^{*}

Edited by: Jin Liang, Shanghai Jiao Tong University, China

Reviewed by: Ming Tian, Civil Aviation University of China, China; Yekini Shehu, University of Nigeria, Nigeria

*Correspondence: Alexander J. Zaslavski, Department of Mathematics, The Technion – Israel Institute of Technology, Amado Mathematics Building, Haifa 32000, Israel

This article was submitted to Fixed Point Theory, a section of the journal Frontiers in Applied Mathematics and Statistics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We study the generic convergence of infinite products of nonexpansive mappings with unbounded domains in hyperbolic metric spaces.

Let (^{1} denote the real line. We say that a mapping ^{1} → ^{1} into ^{1} under a metric embedding will be called a ^{1}:

Assume that (

This point is denoted by (1 −

It is clear that all normed linear spaces are hyperbolic in this sense. A discussion of more examples of hyperbolic spaces and, in particular, of the Hilbert ball can be found, for example, in Goebel and Reich [

Let (^{0}(_{t}}^{∞}_{t = 1} of mappings _{t} :

For each

_{K}(

Fix θ ∈

We equip the set

It is not difficult to see that the uniform space

Denote by _{*} the set of all {_{t}}^{∞}_{t = 1} ∈

Denote by _{*} the closure of the set _{*} in the uniform space _{*} ⊂

In this paper we study the asymptotic behavior of (unrestricted) infinite products of generic sequences of mappings belonging to the space _{*} and obtain convergence to a unique common fixed point. More precisely, we establish the following result, which generalizes the corresponding result in Reich and Zaslavski [

_{*} _{*}, _{t}}^{∞}_{t = 1} ∈

_{t}(

_{t}(

_{t}}^{∞}_{t = 1} _{*} _{t}}^{∞}_{t = 1} ∈ _{K}(θ, _{t}(

_{t}}^{∞}_{t = 1} _{*}, _{t}}^{∞}_{t = 1} ∈ _{i}}^{m}_{i = 0} ⊂

Elements of the space _{t}}^{∞}_{t = 1}, _{t}}^{∞}_{t = 1}, _{t}}^{∞}_{t = 1}, respectively.

Let _{t}}^{∞}_{t = 1} ∈ _{*} and γ ∈ (0, 1). There exists a point _{A} ∈

For each integer

By (1.1), (2.1), and (2.2), for all integers

In view of (2.2–2.4),

Let

There exists an open neighborhood _{γ, t}}^{∞}_{t = 1} in _{*} such that

Assume that
_{i}}^{m}_{i = 0} ⊂

We now show by induction that for all integers

Assume that

By (2.17), which holds for

Relations (2.19) and (2.20) imply that

Thus, (2.18) holds for

By the above relation and (2.7),

Hence (2.16) and (2.17) hold for

We claim that for all

First we show that there exists

Assume the contrary. Then

By (2.8), (2.18), and (2.23), for all integers

In view of the above inequality and (2.16),

This contradicts (2.9). The contradiction we have reached proves that there indeed exists an integer

Next we claim that (2.2) holds for all integers

Indeed, by (2.24), inequality (2.22) is true for

Assume now that (2.25) holds. In view of (2.8), (2.18), and (2.25),

Assume that (2.26) holds. Then it follows from (2.8), (2.18), (2.22), and (2.26) that

Thus, in both cases,

This means that we have shown by induction that (2.22) is indeed valid for all

(P) For each
_{i}}^{m}_{i = 0} ⊂

Set

By (1.1), (2.1), and (2.2), for each _{t}}^{∞}_{t = 1} ∈ _{*}, each γ ∈ (0, 1), each integer

In view of (1.2) and (2.28),

When combined with (2.27), this implies that _{*}.

Assume that

and

By (2.27) and (2.29), there exist

Let

let ^{i}_{t}(^{∞}_{i = 0}. By (2.30)–(2.33) and property (P) (applied to {_{s}}^{∞}_{s = 1} = {_{s}}^{∞}_{s = 1} and

Since ϵ is an arbitrary positive number, we conclude that for each point _{K}(θ, ^{i}_{t}(^{∞}_{i = 0} is a Cauchy sequence. Since _{K}(θ,

This implies that for each pair of points _{1}, _{2} ∈ _{K}(θ, _{1}, _{2},

Since ϵ, _{1}, _{2} ≥ 1 and each pair of points _{1}, _{2} ∈

Let

In view of (2.35),

It immediately follows from (2.35) and (2.36) that properties (a) and (b) hold. We claim that property (c) also holds.

Let

Set

It follows from (2.37) and (2.38) that for all integers

By (2.30), (2.31), (2.37–2.39) and property (P) applied to any integer _{i} = _{i},

In view of (2.30), (2.31), (2.34), (2.35), and (2.40),

Thus, property (c) does hold, as claimed.

Finally, we show that property (d) holds too. It follows from (2.34) and (2.35) that

Assume that
_{i}}^{m}_{i = 0} ⊂

By the relations above and property (P),

It now follows from (2.30), (2.31), (2.41), and (2.42) that for all integers

Thus, property (d) indeed holds. This completes the proof of Theorem 1.1.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

SR was partially supported by the Israel Science Foundation (Grant No. 389/12), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.