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

Fixed point theory in the framework of probabilistic metric spaces [ ¹ – ⁴ ] is receiving important research attention. See, for instance, [ ² – ⁴ , ⁷ – 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 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 t t>0 0} \right)$$\end{document}]]> [ 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 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 [ ⁷ ] for contractions based on control φ -functions. See also [ ⁸ ]. It was found that fixed points are unique. In addition, φ -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 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 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 – ²⁷ , ³¹ – 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 α - ψ -type contractions. The existence and uniqueness of fixed points and best proximity points of p -cyclic α - ψ -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 R+=z∈R:z>0 0} \right\}$$\end{document}]]> , R0+=R+∪0 , Z+=z∈Z:z>0 0} \right\}$$\end{document}]]> , Z0+=Z+∪0 , n¯={1,2,…,n} , and denote also by L , the set of distance distribution functions H:R→0,1 , [ ¹ ], which are non-decreasing and left continuous such that H0=0 and supt∈RHt=1 . Let X be a nonempty set and let the probabilistic metric (or distance) F:X×X→L a symmetric mapping from X×X , where X is an abstract set, to the set of distance distribution functions L of the form H:R→0,1 which are functions of elements Fx,y for every x,y∈X×X . Then, the ordered pair X,F is a probabilistic metric space (PM) [ ² , ³ , ²⁹ ] if

• ∀x,y∈X Fx,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

A particular distance distribution function Fx,y∈L is a probabilistic metric (or distance) which takes values Fx,yt identified with a probability distance density function H:R→0,1 in the set of all the distance distribution functions L .

A Menger PM-space is a triplet X,F,Δ , where X,F is a PM-space which satisfies:

1. Δa,1=a

2. Δa,b=Δb,a

3. Δc,d≥Δa,b if c≥a , d≥b

4.

A property which follows from the above ones is Δa,0=0 for a∈0,1 . Typical continuous t -norms are the minimum t -norm defined by ΔMa,b=mina,b , the product t -norm defined by ΔPa,b=a.b and the Lukasiewicz (or nilpotent-minimum) t -norm defined by ΔLa,b=maxa+b-1,0 which are related by the inequalities ΔL≤ΔP≤ΔM .

The (probabilistic) diameter of a subset A of X is a function from R0+ to 0,1 defined by DAz=supt<zinfx,y∈AFx,yt and A is probabilistically bounded if DAp=supz∈R+DAz=1 ( DAp can be defined equivalently as limz→∞DAz ), probabilistically semibounded if 0<DAp<1 and probabilistically unbounded if DAp=0 [ ¹ , ² ]. The diameter of a subset A⊂X in the PM-space X,F , induced by a metric space X,d , refers to maximum real interval measure, where the argument of the probabilistic metric is unity, that is,

Example 1.1

Let X be an abstract nonempty set, X,F be a PM-space and X,d be a companion metric space and let A be a nonempty subset of X with Fx,yt=αx,ytβx,yt+dx,y for t≤t1 , where t1=supx,y∈Adx,yα¯-β¯ , and Fx,yt=1 for t>t1 t_{1}$$\end{document}]]> with some given positive real functions subject to βx,y=β¯,αx,y=α¯≥β¯∈R+ and αx,y=βx,y=1 if dx,y<supx,y∈Adx,y . In this case, diamA=t1≤∞ being, in particular, infinity if α¯=β¯ (i.e., the probability one is reached as a limit as t→∞ ) or if supx,y∈Adx,y is arbitrarily large (i.e., if A is unbounded as a subset of the metric space X,d ). If supx,y∈Adx,y<∞ and α¯>β¯ \bar{\beta }$$\end{document}]]> then diamA<∞ .

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:

Definition 1.1 [7, 8, 16]

Let X,F,Δ be a Menger PM-space. Then:

• A sequence xn in X is said to be convergent to x in X if, for every ε,λ∈R+ , there exists n0=n0ε,λ∈Z0+ such that Fxn,xε>1-λ 1 - \lambda$$\end{document}]]> , whenever n≥n0 .

• A sequence xn in X is said to be a Cauchy sequence if, for every ε,λ∈R+ , there exists n0=n0ε,λ∈Z0+ such that Fxn,xmε>1-λ 1 - \lambda$$\end{document}]]> , whenever n,m≥n0 .

• X,F,Δ is complete if every Cauchy sequence in X is convergent to a point in X .

• A sequence xn is said to be G -Cauchy if, for every ε∈R+ , limn→∞Fxn,xn+mε=1 , ∀m∈Z0+ .

• X,F,Δ is G -complete if every G -Cauchy sequence in X is convergent in X . □

Assertion 1.1

Let X,F,Δ be a Menger PM-space with Δ=ΔM,ΔP or ΔL . The following properties hold:

• xn⊂X convergent ⇒xn is Cauchy ⇒xn is G -Cauchy.

• X,F,Δ G -complete ⇒ X,F,Δ is complete

Proof

Proof of (i) Step 1 We first prove that xn convergent ⇒ xn is Cauchy. Since xn is convergent then for every ε,λ∈0,1∈R+ , there exists n0=n0ε,λ∈Z0+ such that Fxm,xε/2>1-λ/2 1 - \lambda /2$$\end{document}]]> , ∀n,m≥n0∈Z0+ . Then, since F:R→0,1 is non-decreasing, one gets:

Proof of (i) Step 2 We next prove by contradiction that xn is Cauchy ⇒ xn is G -Cauchy. Assume that xn is Cauchy while it is not G -Cauchy. Then, liminfn→∞Fxn,xn+mε1/2>1-λ 1 - \lambda$$\end{document}]]> and limsupn→∞Fxn,xn+mε1<1-2λ , for some m∈Z0+ , some ε1∈R+ and some given λ=λε1∈0,1/2 . Since Fx,yt is non-decreasing in t for all x,y∈X , so that Fx,yε1≥Fx,yε1/2 , then:

∀m∈Z+. But then λ<λ/2 . Then, xn is G -Cauchy.

Proof of (ii): Let X,F,Δ be G -complete and let xn⊂X be any given Cauchy sequence. Then xn is G -Cauchy, from property (1), and convergent to some x∈X since X,F,Δ is G -complete. Since xn is an arbitrary Cauchy sequence convergent in X , it turns out that X,F,Δ is complete. □

The ε,λ -topology in a Menger in a PM-space X,F,Δ is a Hausdorff topology introduced by the family of neighborhoods Nx of a point x∈x given by Nx=Nxε,λ:ε∈R+,λ∈0,1 where Nxε,λ=z∈X:Fx,zε>1-λ 1 - \lambda } \right\}$$\end{document}]]> . In this topology, a function is continuous at x0∈X if and only if fxn→fx0 for every convergent sequence xn→x0 . See [ ¹ , ¹⁶ ] for more details. □

We next denote by φz+ and φz- , respectively, the right and left limits of φt as t→z .

Definition 1.2

A function φ:R→R0+ is said to be a Φxy -function if, for given real constants x,y∈R0+ , with y≥x , it satisfies the following conditions:

• φt is strictly increasing for t∈x,∞

• φx+=y

• φt is everywhere left-continuous

• φt=0 for t∈-∞,x □

The set of functions Φxx is simply denoted by Φx . If φ:R→R0+ is in Φx then φx=φx-=0≤φx+=x and then if φ∈Φ0 , it is continuous at t=0 . Note also that the particular set of functions Φ0 coincides with the set of functions Φ 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 ΦD and ΦDD^ , where D is the distance in-between adjacent subsets of the cyclic disposal in X .

Definition 1.3 [16]

A function ψ:R0+→R0+ is said to be a Ψ -function if it continuous with ψ0=0 and ψnan→0 when an→0 as n→∞ .

The definition of a p≥2 -cyclic α - ψ -type contraction follows:

Definition 2.1

Let X,F be a PM-space and let Ai be nonempty subsets of X , ∀i∈p¯ such that D=dAi,Ai+1 is the common distance in-between adjacent subsets, ∀i∈p¯ . Then T:⋃i∈p¯Ai→⋃i∈p¯Ai is a p≥2 -cyclic α - ψ -type contraction if TAi⊆Ai+1 , ∀i∈p¯ and there exist two functions α:⋃i∈p¯Ai×⋃i∈p¯Ai×R+→R0+ and ψ∈Ψ satisfying the following inequality:

Note that if T:⋃i∈p¯Ai→⋃i∈p¯Ai is p≥2 -cyclic and if x∈Ai for some i∈p¯ then for any j∈p¯ and n∈Z0+ , Tnp+i+jx∈Anp+i+j=Ak for some k∈p¯ since if n,m∈Z+ and n≡mmodp then Am=An . In particular, Ai+np=Ai , ∀i∈p¯ , ∀n∈Z0+ . It can be pointed out that p -cyclic contractions include the case of cyclic self-mappings T on X such that X=⋃i∈p¯Ai . In this case, Aii∈p¯ is said to be a cyclic representation of X,T . On the other hand, note that is an α - ψ -type contraction if (2.1) holds with D=0 for t∈R+ [ 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 in-between adjacent subsets in non-expansive cyclic self-mappings are identical in uniformly convex Banach spaces [ ²⁷ ].

An equivalent constraint to ( ^2.1 ) is now discussed:

Proposition 2.1

The constraint ( ^2.1 ) is identical to:

Proof

Note that, given K∈0,1 and D≥0 , the function βK,D:D,∞→D,∞ defined by βK,Dt=Kt+1-KD for t∈D,∞ is a strictly increasing, bijective and bicontinuous function of (then continuous) inverse βK,D-1t=K-1t-D+D for t∈D,∞ , which can be extended by continuity to t=D by defining βD=D .Then, one gets from ( ^2.1 ) that

Note from ( ^2.2 ) that, if αx,y,t=1 and φt=ψt=t , ∀x,y∈X , ∀t∈R , then the p -cyclic α - ψ -type contraction T:⋃i∈p¯Ai→⋃i∈p¯Ai becomes a p -cyclic B -contraction since one gets FTx,Tyt≥Fx,yK-1t-D+D , ∀x,y∈X , ∀t∈R+ from ( ^2.2 ) if αx,y,t=1 and φt=ψt=t . Thus, a p -cyclic B -contraction is a particular type of p -cyclic α - ψ -type contraction. See [ ² ] for the case D=0 . Some basic properties of a p -cyclic α - ψ -type contraction are now given:

Proposition 2.2

Let X,F be a PM-space. Let T:⋃i∈p¯Ai→⋃i∈p¯Ai be a p - cyclic α - ψ - type contraction with Ai⊂X being bounded , ∀i∈p¯ with D=dAi,Ai+1 , ∀i∈p¯ and D¯=maxi∈p¯diamAi being the distance in-between adjacent subsets , ∀i∈p¯ . Then, the following properties hold provided that φ∈ΦDD^ for any given D^≥D :

• FTnx,Tnyφt=1 , ∀x,y∈Ai×Ai+1 , ∀i∈p¯ , ∀t>D+2D¯∈R+ D + 2\bar{D}} \right) \in {\mathbf{R}}_{ + }$$\end{document}]]> , ∀n∈Z0+ .

• limt→∞FTnx,Tnyφt=1 , ∀x,y∈Ai×Ai+1 , ∀i∈p¯ , ∀n∈Z0+ .

• Fx,yφt=Fx,yτ=0 , ∀x,y∈Ai×Ai+1 , ∀i∈p¯ , t∈-∞,D , τ∈-∞,D^ .

• Fx,yφD+=Fx,yD^=0;∀x,y∈Ai×Ai+1,∀i∈p¯ifD^=D,Fx,yφD++=Fx,yD^+=1;∀x,y∈Ai×Ai+1,∀i∈p¯ifD^=D,Fx,yφD+=Fx,yD^=1;∀x,y∈Ai×Ai+1,∀i∈p¯ifD^>D D \hfill \\ \end{aligned}$$\end{document}]]>

Proof

Since Ai and Ai+1 are bounded then the maximum distance in-between any two points of adjacent subsets is not larger than D+2D¯ . Then, lims→∞Fx,ys=maxi∈p¯maxz∈Ai,ω∈Ai+1Fz,ωφt=1, ∀x,y∈Ai×Ai+1 , ∀i∈p¯ , if t∈D+2D¯,∞ , since the distance distribution function F:R→0,1 is non-decreasing and left-continuous, and TAi⊆Ai+1 with D=dAi,Ai+1 , ∀i∈p¯ . Then,

Then, from Proposition 2.1 [Eq. ( ^2.3 )], FTn+1x,Tn+1yφt=FTn+1x,Tn+1y-1φt=1 for t∈D+2D¯,∞ . Hence, the proofs of Properties (1) and (2) follow by complete induction.

Properties (3)–(5) follow directly from the definitions of the sets ΦDD^ for τ=φt , being equivalent to t=tτ=argz∈R:φz-=τforτ∈φD,D^ which is point-wise unique for any τ∈φD,D^ , zero for t∈R0- if ΦDD^ and, left-continuous and strictly increasing for t∈R>D D$$\end{document}]]> . □

It turns out that Proposition 2.2 holds, in particular, for φ∈ΦD . The α -admissibility of α - ψ -type contractions is defined to state layer on the main result:

Definitions 2.2

Let X,F a PM-space. Then:

• an α - ψ -type contraction T:X→X is α -admissible for a given function α:X×X×R+→R0+ [ ¹⁶ ] if

  ∀x,y∈X,∀t∈R+αx,y,t≥1⇒αTx,Ty,t≥1,

• a p -cyclic α - ψ -type contraction T:⋃i∈p¯Ai→⋃i∈p¯Ai is α -admissible for a given function α:⋃i∈p¯Ai×⋃i∈p¯Ai×R+→R0+ if

  ∀x,y∈Ai×Ai+1,∀i∈p¯,∀t∈R+αx,y,t≥1⇒αTx,Ty,t≥1.
  □