tpzi≠zi for such an i∈p-, a contradiction. Thus, zi=Tpzi, ∀i∈p-. It is now proved by contradiction that zi=Tpzi is unique, ∀i∈p-. Assume that zi=Tpzi≠z-i=Tpz-i for some i∈p-. Then,(24)1=FTnpzi,Tnpz-it=limn→∞FTnTpzi,TnTpzit=Fzi,z-it;∀t∈R+;∀i∈p-⟺zi=z-i;∀i∈p-which contradicts zi≠z-i for some i∈p-. Thus, the fixed points of the restricted composite mapping Tp to each Ai are all unique, ∀i∈p-. Now, assume that zi+1=Tzi=Tp+1zi, ∀i∈p-, is false; that is, zi+1≠Tzi for zi+1 and Tzi is some Ai+1 for some i∈p-. Then, since T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction in X,F,Δ, one has the following contradiction for some t1∈R+, positive reals:(25)1>FTzi,zi+1t1=FTnp+1zi,Tnpzi+1t1≥limn→∞limx→D+FTnpTzi,Tnpzi+1K-npt-D+x=Fzi+1,Tzit=Fzi+1,TziD+=1;∀t>D∈R+.Then, zi+1≠Tzi, ∀i∈p-.

Corollary 8.

Assume that T:A→A is a contraction in a PM-space X,F, where A⊂X is nonempty subsets of X, endowed with a probability density function F:R→0,1 such that Fx,yt=0 for t∈-∞,0 and Fx,yt=1 for t>0 for any x,y∈A. Then, the following properties hold:(26)ilimn→∞limt→0+FTnx,TnyKnt=1,∀x,y∈A,∀t>0,limt→0-FTnx,TnyKnt=FTnx,Tnyt=0,∀x,y∈A,∀n∈Z0+,t∈-∞,0.Assume, furthermore, that X,d≡X, is a uniformly convex Banach space under a complete norm-induced metric d:X×X→R0+, so that X,F,Δ is a complete Menger PM-space under the distribution function Fx,yt=Ht-dx,y, ∀x,y∈X for any t∈R subject to(27)Fx,yt=0ift≤0,Fx,yt=1ift>0,∀x,y∈A.Assume also that A is closed and convex. Then consider the following.

Consider sequences Tnx,Tnz,Tny⊂A for any x,y,z∈A such that

FTnx,Tnz0+→1 as n→∞;

FTmx,Tnyε>1-λ for all m>n≥N1 and each given ε,λ<1∈R+ and some N1=N1ε,λ.

Then, for each given ε,λ<1∈R+, there exists N∈Z0+ such that FTmx,Tnzε>1-λ for all m>n≥N and limn,m>n→∞limε→0+FTmx,Tnzε=1.

Tnx is a Cauchy sequence for any x∈A and then bounded, and convergent to a unique fixed point z=Tz.

Remark 9.

Note that if T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction in X,F then the composite mappings Tp from ⋃i∈p-Ai∣Aj into Aj, ∀j∈p-, satisfy from (4) for each x,y∈Aj; ∀j∈p- that:(28)FTpx,TpyK-t+1-K-D≥Fx,yt;∀t∈Rwith K-=Kp∈0,1. However, each composite mapping is not, in general, a contraction on each Ai, ∀i∈p, although any iterated sequences Tnpx⊂Ai are Cauchy sequences, ∀i∈p, in a complete Menger PM-space X,F,Δ.

The following result is a direct consequence of Theorem 7 if the subsets of the cyclic disposal intersect.

Corollary 10.

Assume that all the hypothesis of Theorem 7 hold while ⋂i∈p-Ai≠Ø; that is, the sets Ai, ∀i∈p-, are not disjoint. Then, the following properties hold:(29)iT:⋃i∈p-Ai⟶⋃i∈p-Ai,Tp:⋃i∈p-Ai∣Aj⟶Aj;∀j∈p-arecontractionsinX,F,limn→∞limt→0+FTnx,TnyKnt=1;∀x,y∈Ai×Ai+1,∀i∈p-,∀t>0,Limt→0-FTnx,TnyKnt=FTnx,Tnyt=0;∀x,y∈Ai×Ai+1,∀i∈p-,∀n∈Z0+,t∈-∞,0.Assume, furthermore, that X,d≡X, is a uniformly convex Banach space under a norm-induced metric d:X×X→R0+, so that X,F,Δ is a complete Menger PM-space under the distribution function Fx,yt=Ht-dx,y, ∀x,y∈X, for any t∈R subject to(30)Fx,yt=0ift≤0,Fx,yt=1ift>0;∀x,y∈Ai×Ai+1,∀i∈p-.Assume also that the subsets Ai of X are closed and convex, ∀i∈p-. Then, the following two further properties hold.

Consider the sequences Tnx,Tnz,Tny⊂⋃i∈p-Ai for any x,z∈Ai and y∈Ai+1 for some arbitrarily given i∈p- such that:

FTnpx,Tnpz0+→1 as n→∞,

FTmpx,Tnpyε>1-λ for all m>n≥N1 and each given ε,λ<1∈R+ and some N1=N1ε,λ.

Then, for each given ε,λ<1∈R+, there exists N∈Z0+ such that FTmpx,Tnpzε>1-λ for all m>n≥N; then limn,m>n→∞limε→0+FTmpx,Tnpzε=1.

Tnpx⊂Ai, Tnpz⊂Ai, and Tnpy⊂Ai+1 are Cauchy sequences for any x,z∈Ai, y∈Ai+1, and any given i∈p-, then bounded, and convergent to a unique fixed point z∈⋂i∈p-Ai of T:⋃i∈p-Ai→⋃i∈p-Ai and Tp:⋃i∈p-Ai∣Aj→Aj, ∀j∈p-.

Proof.

The proof of Theorem 7(i) and the fact that the sequences of Theorem 7(iii) are Cauchy sequences are particular cases of Theorem 11 and do not need a proof. Now, assume that two adjacent best proximity points are distinct, namely, zi+1≠Tzi for some i∈p-. Then, in the same way as in the last part of the proof of Theorem 7, one gets the contradiction(31)1>FTzi,zi+1t≥limn→∞limx→D+FTnpTzi,Tnpzi+1K-npt+x=Fzi+1,Tzit=Fzi+1,Tzi0+=1;∀t∈R+,together with the parallel contradiction to Tzi≠zi for some i∈p-:(32)1>FTzi,zit≥limn→∞limx→D+FTnpTzi,TnpziK-npt+x=Fzi,Tzit=Fzi,Tzi0+=1;∀t∈R+.Thus, zi+1=zi=Tzi, ∀i∈p-, which is in ⋂i∈p-Ai, since ⋂i∈p-Ai is nonempty and closed.

The subsequent result formulates a connection between metric spaces and their induced Menger PM-spaces [3] concerning some basic properties of p-cyclic contractions whose contractive condition is defined on the metric space X,d rather than on the Menger PM-space.

Theorem 11.

Let X,d be a metric space, let X,F be an induced PM-space, where F:X×X→L, H:R→0,1 in L being defined by Fx,yt=Ht-dx,y, ∀x,y∈X, ∀t∈R. Let T:⋃i∈p-Ai→⋃i∈p-Ai be a p-cyclic contraction in X,d; that is, (33)dTx,Ty≤Kdx,y+1-KD;∀x,y∈Ai∪Ai+1;∀i∈p-for some real constant K∈0,1, where Ai are nonempty subsets of X, ∀i∈p-, such that D=dAi,Ai+1, ∀i∈p-.

Then, the following properties hold:(34)(i)limn→∞limt→D+FTnx,Tnyt=HD+=1;∀x,y∈Ai∪Ai+1,∀i∈p-,limn→∞limt→D-FTnx,Tnyt=HD-=0;∀x,y∈Ai∪Ai+1,∀i∈p-,Fx,yt=0;∀x,y∈Ai∪Ai+1,∀i∈p-;∀t∈-∞,D.

Assume, in addition, that X,d≡X, is a uniformly convex Banach space under a norm-induced metric, so that X,F,Δ is an induced complete probabilistic Menger PM-space under the Δ-norm of the minimum. Assume also that the subsets Ai of X are closed and convex, ∀i∈p-. Then, Tnpx⊂Ai is a Cauchy sequence and then bounded and convergent to a unique fixed point zi=Tpzi of the composite mapping Tp restricted to Ai into itself, ∀i∈p-, while zi and Tzi are the unique best proximity points in the adjacent subsets Ai and Ai+1 of X, ∀i∈p-. There is a unique fixed point z=Tz which is coincident with all the p best proximity points if the subsets Ai, ∀i∈p-, have a nonempty intersection.

Proof.

If F:X×X→F is the mapping induced by the metric d:X×X→R0+ and T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction then there is a real constant K∈0,1 such that (35)dTx,Ty≤Kdx,y+1-KD;∀x,y∈Ai∪Ai+1;∀i∈p-and X,F is an induced PM-space by the metric space X,d. Thus, one gets from (35) in accordance with the definition Fx,yt=Ht-dx,y, ∀x,y∈Ai∪Ai+1, ∀i∈p-, for any t∈R, and since H:R→R0+ is nondecreasing and left-continuous and dx,y≥D that(36)FTx,TyKt+1-KD=HKt+1-KD-dTx,Ty≥HKt-dx,y≥HKt-D=Fx,yKt-D+dx,y≥Fx,yKt-D+D≥Fx,yD,∀x,y∈Ai∪Ai+1, ∀i∈p-, ∀t>D∈R+. Then, by proceeding recursively with (36) since TAi⊆Ai+1, ∀i∈p-, one gets(37)FTnx,TnyKt+1-KD≥Fx,yD;∀x,y∈Ai∪Ai+1,∀i∈p-,∀n∈Z0+,∀t>D∈R+liminfn→∞limt→D+FTnx,TnyKnt+1-Knt-Fx,yt≥0,∀x,y∈Ai∪Ai+1, ∀i∈p-, ∀t>D∈R+. Then, there exists the limit limn→∞limt→D+FTnx,Tnyt=HD+=1, ∀x,y∈Ai∪Ai+1, ∀i∈p-. In the same way, and since Fx,yD-=HD-=0; ∀x,y∈Ai∪Ai+1, one gets(38)liminfn→∞limt→D-FTnx,TnyKnt+1-Knt-Fx,yt=HD-;∀x,y∈Ai∪Ai+1,∀i∈p-.Note from (1) that if t≤D then(39)FTx,TyKt+1-KD≤HD-dTx,Ty=H0=0;∀x,y∈Ai∪Ai+1,∀i∈p-so that Ht=HD-=H0=0 for t∈0,D and property (i) has been proved. Property (ii) follows if X,F,Δ is now a Menger PM-space under the triangular norm Δ of the minimum induced by the metric space X,d. Note that T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction, X,≡X,d is a uniformly convex Banach space, and then a complete metric space and the subsets Ai of X are closed and convex, ∀i∈p-. Thus, from Theorem 7 (see also [4] for the deterministic framework), X,F,Δ is a complete Menger PM-space under the Δt-norm of the minimum defined by Δa,b=mina,b. Then, for any given ε∈R+ and λ∈0,1 there is N=Nε,λ, FTnpx,Tnp+1xD+ε≥1-λ, FTnpx,Tn+mpxε≥1-λ, and FTnpx,ziε≥1-λ, ∀n>N∈Z0+, some unique zi∈Ai, any given x∈Ai, and i∈p-. Thus, Tnpx⊂Ai is a Cauchy sequence, then bounded, and convergent to a unique fixed point zi=Tpzi of the composite mapping Tp restricted to Ai into itself, ∀i∈p-, while zi and Tzi are the unique best proximity points in the adjacent subsets Ai and Ai+1 of X, ∀i∈p-.

3. ExamplesExample 1.

Assume that the PM-space X,F is induced by a metric space X,d and that T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic contraction in X,d with contractive constant K∈0,1, then in X,F with D being the distance in-between adjacent subsets Ai and Ai+1, ∀i∈p-. If x∈⋃j∈p-Aj then for any t>dx,Tx∈R+ and any given ε∈R+ and any given λ∈0,1, (40)FTnx,Tn+1xD+ε=FTnx,Tn+1xKnt+1-KnD=HKnt+1-KnD-dTnx,Tn+1x≥HKnt-dx,Tx=1>1-λfor all integer n≥n0 and some integer n0=n0ε,λ. Since dTnx,Tn+1x≤Kndx,Tx-D+D, the condition t>dx,Tx implies that HKnt-dx,Tx≥H0+=1 and then Knt+1-KnD≤D+ε for n≥n0≥lnt-D/ε/lnK. Assume that x∈Ai for some i∈p-. If the probability of the distance in-between x∈Ai and Tx∈Ai+1 is less than t for some real t>dx,y then the probability of the distance between Tn+lx∈Ai+j and Tn+l+1x∈Ai+j+1 being less than D+ε in the PM-space X,F, for some j∈p-1¯∪0 and any l∈Z0+, holds for any integer n≥lnt-D/ε/lnK. Such an integerj∈p-1¯∪0 satisfies uniquely the constraints n=mp+i+j≤m+1p+i-1 for the given integers i∈p-,n∈Z0+; that is, j≡n-imodp. The (deterministic) distances dTn+lx,Tn+l+1x≤D+ε in the metric space X,d for any integer n≥n0≥lndx,Tx-D/ε/lnK provided that dx,Tx>D. For any x∈Ai and some arbitrary given i∈p-, the subsequences Tnpx⊂Ai and Tnp+1x⊂Ai+1 of Tnx,Tn+1x⊂⋃j∈p-Aj, which are Cauchy convergent sequences, that is Tnpx→zi∈Ai and Tnp+1x→zi+1∈Ai+1, satisfy the same constraints if n≥n0≥(1/plnK)lnt-D/ε, respectively, n≥n0≥1/plnKlnt-D/ε for any t>dx,Tx≥D (see Theorem 5). The points zi and zi+1 are adjacent to best proximity points of T (being identical and a unique fixed point of T if the subsets Ai intersect) and fixed points of the composite map Tp restricted to Ai and of Tp restricted to Ai+1, respectively.

Also, if X,F,Δ is a complete Menger PM-space, then the subsequences Tnpx⊂Ai, and Tnp+1x⊂Ai+1 are Cauchy and convergent sequences. Note that(41)dTnpx,Tn+1px≤∑j=0p-1dTnp+jx,Tnp+j-1x≤∑j=0p-1KjdTnp+jx,Tnp+j+1x-D+pD≤∑j=0p-1KjdTnpx,Tnp+1x-D+pD=1-Kp1-KdTnpx,Tnp+1x-D+pD≤1-Kp1-KKnpdx,Tx-D+D+pD≤1-Kp1-KKnpdx,Tx+1-Kp1-K1-Knp+pDfor all integer n≥n01 and some integer n01=n01ε,λ, for any given ε∈R+ and λ∈0,1. Then, (42)FTnpx,Tn+1pxD+ε=H1-Kp1-KKnpt-D+D+pD-dTnx,Tn+1x=H1-Kp1-KKnpt-D=1>1-λ,such that D+ε≥(1-Kp/1-K)Knpt-D+D+pD which holds for n≥n01 with(43)n01≥1lnKln1-Kp1-KεKnpt-D+Dεp+K1-Kp-11-K.This can be refined with Theorem 7(ii) (see (12)) leading to the particular case for m=1:(44)FTnpx,Tn+1pxt≥Fx,TxK-np1-Kt-D+D;∀t>D∈R+implying limn→∞FTnpx,Tn+1pxε=limt→∞Fx,Txt=1 and FTnpx,Tn+1pxt>1-λ, ∀t∈R+ for any given arbitrary ε∈R+, λ∈0,1, all integer n≥n02 and a sufficiently large integer number n02=n02ε. This implies that the subsequences Tnpx⊂Ai, and Tnp+1x⊂Ai+1 are Cauchy and convergent and contained in ε,λ-neighborhoods Upjε,λ=qj∈X:Fpj,qjε>1-λ for n≥n02 which are centred at some pj=pjx∈clAj for j=i,i+1.

Example 2.

Assume that the PM- space X,F is induced by a metric space X,d and that Tn are possible sequences of p-cyclic mappings with Tn:⋃i∈p-Ai→⋃i∈p-Ai in X,d with an associate sequence of constants Kn∈0,∞ with D being the distance in-between adjacent subsets Ai and Ai+1, ∀i∈p-.

Furthermore, assume that Kn∈Kc=Kcs∪Kcu where Kcs=L1,L2,…,Lns and Kcu=Lns+1,Lns+2,…,Lns+nu are disjoint and Kcs is nonempty. It is assumed that Li∈0,γ0⊂0,1 for i∈n-s, with at least one K∈Kcu satisfying K≥p-1 for some prefixed real constant p∈0,1 and Li∈1,γ1 for i∈ns+1,ns+nu. Note that there is at least one member of Kc which is also a member of the set Kcs of value in 0,p⊂0,1.

Now, construct sequences xn defined by xn=Txn-1 for x0∈⋃i∈p-Ai and all n∈Z+ subject to(45)Fxn+2,xn+1Kn+1t+1-Kn+1D≥Fxn+1,xnt,∀t>D∈R+,∀n∈Z0+,equivalently, (46)Fxn+2,xn+1t≥Fxn+1,xnKn+1-1t-D+D,∀t>D∈R+,∀n∈Z0+.Assume that the objective is to construct admissible sequences of cyclic self-mappings of the given class so as to keep a distance D0∈D,D+pγε for some prefixed real constant γ>1 with guaranteed probability in-between consecutive sequence points of value at least 1-λ for some given real constants λ,ε∈0,1. It follows from the theoretical framework that for composite p-cyclic mappings T^n:⋃i∈p-Ai→⋃i∈p-Ai, ∀n∈Z+ of the form T^n=Tn∘T^n-1 for n∈Z+ with contractive constants in the set Kcs=L1,L2,…,Lns, that is, of values less than one, one has that limn→∞Fxn+2,xn+1t=limt→∞Fx1,x0t=1, ∀t>D∈R+. Thus, there is n0=n0ε,λ∈Z0+ such that Fxn+2,xn+1t≥Fx1,x0D+ε>1-λ, ∀t>D+ε∈R+, ∀n≥n0.

Then, a composite mapping T^n=Tn∘T^n-1 for n∈Z+ is built with any sequence Tn of p-cyclic mappings Tn:⋃i∈p-Ai→⋃i∈p-Ai that satisfies the property Fxn+2,xn+1t>1-λ, ∀t>D∈R+, ∀n≥n0, if the following steps are followed for its construction.

Tn are taken with contractive constants in the set Kcs for n∈n-0∪0 and some n0∈Z0+. Then, Fxn0+1,xn0t>1-λ, ∀t>D+ε∈R+. It suffices that Fxn0+1,xn0D+ε>1-λ since the distribution function is nondecreasing.

Tn are taken with constants in the set Kc as being contractive, expansive or nonexpansive depending on the values of their respective constants in the set Kc for any integer n>n0 such that the inequalities below hold:(47)Fxn+2,xn+1t≥Fxn+1,xnD+ρ≥Fxn0+1,xn0D+ρ≥Fx1,x0D+ε>1-λ;ssssssss∀t>D+ρ∈R+

for any ρ∈ε,γε. If the above chain of inequalities fails for t=D+γε, some tentative self-mapping Tn+1' and some n1>n0+1∈Z0+, then Tn+1' is changed to Tn+1 with contractive constant Kn+1 so that to guarantee(48)Fxn+2,xn+1t≥Fxn+1,xnD+pγε≥Fxn0+1,xn0D+γε≥Fx1,x0D+ε>1-λ;ssss∀t>D+pγε∈R+provided from (46) that Kn+1-1t-D+D≤D+pγε for t=D+γε; that is, Tn+1 is chosen to satisfy 1>Kn+1≥p-1, ∀n∈Z0+.

For n∈n1+1,n2, repeat Step (a) for some n2>n1+1 such that Fxn2+1,xn2D+ε>1-λ.

Example 3.

Let X,F,Δ be a Menger PM-space, where X=α,β,γ,δ with subsets A=α,β,δ and B=γ,δ, Δ is the minimum t-norm and define the 2-cyclic mapping T:A∪B→A∪B as follows: Tα=γ, Tβ=Tδ=Tγ=δ, and let F be defined by(49)Fβ,δt=Fβ,γt=0,ift≤0,1,ift>0,Fγ,αt=Fδ,γt=Fδ,αt=0,ift≤0,0.5,if0<t<4,1,ift≥3.Note that A∩B=δ so that these two sets intersect; then D=dA,B=0, T:A∪B→A∪B is a 2-cyclic contraction for any real constant K∈0,1, since Fδ,δt=1, ∀t∈R+, and (50)FT2α,T2γt≥FTγ,TδK-1t≥Fδ,δK-2t,FT2α,T2δt≥FTγ,TδK-1t≥Fδ,δK-2t,FT2β,T2γt≥FTδ,TδK-1t=Fδ,δK-2t,FT2β,T2δt≥FTδ,TδK-1t=Fδ,δK-2t,FT2δ,T2γt=FTδ,TδK-1t=Fδ,δK-2tfor any real ∀t∈R+. Thus, X,F,Δ is a complete Menger PM-space and δ∈A∩B is the unique fixed point of T:A∪B→A∪B to which any sequences generated through T converge in accordance with Theorem 5. A close example has been discussed in [6] related to a cyclic contraction in a 2-Menger PM-space.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors are grateful to the Spanish Government for its support of this research with Grant DPI2012-30651 and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. Finally, the authors thank the referees for their useful comments.

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