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

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.[PUBLICATION ABSTRACT]

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R E S E A R C H Open Access

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

M De la Sen1* and RP Agarwal2,3

*Correspondence: [email protected]

1Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia), Aptdo. 644, Bilbao, Bilbao 48080, SpainFull 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

dened 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 xed 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

xed 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 xed 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 Banachs principle [] and Kannans xed 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 xed point theory for Kannans mappings is extended in [] by the use of a non-increasing function aecting the contractive condition and the best constant to ensure a xed point is also obtained. Three xed point theorems which extended the xed point theory for Kannans mappings have been stated and proved in []. Also, signicant attention has been paid to the investigation of standard contractive and Meir-Keeler-type contractive -cyclic self-mappings T : A B A B dened on

subsets A, B X and, in general, p-cyclic self-mappings T :

ip Ai

ip Ai dened on any number of subsets Ai 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

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best proximity points, to which all the sequences of iterates of composite self-mappings T : AB AB converge, is proved in [] for the extension of the contractive principle

for cyclic self-mappings in uniformly convex Banach spaces (then being strictly convex and reexive, []) if the subsets A, B X in the metric space (X, d), or in the Banach

space (X, ), where the -cyclic self-mappings are dened are both nonempty, convex

and closed. The research in [] is centred on the case of the cyclic self-mapping being dened 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-mapping T :

ip Ai

ip Ai be a p( )-cyclic self-mapping being dened on any number of subsets of the metric space with p := {, , . . . , p}.

A relevant problem is when self-mappings from a metric space into itself or from a set into itself have common xed points, [, ]. A related problem is when composite self-maps built with those self-mappings have common xed 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 extensions have removed the need for the continuity of one of the self-mappings [, ]. Some recent papers have investigated the existence of common xed points in cone metric spaces [, ] and in fuzzy metric 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 xed points/common best proximity points of two -cyclic self-mappings (refereed to simply as cyclic self-mappings) on the union of two subsets A and B of a metric space under three contractive conditions. Section is devoted to the convergence properties of distances for such contractive conditions which involve both cyclic self-mappings. Further results obtained in this section are concerned with the existence and uniqueness of common xed 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 that A and B intersect in the sense that the role of common xed points is played instead by common best proximity points under the assumption that the subsets A and B belong to a uniformly convex Banach space.

2 Convergence properties and common xed points under three contractive conditions if A and B intersect

Let (X, d) be a metric space and consider two nonempty subsets A and B of X. It is assumed through the manuscript that S, T : A B A B are cyclic self-mappings, i.e. T(A) B,

S(A) B, T(B) A and S(B) A. Suppose, in addition, that T : A B A B satises

the constraint

d(Sx, TSy) d(x, Sy) +

x, y( = x) A B, (.)

d(x, y) : x A, y B[bracerightbig] , (.)

d(x, Sx) + d(Sy, TSy)

[parenrightbig]+

d(x, Sy) + d(Sx, TSy)

[parenrightbig]

+ D;

where

D := dist(A, B) := inf

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where , , , . Note that if x A and y B or conversely, then the

various point-to-point distances in (.) are not less than D so that the parametrical constraint + ( + ) + has to be fullled from (.) if D = . The following result can

be stated:

Lemma . Assume that d(x, Sx) d(x, Tx); x A B and that the constraints

+ ( + ) < if D = and + ( + ) < if D = both hold. Then, the following

properties hold:(i)

D lim sup

n

d

Snx, Sn+x[parenrightbig]D ( + ); x A B. (.)

If = ( + ) then

lim

n d[parenleftbig]Snx,

Sn+x

[parenrightbig]

= D. (.)

(ii) d(Snx, Sn+m+x) D + Gm(n) for any x A B, n N, m N = N {}, where {Gm(n)}mN is a nonnegative strictly decreasing real sequence for any n N, then

being convergent to zero as m , and

lim sup

m

d

Snx, Sn+m+x[parenrightbig] D.

Proof Take y = x A B and replace x Snx in (.) to yield

d Snx, Sn+x

[parenrightbig]

d

Snx, TSnx

[parenrightbig]

d

Snx, Snx[parenrightbig]+

d

Snx, Snx[parenrightbig]+ d

Snx, TSnx

[parenrightbig][parenrightbig]

+

d

Snx, Snx

[parenrightbig]

+ d

TSnx, Snx[parenrightbig][parenrightbig]+ D

d

Snx, Snx

[parenrightbig]

+

d

Snx, Snx[parenrightbig]+ d

Snx, TSnx

[parenrightbig][parenrightbig]

TSnx, Snx[parenrightbig][parenrightbig]+ D; x A B, n N (.)

since d(x, Sx) d(x, Tx), Snx A (TSnx B Sn+x B) and Snx B (TSnx A

Sn+x A) for any x A B and, equivalently,

d

Snx, Sn+x

[parenrightbig]

d

+

d

Snx, Snx

[parenrightbig]

+ d

Snx, TSnx

[parenrightbig]

+ + d

Snx, Snx

[parenrightbig]

+

D ; x A B, n

N (.)

so that

D lim sup

n

d

Snx, Sn+x

[parenrightbig]

lim

n

[bracketleftBigg][parenleftbigg]

+ +

ni

[parenrightBigg][bracketrightBigg]

n d

Snx, Snx[parenrightbig]+

D

[parenrightbigg][parenleftBigg]

n

i=

+ +

D ( + ) (.)

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since + < , ( + ), and + ( + ) < if D = and

+( + ) < if D = imply that the contraction constant k := ++ < . Hence, (.)

holds and the limit (.) exists for = ( + ). Hence, Property (i) is proved. To prove Property (ii), note that for any natural numbers m and n, one gets from (.) and the above denition of the contraction constant k < that

d Snx, Sn+m+x

[parenrightbig]

Snx, TSnx

Snx, Sn+m+x[parenrightbig] D; x A B, n N. (.b)

Hence, Property (ii) has been proved.

Note that the contractive condition

d(Sx, TSy) d(x, Sy) +

d

= D. (.)

(ii) d(Snx, Sn+m+x) D + Fm(n) for any x A B, n N, m N = N {}, where {Fm(n)}mN is a nonnegative strictly decreasing real sequence for any n N, then

being convergent to zero as m , and

lim sup

m

Snx, Sn+m+x[parenrightbig] D.

Proof The contractive condition (.) removes the additive term d(Sx, TSy) from (.) so that d(TSnx, Snx) is correspondingly removed in the resulting modied counterpart of (.) by taking y = x A B and performing the replacement x Snx. The resulting

contractive constant now becomes k :=

+ < , subject to

+ , which, on

the other hand, results in the needed constraint + + < to reformulate the

results of Lemma . leading to the modied Properties (i), with (.)-(.), and (ii).

The following two results are concerned with the existence and eventual uniqueness of xed points of the self-mappings S, T, S, T S : A B A B (with (T S)x = TSx, x A B) if A and B intersect, are nonempty and closed. If they are also convex then the

xed point is unique, fullling the property Fix(S) Fix(T) Fix(T S) = {z} A B.

D

d

kmd

Snx, TSn+mx[parenrightbig] kmd

Snx, TSnx[parenrightbig]+ D

Sn+x, Snx[parenrightbig]+ D

km

[parenrightbig]

km

[parenrightbig]

d

[parenrightbig]

+ D; x A B, n N, m N, (.a)

lim sup

d

m

d(x, Sx) + d(Sy, TSy)

[parenrightbig]+ d(x, Sy) + D (.)

is distinct from (.), while it modies Lemma ., resulting in the subsequent result.

Lemma . Assume that d(x, Sx) d(x, Tx); x A B and that the constraint

( + + ) > holds. Then, the following properties hold:(i)

D lim sup

n

Snx, Sn+x[parenrightbig]D ( + + ); x A B. (.)

If = ( + + ) then

lim

n d[parenleftbig]Snx,

Sn+x

[parenrightbig]

d

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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 + ( + ) < . Then, there exists a xed

point of S : A B A B in A B which is also a xed point of the composite self-mapping

T S : A B A B and a xed point of T : A B A B. If, in addition, A and B are

convex, then Fix(S) Fix(T S) Fix(T) consists of a single point.

Proof From Lemma . and D = (since A B = ), it follows that

lim

n d[parenleftbig]Snx,

= ; x A B.

Thus, limn,m d(Sn+mx, Sn+m+x) = limn,m d(Sn+mx, TSn+mx) = ; x A B so that {Snx}nN 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) with x

A B and then {Snx}nN is in A B. Thus, Sn+x = S(Snx) z = Sz; x A B and

z Fix(S) A B. Since Snx z and limn d(Snx, TSnx) = ; x A B then

TSn+x TSnx z (TS)(Snx) TSz = z. Thus, z Fix(S) Fix(T S) A B and

TSz = Sz = z TSz = T(TSz) = TSz = z = Tz. Then, z Fix(S) Fix(T) Fix(T S)

A B.

Finally assume, in addition, that A and B are also convex and z, z( = z) Fix(S T)

Fix(S) Fix(T) so that

z, z, Snz = z, TSnz = z Fix(T S) Fix(S) A B

since AB is convex. Using (.) with x = z, y = z and D = , the following contradictions

lead to z = z from the contractive assumption in Lemma . since < ( + ) < :

< d(z, z) nd(z, z) < d(z, z); n N, < d(z, z) lim

n

+ d(x, Sx) + d

Sn+x

TSnx

[parenrightbig]

[parenrightbig]

d(z, z) =

so that z = z and Fix(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 to ( + + ) > .

The proof of Theorem . is omitted since it is similar to that of Theorem .. Assume now that the contractive condition (.) is modied as follows to give relevance to the composite self-mapping S T : A B A B:

max

d

Tx, Tx , d Sx, Sx

[parenrightbig][parenrightbig]

d(Sx, STx)

d(x, Tx) + d

Tx, Tx

= lim

n d[parenleftbig]Snx,

n

[parenrightbig]

Sx, Sx[parenrightbig]+ D; x A B (.)

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for some real constants , , , and , so that by using the lower-bound of

(.) to build a further upper-bounding condition of it, one gets

d(Sx, STx) d(x, Tx) + d

Tx, Tx

[parenrightbig]

+ d(x, Sx) + d

Sx, Sx[parenrightbig]+ D

d(x, Tx) + d(Sx, STx) + d(x, Tx)+ d(Sx, STx) + D; x A B, (.)

which is identical to

d(Sx, STx)

+ d(x, Tx) +

D ; x A B (.)

if + + + < . The following two results hold under the contractive condition (.).

Lemma . Assume that the contractive condition (.) holds subject to +++ < and ( + ). Then

d Snx, SnTx

[parenrightbig]

knd(x, Tx) +

D

kn

[parenrightbig]

d(x, Tx) +

D ; x A B, n

N, (.)

lim sup

n

d

Snx, SnTx[parenrightbig]D ; x A B (.)

and

lim

= D; x A B. (.)

Proof Redene the contractive constant as k :=

+ < so that

n d[parenleftbig]Snx,

SnTx

[parenrightbig]

D k if . 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 to + ++ < , where A and B intersect and are nonempty and closed. Then, there exists a xed point of S : A B A B in A B which is also a xed point of the composite self-mapping

T S : A B A B. If, in addition, A and B are convex then Fix(S) Fix(T S) consists

of a single point.

Outline of proof Let (X, d) be the complete metric space where d : X X R+ is the norm-induced metric by the norm on the Banach space (X, ). If A and B intersect, then

limn d(Snx, SnTx) = ; x A B from Lemma . with S : A B A B satisfying

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

xed point; i.e. Sn+x = S(Snx), Snx, SnTx z so that z Fix(S) Fix(S T) A B. The

uniqueness of the xed point is proved by using the convexity of A B as follows. Assume

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that z, z ( = z) are xed points of S, S T : A B A B in A B. Then, the following

contradictions lead to z = z in terms that either

<

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]S

z + z

[parenrightbigg]

ST

z + z

[parenrightbigg][vextenddouble][vextenddouble][vextenddouble][vextenddouble]

= d

S z + z , STz + z

[parenrightbigg]=

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

STz Sz

STz Sz

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

STz Sz +

STz Sz =

d(Sz, STz) + d(Sz, STz)

[parenrightbig]

= (.)

since z+z A B, since A B is convex, or all points in the segment [z, z] A B, again

since A B is convex, are xed points of the self-mappings S, S T : A B A B.

Now, take arbitrarily closed points z , z = z + (z, z), which are also xed points. Then

the continuity of S : A B A B leads to a further contradiction limz z (Sz ) = z = z . Then, no segment [z, z] A B can consist of xed points of S, S T : A B A B.

3 Relaxing a hypothesis of Section 2

The assumption d(x, Sx) d(x, Tx); x A B in Lemma . and Lemma ., then in The

orem . and Theorem ., 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

d Snx, Sn+x

d

Snx, TSnx[parenrightbig]+ d

TSnx, Sn+x

; x A B. (.)

The contractive condition (.) becomes equivalent to

d Sn+x, TSnx

[parenrightbig]

d

Snx, Sn+x[parenrightbig]+ d

Snx, TSnx

[parenrightbig][parenrightbig]

+ d

Sn+x, TSnx

[parenrightbig]

+ D; x A B (.)

with the replacements x, Sy Snx, y Snx. The inequality (.) is equivalent to

d Sn+x, TSnx

[parenrightbig]

d

Snx, TSnx[parenrightbig][parenrightbig]+D ; x A B (.)

if < . The substitution of (.) into (.) yields

d

Snx, Sn+x

[parenrightbig]

Snx, Sn+x

[parenrightbig]

+ d

d

Snx, Sn+x

[parenrightbig]

+ d

Snx, TSnx

[parenrightbig][parenrightbig]

+ d

Snx, TSnx

[parenrightbig]

+

D ; x A B (.)

and using (.) in (.)

d

Sn+x, Snx

[parenrightbig]

+ d

Snx, TSnx

[parenrightbig]

+

D

+

+ + d

Snx, Snx

[parenrightbig]

+

D

[parenrightbigg]

+

D

= ( +

)( + + )

( ) d

Snx, Snx[parenrightbig]+ ( )D( ) ; x A B (.)

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and S : A B A B is cyclic contractive if (+)(++)() < , that is, if

<

( ) + (

+ ); <

+ ( ) ;

(/, ), (.)

( ) ( + )( + )( ) , (.)

where the second constraint of (.) guarantees that + < ()

+ ; i.e. and the third one that is nonnegative. Lemma .(i) is modied by using (.)-(.) as follows without using the assumption d(x, Sx) d(x, Tx); x A B.

Lemma . Assume that (.) holds subject to the constraints (.)-(.). Then

D lim sup

n

d

Snx, Sn+x[parenrightbig]( )D( ) ; x A B. (.)

If = ()

(+ )(+ )( ) , then

lim

n d[parenleftbig]Snx,

Sn+x

[parenrightbig]

= D. (.)

An ad-hoc modied 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

<

( )

+ (

+ ); <

+ ( )

;

(/, ), (.)

) ( + )( + )

( ) . (.)

Then, there exists a xed point z of S : AB AB in AB. If, in addition, <

and / then z is a xed 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 if D = then {Snx}nN is a Cauchy sequence conver

gent in the closed set A B since S : A B A B is globally Lipschitz continuous for

any x A B from (.). Thus, z Fix(S) A B, which satises Snx z = Sz for any given x A B. If, in addition, we take x = y = z = Sz and D = in (.) with <

and /, one gets

d(Sz, TSz) = d(Sz, Tz)

= (

d(z, Sz), (.)

which only holds if and only if Sz = TSz = Tz = z so that z Fix(S) Fix(T S) Fix(T)

A B. The uniqueness of the xed point follows as in the proof of Theorem . by using

the convexity assumption.

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Reformulations of Lemma . and Theorem . without using d(x, Sx) d(x, Tx); x

A B could be made in a similar way.

4 Properties of convergence and common best proximity points for the case when A and B do 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 xed points in A 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 if T : A B A B is a cyclic -contraction with A and B

being weakly closed subsets of a reexive Banach space (X, ), then (x, y) A B such

that D = d(x, y) = x y where d : R+ R+ is a norm-induced metric, i.e. x and y are

best proximity points. Also, if A and B are nonempty subsets of a metric space (X, d), A is

compact, B is approximatively compact with respect to A and T : A B A B is a cyclic

contraction, then (x, y) A B such that D = d(x, y) (i.e. if limn d(Tnx, y) = d(B, y) :=

infzB d(z, y) for some y A and all x B then the sequence {Tnx}nN has a convergent

subsequence, []). Theorem . extends as follows, via Lemma ., for the general case when A and B do 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 to min(, , ) , +( +

) < and d(x, Sx) d(x, Tx); x AB. 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 = TSy = TSz = Sz = STy = Sy, (.)

y = Sz = Tz = TSz = TSy = Sy = STz = Sz. (.)

If A B = , then z = y A B is the unique xed point of S, T, T S : A B A B which

is in A B.

Proof If D = , i.e. A and B intersect, then this result reduces to Theorem ., with the best proximity points being coincident and equal to the unique xed point. Consider the case that A and B do not intersect, that is, D > , and take x AB. Assume that x A. Since A

and B are nonempty and closed, A is convex and Lemma .(i) holds; since min(, , )

, + ( + ) < and d(x, Sx) d(x, Tx); x A B, it follows that

d Sn+x, Snx

[parenrightbig]

Snx, Snx

D; d

Sn+x, Sn+x[parenrightbig] D [bracketrightbig]

d

[parenrightbig]

as n (.)

(which was proved in Lemma . []). The same conclusion arises if x B since B is

convex. Thus, {Snx}nN is bounded and converges to some point z = z(x), being poten

tially dependent on the initial point x which is in A if x A, since A is closed, and in B

if x B, since B is closed. Take, with no loss in generality, the norm-induced metric and

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consider the associate metric space (X, d) which can be identied with (X, ) in this con

text. It is now proved by contradiction that for every R+, there exists n N such that

d(Smx, Sn+x) D + for all m > n n. Assume the contrary; that is, given some R+,

there exists n N such that d(Sm

k x, Snk+x) > D + for all mk > nk n, k N. Then,

by using the triangle inequality for distances

D + < d Smkx, Snk+x

[parenrightbig]

d

Smkx, Smk+x

[parenrightbig]

+ d

Smk+x, Snk+x[parenrightbig]as n (.)

one gets from (.)-(.) that

lim inf

k

d

Smkx, Smk+x

[parenrightbig]

+ d

Smk+x, Snk+x

[parenrightbig][parenrightbig]

= lim inf

k d[parenleftbig]Smk+x,

Snk+x

[parenrightbig]

> D + . (.)

Now, one gets from (.), (.) and Lemma .(i) the following contradiction:

D + < lim sup

k

d

Smk+x, Snk+x

[parenrightbig]

lim sup

nk

d

Snk+x, Snk+x[parenrightbig]+ lim sup kd

Smk+x, Snk+x

[parenrightbig]

Snk+x, Snk+x[parenrightbig]= D. (.)

As a result, d(Smx, Sn+x) D + for every given R+ and all m > n n for some

existing n N. This leads by a choice of arbitrarily small to

D lim sup

n

d

= lim sup

nk

d

Smx, Sn+x[parenrightbig] D limn d[parenleftbig]Smx,Sn+x[parenrightbig]= D. (.)

But {Snx}nNis a Cauchy sequence with a limit z = Sz in A (respectively, with a limit y =

Sy in B) if x A (respectively, if x B) such that D = Sz z = d(z, Sz) (Proposition .,

[]). Assume on the contrary that x A and {Snx}nN z = Sz as n so that Sz Sz = z Sz = z y; so that since A is convex and (X, ) is a uniformly convex Banach

space, then being strictly convex, one has

D = d(z, Sz) = d

Sz + z Sz[parenrightbigg]=

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

Sz Sz

+

z Sz

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

Sz Sz

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

+

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

z Sz

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

= D, (.)

which is a contradiction and z = Sz is the best proximity point in A of S : A B A B.

In the same way, {Snx}nN is a Cauchy sequence with a limit Sy = y B which is the

best proximity point in B of S : A B A B if x B since B is convex and (X, )

is strictly convex. We prove now that y = Sz. Assume, on the contrary, that y = Sz with

y = Sy, Sz = Sz B, z = Sz A, d(z, y) > D, d(Sz, Sy) D, d(Sz, z) = d(Sy, y) = D. One

gets the following contradiction from (.), which is obtained from (.) provided that d(x, Sx) d(x, Tx); x A B, since S : A B A B is globally Lipschitz continuous

< D

+

D

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from Banach contraction principle since all composite self-mappings Sn : A B A B; n N are contractive:

d Snx, Sn+x

[parenrightbig]

y Sn+x = S

= d

D = d(z, y); x A B as n

Snx[parenrightbig] Sz as n

. (.)

Thus, z = Sy = Sz = Sy and y = Sz = Sy = Sz are the best proximity points of S : A B A B in A and B. Finally, we prove that the best proximity points z A and

y B are unique. Assume that z( = z) A are two distinct best proximity points of S :

A B A B in A. Thus, Sz( = Sz) B are two distinct best proximity points in B. Oth

erwise, Sz = Sz Sz = Sz z = z, since z and z are best proximity points, contra

dicts z = z. From Lemma .(i) and d(Sz, Sz) = d(Sz, Sz) = d(z, Sz) = d(z, Sz) = D

through a similar argument to that concluding with (.) with the convexity of A and the strict convexity of (X, ), guaranteed by its uniform convexity, one gets the following

contradiction:

D = d Sz, Sz

[parenrightbig]

[bracketrightbig]

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

[vextenddouble][vextenddouble][vextenddouble][vextenddouble]

Sz Sz

+

z Sz

= D (.)

since Sz Sz = Sz z. Thus, z is the unique best proximity point of S : A B A B

in A while Sz is its unique best proximity point in B.

Now, note that the condition d(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)

= d

z, TSz[parenrightbig]= d y, TSy[parenrightbig]= 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)

= d

z, TSz

[parenrightbig]

< D

+

D

y, TSy[parenrightbig]= d(y, TSz) = D. (.)

If A B = , then y = z is the unique xed point of S, T, T S : A B A B from Theo

rem ..

In a similar way, Theorem . might be directly extended via Lemma . for the modication (.) 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 constraints min(, , , , ) , = ( + ) and + + + < 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.

If A B = , then z = y A B is the unique xed point of S, T, T S : A B A B

which is in A B.

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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 constraint d(x, Sx) d(x, Tx); x AB. 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 constraints min(, , , , )

and

<

+ ( ) ,

) ( + )( + )

( ) . (.)

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 xed point of S, T, T S : A B A B

which is in A B.

It has to be pointed out that if in Theorems .-. is not given by the corresponding denitions but instead their respective equality right-hand sides are strict lower-bounds of , then the distances in Lemmas ., ., . and . do not converge to D but to some D > D. The iterates Snx and Sn+x are always in A and B for any x A and, respectively, in

B and A for any x B, and they are as a result in some nonempty subsets A A and B B

such that D := dist(A , B ) > D or, conversely, as n by construction since S(A) B,

T(A) B, S(B) A and T(B) A. Lemmas . and . of [] still hold. Then, {Snx}nN and {Sn+x}nN are Cauchy sequences which converge to some z A and Sz B such that

d(z, Sz) = D if x A and to Sz and z if x B which are unique since A and B are closed and

convex and (X, ) is a uniformly convex Banach space. The sets A and B are non-unique

but they are in families A and B of the subsets of A and B which contain by construction the two above unique convergence points. Then, the convergence points of the Cauchy sequences z = Sz A and Sz B are the unique best proximity points of all the closed

convex sets in the families A and B of the subsets of A and B if D > D > . Then, Theorems .-. extend as follows.

Corollary . Assume that Theorems .-. are reformulated under respective identical assumptions except that D > and the respective denitions of are replaced with strict lower-bounds for their respective right-hand sides. Then, there exist two unique best proximity points z = Sz A and Sz = Tz = TSz B of all sets in two families A and B

of nonempty, closed and convex subsets of A and B which are convergence points of the sequences {Snx}nN and {Sn+x}nN for x A so that Sx B.

( ) + (

+ );

< min

; (/, ), (.)

= (

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5 Examples

Example . Dene the discrete time-invariant scalar positive dynamic systems

xn+ = Sxn := axn + b; yn+ = Tyn := cyn + d; n N, (.)

where N = N {} with a, c [, ) and min(b, d) subject to initial conditions satis

fying min(x, y) dbac. The respective solutions converge asymptotically to the globally

stable equilibrium points x* = b

a and y* =

dc which are both identical if d = cab for

aba + b =

ba is a common unique xed point of S, T : [ ba, ) [ ba, ). This result also follows from Lemma ., and Theorem . with xn, yn(= xn) A B := [ ba, ); n N

yields for x = y A and D = dist(A, B) = since A B = A = under the contractive con

straint (.):

( )d

Sx, Sx

[parenrightbig]

( )d(Sx, TSx) ( + + )d(x, Sx) (.)

if + ( + ) < from the necessary condition of Lemma .

d(x, Sx) d(x, Tx) d(Sx, TSx) d

+ +

<

ba A(= A B) since it is continuous. It also holds that x* is a xed point of the composite mappings

T S : [ ba, ) [ ba, ) (Theorem .) and S T : [ ba, ) [ ba, ). Note that

the necessary condition of Lemma . d(x, Sx) d(x, Tx) x dbac = x* justies to x

A B = [ ba, ) as denition domain of the self-mappings S and T.

Example . The results of Example . also hold from Lemma . and Theorem . for the contractive constraint (.) subject to + + < .

Example . Consider the following dynamic systems:

xn+ = Sxn := axn + b; xn+ = Sxn+ := xn+; n N,

yn+ = Tyn := ayn + b; yn+ = Tyn+ := yn+; n N

under the same constraints of Example .. Dene real subsets A := [ ba, ) and B :=

(,

ba ] of empty intersection whose Euclidean distance is D =

ba ] (,

ba ] being associated with the solutions of both dynamic systems which full the necessary condition of Lemma . d(x, Sx) d(x, Tx) everywhere

in their denition domain. It follows the convergence to unique best proximity points x*

maps S, T : (,

min(x, y)

cdc + d =

ba . Then, note also that x* = Sx* =

ba and y* = Ty* = x* = Sx* =

. (.)

Then, (.) is re-arranged as d(Sx, Sx) kd(Sx, x) being contractive provided that

k =

Sx, Sx

+ ( + ) <

since the necessary condition d(x, Sx) d(x, Tx) x dbac = x*; x A holds for the Euclidean metric d(x, y) = |x y|. Then, S : [ ba, ) [ ba, ) is contractive, so that

d(Sn+x, Snx) as n , and has a unique xed point x* = y* =

ba and consider

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being the limit of the sequences {xn} and {yn} and z* = x* being that of the sequences {xn+} and {yn+} if x, y A and, conversely, if x, y B which is a unique common

xed point x* = of S, T, T S, T S : R R if b = d = with A B := R. The conclusion

also follows directly from Lemma ., under the constraint (.), with := + ( + ) < and = , and Lemma ., under the constraint (.) with + + < and = , 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, (.)

where M( = ), Cn( = ) Rpp+, m, Gn Rp+, d R+ and the real sequences {Cn}, {Gn} are

bounded. Assume that M, Gn are convergent matrices for n N and that

q+ j=q [Cj] K ; q N, N for some, in general, norm-dependent K [, ) and norm-

independent [, ) 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 for Gn Rpp+ and d Rp+. We can take the Euclidean norm (and metric) in Rp

for the subsequent discussion and the corresponding vector-induced spectral matrix norm in Rpp which is compatible for well-posed mixed vector/matrix norm computations with the Euclidean vector norm. The xed point x* = y* = (I M)m =

i=(

j=i+[Cj]Gi)d

exists for any d R+ which satises

m d

[parenrightBigg][bracketrightBigg], (.)

where I denotes the pth identity matrix, since the above null-space is nonempty which holds from Rouch-Froebenius theorem from linear algebra, since

rank

(I M) ...

i=

[parenleftBigg]

[parenrightBigg]

Ker

(I M) ...

i=

j=i+ [Cj]Gi

j=i+ [Cj]Gi

[parenrightBigg][bracketrightBigg] = p

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

(I M) ...

i=

[parenleftBigg]

j=i+ [Cj]Gi

[parenrightBigg][bracketrightBigg][parenleftbig]

bT ... dT

T = .

j=i+[Cj]Gi)d < and M is 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 xed points which are also stable equilibrium points of both mappings if

m = (I M)

On the other hand, note that if

i=(

[parenleftBigg]

i=

[parenleftBigg]

j=i+ [Cj]Gi

[parenrightBigg][parenrightBigg]d = (I M)y* (.)

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since the algebraic linear system (I M)x* = (I M)y* = m is indeterminate compatible since

p = rank[I M, m] = rank

I M, (I M)

[parenleftBigg]

i=

[parenleftBigg]

j=i+ [Cj]Gi

[parenrightBigg][parenrightBigg]d[bracketrightBigg]

= rank

[I M ... I M] Block Diag(I ... m)

[parenrightbig] rank(I M

... I M) = rank(I M) = p.

A particular interesting case of both mappings having the same unique xed point, so that both dynamic systems have the same stable equilibrium point being identical to such a xed point, is when the second dynamic system is a perturbation of the rst one considered to be the nominal one, that is Cn = M + Cn; d = Gm and Gnd = (G + Gn)d = m + Gnd with

G, Gn being real square p-matrices, provided that the following equations have a solution in G irrespective of the p-vector m:

x* = (I M)m =

i=

[parenleftBigg]

j=i+[Cj](G + Gi)

Gm =

i=

[parenleftBigg]

j=i+ [Cj]

I + GiG

[parenrightbig][parenrightBigg]

m,

which is

G =

(I M)

i=

[parenrightBigg] (.)

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

i=(

[parenleftBigg]

[parenleftBigg]

j=i+ [Cj]

[parenrightBigg][bracketrightBigg]

i=

j=i+[Cj] Gi

j=i+[Cj])] and

j=i+[Cj] Gi are non-singular.

The discussion of the existence of common xed points from Lemma . and Theorem . under the constraint (.) imply

d

i=

Sx, Sx

[parenrightbig]

+ +

<

provided that d(x, Sx) d(x, Tx) d(Sx, TSx) d(Sx, Sx) leading together to the con

tractive condition d(Sx, Sx) kd(Sx, x) for the self-mappings S, T : A A with

A B :=

z = (z, z, . . . , zp) Rp : zi eTi(I M)m

kd(Sx, TSx); k =

, (.)

where ei is the ith unit Euclidean vector in Rp whose ith component is unit provided that {Gn} is such that

p

i=

[bracketleftbig][parenleftbig][parenleftbig]MT

i 1

xin + mi

; n N (.)

for x A satisfying the sequences (.) and (.), where MTi, CTin and GTin; n N are

the ith row of the matrices M, Cn and GTn; n N respectively; xin and mi are the ith

components of xn; n N and m, respectively; 1 is a Euclidean p-vector with all its com

ponents being one. It can be easily seen that (.) is equivalent to the necessary condition (I S)x = d(x, Sx) d(x, Tx) = (I T)x of Lemma . for the Euclidean metric.

CTin 1

xin + GTind)

[bracketrightbig]

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Competing interests

The authors declare that they have no competing interests.

Authors contributions

Both authors contributed equally and signicantly in writing this paper. Both authors read and approved the nal manuscript.

Author details

1Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia), Aptdo. 644, Bilbao, Bilbao 48080, Spain. 2Department of Mathematics, Texas A&M University- Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA. 3Department 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 rst version of the manuscript.

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

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doi:10.1186/1687-1812-2012-136Cite this article as: Sen and Agarwal: Common xed points and best proximity points of two cyclic self-mappings. Fixed Point Theory and Applications 2012 2012:136.