Open Access

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

Fixed Point Theory and Applications20142014:169

https://doi.org/10.1186/1687-1812-2014-169

Received: 4 May 2014

Accepted: 22 July 2014

Published: 18 August 2014

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 T n : A n B n for n Z 0 + and T : j Z 0 + A 0 j j Z 0 + B 0 j , or T : A ( B n ) , subject to T ( A 0 n ) B 0 n and T n ( A n ) B n , such that T n converges uniformly to T, and the distances D n = d ( A n , B n ) are iteration-dependent, where A 0 n , A n , B 0 n and B n are non-empty subsets of X, for n Z 0 + , where ( X , d ) is a metric space, provided that the set-theoretic limit of the sequences of closed sets { A n } and { B n } exist as n 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.

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 fixed 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 [115]. The application of the theory of fixed points in stability issues has been proved to be a very useful tool. See, for instance, [1618] 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 { T n } { T } of the form T n : A n B n ; n Z 0 + , where A n and B n are in general distinct, with a set-theoretic limit of the form T : j Z 0 + A 0 j j Z 0 + B 0 j , provided that the set-theoretic limits of the involved set exist and that the infinite unions of the involved closed sets are also closed. Further related results are obtained for generalized weak proximal and proximal contractions in metric spaces [2, 3, 19], which are subject to certain parametrical 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 [13]. Some related generalizations are also given for non-self-mappings of the form T : A ( B n ) , subject to a set distance D n = d ( A , B n ) , where A and B n are non-empty and closed subsets of a metric space ( X , d ) for n Z 0 + , provided that the set-theoretic limit of the sequence of sets { B n } exists as n . The properties of 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 for the different 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 identification of dynamic systems [811].

1.1 Notation

{ T n } { T } denotes uniform convergence to a limit T of the sequence { T n } of, in general, non-self-mappings T n from A to B; n Z 0 + .

Z 0 + and Z + are, respectively, the sets of non-negative and positive integer numbers and R 0 + and R + are, respectively, the sets of non-negative and positive real numbers.

The notation { x n } stands for a sequence with n running on Z 0 + simplifying the more involved usual notation { x n } n Z 0 + . A subsequence for indexing subscripts larger than (respectively, larger than or equal to) n 0 is denoted as { x n } n > n 0 (respectively, as { x n } n n 0 ).

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 definitions of usefulness for the main results of this section.

Definition 2.1 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:
  1. (1)
    A proximal contraction if there exists a non-negative real number α < 1 such that, for all u 1 , u 2 , x 1 , x 2 A , one has
    d ( u i , T x i ) = D ( i = 1 , 2 ) d ( u 1 , u 2 ) α d ( x 1 , x 2 ) ,

    where D = d ( A , B ) = inf x A , y B d ( x , y ) .

     
  2. (2)
    An asymptotic proximal contraction if there exists a sequence of non-negative real numbers { α n } , with α n < 1 ; n Z 0 + , with α n α ( [ 0 , 1 ) ) as n such that, for all sequences { u 1 n } , { u 2 n } , { x 1 n } , { x 2 n } A ,
    d ( u n + 1 , T x n ) = D ( n Z 0 + ) d ( u n + 2 , u n + 1 ) α n d ( x n + 1 , x n ) ; n Z 0 + .
     

If x , u A satisfy d ( u , T x ) = D then u A and T x 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 { x n } A is such that ( x n + 1 , T x n ) A × B ; n Z 0 + is a best proximity pair, then there is a subsequence { x n } n n 0 of { x n } such that the relation d ( x n + 2 , x n + 1 ) α n d ( x n + 1 , x n ) α ¯ d ( x n + 1 , x n ) ; n n 0 , holds for some real constant α ¯ [ α , 1 ) .

Some asymptotic properties of the distances between the sequences of domains and images of the sequences of non-self-mappings T n : A n B n ; n Z 0 + , which converge uniformly to a limit non-self-mapping T : j Z 0 + A 0 j j Z 0 + B 0 j , are given and proved related to the distance between the domain and image of the limit non-self-mapping.

Lemma 2.2 Let ( X , d ) be a metric space endowed with a homogeneous translation-invariant metric d : X × X R 0 + . Consider also a proximal non-self-mapping and a sequence of proximal non-self-mappings T and { T n } defined, respectively, by T : j Z 0 + A 0 j j Z 0 + B 0 j , having non-empty images of its restrictions T : j Z 0 + A 0 j | A 0 n j Z 0 + B 0 j | B 0 n ; n Z 0 + , and T n : A n B n ; n Z 0 + , where A n ( ) A 0 n and A 0 n are non-empty subsets of X subject to T ( A 0 n ) B 0 n and T n ( A n ) B n ; n Z 0 + such that the sets of best proximity points:
A 0 n 0 = { x A 0 n : d ( x , y ) = D n 0  for some  y B 0 n } , B 0 n 0 = { y B 0 n : d ( x , y ) = D n 0  for some  x A 0 n } , A n 0 = { x A n : d ( x , y ) = D n  for some  y B n } , B n 0 = { y B n : d ( x , y ) = D n  for some  x A n }
are non-empty, where D n 0 = d ( A 0 n , B 0 n ) and D n = d ( A n , B n ) ; n Z 0 + . Let { x n } and { y n } be proximal sequences built in such a way that x 0 A 01 , y 0 A 1 , d ( x n + 1 , T x n ) = D n 0 and d ( y n + 1 , T n y n ) = D n ; n Z 0 + and define also the error sequence { x ˜ n } by x ˜ n = y n x n ; n Z 0 + . Then the following properties hold:
  1. (i)
    lim sup n ( | D n + 1 D n + 1 0 | g ¯ n + 1 ) lim sup n ( | D n + 1 D n + 1 0 | g n + 1 ) 0 , g n + 1 D n + 1 + D n + 1 0 ; n Z 0 + , lim sup n ( g n + 1 D n + 1 D n + 1 0 ) 0 ,
    where
    g n + 1 = d ( x n + 1 T x n , y n + 1 T n y n ) ; n Z 0 + , g ¯ n + 1 = d ( T x n , T y n ) + d ( T y n , T n y n ) + d ( y n + 1 , x n + 1 ) ; n Z 0 + .
     
  2. (ii)

    If { g n } g then lim sup n ( | D n D n 0 | g ) 0 .

     
  3. (iii)

    If { T n } { T } then lim sup n ( | D n D n 0 | d ( T x n , T y n ) d ( y n + 1 , x n + 1 ) ) 0 .

     
  4. (iv)

    If { T n } { T } , { x ˜ n } = { x n y n } 0 and T : j Z 0 + A 0 j j Z 0 + B 0 j is uniformly Lipschitzian then lim n | D n D n 0 | = 0 .

     
  5. (v)
    If { g n } 0 at exponential rate ρ [ 0 , 1 ) , such that g n C ρ n g 0 for some real constant C 1 , then
    lim sup n j = 0 n | D j D j 0 | C g 0 1 ρ , lim sup m j = m n | D j D j 0 | C ρ m g 0 1 ρ ; m Z 0 + , lim n , m j = m n | D j D j 0 | = 0 ; n Z 0 + .
     
  6. (vi)
    Assume that { g n } is non-increasing and it converges linearly to g at rate μ ( 0 , 1 ) . Then
    lim sup n j = 0 n ( | D j D j 0 | n ) g 0 1 μ lim sup n j = 0 n ( | D j D j 0 | g ( 1 μ j ) ) g 0 1 μ lim sup n j = 0 n ( | D j D j 0 | g n ) 0 .
     
  7. (vii)
    Assume that { g n } is non-increasing and converges linearly to g at rate μ > 0 with order q 1 . Then
    lim sup n j = 0 n ( | D j D j 0 | μ ( g n g ) q ) g lim sup n j = 0 n ( | D j D j 0 | g n ) 0 .
     
Proof Note that, since d : X × X R 0 + is a homogeneous translation-invariant metric, x 0 A 01 , y 0 A 1 , d ( x n + 1 , T x n ) = D n 0 and d ( y n + 1 , T n y n ) = D n ; n Z 0 + , one has via induction by using the constraints that x n A 0 n 0 ( A 0 n ), y n A n 0 ( A n ); n Z + according to
d ( y n + 1 , T n y n ) = D n + 1 = d ( x n + 1 + x ˜ n + 1 , T n y n ) d ( x n + 1 , T x n ) + d ( T x n x n + 1 , T n y n y n + 1 ) = D n + 1 0 + g n + 1 D n + 1 0 + d ( T x n , T y n ) + d ( T y n , T n y n ) + d ( T n y n , T n y n + x n + 1 y n + 1 ) = D n + 1 0 + g ¯ n + 1 ; n Z 0 + ,
(2.1)
and also one gets in a similar way
d ( x n + 1 , T x n ) = D n + 1 0 = d ( y n + 1 x ˜ n + 1 , T x n ) d ( y n + 1 , T n y n ) + d ( T n y n , T x n + y n + 1 x n + 1 ) = D n + 1 + g n + 1 D n + 1 + d ( T y n , T n y n ) + d ( T x n , T y n ) + d ( T x n , T x n + y n + 1 x n + 1 ) D n + 1 + g ¯ n + 1 ; n Z 0 + .
(2.2)
Furthermore,
g n + 1 = d ( x n + 1 T x n , y n + 1 T n y n ) d ( x n + 1 , T x n ) + d ( T n y n , y n + 1 ) = D n + 1 0 + D n + 1 ; n Z 0 +
(2.3)
and properties (i)-(ii) follow directly. Also, if { T n } { T } then { d ( T y n , T n y n ) } 0 and then property (iii) is proved from property (i) since
g n + 1 = d ( x n + 1 T x n , y n + 1 T y n ) d ( x n + 1 , y n + 1 ) d ( y n + 1 , y n + 1 T y n + T x n ) = d ( x n + 1 , y n + 1 ) d ( T y n , T x n ) ;
(2.4)
we have
lim sup n ( | D n + 1 D n + 1 0 | d ( x n + 1 , y n + 1 ) d ( T y n , T x n ) ) lim sup n ( | D n + 1 D n + 1 0 | g n + 1 ) 0 .
(2.5)
If, furthermore, { x n y n } 0 and T : j Z 0 + A 0 j j Z 0 + B 0 j is uniformly Lipschitzian in its definition domain then there is a positive real constant k T such that d ( T x n , T y n ) k T d ( x n , y n ) ; and then property (iv) follows from property (iii) since
lim n | D n D n 0 | = lim sup n ( | D n D n 0 | ( 1 + k T ) max ( d ( x n , y n ) , d ( y n + 1 , x n + 1 ) ) ) = 0 .
(2.6)
Property (v) follows from property (i) by using
j = m n | D j D j 0 | j = m n g j C g 0 j = m n ρ j C ρ m g 0 1 ρ .
(2.7)
Properties (vi)-(vii) follow from property (i) with
lim n | g n + 1 g ( g n g ) q | = lim n g n + 1 g ( g n g ) q = μ
(2.8)
with q = 1 and μ ( 0 , 1 ) for property (vi) since { g n g } is non-negative and converges to zero which leads for both (vi)-(vii) to
lim sup n ( g n + 1 μ ( g n g ) q g ) 0 ,
(2.9)
and also one gets for q = 1 and μ ( 0 , 1 )
lim sup n ( g n + 1 μ n g ( 1 μ ) g ) lim sup n ( j = 0 n g j g j = 0 n ( 1 μ j ) 1 μ n + 1 1 μ g 0 ) 0 ,
(2.10)

which together with (i) yields (vii). □

It turns out that Lemma 2.2 is extendable to the condition { g ¯ n } g ¯ with the replacements g n g ¯ n and g g ¯ . Some results on boundedness of distances from points of the domains and their images of T n : A n B n ; n Z 0 + , and T : j Z 0 + A 0 j j Z 0 + B 0 j and their asymptotic closeness to the set distance are given in the subsequent result.

Lemma 2.3 Let ( X , d ) be a metric space. Consider two sequences { x n } and { y n } built, respectively, under the proximal non-self-mapping { T } and under the sequence of proximal non-self-mappings { T n } of Lemma  2.2 and assume that { T n } { T } . Then, for any given ε R + , there is N = N ( ε ) Z 0 + such that the following properties hold:
  1. (i)

    d ( x n + 1 , T n x n ) < D n 0 + ε , d ( y n + 1 , T y n ) < D n + ε ; n ( Z 0 + ) > N .

     
  2. (ii)
    If, furthermore,
    d ( T y n , T x n ) K 0 max n m n j n d ( y j , x j ) + M ; n Z 0 +
    (2.11)
     
for some non-negative sequences of integers { m n } with m n n ; n Z 0 + and some real constants K 0 [ 0 , 1 ) and M R 0 + , then
lim sup n max N < j n + 1 d ( x j , y j ) D ¯ + M + K 0 M 1 K 0 , lim sup n d ( T x n , T n y n ) 1 1 K 0 [ ( 2 K 0 ) D ¯ + M + K 0 M ] ,

where D ¯ = D ¯ ( N ) = max N < j < ( D j 0 + D j ) , and M = M ( N ) = max 1 j N d ( x j , y j ) for any given arbitrary finite N Z + .

Proof Assume that the first assertion fails. Then ε R + such that d ( x n + 1 , T n x n ) D n 0 + ε ; n Z 0 + . As a result, since d ( x n + 1 , T x n ) = D n 0 by construction and { T n } { T } , one gets
D n 0 + ε d ( x n + 1 , T n x n ) d ( x n + 1 , T x n ) + d ( T x n , T n x n ) = D n 0 + d ( T x n , T n x n ) < D n 0 + ε / 2 ;
(2.12)

n ( Z 0 + ) > N and some N Z 0 + , a contradiction. Then the first assertion is true.

Now, assume that the second assertion fails. Then ε R + such that d ( y n + 1 , T y n ) D n + ε ; n Z 0 + . As a result, since d ( y n + 1 , T n y n ) = D n by construction and { T n } { T } , one gets
D n + ε d ( y n + 1 , T y n ) d ( y n + 1 , T n y n ) + d ( T n y n , T y n ) = D n + d ( T y n , T n y n ) < D n + ε / 2 ;
(2.13)
n ( Z 0 + ) > N and some N Z 0 + , a contradiction. Then the second assertion is also true and property (i) has been proved. To prove (ii) note that, since { T n } { T } , for any given ε R + , there are N i = N i ( ε ) Z 0 + for i = 1 , 2 such that
d ( x n + 1 , y n + 1 ) D n 0 + d ( T x n , T n x n ) + d ( T n y n , T n x n ) + D n + 1 < D n + 1 0 + D n + 1 + ε / 3 + d ( T n x n , T n y n ) < D n + 1 0 + D n + 1 + ε + d ( T x n , T y n ) D n + 1 0 + D n + 1 + ε + K 0 max m n j n d ( x j , y j ) + M D n + 1 0 + D n + 1 + ε + K 0 max m n j n + 1 d ( x j , y j ) + M D ¯ + M + ε + K 0 max m n j n + 1 d ( x j , y j ) D ¯ + M + ε + K 0 ( max m n < j n + 1 d ( x j , y j ) max N < j n + 1 d ( x j , y j ) ) + K 0 max N < j n + 1 d ( x j , y j ) = D ¯ + M + ε + K 0 [ M + max N < j n + 1 d ( x j , y j ) ] ; n ( Z 0 + ) > N
(2.14)
for some non-negative sequences of integers { m n } with m n n ; n Z 0 + , where d ( T x n , T n x n ) < ε / 3 ; n ( Z 0 + ) > N 1 , d ( T y n , T n y n ) < ε / 3 ; n ( Z 0 + ) > N 2 , N = max ( N 1 , N 2 ) , D ¯ = D ¯ ( N ) = max N < j < ( D j 0 + D j ) , and M is a non-negative real constant, which is not dependent on n, defined by M = M ( N ) = max 1 j N d ( x j , y j ) . Since K 0 [ 0 , 1 ) , one gets
max N < j n + 1 d ( x j , y j ) < D ¯ + M + ε + K 0 M 1 K 0 ; n ( Z 0 + ) > N ,
(2.15)
lim sup n max N < j n + 1 d ( x j , y j ) D ¯ + M + K 0 M 1 K 0 .
(2.16)
On the other hand, note that
d ( T x n , T n y n ) d ( T x n , y n + 1 ) + d ( y n + 1 , T n y n ) d ( T x n , T n y n ) d ( x n + 1 , y n + 1 ) + d ( x n + 1 , T x n ) + D n = D n 0 + D n + d ( x n + 1 , y n + 1 )
(2.17)
d ( T x n , T n y n ) < 1 1 K 0 [ ( 2 K 0 ) D ¯ + M + ε + K 0 M ] ; n ( Z 0 + ) > N ,
(2.18)
lim sup n d ( T x n , T n y n ) 1 1 K 0 [ ( 2 K 0 ) D ¯ + M + K 0 M ] ,
(2.19)

and property (ii) has been proved. □

By interchanging the positions of d ( x n + 1 , y n + 1 ) and d ( T x n , T n y n ) in the triangle inequality of (2.18), it follows that
| d ( T x n , T n y n ) d ( y n + 1 , x n + 1 ) | D n 0 + D n ; n Z 0 + ,
(2.20)

so that if either n = 1 A 0 n = n = 1 ( A n A 0 n ) , since A n A 0 n ; n Z 0 + is bounded, or n = 1 ( B n B 0 n ) is bounded, both sequences { d ( T x n , T n y n ) } and { d ( x n , y n ) } are bounded. On the other hand, note that the non-negative sequence of integers { m n } might imply the use of an infinite memory in the upper-bounding term of (2.11) if m n a Z 0 + ; n Z 0 + or a finite memory of such a bound if m m n m ¯ ; n Z 0 + .

Definition 2.4 [3, 5]

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 { x n } A such that d ( y , x n ) d ( y , A ) for some y B has a convergent subsequence.

Theorem 2.5 Let a proximal contraction and a sequence of proximal mappings { T } and { T n } be defined, respectively, by T : n Z 0 + A 0 n n Z 0 + B 0 n having non-empty images of its restrictions T : j Z 0 + A 0 j | A 0 n j Z 0 + B 0 j | B 0 n ; n Z 0 + and T n : A n B n ; n Z 0 + , where A n ( ) A 0 n and A n are subsets of X, where ( X , d ) is a metric space, subject to T ( A 0 n ) B 0 n and T n ( A n ) B n ; n Z 0 + and the set-theoretic limits lim n A n , lim n A 0 n , lim n B n and lim n B 0 n of the sequences of the sets { A n } , { A 0 n } , { B n } , { B 0 n } , respectively, exist and are non-empty being defined in the usual way for any sequence { Z n } of subsets Z n X as:
lim n Z n = { z X : ( z n = 1  if  z Z n ) ( z n = 0  if  z X n ) ( lim n z n = 1 ) }
via the binary set indicator sequences { z n } , satisfy the improper set inclusion condition lim n A n lim n A 0 n . Assume that the sets of best proximity points A 0 n 0 , B 0 n 0 , A n 0 , and B n 0 are non-empty, where D n 0 = d ( A 0 n , B 0 n ) = d ( A 0 n 0 , B 0 n 0 ) and D n = d ( A n , B n ) = d ( A n 0 , B n 0 ) ; n Z 0 + . Then the following properties hold:
  1. (i)

    lim inf n T ( A 0 n ) lim sup n T ( A 0 n ) lim n B 0 n .

    The set-theoretic limits ( lim n A 0 n ) and T ( lim n A 0 n ) exist and they are non-empty and closed if the subsets A 0 n of X are all closed and n = 0 A 0 n is closed while it satisfies the following set inclusion constraints:
    T ( lim n A 0 n ) lim sup n T ( A 0 n ) lim sup n T ( A 0 n ) lim n B 0 n .
    If, furthermore, lim n T ( A 0 n ) exists then it is non-empty and then
    T ( lim n A 0 n ) = lim n T ( A 0 n ) lim n B 0 n .
    The set-theoretic limits ( lim n A n ) and T ( lim n A n ) exist and they are non-empty and closed if the subsets A n of X; n Z 0 + are all closed and n = 0 A n is closed, and then
    T ( lim n A n ) lim sup n T ( A n ) lim sup n T ( A n ) lim n B n .
    If, furthermore, lim n T ( A n ) exists then it is non-empty and
    T ( lim n A n ) = lim n T ( A n ) lim n B n .
     
  2. (ii)

    Assume that T : j Z 0 + A 0 j j Z 0 + B 0 j is a proximal contraction and that { T n } { T } , where T n : A n B n ; n Z 0 + . Then T n : A n B n ; n Z 0 + , is a sequence of asymptotic proximal contractions.

     
  3. (iii)
    If the sets in sequence { A 0 n } n m , for some m Z 0 + , are closed then lim n A 0 n is closed and also
    lim n d ( x n + 1 , T x n ) = D 0 , lim n d ( x n + 1 , x n ) = 0 , { x n } x ( lim n A 0 n 0 ) ,
    (2.21)
     
lim n d ( y n + 1 , T y n ) = D , lim n d ( y n + 1 , y n ) = 0 , { y n } y ( lim n A n 0 ) ,
(2.22)
| D 0 D | d ( x , y ) + lim inf n d ( T x n , T y n ) | D 0 D | d ( x , y ) + min ( d ( ( T x ) + , ( T y ) + ) , d ( ( T x ) , ( T y ) ) )
(2.23)
for any sequences { x n } A 0 n , { y n } A n satisfying d ( x n + 1 , T x n ) = D n + 1 0 ; n Z 0 + and d ( y n + 1 , T y n ) = D n + 1 ; n ( n 0 ) Z 0 + for any sequences { x n } A 0 n , { y n } A n being built in such a way that d ( x n + 1 , T x n ) = D n + 1 0 ; n Z 0 + and d ( y n + 1 , T y n ) = D n + 1 ; n ( n 0 ) Z 0 + .
  1. (iv)

    If ( X , d ) is complete and lim n B 0 n is approximatively compact with respect to lim n A 0 n then there is a convergent subsequence { T x n k } ( lim n B 0 n ) T x where x lim n A 0 n 0 is the limit of { x n } . If lim n B n is approximatively compact with respect to lim n A n then there is a convergent subsequence { T y n k } ( lim n B n ) T y where y lim n A n 0 is the limit of { y n } . If D = D 0 , where D n 0 D 0 and D n D as n , and the above two approximative compactness conditions hold and, furthermore, A n and A 0 n are closed; n Z 0 + and n = 0 A 0 n and n = 0 A n (it suffices that n = m A 0 n and n = m A n be closed for some m Z 0 + ) are closed then x = y , which is then the unique best proximity point of T in the limit set lim n A n 0 .

     
Proof Note that since lim n A n and lim n A 0 n exist, we have
lim n A n = lim inf n A n = n = 0 ( m = n A m ) = lim sup n A n = n = 0 ( m = n A m ) , lim n A 0 n = lim inf n A 0 n = n = 0 ( m = n A 0 m ) = lim sup n A 0 n = n = 0 ( m = n A 0 m ) .
Note also that, since lim n B 0 n exists, we have lim inf n T ( A 0 n ) lim sup n T ( A 0 n ) lim n B 0 n since
lim inf n T ( A 0 n ) = n = 0 ( m = n T ( A 0 m ) ) lim inf n T ( A 0 n ) n = 0 ( m = n B 0 m ) = lim inf n B 0 n = lim n B 0 n , lim sup n T ( A 0 n ) = n = 0 ( m = n T ( A 0 m ) ) lim sup n T ( A 0 n ) n = 0 ( m = n B 0 m ) = lim sup n B 0 n = lim n B 0 n .
Also, since lim n B n exists, we have lim inf n T ( A n ) lim sup T ( A n ) lim n B n , which follows under a similar reasoning. On the other hand, the existence and non-emptiness of lim n A 0 n , by hypothesis, implies that T ( lim n A 0 n ) exists since A 0 n dom T ; n Z 0 + so that
T ( lim n A 0 n ) = T ( n = 0 ( m = n A 0 m ) ) T ( lim n A 0 n ) n = 0 ( m = n T ( A 0 m ) ) = lim sup n T ( A 0 n ) lim n B 0 n , T ( lim n A 0 n ) = T ( n = 0 ( m = n A 0 m ) ) = n = 0 T ( m = n A 0 m ) T ( lim n A 0 n ) n = 0 ( m = n T ( A 0 m ) ) T ( lim n A 0 n ) = lim inf n T ( A 0 n ) lim n B 0 n .
From the above set inclusion conditions it also holds that
T ( lim n A 0 n ) = lim n T ( A 0 n ) lim n B 0 n
if lim n T ( A 0 n ) exists. If the subsets A 0 n ; n Z 0 + are all closed and n = 0 A 0 n is closed then n = k A 0 n is closed for any k Z + . In order to prove that lim n A 0 n = n = 0 ( m = n A 0 m ) is closed, that is, an infinite intersection of unions of infinitely many closed sets is closed, it is first proved that all those unions are closed under the given assumption that n = 0 A 0 n is closed. Assume this not to be the case, so that there is k Z + such that n = 0 A 0 n = ( n = 0 k 1 A 0 n ) ( n = k A 0 n ) is closed with n = k A 0 n not being closed and such that n = 0 k A 0 n is closed, since it is the union of a finite number of closed sets. Then either n = k 1 A 0 n = A 0 , k 1 ( n = k A 0 n ) is not closed or it is closed. In the second case, n = k A 0 n A 0 , k 1 since if n = k A 0 n A 0 , k 1 is not closed then n = k 1 A 0 n cannot be closed by construction. But, if n = k A 0 n A 0 , k 1 then n = k A 0 n is closed, which contradicts that it is not closed, since A 0 , k 1 is closed for any k Z + . As a result n = k A 0 n being no closed for any k Z + implies that n = k 1 A 0 n is not closed. By complete induction, it follows that n = j A 0 n is not closed for any j ( Z 0 + ) k . Thus, n = 0 A 0 n is not closed what contradicts that it is closed. As a result, since n = 0 A 0 n is non-empty and closed, any infinite union of non-empty closed sets n = k A 0 n is also non-empty and closed for any k Z 0 + since n = 0 A 0 n is assumed to be closed by hypothesis and it is trivially non-empty. Then lim n A 0 n = n = 0 ( m = n A 0 m ) is non-empty and closed since it is the infinite intersection of infinitely many unions of non-empty closed sets (but already proved to be non-empty and closed) and there exists the set limit T ( lim n A 0 n ) = T ( n = 0 ( m = n A 0 m ) ) which is now proved to be non-empty. Proceed by contradiction by assuming that it is empty. Then there is some x lim n A 0 n such that x dom T . But then, since x lim n A 0 n , lim n A 0 n = n = 0 ( m = n A 0 n ) is non-empty and closed and all the sets in the sequence { A 0 n } are closed we have x A 0 n for some n Z 0 + so that x dom T from the definition of the non-self-mapping T which contradicts the existence of x lim n A 0 n such that x dom T . It has been proved that lim n A 0 n is closed and T ( lim n A 0 n ) exists and it is non-empty and closed. The proof that T ( lim n A n ) lim n B n is similar to the above one. Now, one proves by contradiction that T ( lim n A n ) is non-empty if lim n A n is non-empty. The limit set lim n A n is non-empty since the subsets A n of X; n Z 0 + are all closed and n = 0 A n is closed under similar arguments that those used above to prove those properties for lim n A 0 n . Assume that T ( lim n A n ) is empty. Since lim n A n is non-empty, there is x lim n A n such that x dom T . Since the sets in the sequence { A n } are closed, x A n and x A 0 n , since A n A 0 n , for some n Z 0 + . Thus, x dom T , from the definition of T and T x T ( A 0 n ) T ( A n ) B 0 n B n since T ( A n ) T ( A 0 n ) B n B 0 n B n B 0 n . Then T ( lim n A n ) is non-empty since lim n A n is non-empty. On the other hand, since lim n A n lim n A 0 n we have
T ( lim n A n ) = T ( n = 0 ( m = n A m ) ) T ( n = 0 ( m = n A 0 m ) ) n = 0 ( m = n T ( A 0 m ) ) = n = 0 ( m = n B 0 m ) = lim n B 0 n ,
so that
T ( lim n A 0 n ) lim n B n lim n B 0 n ; lim n T ( A n ) T ( lim n A n ) lim n B n lim n B 0 n .
In the same way, it follows that T ( lim n A n ) exists and if, in addition, lim n T ( A n ) exists then
lim n T ( A n ) = T ( lim n A n ) lim n B n .
Hence, property (i) has been proved. Now, build sequences { x n } and { x n } , and { y n } and { y n } in X sequences built in such a way that x 0 , x 0 A 01 , y 0 , y 0 A 1 , d ( x n + 1 , T x n ) = D n 0 , d ( x n + 1 , T x n ) = D n + 1 0 d ( y n + 1 , T n y n ) = D n + 1 and d ( y n + 1 , T y n ) = D n ; n Z 0 + . It follows inductively by using those distance constraints that x n , x n A 0 n 0 ( A 0 n ), y n , y n A n 0 ( A n ); n Z + . Note that lim n A n lim n A 0 n implies the following inclusion of limit sets lim n A n 0 lim n A 0 n 0 of the sets of best proximity points and the four above limit sets are trivially non-empty. Since T : j Z 0 + A 0 j j Z 0 + B 0 j is a proximal contraction:
( d ( x n + 1 , T x n ) = D n + 1 0 ; n Z 0 + ) ( d ( x n + 1 , x n + 2 ) K d ( x n , x n + 1 ) ; n Z 0 + )
(2.24)
for { x n } A 0 n 0 and some real constant K [ 0 , 1 ) and D n 0 D as n , since the limit set lim n A 0 n exists. On the other hand, if { T n } is a sequence of asymptotic proximal contractions T n : A n B n ; n Z 0 + then there is a real sequence { K n } with a subsequence { K n } n n 0 [ 0 , K ) [ 0 , 1 ) such that
( d ( y n + 1 , T n y n ) = D n + 1 ; n ( n 0 ) Z 0 + ) ( d ( y n + 1 , y n + 2 ) K d ( y n , y n + 1 ) ; n ( n 0 ) )
(2.25)
for { y n } n n 0 A 0 n 0 and some n 0 Z + and some real constant K [ 0 , 1 ) . Recall the following properties for logical assertions then to be used related to (2.24). Consider the logical propositions P i ; i = 1 , 2 . Then
( P 2 P 1 ) ( P 1 ¬ P 2 ) ,
(2.26a)
( ¬ ( P 2 P 1 ) ) ( ¬ P 1 P 2 ) .
(2.26b)
The condition (2.25) corresponds to (2.26a) with P 2 ( d ( y n + 1 , T y n ) = D n + 1 ; n Z + ) and P 1 ( d ( y n + 1 , y n + 2 ) K d ( y n + 1 , y n + 2 ) ; n Z + ) . It is now proved by contradiction that { T n } is a sequence of asymptotic proximal contractions, that is, by assuming that (2.25) is false so that its logical negation
[ ( d ( y n + 1 , T n y n ) > D n + 1 ) ( d ( y n + 1 , y n + 2 ) K d ( y n , y n + 1 ) ) ; n ( n 0 ) Z + ]
(2.27)
for { y n } n n 0 A 0 n 0 and some n 0 Z + , obtained from (2.26a)-(2.26b) versus to (2.26a), is true. Then it follows from (2.24) that for any given arbitrary ε R + , there are n 0 , n 1 = n 1 ( ε ) Z 0 + such that d ( T y n + 1 , T n y n + 2 ) < ε ; n n 1 (since { T n } { T } ) and (2.27) hold. Then
| D n + 2 D n + 1 | < d ( y n + 2 , T n y n + 1 ) d ( y n + 2 , y n + 1 ) + d ( T n y n + 1 , T y n + 1 ) < K d ( y n + 1 , y n ) + d ( T y n + 1 , T y n ) + ε K n n 0 n 1 d ( y n 0 + n 1 + 1 , y n 0 + n 1 ) + d ( T y n + 1 , T y n ) + ε ; n n 0 + n 1 .
(2.28)
Since ε R + is arbitrary, K [ 0 , 1 ) , and D n D as n since the limit set lim n A n exists and D D 0 , since lim n A n lim n A 0 n , one concludes from (2.28) that
lim n d ( y n + 1 , y n + 2 ) = 0 , { y n } y ( lim n A n ) , lim inf n d ( T y n + 1 , T y n ) > 0 .
(2.29)

Since T : n Z 0 + A 0 n n Z 0 + B 0 n and lim n A n lim n A 0 n , so that y lim n A 0 n , since n m A 0 n is closed by hypothesis for some m Z 0 + and lim n A 0 n is also closed. Then lim n A 0 n dom T so that the possibility of Ty being undefined is excluded and then Ty is defined provided that there is no finite or infinite jump discontinuities at y. If T is continuous at y, then (2.29) leads to the contradiction d ( T y , T y ) > 0 . Otherwise, if T has a (finite or infinity) jump discontinuity at y, with left and right limits ( T y ) and ( T y ) + ( ( T y ) ), then min ( d ( ( T y ) , ( T y ) ) , d ( ( T y ) + , ( T y ) + ) ) > 0 , again a contradiction is got. Thus, (2.27) is false so that its negation (2.22) is true. Then { T n } 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 (2.24) and the constraints lim n A 0 n 0 lim n A 0 n and lim n A n 0 lim n A n lim n A 0 n together with the fact that, since n m A 0 n is closed, lim n A 0 n is closed. Note that the conditions of A 0 n , A n , n Z 0 + being closed and n = 0 A 0 n and n = 0 A n being closed can be relaxed in property (i) to closeness of the sets A 0 n , A n , n ( m ) Z 0 + and n = m A 0 n and n = m A n for some m Z 0 + being closed while keeping the corresponding result. Assume this not to be the case. Now, take a subsequence { A 0 n k } of (non-empty closed) subsets of X such that there is x X such that x fr ( lim n A 0 n ) , x lim n A 0 n , since lim n A 0 n is not closed, and x A 0 n k , the indicator variable of x in any subset in the sequence { A 0 n k } is x 0 n k = 1 then x lim k A 0 n k , since lim k x 0 n k = 1 , and x lim n A 0 n which contradicts lim n A 0 n = lim k A 0 n k . Thus, lim n A 0 n is non-empty and closed. Then one obtains (2.21)-(2.23).

On the other hand, since D n 0 D 0 as n and lim n B 0 n is approximatively compact with respect to lim n A 0 n for some sequence { x n 0 } lim n A 0 n
d ( T x n 0 , x ) d ( lim n B 0 n , x ) = D 0 as  n
with x lim n B 0 n . From property (i), T ( lim n A n ) lim n B n lim n B 0 n T ( lim n A 0 n ) lim n B 0 n , T ( lim n A n ) lim n B n . Then there is a subsequence { T x n k 0 } ( { T x n 0 } ) z x cl lim n B 0 n since lim n B 0 n is closed and it is obvious that z x cl lim n B 0 n 0 ( = lim n B 0 n ) so that cl lim n B 0 n 0 . Assume that lim n B 0 n 0 is empty then the best proximity point z x lim n B 0 n 0 , a contradiction so that lim n B 0 n 0 is non-empty and it is closed since it is on the boundary of lim n B 0 n 0 which is then non-empty and closed. But { x n 0 } x ( lim n A 0 n ). Since { x n 0 } is convergent all its subsequences are convergent to the same limit so that { x n k 0 } x . Thus, z x = T x . Under a similar reasoning, it can be proved under the second given approximative compactness condition that { T y n k 0 } ( { T y n 0 } ) z y = T y ( cl lim n B n ) with lim n B n 0 being non-empty and closed. If the above both approximative compactness conditions jointly hold and D = D 0 then one gets from the contractive condition for the proximal non-self-mapping T that for any given n Z 0 + :
D = d ( x n + 1 0 , T x n 0 ) = ( y n + 1 0 , T y n 0 ) d ( x n + 1 0 , y n + 1 0 ) K d ( x n 0 , y n 0 )

so that { x n 0 } x and { y n 0 } y implies { d ( x n 0 , y n 0 ) } d ( x , y ) so that ( 1 K ) d ( x , y ) 0 holds what implies x = y , which has to be necessarily unique from the identity, itself. Hence, property (iv) has been proved. □

Remark 2.6 Note that the conditions of A 0 n , A n , n Z 0 + being closed and the infinite countable unions n = 0 A 0 n and n = 0 A n being closed can be relaxed in Theorem 2.5 (property (i)) to the closeness of the sets A 0 n , A n , n ( m ) Z 0 + and n = m A 0 n and n = m A n for some m Z 0 + being closed while keeping the corresponding result.

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 2.5.

Remark 2.7 It can be pointed out that a sufficient condition for the infinite countable union of closed subsets n = 0 A 0 n (respectively, n = 0 A n ) of a topological space ( X , { B i } ) to be closed is that X has the local finiteness property, that is, each point in X has a neighborhood which intersects only finitely many of the closed sets in { A 0 n } (respectively, in { A n } ) ([20], pp.29-31). 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 define the open balls of the topology. More general results guaranteeing that the infinite countable union of closed sets is closed, and equivalently that the infinite 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 definitions of usefulness for the main results of this section.

Definition 3.1 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:
  1. (1)
    A generalized asymptotic weak proximal contraction if there are sequences of non-negative real numbers { α n } , with α n [ 0 , ) ; n Z 0 + and α n α ( [ 0 , 1 ) ) as n ; and { β n } , with β n [ 0 , ) ; n Z 0 + and β n β ( [ 0 , ) ) as n such that, for all sequences { u 1 } , { u 2 } , { x 1 } , { x 2 } A :
    [ d ( u i , T x i ) = D ( i = 1 , 2 ) ] [ d ( x 1 , T x 1 ) ( 1 + α n + β n ) d ( x 1 , x 2 ) + D ] d ( u 1 , u 2 ) α n d ( x 1 , x 2 ) + β n [ d ( T x 1 , x 2 ) D ] ; n Z 0 + .
    (3.1)
     
  2. (2)

    A generalized weak proximal contraction [2, 3], if (3.1) holds with non-negative real constants α n = α < 1 and β n = β < ; n Z 0 + .

     
  3. (3)

    T : A ( B n ) is a strongly generalized asymptotic weak proximal contraction if:

     
  4. (a)

    A 0 and B n with D n = d ( A , B n ) ; n Z 0 + are non-empty and T ( A 0 ) B 0 n ; n Z 0 + .

     
  5. (b)
    d ( u 1 n , u 2 n ) α n d ( x 1 n , x 2 n ) + β n [ d ( T x 1 n , x 2 n ) D n ] ; n Z 0 +
    (3.2a)
     
for any sequences { x i n } A and { u i n } B n such that d ( u i n , T x i n ) = D n ( i = 1 , 2 ); n Z 0 + and { α n } , { β n } are real non-negative sequences which satisfy α = lim sup n α n < 1 and
β = lim sup n β n < with  μ = ( α + β ) ( 1 + β ) + β < 1 .
  1. (c)
    The sequence of set distances { D n } converges and { D n D n 0 } 0 , where
    D n 0 d ( x 1 n , T x 1 n ) ( 1 + α n + β n ) d ( x 1 n , x 2 n ) ; n Z 0 + .
    (3.2b)
     

Note if β = 0 in Definition 3.1(2), one has the subclass of weak proximal contractions. In this case, one gets by making x n + 1 = u n ; n Z 0 + that d ( x n , x n + 1 ) α n d ( x 0 , x 1 ) ; n Z 0 + , so that d ( x n , x n + 1 ) 0 as n , since α < 1 , provided that d ( u n , T x n ) = D and d ( x n , T x n ) ( 1 + α ) d ( x n , x n + 1 ) + d ( u n , T x n ) ; n Z 0 + implying d ( x n , T x n ) D as n and under the conditions that { x n } A and { T x n } B converge they should necessarily converge to best proximity points. Note that Definition 3.1(1) relaxes Definition 3.1(2) and Definition 3.1(3) allows considering weak proximal contractions with sequences built from non-self-mappings which have iteration-dependent image sets.

The following results hold.

Proposition 3.2 Let A and B n ; n Z 0 + be non-empty subsets of a metric space ( X , d ) . Assume that T : A ( B n ) , such that A 0 is non-empty and T ( A 0 ) B 0 n ; n Z 0 + , with D n = d ( A , B n ) ; n Z 0 + satisfies the contractive condition:
d ( x n + 1 , x n + 2 ) α d ( x n + 1 , x n ) + β ( d ( T x n , x n + 1 ) D n ) ; n Z + .
(3.3)
Then the following properties hold:
  1. (i)
    d ( x n + 1 , x n + 2 ) μ n + 1 d ( x 0 , x 1 ) + β k = 0 n μ n k ( D k 0 D k ) ; n Z + ,
    (3.4)
    which is bounded for any finite x 0 A if μ 1 and k = 0 n μ n k ( | D k D k 0 | ) < , where μ = ( α + β ) ( 1 + β ) + β , and the sequence { D n 0 } satisfies (3.2b) which can be relaxed to (3.5) below
    D n 0 d ( x n , T x n ) ( 1 + α + β ) d ( x n , x n + 1 ) ; n Z 0 + .
    (3.5)
     
  2. (ii)
    If (3.3) holds, subject to (3.5), with μ < 1 then
    d ( x n , x n + 1 ) μ n d ( x 0 , x 1 ) + 1 μ n 1 μ β sup 0 k n + 1 | D k D k 0 | ,
    (3.6)
     
lim sup n ( d ( x n , x n + 1 ) β 1 μ ( sup 0 k n | D k D k 0 | ) ) 0 ,
(3.7)
lim sup n ( | d ( x n + 1 , T x n ) d ( x n , T x n ) | d ( x n , x n + 1 ) ) 0 ,
(3.8)
lim sup n ( | d ( x n + 1 , T x n ) d ( x n , T x n ) | β 1 μ sup 0 k n | D k D k 0 | ) 0 ,
(3.9)
lim n d ( x n , x n + 1 ) = 0 ; lim n ( | d ( x n + 1 , T x n ) d ( x n , T x n ) | ) = 0
(3.10)
if | D n D n 0 | 0 as n .
  1. (iii)
    If (3.3) holds subject to (3.5) with μ < 1 then
    lim inf n [ min ( d ( x n , T x n ) D n , d ( x n + 1 , T x n ) D n + 1 ) ] 0 ,
    (3.11)
    lim sup n ( d ( x n , T x n ) D n 0 β ( 1 + α + β ) 1 μ ( sup 0 k n | D k D k 0 | ) ) 0 ,
    (3.12)
    lim sup n ( d ( x n + 1 , T x n ) D n 0 β ( 2 + α + β ) 1 μ ( sup 0 k n | D k D k 0 | ) ) 0 .
    (3.13)
     
  2. (iv)
    If μ < 1 and | D n D n 0 | 0 as n then the limits below exist and are identical:
    lim n ( d ( x n , T x n ) D n 0 ) = lim n ( d ( x n , T x n ) D n ) = lim n ( d ( x n , T x n ) D n 0 ) = lim n ( d ( x n + 1 , T x n ) D n ) .
    (3.14)
     

If (3.3) holds, subject to (3.5), with μ < 1 , { D n 0 } D and { D n } D then lim n d ( x n , T x n ) = lim n d ( x n + 1 , T x n ) = D .

Proof Since d ( A , B n ) = D n we have d ( T x n , x n + 1 ) D n ; n Z 0 + . Also, if follows from (3.5) and (3.3) with μ = ( α + β ) ( 1 + β ) + β that (3.4) holds since
d ( x n + 2 , x n + 1 ) α d ( x n + 1 , x n ) + β ( d ( T x n , x n ) + d ( x n , x n + 1 ) D n ) = ( α + β ) d ( x n + 1 , x n ) + β ( d ( T x n , x n ) D n ) μ d ( x n + 1 , x n ) + β ( D n 0 D n ) μ n + 1 d ( x 0 , x 1 ) + β k = 0 n μ n k ( D k 0 D k ) ; n Z +
(3.15)
for any given x 0 A . If μ 1 and k = 0 n μ n k ( | D k D k 0 | ) < then the sequence { d ( x n , x n + 1 ) } is bounded. Thus, property (i) has been proved. The relations (3.6) and (3.7) of property (ii) follow directly from (3.4) of property (i) if μ < 1 . On the other hand, the triangle inequality and (3.5) lead to (3.11) since
| d ( x n + 1 , T x n ) d ( x n , T x n ) | d ( x n , x n + 1 ) .
The relation (3.9) follows from (3.7) and (3.8). To prove (3.10), note that if lim n ( D n D n 0 ) = 0 , then for any given ε R + , there is m = m ( ε ) Z 0 + such that sup m k n + m + 1 | D k D k 0 | < ε for any n ( Z + ) m and then, from (3.10), lim sup n d ( x n + 1 , x n ) β ε 1 μ . Since ε is arbitrary, the limit lim n d ( x n , x n + 1 ) exists and lim n d ( x n , x n + 1 ) = 0 . This property and (3.8) yield directly lim n ( | d ( x n + 1 , T x n ) d ( x n , T x n ) | ) = 0 and then property (ii) has been fully proved. To prove property (iii), note that (3.11) holds directly from d ( A , B n ) = D n ; n Z 0 + and { x n } A . Also, (3.5) leads to (3.12) by taking into account (3.7). The relation (3.13) follows from (3.7), (3.12), and the relation
d ( x n + 1 , T x n ) d ( x n , T x n ) + d ( x n + 1 , x n ) ; n Z 0 + .

Hence, property (iii) has been proved. Property (iv) is a direct consequence of property (iii) for the case when d ( x n , x n + 1 ) 0 and | D n D n 0 | 0 as n including its particular sub-case when { D n 0 } D and { D n } D . □

Note that Proposition 3.2 is applicable to the strongly generalized asymptotic weak proximal contraction of Definition 3.1(3) which do not need the fulfilment of the implying part of the logic proposition of Definition 3.1(1)-(2) but the distances of sequences of sets satisfy (3.2b) or, at least, (3.5). The subsequent result is concerned with the existence and uniqueness of best proximity points if, in addition to the assumptions of Proposition 3.2, the set-theoretic limit of the sequence { B n } exists and is closed and approximatively compact with respect to A.

Theorem 3.3 Under all the assumptions of Proposition  3.2 and property (iii), equation (3.10), assume also that ( X , d ) is complete, that A and B n ; n Z 0 + , are non-empty subsets of X such that A is closed, A 0 is non-empty, the set-theoretic limit B : = lim n B n exists, is closed and approximatively compact with respect to A (or the weaker condition that A 0 is closed) and T ( A 0 ) B 0 . Assume also that the non-self-mapping restriction T : A | A 1 ( B n ) | B , for some subset A 1 A , which contains the set of best proximity points A 0 , is a strongly generalized asymptotic weak proximal contraction. Then T : A | A 1 B has a unique best proximity point if μ < 1 .

Proof Since d ( x n + 1 , x n ) 0 as n , { x n } x from (3.10). Since { x n } A and A is closed we have x A . Since D n d ( x n + 1 , T x n ) = { d ( x , T x n ) } (→D) since { D n } D , because { B n } B ; and | D n d ( x n , T x n ) | d ( x n , x n + 1 ) ; n Z 0 + . Since B is approximatively compact with respect to A and { x n } A , there are y A 0 and a sequence { x ¯ n } A , such that { x ¯ n x n } 0 , then { x ¯ n } x since { x n } x , and { T x n } B such that one gets as n :
d ( x ¯ n , T x ¯ n ) d ( y , B ) = D ; d ( x ¯ n , B ) d ( y , B ) = D ; d ( x n , B ) d ( y , B ) = D ,
{ d ( x n , T x n ) } D , { d ( x n + 1 , T x n ) } D and { d ( x , T x n ) } D . Since B is approximatively compact with respect to A and { x n } A , the sequence { T x n } B n , such that T x n B n , has a convergent subsequence { T x n k } z B , since B is also closed and both { T x n k } and { T x n k } have the same limit z B . Also, z cl B 0 and d ( x , T x n k ) D ( = d ( x , z ) ) as k so that x is a best proximity point of T : A | A 1 B . Note that since the limit set B exists, it is by construction the infinite union of intersections of the form B = n = 0 m n B m so that B 0 B B n so that there is a restriction T : A | A 1 ( B n ) | B for some A 1 A which is non-empty. Assume not so that A 1 = . If A 1 = then A 0 is also empty which is impossible then A 1 . 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 { x n } x and { y n } y contained in A. Since T : A | A 1 ( B n ) | B is a strongly generalized asymptotic weak proximal contraction, one gets from the implied logic proposition of (3.1) with u = x and v = y and D n = D that
( 1 α ) d ( x , y ) β ( d ( T x , y ) D ) β d ( T x , x ) + β d ( x , y ) β D = β d ( x , y ) ,

which fails for 0 β < 1 α if x y . Thus, x = y . □

Closely to Theorem 3.3, the following result can be proved.

Theorem 3.4 Let A and B n ; n Z 0 + be non-empty closed subsets of a complete metric space ( X , d ) . Assume that T : A ( B n ) is a weakly generalized asymptotic weak proximal contraction, such that A 0 is non-empty and T ( A 0 ) B 0 n ; n Z 0 + , with D n = d ( A , B n ) and T ( A ) B n ; n Z 0 + . Assume also that the set limit B : = lim n B n exists, is closed and approximatively compact with respect to A, or instead, the weaker condition that A 0 is closed. Then T : A ( B n ) has a unique best proximity point.

4 Examples

Two examples are described to the light of proximal contractions. The first 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 identification 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 identification.

Example 4.1 (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 scientific 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 C x = e where C R n × p (a real matrix of order n × p ) and e R p . It is known from the Rouché-Frobenius theorem from Linear Algebra that a solution x R p exists if rank ( C , e ) = rank C . The solution is unique given by x = C 1 e 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 infinitely 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 C X = E , where C R n × p , E R n × q are given and X R p × q is a solution which exists if and only if rank C = rank ( C E ) . The following cases hold:
  1. (a)
    If rank C = rank ( C E ) p , then C + C E E , and there are infinitely 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 T C : A B , represented by the matrix C, can be restricted to A = { X = C + E + ( I C + C ) W : W R p × q } . To close a proper formalism we extend the matrices in A to matrices A ¯ R n × ( max ( p , q ) q ) and those in B to B ¯ R n × ( 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 R n × ( max ( p , q ) q ) and consider the metric space ( X , d ) with d being the Euclidean metric so that D = 0 with the sets of best proximity points of A ¯ and B ¯ being:
    A ¯ 0 = A ¯ = { X ¯ = ( X 0 ) : X = C + E + ( I C + C ) W , W R n × q , 0 R n × ( max ( p , q ) q ) } , B ¯ 0 = B ¯ = { ( E 0 ) : 0 R n × ( max ( p , q ) p ) } ;

    A ¯ 0 is the set of solutions of the compatible indeterminate algebraic system.

     
  2. (b)

    If rank C = rank ( C E ) = p then the solution X = C + E is unique and A ¯ 0 = A ¯ = { ( C + E 0 ) : 0 R n × ( max ( p , q ) q ) } consist of one element which is the unique solution.

     
  3. (c)
    If min ( p , n ) rank C < rank ( C E ) then C + C E 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 C X E over X R p × q if it exists. Such a solution exists in a least-squares sense and it is unique if n rank C = p < rank ( C E ) , since C T C is non-singular, of order p, and C + = ( C T C ) 1 C T , and X ˆ = ( C T C ) 1 C T E minimizes C X E over the set of matrices X R p × q so that D = d ( A ¯ , B ¯ ) = ( ( C T C ) 1 C T I ) E , the sets of best proximity points of A ¯ being
    A ¯ 0 = A ¯ = { X ¯ = ( X 0 ) : X = ( C T C ) 1 C T E , 0 R n × ( max ( p , q ) q ) } ,
     

since ( I C + C ) = 0 and B ¯ 0 is as the above one of case (a). A ¯ 0 is the best solution of the incompatible algebraic system.

The pseudo-inverse can be calculated without inverting C T C by the iterative process:
C n + 1 + = ( 2 I C n + C ) C n + ; n Z + ; C 0 + such that  C 0 + C = ( C 0 + C )
(so-called Ben-Israel-Cohen, or hyper-power sequence, iterative method [21]). It follows that C n + C + as n since (1) C + is unique; and (2) the iterative process satisfies the pseudo-inverse properties C + = C + C C + and C + C = ( C + C ) under the replacement C n + C + ; n Z 0 + . By using the iterative process, we can also define sequences of sets and associate sequences of distances by:
D n = d ( A ¯ n , B ¯ ) = ( C n + I ) E D = ( C + I ) E as  n , B ¯ 0 = B ¯ = { ( E 0 ) : 0 R n × ( max ( p , q ) p ) } , A ¯ 0 n = A ¯ n = { ( C n + E 0 ) : 0 R n × ( max ( p , q ) q ) } A ¯ 0 A ¯ 0 n = A ¯ = { ( C + E 0 ) : 0 R n × ( max ( p , q ) q ) } as  n ;
B ¯ 0 and A ¯ 0 n are unique, if the pseudo-inverse C + = ( C T C ) 1 C T exists for the given initial C 0 + . However, if rank C = rank ( C E ) p , so that E C n + C n E (case (a) - incompatible algebraic system) then if we use the iterative procedure:
C n + 1 + = ( 2 I C n + C ) C n + ; n Z + ; C 0 + such that  C 0 + C = ( C 0 + C ) , A ¯ 0 n = A ¯ 0 n ( W n ) A ¯ 0 n