Open Access

An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces

  • Abdul Rahim Khan1Email author,
  • Hafiz Fukhar-ud-din1, 2 and
  • Muhammad Aqeel Ahmad Khan2
Fixed Point Theory and Applications20122012:54

https://doi.org/10.1186/1687-1812-2012-54

Received: 25 May 2011

Accepted: 2 April 2012

Published: 2 April 2012

Abstract

In this article, we propose and analyze an implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces. Results concerning Δ-convergence as well as strong convergence of the proposed algorithm are proved. Our results are refinement and generalization of several recent results in CAT(0) spaces and uniformly convex Banach spaces.

Mathematics Subject Classification (2010): Primary: 47H09; 47H10; Secondary: 49M05.

Keywords

hyperbolic space nonexpansive map common fixed point implicit algorithm condition(A) semi-compactness Δ-convergence

1. Introduction

Most of the problems in various disciplines of science are nonlinear in nature. Therefore, translating linear version of a known problem into its equivalent nonlinear version is of paramount interest. Furthermore, investigation of numerous problems in spaces without linear structure has its own importance in pure and applied sciences. Several attempts have been made to introduce a convex structure on a metric space. One such convex structure is available in a hyperbolic space. Throughout the article, we work in the setting of hyperbolic spaces introduced by Kohlenbach [1], which is restrictive than the hyperbolic type introduced in [2] and more general than the concept of hyperbolic space in [3]. Spaces like CAT(0) and Banach are special cases of hyperbolic space. The class of hyperbolic spaces also contains Hadamard manifolds, Hilbert ball equipped with the hyperbolic metric [4], -trees and Cartesian products of Hilbert balls, as special cases.

Recent developments in fixed point theory reflect that the iterative construction of fixed points is vigorously proposed and analyzed for various classes of maps in different spaces. Implicit algorithms provide better approximation of fixed points than explicit algorithms. The number of steps of an algorithm also plays an important role in iterative approximation methods. The case of two maps has a direct link with the minimization problem [5].

The pioneering work of Xu and Ori [6] deals with weak convergence of one-step implicit algorithm for a finite family of nonexpansive maps. They also posed an open question about necessary and sufficient conditions required for strong convergence of the algorithm. Since then many articles have been published on weak and strong convergence of implicit algorithms (see [710] and references therein).

It is worth mentioning that introducing and analyzing a general iterative algorithm in a more general setup is a problem of interest in theoretical numerical analysis. Very recently, Khan et al. [11] proposed and analyzed a general algorithm for strong convergence results in CAT(0) spaces. We do not know whether their work can be extended to hyperbolic spaces. The purpose of this article is to investigate Δ- convergence as well as strong convergence through a two-step implicit algorithm for two finite families of nonexpansive maps in the more general setup of hyperbolic spaces. Our results can be viewed as refinement and generalization of several well-known results in CAT(0) spaces and uniformly convex Banach spaces.

2. Preliminaries and lemmas

Let (X, d) be a metric space and K be a nonempty subset of X. Let T be a selfmap on K. Denote by F(T) = {x K : T(x) = x}, the set of fixed points of T. A selfmap T on K is said to be nonexpansive if d(Tx, Ty) ≤ d(x, y). Takahashi [12] introduced a convex structure on a metric space to obtain nonlinear version of some known fixed point results on Banach spaces.

We now describe an other convex structure on a metric space.

A hyperbolic space [1] is a metric space (X, d) together with a map W : X2 × [0, 1] → X satisfying:
  1. (1)

    d ( u , W ( x , y , α ) ) ( 1 - α ) d ( u , x ) + α d ( u , y )

     
  2. (2)

    d ( W ( x , y , α ) , W ( x , y , β ) ) = | α - β | d ( x , y )

     
  3. (3)

    W ( x , y , α ) = W ( y , x , ( 1 - α ) )

     
  4. (4)

    d ( W ( x , z , α ) , W ( y , w , α ) ) ( 1 - α ) d ( x , y ) + α d ( z , w )

     

for all x, y, z, w X and α, β [0, 1]. We denote the above defined hyperbolic space by (X, d, W); if it satisfies only (1), then it coincides with the convex metric space introduced by Takahashi [12]. A subset K of a hyperbolic space X is convex if W(x, y, α) K for all x, y K and α [0, 1].

A hyperbolic space (X, d, W) is said to be:
  1. (i)
    strictly convex [12] if for any x, y X and λ [0, 1], there exists a unique element z X such that
    d ( z , x ) = λ d ( x , y ) and d ( z , y ) = ( 1 - λ ) d ( x , y ) ;
     
  2. (ii)
    uniformly convex [13] if for all u, x, y X, r > 0 and ε (0, 2], there exists a δ (0, 1] such that
    d ( x , u ) r d ( y , u ) r d ( x , y ) ε r d W ( x , y , 1 2 ) , u ( 1 - δ ) r .
     

A map η : (0, ∞) × (0, 2] → (0, 1] which provides such a δ = η(r, ε) for given r > 0 and ε (0, 2], is called modulus of uniform convexity. We call η monotone if it decreases with r (for a fixed ε). A uniformly convex hyperbolic space is strictly convex (see [14]).

Lemma 2.1. [15] Let (X, d, W) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. For r > 0, ε (0, 2], a, x, y X, the inequalities
d ( x , a ) r , d ( y , a ) r and d ( x , y ) ε r
imply
d ( W ( x , y , λ ) , a ) ( 1 - 2 λ ( 1 - λ ) η ( s , ε ) ) r ,

where λ [0, 1] and sr.

The concept of Δ-convergence in a metric space was introduced by Lim [16] and its analogue in CAT(0) spaces has been investigated by Dhompongsa and Panyanak [17]. In this article, we continue the investigation of Δ-convergence in the general setup of hyperbolic spaces.

For this, we collect some basic concepts.

Let {x n } be a bounded sequence in a hyperbolic space X. For x X, define a continuous functional r(., {x n }): X → [0, ∞) by:
r ( x , { x n } ) = lim sup n d ( x , x n ) .
The asymptotic radius ρ = r({x n }) of {x n } is given by:
ρ = inf { r ( x , { x n } ) : x X } .
The asymptotic center of a bounded sequence {x n } with respect to a subset K of X is defined as follows:
A K ( { x n } ) = { x X : r ( x , { x n } ) r ( y , { x n } ) for any  y K } .

If the asymptotic center is taken with respect to X, then it is simply denoted by A({x n }). It is known that uniformly convex Banach spaces and even CAT(0) spaces enjoy the property that "bounded sequences have unique asymptotic centers with respect to closed convex subsets". The following lemma is due to Leustean [15] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

Lemma 2.2. [15] Let (X, d, W) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then every bounded sequence {x n } in × has a unique asymptotic center with respect to any nonempty closed convex subset K of X. Recall that a sequence {x n } in X is said to Δ-converge to x X if x is the unique asymptotic center of {u n } for every subsequence {u n } of {x n }. In this case, we write Δ - lim n x n = x and call x as Δ - limit of {x n }.

Iterative construction by means of classical algorithms like:
  1. (i)

    x n + 1 = α n x n + ( 1 - α n ) T x n , n 0 , [18],

     
  2. (ii)

    x n + 1 = α n x n + ( 1 - α n ) T ( β n x n + ( 1 - β n ) T x n , ) , n 0 , [19],

     

are vigorously analyzed for approximation of fixed points of various maps under suitable conditions imposed on the control sequences. The algorithm (i) exhibits weak convergence even in the setting of Hilbert space. Moreover, Chidume and Mutangadura [20] constructed an example for Lipschitz pseudocontractive map with a unique fixed point for which the algorithm (i) fails to converge.

Kirk [21] proved a fixed point theorem using Browder's type implicit algorithm (i.e., x t = (1 - t)x + tT(x t )) in a complete CAT (0) space. More precisely, he proved the following result:

Theorem 2.3. [21] Let K be a bounded closed convex subset of a complete CAT(0) space X and f : KK be a nonexpansive map. Fix x K and for each t [0, 1) let x t be the unique fixed point such that
x t [ x , T x t ] and d ( x , x t ) = t d ( x , T x t ) .

Then {x t } converges as t → 1- to the unique fixed point of f which is nearest to x.

Furthermore, he posed an open question: whether Theorem 2.3 can be extended to spaces of nonpositive curvature.

Denote the set {1, 2, 3, . . . N} by I.

In 2001, Xu and Ori [6] obtained weak convergence result using an implicit algorithm for a finite family of nonexpansive maps as follows:

Theorem 2.4. [6] Let {T i : i I} be a family of nonexpansive selfmaps on a closed convex subset C of a Hilbert space with i = 1 N F ( T i ) , let x0 C and let {α n } be a sequence in (0, 1) such that limn→∞ a n = 0. Then the sequence x n = α n xn-1+ (1 - α n )T n x n , where n ≥ 1 and T n = Tn(mod N)(here the mod N function takes values in I), converges weakly to a point in F.

In 2007, Plubtieng et al. [9] generalized the algorithm of Xu and Ori [6] for two finite families {T i : i I} and {S i : i I} of nonexpansive maps and studied its weak and strong convergence. Given x0 in K(a subset of Banach space), their algorithm reads as follows:
x n = α n x n - 1 + ( 1 - α n ) T n [ β n x n + ( 1 - β n ) S n x n ]
(2.1)

where {α n } and {β n } are two sequences in (0, 1).

Inspired and motivated by the work of Kirk [21], Xu and Ori [6] and Plubtieng et al. [9], we investigate Δ-convergence as well as strong convergence through a two-step implicit algorithm for two finite families of nonexpansive maps in the more general setup of hyperbolic spaces.

The two-step algorithm (2.1) can be defined in a hyperbolic space as:
x n = W ( x n - 1 , T n y n , α n ) , y n = W ( x n , S n x n , β n ) , n 1
(2.2)

where T n = Tn(mod N)and S n = Sn(mod N).

In order to establish that algorithm (2.2) exists, we define a map G1 : KK by: G1x = W(x0, T1W (x, S1x, β i ), α i ). For a given x0 K, the existence of x1 = W(x0, T1W (x1, S1x1, β1), α1) is guaranteed if G1 has a fixed point. Now for any u, v K and making use of (4), we have
d ( G 1 u , G 1 v ) = d ( W ( x 0 , T 1 W ( u , S 1 u , β 1 ) , α 1 ) , W ( x 0 , T 1 W ( v , S 1 v , β 1 ) , α 1 ) α 1 d ( T 1 W ( u , S 1 u , β 1 ) , T 1 W ( v , S 1 v , β 1 ) ) α 1 d ( W ( u , S 1 u , β 1 ) , W ( v , S 1 v , β 1 ) ) α 1 [ ( 1 - β 1 ) d ( u , v ) + β 1 d ( S 1 u , S 1 v ) ] α 1 [ ( 1 - β 1 ) d ( u , v ) + β 1 d ( u , v ) ] α 1 d ( u , v ) .

Since a1 (0, 1), therefore G1 is a contraction. By Banach contraction principle, G1 has a unique fixed point. Thus the existence of x1 is established. Continuing in this way, we can establish the existence of x2, x3 and so on. Thus the implicit algorithm (2.2) is well defined.

In 2010, Laowang and Panyanak [22] obtained a generalized version of Lemma 1.3 of Schu [23] in a uniformly convex hyperbolic space where the proof relies on the fact that modulus of uniform convexity η increases with r (for a fixed ε).

We prove the generalized version of Lemma 1.3 of Schu [23] in a uniformly convex hyperbolic space with monotone modulus of uniform convexity.

Lemma 2.5. Let (X, d, W) be a uniformly convex hyperbolic space with monotone modulus of uniform convexity η. Let × X and {a n } be a sequence in [b, c] for some b, c (0, 1). If {x n } and {y n } are sequences in × such that lim supn→∞ d(x n , x) ≤ r, lim supn→∞ d(y n , x) ≤ r and limn→∞ d(W (x n , y n , α n ), x) = r for some r ≥ 0, then limn→∞ d(x n , y n ) = 0.

Proof. The case r = 0 is trivial. Suppose r > 0 and assume limn→∞ d(x n , y n ) ≠ 0. If n1 , then d ( x n , y n ) λ 2 > 0 for some λ > 0 and for nn1. Since lim supn→∞ d(x n , x) ≤ r and lim supn→∞ d(y n , x) ≤ r, we have:
  1. (i)

    d ( x n , x ) r + 1 n ;

     
  2. (ii)

    d ( y n , x ) r + 1 n for each n ≥ 1.

     
Moreover, d ( x n , y n ) λ 2 r + 1 n λ 2 ( r + 1 ) , where λ 2 ( r + 1 ) 1 . So it follows from Lemma 2.1, that
d ( W ( x n , y n , α n ) , x ) 1 - 2 α n ( 1 - α n ) η r + 1 n , λ 2 ( r + 1 ) r + 1 n 1 - 2 α n ( 1 - α n ) η r + 1 , λ 2 ( r + 1 ) r + 1 n 1 - 2 b ( 1 - c ) η r + 1 , λ 2 ( r + 1 ) r + 1 n .
Thus, by letting n → ∞, we obtain
lim n d ( W ( x n , y n , α n ) , x ) 1 - 2 b ( 1 - c ) η r + 1 , λ 2 ( r + 1 ) r < r ,

a contradiction to the fact that limn→∞ d(W (x n , y n , α n ), x) = r for some r ≥ 0.   □

We now prove a metric version of a result due to Bose and Laskar [24] which plays a crucial role in proving Δ-convergence of the algorithm (2.2).

Lemma 2.6. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space and {x n } a bounded sequence in K such that A({x n }) = {y} and r({x n }) = ρ. If {y m } is another sequence in K such that

limm→∞r(y m , {x n }) = ρ, then limm→∞y m = y.

Proof. If y m y, then there exist a subsequence { y m j } of {y m } and M > 0 such that
d ( y m j , y ) M 2 for all  j .
Observe that the inequality:
( ρ + ε ) 1 - η ρ + 1 , M 2 ( ρ + 1 ) < ρ
(2.3)

holds when ε → 0, where ε (0, 1] and ρ is the asymptotic radius of {x n }.

Since A({x n }) = {y}, so for every ε (0, 1] there exists an integer N1 such that d(y, x n ) ≤ ρ+ε, for all nN1. Since lim m r ( y m , { x n } ) = ρ = lim j r ( y m j , { x n } ) , so there exists an integer j* such that r ( y m j , { x n } ) ρ + ε 2 for all jj*. Hence there exists an integer N2 such that d ( y m j , x n ) ρ + ε for all nN2.

That is,
d ( y , x n ) ρ + ε ρ + 1 and d ( y m j , x n ) ρ + ε ρ + 1 ,

for all nN = max{N1, N2}.

Using Lemma 2.1, we have
d W y , y m j , 1 2 , x n 1 - η ρ + ε , M 2 ( ρ + 1 ) ( ρ + ε ) 1 - η ρ + 1 , M 2 ( ρ + 1 ) ( ρ + ε ) ,
so that letting n → ∞, we have
r W ( y , y m j , 1 2 ) , { x n } 1 - η ρ + 1 , M 2 ( ρ + 1 ) ( ρ + ε ) .

Now let ε → 0 and use (2.3) to conclude that r ( W ( y , y m j , 1 2 ) , {x n }) < ρ which contradicts the fact that ρ is the asymptotic radius of {x n }. Hence limm→∞y m = y.   □

From now on for two finite families {T i : i I} and {S i : i I} of maps, we set F = i = 1 N ( F ( T i ) F ( S i ) ) ϕ

Lemma 2.7. Let K be a nonempty closed convex subset of a hyperbolic space × and let {T i : i I} and {S i : i I} be two finite families of nonexpansive selfmaps on K such that Fϕ. Then for the sequence {x n } defined implicitly in (2.2), we have limn→∞ d(x n , p) exists for each p F.

Proof. For any p F, it follows from (2.2) that
d ( x n , p ) = d ( W ( x n - 1 , T n y n , α n ) , p ) ( 1 - α n ) d ( x n - 1 , p ) + α n d ( T n y n , p ) ( 1 - α n ) d ( x n - 1 , p ) + α n d ( y n , p ) = ( 1 - α n ) d ( x n - 1 , p ) + α n d ( W ( x n , S n x n , β n ) , p ) ( 1 - α n ) d ( x n - 1 , p ) + α n ( 1 - β n ) d ( x n , p ) + α n β n d ( S n x n , p ) ( 1 - α n ) d ( x n - 1 , p ) + α n ( 1 - β n ) d ( x n , p ) + α n β n d ( x n , p ) ( 1 - α n ) d ( x n - 1 , p ) + α n d ( x n , p ) .
That is
d ( x n , p ) d ( x n - 1 , p ) .
(2.4)

It follows from (2.4) that limn→∞ d(x n , p) exists for each p F. Consequently, limn→∞ d(x n , F)

exists.   □

Lemma 2.8. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space × with monotone modulus of uniform convexity η and let {T i : i I} and{S i : i I} be two finite families of nonexpansive selfmaps of K such that Fϕ. Then for the sequence {x n } defined implicitly in (2.2), we have
lim n d ( x n , T l x n ) = lim n d ( x n , S l x n ) = 0 f o r e a c h l = 1 , 2 , , N .
Proof. It follows from Lemma 2.7 that, limn→∞ d(x n , p) exists for each p F. Assume that limn→∞ d(x n , p) = c. The case c = 0 is trivial. Next, we deal with the case c > 0. Note that
d ( y n , p ) = d ( W ( x n , S n x n , β n ) , p ) ( 1 - β n ) d ( x n , p ) + β n d ( S n x n , p ) d ( x n , p ) .
Taking lim sup on both sides in the above estimate, we have
lim sup n d ( y n , p ) c .

Since T n is nonexpansive, so lim supn→∞ d(T n y n , p) ≤ c. Further, lim supn→∞ d(xn-1, p) ≤ c.

Moreover,
lim n d ( x n , p ) = lim n d ( W ( x n - 1 , T n y n , α n ) , p ) = c .
So, by Lemma 2.5, we have
lim n d ( x n - 1 , T n y n ) = 0 .
(2.5)
Next, taking lim sup on both sides in the inequality
d ( x n , x n - 1 ) = d ( W ( x n - 1 , T n y n , α n ) , x n - 1 ) α n d ( T n y n , x n - 1 ) ,
we have
lim sup n d ( x n , x n - 1 ) 0 .
Hence,
lim sup n d ( x n , x n - 1 ) = 0 .
(2.6)
Clearly,
d ( x n , x n + l ) d ( x n , x n + 1 ) + d ( x n + 1 , x n + 2 ) + + d ( x n + l - 1 , x n + l ) .
Taking lim sup on both sides of the above inequality and using (2.6), we have
lim n d ( x n , x n + l ) = 0 for  l < N .
Further, observe that
d ( x n , p ) ( 1 - α n ) d ( x n - 1 , p ) + α n d ( T n y n , p ) ( 1 - α n ) d ( x n - 1 , T n y n ) + ( 1 - α n ) d ( T n y n , p ) + α n d ( y n , p ) ( 1 - α n ) d ( x n - 1 , T n y n ) + ( 1 - α n ) d ( y n , p ) + α n d ( y n , p ) ( 1 - α n ) d ( x n - 1 , T n y n ) + d ( y n , p ) .
Combining the inequalities after applying lim inf and lim sup on both sides in the above estimate and using (2.5), we get
c lim inf n d ( y n , p ) lim sup n d ( y n , p ) c .
That is,
lim n d ( W ( x n , S n x n , β n ) , p ) = lim n d ( y n , p ) = c .
Finally, by Lemma 2.5, we have
lim n d ( x n , S n x n ) = 0 .
Moreover,
d ( x n , T n x n ) d ( x n , T n y n ) + d ( T n y n , T n x n - 1 ) + d ( T n x n - 1 , T n x n ) ( 1 - α n ) d ( x n - 1 , T n y n ) + d ( x n - 1 , y n ) + d ( x n - 1 , x n ) ( 1 - α n ) d ( x n - 1 , T n y n ) + β n d ( x n , S n x n ) + 2 d ( x n - 1 , x n )
gives that
lim n d ( x n , T n x n ) = 0 .
For each l I, we have
d ( x n , T n + l x n ) d ( x n , x n + l ) + d ( x n + l , T n + l x n + l ) + d ( T n + l x n + l , T n + l x n ) 2 d ( x n , x n + l ) + d ( x n + l , T n + l x n + l ) .
Therefore
lim n d ( x n , T n + l x n ) = 0 for each  l I .
Since for each l I, the sequence {d(x n , T l x n )} is a subsequence of i = 1 N { d ( x n , T n + l x n ) } and lim n d ( x n , T n + l x n ) = 0 for each l I, therefore
lim n d ( x n , T l x n ) = 0 for each  l I .
Similarly, we have
lim n d ( x n , S n + l x n ) = 0 for each  l I ,

and hence

lim n d ( x n , S l x n ) = 0 for each  l I .    □

3. Convergence in hyperbolic spaces

In this section, we establish Δ- convergence and strong convergence of the implicit algorithm (2.2).

Theorem 3.1. Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space × with monotone modulus of uniform convexity η and let {T i : i I} and {S i : i I} be two finite families of nonexpansive selfmaps on K such that Fϕ.

Then the sequence {x n } defined implicitly in (2.2), Δ-converges to a common fixed point of {T i : i I} and {S i : i I}.

Proof. It follows from Lemma 2.7 that {x n } is bounded. Therefore by Lemma 2.2, {x n } has a unique asymptotic center, that is, A({x n }) = {x}. Let {u n } be any subsequence of {x n } such that A({u n }) = {u}. Then by Lemma 2.8, we have limn→∞ d(u n , T l u n ) = 0 = limn→∞ d(u n , S l u n ) for each l I. We claim that u is the common fixed point of {T i : i I} and {S i : i I}.

Now, we define a sequence {z m } in K by z m = T m u where T m = Tm(mod N).

Observe that
d ( z m , u n ) d ( T m u , T m u n ) + d ( T m u n , T m - 1 u n ) + + d ( T u n , u n ) d ( u , u n ) + i = 1 m - 1 d ( u n , T i u n ) .
Therefore, we have
r ( z m , { u n } ) = lim sup n d ( z m , u n ) lim sup n d ( u , u n ) = r ( u , { u n } ) .

This implies that |r(z m , {u n }) - r(u, {u n })| → 0 as m → ∞. It follows from Lemma 2.6 that Tm(mod N)u = u. Hence u is the common fixed point of {T i : i I}. Similarly, we can show that u is the common fixed point of {S i : i I}. Therefore u is the common fixed point of {T i : i I} and {S i : i I}. Moreover, limn→∞ d(x n , u) exists by Lemma 2.7.

Suppose x ≠ u. By the uniqueness of asymptotic centers,
lim sup n d ( u n , u ) < lim sup n d ( u n , x ) lim sup n d ( x n , x ) < lim sup n d ( x n , u ) = lim sup n d ( u n , u ) ,

a contradiction. Hence x = u. Since {u n } is an arbitrary subsequence of {x n }, therefore A({u n }) = {u} for all subsequences {u n } of {x n }. This proves that {x n } Δ- converges to a common fixed point of {T i : i I} and {S i : i I}.   □

Recall that a sequence {x n } in a metric space X is said to be Fejér monotone with respect to K (a subset of X) if d(xn+1, p) ≤ d(x n , p) for all p K and for all n ≥ 1. A map T : KK is semi-compact if any bounded sequence {x n } satisfying d(x n , Tx n ) → 0 as n → ∞, has a convergent subsequence.

Let f be a nondecreasing selfmap on [0, ∞) with f(0) = 0 and f(t) > 0 for all t (0, ∞) and let d(x, H) = inf{d(x, y): y H}. Then a family {T i : i I} of selfmaps on K with F 1 = i = 1 N F ( T i ) ϕ , satisfies condition (A) if
d ( x , T x ) f ( d ( x , F 1 ) ) for all  x K ,
holds for at least one T {T i : i I} or
max i I d ( x , T i x ) f ( d ( x , F 1 ) ) for all  x K ,

holds.

Different modifications of the condition (A) for two finite families of selfmaps have been made recently in the literature [25], [9] as follows:

Let {T i : i I} and {S i : i I} be two finite families of nonexpansive selfmaps on K with Fϕ. Then the two families are said to satisfy:
  1. (i)
    condition (B) on K if
    d ( x , T x ) f ( d ( x , F ) ) or d ( x , S x ) f ( d ( x , F ) ) for all  x K ,
     
holds for at least one T {T i : i I} or one S {S i : i I};
  1. (ii)
    condition (C) on K if
    1 2 { d ( x , T i x ) + d ( x , S i x ) } f ( d ( x , F ) ) for all  x K .
     

Note that the condition (B) and the condition (C) are equivalent to the condition (A) if T i = S i for all i I. We shall use condition (B) to study strong convergence of the algorithm (2.2).

For further development, we need the following technical result.

Lemma 3.2. [26] Let K be a nonempty closed subset of a complete metric space (X, d) and {x n } be Fejér monotone with respect to K. Then {x n } converges to some p K if and only if limn→∞d(x n , K) = 0.

Lemma 3.3. Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space × with monotone modulus of uniform convexity η and let {T i : i I} and {S i : i I} be two finite families of nonexpansive selfmaps on K such that Fϕ. Then the sequence {x n } defined implicitly in (2.2) converges strongly to p F if and only if limn→∞ d(x n , F) = 0.

Proof. It follows from (2.4) that {x n } is Fejér monotone with respect to F and limn→∞ d(x n , F) exists. Hence, the result follows from Lemma 3.2.   □

We now establish strong convergence of the algorithm (2.2) based on Lemma 3.3.

Theorem 3.4. Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space × with monotone modulus of uniform convexity η and let {T i : i I} and {S i : i I} be two finite families of nonexpansive selfmaps on K such that Fϕ. Suppose that a pair of maps T and S in {T i : i I} and {S i : i I}, respectively, satisfies condition (B). Then the sequence {x n } defined implicitly in (2.2) converges strongly to p F.

Proof. It follows from Lemma 2.7 that limn→∞ d(x n , F) exists. Moreover, Lemma 2.8 implies that limn→∞ d(x n , T l x n ) = d(x n , S l x n ) = 0 for each l I. So condition (B) guarantees that limn→∞ f(d(x n , F)) = 0. Since f is nondecreasing with f (0) = 0, it follows that limn→∞ d(x n , F) = 0. Therefore, Lemma 3.3 implies that {x n } converges strongly to a point p in F.   □

Theorem 3.5. Let K be a nonempty closed convex subset of a complete uniformly convex

hyperbolic space × with monotone modulus of uniform convexity η and let {T i : i I} and {S i : i I} be two finite families of nonexpansive selfmaps on K such that Fϕ. Suppose that one of the map in {T i : i I} and {S i : i I} is semi-compact. Then the sequence {x n } defined implicitly in (2.2) converges strongly to p F.

Proof. Use Lemma 2.8 and the line of action given in the proof of Theorem 3.4 in [9].   □

Remark 3.6. (1) Theorem 3.1 sets analogue of [ 17, Theorem 3.3], for two finite families

of nonexpansive maps on unbounded domain in a uniformly convex hyperbolic space X;
  1. (2)

    Lemma 3.3 improves [ 8, Theorem 1] and [ 10, Theorem 3.1] for two finite families of nonexpansive maps on X;

     
  2. (3)

    Theorem 3.4 extends and improves Theorem 3.3 (Theorem 3.4) of [9] from uniformly convex Banach space setting to the general setup of uniformly convex hyperbolic space;

     
  3. (4)

    Theorem 3.5 improves and extends [ 8, Theorem 2] for two finite families of nonexpansive maps on X.

     

Declarations

Acknowledgements

The authors wish to thank an anonymous referee for careful reading and valuable suggestions which led the manuscript to the present form. The authors A. R. Khan and H. Fukhar-ud-din are grateful to King Fahd University of Petroleum & Minerals for supporting this research. The author M. A. A. Khan gratefully acknowledges Higher Education Commission(HEC) of Pakistan for financial support during this research project.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals
(2)
Department of Mathematics, The Islamia University of Bahawalpur

References

  1. Kohlenbach U: Some logical metatheorems with applications in functional analysis. Trans Am Math Soc 2005, 357: 89–128.MATHMathSciNetView ArticleGoogle Scholar
  2. Goebel K, Kirk WA: Iteration processes for nonexpansive mappings Topological Methods in Nonlinear Functional Analysis. In Contemp Math Am Math Soc AMS, Providence, RI Edited by: Singh SP, Thomeier S, Watson B. 1983, 21: 115–123.Google Scholar
  3. Reich S, Shafrir I: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal 1990, 15: 537–558.MATHMathSciNetView ArticleGoogle Scholar
  4. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York 1984.Google Scholar
  5. Takahashi W: Iterative methods for approximation of fixed points and thier applications. J Oper Res Soc Jpn 2000, 43(1):87–108.MATHView ArticleGoogle Scholar
  6. Xu HK, Ori RG: An implicit iteration process for nonexpansive mappings. Num Funct Anal Optim 2001, 22(5–6):767–773.MATHMathSciNetView ArticleGoogle Scholar
  7. Fukhar-ud-din H, Khan AR: Convergence of implicit iterates with errors for mappings with unbounded domain in Banach spaces. Int J Math Math Sci 2005, 10: 1643–1653.MathSciNetView ArticleGoogle Scholar
  8. Liu JA: Some convergence theorems of implicit iterative process for nonexpansive mappings in Banach spaces. Math Commun 2002, 7: 113–118.MATHMathSciNetGoogle Scholar
  9. Plubtieng S, Ungchittrakool K, Wangkeeree R: Implicit iteration of two finite families for nonexpansive mappings in Banach spaces. Numer Funct Anal Optim 2007, 28(56):737–749.MATHMathSciNetView ArticleGoogle Scholar
  10. Sun ZH, He C, Ni YQ: Strong convergence of an implicit iteration process for nonexpansive mappings in Banach space. Nonlinear Funct Anal Appl 2003, 8(4):595–602.MATHMathSciNetGoogle Scholar
  11. Khan AR, Khamsi MA, Fukhar-ud-din H: Strong convergence of a general iteration scheme in CAT(0)-spaces. Nonlinear Anal 2011, 74: 783–791.MATHMathSciNetView ArticleGoogle Scholar
  12. Takahashi W: A convexity in metric spaces and nonexpansive mappings. Kodai Math Sem Rep 1970, 22: 142–149.MATHMathSciNetView ArticleGoogle Scholar
  13. Shimizu T, Takahashi W: Fixed points of multivalued mappings in certain convex metric spaces. Topol Methods Nonlinear Anal 1996, 8: 197–203.MATHMathSciNetGoogle Scholar
  14. Leustean L: A quadratic rate of asymptotic regularity for CAT(0)-spaces. J Math Anal Appl 2007, 325: 386–399.MATHMathSciNetView ArticleGoogle Scholar
  15. Leustean L: Nonexpansive iterations in uniformly convex W -hyperbolic spaces. In Contemp Math Am Math Soc AMS Edited by: Leizarowitz A, Mordukhovich BS, Shafrir I, Zaslavski A. 2010, 513: 193–209. Nonlinear Analysis and Optimization I: Nonlinear AnalysisGoogle Scholar
  16. Lim TC: Remarks on some FIxed point theorems. Proc Am Math Soc 1976, 60: 179–182.View ArticleGoogle Scholar
  17. Dhompongsa S, Panyanak B: On Δ-convergence theorems in CAT(0)-spaces. Comp Math Appl 2008, 56(10):2572–2579.MATHMathSciNetView ArticleGoogle Scholar
  18. Mann WR: Mean value methods in iteration. Proc Am Math Soc 1953, 4: 506–510.MATHView ArticleGoogle Scholar
  19. Ishikawa S: Fixed points by a new iteration method. Proc Am Math Soc 1974, 44: 147–150.MATHMathSciNetView ArticleGoogle Scholar
  20. Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschits pseudo-contarctions. Proc Am Math Soc 2001, 129: 2359–2363.MATHMathSciNetView ArticleGoogle Scholar
  21. Kirk WA: Geodesic geometry and FIxed point theory Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003). Univ Sevilla Secr Publ, Seville; 2003:195–225.Google Scholar
  22. Laowang W, Panyanak B: Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces. Fixed Point Theory Appl 2010., 2010: 367274, 11Google Scholar
  23. Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull Aust Math Soc 1991, 43: 153–159.MATHMathSciNetView ArticleGoogle Scholar
  24. Bose SC, Laskar SK: Fixed point theorems for certain class of mappings. J Math Phys Sci 1985, 19: 503–509.MATHMathSciNetGoogle Scholar
  25. Khan SH, Fukhar-ud-din H: Weak and strong convergence of a scheme for two nonexpansive mappings. Nonlinear Anal 2005, 8: 1295–1301.MathSciNetView ArticleGoogle Scholar
  26. Bauschke HH, Combettes PL: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer-Verlag, New York; 2011.View ArticleGoogle Scholar

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