Common fixed points and best proximity points of two cyclic self-mappings

R E S E A R C H

Open Access

Common fixed points and best proximity

points of two cyclic self-mappings

M De la Sen

_{and RP Agarwal}

*_{Correspondence:}

[email protected] 1_{Instituto de Investigacion y}

Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia), Aptdo. 644, Bilbao, Bilbao 48080, Spain

Full list of author information is available at the end of the article

Abstract

This paper discusses three contractive conditions for two 2-cyclic self-mappings defined on the union of two nonempty subsets of a metric space to itself. Such self-mappings are not assumed to commute. The properties of convergence of distances to the distance between such sets are investigated. The presence and uniqueness of common fixed points for the two self-mappings and the composite mapping are discussed for the case when such sets are nonempty and intersect. If the space is uniformly convex and the subsets are nonempty, closed and convex, then the iterates of points obtained through the self-mapping converge to unique best proximity points in each of the subsets. Those best proximity points coincide with the fixed point if such sets intersect.

1 Introduction

General rational contractive relations for self-mappings from certain sets into themselves have received important interest in the last years. The related background literature is very rich and, in particular, a very general rational contractive condition has been discussed in [, ]. Relevant results about the existence of fixed points and their uniqueness under supplementary conditions have also been investigated in those papers. On the other hand, the rational contractive condition proposed in [] is proved to include as particular cases several of the previously proposed ones [, –], including Banach’s principle [] and Kannan’s fixed point theorems [, , , ]. Fixed point theory is also useful to investigate the stability of iterative sequences and discrete dynamic systems [, , ]. The rational contractive conditions of [, ] are applicable only on distinct points of the considered metric spaces. In particular, the fixed point theory for Kannan’s mappings is extended in [] by the use of a non-increasing function affecting the contractive condition and the best constant to ensure a fixed point is also obtained. Three fixed point theorems which extended the fixed point theory for Kannan’s mappings have been stated and proved in []. Also, significant attention has been paid to the investigation of standard contractive and Meir-Keeler-type contractive -cyclic self-mappingsT:A∪B→A∪Bdefined on subsetsA,B⊆Xand, in general,p-cyclic self-mappingsT :_i∈¯pAi→

i∈¯pAidefined

on any number of subsetsAi⊂X,i∈ ¯p:={, , . . . ,p}, where (X,d) is a metric space (see,

for instance, [–] and [–]). More recent investigation of cyclic self-mappings has been devoted to its characterisation in partially ordered spaces and also to the formal extension of the contractive condition through the use of more general strictly increasing functions of the distance between adjacent subsets. In particular, the uniqueness of the

best proximity points, to which all the sequences of iterates of composite self-mappings T_{:A∪B→A∪Bconverge, is proved in [] for the extension of the contractive principle} for cyclic self-mappings in uniformly convex Banach spaces (then being strictly convex and reflexive, []) if the subsetsA,B⊂Xin the metric space (X,d), or in the Banach space (X, ), where the-cyclic self-mappings are defined are both nonempty, convex and closed. The research in [] is centred on the case of the cyclic self-mapping being defined on the union of two subsets of the metric space. Those results have been extended in [] for Meir-Keeler cyclic contraction maps and, in general, for the self-mappingT :

i∈¯pAi→i∈¯pAibe ap(≥)-cyclic self-mapping being defined on any number of subsets

of the metric space withp¯:={, , . . . ,p}.

A relevant problem is when self-mappings from a metric space into itself or from a set into itself have common fixed points, [, –]. A related problem is when compos-ite maps built with those mappings have common fixed points with such self-mappings. There are some classical results available concerning the case when one of the self-mappings is continuous or when both self-mappings commute []. Some later ex-tensions have removed the need for the continuity of one of the self-mappings [, ]. Some recent papers have investigated the existence of common fixed points in cone met-ric spaces [, ] and in fuzzy metmet-ric spaces and under contractive conditions of integral type [, ]. This paper is concerned with the investigation of convergence properties of distances and the existence/uniqueness of common fixed points/common best proximity points of two -cyclic self-mappings (refereed to simply as cyclic self-mappings) on the union of two subsetsAandBof a metric space under three contractive conditions. Sec-tion  is devoted to the convergence properties of distances for such contractive condi-tions which involve both cyclic self-mappings. Further results obtained in this section are concerned with the existence and uniqueness of common fixed points for the two cyclic self-mappings and their composite self-mapping if the involved subsets intersect and are closed and convex. Section  gives some direct extensions of the results in Section  when the most restrictive assumption in the section is removed. Finally, Section  extends the relevant results of the former sections to the case thatAandBintersect in the sense that the role of common fixed points is played instead by common best proximity points under the assumption that the subsetsAandBbelong to a uniformly convex Banach space.

2 Convergence properties and common fixed points under three contractive conditions ifAandBintersect

Let (X,d) be a metric space and consider two nonempty subsetsAandBofX. It is assumed through the manuscript thatS,T:A∪B→A∪Bare cyclic self-mappings,i.e. T(A)⊆B, S(A)⊆B,T(B)⊆AandS(B)⊆A. Suppose, in addition, thatT:A∪B→A∪Bsatisfies the constraint

d(Sx,TSy)≤αd(x,Sy) +βd(x,Sx) +d(Sy,TSy)+γd(x,Sy) +d(Sx,TSy)+ωD;

∀x,y(=x)∈A∪B, (.)

where

whereα≥,β≥,γ ≥,ω≥. Note that ifx∈Aandy∈Bor conversely, then the various point-to-point distances in (.) are not less thanDso that the parametrical con-straintα+ (β+γ) +ω≥ has to be fulfilled from (.) ifD= . The following result can be stated:

Lemma . Assume that d(x,Sx)≤d(x,Tx);∀x∈A∪B and that the constraints≤ –

ω≤α+ (β+γ) < if D= and≤α+ (β+γ) < if D= both hold. Then, the following properties hold:

(i)

D≤lim sup

n→∞ d

Snx,Sn+x≤ ωD

 –α– (β+γ); ∀x∈A∪B. (.)

Ifω=  –α– (β+γ)then

∃lim

n→∞d

Snx,Sn+x=D. (.)

(ii) d(Sn_x,Sn+m+_x)≤ωD+G

m(n)for anyx∈A∪B,∀n∈N,∀m∈N=N∪ {}, where {Gm(n)}m∈Nis a nonnegative strictly decreasing real sequence for anyn∈N, then being convergent to zero asm→ ∞, and

lim sup

m→∞

dSnx,Sn+m+x≤ωD.

Proof Takey=x∈A∪Band replacex→Sn–xin (.) to yield

dSn_x,Sn+_x

≤dSnx,TSnx≤αdSn–x,Snx+βdSn–x,Snx+dSnx,TSnx

+γdSn–x,Snx+dTSnx,Snx+ωD

≤αdSn–x,Snx+βdSn–x,Snx+dSnx,TSnx

+γdSn–x,Snx+dTSnx,Snx+ωD; ∀x∈A∪B,∀n∈N (.)

sinced(x,Sx)≤d(x,Tx),Sn_x∈A⇒(TSn_x∈B∧Sn+_{x∈B) andS}n_x∈B⇒(TSn_x∈A∧

Sn+_{x∈A) for anyx∈A∪Band, equivalently,}

dSnx,Sn+x≤dSnx,TSnx

≤α+β+γ  –β–γ d

Sn–x,Snx+ ωD

 –β–γ; ∀x∈A∪B,∀n∈N (.)

so that

D≤lim sup

n→∞ d

Snx,Sn+x

≤ lim

n→∞

α+β+γ

 –β–γ n

dSn–x,Snx+

ωD

 –β–γ n

i=

α+β+γ

 –β–γ n–i

≤ ωD

since ≤β+γ< , ≥ω≥ –α– (β+γ), and ≤ –ω≤α+ (β+γ) <  ifD=  and ≤α+ (β+γ) <  ifD=  imply that the contraction constantk:=α+β+γ_{–β–γ} < . Hence, (.) holds and the limit (.) exists forω=  –α– (β+γ). Hence, Property (i) is proved. To prove Property (ii), note that for any natural numbersmandn, one gets from (.) and the above definition of the contraction constantk<  that

dSnx,Sn+m+x≤dSnx,TSn+mx≤kmdSn+x,Snx+ωD –km

≤kmdSnx,TSnx+ωD –km

≤dSnx,TSnx+ωD; ∀x∈A∪B,∀n∈N,∀m∈N, (.a)

lim sup

m→∞

dSnx,Sn+m+x≤ωD; ∀x∈A∪B,∀n∈N. (.b)

Hence, Property (ii) has been proved.

Note that the contractive condition

d(Sx,TSy)≤αd(x,Sy) +βd(x,Sx) +d(Sy,TSy)+γd(x,Sy) +ωD (.)

is distinct from (.), while it modifies Lemma ., resulting in the subsequent result.

Lemma . Assume that d(x,Sx)≤d(x,Tx);∀x∈A∪B and that the constraintω≥ – (α+γ + β) > holds. Then, the following properties hold:

(i)

D≤lim sup

n→∞

dSnx,Sn+x≤ ωD

 – (α+ β+γ); ∀x∈A∪B. (.)

Ifω=  – (α+ β+γ)then

∃lim

n→∞d

Snx,Sn+x=D. (.)

(ii) d(Sn_x,Sn+m+_x)≤ωD+F

m(n)for anyx∈A∪B,∀n∈N,∀m∈N=N∪ {}, where {Fm(n)}m∈Nis a nonnegative strictly decreasing real sequence for anyn∈N, then being convergent to zero asm→ ∞, and

lim sup

m→∞

dSnx,Sn+m+x≤ωD.

Proof The contractive condition (.) removes the additive termd(Sx,TSy) from (.) so thatd(TSn_x,Sn_{x) is correspondingly removed in the resulting modified counterpart}

of (.) by takingy=x∈A∪Band performing the replacementx→Sn–_{x. The resulting} contractive constant now becomesk:=_{–β–γ}α+β < , subject to _{–β–γ}ωD ≥ –_{–β–γ}α+β , which, on the other hand, results in the needed constraint  –ω≤α+γ + β<  to reformulate the results of Lemma . leading to the modified Properties (i), with (.)-(.), and (ii).

Theorem . Let(X,d)be a complete metric space and assume that S,T:A∪B→A∪B are cyclic self-mappings, where A and B intersect, are nonempty and closed, and that the contractive condition (.) holds subject to≤α+ (β+γ) < . Then, there exists a fixed point of S:A∪B→A∪B in A∩B which is also a fixed point of the composite self-mapping T◦S:A∪B→A∪B and a fixed point of T:A∪B→A∪B. If, in addition, A and B are convex, thenFix(S)≡Fix(T◦S)≡Fix(T)consists of a single point.

Proof From Lemma . andD=  (sinceA∩B=∅), it follows that

lim

n→∞d

Snx,Sn+x= lim

n→∞d

Snx,TSnx= ; ∀x∈A∪B.

Thus, limn,m→∞d(Sn+mx,Sn+m+x) =limn,m→∞d(Sn+mx,TSn+mx) = ;∀x∈A∪Bso that

{Sn_x}

n∈N is a Cauchy sequence; ∀x∈A∪B, then it is convergent to some z∈ A∩B

since A∩B is nonempty and closed. Also, since S : A∪B→ A∪B is contractive from Lemma ., then it is globally Lipschitz-continuous for any pair (x,Sx) withx∈ A∪Band then {Sn_x}

n∈N is in A∩B. Thus, Sn+x=S(Snx)→z=Sz; ∀x∈A∪Band

z ∈ Fix(S)⊂ A∩ B. Since Sn_{x→ z and lim}

n→∞d(Snx,TSnx) = ; ∀x ∈ A∪B then

TSn+x→TSnx→z⇒(TS)(Snx)→TSz=z. Thus,z∈Fix(S)∩Fix(T◦S)⊂A∩Band TSz=Sz=z⇒T_{Sz=T(TSz) =TSz=z=Tz. Then,z∈Fix(S)∩Fix(T)∩Fix(T ◦S)⊂} A∩B.

Finally assume, in addition, thatAandBare also convex andz,z(= z)∈Fix(S◦T)∩

Fix(S)∩Fix(T) so that

z,z, Snz=z, TSnz=z∈Fix(T◦S)∩Fix(S)⊂A∩B

sinceA∩Bis convex. Using (.) withx=z,y=zandD= , the following contradictions lead toz=zfrom the contractive assumption in Lemma . since ≤α<  – (β+γ) < :

 <d(z,z)≤αnd(z,z) <d(z,z); ∀n∈N,

 <d(z,z)≤ lim

n→∞

αnd(z,z) = 

so thatz=zandFix(S)≡Fix(T)≡Fix(T◦S) ={z} ⊂A∩B. Hence, the proof is

com-plete.

Theorem . Theorem . applies “mutatis-mutandis” for the contractive constraint (.) subject toω≥ – (α+γ+ β) > .

The proof of Theorem . is omitted since it is similar to that of Theorem ..

Assume now that the contractive condition (.) is modified as follows to give relevance to the composite self-mappingS◦T:A∪B→A∪B:

maxμ–dTx,Tx,ν–dSx,Sx

≤d(Sx,STx)

for some real constants α,β,δ,γ ≥ andμ,ν≥ so that by using the lower-bound of (.) to build a further upper-bounding condition of it, one gets

d(Sx,STx)≤αd(x,Tx) +βdTx,Tx+δd(x,Sx) +γdSx,Sx+ωD

≤αd(x,Tx) +βμd(Sx,STx) +δνd(x,Tx)

+γ νd(Sx,STx) +ωD; ∀x∈A∪B, (.)

which is identical to

d(Sx,STx)≤ α+δ

 –βμ–γ νd(x,Tx) +

ωD

 –βμ–γ ν; ∀x∈A∪B (.)

if α+δ+βμ+γ ν< . The following two results hold under the contractive condition (.).

Lemma . Assume that the contractive condition (.) holds subject toα+δ+βμ+γ ν<  andω≥ –α– (β+γ). Then

dSnx,SnTx≤knd(x,Tx) + ωD  –βμ–γ ν

 –kn

≤d(x,Tx) + ωD

 –βμ–γ ν; ∀x∈A∪B,∀n∈N, (.)

lim sup

n→∞

dSnx,SnTx≤ ωD

 –βμ–γ ν; ∀x∈A∪B (.)

and

∃ lim

n→∞d

Snx,SnTx=D; ∀x∈A∪B. (.)

Proof Redefine the contractive constant ask:=_{–βμ–γ ν}α+δ <  so that _{–βμ–γ ν}ωD ≥ –kifω≥

 –βμ–γ ν–α–δ. One gets (.)-(.) directly from (.).

Theorem . Let(X, )be a Banach space and assume that S,T:A∪B→A∪B are cyclic self-mappings satisfying the contractive condition (.) subject toα+δ+βμ+γ ν< , where A and B intersect and are nonempty and closed. Then, there exists a fixed point of S:A∪B→A∪B in A∩B which is also a fixed point of the composite self-mapping T◦S:A∪B→A∪B. If, in addition, A and B are convex thenFix(S)≡Fix(T◦S)consists of a single point.

Outline of proof Let (X,d) be the complete metric space whered:X×X→R_+is the norm-induced metric by the norm on the Banach space (X, ). IfAandBintersect, then

limn→∞d(Snx,SnTx) = ;∀x∈A∪Bfrom Lemma . withS:A∪B→A∪Bsatisfying

a contractive condition and then being globally Lipschitz continuous for any pair (x,Tx) with x∈A∪B. Thus, the following general terms of Cauchy sequences converge to a fixed point;i.e. Sn+_x=S(Sn_x),Sn_x,Sn_{Tx→zso thatz∈Fix(S)≡Fix(S◦T)⊂A∩B. The}

thatz,z (=z) are fixed points ofS,S◦T :A∪B→A∪BinA∩B. Then, the following contradictions lead toz=zin terms that either

 < S

z+z  –ST

z+z  =d

Sz+z  ,ST

z+z

 =

STz_–Sz–STz–Sz 

≤ 

STz–Sz+ 

STz–Sz=  

d(Sz,STz) +d(Sz,STz)=  (.)

since z+z

 ∈A∩B, sinceA∩Bis convex, or all points in the segment [z,z]⊂A∩B, again since A∩Bis convex, are fixed points of the self-mappings S,S◦T :A∪B→A∪B. Now, take arbitrarily closed pointsz,z=z+∈(z,z), which are also fixed points. Then the continuity ofS:A∪B→A∪Bleads to a further contradictionlimz→z(Sz) =z=z.

Then, no segment [z,z]⊂A∩Bcan consist of fixed points ofS,S◦T:A∪B→A∪B.

3 Relaxing a hypothesis of Section 2

The assumptiond(x,Sx)≤d(x,Tx);∀x∈A∪Bin Lemma . and Lemma ., then in The-orem . and TheThe-orem ., can be removed at the expense of more restrictive constraints on the corresponding contractive conditions on the parameters. For instance, the triangle inequality for distances yields

dSnx,Sn+x≤dSnx,TSnx+dTSnx,Sn+x; ∀x∈A∪B. (.)

The contractive condition (.) becomes equivalent to

dSn+x,TSnx≤βdSnx,Sn+x+dSnx,TSnx

+γdSn+x,TSnx+ωD; ∀x∈A∪B (.)

with the replacementsx,Sy→Sn_x,y→Sn–_{x. The inequality (.) is equivalent to}

dSn+x,TSnx≤ β  –γ

dSnx,Sn+x+dSnx,TSnx+ ωD

 –γ; ∀x∈A∪B (.)

if ≤γ < . The substitution of (.) into (.) yields

dSnx,Sn+x≤ β  –γ

dSnx,Sn+x+dSnx,TSnx

+dSnx,TSnx+ ωD

 –γ; ∀x∈A∪B (.)

and using (.) in (.)

dSn+x,Snx

≤ +β–γ  –β–γd

Snx,TSnx+ ωD  –β–γ

≤ +β–γ  –β–γ

α+β+γ

 –β–γ d

Sn–x,Snx+ ωD  –β–γ +

ωD

 –β–γ

=( +β–γ)(α+β+γ) ( –β–γ) d

Sn–x,Snx+( –γ)ωD

andS:A∪B→A∪Bis cyclic contractive if (+β–γ_{(–β–γ})(α+β+γ)₎ < , that is, if

≤α<( –β–γ) 

 +β–γ – (β+γ); ≤β<

 +γ(γ– )

 – γ ; γ ∈(/, ), (.)

ω≥( –β–γ)

_{– ( +β–γ)(α+β–γ)}

( –γ) , (.)

where the second constraint of (.) guarantees thatβ+γ < (–β–γ_+β–γ);i.e.α≥ and the third one thatβ is nonnegative. Lemma .(i) is modified by using (.)-(.) as follows without using the assumptiond(x,Sx)≤d(x,Tx);∀x∈A∪B.

Lemma . Assume that (.) holds subject to the constraints (.)-(.). Then

D≤lim sup

n→∞

dSnx,Sn+x≤ ( –γ)ωD

( –β–γ); ∀x∈A∪B. (.)

Ifω=(–β–γ)–(+β–γ_{(–γ)} )(α+β–γ), then

∃ lim

n→∞d

Snx,Sn+x=D. (.)

An “ad-hoc” modified version of Theorem . follows.

Theorem . Let(X,d)be a complete metric space and assume that S,T:A∪B→A∪B are cyclic self-mappings where A and B intersect and are nonempty and closed, and that the contractive condition (.) holds subject to

≤α<( –β–γ) 

 +β–γ – (β+γ); ≤β<

 +γ(γ– )

 – γ ; γ ∈(/, ), (.)

ω=( –β–γ)

_{– ( +β–γ)(α+β–γ)}

( –γ) . (.)

Then, there exists a fixed point z of S:A∪B→A∪B in A∩B. If, in addition,≤β<  – γ

and≤γ ≤/then z is a fixed point of the composite self-mapping T◦S:A∪B→A∪B and also of the self-mapping T:A∪B→A∪B. If, furthermore, A and B are convex, then

Fix(S)≡Fix(T◦S)≡Fix(T)consists of a single point.

Proof It follows from Lemma . that ifD=  then{Sn_x}

n∈Nis a Cauchy sequence

conver-gent in the closed setA∩BsinceS:A∪B→A∪Bis globally Lipschitz continuous for anyx∈A∪Bfrom (.). Thus,∃z∈Fix(S)⊂A∩B, which satisfiesSnx→z=Szfor any givenx∈A∪B. If, in addition, we takex=y=z=SzandD=  in (.) with ≤β<  – γ

and ≤γ ≤/, one gets

d(Sz,TSz) =d(Sz,Tz)≤ γ

 –β–γd(z,Sz), (.)

which only holds if and only ifSz=TSz=Tz=zso thatz∈Fix(S)∩Fix(T◦S)∩Fix(T)⊂ A∩B. The uniqueness of the fixed point follows as in the proof of Theorem . by using

Reformulations of Lemma . and Theorem . without usingd(x,Sx)≤d(x,Tx);∀x∈ A∪Bcould be made in a similar way.

4 Properties of convergence and common best proximity points for the case whenAandBdo not intersect

This section extends some relevant results from the previous sections to the case that the subsets do not intersect provided they are subsets of a uniformly convex Banach space. For such a case, Lemmas ., ., . and . still hold. However, Theorems ., ., . and . do not further hold since fixed points inA∩B=∅cannot exist. Thus, the further investigation is centred on the existence and potential uniqueness of best proximity points. It has been proved in [] that ifT:A∪B→A∪Bis a cyclicϕ-contraction withAandB being weakly closed subsets of a reflexive Banach space (X, ), then∃(x,y)∈A×Bsuch thatD=d(x,y) =x–ywhered:R_+→R_+is a norm-induced metric,i.e. xandyare best proximity points. Also, ifAandBare nonempty subsets of a metric space (X,d),Ais compact,Bis approximatively compact with respect toAandT:A∪B→A∪Bis a cyclic contraction, then∃(x,y)∈A×Bsuch thatD=d(x,y) (i.e.iflimn→∞d(Tnx,y) =d(B,y) :=

infz∈Bd(z,y) for somey∈Aand allx∈Bthen the sequence{Tnx}n∈Nhas a convergent

subsequence, []). Theorem . extends as follows, via Lemma ., for the general case whenAandBdo not intersect.

Theorem . Assume that A and B are nonempty closed and convex subsets of a uniformly convex Banach space(X, ). Assume also that S,T:A∪B→A∪B are both cyclic self-mappings and that the contractive condition (.) holds subject tomin(α,β,γ)≥,α+(β+

γ) < and d(x,Sx)≤d(x,Tx);∀x∈A∪B. Then, there exist two unique best proximity points z∈A, y∈B of the self-mappings S,T,T◦S:A∪B→A∪B such that

z=Sy=Ty=TSy=TSz=Sz=STy=Sy, (.)

y=Sz=Tz=TSz=TSy=Sy=STz=Sz. (.)

If A∩B=∅, then z=y∈A∩B is the unique fixed point of S,T,T◦S:A∪B→A∪B which is in A∩B.

Proof IfD= ,i.e. AandBintersect, then this result reduces to Theorem ., with the best proximity points being coincident and equal to the unique fixed point. Consider the case thatAandBdo not intersect, that is,D> , and takex∈A∪B. Assume thatx∈A. SinceA andBare nonempty and closed,Ais convex and Lemma .(i) holds; sincemin(α,β,γ)≥ ,α+ (β+γ) <  andd(x,Sx)≤d(x,Tx);∀x∈A∪B, it follows that

dSn+x,Snx→D;dSn+x,Sn+x→D

⇒ dSnx,Snx→ asn→ ∞ (.)

(which was proved in Lemma . []). The same conclusion arises ifx∈Bsince B is convex. Thus,{Sn_x}

n∈Nis bounded and converges to some pointz=z(x), being

consider the associate metric space (X,d) which can be identified with (X, ) in this con-text. It is now proved by contradiction that for everyε∈R+, there existsn∈Nsuch that d(Sm_x,Sn+_{x)≤D+εfor allm>n≥n. Assume the contrary; that is, given someε∈R}

+, there existsn∈Nsuch thatd(Smkx,Snk+x) >D+εfor allmk>nk≥n,∀k∈N. Then, by using the triangle inequality for distances

D+ε<dSmk_x,Snk+_x

≤dSmk_x,Smk+_x+dSmk+_x,Snk+_{x asn}→ ∞ _(.)

one gets from (.)-(.) that

lim inf

k→∞

dSmk_x,Smk+_x+dSmk+_x,Snk+_x

=lim inf

k→∞ d

Smk+_x,Snk+_{x>D+ε. (.)}

Now, one gets from (.), (.) and Lemma .(i) the following contradiction:

D+ε<lim sup

k→∞

dSmk+_x,Snk+_x

≤lim sup

nk→∞

dSnk+_x,Snk+_{x+lim sup}

k→∞

dSmk+_x,Snk+_x

=lim sup

nk→∞

dSnk+_x,Snk+_{x=D. (.)}

As a result,d(Sm_x,Sn+_{x)≤D+εfor every givenε∈R}

+and allm>n≥nfor some existingn∈N. This leads by a choice of arbitrarily smallεto

D≤lim sup

n→∞ d

Smx,Sn+x≤D ⇒ ∃lim

n→∞d

Smx,Sn+x=D. (.)

But{Sn_x}

n∈Nis a Cauchy sequence with a limitz=SzinA(respectively, with a limity=

S_{yinB) ifx∈A(respectively, ifx∈B) such thatD=Sz–z=d(z,Sz) (Proposition .,} []). Assume on the contrary that x∈Aand{Snx}n∈N →z=S

_{z asn→ ∞so that}

S_{z–Sz=z–Sz= z–y; so that sinceAis convex and (X, ) is a uniformly convex Banach} space, then being strictly convex, one has

D=d(z,Sz) =d

S_z+z

 –Sz =

Sz_–Sz+z–Sz 

≤Sz–Sz 

+z–Sz 

<D

 + D

 =D, (.)

which is a contradiction andz=Szis the best proximity point inAofS:A∪B→A∪B. In the same way,{Sn_x}

n∈N is a Cauchy sequence with a limitSy=y∈Bwhich is the

from Banach contraction principle since all composite self-mappingsSn_{:A∪B→A∪B;}

n∈Nare contractive:

dSnx,Sn+x→D=d(z,y);∀x∈A∪Basn→ ∞

⇒ y←Sn+x=SSnx→Szasn→ ∞. (.)

Thus, z=Sy=Sz=Syandy=Sz=Sy=Sz are the best proximity points ofS : A∪B→A∪BinAandB. Finally, we prove that the best proximity points z∈Aand y∈Bare unique. Assume thatz(=z)∈Aare two distinct best proximity points ofS: A∪B→A∪BinA. Thus,Sz(=Sz)∈Bare two distinct best proximity points inB. Oth-erwise,Sz=Sz⇒Sz=Sz⇒z=z, sincezandzare best proximity points, contra-dictsz=z. From Lemma .(i) andd(Sz,Sz) =d(Sz,Sz) =d(z,Sz) =d(z,Sz) =D through a similar argument to that concluding with (.) with the convexity ofAand the strict convexity of (X, ), guaranteed by its uniform convexity, one gets the following contradiction:

D=dSz,Sz≤S _z–Sz



+z–Sz 

<D

 + D

 =D (.)

sinceS_z

–Sz=Sz–z. Thus,zis the unique best proximity point ofS:A∪B→A∪B inAwhileSzis its unique best proximity point inB.

Now, note that the conditiond(x,Sx)≤d(x,Tx) applied to the best proximity points yields

D=d(z,y) =d(z,Sz) =d(y,Sy)≤d=d(y,Sy)

=dz,TSz=dy,TSy=d(y,TSz) =D, (.)

which implies strict equalities in (.),i.e.

D=d(z,y) =d(z,Sz) =d(y,Sy) =d(z,Tz) =d(z,TSy)

=dz,TSz=dy,TSy=d(y,TSz) =D. (.)

IfA∩B=∅, theny=zis the unique fixed point ofS,T,T◦S:A∪B→A∪Bfrom

Theo-rem ..

In a similar way, Theorem . might be directly extended via Lemma . for the modifi-cation (.) of the contractive condition (.). On the other hand, Theorem . is extended via Lemma . under the contractive constraint (.) if, in general,D=  as follows.

Theorem . Let(X, )be a uniformly convex Banach space and assume that S,T : A∪B→A∪B are cyclic self-mappings satisfying the contractive condition (.), subject to the constraintsmin(α,β,γ,μ,ν)≥,ω=  –α– (β+γ)andα+δ+βμ+γ ν< and d(x,Sx)≤d(x,Tx);∀x∈A∪B, where the subsets A and B are nonempty, closed and convex. Then, there exist two unique best proximity points z∈A, y∈B of the self-mappings S,T,T◦ S:A∪B→A∪B such that (.)-(.) hold.

The proof of Theorem . is very similar to that of Theorem . by using Lemma .. In a similar way, Theorem . is extended as follows under Lemma . allowing the removal of the constraintd(x,Sx)≤d(x,Tx);∀x∈A∪B. The proof is very close to that of Theorem . and is, therefore, omitted.

Theorem . Let(X, )be a uniformly convex Banach space and assume that S,T:A∪

B→A∪B are cyclic self-mappings where A and B intersect and are nonempty and closed, and that the contractive condition (.) holds subject to the constraintsmin(α,β,γ,μ,ν)≥ and

≤α<( –β–γ) 

 +β–γ – (β+γ);

≤β<min

 +γ(γ– )

 – γ ,  – γ ; γ ∈(/, ), (.)

ω=( –β–γ)

_{– ( +β–γ)(α+β–γ)}

( –γ) . (.)

Then, there exist two unique best proximity points z∈A, y∈B of the self-mappings S,T,T◦ S:A∪B→A∪B such that (.)-(.) hold.

If A∩B=∅, then z=y∈A∩B is the unique fixed point of S,T,T◦S:A∪B→A∪B which is in A∩B.

It has to be pointed out that ifωin Theorems .-. is not given by the corresponding definitions but instead their respective equality right-hand sides are strict lower-bounds ofω, then the distances in Lemmas ., ., . and . do not converge toDbut to some

¯

D>D. The iteratesSnxandSn+xare always inAandBfor anyx∈Aand, respectively, in BandAfor anyx∈B, and they are as a result in some nonempty subsetsA⊆AandB⊆B such thatD¯ :=dist(A,B) >Dor, conversely, asn→ ∞by construction sinceS(A)⊆B, T(A)⊆B,S(B)⊆AandT(B)⊆A. Lemmas . and . of [] still hold. Then,{Sn_x}

n∈N

and{Sn+_x}

n∈Nare Cauchy sequences which converge to somez∈AandSz∈Bsuch that

d(z,Sz) =D¯ ifx∈Aand toSzandzifx∈Bwhich are unique sinceAandBare closed and convex and (X, ) is a uniformly convex Banach space. The setsAandBare non-unique but they are in familiesAandBof the subsets ofAandBwhich contain by construction the two above unique convergence points. Then, the convergence points of the Cauchy sequencesz=Sz∈AandSz∈Bare the unique best proximity points of all the closed convex sets in the familiesAandBof the subsets ofAandBifD¯ >D> . Then, Theorems .-. extend as follows.

Corollary . Assume that Theorems .-. are reformulated under respective

identi-cal assumptions except that D> and the respective definitions ofωare replaced with strict lower-bounds for their respective right-hand sides. Then, there exist two unique best proximity points z=S_{z∈A and Sz=Tz=TS}_{z∈B of all sets in two familiesAandB} of nonempty, closed and convex subsets of A and B which are convergence points of the sequences{Sn_x}

5 Examples

Example . Define the discrete time-invariant scalar positive dynamic systems

xn+=Sxn:=axn+b; yn+=Tyn:=cyn+d; ∀n∈N, (.)

whereN=N∪ {}witha,c∈[, ) andmin(b,d)≥ subject to initial conditions satis-fyingmin(x,y)≥ d_a–_–b_c. The respective solutions converge asymptotically to the globally stable equilibrium pointsx*₌ b

–a andy*= d

–c which are both identical if d=

–c

–abfor

min(x,y)≥ _–b_a. Then, note also thatx*_=Sx*₌ ab

–a +b= b

–a andy*=Ty*=x*=Sx*= cd

–c+d= b

–a is a common unique fixed point ofS,T: [ b

–a,∞)→[ b

–a,∞). This result also

follows from Lemma ., and Theorem . withxn,yn(=xn)∈A≡B:= [_–b_a,∞);∀n∈N yields forx=y∈AandD=dist(A,B) =  sinceA∩B=A=∅under the contractive con-straint (.):

( –γ–β)dSx,Sx≤( –γ–β)d(Sx,TSx)≤(α+β+γ)d(x,Sx) (.)

ifα+ (β+γ) <  from the necessary condition of Lemma .

d(x,Sx)≤d(x,Tx) ⇒ d(Sx,TSx)≥dSx,Sx. (.)

Then, (.) is re-arranged asd(S_{x,Sx)≤kd(Sx,x) being contractive provided that}

k=α+β+γ

 –β–γ <  ⇔ α+ (β+γ) < 

since the necessary condition d(x,Sx)≤d(x,Tx)⇔x≥ d_a–_–b_c =x*;∀x∈Aholds for the Euclidean metric d(x,y) =|x–y|. Then,S: [_–b_a,∞)→[_–b_a,∞) is contractive, so that d(Sn+_x,Sn_{x)→ asn→ ∞, and has a unique fixed pointx}*_=y*₌ b

–a ∈A(=A∩B)

since it is continuous. It also holds that x* is a fixed point of the composite mappings T ◦S: [_–b_a,∞)→[_–b_a,∞) (Theorem .) andS◦T : [_–b_a,∞)→[_–b_a,∞). Note that the necessary condition of Lemma .d(x,Sx)≤d(x,Tx)⇔x≥d_a–_–b_c =x* _{justifies to fix} A≡B= [ b

–a,∞) as definition domain of the self-mappingsSandT.

Example . The results of Example . also hold from Lemma . and Theorem . for the contractive constraint (.) subject toα+ β+γ< .

Example . Consider the following dynamic systems:

xn+=Sxn:=axn+b; xn+=Sxn+:= –xn+; ∀n∈N,

yn+=Tyn:=ayn+b; yn+=Tyn+:= –yn+; ∀n∈N

being the limit of the sequences{xn}and{yn}andz*= –x*being that of the sequences

{xn+}and{yn+} ifx,y∈Aand, conversely, ifx,y∈Bwhich is a unique common

fixed pointx*_{=  ofS,T,T◦S,T◦S:R→Rifb=d=  withA≡B:=R. The conclusion} also follows directly from Lemma ., under the constraint (.), withλ:=α+ (β+γ) <  andω=  –λ, and Lemma ., under the constraint (.) withα+ β+γ <  andω=  –λ, and Theorem ..

Example . The extension of the above examples to the non-scalar case is direct. For instance, consider the discrete dynamic systems:

xn+=Sxn:=Mxn+m; ∀n∈N, (.)

yn+=Tyn:=Cnyn+Gnd; ∀n∈N, (.)

whereM(= ),Cn(= ) ∈Rp+×p,m,Gn∈Rp+,d∈R+and the real sequences{Cn},{Gn}are

bounded. Assume thatM,Gnare convergent matrices forn∈Nand that

q+–

j=q [Cj] ≤

Kρ; ∀q ∈N_, ∀∈N for some, in general, norm-dependent K ∈ [,∞) and norm-independent ρ ∈[, ) real constants. Thus, {xn} →x* = (I–M)–m and{yn} →y*= ∞

i=(

∞

j=i+[Cj]Gi)d. Note that the dynamic systems (.)-(.) can be easily described in

a close way forGn∈Rp+×pandd∈R

p

+. We can take the Euclidean norm (and metric) inRp for the subsequent discussion and the corresponding vector-induced spectral matrix norm inRp×pwhich is compatible for well-posed mixed vector/matrix norm computations with the Euclidean vector norm. The fixed pointx*_=y*_{= (I–M)}–_m=∞

i=(

∞

j=i+[Cj]Gi)d

exists for anyd∈R_+which satisfies

m d ∈Ker

–(I–M)–...

∞

i=

_∞

j=i+ [Cj]Gi

, (.)

whereI denotes thepth identity matrix, since the above null-space is nonempty which holds from Rouché-Froebenius theorem from linear algebra, since

rank

–(I–M)–...

∞

i=

_∞

j=i+ [Cj]Gi

=p

holds from (I–M) being non-singular, which implies the compatibility of the subsequent algebraic system of linear equations:

–(I–M)–...

∞

i=

_∞

j=i+ [Cj]Gi

bT...dTT= .

On the other hand, note that if∞_i=(∞_j=i+[Cj]Gi)d<∞andMis critically stable (i.e.

it is singular with at least one eigenvalue of modulus one, while it has no eigenvalue with modulus larger than one), then there are still non-unique common fixed points which are also stable equilibrium points of both mappings if

m= (I–M)

_∞

i=

_∞

j=i+ [Cj]Gi

since the algebraic linear system (I–M)x*_{= (I–M)y}*_{=mis indeterminate compatible} since

p=rank[I–M,m] =rank

I–M, (I–M)

_∞

i=

_∞

j=i+ [Cj]Gi

d

=rank[I–M...I–M]Block Diag(I...m)≤rank(I–M...I–M) =rank(I–M) =p.

A particular interesting case of both mappings having the same unique fixed point, so that both dynamic systems have the same stable equilibrium point being identical to such a fixed point, is when the second dynamic system is a perturbation of the first one considered to be the nominal one, that isCn=M+C˜n;d=G–mandGnd= (G+G˜n)d=m+G˜ndwith

G,Gnbeing real squarep-matrices, provided that the following equations have a solution

inGirrespective of thep-vectorm:

x*= (I–M)–m=

∞

i=

_∞

j=i+

[Cj](G+G˜i)

G–m=

∞

i=

_∞

j=i+ [Cj]

I+G˜iG–

m,

which is

G=

(I–M)––

∞

i=

_∞

j=i+ [Cj]

– ∞ i= ∞

j=i+ [Cj]G˜i

(.)

provided that the sequences{Cn}and{ ˜Gn}are such that [(I–M)–– ∞

i=(

∞

j=i+[Cj])] and ∞

i=

∞

j=i+[Cj]G˜iare non-singular.

The discussion of the existence of common fixed points from Lemma . and Theo-rem . under the constraint (.) imply

dSx,Sx≤kd(Sx,TSx); k=α+β+γ  –β–γ < 

provided thatd(x,Sx)≤d(x,Tx)⇒d(Sx,TSx)≥d(Sx,S_{x) leading together to the} con-tractive conditiond(S_{x,Sx)≤kd(Sx,x) for the self-mappingsS,T:A→Awith}

A≡B:=z= (z,z, . . . ,zp)∈Rp:zi≥eTi(I–M)–m

, (.)

whereeiis theith unit Euclidean vector inRpwhoseith component is unit provided that

{Gn}is such that

p

i=

MT_i –1xin+mi 

–CT_in–1xin+GTind)

≤; ∀n∈N (.)

forx∈Asatisfying the sequences (.) and (.), whereMT

i, CTin andGTin;∀n∈Nare the ith row of the matricesM,CnandGTn;∀n∈Nrespectively;xin andmi are theith

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

Author details

1_{Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia), Aptdo. 644,}

Bilbao, Bilbao 48080, Spain. 2_{Department of Mathematics, Texas A&M University- Kingsville, 700 University Blvd.,} Kingsville, TX 78363-8202, USA.3_{Department of Mathematics, Faculty of Science, King Abdulazid University, Jeddah,} 21589, Saudi Arabia.

Acknowledgements

The authors are grateful to the Spanish Ministry of Education for its partial support of this work through Grant DPI2009-07197. They are also grateful to the Basque Government for its support through Grants IT378-10 and SAIOTEK S-PE08UN15 and 09UN12. Finally, the authors are very grateful to the reviewers for their comments which have been very useful when improving the first version of the manuscript.

Received: 14 May 2012 Accepted: 15 August 2012 Published: 31 August 2012 References

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Cite this article as:Sen and Agarwal:Common fixed points and best proximity points of two cyclic self-mappings.