Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets

Abstract

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms [InlineEquation not available: see fulltext.] for [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], or [InlineEquation not available: see fulltext.], subject to [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.], such that [InlineEquation not available: see fulltext.] converges uniformly to T, and the distances [InlineEquation not available: see fulltext.] are iteration-dependent, where [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] are non-empty subsets of X, for [InlineEquation not available: see fulltext.], where [InlineEquation not available: see fulltext.] is a metric space, provided that the set-theoretic limit of the sequences of closed sets [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext.] exist as [InlineEquation not available: see fulltext.] and that the countable infinite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identification of dynamic systems.

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R E S E A R C H Open Access

Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets

Manuel De la Sen1*, Ravi P Agarwal2,3 and Asier Ibeas4

*Correspondence: [email protected]

1Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), Barrio Sarriena, P.O. Box 644, Bilbao, Leioa 48940, SpainFull list of author information is available at the end of the article

Abstract

This paper presents some further results on proximal and asymptotic proximal contractions and on a class of generalized weak proximal contractions in metric spaces. The generalizations are stated for non-self-mappings of the forms Tn : An Bn

for n Z0+ and T : jZ0+ A0j jZ0+ B0j, or T : A ( Bn), subject to T(A0n) B0n and Tn(An) Bn, such that Tn converges uniformly to T, and the distances

Dn = d(An, Bn) are iteration-dependent, where A0n, An, B0n and Bn are non-empty subsets of X, for n Z0+, where (X, d) is a metric space, provided that the set-theoretic

limit of the sequences of closed sets {An} and {Bn} exist as n and that the

countable innite unions of the closed sets are closed. The convergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated if the metric space is complete. Two application examples are also given, being concerned, respectively, with the solutions through pseudo-inverses of both compatible and incompatible linear algebraic systems and with the parametrical identication of dynamic systems.

Keywords: proximal contraction; weak proximal contraction; best proximity point; set-theoretic limit; Moore-Penrose pseudo-inverse

1 Introduction

The characterization and study of existence and uniqueness of best proximity points is an important tool in xed point theory concerning cyclic nonexpansive mappings including the problems of (strict) contractions, asymptotic contractions, contractive and weak contractive mappings and also in related problems of proximal contractions, weak proximal contractions and approximation results and methods []. The application of the theory of xed points in stability issues has been proved to be a very useful tool. See, for instance, [] and references therein. This paper is devoted to formulating and proving some further results for more general classes of proximal contractions. The problem of proximal contractions associated with uniformly converging non-self-mappings {Tn} {T}

of the form Tn : An Bn; n Z+, where An and Bn are in general distinct, with a set-

theoretic limit of the form T : jZ+ Aj

jZ+ Bj, provided that the set-theoretic limits of the involved set exist and that the innite unions of the involved closed sets are also closed. Further related results are obtained for generalized weak proximal and proximal contractions in metric spaces [, , ], which are subject to certain parametrical

2014 De la Sen et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0

Web End =http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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constraints on the contractive conditions. Such constraints guarantee that the implying condition of the proximal contraction holds for the proximal sequences so that it can be removed from the analysis []. Some related generalizations are also given for non-self-mappings of the form T : A (

Bn), subject to a set distance Dn = d(A, Bn), where A and

Bn are non-empty and closed subsets of a metric space (X, d) for n Z+, provided that

the set-theoretic limit of the sequence of sets {Bn} exists as n . The properties of con

vergence of the sequences in the domain and the image sets of the non-self-mapping, as well as the existence and uniqueness of the best proximity points, are also investigated for the dierent constraints and the given extension. Application examples are given related to the exact and approximate solutions of compatible and incompatible linear algebraic systems and with the parametrical identication of dynamic systems [].

1.1 Notation

{Tn} {T} denotes uniform convergence to a limit T of the sequence {Tn} of, in general,

non-self-mappings Tn from A to B; n Z+.

Z+ and Z+ are, respectively, the sets of non-negative and positive integer numbers and R+ and R+ are, respectively, the sets of non-negative and positive real numbers.

The notation {xn} stands for a sequence with n running on Z+ simplifying the more

involved usual notation {xn}nZ+. A subsequence for indexing subscripts larger than (re

spectively, larger than or equal to) n is denoted as {xn}n>n (respectively, as {xn}nn).

The symbols , , stand, respectively, for logic negation, disjunction, and conjunction.

2 Proximal and asymptotic proximal contractions of uniformly converging non-self-mappings

Let us establish two denitions of usefulness for the main results of this section.

Denition . Let (A, B) be a pair of non-empty subsets of a metric space (X, d). A mapping T : A B is said to be:

() A proximal contraction if there exists a non-negative real number < such that, for all u, u, x, x A, one has

d(ui, Txi) = D (i = , ) d(u, u) d(x, x),

where D = d(A, B) = infxA,yB d(x, y).() An asymptotic proximal contraction if there exists a sequence of non-negative real numbers {n}, with n < ; n Z+, with n ( [, )) as n such that, for

all sequences {un}, {un}, {xn}, {xn} A,

d(un+, Txn) = D (n Z+) d(un+, un+) nd(xn+, xn); n Z+.

If x, u A satisfy d(u, Tx) = D then u A and Tx B are a pair of best proximity points

of T in A and B, respectively. Note that if T : A B is an asymptotic proximal contraction

and the sequence {xn} A is such that (xn+, Txn) A B; n Z+ is a best proximity

pair, then there is a subsequence {xn}nn of {xn} such that the relation d(xn+, xn+)

nd(xn+, xn) d(xn+, xn); n n, holds for some real constant

[, ).

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Some asymptotic properties of the distances between the sequences of domains and images of the sequences of non-self-mappings Tn : An Bn; n Z+, which converge

uniformly to a limit non-self-mapping T : jZ+ Aj

jZ+ Bj, are given and proved related to the distance between the domain and image of the limit non-self-mapping.

Lemma . Let (X, d) be a metric space endowed with a homogeneous translation-invariant metric d : X X R+. Consider also a proximal non-self-mapping and a se

quence of proximal non-self-mappings T and {Tn} dened, respectively, by T : jZ+ Aj

jZ+ Bj, having non-empty images of its restrictions T : jZ+ Aj|An jZ+ Bj|Bn;

n Z+, and Tn : An Bn; n Z+, where An ( =

) An and An are non-empty sub

sets of X subject to T(An) Bn and Tn(An) Bn; n Z+ such that the sets of best

proximity points:

An =

x An : d(x, y) = Dn for some y Bn

,

Bn =

y Bn : d(x, y) = Dn for some x An

,

An =

x An : d(x, y) = Dn for some y Bn

,

[bracerightbig]

are non-empty, where Dn = d(An, Bn) and Dn = d(An, Bn); n Z+. Let {xn} and {yn}

be proximal sequences built in such a way that x A, y A, d(xn+, Txn) = Dn and d(yn+, Tnyn) = Dn; n Z+ and dene also the error sequence {xn} by xn = yn xn; n

Z+. Then the following properties hold:(i)

lim sup

n[parenleftbig][vextendsingle][vextendsingle]D

n+ Dn+

Bn =

y Bn : d(x, y) = Dn for some x An

[vextendsingle][vextendsingle]

gn+

[parenrightbig]

lim sup

n

[parenleftbig][vextendsingle][vextendsingle]D

n+ Dn+

[vextendsingle][vextendsingle]

gn+

[parenrightbig]

,

gn+ Dn+ + Dn+; n Z+, lim sup

n

gn+ Dn+ Dn+

[parenrightbig] ,

where

gn+ = d(xn+ Txn, yn+ Tnyn); n Z+,

gn+ = d(Txn, Tyn) + d(Tyn, Tnyn) + d(yn+, xn+); n Z+.

(ii) If {gn} g then lim supn(|Dn Dn| g) .

(iii) If {Tn} {T} then lim supn(|Dn Dn| d(Txn, Tyn) d(yn+, xn+)) .

(iv) If {Tn} {T}, {xn} = {xn yn} and T :

jZ+ Aj

jZ+ Bj is uniformly

Lipschitzian then limn |Dn Dn| = .

(v) If {gn} at exponential rate [, ), such that gn Cng for some real

constant C , then

lim sup

n

n

j=

[vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

Cg ,

Cmg

; m Z+,

lim

n,m

lim sup

m

n

j=m

[vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

n

j=m

[vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

= ; n Z+.

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(vi) Assume that {gn} is non-increasing and it converges linearly to g at rate (, ).

Then

lim sup

n

n

j=

[parenleftbig][vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

n

[parenrightbig]

g

lim sup

n

n

j=

[parenleftbig][vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

g

j

[parenrightbig][parenrightbig]

g

lim sup

n

n

j=

[parenleftbig][vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

gn

[parenrightbig]

.

(vii) Assume that {gn} is non-increasing and converges linearly to g at rate > with

order q . Then

lim sup

n

n

j=

[parenleftbig][vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

(gn g)q

[parenrightbig]

g

lim sup

n

n

j=

[parenleftbig][vextendsingle][vextendsingle]D

j Dj

[vextendsingle][vextendsingle]

gn

[parenrightbig]

.

Proof Note that, since d : X X R+ is a homogeneous translation-invariant metric,

x A, y A, d(xn+, Txn) = Dn and d(yn+, Tnyn) = Dn; n Z+, one has via induction

by using the constraints that xn An ( An), yn An ( An); n Z+ according to

d(yn+, Tnyn) = Dn+ = d(xn+ + xn+, Tnyn)

d(xn+, Txn) + d(Txn xn+, Tnyn yn+) = Dn+ + gn+

Dn+ + d(Txn, Tyn) + d(Tyn, Tnyn) + d(Tnyn, Tnyn + xn+ yn+)= Dn+ + gn+; n Z+, (.)

and also one gets in a similar way

d(xn+, Txn) = Dn+ = d(yn+ xn+, Txn)

d(yn+, Tnyn) + d(Tnyn, Txn + yn+ xn+) = Dn+ + gn+

Dn+ + d(Tyn, Tnyn) + d(Txn, Tyn) + d(Txn, Txn + yn+ xn+)

Dn+ + gn+; n Z+. (.)

Furthermore,

gn+ = d(xn+ Txn, yn+ Tnyn) d(xn+, Txn) + d(Tnyn, yn+)

= Dn+ + Dn+; n Z+ (.)

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and properties (i)-(ii) follow directly. Also, if {Tn} {T} then {d(Tyn, Tnyn)} and then

property (iii) is proved from property (i) since

gn+ = d(xn+ Txn, yn+ Tyn)

d(xn+, yn+) d(yn+, yn+ Tyn + Txn)= d(xn+, yn+) d(Tyn, Txn); (.)

we have

lim sup

n[parenleftbig][vextendsingle][vextendsingle]D

n+ Dn+

[vextendsingle][vextendsingle]

d(xn+, yn+) d(Tyn, Txn)

[parenrightbig]

lim sup

n

[parenleftbig][vextendsingle][vextendsingle]D

n+ Dn+

[vextendsingle][vextendsingle]

gn+

[parenrightbig]

. (.)

If, furthermore, {xn yn} and T :

jZ+ Bj is uniformly Lipschitzian in its denition domain then there is a positive real constant kT such that d(Txn, Tyn)

kTd(xn, yn); and then property (iv) follows from property (iii) since

lim

n[vextendsingle][vextendsingle]D

n Dn

jZ+ Aj

[vextendsingle][vextendsingle]

= lim sup

n

[parenleftbig][vextendsingle][vextendsingle]D

n Dn

[vextendsingle][vextendsingle]

( + kT) max

d(xn, yn), d(yn+, xn+)

[parenrightbig][parenrightbig]= . (.)

Property (v) follows from property (i) by using

n

j=m

[vextendsingle][vextendsingle]D

j Dj

n

j=mj

Cmg . (.)

Properties (vi)-(vii) follow from property (i) with

lim

n

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

gn+ g

(gn g)q

[vextendsingle][vextendsingle]

j=mgj Cg

n

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

(.)

with q = and (, ) for property (vi) since {gn g} is non-negative and converges to

zero which leads for both (vi)-(vii) to

lim sup

n

gn+ (gn g)q g

= lim

n

gn+ g

(gn g)q =

[parenrightbig]

, (.)

and also one gets for q = and (, )

lim sup

n

gn+ ng ( )g

[parenrightbig]

j[parenrightbig] n+ g[parenrightBigg] , (.)

which together with (i) yields (vii).

It turns out that Lemma . is extendable to the condition {gn} g with the replace

ments gn gn and g g. Some results on boundedness of distances from points of the

domains and their images of Tn : An Bn; n Z+, and T :

jZ+ Aj

jZ+ Bj and their asymptotic closeness to the set distance are given in the subsequent result.

n

lim sup

n

[parenleftBigg]

j=gj g

n

j=

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Lemma . Let (X, d) be a metric space. Consider two sequences {xn} and {yn} built, re

spectively, under the proximal non-self-mapping {T} and under the sequence of proximal

non-self-mappings {Tn} of Lemma . and assume that {Tn} {T}. Then, for any given

R+, there is N = N() Z+ such that the following properties hold:

(i) d(xn+, Tnxn) < Dn + , d(yn+, Tyn) < Dn + ; n ( Z+) > N.

(ii) If, furthermore,

d(Tyn, Txn) K max

nmnjn d(yj, xj) + M; n Z+ (.)

for some non-negative sequences of integers {mn} with mn n; n Z+ and some

real constants K [, ) and M R+, then

lim sup

n

lim sup

n

where D = D(N) = maxN<j<(Dj + Dj), and M = M (N) = maxjN d(xj, yj) for any given arbitrary nite N Z+.

Proof Assume that the rst assertion fails. Then R+ such that d(xn+, Tnxn) Dn + ; n Z+. As a result, since d(xn+, Txn) = Dn by construction and {Tn} {T}, one gets

Dn + d(xn+, Tnxn) d(xn+, Txn) + d(Txn, Tnxn)

= Dn + d(Txn, Tnxn) < Dn + /; (.)

n ( Z+) > N and some N Z+, a contradiction. Then the rst assertion is true.

Now, assume that the second assertion fails. Then R+ such that d(yn+, Tyn) Dn +

; n Z+. As a result, since d(yn+, Tnyn) = Dn by construction and {Tn} {T}, one gets

Dn + d(yn+, Tyn) d(yn+, Tnyn) + d(Tnyn, Tyn)

= Dn + d(Tyn, Tnyn) < Dn + /; (.)

n ( Z+) > N and some N Z+, a contradiction. Then the second assertion is also true

and property (i) has been proved. To prove (ii) note that, since {Tn} {T}, for any given R+, there are Ni = Ni() Z+ for i = , such that

d(xn+, yn+) Dn + d(Txn, Tnxn) + d(Tnyn, Tnxn) + Dn+

< Dn+ + Dn+ + / + d(Tnxn, Tnyn)

< Dn+ + Dn+ + + d(Txn, Tyn)

Dn+ + Dn+ + + K max

mnjn d(xj, yj) + M

Dn+ + Dn+ + + K max

mnjn+ d(xj, yj) + M

D + M + + K max

mnjn+ d(xj, yj)

max

N<jn+ d(xj, yj)

D + M + KM

K ,

d(Txn, Tnyn)

K

( K)D + M + KM

,

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D + M + + K

[parenleftBig]

max

mn<jn+ d(xj, yj)

N<jn+ d(xj, yj)[parenrightBig]

+ K max

N<jn+ d(xj, yj)

max

M + maxN<jn+ d(xj, yj)[bracketrightBig]; n ( Z+) > N (.)

for some non-negative sequences of integers {mn} with mn n; n Z+, where d(Txn,

Tnxn) < /; n ( Z+) > N, d(Tyn, Tnyn) < /; n ( Z+) > N, N = max(N, N), D =

D(N) = maxN<j<(Dj +Dj), and M is a non-negative real constant, which is not dependent on n, dened by M = M (N) = maxjN d(xj, yj). Since K [, ), one gets

max

N<jn+ d(xj, yj) <

D + M + + KM

K ; n ( Z+) > N, (.)

lim sup

n

= D + M + + K

max

N<jn+ d(xj, yj)

D + M + KM

K . (.)

On the other hand, note that

d(Txn, Tnyn) d(Txn, yn+) + d(yn+, Tnyn)

d(xn+, yn+) + d(xn+, Txn) + Dn = Dn + Dn + d(xn+, yn+) (.)

<

K

( K)D + M + + KM

; n ( Z+) > N, (.)

lim sup

n

d(Txn, Tnyn)

K

( K)D + M + KM

, (.)

and property (ii) has been proved.

By interchanging the positions of d(xn+, yn+) and d(Txn, Tnyn) in the triangle inequality of (.), it follows that

[vextendsingle][vextendsingle]d(Tx

n, Tnyn) d(yn+, xn+)

[vextendsingle][vextendsingle]

Dn + Dn; n Z+, (.)

so that if either

n= An =

n=(An An), since An An; n Z+ is bounded, or

n=(Bn Bn) is bounded, both sequences {d(Txn, Tnyn)} and {d(xn, yn)} are bounded. On the other hand, note that the non-negative sequence of integers {mn} might imply the

use of an innite memory in the upper-bounding term of (.) if mn a Z+; n Z+ or a nite memory of such a bound if m mn m; n Z+.

Denition . [, ] Let (A, B) be a pair of non-empty subsets of a complete metric space (X, d). The set A is said to be approximatively compact with respect to the set B if every sequence {xn} A such that d(y, xn) d(y, A) for some y B has a convergent subsequence.

Theorem . Let a proximal contraction and a sequence of proximal mappings {T} and {Tn} be dened, respectively, by T :

nZ+ An

nZ+ Bn having non-empty images of

its restrictions T :

jZ+ Aj|An

jZ+ Bj|Bn; n Z+ and Tn : An Bn; n Z+,

where An ( =) An and An are subsets of X, where (X, d) is a metric space, sub-

ject to T(An) Bn and Tn(An) Bn; n Z+ and the set-theoretic limits limn An,

limn An, limn Bn and limn Bn of the sequences of the sets {An}, {An}, {Bn}, {Bn},

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respectively, exist and are non-empty being dened in the usual way for any sequence {Zn} of subsets Zn X as:

lim

n Zn = [braceleftBig]

z X : (zn = if z Zn) (zn = if z /

[parenleftBig]

n zn = [parenrightBig][bracerightBig]

via the binary set indicator sequences {zn}, satisfy the improper set inclusion condition

limn An limn An. Assume that the sets of best proximity points An, Bn, An, and

Bn are non-empty, where Dn = d(An, Bn) = d(An, Bn) and Dn = d(An, Bn) = d(An, Bn); n Z+. Then the following properties hold:

(i) lim infn T(An) lim supn T(An) limn Bn.

The set-theoretic limits (limn An) and T(limn An) exist and they are non-empty and closed if the subsets An of X are all closed and

n= An is closed

Xn)

lim

while it satises the following set inclusion constraints:

T

[parenleftBig]

n An[parenrightBig]

lim sup

n

lim

T(An) lim sup

n

T(An) lim

n Bn.

If, furthermore, limn T(An) exists then it is non-empty and then

T [parenleftBig]

lim

n An[parenrightBig]

= lim

n T(An)

n Bn.

The set-theoretic limits (limn An) and T(limn An) exist and they are non-empty and closed if the subsets An of X; n Z+ are all closed and

n= An is

lim

closed, and then

T

[parenleftBig]

n An[parenrightBig]

lim sup

n

lim

T(An) lim sup

n

T(An) lim

n Bn.

If, furthermore, limn T(An) exists then it is non-empty and

T [parenleftBig]

lim

n An[parenrightBig]

= lim

n T(An)

lim

n Bn.

(ii) Assume that T :

jZ+ Aj

jZ+ Bj is a proximal contraction and that {Tn} {T}, where Tn : An Bn; n Z+. Then Tn : An Bn; n Z+, is a

sequence of asymptotic proximal contractions.(iii) If the sets in sequence {An}nm, for some m Z+, are closed then limn An is

closed and also

lim

n d(xn+, Txn) = D,

lim

n d(xn+, xn) = ,

{xn} x

(.)

limn An[parenrightBig] ,

lim

n d(yn+, Tyn) = D,

lim

n d(yn+, yn) = ,

{yn} y

(.)

limn An[parenrightBig] ,

[vextendsingle][vextendsingle]D

D

[vextendsingle][vextendsingle]

d(x, y) + lim inf

n d(Txn, Tyn)

d(x, y) + min

d

(Tx)+, (Ty)+

, d

(Tx), (Ty) [parenrightbig][parenrightbig]

(.)

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for any sequences {xn} An, {yn} An satisfying d(xn+, Txn) = Dn+; n Z+ and

d(yn+, Tyn) = Dn+; n ( n) Z+ for any sequences {xn} An, {yn} An being

built in such a way that d(xn+, Txn) = Dn+; n Z+ and d(yn+, Tyn) = Dn+; n ( n) Z+.

(iv) If (X, d) is complete and limn Bn is approximatively compact with respect to

limn An then there is a convergent subsequence {Txnk} ( limn Bn) Tx

where x limn An is the limit of {xn}. If limn Bn is approximatively compact

with respect to limn An then there is a convergent subsequence{Tynk} ( limn Bn) Ty where y limn An is the limit of {yn}. If D = D,

where Dn D and Dn D as n , and the above two approximative

compactness conditions hold and, furthermore, An and An are closed; n Z+ and

n= An and

n= An (it suces that

n=m An be closed for some m Z+) are closed then x = y, which is then the unique best proximity point of T in

the limit set limn An.

Proof Note that since limn An and limn An exist, we have

lim

n An =

lim inf

n An =

n=m An and

n=

[parenleftBigg]

m=n Am

[parenrightBigg] =

lim sup

n

An =

n=

[parenleftBigg]

m=n Am

,

lim inf

n An =

n=

[parenleftBigg]

lim

n An =

m=n Am

[parenrightBigg] =

lim sup

n

An =

n=

[parenleftBigg]

m=n Am

.

Note also that, since limn Bn exists, we have lim infn T(An)lim supn T(An)

limn Bn since

lim inf

n T(An) =

n=

[parenleftBigg]

m=n T(Am)

[parenrightBigg]

n=

[parenleftBigg]

m=n Bm

[parenrightBigg] =

lim inf

n Bn =

lim

n Bn,

lim sup

n

T(An) =

n=

[parenleftBigg]

m=n T(Am)

[parenrightBigg]

n=

[parenleftBigg]

n Bn.

Also, since limn Bn exists, we have lim infn T(An) lim sup T(An) limn Bn,

which follows under a similar reasoning. On the other hand, the existence and nonemptiness of limn An, by hypothesis, implies that T(limn An) exists since An

dom T; n Z+ so that

T

[parenleftBig]

lim

m=n Bm

[parenrightBigg] =

lim sup

n

Bn = lim

n An[parenrightBig]

= T

[parenleftBigg]

n=

[parenleftBigg]

m=n Am

[parenrightBigg][parenrightBigg]

n=

[parenleftBigg]

m=n T(Am)

[parenrightBigg] =

lim sup

n

T(An) lim

n Bn,

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T

[parenleftBig]

n An[parenrightBig]

= T

lim

[parenleftBigg]

n=

[parenleftBigg]

m=n Am

[parenrightBigg][parenrightBigg] =

n=

T

[parenleftBigg]

m=n Am

[parenrightBigg]

n=

[parenleftBigg]

m=n T(Am)

[parenrightBigg]

n Bn.

From the above set inclusion conditions it also holds that

T [parenleftBig]

lim

= lim inf

n T(An)

lim

lim

n Bn

if limn T(An) exists. If the subsets An; n Z+ are all closed and

n= An is

n An[parenrightBig]

= lim

n T(An)

closed then

n=k An is closed for any k Z+. In order to prove that limn An =

n=(

m=n Am) is closed, that is, an innite intersection of unions of innitely many closed sets is closed, it is rst proved that all those unions are closed under the given assumption that

n= An is closed. Assume this not to be the case, so that there is k Z+ such that

n= An = (

kn= An) (

n=k An) is closed with

n=k An not being closed

and such that

kn= An is closed, since it is the union of a nite number of closed sets. Then either

n=k An = A,k (

n=k An) is not closed or it is closed. In the second

case,

n=k An A,k since if

n=k An A,k is not closed then

n=k An cannot be

closed by construction. But, if

n=k An A,k then

n=k An is closed, which contradicts that it is not closed, since A,k is closed for any k Z+. As a result

n=k An being

no closed for any k Z+ implies that

n=k An is not closed. By complete induction, it

follows that

n=j An is not closed for any j ( Z+) k. Thus,

n= An is not closed what

contradicts that it is closed. As a result, since

n= An is non-empty and closed, any in-

nite union of non-empty closed sets

n=k An is also non-empty and closed for any k Z+

since

n= An is assumed to be closed by hypothesis and it is trivially non-empty. Then limn An =

n=(

m=n Am) is non-empty and closed since it is the innite intersection of innitely many unions of non-empty closed sets (but already proved to be non-empty and closed) and there exists the set limit T(limn An) = T(

n=(

m=n Am)) which is now proved to be non-empty. Proceed by contradiction by assuming that it is empty.

Then there is some x limn An such that x /

dom T. But then, since x limn An,

limn An =

m=n An) is non-empty and closed and all the sets in the sequence

{An} are closed we have x An for some n Z+ so that x dom T from the deni

tion of the non-self-mapping T which contradicts the existence of x limn An such that x /

dom T. It has been proved that limn An is closed and T(limn An) exists

and it is non-empty and closed. The proof that T(limn An) limn Bn is similar

to the above one. Now, one proves by contradiction that T(limn An) is non-empty if

limn An is non-empty. The limit set limn An is non-empty since the subsets An of X; n Z+ are all closed and

n= An is closed under similar arguments that those used above to prove those properties for limn An. Assume that T(limn An) is empty.

Since limn An is non-empty, there is x limn An such that x /

dom T. Since the sets

in the sequence {An} are closed, x An and x An, since An An, for some n Z+.

Thus, x dom T, from the denition of T and Tx T(An) T(An) Bn Bn since T(An) T(An)Bn Bn Bn Bn. Then T(limn An) is non-empty since limn An

n=(

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is non-empty. On the other hand, since limn An limn An we have

T [parenleftBig]

lim

n An[parenrightBig]

= T

[parenleftBigg]

[parenleftBigg]

[parenrightBigg][parenrightBigg] T[parenleftBigg]

n=

[parenleftBigg]

m=n Am

[parenrightBigg][parenrightBigg]

[parenleftBigg]

n=

m=n T(Am)

[parenrightBigg] =

n=

[parenleftBigg]

lim

m=n Bm

[parenrightBigg] =

n Bn,

so that

T

[parenleftBig]

n An[parenrightBig]

lim

n Bn

lim

lim

n Bn;

n T(An) T[parenleftBig]

lim

lim

n An[parenrightBig]

lim

n Bn

n Bn.

In the same way, it follows that T(limn An) exists and if, in addition, limn T(An) exists then

lim

n T(An) = T[parenleftBig]

lim

lim

n An[parenrightBig]

lim

n Bn.

Hence, property (i) has been proved. Now, build sequences {xn} and {x n}, and {yn} and {y n} in X sequences built in such a way that x, x A, y, y A, d(xn+, Txn) = Dn,

d(x n+, Tx n) = Dn+d(yn+, Tnyn) = Dn+ and d(y n+, Ty n) = Dn; n Z+. It follows induc

tively by using those distance constraints that xn, x n An ( An), yn, y n An ( An); n Z+. Note that limn An limn An implies the following inclusion of limit sets

limn An limn An of the sets of best proximity points and the four above limit sets

are trivially non-empty. Since T :

jZ+ Aj

jZ+ Bj is a proximal contraction:

d(xn+, Txn) = Dn+; n Z+

[parenrightbig]

d(xn+, xn+) Kd(xn, xn+); n Z+[parenrightbig](.)

for {xn} An and some real constant K [, ) and Dn D as n , since the limit

set limn An exists. On the other hand, if {Tn} is a sequence of asymptotic proximal

contractions Tn : An Bn; n Z+ then there is a real sequence {Kn} with a subsequence {Kn}nn [, K ) [, ) such that

d(yn+, Tnyn) = Dn+; n ( n) Z+

[parenrightbig]

d(yn+, yn+) K d(yn, yn+); n ( n)

[parenrightbig]

(.)

for {yn}nn An and some n Z+ and some real constant K [, ). Recall the following

properties for logical assertions then to be used related to (.). Consider the logical propositions Pi; i = , . Then

(P P) (P P), (.a)

(P P)

[parenrightbig]

(P P). (.b)

The condition (.) corresponds to (.a) with P (d(yn+, Tyn) = Dn+; n Z+) and

P (d(yn+, yn+) K d(yn+, yn+); n Z+). It is now proved by contradiction that {Tn}

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is a sequence of asymptotic proximal contractions, that is, by assuming that (.) is false so that its logical negation

[bracketleftbig][parenleftbig]d(y

(.)

for {yn}nn An and some n Z+, obtained from (.a)-(.b) versus to (.a), is

true. Then it follows from (.) that for any given arbitrary R+, there are n, n =

n() Z+ such that d(Tyn+, Tnyn+) < ; n n (since {Tn} {T}) and (.) hold.

Then

|Dn+ Dn+| < d(yn+, Tnyn+) d(yn+, yn+) + d(Tnyn+, Tyn+)

< K d(yn+, yn) + d(Tyn+, Tyn) +

K nnnd(yn+n+, yn+n) + d(Tyn+, Tyn) + ; n n + n. (.)

Since R+ is arbitrary, K [, ), and Dn D as n since the limit set limn An

exists and D D, since limn An limn An, one concludes from (.) that

lim

n d(yn+, yn+) = , {yn} y [parenleftBig]

n An[parenrightBig]

nZ+ Bn and limn An limn An, so that y limn An,

nm An is closed by hypothesis for some m Z+ and limn An is also closed. Then limn An dom T so that the possibility of Ty being undened is excluded and

then Ty is dened provided that there is no nite or innite jump discontinuities at y. If T is continuous at y, then (.) leads to the contradiction d(Ty, Ty) > . Otherwise, if T has a (nite or innity) jump discontinuity at y, with left and right limits (Ty) and (Ty)+ ( = (Ty)), then min(d((Ty), (Ty)), d((Ty)+, (Ty)+)) > , again a contradiction is got. Thus,

(.) is false so that its negation (.) is true. Then {Tn} is a sequence of asymptotic

proximal contractions which converge uniformly to the proximal contraction T. Hence, property (ii) has been proved.

Property (iii) is proved by taking into account also (.) and the constraints limn An limn An and limn An limn An limn An together with the

fact that, since

nm An is closed, limn An is closed. Note that the conditions of An, An, n Z+ being closed and

n= An and

n= An being closed can be relaxed in property (i) to closeness of the sets An, An, n ( m) Z+ and

n=m An and

n=m An for some m Z+ being closed while keeping the corresponding result. Assume this not to

be the case. Now, take a subsequence {Ank} of (non-empty closed) subsets of X such

that there is x X such that x fr(limn An), x /

limn An, since limn An is not

closed, and x Ank , the indicator variable of x in any subset in the sequence {Ank} is

xnk = then x limk Ank , since limk xnk = , and x /

limn An which contra

dicts limn An = limk Ank . Thus, limn An is non-empty and closed. Then one obtains (.)-(.).

On the other hand, since Dn D as n and limn Bn is approximatively com

pact with respect to limn An for some sequence {xn} limn An

d

Txn, x[parenrightbig] d

n Bn, x[parenrightBig]

n+, Tnyn) > Dn+

d(yn+, yn+) K d(yn, yn+)

lim

; n ( n) Z+

[bracketrightbig]

, lim inf

n d(Tyn+, Tyn) > . (.)

Since T :

nZ+ An

since

[parenleftBig]

lim

= D as n

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with x limn Bn. From property (i), T(limn An) limn Bn limn Bn

T(limn An) limn Bn, T(limn An) limn Bn. Then there is a subsequence {Txnk} ( {Txn}) zx cl limn Bn since limn Bn is closed and it is obvious that

zx cl limn Bn (= limn Bn) so that cl limn Bn =

. Assume that limn Bn is

limn Bn, a contradiction so that limn Bn

is non-empty and it is closed since it is on the boundary of limn Bn which is then nonempty and closed. But {xn} x ( limn An). Since {xn} is convergent all its subse

quences are convergent to the same limit so that {xnk} x. Thus, zx = Tx. Under a similar

reasoning, it can be proved under the second given approximative compactness condition that {Tynk} ( {Tyn}) zy = Ty (cl limn Bn) with limn Bn being non-empty and

closed. If the above both approximative compactness conditions jointly hold and D = D then one gets from the contractive condition for the proximal non-self-mapping T that for any given n Z+:

D = d xn+, Txn

[parenrightbig]

so that {xn} x and {yn} y implies {d(xn, yn)} d(x, y) so that ( K)d(x, y) holds

what implies x = y, which has to be necessarily unique from the identity, itself. Hence, property (iv) has been proved.

Remark . Note that the conditions of An, An, n Z+ being closed and the innite

countable unions

n= An and

n= An being closed can be relaxed in Theorem . (property (i)) to the closeness of the sets An, An, n ( m) Z+ and

n=m An and

The following remark is of interest; it concerns a condition of validity of the assumption that a countable union of closed sets is closed used in Theorem ..

Remark . It can be pointed out that a sucient condition for the innite countable union of closed subsets

n= An (respectively,

n= An) of a topological space (X, {Bi}) to be closed is that X has the local niteness property, that is, each point in X has a neighborhood which intersects only nitely many of the closed sets in {An} (respectively, in {An})

([], pp.-). This property can also be applied to a metric space (X, d) since metric spaces are specializations of topological spaces where the metric is used to dene the open balls of the topology. More general results guaranteeing that the innite countable union of closed sets is closed, and equivalently that the innite countable intersection of open sets is open, stand also for Alexandrov spaces (topological spaces under topologies which are uniquely determined by their specialization preorders) and for P-spaces (the intersection of countably many neighborhoods of each point of the space is also a neighborhood of such a point).

3 Weak proximal contractions of uniformly converging non-self-mappings

Let us establish two denitions of usefulness for the main results of this section.

Denition . Let (A, B) be a pair of non-empty subsets of a metric space (X, d). A mapping T : A B is said to be:

empty then the best proximity point zx /

= yn+, Tyn[parenrightbig] d

xn+, yn+

[parenrightbig] Kd

xn, yn [parenrightbig]

n=m An for

some m Z+ being closed while keeping the corresponding result.

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() A generalized asymptotic weak proximal contraction if there are sequences of non-negative real numbers {n}, with n [, ); n Z+ and n ( [, )) as

n ; and {n}, with n [, ); n Z+ and n ( [, )) as n such

that, for all sequences {u}, {u}, {x}, {x} A:

d(ui, Txi) = D (i = , ) [bracketrightbig]

d(x, Tx) ( + n + n)d(x, x) + D

[bracketrightbig]

d(u, u) nd(x, x) + n

; n Z+. (.)

() A generalized weak proximal contraction [, ], if (.) holds with non-negative real constants n = < and n = < ; n Z+.

() T : A (

Bn) is a strongly generalized asymptotic weak proximal contraction if:(a) A and Bn with Dn = d(A, Bn); n Z+ are non-empty and T(A) Bn;

n Z+.(b)

d(un, un) nd(xn, xn) + n

n < with = ( + )( + ) + < .

(c) The sequence of set distances {Dn} converges and {Dn Dn} , where

Dn d(xn, Txn) ( + n + n)d(xn, xn); n Z+. (.b)

Note if = in Denition .(), one has the subclass of weak proximal contractions. In this case, one gets by making xn+ = un; n Z+ that d(xn, xn+) nd(x, x);

n Z+, so that d(xn, xn+) as n , since < , provided that d(un, Txn) = D and

d(xn, Txn) ( + )d(xn, xn+) + d(un, Txn); n Z+ implying d(xn, Txn) D as n

and under the conditions that {xn} A and {Txn} B converge they should necessar

ily converge to best proximity points. Note that Denition .() relaxes Denition .() and Denition .() allows considering weak proximal contractions with sequences built from non-self-mappings which have iteration-dependent image sets.

The following results hold.

Proposition . Let A and Bn; n Z+ be non-empty subsets of a metric space (X, d).

Assume that T : A (

Bn), such that A is non-empty and T(A) Bn; n Z+, with Dn = d(A, Bn); n Z+ satises the contractive condition:

d(xn+, xn+) d(xn+, xn) +

Then the following properties hold:

d(Tx, x) D

; n Z+ (.a)

for any sequences {xin} A and {uin} Bn such that d(uin, Txin) = Dn (i = , ); n Z+ and {n}, {n} are real non-negative sequences which satisfy

= lim supn n < and

= lim sup

n

d(Txn, xn) Dn

d(Txn, xn+) Dn

; n Z+. (.)

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(i)

d(xn+, xn+) n+d(x, x) +

n

k= nk

Dk Dk

; n Z+, (.)

which is bounded for any nite x A if and

nk= nk(|Dk Dk|) < , where = ( + )( + ) + , and the sequence {Dn} satises (.b) which can be

relaxed to (.) below

Dn d(xn, Txn) ( + + )d(xn, xn+); n Z+. (.)

(ii) If (.) holds, subject to (.), with < then

d(xn, xn+) nd(x, x) +

n

sup

kn+

[vextendsingle][vextendsingle]D

k Dk

[vextendsingle][vextendsingle],

(.)

lim sup

n

d(xn, xn+)

[parenleftBig]

sup

kn

[vextendsingle][vextendsingle]D

k Dk

[vextendsingle][vextendsingle][parenrightBig]

, (.)

lim sup

n

[parenleftbig][vextendsingle][vextendsingle]d(x

n+, Txn) d(xn, Txn)

[vextendsingle][vextendsingle]

d(xn, xn+)

[parenrightbig]

, (.)

lim sup

n

[parenleftbigg][vextendsingle][vextendsingle]d(xn+,

Txn) d(xn, Txn)

[vextendsingle][vextendsingle]

sup

kn

[vextendsingle][vextendsingle]D

k Dk

[vextendsingle][vextendsingle]

, (.)

lim

n

[parenleftbig][vextendsingle][vextendsingle]d(x

n+, Txn) d(xn, Txn)

[vextendsingle][vextendsingle][parenrightbig]

lim

n d(xn, xn+) = ;

= (.)

if |Dn Dn| as n .

(iii) If (.) holds subject to (.) with < then

lim inf

n

min

d(xn, Txn) Dn, d(xn+, Txn) Dn+

[parenrightbig][bracketrightbig] , (.)

lim sup

n

d(xn, Txn) Dn ( + + )

[parenleftBig]

sup

kn

[vextendsingle][vextendsingle]D

k Dk

[vextendsingle][vextendsingle][parenrightBig]

, (.)

lim sup

n

d(xn+, Txn) Dn ( + + )

[parenleftBig]

sup

kn

[vextendsingle][vextendsingle]D

k Dk

[vextendsingle][vextendsingle][parenrightBig]

. (.)

(iv) If < and |Dn Dn| as n then the limits below exist and are identical:

lim

n

d(xn, Txn) Dn[parenrightbig]= lim n

d(xn, Txn) Dn[parenrightbig]= lim n

d(xn, Txn) Dn [parenrightbig]

= lim

n

. (.)

If (.) holds, subject to (.), with < , {Dn} D and {Dn} D then

limn d(xn, Txn) = limn d(xn+, Txn) = D.

Proof Since d(A, Bn) = Dn we have d(Txn, xn+) Dn; n Z+. Also, if follows from (.)

and (.) with = ( + )( + ) + that (.) holds since

d(xn+, xn+) d(xn+, xn) +

d(Txn, xn) + d(xn, xn+) Dn

d(xn+, Txn) Dn

[parenrightbig]

= ( + )d(xn+, xn) +

d(Txn, xn) Dn

[parenrightbig]

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Dn Dn

n

nk= nk(|Dk Dk|) < then the sequence {d(xn, xn+)} is bounded. Thus, property (i) has been proved. The relations (.) and (.) of property (ii)

follow directly from (.) of property (i) if < . On the other hand, the triangle inequality and (.) lead to (.) since

[vextendsingle][vextendsingle]d(x

for any given x A. If and

n+, Txn) d(xn, Txn)

. Since is

arbitrary, the limit limn d(xn, xn+) exists and limn d(xn, xn+) = . This property and (.) yield directly limn(|d(xn+, Txn) d(xn, Txn)|) = and then property (ii) has been

fully proved. To prove property (iii), note that (.) holds directly from d(A, Bn) = Dn;

n Z+ and {xn} A. Also, (.) leads to (.) by taking into account (.). The relation

(.) follows from (.), (.), and the relation

d(xn+, Txn) d(xn, Txn) + d(xn+, xn); n Z+.

Hence, property (iii) has been proved. Property (iv) is a direct consequence of property (iii) for the case when d(xn, xn+) and |Dn Dn| as n including its particular

sub-case when {Dn} D and {Dn} D.

Note that Proposition . is applicable to the strongly generalized asymptotic weak proximal contraction of Denition .() which do not need the fullment of the implying part of the logic proposition of Denition .()-() but the distances of sequences of sets satisfy (.b) or, at least, (.). The subsequent result is concerned with the existence and uniqueness of best proximity points if, in addition to the assumptions of Proposition ., the set-theoretic limit of the sequence {Bn} exists and is closed and approximatively com

pact with respect to A.

Theorem . Under all the assumptions of Proposition . and property (iii), equation (.), assume also that (X, d) is complete, that A and Bn; n Z+, are non-empty subsets

of X such that A is closed, A is non-empty, the set-theoretic limit B := limn Bn exists, is closed and approximatively compact with respect to A (or the weaker condition that A

is closed) and T(A) B. Assume also that the non-self-mapping restriction T : A|A

(

Bn)|B, for some subset A A, which contains the set of best proximity points A, is a strongly generalized asymptotic weak proximal contraction. Then T : A|A B has a

unique best proximity point if < .

Proof Since d(xn+, xn) as n , {xn} x from (.). Since {xn} A and A is

closed we have x A. Since Dn d(xn+, Txn) = {d(x, Txn)} ( D) since {Dn} D, because {Bn} B; and |Dn d(xn, Txn)| d(xn, xn+); n Z+. Since B is approximatively compact

d(xn+, xn) +

[parenrightbig]

n+d(x, x) +

k= nk

Dk Dk

; n Z+ (.)

[vextendsingle][vextendsingle]

d(xn, xn+).

The relation (.) follows from (.) and (.). To prove (.), note that if limn(Dn Dn) = , then for any given R+, there is m = m() Z+ such that supmkn+m+ |Dk

Dk| < for any n ( Z+) m and then, from (.), lim supn d(xn+, xn)

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with respect to A and {xn} A, there are y A and a sequence {xn} A, such that {xn

xn} , then {xn} x since {xn} x, and {Txn} B such that one gets as n :

d(xn, T xn) d(y, B) = D; d(xn, B) d(y, B) = D; d(xn, B) d(y, B) = D,

{d(xn, Txn)} D, {d(xn+, Txn)} D and {d(x, Txn)} D. Since B is approximatively

compact with respect to A and {xn} A, the sequence {Txn}

Bn, such that Txn Bn, has a convergent subsequence {Txnk} z B, since B is also closed and both {Txnk} and {Txnk} have the same limit z B. Also, z cl B and d(x, Txnk) D (= d(x, z)) as k so

that x is a best proximity point of T : A|A B. Note that since the limit set B exists, it is

by construction the innite union of intersections of the form B =

n=

Bn so that there is a restriction T : A|A ( Bn)|B for some A A which is non-empty. Assume not so that A = . If A = then A is also empty which is impossible then A =

. It is now proved by contradiction that the best proximity point is unique. Assume this not to be the case so that there are two best proximity points x, y such that there are two sequences {xn} x and {yn} y contained in A. Since T : A|A (

Bn)|B is a strongly generalized asymptotic weak proximal contraction, one gets from the implied logic proposition of (.) with u = x and v = y and Dn = D that

( )d(x, y)

Bn) is a weakly generalized asymptotic weak proximal contraction, such that A is non-empty and T(A) Bn; n Z+, with Dn = d(A, Bn)

and T(A) Bn =

; n Z+. Assume also that the set limit B := limn Bn exists, is closed

and approximatively compact with respect to A, or instead, the weaker condition that A is closed. Then T : A (

Bn) has a unique best proximity point.

4 Examples

Two examples are described to the light of proximal contractions. The rst one is concerned with the solution of algebraic systems which can have more or less unknown than equations and which can be compatible or not. The second one is referred to an identication problem of a discrete dynamic system whose parameters are unknown and which can be subject to unmodeled dynamics and/or exogenous noise which makes not possible, in general, an exact identication.

Example . (Moore-Penrose pseudo-inverse) The problem of solving either exactly or approximately a linear system of algebraic equations is very important and it appears in many engineering and scientic applications. It is possible to focus it to the light of best proximity points of non-self-mappings as follows. Consider the linear algebraic system Cx = e where C Rnp (a real matrix of order n p) and e Rp. It is known from the

mn Bm so that

B B

d(Tx, y) D[parenrightbig] d(Tx, x) + d(x, y) D

= d(x, y),

which fails for < if x = y. Thus, x = y.

Closely to Theorem ., the following result can be proved.

Theorem . Let A and Bn; n Z+ be non-empty closed subsets of a complete metric

space (X, d). Assume that T : A (

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Rouch-Frobenius theorem from Linear Algebra that a solution x Rp exists if rank(C, e) =

rank C. The solution is unique given by x = Ce if p = n, and rank C = p with the algebraic

system being determined compatible. If rank(C, e) = rank C = q min(n, p) = n then there

are innitely many solutions and the algebraic system is indetermined compatible being, in particular, overdetermined if n > p and undetermined if n < p. If rank(C, e) > rank C then the algebraic system is incompatible. A more general setting is CX = E, where C Rnp,

E Rnq are given and X Rpq is a solution which exists if and only if rank C = rank(C

... E).

,

A is the set of solutions of the compatible indeterminate algebraic system.(b) If rank C = rank(C ... E) = p then the solution X = C+E is unique and A = A = {(C+E

... ) : Rn(max(p,q)q)} consist of one element which is the unique

solution.(c) If min(p, n) rank C < rank(C

... E) then C+CE = E and the algebraic system is

incompatible. By considering (X, d) as a Banach space endowed with the Euclidean norm, we can check for the best solution which minimizes CX E over X Rpq if it exists. Such a solution exists in a least-squares sense and it is unique ifn rank C = p < rank(C

... E), since CTC is non-singular, of order p, and

C+ = (CTC)CT, and X = (CTC)CTE minimizes CX E over the set of

matrices X Rpq so that D = d(A, B) = ((CTC)CT I)E , the sets of best

proximity points of A being

A = A =

since (I C+C) = and B is as the above one of case (a). A is the best solution of the incompatible algebraic system.

The pseudo-inverse can be calculated without inverting CTC by the iterative process:

C+n+ =

I C+nC C+n; n Z+; C+ such that C+C =

(so-called Ben-Israel-Cohen, or hyper-power sequence, iterative method []). It follows that C+n C+ as n since () C+ is unique; and () the iterative process satises the

pseudo-inverse properties C+ = C+CC+ and C+C = (C+C) under the replacement C+n

The following cases hold:(a) If rank C = rank(C ... E) = p, then C+CE = E, and there are innitely many solutions of

the form X = C+E + (I C+C)W with C+ being the Moore-Penrose pseudo-inverse of C. The domain of the non-self-mapping TC : A B, represented by the matrix

C, can be restricted to A = {X = C+E + (I C+C)W : W Rpq}. To close a proper

formalism we extend the matrices in A to matrices A Rn(max(p,q)q) and those in B to B Rn(max(p,q)p) by adding zero columns (if p = q either A = A or B = B) and we

consider them as subsets of X Rn(max(p,q)q) and consider the metric space (X, d)

with d being the Euclidean metric so that D = with the sets of best proximity points of A and B being:

A = A =

X = (X ... ) : X = C+E +

I C+C

;

W, W Rnq, Rn(max(p,q)q)

B = B =

(E ... ) : Rn(max(p,q)p)

X = (X ... ) : X =

CTC

CTE, Rn(max(p,q)q)

,

C+C

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C+; n Z+. By using the iterative process, we can also dene sequences of sets and

associate sequences of distances by:

Dn = d(An, B) =

[parenleftbig]C+

n I

E

D =

[parenleftbig]C+

I

E

as n ,

B = B =

(E ... ) : Rn(max(p,q)p)

,

An = An =

[braceleftbig][parenleftbig]C+

n E

...

[parenrightbig]

: Rn(max(p,q)q)

[bracerightbig]

A

= A =

[braceleftbig][parenleftbig]C+E

...

[parenrightbig]

: Rn(max(p,q)q)

[bracerightbig]

as n ;

B and An are unique, if the pseudo-inverse C+ = (CTC)CT exists for the given initial C+. However, if rank C = rank(C ... E) = p, so that E = C+nCnE (case (a) - incompatible algebraic

system) then if we use the iterative procedure:

C+n+ =

I C+nC C+n; n Z+; C+ such that C+C =

C+C

,

An = An(Wn)

= Xn = (Xn ... ) : Xn = C+nE +

I C+nCn

Wn, Wn Rnq, Rn(max(p,q)q) [bracerightbig]

is a non-unique set of arbitrary solutions of the incompatible algebraic system of which the best (error-norm minimizing) solution is the unique one described above. For initial conditions satisfying C+C = (C+C), for instance, C+ = (CC + I)C, R+, the

pseudo-inverse converges quadratically to its limit, that is, limn C

+ n+C+

C

+ n+C+

= > . Fur-

thermore,

Dn = xn+ Cxn =

x

n+ CC+nE

;

n Z+

and, since the convergence of the pseudo-inverse is quadratic, there is a bounded positive sequence {n} such that since {xn}, {yn} x = C+E, since the pseudo-inverse is

unique when it exists; we have

xn+ yn+ [parenleftbig]C+

n+(x) C+ + C+ C+n+(y)

b

n

[parenleftbig]C+

n (x) C+ + C+ C+n(y)

b

[parenrightbig][parenleftbig]C+

n (x) C+ + C+ C+n(y)

b

n

[parenleftbig]C+

n (x) C+n(y)

b

[parenrightbig][parenleftbig]x

n x yn + x

[parenrightbig]

; n ( n) Z+

for any given R+ any n ( n) Z+ and some n = n() Z+. Since the choice of is

arbitrary, the iterative process is an asymptotic proximal contraction since, for any given real (, ) there is n = n(, ) Z+ such that n ; n ( n) Z+.

Example . (Parametrical estimation of an uncertain discrete dynamic system) Consider d : X X R+ to be a homogeneous translation-invariant metric and two non

empty sequences of closed subsets {An} and {Bn} of X, with An Bn =

n

xn yn

with mu

tual distances Dn = d(An, Bn); n Z+ such that each proximal set An An of An to

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Bn is non-empty; n Z+. Now, built {xn} An being a nominal proximal sequence

to Bn constructed such that, given a proximal mapping T : jZ+ Aj

jZ+ Bj with

jZ+ Aj|An

jZ+ Bj|Bn; n Z+, that is

T(An) Bn for any n Z+ is non-empty, then d(xn+, Txn) = Dn; n Z+. We also con

sider a sequence of proximal mappings Tn :

jZ+ Aj

jZ+ Bj for sequences of closed

subsets {An} and {Bn} of X, with An Bn = with mutual distances Dn = d(An, Bn);

n Z+ such that each set of best proximity points An An An of An to Bn is non

empty; n Z+. Build a proximal sequence {yn} An to Bn constructed such that, given

a sequence {Tn} of proximal mappings Tn :

jZ+ Bj with non-empty images

jZ+ Aj|An

jZ+ Bj|Bn, then d(yn+, Tnyn) = Dn+; n Z+. Dene sequences of measured and operator errors as follows:

xn = yn xn;

Tn = Tn T; n Z+, (.)

which can be associated with a parametrical identication problem of a discrete dynamic system, where {xn} is the sequence of measured data, or measurable output, of the iden

tied system and {yn} is the corresponding data given by the identier, i.e. the sequence

of adjustable data, which is a recursive parameter estimator []. Then the following distance constraints for the best proximity points of the identied system and its associate estimator, operators, parameters, and sequences are relevant to the studied problem:

d(xn+, Txn) = Dn+, d(yn+, Tnyn) = Dn+, (.)

Tn = T +

Tn; yn+ = xn+ + xn+, xn+ =n+ yn+, (.)

n+ =

Tnxn =

Txn =

n} is an estimated se

quence of the parameter vector of the modeled part, dn is the internal noise, T is the auto-regression operator which provides the nominal output; Tn is the auto-regression operator which provides the nominal value of the output from the nominal part xn of the

whole process regressor xn, where xn = (xTn, x Tn)T, such that xn is the nominal regressor (that is, the regressor of the modeled dynamics), xn is the regressor associated with the unmodeled dynamics, and T is the auto-regressor operator which supplies the contribution of the unmodeled dynamics of the system to the measured output. Note that the above metric space can also be considered a normed space when endowed with the metric-induced norm since the metric is homogeneous and translation-invariant.

Remark . The subsequent equations describe the identication process when the identied process is linear in the parameters as well as the estimation parallel process. Thus, the measured output and estimated output sequences, respectively, yn,n are real scalar

products of a parameter vector, respectively, ,

n by their associated regressors, respectively, xn, x n which are associated with previous values which depend on the order of the modeled and unmodeled parts of system. All the parameters (parameter vector

non-empty images of its restrictions T :

jZ+ Aj

of its restrictions T :

[parenleftbig]

Tn, T

n , x Tn

T = Tnxn = Tnxn, (.)

[parenrightbig][parenleftbig]xT

yn+ =

T, T

[parenrightbig][parenleftbig]xT

n , x Tn

T = Txn + T x n + dn = Txn + dn; (.)

n Z+, where the superscript T stands for transposition, is the measured output of

the real process, and , are parameters of the modeled and unmodeled dynamics of the discrete dynamic system provided it is linear in the parameters; {

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of the modeled part) and (parameter vector of the unmodeled part) are assumed to be unknown in the most general framework, the rst one being estimated by the estimation algorithm. The dimension of , so that its estimation

n, are known while that of is unknown, in general. The updating process of the estimation is of the form n+ = n+( n, yn+,n+, xn); n Z+ and it is performed through an algorithm, like for in

stance, recursive or batch, least-squares type algorithms. The unmodeled parameters, i.e. the components of are not estimated. Thus, there are three sources of identication error in the process, that is, the error between the identied system and its identied counterpart (or parametrical identication algorithm), namely, (a) the fact that is unknown so that the nominal parametrical error

n (n Z+) is nonzero, in general;

(b) the eventual presence of output exogenous additive noise dn (n Z+); (c) the presence of unmodeled dynamics for which no parametrical estimation algorithm is included. The whole identication process can also be described through linear operators dened to the light of the formalism of Section . Such a description is also incorporated in the sequel under the form of equalities to the linear parametrical description. It can be pointed out that this alternative parallel description is also useful to describe nonlinear processes by the appropriate denitions of the operators describing the trajectory sequences as T, Tn,

and their auxiliary ones for the modeled and unmodeled dynamics.

The identied process and its estimation algorithm are described through the equations below in the particular case that the identied process and the estimator are in a parallel operation so that they share a common regressor xn:

xn+

xTn, x Tn

n =

T =

yn+, E

xn, x n

T

T

= Txn + T x n + dn, E

xn, x n

T

T Txn + dn, (.)

yn+ =

T

T =

n+, E

xn, x n

Tnxn, E

xn, x n

T

T Tnxn, (.)

xn+ =

n+ yn+, T

T =

Tnxn Tx n, T

T ; n = n (.a)

T (.b)

= Tnxn Txn dn = Tnxn dn, (.c) dn =

dn, T

T ; n = n ;

Tn = Tn T;

= ( Tn T)xn T x n dn, T

T =

[parenleftbig]T

nxn T x n dn, T

Tn = Tn T, (.)

Tnxn dn = (Tn T)xn =

( Tn T)xn T x n dn, T

T (.a)

[parenleftbig]

= Tnxn Tx n dn, T

T , (.b)

Tn+xn+ + dn+ = (Tn+ T)xn+ = (Tn+ T)(Txn + dn) = (Tn+ T)

[parenleftbig]

Txn + T x n + dn, E

xn, x n

T

T ; (.)

... ) is a rectangular matrix with a number of rows equalizing that of the columns less one which loses the last value of the regressor of the preceding sample. In real identication situations, we see that:(a) Tn, xn - are both known;(b) xn - the whole regressor, which includes the contribution of the unmodeled dynamics, is partially unknown, since the sub-vector x n of xn is of unknown dimension since the order of the unmodeled dynamics contribution is unknown;

n Z+ and E = (I

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(c) T, T, T - are unknown or not precisely known;(d) Tn - is known; dn - is unknown;(e) yn,n - are known, the rst one by direct measurement, and the second one since it

is generated by the estimation algorithm.

The following sequences are relevant for the identication problem xn = yn xn;

Tn =

Tn T; n Z+ so that we also dene

gn+ = d(,

Tnxn) + d(xn+, T xn) + d(,

Tnxn)

= d(,

Tnxn) + d(Tnxn, T xn + dn) + d(,

Tnxn) gn+

Tnxn); n Z+ (.)

so that, related to the results of Section , Lemma ., we can adjust the estimation algorithm sequences to the real data according to the following distance best proximity constraints:

d(yn+, Tnyn) = Dn+ = d(xn+ + xn+, Tnxn + Tnxn)

d(xn+, Tnxn) + d(Tnxn, Tnxn + Tnxn xn+)

d(xn+, Txn) + d(Txn, Txn +

Tnxn) + d(xn+, Tnxn)

d(xn+, Txn) + d(,

= d(,

Tnxn) + d(Tnxn, T xn) + d(T xn, T xn + dn) + d(,

Tnxn) + d(xn+, T xn) + d(T xn, T xn +

Tnxn)

Dn+ + d(,

Tnxn) + d(, Tnxn) + d(Tnxn, T xn) + d(T xn, T xn + dn)= Dn+ + gn+; n Z+, (.)

d(xn+, Txn) = Dn+ = d(yn+ xn+, Tyn T xn)

d(yn+, Tyn) + d(T xn, xn+)

Dn+ + d(

Tnxn, ) + d(

Tnxn, ) + d(T xn, Tnxn dn)

Dn+ + gn+; n Z+, (.)

then

max(Dn gn, ) Dn Dn + gn, max

Dn gn,

[parenrightbig]

Dn Dn + gn; n Z+.

(.)

If gn as n , i.e. if all the uncertainty sources of noise/unmodeled dynamics and

the identication error between the true process vanish asymptotically and the identier

{

Tn} uniformly, then { Tn} T uniformly, T = , then |Dn Dn| as n and {Dn}, {Dn} D = . For {Dn} D, it suces { dn } D from (.) with D = if and only

if {dn} , that is, if {dn} . However, if T is not the zero operator (in the linear case,

if the parameter vector of the unmodeled dynamics is nonzero), then even if { Tn} T

uniformly (perfect asymptotic identication of the modeled dynamics - in the linear case

{

n} , equivalently {

n} ) and { dn } converges, it can happen that {Dn} does not

converge in the sense that the best proximity points of the identier are not subject to a set

distance which converges. Typically, in the linear case with a scalar measurement, there is

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an identication error of the form

d(xn+, yn+) = xn+ yn+ =

(a) {dn} d.

(b) {T x n} x e, that is, the contribution of the unmodeled dynamics vanishes

asymptotically to an equilibrium. There are typically two situations for that to happen, namely, (b) T , that is, the process is perfectly modeled so that the

order of its whole dynamics is known. This occurs very seldom in controlled processes which have to work under very dierent or changing operation conditions along large time intervals, (b) the unmodeled dynamics process converges to an asymptotically stable partial equilibrium x e.

(c) { Tn} T uniformly under the updating estimation algorithmTn+ = Tn+( Tnxn, yn+(dn+)); n Z+ employed under a given initialization, i.e.,

there is a perfect asymptotic identication of the modeled dynamics achieved. Again, it is dicult in practice to achieve the perfect identication objective of the modeled dynamics, even if {dn} d and {T x n} x e unless d = and x e = .

Note that the property {

Tnxn

Tn} converges to a proximal contraction but the result is obtained that the equations of best proximity points (.) hold with asymptotic convergence and asymptotic convergence of the distance sequences {Dn}, {Dn} .

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the nal manuscript.

Author details

1Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), Barrio Sarriena, P.O. Box 644, Bilbao, Leioa 48940, Spain. 2Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA. 3Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia. 4Department of Telecommunications and Systems Engineering, Universitat Autnoma de Barcelona, Bellaterra, Barcelona, 08193, Spain.

Acknowledgements

Professor De la Sen and Professor Ibeas are grateful to the Spanish Government for its support of this research with Grant DPI2012-30651, and to the Basque Government for its support of this research trough Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its nancial support through Grant UFI 2011/07. The author Manuel De la Sen is very grateful to Dr. S Born, Dr. D Grubb, Dr. M Cichon, Dr. A Szaz, Dr. G Oman, Dr. U Mutze and Dr. P Bankston for their useful comments, opinions and mutual discussions concerning the local niteness property, Alexandrov spaces, P-spaces and other related points of view about the closeness of countable unions of closed sets. Finally, the authors thank the referees for their suggestions to improve the rst version of the manuscript.

Received: 4 May 2014 Accepted: 22 July 2014 Published: 18 August 2014

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doi:10.1186/1687-1812-2014-169Cite this article as: De la Sen et al.: Results on proximal and generalized weak proximal contractions including the case of iteration-dependent range sets. Fixed Point Theory and Applications 2014 2014:169.