Theorem 2.1 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let {T}_{i}:K\to K, i=1,2, be a uniformly {L}_{i}Lipschitzian and (\{{\mu}_{n}^{i}\},\{{\xi}_{n}^{i}\},{\rho}^{i})total asymptotically nonexpansive mapping with sequence \{{\mu}_{n}^{i}\} and \{{\xi}_{n}^{i}\} satisfying {lim}_{n\to \mathrm{\infty}}{\mu}_{n}^{i}=0, {lim}_{n\to \mathrm{\infty}}{\xi}_{n}^{i}=0 and a strictly increasing function {\rho}^{i}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) with {\rho}^{i}(0)=0, i=1,2, let {S}_{i}:K\to K, i=1,2, be a uniformly \tilde{{L}_{i}}Lipschitzian and asymptotically nonexpansive mapping with sequence \{{k}_{n}^{i}\} satisfying {lim}_{n\to \mathrm{\infty}}{k}_{n}^{i}=0. Assume that \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i})\ne \mathrm{\varnothing}, and for arbitrarily chosen {x}_{1}\in K, \{{x}_{n}\} is defined as follows:
\{\begin{array}{c}{x}_{n+1}=W({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n},{\alpha}_{n}),\hfill \\ {y}_{n}=W({S}_{2}^{n}{x}_{n},{T}_{2}^{n}{x}_{n},{\beta}_{n}),\hfill \end{array}
(2.1)
where \{{\mu}_{n}^{i}\}, \{{\xi}_{n}^{i}\}, {\rho}^{i}, {k}_{n}^{i}, i=1,2, \{{\alpha}_{n}\} and \{{\beta}_{n}\} satisfy the following conditions:

(1)
{\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}^{i}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}^{i}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{k}_{n}^{i}<\mathrm{\infty}, i=1,2;

(2)
There exist constants a,b\in (0,1) with 0<b(1a)\le \frac{1}{2} such that \{{\alpha}_{n}\}\subset [a,b] and \{{\beta}_{n}\}\subset [a,b];

(3)
There exists a constant {M}^{\ast}>0 such that {\rho}^{i}(r)\le {M}^{\ast}r, r>0, i=1,2;

(4)
d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y) for all x,y\in K and i=1,2.
Then the sequence \{{x}_{n}\} defined by (2.1) Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
Proof Set L=max\{{L}_{i},\tilde{{L}_{i}},i=1,2\}, {\mu}_{n}=max\{{\mu}_{n}^{i},{k}_{n}^{i},i=1,2\} and {\xi}_{n}=max\{{\xi}_{n}^{i},i=1,2\}, \rho =max\{{\rho}^{i},i=1,2\}. By condition (1), we know that {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}<\mathrm{\infty}. The proof of Theorem 2.1 is divided into four steps.
Step 1. First we prove that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for each p\in \mathcal{F}.
For any given p\in \mathcal{F}, since {T}_{i}, i=1,2, is a total asymptotically nonexpansive mapping and {S}_{i}, i=1,2, is an asymptotically nonexpansive mapping, by condition (3) and (2.1), we have
\begin{array}{rl}d({x}_{n+1},p)=& d(W({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n},{\alpha}_{n}),p)\\ \le & (1{\alpha}_{n})d({S}_{1}^{n}{x}_{n},p)+{\alpha}_{n}d({T}_{1}^{n}{y}_{n},p)\\ \le & (1{\alpha}_{n})[(1+{\mu}_{n})d({x}_{n},p)]\\ +{\alpha}_{n}[d({y}_{n},p)+{\mu}_{n}\rho (d({y}_{n},p))+{\xi}_{n}]\\ \le & (1{\alpha}_{n})[(1+{\mu}_{n})d({x}_{n},p)]+{\alpha}_{n}[(1+{\mu}_{n}{M}^{\ast})d({y}_{n},p)+{\xi}_{n}],\end{array}
(2.2)
where
\begin{array}{rl}d({y}_{n},p)=& d(W({S}_{2}^{n}{x}_{n},{T}_{2}^{n}{x}_{n},{\beta}_{n}),p)\\ \le & (1{\beta}_{n})d({S}_{2}^{n}{x}_{n},p)+{\beta}_{n}d({T}_{2}^{n}{x}_{n},p)\\ \le & (1{\beta}_{n})[(1+{\mu}_{n})d({x}_{n},p)]\\ +{\beta}_{n}[d({x}_{n},p)+{\mu}_{n}\rho (d({x}_{n},p))+{\xi}_{n}]\\ \le & (1{\beta}_{n})[(1+{\mu}_{n})d({x}_{n},p)]+{\beta}_{n}[(1+{\mu}_{n}{M}^{\ast})d({x}_{n},p)+{\xi}_{n}]\\ =& [1+(1{\beta}_{n}){\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast}]d({x}_{n},p)+{\beta}_{n}{\xi}_{n}\\ \le & (1+{\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast})d({x}_{n},p)+{\beta}_{n}{\xi}_{n}.\end{array}
(2.3)
Substituting (2.3) into (2.2) and simplifying it, we have
\begin{array}{rl}d({x}_{n+1},p)\le & (1{\alpha}_{n})[(1+{\mu}_{n})d({x}_{n},p)]\\ +{\alpha}_{n}[(1+{\mu}_{n}{M}^{\ast})((1+{\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast})d({x}_{n},p)+{\beta}_{n}{\xi}_{n})+{\xi}_{n}]\\ =& \{1+[1+{\alpha}_{n}{M}^{\ast}(1+{\beta}_{n}+{\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast})]{\mu}_{n}\}d({x}_{n},p)\\ +(1+{\beta}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast}){\alpha}_{n}{\xi}_{n}\\ =& (1+{\delta}_{n})d({x}_{n},p)+{b}_{n},\end{array}
(2.4)
where {\delta}_{n}={\mu}_{n}(1+{\alpha}_{n}{M}^{\ast}(1+{\beta}_{n}+{\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast})), {b}_{n}=(1+{\beta}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast}){\alpha}_{n}{\xi}_{n}. Since {\sum}_{n=1}^{\mathrm{\infty}}{\mu}_{n}<\mathrm{\infty}, {\sum}_{n=1}^{\mathrm{\infty}}{\xi}_{n}<\mathrm{\infty} and condition (2), it follows from Lemma 1.2 that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists for p\in \mathcal{F}.
Step 2. We show that
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{T}_{i}{x}_{n})=0,\phantom{\rule{2em}{0ex}}\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{S}_{i}{x}_{n})=0,\phantom{\rule{1em}{0ex}}i=1,2.
(2.5)
For each p\in \mathcal{F}, from the proof of Step 1, we know that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists. We may assume that {lim}_{n\to \mathrm{\infty}}d({x}_{n},p)=c\ge 0. If c=0, then the conclusion is trivial. Next, we deal with the case c>0. From (2.3), we have
d({y}_{n},p)\le (1+{\mu}_{n}+{\beta}_{n}{\mu}_{n}{M}^{\ast})d({x}_{n},p)+{\beta}_{n}{\xi}_{n}.
(2.6)
Taking lim sup on both sides in (2.6), we have
\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({y}_{n},p)\le c.
(2.7)
In addition, since
d({T}_{1}^{n}{y}_{n},p)\le d({y}_{n},p)+{\mu}_{n}\rho (d({y}_{n},p))+{\xi}_{n}\le (1+{\mu}_{n}{M}^{\ast})d({y}_{n},p)+{\xi}_{n},
and
d({S}_{1}^{n}{x}_{n},p)\le (1+{\mu}_{n})d({x}_{n},p),
then we have
\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({T}_{1}^{n}{y}_{n},p)\le c
(2.8)
and
\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({S}_{1}^{n}{x}_{n},p)\le c.
(2.9)
Since {lim}_{n\to \mathrm{\infty}}d({x}_{n+1},p)=c, it is easy to prove that
\underset{n\to \mathrm{\infty}}{lim}d(W({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n},{\alpha}_{n}),p)=c.
(2.10)
It follows from (2.8)(2.10) and Lemma 1.3 that
\underset{n\to \mathrm{\infty}}{lim}d({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n})=0.
(2.11)
By the same method, we can also prove that
\underset{n\to \mathrm{\infty}}{lim}d({S}_{2}^{n}{x}_{n},{T}_{2}^{n}{x}_{n})=0.
(2.12)
By virtue of condition (4), it follows from (2.11) and (2.12) that
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{T}_{1}^{n}{y}_{n})\le \underset{n\to \mathrm{\infty}}{lim}d({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n})=0
(2.13)
and
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{T}_{2}^{n}{x}_{n})\le \underset{n\to \mathrm{\infty}}{lim}d({S}_{2}^{n}{x}_{n},{T}_{2}^{n}{x}_{n})=0.
(2.14)
From (2.1) and (2.12) we have
\begin{array}{rl}d({y}_{n},{S}_{2}^{n}{x}_{n})& =d(W({S}_{2}^{n}{x}_{n},{T}_{2}^{n}{x}_{n},{\beta}_{n}),{S}_{2}^{n}{x}_{n})\\ \le {\beta}_{n}d({T}_{2}^{n}{x}_{n},{S}_{2}^{n}{x}_{n})\to 0\phantom{\rule{1em}{0ex}}(\text{as}n\to \mathrm{\infty}).\end{array}
(2.15)
Observe that
d({x}_{n},{y}_{n})=d({x}_{n},{T}_{2}^{n}{x}_{n})+d({T}_{2}^{n}{x}_{n},{S}_{2}^{n}{x}_{n})+d({S}_{2}^{n}{x}_{n},{y}_{n}).
It follows from (2.14) and (2.15) that
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{y}_{n})=0.
(2.16)
This together with (2.13) implies that
\begin{array}{rl}d({x}_{n},{T}_{1}^{n}{x}_{n})& \le d({x}_{n},{T}_{1}^{n}{y}_{n})+d({T}_{1}^{n}{y}_{n},{T}_{1}^{n}{x}_{n})\\ \le d({x}_{n},{T}_{1}^{n}{y}_{n})+Ld({y}_{n},{x}_{n})\to 0\phantom{\rule{1em}{0ex}}(n\to \mathrm{\infty}).\end{array}
(2.17)
On the other hand, from (2.11) and (2.16), we have
\begin{array}{rl}d({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{x}_{n})& \le d({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n})+d({T}_{1}^{n}{y}_{n},{T}_{1}^{n}{x}_{n})\\ \le d({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n})+Ld({y}_{n},{x}_{n})\to 0\phantom{\rule{1em}{0ex}}(n\to \mathrm{\infty}).\end{array}
(2.18)
Hence from (2.17) and (2.18), we have that
d({S}_{1}^{n}{x}_{n},{x}_{n})\le d({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{x}_{n})+d({T}_{1}^{n}{x}_{n},{x}_{n})\to 0\phantom{\rule{1em}{0ex}}(n\to \mathrm{\infty}).
(2.19)
In addition, since
\begin{array}{rl}d({x}_{n+1},{x}_{n})& =d(W({S}_{1}^{n}{x}_{n},{T}_{1}^{n}{y}_{n},{\alpha}_{n}),{x}_{n})\\ \le (1{\alpha}_{n})d({S}_{1}^{n}{x}_{n},{x}_{n})+{\alpha}_{n}d({T}_{1}^{n}{y}_{n},{x}_{n}),\end{array}
from (2.13) and (2.19), we have
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n+1},{x}_{n})=0.
(2.20)
Finally, for all i=1,2, we have
\begin{array}{rl}d({x}_{n},{T}_{i}{x}_{n})\le & d({x}_{n},{x}_{n+1})+d({x}_{n+1},{T}_{i}^{n+1}{x}_{n+1})\\ +d({T}_{i}^{n+1}{x}_{n+1},{T}_{i}^{n+1}{x}_{n})+d({T}_{i}^{n+1}{x}_{n},{T}_{i}{x}_{n})\\ \le & (1+L)d({x}_{n},{x}_{n+1})+d({x}_{n+1},{T}_{i}^{n+1}{x}_{n+1})+Ld({T}_{i}^{n}{x}_{n},{x}_{n}).\end{array}
It follows from (2.14), (2.17) and (2.20) that
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{T}_{i}{x}_{n})=0,\phantom{\rule{1em}{0ex}}i=1,2.
By virtue of condition (4), d({S}_{i}{x}_{n},{T}_{i}^{n}{x}_{n})\le d({S}_{i}^{n}{x}_{n},{T}_{i}^{n}{x}_{n}), we have
\begin{array}{rl}d({x}_{n},{S}_{i}{x}_{n})& \le d({x}_{n},{T}_{i}^{n}{x}_{n})+d({S}_{i}{x}_{n},{T}_{i}^{n}{x}_{n})\\ \le d({x}_{n},{T}_{i}^{n}{x}_{n})+d({S}_{i}^{n}{x}_{n},{T}_{i}^{n}{x}_{n}),\end{array}
it follows from (2.12), (2.14), (2.17) and (2.18) that
\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},{S}_{i}{x}_{n})=0,\phantom{\rule{1em}{0ex}}i=1,2.
Step 3. Now we prove that the sequence \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
In fact, for each p\in F, {lim}_{n\to \mathrm{\infty}}d({x}_{n},p) exists. This implies that the sequence \{d({x}_{n},p)\} is bounded, so is the sequence \{{x}_{n}\}. Hence, by virtue of Lemma 1.1, \{{x}_{n}\} has a unique asymptotic center {A}_{K}(\{{x}_{n}\})=\{x\}.
Let \{{u}_{n}\} be any subsequence of \{{x}_{n}\} with {A}_{K}(\{{u}_{n}\})=\{u\}. It follows from (2.5) that
\underset{n\to \mathrm{\infty}}{lim}d({u}_{n},{T}_{i}{u}_{n})=0.
(2.21)
Now, we show that u\in F({T}_{i}). For this, we define a sequence \{{z}_{n}\} in K by {z}_{j}={T}_{i}^{j}u. So we calculate
\begin{array}{rl}d({z}_{j},{u}_{n})& \le d({T}_{i}^{j}u,{T}_{i}^{j}{u}_{n})+d({T}_{i}^{j}{u}_{n},{T}_{i}^{j1}{u}_{n})+\cdots +d({T}_{i}{u}_{n},{u}_{n})\\ \le d(u,{u}_{n})+{\mu}_{j}\rho (d(u,{u}_{n}))+{\xi}_{j}+\sum _{k=1}^{j}d({T}_{i}^{k}{u}_{n},{T}_{i}^{k1}{u}_{n})\\ \le (1+{\mu}_{j}{M}^{\ast})d(u,{u}_{n})+{\xi}_{j}+\sum _{k=1}^{j}d({T}_{i}^{k}{u}_{n},{T}_{i}^{k1}{u}_{n}).\end{array}
(2.22)
Since {T}_{i} is uniformly LLipschitzian, from (2.22) we have
d({z}_{j},{u}_{n})\le (1+{\mu}_{j}{M}^{\ast})d(u,{u}_{n})+{\xi}_{j}+jLd({T}_{i}{u}_{n},{u}_{n}).
Taking lim sup on both sides of the above estimate and using (2.21), we have
\begin{array}{rl}r({z}_{j},\{{u}_{n}\})& =\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({z}_{j},{u}_{n})\\ \le (1+{\mu}_{j}{M}^{\ast})\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d(u,{u}_{n})+{\xi}_{j}\\ =(1+{\mu}_{j}{M}^{\ast})r(u,\{{u}_{n}\})+{\xi}_{j}.\end{array}
And so
\underset{j\to \mathrm{\infty}}{lim\hspace{0.17em}sup}r({z}_{j},\{{u}_{n}\})\le r(u,\{{u}_{n}\}).
Since {A}_{K}(\{{u}_{n}\})=\{u\}, by the definition of asymptotic center {A}_{K}(\{{u}_{n}\}) of a bounded sequence \{{u}_{n}\} with respect to K\subset X, we have
r(u,\{{u}_{n}\})\le r(y,\{{u}_{n}\}),\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in K.
This implies that
\underset{j\to \mathrm{\infty}}{lim\hspace{0.17em}inf}r({z}_{j},\{{u}_{n}\})\ge r(u,\{{u}_{n}\}).
Therefore we have
\underset{j\to \mathrm{\infty}}{lim}r({z}_{j},\{{u}_{n}\})=r(u,\{{u}_{n}\}).
It follows from Lemma 1.4 that {lim}_{j\to \mathrm{\infty}}{T}_{i}^{j}u=u. As {T}_{i} is uniformly continuous, so that {T}_{i}u={T}_{i}({lim}_{j\to \mathrm{\infty}}{T}_{i}^{j}u)={lim}_{j\to \mathrm{\infty}}{T}_{i}^{j+1}u=u. That is, u\in F({T}_{i}). Similarly, we also can show that u\in F({S}_{i}). Hence, u is the common fixed point of {T}_{i} and {S}_{i}. Reasoning as above by utilizing the uniqueness of asymptotic centers, we get that x=u. Since \{{u}_{n}\} is an arbitrary subsequence of \{{x}_{n}\}, therefore A(\{{u}_{n}\})=\{u\} for all subsequence \{{u}_{n}\} of \{{x}_{n}\}. This proves that \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}). This completes the proof. □
The following theorem can be obtained from Theorem 2.1 immediately.
Theorem 2.2 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let {T}_{i}:K\to K, i=1,2, be a uniformly {L}_{i}Lipschitzian and asymptotically nonexpansive mapping with sequence \{{t}_{n}^{i}\}\subset [1,\mathrm{\infty}) satisfying {lim}_{n\to \mathrm{\infty}}{t}_{n}^{i}=1, and {S}_{i}:K\to K, i=1,2, be a nonexpansive mapping. Assume that \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}), for arbitrarily chosen {x}_{1}\in K, \{{x}_{n}\} is defined as follows:
\{\begin{array}{c}{x}_{n+1}=W({S}_{1}{x}_{n},{T}_{1}^{n}{y}_{n},{\alpha}_{n}),\hfill \\ {y}_{n}=W({S}_{2}{x}_{n},{T}_{2}^{n}{x}_{n},{\beta}_{n}),\hfill \end{array}
(2.23)
where \{{t}_{n}^{i}\}, i=1,2, \{{\alpha}_{n}\} and {\beta}_{n} satisfy the following conditions:

(1)
{\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}^{i}1)<\mathrm{\infty}, i=1,2;

(2)
There exist constants a,b\in (0,1) with 0<b(1a)\le \frac{1}{2} such that \{{\alpha}_{n}\}\subset [a,b] and \{{\beta}_{n}\}\subset [a,b];

(3)
d(x,{T}_{i}y)\le d({S}_{i}x,{T}_{i}y) for all x,y\in K and i=1,2.
Then the sequence \{{x}_{n}\} defined in (2.23) Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
Proof Take {\rho}^{i}(t)=t, t\ge 0, {\xi}_{n}^{i}=0, {\mu}_{n}^{i}={t}_{n}^{i}1, {k}_{n}^{i}=0, i=1,2, in Theorem 2.1. Since all the conditions in Theorem 2.1 are satisfied, it follows from Theorem 2.1 that the sequence \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i})\cap F({S}_{i}).
This completes the proof of Theorem 2.2. □
Theorem 2.3 Let K be a nonempty closed convex subset of a complete uniformly convex hyperbolic space X with monotone modulus of uniform convexity η. Let {T}_{i}:K\to K, i=1,2, be a uniformly {L}_{i}Lipschitzian and asymptotically nonexpansive mapping with sequence \{{t}_{n}^{i}\}\subset [1,\mathrm{\infty}) satisfying {lim}_{n\to \mathrm{\infty}}{t}_{n}^{i}=1. Assume that \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i}), for arbitrarily chosen {x}_{1}\in K, \{{x}_{n}\} is defined as follows:
\{\begin{array}{c}{x}_{n+1}=W({x}_{n},{T}_{1}^{n}{y}_{n},{\alpha}_{n}),\hfill \\ {y}_{n}=W({x}_{n},{T}_{2}^{n}{x}_{n},{\beta}_{n}),\hfill \end{array}
(2.24)
where \{{t}_{n}^{i}\}, i=1,2, \{{\alpha}_{n}\} and {\beta}_{n} satisfy the following conditions:

(1)
{\sum}_{n=1}^{\mathrm{\infty}}({t}_{n}^{i}1)<\mathrm{\infty}, i=1,2;

(2)
There exist constants a,b\in (0,1) with 0<b(1a)\le \frac{1}{2} such that \{{\alpha}_{n}\}\subset [a,b] and \{{\beta}_{n}\}\subset [a,b].
Then the sequence \{{x}_{n}\} defined in (2.24) Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i}).
Proof Take {\rho}^{i}(t)=t, t\ge 0, {\xi}_{n}^{i}=0, {\mu}_{n}^{i}={t}_{n}^{i}1, {S}_{i}=I, i=1,2, in Theorem 2.1. Since all the conditions in Theorem 2.1 are satisfied, it follows from Theorem 2.1 that the sequence \{{x}_{n}\} Δconverges to a common fixed point of \mathcal{F}:={\bigcap}_{i=1}^{2}F({T}_{i}).
This completes the proof of Theorem 2.3. □