Research

# On best proximity points for pseudocontractions in the intermediate sense for non-cyclic and cyclic self-mappings in metric spaces

Manuel De la Sen

Author Affiliations

Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, 48080, Spain

Fixed Point Theory and Applications 2013, 2013:146  doi:10.1186/1687-1812-2013-146

 Received: 17 September 2012 Accepted: 17 May 2013 Published: 5 June 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

### 1 Introduction

Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated in the framework of Hilbert spaces have received a certain attention in the last years concerning their convergence properties and the existence of fixed points. See, for instance, [1-4] and references therein. Results about the existence of a fixed point are discussed in those papers. On the other hand, important attention has been paid during the last decades to the study of the convergence properties of distances in cyclic contractive self-mappings on p subsets A i X of a metric space ( X , d ) , or a Banach space ( X , ) . The cyclic self-mappings under study have been of standard contractive or weakly contractive types and of Meir-Keeler type. The convergence of sequences to fixed points and best proximity points of the involved sets has been investigated in the last years. See, for instance, [5-20] and references therein. It has to be noticed that every nonexpansive mapping [21,22] is a 0-strict pseudocontraction and also that strict pseudocontractions in the intermediate sense are asymptotically nonexpansive [2]. The uniqueness of the best proximity points to which all the sequences of iterations converge is proven in [6] for the extension of the contractive principle for cyclic self-mappings in either uniformly convex Banach spaces (then being strictly convex and reflexive [23]) or in reflexive Banach spaces [13]. The p subsets A i X of the metric space ( X , d ) , or the Banach space ( X , ) , where the cyclic self-mappings are defined, are supposed to be nonempty, convex and closed. If the involved subsets have nonempty intersections, then all best proximity points coincide, with a unique fixed point being allocated in the intersection of all the subsets, and framework can be simply given on complete metric spaces. The research in [6] is centered on the case of the 2-cyclic self-mapping being defined on the union of two subsets of the metric space. Those results are extended in [7] for Meir-Keeler cyclic contraction maps and, in general, with the p ( 2 ) -cyclic self-mapping T : i p ¯ A i i p ¯ A i defined on any number of subsets of the metric space with p ¯ : = { 1 , 2 , , p } . Other recent research which has been performed in the field of cyclic maps is related to the introduction and discussion of the so-called cyclic representation of a set M, as the union of a set of nonempty sets as M = i = 1 m M i , with respect to an operator f : M M [14]. Subsequently, cyclic representations have been used in [15] to investigate operators from M to M which are cyclic φ-contractions, where φ : R 0 + R 0 + is a given comparison function, M X and ( X , d ) is a metric space. The above cyclic representation has also been used in [16] to prove the existence of a fixed point for a self-mapping defined on a complete metric space which satisfies a cyclic weak φ-contraction. In [18], a characterization of best proximity points is studied for individual and pairs of non-self-mappings S , T : A B , where A and B are nonempty subsets of a metric space. The existence of common fixed points of self-mappings is investigated in [24] for a class of nonlinear integral equations, while fixed point theory is investigated in locally convex spaces and non-convex sets in [25-28]. More recently, the existence and uniqueness of best proximity points of more general cyclic contractions have been investigated in [29,30] and a study of best proximity points for generalized proximal contractions, a concept referred to non-self-mappings, has been proposed and reported in detail in [31]. Also, the study and characterization of best proximity points for cyclic weaker Meir-Keeler contractions have been performed in [32] and recent contributions on the study of best proximity and proximal points can be found in [33-38] and references therein. In general, best proximity points do not fulfill the usual ‘best proximity’ condition x = S x = T x under this framework. However, best proximity points are proven to jointly globally optimize the mappings from x to the distances d ( x , T x ) and d ( x , S x ) . Furthermore, a class of cyclic φ-contractions, which contains the cyclic contraction maps as a subclass, has been proposed in [18] in order to investigate the convergence and existence results of best proximity points in reflexive Banach spaces completing previous related results in [6]. Also, the existence and uniqueness of best proximity points of cyclic φ-contractive self-mappings in reflexive Banach spaces have been investigated in [19]. This paper is devoted to the convergence properties and the existence of fixed points of a generalized version of pseudocontractive, strict pseudocontractive and asymptotically pseudocontractive in the intermediate sense in the more general framework of metric spaces. The case of 2-cyclic pseudocontractive self-mappings is also considered. The combination of constants defining the contraction may be different on each of the subsets and only the product of all the constants is requested to be less than unity. It is assumed that the considered self-mapping can perform a number of iterations on each of the subsets before transferring its image to the next adjacent subset of the 2-cyclic self-mapping. The existence of a unique closed finite limiting sequence on any sequence of iterations from any initial point in the union of the subsets is proven if X is a uniformly convex Banach space and all the subsets of X are nonempty, convex and closed. Such a limiting sequence is of size q p (with the inequality being strict if there is at least one iteration with image in the same subset as its domain), where p of its elements (all of them if q = p ) are best proximity points between adjacent subsets. In the case that all the subsets A i X intersect, the above limit sequence reduces to a unique fixed point allocated within the intersection of all such subsets.

### 2 Asymptotic contractions and pseudocontractions in the intermediate sense in metric spaces

If H is a real Hilbert space with an inner product , and a norm and A is a nonempty closed convex subset of H, then T : A A is said to be an asymptotically β-strictly pseudocontractive self-mapping in the intermediate sense for some β [ 0 , 1 ) if

lim sup n sup x , y A ( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) 0 (2.1)

for some sequence { α n } [ 1 , ) , α n 1 as n [1-4,23]. Such a concept was firstly introduced in [1]. If (2.1) holds for β = 1 , then T : A A is said to be an asymptotically pseudocontractive self-mapping in the intermediate sense. Finally, if α n α [ 0 , 1 ) as n , then T : A A is asymptotically β-strictly contractive in the intermediate sense, respectively, asymptotically contractive in the intermediate sense if β = 1 . If (2.1) is changed to the stronger condition

( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) 0 ; x , y A , n N , (2.2)

then the above concepts translate into T : A A being an asymptotically β-strictly pseudocontractive self-mapping, an asymptotically pseudocontractive self-mapping and asymptotically contractive one, respectively. Note that (2.1) is equivalent to

T n x T n y 2 α n x y 2 + β ( I T n ) x ( I T n ) y 2 + ξ n ; x , y A , n N (2.3)

or, equivalently,

T n x T n y , x y 1 2 β [ ( α n + β ) x y 2 + ( β 1 ) T n x T n y 2 + ξ n ] ; x , y A , n N , (2.4)

where

ξ n : = max { 0 , sup x , y A ( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) } ; n N . (2.5)

Note that the high-right-hand-side term ( I T n ) x ( I T n ) y 2 of (2.3) is expanded as follows for any x , y A :

(2.6)

The objective of this paper is to discuss the various pseudocontractive in the intermediate sense concepts in the framework of metric spaces endowed with a homogeneous and translation-invariant metric and also to generalize them to the β-parameter to eventually be replaced with a sequence { β n } in ( 0 , 1 ) . Now, if instead of a real Hilbert space H endowed with an inner product , and a norm , we deal with any generic Banach space ( X , ) , then its norm induces a homogeneous and translation invariant metric d : X × X R 0 + defined by d ( x , y ) = d ( x y , 0 ) = x y 1 / 2 ; x , y A so that (2.6) takes the form

(2.7)

Define

μ n ( x , y ) : = min ( ρ [ 1 , 1 ] : d 2 ( x y , T n x T n y ) d 2 ( x , y ) + d 2 ( T n x , T n y ) + 2 ρ d ( x , y ) d ( T n x , T n y ) ) ; x , y A , n N , (2.8)

which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that

{ 1 } { ρ R : ( I T n ) x ( I T n ) y 2 d 2 ( x , y ) + d 2 ( T n x , T n y ) + 2 ρ d ( x , y ) d ( T n x , T n y ) } ( ) . (2.9)

The following result holds related to the discussion (2.7)-(2.9) in metric spaces.

Theorem 2.1Let ( X , d ) be a metric space and consider a self-mapping T : X X . Assume that the following constraint holds:

d 2 ( T n x , T n y ) α n ( x , y ) d 2 ( x , y ) + β n ( x , y ) ( d 2 ( x , y ) + d 2 ( T n x , T n y ) ) + 2 μ n ( x , y ) β n ( x , y ) d ( x , y ) d ( T n x , T n y ) + ξ n ( x , y ) ; x , y X , n N (2.10)

with

ξ n = ξ n ( x , y ) : = max ( 0 , ( 1 β n ( x , y ) ) d 2 ( T n x , T n y ) ( α n ( x , y ) + β n ( x , y ) ) d 2 ( x , y ) 2 μ n ( x , y ) β n ( x , y ) d ( x , y ) d ( T n x , T n y ) ) 0 ; x , y X   as   n (2.11)

for some parameterizing bounded real sequences { α n ( x , y ) } , { β n ( x , y ) } and { μ n ( x , y ) } of general terms α n = α n ( x , y ) , β n = β n ( x , y ) , μ n = μ n ( x , y ) satisfying the following constraints:

(2.12)

with lim sup n [ β n ( x , y ) max ( 1 , 1 + 2 μ n ( x , y ) ) ] < 1 and, furthermore, the following condition is satisfied:

( μ n ( x , y ) 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) ) 0 ; x , y X   as   n (2.13)

if and only if α n + 2 β n ( 1 + μ n ) 1 ; x , y X as n .

Then the following properties hold:

(i) lim n d ( T n x , T n y ) d ( x , y ) for any x , y X so that T : X X is asymptotically nonexpansive.

(ii) Let ( X , d ) be complete, d : X × X R 0 + be, in addition, a translation-invariant homogeneous norm and let ( X , ) ( X , d ) , with being the metric-induced norm from d : X × X R 0 + , be a uniformly convex Banach space. Assume also that T : X X is continuous. Then any sequence { T n x } ; x A is bounded and convergent to some point z x = z x ( x ) C , being in general dependent onx, in some nonempty bounded, closed and convex subsetCofA, whereAis any nonempty bounded subset ofX. Also, d ( T n x , T n + m x ) is bounded; n , m N , lim n d ( T n x , T n + m x ) = 0 ; x A , m N and z x = z x ( x ) = T z x C is a fixed point of the restricted self-mapping T : C C ; x A . Furthermore,

lim n ( d 2 ( T n + 1 x , T n + 1 y ) d 2 ( T n x , T n y ) ) = 0 ; x , y A . (2.14)

Proof Consider two possibilities for the constraint (2.10), subject to (2.11), to hold for each given x , y X and n N as follows:

(A) d ( T n x , T n y ) d ( x , y ) for any x , y X , n N . Then one gets from (2.10)

d 2 ( T n x , T n y ) ( α n + β n ) d 2 ( x , y ) + β n d 2 ( T n x , T n y ) + 2 μ n β n d 2 ( x , y ) + ξ n d ( T n x , T n y ) k a n d 2 ( x , y ) + ξ n 1 β n ; (2.15)

x , y A , n N , where

k a n = k a n ( x , y ) = α n + β n ( 1 + 2 μ n ) 1 β n 1 ; x , y X  as  n , (2.16)

which holds from (2.12)-(2.13) if lim sup n β n ( x , y ) < 1 since

( μ n ( x , y ) 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) ) 0 ;

x , y X as n in (2.13) is equivalent to (2.16). Note that 0 k a n 1 is ensured either with min ( α n + β n ( 1 + 2 μ n ) , 1 β n ) 0 or with max ( α n + β n ( 1 + 2 μ n ) , 1 β n ) 0 if

(2.17)

However, β n > 1 with ξ n > 0 has to be excluded because of the unboundedness or nonnegativity of the second right-hand-side term of (2.15).

(B) d ( T n x , T n y ) d ( x , y ) for some x , y X , n N . Then one gets from (2.10)

d ( T n x , T n y ) 2 ( α n + β n ) d 2 ( x , y ) + β n d 2 ( T n x , T n y ) + 2 μ n β n d 2 ( T n x , T n y ) + ξ n d ( T n x , T n y ) 2 k b n d 2 ( x , y ) + ξ n 1 β n ( 1 + 2 μ n ) , (2.18)

where

k b n = k b n ( x , y ) = α n + β n 1 β n ( 1 + 2 μ n ) 1 as  n , (2.19)

which holds from (2.12) and k b n 1 if lim sup n [ β n ( x , y ) max ( 1 , 1 + 2 μ n ( x , y ) ) ] < 1 , and

μ n ( x , y ) [ 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) , 1 β n ( x , y ) 2 β n ( x , y ) ) . (2.20)

Thus, (2.15)-(2.16), with the second option in the logic disjunction being true if and only if ξ n = 0 together with (2.18)-(2.20), are equivalent to (2.12)-(2.13) by taking k n = k n ( x , y ) to be either k a n or k b n for each n N . It then follows that lim sup n ( d ( T n x , T n y ) d ( x , y ) ) 0 ; x , y X from (2.15)-(2.19) since 0 k n = k n ( x , y ) 1 and k n ( x , y ) 1 ; x , y X as n . Thus, T : X X is asymptotically nonexpansive. Thus, Property (i) has been proven. Property (ii) is proven as follows. Consider the metric-induced norm equivalent to the translation-invariant homogeneous metric d : X × X R 0 + . Such a norm exists since the metric is homogeneous and translation-invariant so that norm and metric are formally equivalent. Rename A 0 A and define a sequence of subsets A j : = { T j x : x A 0 } of X. From Property (i), { d ( T n x , T n y ) } is bounded; x , y X if d ( x , y ) is finite, since it is bounded for any finite n N and, furthermore, it has a finite limit as n . Thus, all the collections of subsets i = 1 k A i ; k N are bounded since A 0 is bounded. Define the set C = C ( A 0 ) : = cl [ convex ( i = 1 A k ) ] which is nonempty bounded, closed and convex by construction. Since ( X , d ) is complete, ( X , ) ( X , d ) is a uniformly convex Banach space and T : C C is asymptotically nonexpansive from Property (i), then it has a fixed point z = T z C [1,23]. Since the restricted self-mapping T : C C is also continuous, one gets from Property (i)

lim n d ( T n x , T n z ) = lim n d ( T n x , z ) = d ( lim n T n x , z ) d ( x , z ) < ; x A . (2.21)

Then any sequence { T n x } is convergent (otherwise, the above limit would not exist contradicting Property (i)), and then bounded in C; x A . This also implies d ( T n x , T n + m x ) is bounded; x A , n , m N and lim n d ( T n x , T n + m x ) = 0 ; x A , m N . This implies also T n x z x ( x ) as n ; x A such that z x ( x ) = T z x ; x A which is then a fixed point of T : C C (otherwise, the above property lim n d ( T n x , T n + m x ) = 0 ; x A , m N would be contradicted). Hence, Property (ii) is proven. □

First of all, note that Property (ii) of Theorem 2.1 applies to a uniformly convex space which is also a complete metric space. Since the metric is homogeneous and translation-invariant, a norm can be induced by such a metric. Alternatively, the property could be established on any uniformly convex Banach space by taking a norm-induced metric which always exists. Conceptually similar arguments are used in later parallel results throughout the paper. Note that the proof of Theorem 2.1(i) has two parts: Case (A) refers to an asymptotically nonexpansive self-mapping which is contractive for any number of finite iteration steps and Case (B) refers to an asymptotically nonexpansive self-mapping which is allowed to be expansive for a finite number of iteration steps. It has to be pointed out concerning such a Theorem 2.1(ii) that the given conditions guarantee the existence of at least a fixed point but not its uniqueness. Therefore, the proof is outlined with the existence of a z Fix ( T | C ) for any nonempty, bounded and closed subset A of X. Note that the set C, being in general dependent on the initial set A, is bounded, convex and closed by construction while any taken nonempty set of initial conditions A X is not required to be convex. However, the property that all the sequences converge to fixed points opens two potential possibilities depending on particular extra restrictions on the self-mapping T : C C , namely: (1) the fixed point is not unique so that z x z for any x A (and any A in X) so that some set Fix ( T | C ) for some C = C ( A ) X contains more than one point. In other words, d 2 ( T n x , T n y ) 0 as n ; x , y A has not been proven although it is true that lim n ( d 2 ( T n + 1 x , T n + 1 y ) d 2 ( T n x , T n y ) ) = 0 ; x , y A ; (2) there is only a fixed point in X. The following result extends Theorem 2.1 for a modification of the asymptotically nonexpansive condition (2.10).

Theorem 2.2Let ( X , d ) be a metric space and consider the self-mapping T : X X . Assume that the constraint below holds:

d 2 ( T n x , T n y ) α n ( x , y ) d 2 ( x , y ) + β n ( x , y ) ( d 2 ( x , y ) + d 2 ( T n x , T n y ) ) + 2 μ n ( x , y ) β n ( x , y ) d 2 ( T n x , T n y ) + ξ n ( x , y ) ; x , y X , n N (2.22)

with

ξ n = ξ n ( x , y ) : = max ( 0 , ( 1 β n ( x , y ) ) d 2 ( T n x , T n y ) ( α n ( x , y ) + β n ( x , y ) ) d 2 ( x , y ) 2 μ n ( x , y ) β n ( x , y ) d 2 ( T n x , T n y ) ) 0 ; x , y X   as   n (2.23)

for some parameterizing real sequences α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) satisfying, for any n N ,

{ α n ( x , y ) } [ 0 , ) , { μ n ( x , y ) } [ 1 , 1 β n ( x , y ) 2 β n ( x , y ) ) , { β n ( x , y ) } [ 0 , 1 ] ; x , y X , n N . (2.24)

Then the following properties hold:

(i) lim n d ( T n x , T n y ) d ( x , y ) so that T : X X is asymptotically nonexpansive, and then lim n d ( T n x , T n y ) d ( x , y ) ; x , y X if

α n ( x , y ) + β n ( x , y ) 1 β n ( x , y ) ( 1 + 2 μ n ( x , y ) ) 1 μ n ( x , y ) [ 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) , 1 β n ( x , y ) 2 β n ( x , y ) ) ; x , y X , n N (2.25)

and the following limit exists:

α n ( x , y ) + 2 β n ( x , y ) ( 1 + μ n ( x , y ) ) 1 ; x , y X   as   n . (2.26)

(ii) Property (ii) of Theorem 2.1 if ( X , d ) is complete and ( X , ) ( X , d ) is a uniformly convex Banach space under the metric-induced norm .

Sketch of the proof Property (i) follows in the same way as the proof of Property (i) of Theorem 2.1 for Case (B). Using proving arguments similar to those used to prove Theorem 2.1, one proves Property (ii). □

The relevant part in Theorem 2.1 being of usefulness concerning the asymptotic pseudocontractions in the intermediate sense and the asymptotic strict contractions in the intermediate sense relies on Case (B) in the proof of Property (i) with the sequence of constants k n ( x , y ) 1 ; x , y X , n N and k n ( x , y ) 1 ; as n , x , y X . The concepts of an asymptotic pseudocontraction and an asymptotic strict pseudocontraction in the intermediate sense motivated in Theorem 2.1 by (2.7)-(2.9), under the asymptotically nonexpansive constraints (2.10) subject to (2.11) and in Theorem 2.2 by (2.22) subject to (2.23) are revisited as follows in the context of metric spaces.

Definition 2.3 Assume that ( X , d ) is a complete metric space with d : X × X R 0 + being a homogeneous translation-invariant metric. Thus, T : A A is asymptotically β-strictly pseudocontractive in the intermediate sense if

lim sup n ( ( 1 β n ( 1 + 2 μ n ) ) d 2 ( T n x , T n y ) ( α n + β n ) d 2 ( x , y ) ) 0 ; x , y A (2.27)

for β n = β [ 0 , 1 ) ; n N and some real sequences { α n } , { μ n } being, in general, dependent on the initial points, i.e., α n = α n ( x , y ) , μ n = μ n ( x , y ) and

{ μ n } [ 1 , 1 β 2 β ) and { α n } [ 1 , ) ; n N , α n 1  and  μ n 1  as  n ; x , y A , n N . (2.28)

Definition 2.4 T : A A is asymptotically pseudocontractive in the intermediate sense if (2.30) holds with { μ n } [ 1 , 1 β n 2 β n ) , { β n } [ 0 , 1 ] , { α n } [ 1 , ) , α n 1 , β n 1 , μ n 1 as n and the remaining conditions as in Definition 2.3 with α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) .

Definition 2.5 T : A A is asymptotically β-strictly contractive in the intermediate sense if α n [ 0 , ) , β n = β [ 0 , 1 ) , μ n [ 1 , 1 β 2 β ) ; n N , μ n μ [ 1 , 1 β 2 β min ( 1 , 1 α + β ) ) , α n α [ 0 , 1 ) as n , in Definition 2.3 with α n = α n ( x , y ) , μ n = μ n ( x , y ) .

Definition 2.6 T : A A is asymptotically contractive in the intermediate sense if α n [ 0 , ) , { β n } [ 0 , 1 ) , μ n [ 1 , 1 β n 2 β n ) ; n N , μ n μ [ 1 , 1 + α 2 ) , α n α [ 0 , 1 ) , and β n β = 1 as n in Definition 2.3 with α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) .

Remark 2.7 Note that Definitions 2.3-2.5 lead to direct interpretations of their role in the convergence properties under the constraint (2.22), subject to (2.23), by noting the following:

(1) If T : A A is asymptotically β-strictly pseudocontractive in the intermediate sense (Definition 2.3), then the real sequence { k 1 n } of asymptotically nonexpansive constants has a general term k 1 n : = ( α n + β 1 β ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β 1 + β ) 1 / 2 , ) [ 1 , ) ; n N , and it converges to a limit k 1 = 1 since ξ n 0 and α n 1 as n ; x , y A from (2.22) since μ n 1 from (2.27). Then T : A A is trivially asymptotically nonexpansive as expected.

(2) If T : A A is asymptotically pseudocontractive in the intermediate sense (Definition 2.4), then the sequence { k 2 n } of asymptotically nonexpansive constants has the general term: k 2 n : = ( α n + β n 1 β n ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β n 1 + β n ) 1 / 2 , ) [ 1 , ) ; n N , and it converges to a limit k 2 = 1 since α n 1 , β n 1 as n . Then T : A A is also trivially asymptotically nonexpansive as expected. Since α n 1 , note that β n > β k 2 n > k 1 n and β n < β k 2 n < k 1 n for any n N , while k 1 n 1 , k 2 n 1 as n since ξ n 0 as n ; x , y A from (2.22)-(2.23).

(3) If T : A A is asymptotically β-strictly contractive in the intermediate sense (Definition 2.5), then the sequence of asymptotically contractive constants is defined by k 3 n : = ( α n + β 1 β ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β 1 + β ) 1 / 2 , ) [ ( β 1 + β ) 1 / 2 , ) ; n N and k 3 n k 3 = ( α + β 1 β ( 1 + 2 μ ) ) 1 / 2 [ 0 , 1 ) as n for any μ [ 1 , 1 α 2 β 2 β ) such that μ n μ as n , since α + 2 β ( 1 + μ ) < 1 . Then T : A A is an asymptotically strict contraction as expected since ξ n 0 as n ; x , y A from (2.22)-(2.23). Note that the asymptotic convergence rate is arbitrarily fast as α and β are arbitrarily close to zero, since k 3 = O ( α + β ) = o ( α + β ) = o ( max ( α , β ) ) becomes also arbitrarily close to zero, and k 3 2 K ( α + β ) with K = K ( β , μ ) = 1 1 β ( 1 + 2 μ ) ( 0 , ) .

(4) If T : A A is asymptotically contractive in the intermediate sense (Definition 2.6), then the sequence of asymptotically contractive constants is defined by

k 4 n : = ( α n + β n 1 β n ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β n 1 + β n ) 1 / 2 , ) [ ( β n 1 + β n ) 1 / 2 , ) ; n N

with β n β = 1 and k 4 n k 4 = ( 1 + α 2 | μ | ) 1 / 2 [ 1 2 , 1 ) as n for some μ [ 1 , 1 + α 2 ) since μ < 0 with | μ | > 1 + α 2 so that k 4 [ ( 1 + α 2 ) 1 / 2 , 1 ) . Then T : A A is an asymptotically strict contraction as expected since ξ n 0 as n ; x , y A from (2.23). Note that k 3 = k 4 if μ < 0 and | μ | = 1 + α 2 α and k 4 2 1 2 + o ( α ) 1 2 . Note also that k 3 < k 4 if μ 0 and | μ | < 1 + α 2 α , while k 3 k 4 if μ 0 and | μ | 1 + α 2 α . In the first case, the convergence to fixed points (see Theorem 2.8 below) is guaranteed to be asymptotically faster if the self-mapping is asymptotically β-strictly contractive in the intermediate sense than if it is just asymptotically contractive in the intermediate sense if β n > β , n N . Note also that if the sequences { α n } and { μ n } are identical in both cases, then k 3 n < k 4 n for any n N such that β n > β and k 3 n k 4 n for any n N such that β n β .

(5) The above considerations could also be applied to Theorem 2.1 for the case d ( T n x , T n y ) d ( x , y ) (Case (B) in the proof of Property (i)) being asymptotically nonexpansive for the asymptotically nonexpansive condition (2.10) subject to (2.11).

The subsequent result, being supported by Theorem 2.2, relies on the concepts of asymptotically contractive and pseudocontractive self-mappings in the intermediate sense. Therefore, it is assumed that { α n ( x , y ) } [ 1 , ) .

Theorem 2.8Let ( X , d ) be a complete metric space endowed with a homogeneous translation-invariant metric d : X × X R 0 + and consider the self-mapping T : X X . Assume that ( X , ) ( X , d ) is a uniformly convex Banach space endowed with a metric-induced norm from the metric d : X × X R 0 + . Assume that the asymptotically nonexpansive condition (2.22), subject to (2.23), holds for some parameterizing real sequences α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) satisfying, for any n N ,

{ α n ( x , y ) } [ 1 , ) , { μ n ( x , y ) } [ 1 , 1 β n ( x , y ) 2 β n ( x , y ) ) , { β n ( x , y ) } [ 0 , β ) [ 0 , 1 ] ; (2.29)

x , y X , n N . Then lim n d ( T n x , T n y ) d ( x , y ) for any x , y X satisfying the conditions

α n ( x , y ) + β n ( x , y ) 1 β n ( x , y ) ( 1 + 2 μ n ( x , y ) d ( x , y ) ) 1 ; α n ( x , y ) + 2 β n ( x , y ) ( 1 + μ n ( x , y ) ) 1 ; x , y X   as   n . (2.30)

Furthermore, the following properties hold:

(i) T : C C is asymptoticallyβ-strictly pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set C = C ( A ) X and any given nonempty, bounded and closed subset A X of initial conditions if (2.29) hold with 0 β n = β < 1 , { μ n } [ 1 , 1 β 2 β ) , { α n } [ 1 , ) , α n 1 and μ n 1 as n ; x , y A , n N . Also, T : C C has a fixed point for any such setCif T : X X is continuous.

(ii) T : C C is asymptotically pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set C = C ( A ) X and any given nonempty, bounded and closed subset A X of initial conditions if (2.29) hold with { β n } [ 0 , 1 ] , { μ n } [ 1 , 1 β n 2 β n ) , { α n } [ 1 , ) , β n 1 , α n 1 and μ n 1 as n ; x , y A , n N . Also, T : C C has a fixed point for any such setCif T : X X is continuous.

(iii) If (2.29) hold with α n [ 0 , ) , β n = β [ 0 , 1 ) , μ n [ 1 , 1 β 2 β ) , μ n μ [ 1 , 1 α 2 β 2 β ) ; n N and α n α [ 0 , 1 ) as n , then T : X X is asymptoticallyβ-strictly contractive in the intermediate sense. Also, T : X X has a unique fixed point.

(iv) If (2.29) hold with α n [ 0 , ) , { β n } [ 0 , 1 ) , μ n [ 1 , 1 β n 2 β n ) , μ n μ [ 1 , 1 + α 2 ) ; n N , β n 1 and α n α [ 0 , 1 ) as n , then T : X X is asymptotically strictly contractive in the intermediate sense. Also, T : X X has a unique fixed point.

Proof (i) It follows from Definition 2.3 and the fact that Theorem 2.2 holds under the particular nonexpansive condition (2.22), subject to (2.23), so that T : A A is asymptotically nonexpansive (see Remark 2.7(1)). Property (ii) follows in a similar way from Definition 2.4 (see Remark 2.7(2)). Properties (iii)-(iv) follow from Theorem 2.2 and Definitions 2.5-2.6 implying also that the asymptotically nonexpansive self-mapping T : X X is also a strict contraction, then continuous with a unique fixed point, since α + 2 β ( 1 + μ ) < 1 (see Remark 2.7(3)) and μ < 0 with | μ | > 1 + α 2 (see Remark 2.7(4)), respectively. (The above properties could also be got from Theorem 2.1 for Case (B) of the proof of Theorem 2.1(ii) - see Remark 2.7(5).) □

Example 2.9 Consider the time-varying pth order nonlinear discrete dynamic system

x k + n = T n x k x k + 1 + F n ( x k ) ( x k + 1 x k ) + η n ( x k ) ; x 1 A 1 R p , k N ; (2.31)

k , n N for some given nonempty bounded set A 1 , where { F n k } is a R p × p matrix sequence of elements F n k : R p R p × p with F n k = F n ( x k ) and η n k : R p R p with η n k = η n ( x k ) ; k N , and T : R p R p defines the state-sequence trajectory solution { x k ( x 1 , x 2 ) } . Equation (2.13) requires the consistency constraint F 1 1 to calculate x 2 . However, other discrete systems being dealt with in the same way as, for instance, that obtained by replacing x k + 1 x k 1 in (2.31) with the initial condition x 0 A 1 (and appropriate ad hoc re-definition of the mapping which generates the trajectory solution from given initial conditions) do not require such a consistency constraint. The dynamic system (2.31) is asymptotically linear if η n ( x ) 0 as n ; x R p . Note that for the Euclidean distance (and norm), d ( x k + n , x k + 1 ) F n k d ( x k , x k + 1 ) + η n k ; k N . Assume that the squared spectral norm of F n k is upper-bounded by k n k 2 = k n 2 ( x k ) = α n k + β n k 1 β n k ( 1 + 2 μ n k ) for some parameterizing scalar sequences { α n k } , { β n k } and { μ n } which can be dependent, in a more general case, on the state x k . This holds, for instance, if F n k = F n ( x k ) = 1 a n k ( α n k + β n k 1 β n k ( 1 + 2 μ n k ) ) 1 / 2 P n k , where { a n k } is a real positive sequence satisfying a n k P n k and P n k ( = P n ( x k ) ) : N × R p R p × p both being potentially dependent on the state as the rest of the parameterizing sequences. Since the spectral norm equalizes the spectral radius if the matrix is symmetric, then k n can be taken exactly as the spectral radius of F n k in such a case, i.e., it equalizes the absolute value of its dominant eigenvalue. We have to check the condition

lim sup n ( ( 1 + 2 μ n k β n k ) d 2 ( T n x k , T n y k ) ( α n k + β n k ) d 2 ( x k , y k ) ) 0 ; k N (2.32)

provided, for instance, that the distance is the Euclidean distance, induced by the Euclidean norm, then both being coincident, and provided also that we take the metric space ( R p , d ) which holds, in particular, if

(a) { | η n k | } R 0 + , { α n k } [ 1 , ) , β n k = β [ 0 , 1 ) , { μ n k } [ 1 , 1 β 2 β ) ; n , k N , α n k 1 and η n k 0 , μ n k 1 , as n ; k N . This implies that k 1 n k 2 = α n k + β 1 β n k ( 1 + 2 μ n k ) 1 ; n , k N and k 1 n k 1 as n ; k N . Thus, T : R p R p is asymptotically nonexpansive being also an asymptotic strict β-pseudocontraction in the intermediate sense. This also implies that (2.31) is globally stable as it is proven as follows. Assume the contrary so that there is an infinite subsequence L u of { x n k } which is unbounded, and then there is also an infinite subsequence L u a which is strictly increasing. Since η n k = η n ( x k ) 0 and k 1 n k = k 1 n ( x k ) 1 as n ; k N , one has that for x 1 A 1 , any given k N and some sufficiently large m 01 = m 01 ( x k ) , m 02 = m 02 ( x k ) N , ε 1 = ε 1 ( m 01 ) R + , ε 2 = ε 2 ( m 02 ) R + such that k m 1 = k m 1 ( x k ) 1 + ε 1 and η m 2 ( x k ) ε 2 ; m 1 m 01 , m 2 m 02 . Now, take m 0 = max ( m 01 , m 02 ) and ε = max ( ε 1 , ε 2 ) . Then x k + m / x k 1 + ε + ε / x k ; m ( N ) m 0 and any given k N . If x k 0 , then stability holds trivially. Assume not, and there are unbounded solutions. Thus, take x k ( 0 ) , x k + m L u a such that x k + m x k M for any given M R + , m ( N ) m ¯ and some m ¯ = m ¯ ( M ) m 0 . Note that since L u a is a strictly increasing real sequence { M ( m ¯ ) } implying M ( m ¯ ) as m ¯ , which leads to a contradiction to the inequality M 1 + ε ( 1 + 1 x k ) for ε 0 + for some sufficiently large m ¯ , then for some sufficiently large M, if such a strictly increasing sequence L u a exists. Hence, there is no such sequence, and then no unbounded sequence L u for any initial condition in A 0 . As a result, for any initial condition in any given subset A 1 of R p (even if it is unbounded), any solution sequence of (2.31) is bounded, and then (2.31) is globally stable. The above reasoning implies that there is an infinite collection of numerable nonempty bounded closed sets { A i R p : i N } , which are not necessarily connected, such that x k A k ; k N and any given x 0 A 0 . Assume that the set A 0 of initial conditions is bounded, convex and closed and consider the collection of convex envelopes { convex A i R p : i N } , define constructively the closure convex set C ( A 0 ) = cl ( convex ( i = 1 convex A i ) ) which is trivially bounded, convex and closed. Note that it is not guaranteed that i = 1 convex A i is either open or closed since there is a union of infinitely many closed sets involved. Note also that the convex hull of all the convex envelopes of the collection of sets is involved to ensure that A is convex since the union of convex sets is not necessarily convex (so that i = 1 convex A i is not guaranteed to be convex while A is convex). Consider now the self-mapping T ¯ : C C which defines exactly the same solution as T : R p R p for initial conditions in A 1 so that T ¯ is identified with the restricted self-mapping T : R p | C C from a nonempty bounded, convex and closed set to itself. Note that ( R p , d ) for the Euclidean distance is a convex metric space which is also complete since it is finite dimensional. Then F n k : A R p R p × p and η n k : A R p R p are both continuous, then T ¯ : C C is also continuous and has a fixed point in A from Theorem 2.8(i).

(b) If the self-mapping is asymptotically pseudocontractive in the intermediate sense, then the above conclusions still hold with the modification k 2 n k 2 = α n k + β n k 1 β n k ( 1 + 2 μ n k ) ( 1 ) 1 and μ n k α n k β n k 1 as n ; k N . From Remark 2.7(2), β n k > β k 2 n k > k 1 n k and β n k < β k 2 n k < k 1 n k for any n , k N . Thus the convergence is guaranteed to be faster for an asymptotic β-strict pseudocontraction in the intermediate sense than for an asymptotic pseudocontraction in the intermediate sense with a sequence { β n k } such that β n k > β ; n N with the remaining parameters and parametrical sequences being identical in both cases. If F n k : A R p R p × p and η n k : A R p R p ; n , k N are both continuous, then T ¯ : A A is continuous and has a fixed point in A from Theorem 2.8(ii).

(c) If T : X X is asymptotically β-strictly contractive in the intermediate sense, then k 3 n k = α n k + β n k 1 β n k ( 1 + 2 μ n k ) k 3 = o ( max ( α , β ) ) [ 0 , 1 ) ; k N so that it is asymptotically strictly contractive and has a unique fixed point from Theorem 2.8(iii).

(d) If T : X X is asymptotically contractive in the intermediate sense, k 4 n k = ( α n k + β n k 1 β n k ( 1 + 2 μ n k ) ) 1 / 2 k 4 1 2 + o ( α ) 1 2 ; k N . Thus, T : X X is an asymptotic strict contraction and has a unique fixed point from Theorem 2.8(iv).

Remark 2.10 Note that conditions like (2.32) can be tested on dynamic systems being different from (2.31) by redefining, in an appropriate way, the self-mapping which generates the solution sequence from given initial conditions. This allows to investigate the asymptotic properties of the self-mapping, the convergence of the solution to fixed points, then the system stability, etc. in a unified way for different dynamic systems. Close considerations can be discussed for different dynamic systems and convergence of the solutions generated by the different cyclic self-mappings defined on the union of several subsets to the best proximity points of each of the involved subsets.

### 3 Asymptotic contractions and pseudocontractions of cyclic self-mappings in the intermediate sense

Let A , B X be nonempty subsets of X. T : A B A B is a cyclic self-mapping if T ( A ) B and T ( B ) A . Assume that the asymptotically nonexpansive condition (2.10), subject to (2.11), is modified as follows:

(3.1)

(3.2)

with ( ξ n γ n ( x , y ) D 2 ) 0 ; x , y X as n , and that the asymptotically nonexpansive condition (2.22), subject to (2.23), is modified as follows:

(3.3)

(3.4)

with ( ξ n γ n ( x , y ) D 2 ) 0 ; x , y X as n , where { γ n ( x , y ) } [ 0 , ) and D = dist ( A , B ) 0 . If A B , then D = 0 and Theorems 2.1, 2.2 and 2.8 hold with the replacement A A B . Then if A and B are closed and convex, then there is a unique fixed point of T : A B A B in A B . In the following, we consider the case that A B = so that D > 0 . The subsequent result based on Theorems 2.1, 2.2 and 2.8 holds.

Theorem 3.1Let ( X , d ) be a metric space and let T : A B A B be a cyclic self-mapping, i.e., T ( A ) B and T ( B ) A , whereAandBare nonempty subsets ofX. Define the sequence { k n } n N [ 0 , ) of asymptotically nonexpansive iteration-dependent constants as follows:

( x , y ) ( A × B ) ( B × A ) , n N provided that T : A B A B satisfies the constraint (3.1), subject to (3.2), and

[ ( d ( T n x , T n y ) d ( x , y ) β n = 1 ) ( γ n = 0 ) , ( x , y ) ( A × B ) ( B × A ) , n N ] (3.6)

and

k n = k n ( x , y ) = α n + β n 1 β n ( 1 + 2 μ n ) 1 ; (3.7)

n N for x A ( y B ) and for x B ( y A ) provided that T : A B A B satisfies the constraint (3.3) subject to (3.4) provided that the parameterizing bounded real sequences { α n ( x , y ) } , { β n ( x , y ) } , { μ n ( x , y ) } and { γ n ( x , y ) } of general terms α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) fulfill the following constraints:

(3.8)

γ n = γ n ( x , y ) max ( 0 , 1 k n ) and assuming that the following limits exist:

(3.9)

Then, the following properties hold:

(i) T : A B A B satisfies (3.3) subject to (3.4)-(3.9); ( x , y ) ( A × B ) ( B × A ) . Then

lim n d ( T n x , T n y ) [ D , d ( x , y ) ] ; ( x , y ) ( A × B ) ( B × A )

so that T : A B A B is a cyclic asymptotically nonexpansive self-mapping. If x A is a best proximity point ofAand y B is a best proximity point ofB, then lim n d ( T n x , T n y ) = D and T 2 n x z x = z ( x ) and T 2 n y z y = z ( y ) , which are best proximity points ofAandB (not being necessarily identical toxandy), respectively if T : A B A B is continuous.

(ii) Property (i) also holds if T : A B A B satisfies (3.1) subject to (3.2), (3.7), (3.8)-(3.9) and (3.5b) provided that d ( T n x , T n y ) d ( x , y ) ; ( x , y ) ( A × B ) ( B × A ) .

Proof The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)

(3.10)

and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)

d ( T n x , T n y ) 2 k b n d 2 ( x , y ) + ξ n + γ n D 2 1 β n ( 1 + 2 μ n ) lim n d ( T n x , T n y ) [ D , d ( x , y ) ] ; ( x , y ) ( A × B ) ( B × A ) (3.11)

since T n x , T n x A B ; n N since T ( A ) B and T ( B ) A , and k n 1 and γ n ( x , y ) ( 1 k n ) 0 as n ; ( x , y ) ( A × B ) ( B × A ) . Note that (3.8) implies that there is no division by zero in (3.11). Now, assume that (3.10) holds with β n = 1 . From (3.8) and (3.2), μ n [ 1 + α n 2 , 0 ] , equivalently, | μ n | 1 + α n 2 and d ( T n x , T n y ) > α n + 1 2 | μ n | d 2 ( x , y ) d 2 ( x , y ) , which contradicts (3.5a) if ξ n > 0 so that β n = 1 in (3.5a) under (3.7) implies that ξ n = 0 and, since γ n = 0 from (3.6), there is no division by zero on the right-hand side of (3.10) if β n = 1 .

Also, if T : A B A B is continuous, then lim n d ( T 2 n x , T 2 n y ) = d ( lim n T 2 n x , lim n T 2 n y ) = D so that T 2 n x A ; n N , lim n T 2 n x cl A , T 2 n y B and lim n T 2 n x cl B since T ( A ) B and T ( B ) A . This proves Properties (i)-(ii). □

Remark 3.2 Note that Theorem 3.1 does not guarantee the convergence of { T 2 n x } and { T 2 n y } to best proximity points if the initial points for the iterations x A and y B are not best proximity points if T : A B A B is not contractive.

The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings with k n = α n + β n ( 1 + 2 μ n ) 1 β n < 1 ; n N subject to lim n k n = k c 1 .

Theorem 3.3Let ( X , d ) be a metric space and let T : A B A B be a cyclic self-mapping which satisfies the asymptotically nonexpansive constraint (3.1), subject to (3.2), whereAandBare nonempty subsets ofX. Let the sequence { k n } n N [ 0 , 1 ) of asymptotically nonexpansive iteration-dependent constants be defined by a general term k n ( x , y ) = k n : = α n + β n ( 1 + 2 μ n ) 1 β n [ 0 , 1 ) under the constraints γ n ( x , y ) = γ n : = δ j ( 1 k n ) ( 1 β n ) = o ( 1 β n ) , β n 1 μ n 1 + α n 2 , n N and lim n k n = 1 . Then the subsequent properties hold:

(i) The following limits exist:

lim n d ( T n x , T n y ) = D ; ( x , y ) ( A × B ) ( B × A ) ; lim n d ( T n x , T n + 1 x ) = D ; x A B . (3.12)

(ii) Assume, furthermore, that ( X , d ) is complete, AandBare closed and convex and d : X × X is translation-invariant and homogeneous and ( X , d ) ( X , ) is uniformly convex where is the metric-induced norm. Then

lim n d ( T 2 n x , T 2 n + 2 x ) = lim n d ( T 2 n + 1 x , T 2 n + 3 x ) = 0 ; x A B , (3.13)

{ T 2 n x } z A , { T 2 n + 1 x } T z B ; x A , and { T 2 n y } T z B , { T 2 n + 1 x } z A ; x A , y B , wherezandTzare unique best proximity points inAandB, respectively. If A B , then z = T z is the unique fixed point of T : A B A B .

Proof Note from (3.9), under (3.6) and (3.7), that there is no division by zero on the right-hand side of (3.10) and ξ n = γ n = 0 if β n = 1 . Then one has from (3.1)-(3.2), (3.5a), (3.6) and (3.7) that

d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) k j n d 2 ( T j n x , T j n y ) + ( 1 k j n ) D 2 + ξ j n 1 β j n ; j , n N . (3.14)

There are several possible cases as follows.

Case A: { d ( T j n x , T j n y ) } is non-increasing. Then d ( T j n x , T j n y ) g = g ( x , y ) D as n ; ( x , y ) ( A × B ) ( B × A ) . Since { k n } [ 0 , 1 ) , one gets (3.12).

Case B: { d ( T j n x , T j n y ) } is non-decreasing. Then either { d ( T j n x , T j n y ) } d ( T j n x , T j n y ) g = g ( x , y ) D as n ; ( x , y ) ( A × B ) ( B × A ) or it is unbounded. Then it has a subsequence which diverges, from which a strictly increasing subsequence can be taken. But this contradicts lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 following from (3.14) subject to the given parametrical constraints. Thus, if { d ( T j n x , T j n y ) } is non-decreasing, it cannot have a strictly increasing subsequence so that it is bounded and has a finite limit as in Case A.

Case C: { d ( T j n x , T j n y ) } has an oscillating subsequence. It is proven that such a subsequence is finite. Assume not, then if lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 , there is an integer sequence { p n } of general term subject to p n ( n , 2 n ) such that

lim sup n ( d 2 ( T ( j + 1 ) n + p n x , T ( j + 1 ) n + p n y ) d 2 ( T j n + p n x , T j n + p n y ) ) > 0 ,

but the above expression is equivalent, for x p n = T p n x and y p n = T p n y which are in A B , but not jointly in either A or B, to

lim sup n ( d 2 ( T ( j + 1 ) n x p n , T ( j + 1 ) n y p n ) d 2 ( T j n + p n x p n , T j n + p n y p n ) ) > 0 ,

which contradicts lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 since both sequences {