Best proximity point theorems for probabilistic proximal cyclic contraction with applications in nonlinear programming

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R E S E A R C H

Open Access

Best proximity point theorems for

probabilistic proximal cyclic contraction with

applications in nonlinear programming

Reza Saadati

*

*_{Correspondence: [email protected]}

Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

Abstract

In this paper, we derive a best proximity point theorem for non-self-mappings satisﬁed proximal cyclic contraction in PM-spaces and this shows the existence of optimal approximate solutions of certain simultaneous ﬁxed point equations in the event that there is no solution. As an application we consider a nonlinear

programming problem. Our results extend and improve the recent results of (Sadiq Basha in Nonlinear Anal. 74(17):5844-5850, 2011).

Keywords: optimal approximate solution; ﬁxed point; best proximity point; proximal contraction; proximal cyclic contraction

1 Introduction

Best proximity point theorems are those results that provide sufficient conditions for the existence of a best proximity point and algorithms for finding best proximity points. It is interesting to note that best proximity point theorems generalized fixed point theorems in a natural fashion. Indeed, if the mapping under consideration is a self-mapping, a best proximity point becomes a fixed point.

One of the most interesting is the study of the extension of Banach contraction princi-ple to the case of non-self-mappings. In fact, given nonempty closed subsetsAandBof a complete PM-space (X,F,∗), a contraction non-self-mappingT:A→Bdoes not neces-sarily has a ﬁxed point. Eventually, it is quite natural to ﬁnd an elementxsuch thatFx,Tx(t)

is maximum for a given problem which implies thatxandTxare in close proximity to each other.

Many problems can be formulated as equations of the formTx=x, whereT is a self-mapping in some suitable framework. Fixed point theory ﬁnds the existence of a solution to such generic equations and brings out the iterative algorithms to compute a solution to such equations.

However, in the case thatT is non-self-mapping, the aforementioned equation does not necessarily have a solution. In such a case, it is worthy to determine an approximate solutionxsuch that the errorFx,Tx(t) is maximum.

2 Preliminaries

Throughout this paper, the space of all probability distribution functions (brieﬂy, d.f.’s) is denoted by+₌_{_F_{: R}_{∪ {}_–_∞_{, +}_{∞} →}_{[, ] :}_F_{is left-continuous and non-decreasing on}

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R,F() = , andF(+∞) = }and the subsetD+_⊆+_{is the set}_D+₌_{_F_∈+_:_l–_F(+_∞_{) = }_}_.

Herel–_f_{(x) denotes the left limit of the function}_f _{at the point}_x,_l–_f_{(x) =}_lim

t→x–f(t). The

space+is partially ordered by the usual point-wise ordering of functions,i.e.,F≤Gif and only ifF(t)≤G(t) for alltin R. The maximal element for+_{in this order is the d.f.}

given by

ε(t) =

, t≤, , t> .

Deﬁnition .([]) A mapping∗: [, ]×[, ]→[, ] is a continuoust-norm if∗ satis-ﬁes the following conditions:

(a) ∗is commutative and associative; (b) ∗is continuous;

(c) a∗ =afor alla∈[, ];

(d) a∗b≤c∗dwhenevera≤candb≤d, anda,b,c,d∈[, ].

Two typical examples of a continuoust-norm area∗b=abanda∗b=min(a,b). At-norm∗is said to beof Hadžić typeif

∀∈(, )∃δ∈(, ): a>  –δ ⇒

n

a∗a∗ · · · ∗a>  – (n≥).

Thet-norm minimum is a trivial example of at-norm of Hadžić type, but there exists a t-norm of Hadžić type weaker than minimum (see []).

Deﬁnition . Aprobabilistic metric space(brieﬂy, PM-space) is a triple (X,F,∗), where Xis a nonempty set,∗is a continuoust-norm, andFis a mapping fromX×XintoD+

such that, ifFx,ydenotes the value ofFat the pair (x,y), the following conditions hold: for

allx,y,zinX,

(PM) Fx,y(t) =ε(t)for allt> if and only ifx=y;

(PM) Fx,y(t) =Fy,x(t);

(PM) Fx,z(t+s)≥Fx,y(t)∗Fy,z(s)for allx,y,z∈Xandt,s≥.

For more details and examples of these spaces see also [–].

Deﬁnition . Let (X,F,∗) be a PM-space.

() A sequence{xn}ninXis said to beconvergenttoxinXif, for every> andλ> ,

there exists a positive integerNsuch thatFxn,x() >  –λwhenevern≥N.

() A sequence{xn}ninXis calledCauchy sequenceif, for every> andλ> , there

exists a positive integerNsuch thatFxn,xm() >  –λwhenevern,m≥N.

() A PM-space(X,F,∗)is said to becompleteif and only if every Cauchy sequence inX is convergent to a point inX.

Deﬁnition . Let (X,F,∗) be a PM-space. For eachpinXandλ> , the strongλ -neigh-borhoodofpis the set

Np(λ) =

q∈X:Fp,q(λ) >  –λ

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and the strong neighborhood system forX is the union _p_∈_VNp whereNp ={Np(λ) :

λ> }.

The strong neighborhood system forXdetermines a Hausdorﬀ topology forX.

Theorem .([]) If(X,F,∗)is a PM-space and{pn}and{qn}are sequences such that

pn→p and qn→q,thenlimn→∞Fpn,qn(t) =Fp,q(t)for every continuity point t of Fp,q. Lemma .([]) Let(X,F,∗)be a Menger PM-space with∗of Hadžić-type and{xn}be a

sequence in X such that,for some k∈(, ),

Fxn,xn+(kt)≥Fxn–,xn(t) (n≥,t> ). Then{xn}is a Cauchy sequence.

LetAandBbe two nonempty subsets of a PM-space andt> , the following notions and notations are used in the sequel.

FA,B(t) :=sup{Fx,y(t) :x∈A,y∈B},

A:={x∈A:Fx,y(t) =FA,B(t)for somey∈B},

B:={y∈B:Fx,y(t) =FA,B(t)for somex∈A}.

Deﬁnition . Let (X,F,∗) be a PM-space. Given non-self-mappingsS:A→BandT : B→A, the pair (S,T) is said to form aproximal cyclic contractionif there exists a non-negative numberα<  such that

Fu,Sx(t) =FA,B(t),

Fv,Ty(t) =FA,B(t)

⇒ Fu,v(t)≥min

Fx,y

t

α

,FA,B(t)

for allu,xinAandv,yinBandt> .

Note that, ifSis a self-mapping that is a contraction, then the pair the pair (S,S) forms a proximal cyclic contraction.

Deﬁnition . Let (X,F,∗) be a PM-space. A mappingS:A→Bis said to be aproximal contraction of the ﬁrst kindif there exists a non-negative numberα<  such that

Fu,Sx(t) =FA,B(t),

Fu,Sx(t) =FA,B(t)

⇒ Fu,u(αt)≥Fx,x(t)

for allu,u,x,xinAandt> .

Deﬁnition . Let (X,F,∗) be a PM-space. A mappingS:A→Bis said to be aproximal contraction of the second kindif there exists a non-negative numberα<  such that

Fu,Sx(t) =FA,B(t),

Fu,Sx(t) =FA,B(t)

⇒ FSu,Su(αt)≥FSx,Sx(t)

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Deﬁnition . Let (X,F,∗) be a PM-space. Given a mappingS:A→Band an isometry g:A→A, the mappingSis said to preserve isometric distance with respect togif

FSgx,Sgx(t) =FSx,Sx(t)

for allxandxinA. andt> .

Deﬁnition . Let (X,F,∗) be a PM-space. An elementxinAis said to be a best prox-imity point of the mappingS:A→Bif it satisﬁes the condition that

Fx,Sx(t) =FA,B(t)

for allxinAandt> .

It can be observed that a best proximity reduces to a ﬁxed point if the underlying map-ping is a self-mapmap-ping.

Deﬁnition . Let (X,F,∗) be a PM-space.Bis said to beapproximatively compactwith respect toAif every sequence{yn}ofBsatisfying the condition that for allt> ,Fx,yn(t)→ Fx,B(t) for somexinAhas a convergent subsequence.

It is easy to observe that every set is approximatively compact with respect to itself, and that every compact set is approximatively compact. Moreover,AandBare non-void if

Ais compact andBis approximatively compact with respect toA.

3 Proximal contractions

The following main result is a generalized best proximity point theorem for non-self prox-imal contractions of the ﬁrst kind. Our results extend and improve some results of [].

Theorem . Let A and B be non-void closed subsets of a complete PM-space (X,F,∗) with∗of Hadžić-type such that Aand Bare non-void.Let S:A→B,T :B→A and

g:A∪B→A∪B satisfy the following conditions: (a) SandTare proximal contractions of the ﬁrst kind. (b) S(A)⊆BandT(B)⊆A.

(c) The pair(S,T)forms a proximal cyclic contraction. (d) gis an isometry.

(e) A⊆g(A)andB⊆g(B).

Then there exist a unique element x in A and a unique element y in B satisfying the condi-tions that

Fgx,Sx(t) =FA,B(t),

Fgy,Ty(t) =FA,B(t),

Fx,y(t) =FA,B(t).

Further,for any ﬁxed element xin A,the sequence{xn},deﬁned by

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converges to the element x.For any ﬁxed element yin B,the sequence{yn},deﬁned by

Fgyn+,Tyn(t) =FA,B(t), converges to the element y.

On the other hand,a sequence{un}of elements in A converges to x if there is a sequence

{n}of positive numbers for which

lim

n→∞

n

i=

( –i) = 

in which

n

i=

ai=a∗ · · · ∗an

for ai∈(, ]and

Fun+,zn+(t)≥ –n,

where zn+∈A satisﬁes the condition that

Fzn+,Sun(t) =FA,B(t) for t> .

Proof Letxbe an element inA. In view of the facts thatS(A) is contained inBand

thatAis contained ing(A), it is ascertained that there is an elementxinAsuch that

Fgx,Sx(t) =FA,B(t)

fort> . Again, sinceS(A) is contained inB, andAis contained ing(A), there exists

an elementxinAsuch that

Fgx,Sx(t) =FA,B(t)

fort> . One can proceed further in a similar fashion to ﬁndxninA. Having chosenxn,

one can determine an elementxn+inAsuch that

Fgxn+,Sxn(t) =FA,B(t),

because of the facts thatS(A) is contained inBand thatAis contained ing(A). In light

of the facts thatgis an isometry and thatSis a proximal contraction of the ﬁrst kind,

Fxn,xn+(αt) =Fgxn,gxn+(αt)≥Fxn–,xn(t)

fort> . Therefore, by Lemma .,{xn}is a Cauchy sequence and hence converges to

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Bis contained ing(B), it is guaranteed that there is a sequence{yn}of elements inB

such that

Fgyn+,Tyn(t) =FA,B(t)

fort> . Becauseg is an isometry andT is a proximal contraction of the ﬁrst kind, it follows that

Fyn,yn+(αt) =Fgyn,gyn+(αt)≥Fyn–,yn(t)

fort> . Therefore, by Lemma .,{yn} is a Cauchy sequence and hence converges to

some elementyinB. Since the pair (S,T) forms a proximal cyclic contraction andgis an isometry, it follows that

Fxn+,yn+(t) =Fgxn+,gyn+(t)≥min

Fxn,yn

t

α

,FA,B(t)

fort> .

Lettingn→ ∞, sinceFx,y(t)≤Fx,y(t/α) we have,

Fx,y(t) =FA,B(t)

fort> . Thus, it can be concluded thatxis a member ofAand thatyis a member ofB.

SinceS(A) is contained inB, andT(B) is contained inA, there exist an elementuinA

and an elementvinBsuch that

Fu,Sx(t) =FA,B(t),

Fv,Ty(t) =FA,B(t)

fort> . BecauseSis a proximal contraction of the ﬁrst kind,

Fu,gxn+(αt)≥Fx,xn(t)

fort> . Lettingn→ ∞, we have the result thatu=gx. Thus, it follows that

Fgx,Sx(t) =FA,B(t)

fort> . Similarly, it can be shown thatv=gyand hence

Fgy,Ty(t) =FA,B(t)

fort> . To prove the uniqueness, let us suppose that there exist elementsx∗inAandy∗ inBsuch that

Fgx∗,Sx∗(t) =FA,B(t),

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fort> . Sincegis an isometry, and the non-self-mappingsSandTare proximal contrac-tions of the ﬁrst kind, it follows that

Fx,x∗(αt) =Fgx,gx∗(αt)≥Fx,x∗(t),

Fy,y∗(αt) =Fgy,gy∗(αt)≥Fy,y∗(t)

fort> . Therefore,xandx∗are identical, andyandy∗are identical.

On the other hand, let{un}∞n=in whichu=xbe a sequence of elements inAand{n}

a sequence (, ) such that

lim

n→∞

n

i=

( –i) = 

and

Fun+,zn+(t)≥ –n,

wherezn+∈Asatisﬁes the condition that

Fzn+,Sun(t) =FA,B(t)

fort> . SinceSis a proximal contraction of the ﬁrst kind,

Fxn+,zn+(αt)≥Fxn,un(t).

Givenδ∈(, ), for alln≥Nwe have

Fxn+,un+(t+δ)≥Fxn+,zn+(t)∗Fzn+,un+(δ) ≥Fxn,un

t

α

∗( –n)

≥Fxn,un

t

α

∗( –n–)∗( –n)

≥ · · · ≥Fx,u

t

αn+

∗

n

i=

( –i)

fort> . Sinceδ∈(, ) was arbitrary, we have

Fxn+,un+(t)≥

n

i=

( –i)

fort> . Now,

Fun+,x(t)≥Fun+,xn+(t)∗Fxn+,x(t) ≥

n

i=

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fort> . Then

lim

n→∞Fun+,x(t)→

fort> , and it can be concluded that{un}converges tox. This completes the proof of the

theorem.

The following example illustrates the preceding generalized best proximity point theo-rem.

Example . Consider the complete PM-space (R,F, min) where

Fx,y(t) =

t t+|x–y|,

whent>  and

Fx,y(t) = ,

whent≤ forx,yin R. LetA= [–, ] andB= [, ].

LetS:A→B,T:B→A, andg:A∪B→A∪Bbe deﬁned as

S(x) =–x  ,

T(y) = –y  ,

g(x) = –x.

Then it is easy to see that

FA,B(t) = ,

A={}andB={}. The mappingg is an isometry and the non-self-mappingsSand

T are proximal contractions of the ﬁrst kind, and the pair (S,T) forms a proximal cyclic contraction. The other hypotheses of Theorem . are also satisﬁed. Further, it is easy to observe that the element  inAandBsatisfy the conditions in the conclusion of the preceding result.

Ifgis assumed to be the identity mapping, then Theorem . yields the following best proximity point result.

Corollary . Let A and B be non-void closed subsets of a complete PM-space(X,F,∗)with

∗of Hadžić-type such that Aand Bare non-void.Let S:A→B and T :B→A satisfy

the following conditions:

(a) SandTare proximal contractions of the ﬁrst kind. (b) S(A)⊆BandT(B)⊆A.

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Then there exist a unique element x in A and a unique element y in B satisfying the condi-tions that

Fx,Sx(t) =FA,B(t),

Fy,Ty(t) =FA,B(t),

Fx,y(t) =FA,B(t)

for t> .

Theorem . Let A and B be non-void closed subsets of a complete PM-space(X,F,∗)with

∗of Hadžić-type such that Aand Bare non-void.Let S:A→B and g:A→A satisfy the

following conditions:

(a) Sis a proximal contraction of the ﬁrst and second kind. (b) S(A)is contained inB.

(c) gis an isometry.

(d) Spreserves isometric distance with respect tog. (e) Ais contained ing(A).

Then there exists a unique element x in A such that

Fgx,Sx(t) =FA,B(t)

for t> .Further,for any ﬁxed element xin A,the sequence{xn},deﬁned by

Fgxn+,Sxn(t) =FA,B(t),

converges to the element x for t> .

On the other hand,a sequence{un}of elements in A converges to x if there is a sequence

{n}of positive numbers for which

lim

n→∞

n

i=

( –i) = 

and

Fun+,zn+(t)≥ –n,

where zn+∈A satisﬁes the condition that

Fzn+,Sun(t) =FA,B(t) for t> .

Proof Proceeding as in Theorem ., it is possible to ﬁnd a sequence{xn}of elements in

Asuch that

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fort>  and for all non-negative integral values ofn, because of the facts thatS(A) is

contained inBand thatAis contained ing(A). Due to the facts thatSis a proximal

contraction of the ﬁrst kind andgis an isometry,

Fxn,xn+(αt) =Fgxn,gxn+(αt)≥Fxn–,xn(t)

fort> . Therefore, by Lemma .,{xn}is a Cauchy sequence and hence converges to

some elementxinA. Because of the facts thatSis a proximal contraction of the second kind and preserves the isometric distance with respect tog,

FSxn,Sxn+(αt) =FSgxn,Sgxn+(αt)≥FSxn–,Sxn(t)

fort> . Therefore, by Lemma .,{Sxn}is a Cauchy sequence and hence converges to

some elementyinB. Thus, it can be concluded that

Fgx,y(t) = lim

n→∞Fgxn+,Sxn(t) =FA,B(t).

Eventually,gxis an element ofA. Because of the fact thatAis contained ing(A),gx=gz

for some memberzinA. Owing to the fact thatg is an isometry,Fx,z(t) =Fgx,gz(t) = .

Consequently, the elementsxandzmust be identical, and hencexbecomes an element ofA. BecauseS(A) is containedB,

Fu,Sx(t) =FA,B(t)

fort> , for some elementuinA. On account of the fact that the mappingSis a proximal contraction of the ﬁrst kind,

Fu,gxn+(αt)≥Fx,xn(t)

fort> . As a result, the sequence{g(xn)}must converge tou. However, because of the

continuity ofg, the sequence{g(xn)}converges togxas well. Therefore,uandgxmust be

identical. Thus, we have the result that

Fgx,Sx(t) =Fz,Sx(t) =FA,B(t)

fort> . The uniqueness and the remaining part of the proof follow as in Theorem .. This completes the proof of the theorem.

The preceding generalized best proximity point theorem is illustrated by the following example.

Example . Consider the complete PM-space (R,F, min) where

Fx,y(t) =

t t+|x–y|,

whent> , and

Fx,y(t) = ,

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LetA= [–, ] andB= [–, –]∪[, ]. ThenFA,B(t) = _t_+t ,A={–, }, andB={–, }.

LetS:A→Bbe deﬁned as

Sx=

 ifxis rational,  otherwise.

ThenSis a proximal contraction of the ﬁrst and second kind, andS(A)⊆B.

Further, let g :A→ A be deﬁned as gx = –x. Then g is an isometry, S preserves the isometric distance with respect tog, and A⊆g(A). It can also be observed that

Fg(–),S(–)(t) =FA,B(t) fort> .

Ifgis assumed to be the identity mapping, then Theorem . yields the following best proximity point theorem.

Corollary . Let A and B be non-void closed subsets of a complete PM-space(X,F,∗) with∗of Hadžić-type such that Aand Bare non-void.Let S:A→B satisfy the following

conditions:

(a) Sis a proximal contraction of the ﬁrst and second kind. (b) S(A)is contained inB.

Then there exists a unique element x in A such that

Fx,Sx(t) =FA,B(t).

Further,for any ﬁxed element xin A,the sequence{xn},deﬁned by

Fxn+,Sxn(t) =FA,B(t),

converges to the best proximity point x of S.

4 Application

A solution to the nonlinear programming problem

max

x∈AFx,Tx(t)

is fundamentally an ideal optimal approximate solution to the equationTx=xwhich is shifting to have a solution whenTis supposed to be a non-self-mapping.

Considering the fact thatFx,Tx(t) is at leastFA,B(t) for allxinA, a solutionxto the

afore-mentioned nonlinear programming problem becomes an approximate solution with the lowest possible error to the corresponding equationTx=xif it satisﬁes the condition that Fx,Tx(t) =FA,B(t) fort> .

Competing interests

The author declares that they have no competing interests.

Author’s contributions

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Acknowledgements

The author is thankful to the three anonymous referees for giving valuable comments and suggestions, which helped to improve the ﬁnal version of this paper.

Received: 31 January 2015 Accepted: 17 May 2015 References

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