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

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.

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], ℝ-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 [7–10] 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) $d\left(u,W\left(x,y,\alpha \right)\right)\le \left(1-\alpha \right)d\left(u,x\right)+\alpha d\left(u,y\right)$

• (2) $d\left(W\left(x,y,\alpha \right),W\left(x,y,\beta \right)\right)=|\alpha -\beta |d\left(x,y\right)$

• (3) $W\left(x,y,\alpha \right)=W\left(y,x,\left(1-\alpha \right)\right)$

• (4) $d\left(W\left(x,z,\alpha \right),W\left(y,w,\alpha \right)\right)\le \left(1-\alpha \right)d\left(x,y\right)+\alpha d\left(z,w\right)$

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:

• (i) strictly convex [12] if for any x, y ∈ X and λ ∈ [0, 1], there exists a unique element z ∈ X such that

  $$d\left(z,x\right)=\lambda d\left(x,y\right)\mathsf{\text{and}}d\left(z,y\right)=\left(1-\lambda \right)d\left(x,y\right);$$

• (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

  $$\left.\begin{array}{c}\hfill d\left(x,u\right)\le r\hfill \\ \hfill d\left(y,u\right)\le r\hfill \\ \hfill d\left(x,y\right)\ge \epsilon r\hfill \end{array}\right\}\Rightarrow d\left(W\left(x,y,\frac{1}{2}\right),u\right)\le \left(1-\delta \right)r.$$

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

$$d\left(x,a\right)\le r,d\left(y,a\right)\le r\mathsf{\text{and}}d\left(x,y\right)\ge \epsilon r$$

imply

$$d\left(W\left(x,y,\lambda \right),a\right)\le \left(1-2\lambda \left(1-\lambda \right)\eta \left(s,\epsilon \right)\right)r,$$

where λ ∈ [0, 1] and s ≥ r.

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:

$$r\left(x,\left\{x_n\right\}\right)=\underset{n\to \infty }{\text{lim}\text{sup}}d\left(x,x_n\right).$$

The asymptotic radius ρ = r({x _n }) of {x _n } is given by:

$$\rho =\text{inf}\left\{r\left(x,\left\{x_n\right\}\right):x\in X\right\}.$$

The asymptotic center of a bounded sequence {x _n } with respect to a subset K of X is defined as follows:

$$A_K\left(\left\{x_n\right\}\right)=\left\{x\in X:r\left(x,\left\{x_n\right\}\right)\le r\left(y,\left\{x_n\right\}\right)\mathsf{\text{for any }}y\in K\right\}.$$

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:

• (i) $x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)T{x}_n,n\ge 0$, [18],

• (ii) $x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)T\left({\beta }_n{x}_n+\left(1-{\beta }_n\right)T{x}_n,\right),n\ge 0$, [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 : K → K be a nonexpansive map. Fix x ∈ K and for each t ∈ [0, 1) let x _t be the unique fixed point such that

$$x_t\in \left[x,T{x}_t\right]\mathsf{\text{and}}d\left(x,x_t\right)=td\left(x,T{x}_t\right).$$

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 ${\cap }_{i=1}^{N}F\left(T_i\right)\ne \varnothing$, let x₀ ∈ C and let {α _n } be a sequence in (0, 1) such that lim_n→∞ a _n = 0. Then the sequence x _n = α _n x_n-1+ (1 - α _n )T _n x _n , where n ≥ 1 and T _n = T_{n(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:

$$x_n={\alpha }_n{x}_{n-1}+\left(1-{\alpha }_n\right)T_n\left[{\beta }_n{x}_n+\left(1-{\beta }_n\right)S_n{x}_n\right]$$

(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:

$$\begin{array}{c}{x}_n=W\left(x_{n-1},T_n{y}_n,{\alpha }_n\right),\\ y_n=W\left(x_n,S_n{x}_n,{\beta }_n\right),n\ge 1\end{array}$$

(2.2)

where T _n = T_{n(mod N)}and S _n = S_{n(mod N)}.

In order to establish that algorithm (2.2) exists, we define a map G₁ : K → K 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

$$\begin{array}{ll}\hfill d\left(G_{1}u,G_{1}v\right)& =d(W\left(x_0,T_{1}W\left(u,S_{1}u,{\beta }_1\right),{\alpha }_1\right),W\left(x_0,T_{1}W\left(v,S_{1}v,{\beta }_1\right),{\alpha }_1\right)\\ \le {\alpha }_{1}d\left(T_{1}W\left(u,S_{1}u,{\beta }_1\right),T_{1}W\left(v,S_{1}v,{\beta }_1\right)\right)\\ \le {\alpha }_{1}d\left(W\left(u,S_{1}u,{\beta }_1\right),W\left(v,S_{1}v,{\beta }_1\right)\right)\\ \le {\alpha }_1\left[\left(1-{\beta }_1\right)d\left(u,v\right)+{\beta }_{1}d\left(S_{1}u,S_{1}v\right)\right]\\ \le {\alpha }_1\left[\left(1-{\beta }_1\right)d\left(u,v\right)+{\beta }_{1}d\left(u,v\right)\right]\\ \le {\alpha }_{1}d\left(u,v\right).\end{array}$$

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 sup_n→∞ d(x _n , x) ≤ r, lim sup_n→∞ d(y _n , x) ≤ r and lim_n→∞ d(W (x _n , y _n , α _n ), x) = r for some r ≥ 0, then lim_n→∞ d(x _n , y _n ) = 0.

Proof. The case r = 0 is trivial. Suppose r > 0 and assume lim_n→∞ d(x _n , y _n ) ≠ 0. If n₁ ∈ ℕ, then $d\left(x_n,y_n\right)\ge \frac{\lambda }{2}>0$ for some λ > 0 and for n ≥ n₁. Since lim sup_n→∞ d(x _n , x) ≤ r and lim sup_n→∞ d(y _n , x) ≤ r, we have:

• (i) $d\left(x_n,x\right)\le r+\frac{1}{n}$;

• (ii) $d\left(y_n,x\right)\le r+\frac{1}{n}$ for each n ≥ 1.

Moreover, $d\left(x_n,y_n\right)\ge \frac{\lambda }{2}\ge \left(r+\frac{1}{n}\right)\frac{\lambda }{2\left(r+1\right)}$, where $\frac{\lambda }{2\left(r+1\right)}\le 1$. So it follows from Lemma 2.1, that

$$\begin{array}{ll}\hfill d\left(W\left(x_n,y_n,{\alpha }_n\right),x\right)& \le \left(1-2{\alpha }_n\left(1-{\alpha }_n\right)\eta \left(r+\frac{1}{n},\frac{\lambda }{2\left(r+1\right)}\right)\right)\left(r+\frac{1}{n}\right)\\ \le \left(1-2{\alpha }_n\left(1-{\alpha }_n\right)\eta \left(r+1,\frac{\lambda }{2\left(r+1\right)}\right)\right)\left(r+\frac{1}{n}\right)\\ \le \left(1-2b\left(1-c\right)\eta \left(r+1,\frac{\lambda }{2\left(r+1\right)}\right)\right)\left(r+\frac{1}{n}\right).\end{array}$$

Thus, by letting n → ∞, we obtain

$$\underset{n\to \infty }{\text{lim}}d\left(W\left(x_n,y_n,{\alpha }_n\right),x\right)\le \left(1-2b\left(1-c\right)\eta \left(r+1,\frac{\lambda }{2\left(r+1\right)}\right)\right)r<r,$$

a contradiction to the fact that lim_n→∞ 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

lim_m→∞r(y _m , {x _n }) = ρ, then lim_m→∞y _m = y.

Proof. If y _m ↛ y, then there exist a subsequence $\left\{y_{m_j}\right\}$ of {y _m } and M > 0 such that

$$d\left(y_{m_j},y\right)\ge \frac{M}{2}\mathsf{\text{for all }}j.$$

Observe that the inequality:

$$\left(\rho +\epsilon \right)\left(1-\eta \left(\rho +1,\frac{M}{2\left(\rho +1\right)}\right)\right)<\rho$$

(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 n ≥ N₁. Since $\underset{m\to \infty }{\text{lim}}r\left(y_m,\left\{x_n\right\}\right)=\rho =\underset{j\to \infty }{\text{lim}}r\left(y_{m_j},\left\{x_n\right\}\right)$, so there exists an integer j* such that $r\left(y_{m_j},\left\{x_n\right\}\right)\le \rho +\frac{\epsilon }{2}$ for all j ≥ j^*. Hence there exists an integer N₂ such that $d\left(y_{m_j},x_n\right)\le \rho +\epsilon$ for all n ≥ N₂.

That is,

$$d\left(y,x_n\right)\le \rho +\epsilon \le \rho +1\mathsf{\text{and}}d\left(y_{m_j},x_n\right)\le \rho +\epsilon \le \rho +1,$$

for all n ≥ N = max{N₁, N₂}.

Using Lemma 2.1, we have

$$\begin{array}{ll}\hfill d\left(W\left(y,y_{m_j},\frac{1}{2}\right),x_n\right)& \le \left(1-\eta \left(\rho +\epsilon ,\frac{M}{2\left(\rho +1\right)}\right)\right)\left(\rho +\epsilon \right)\\ \le \left(1-\eta \left(\rho +1,\frac{M}{2\left(\rho +1\right)}\right)\right)\left(\rho +\epsilon \right),\end{array}$$

so that letting n → ∞, we have

$$r\left(W\left(y,y_{m_j},\frac{1}{2}\right),\left\{x_n\right\}\right)\le \left(1-\eta \left(\rho +1,\frac{M}{2\left(\rho +1\right)}\right)\right)\left(\rho +\epsilon \right).$$

Now let ε → 0 and use (2.3) to conclude that $r(W\left(y,y_{m_j},\frac{1}{2}\right)$, {x _n }) < ρ which contradicts the fact that ρ is the asymptotic radius of {x _n }. Hence lim_m→∞y _m = y.   □

From now on for two finite families {T _i : i ∈ I} and {S _i : i ∈ I} of maps, we set $F={\cap }_{i=1}^N\left(F\left(T_i\right)\cap F\left(S_i\right)\right)\ne \varphi$

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 F ≠ ϕ. Then for the sequence {x _n } defined implicitly in (2.2), we have lim_n→∞ d(x _n , p) exists for each p ∈ F.

Proof. For any p ∈ F, it follows from (2.2) that

$$\begin{array}{ll}\hfill d\left(x_n,p\right)& =d\left(W\left(x_{n-1},T_n{y}_n,{\alpha }_n\right),p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},p\right)+{\alpha }_{n}d\left(T_n{y}_n,p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},p\right)+{\alpha }_{n}d\left(y_n,p\right)\\ =\left(1-{\alpha }_n\right)d\left(x_{n-1},p\right)+{\alpha }_{n}d\left(W\left(x_n,S_n{x}_n,{\beta }_n\right),p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},p\right)+{\alpha }_n\left(1-{\beta }_n\right)d\left(x_n,p\right)+{\alpha }_n{\beta }_{n}d\left(S_n{x}_n,p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},p\right)+{\alpha }_n\left(1-{\beta }_n\right)d\left(x_n,p\right)+{\alpha }_n{\beta }_{n}d\left(x_n,p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},p\right)+{\alpha }_{n}d\left(x_n,p\right).\end{array}$$

That is

$$d\left(x_n,p\right)\le d\left(x_{n-1},p\right).$$

(2.4)

It follows from (2.4) that lim_n→∞ d(x _n , p) exists for each p ∈ F. Consequently, lim_n→∞ 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 F ≠ ϕ. Then for the sequence {x _n } defined implicitly in (2.2), we have

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,T_l{x}_n\right)=\underset{n\to \infty }{\text{lim}}d\left(x_n,S_l{x}_n\right)=0foreachl=1,2,\cdots ,N.$$

Proof. It follows from Lemma 2.7 that, lim_n→∞ d(x _n , p) exists for each p ∈ F. Assume that lim_n→∞ d(x _n , p) = c. The case c = 0 is trivial. Next, we deal with the case c > 0. Note that

$$\begin{array}{ll}\hfill d\left(y_n,p\right)& =d\left(W\left(x_n,S_n{x}_n,{\beta }_n\right),p\right)\\ \le \left(1-{\beta }_n\right)d\left(x_n,p\right)+{\beta }_{n}d\left(S_n{x}_n,p\right)\\ \le d\left(x_n,p\right).\end{array}$$

Taking lim sup on both sides in the above estimate, we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}d\left(y_n,p\right)\le c.$$

Since T _n is nonexpansive, so lim sup_n→∞ d(T _n y _n , p) ≤ c. Further, lim sup_n→∞ d(x_n-1, p) ≤ c.

Moreover,

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,p\right)=\underset{n\to \infty }{\text{lim}}d\left(W\left(x_{n-1},T_n{y}_n,{\alpha }_n\right),p\right)=c.$$

So, by Lemma 2.5, we have

$$\underset{n\to \infty }{\text{lim}}d\left(x_{n-1},T_n{y}_n\right)=0.$$

(2.5)

Next, taking lim sup on both sides in the inequality

$$\begin{array}{ll}\hfill d\left(x_n,x_{n-1}\right)& =d\left(W\left(x_{n-1},T_n{y}_n,{\alpha }_n\right),x_{n-1}\right)\\ \le {\alpha }_{n}d\left(T_n{y}_n,x_{n-1}\right),\end{array}$$

we have

$$\underset{n\to \infty }{\text{lim}\text{sup}}d\left(x_n,x_{n-1}\right)\le 0.$$

Hence,

$$\underset{n\to \infty }{\text{lim}\text{sup}}d\left(x_n,x_{n-1}\right)=0.$$

(2.6)

Clearly,

$$d\left(x_n,x_{n+l}\right)\le d\left(x_n,x_{n+1}\right)+d\left(x_{n+1},x_{n+2}\right)+\cdots +d\left(x_{n+l-1},x_{n+l}\right).$$

Taking lim sup on both sides of the above inequality and using (2.6), we have

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,x_{n+l}\right)=0\mathsf{\text{for }}l<N.$$

Further, observe that

$$\begin{array}{ll}\hfill d\left(x_n,p\right)& \le \left(1-{\alpha }_n\right)d\left(x_{n-1},p\right)+{\alpha }_{n}d\left(T_n{y}_n,p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},T_n{y}_n\right)+\left(1-{\alpha }_n\right)d\left(T_n{y}_n,p\right)+{\alpha }_{n}d\left(y_n,p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},T_n{y}_n\right)+\left(1-{\alpha }_n\right)d\left(y_n,p\right)+{\alpha }_{n}d\left(y_n,p\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},T_n{y}_n\right)+d\left(y_n,p\right).\end{array}$$

Combining the inequalities after applying lim inf and lim sup on both sides in the above estimate and using (2.5), we get

$$c\le \underset{n\to \infty }{\text{lim}\text{inf}}d\left(y_n,p\right)\le \underset{n\to \infty }{\text{lim}\text{sup}}d\left(y_n,p\right)\le c.$$

That is,

$$\underset{n\to \infty }{\text{lim}}d\left(W\left(x_n,S_n{x}_n,{\beta }_n\right),p\right)=\underset{n\to \infty }{\text{lim}}d\left(y_n,p\right)=c.$$

Finally, by Lemma 2.5, we have

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,S_n{x}_n\right)=0.$$

Moreover,

$$\begin{array}{ll}\hfill d\left(x_n,T_n{x}_n\right)& \le d\left(x_n,T_n{y}_n\right)+d\left(T_n{y}_n,T_n{x}_{n-1}\right)+d\left(T_n{x}_{n-1},T_n{x}_n\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},T_n{y}_n\right)+d\left(x_{n-1},y_n\right)+d\left(x_{n-1},x_n\right)\\ \le \left(1-{\alpha }_n\right)d\left(x_{n-1},T_n{y}_n\right)+{\beta }_{n}d\left(x_n,S_n{x}_n\right)+2d\left(x_{n-1},x_n\right)\end{array}$$

gives that

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,T_n{x}_n\right)=0.$$

For each l ∈ I, we have

$$\begin{array}{ll}\hfill d\left(x_n,T_{n+l}{x}_n\right)& \le d\left(x_n,x_{n+l}\right)+d\left(x_{n+l},T_{n+l}{x}_{n+l}\right)+d\left(T_{n+l}{x}_{n+l},T_{n+l}{x}_n\right)\\ \le 2d\left(x_n,x_{n+l}\right)+d\left(x_{n+l},T_{n+l}{x}_{n+l}\right).\end{array}$$

Therefore

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,T_{n+l}{x}_n\right)=0\mathsf{\text{for each }}l\in I.$$

Since for each l ∈ I, the sequence {d(x _n , T _l x _n )} is a subsequence of ${\cup }_{i=1}^N\left\{d\left(x_n,T_{n+l}{x}_n\right)\right\}$ and $\underset{n\to \infty }{\text{lim}}d\left(x_n,T_{n+l}{x}_n\right)=0$ for each l ∈ I, therefore

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,T_l{x}_n\right)=0\mathsf{\text{for each }}l\in I.$$

Similarly, we have

$$\underset{n\to \infty }{\text{lim}}d\left(x_n,S_{n+l}{x}_n\right)=0\mathsf{\text{for each }}l\in I,$$

and hence

$\underset{n\to \infty }{\text{lim}}d\left(x_n,S_l{x}_n\right)=0\mathsf{\text{for each }}l\in I.$   □

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 F ≠ ϕ.

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 lim_n→∞ d(u _n , T _l u _n ) = 0 = lim_n→∞ 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 = T_{m(mod N)}.

Observe that

$$\begin{array}{ll}\hfill d\left(z_m,u_n\right)& \le d\left(T_{m}u,T_m{u}_n\right)+d\left(T_m{u}_n,T_{m-1}{u}_n\right)+\dots +d\left(T{u}_n,u_n\right)\\ \le d\left(u,u_n\right)+\sum _{i=1}^{m-1}d\left(u_n,T_i{u}_n\right).\end{array}$$

Therefore, we have

$$r\left(z_m,\left\{u_n\right\}\right)=\underset{n\to \infty }{\text{lim}\text{sup}}d\left(z_m,u_n\right)\le \underset{n\to \infty }{\text{lim}\text{sup}}d\left(u,u_n\right)=r\left(u,\left\{u_n\right\}\right).$$

This implies that |r(z _m , {u _n }) - r(u, {u _n })| → 0 as m → ∞. It follows from Lemma 2.6 that T_{m(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, lim_n→∞ d(x _n , u) exists by Lemma 2.7.

Suppose x ≠ u. By the uniqueness of asymptotic centers,

$$\begin{array}{ll}\hfill \underset{n\to \infty }{\text{lim}\text{sup}}d\left(u_n,u\right)& <\underset{n\to \infty }{\text{lim}\text{sup}}d\left(u_n,x\right)\\ \le \underset{n\to \infty }{\text{lim}\text{sup}}d\left(x_n,x\right)\\ <\underset{n\to \infty }{\text{lim}\text{sup}}d\left(x_n,u\right)\\ =\underset{n\to \infty }{\text{lim}\text{sup}}d\left(u_n,u\right),\end{array}$$

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(x_n+1, p) ≤ d(x _n , p) for all p ∈ K and for all n ≥ 1. A map T : K → K 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 $F_1={\cap }_{i=1}^{N}F\left(T_i\right)\ne \varphi$, satisfies condition (A) if

$$d\left(x,Tx\right)\ge f\left(d\left(x,F_1\right)\right)\mathsf{\text{for all }}x\in K,$$

holds for at least one T ∈ {T _i : i ∈ I} or

$$\underset{i\in I}{\text{max}}d\left(x,T_{i}x\right)\ge f\left(d\left(x,F_1\right)\right)\mathsf{\text{for all }}x\in K,$$

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 F ≠ ϕ. Then the two families are said to satisfy:

• (i) condition (B) on K if

  $$d\left(x,Tx\right)\ge f\left(d\left(x,F\right)\right)\mathsf{\text{or}}d\left(x,Sx\right)\ge f\left(d\left(x,F\right)\right)\mathsf{\text{for all }}x\in K,$$

holds for at least one T ∈ {T _i : i ∈ I} or one S ∈ {S _i : i ∈ I};

• (ii) condition (C) on K if

  $$\frac{1}{2}\left\{d\left(x,T_{i}x\right)+d\left(x,S_{i}x\right)\right\}\ge f\left(d\left(x,F\right)\right)\mathsf{\text{for all }}x\in K.$$

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 lim_n→∞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 F ≠ ϕ. Then the sequence {x _n } defined implicitly in (2.2) converges strongly to p ∈ F if and only if lim_n→∞ d(x _n , F) = 0.

Proof. It follows from (2.4) that {x _n } is Fejér monotone with respect to F and lim_n→∞ 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 F ≠ ϕ. 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 lim_n→∞ d(x _n , F) exists. Moreover, Lemma 2.8 implies that lim_n→∞ d(x _n , T _l x _n ) = d(x _n , S _l x _n ) = 0 for each l ∈ I. So condition (B) guarantees that lim_n→∞ f(d(x _n , F)) = 0. Since f is nondecreasing with f (0) = 0, it follows that lim_n→∞ 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 F ≠ ϕ. 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;

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

• (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;

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