Convergence theorems for total asymptotically nonexpansive non-self mappings in $\mathrm{CAT}(0)$ spaces

Abstract

In this paper, we introduce the concept of total asymptotically nonexpansive nonself mappings and prove the demiclosed principle for this kind of mappings in $CAT(0)$ spaces. As a consequence, we obtain a Δ-convergence theorem of total asymptotically nonexpansive nonself mappings in $CAT(0)$ spaces. Our results extend and improve the corresponding recent results announced by many authors.

MSC:47H09, 47J25.

1 Introduction

A metric space X is a $CAT(0)$ space if it is geodesically connected and if every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane. That is to say, let $(X,d)$ be a metric space, and let $x,y\in X$ with $d(x,y)=l$. A geodesic path from x to y is an isometry $c:[0,l]\to X$ such that $c(0)=x$ and $c(l)=y$. The image of a geodesic path is called a geodesic segment. A metric space X is a (uniquely) geodesic space if every two points of X are joined by only one geodesic segment. A geodesic triangle $\mathrm{\Delta }(x_1,x_2,x_3)$ in a geodesic space X consists of three points $x_1$, $x_2$, $x_3$ of X and three geodesic segments joining each pair of vertices. A comparison triangle of the geodesic triangle $\mathrm{\Delta }(x_1,x_2,x_3)$ is the triangle $\overline{\mathrm{\Delta }}(x_1,x_2,x_3):=\mathrm{\Delta }(\overline{x_1},\overline{x_2},\overline{x_3})$ in the Euclidean space ${\mathcal{R}}^2$ such that

$$d(x_i,x_j)=d_{{\mathcal{R}}^2}(\overline{x_i},\overline{x_j}),\mathrm{\forall }i,j=1,2,3.$$

A geodesic space X is a $CAT(0)$ space if for each geodesic triangle $\mathrm{\Delta }(x_1,x_2,x_3)$ in X and its comparison triangle $\overline{\mathrm{\Delta }}:=\mathrm{\Delta }(\overline{x_1},\overline{x_2},\overline{x_3})$ in ${\mathcal{R}}^2$, the $CAT(0)$ inequality

$$d(x,y)\le d_{{\mathcal{R}}^2}(\overline{x},\overline{y})$$

(1.1)

is satisfied for all $x,y\in \mathrm{\Delta }$ and $\overline{x},\overline{y}\in \overline{\mathrm{\Delta }}$.

A thorough discussion on these spaces and their important role in various branches of mathematics is given in [1–5].

In 1976, Lim [6] introduced the concept of Δ-convergence in a general metric space. Fixed point theory in a $CAT(0)$ space was first studied by Kirk [7, 8]. He showed that every nonexpansive (single-valued) mapping defined on a bounded closed convex subset of a complete $CAT(0)$ space always has a fixed point. In 2008, Kirk and Panyanak [9] specialized Lim’s concept to $CAT(0)$ spaces and proved that it is very similar to the weak convergence in the Banach space setting. So, the fixed point and Δ-convergence theorems for single-valued and multivalued mappings in $CAT(0)$ spaces have been rapidly developed and many papers have appeared [10–25].

Let $(X,d)$ be a metric space. Recall that a mapping $T:X\to X$ is said to be nonexpansive if

$$d(Tx,Ty)\le d(x,y),\mathrm{\forall }x,y\in X.$$

(1.2)

T is said to be asymptotically nonexpansive, if there is a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ such that

$$d(T^{n}x,T^{n}y)\le k_{n}d(x,y),\mathrm{\forall }n\ge 1,x,y\in X.$$

(1.3)

T is said to be $(\{{\mu }_n\},\{{\nu }_n\},\zeta )$-total asymptotically nonexpansive, if there exist nonnegative sequences $\{{\mu }_n\}$, $\{{\nu }_n\}$ with ${\mu }_n\to 0$, ${\nu }_n\to 0$ and a strictly increasing continuous function $\zeta :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\zeta (0)=0$ such that

$$d(T^{n}x,T^{n}y)\le d(x,y)+{\nu }_n\zeta (d(x,y))+{\mu }_n,\mathrm{\forall }n\ge 1,x,y\in X.$$

(1.4)

Let $(X,d)$ be a metric space, and let C be a nonempty and closed subset of X. Recall that C is said to be a retract of X if there exists a continuous map $P:X\to C$ such that $Px=x$, $\mathrm{\forall }x\in C$. A map $P:X\to C$ is said to be a retraction if $P^2=P$. If P is a retraction, then $Py=y$ for all y in the range of P.

Definition 1.1 Let X and C be the same as above. A mapping $T:C\to X$ is said to be $(\{{\mu }_n\},\{{\nu }_n\},\zeta )$-total asymptotically nonexpansive nonself mapping if there exist nonnegative sequences $\{{\mu }_n\}$, $\{{\nu }_n\}$ with ${\mu }_n\to 0$, ${\nu }_n\to 0$ and a strictly increasing continuous function $\zeta :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\zeta (0)=0$ such that

$$d(T{(PT)}^{n-1}x,T{(PT)}^{n-1}y)\le d(x,y)+{\nu }_n\zeta (d(x,y))+{\mu }_n,\mathrm{\forall }n\ge 1,x,y\in C,$$

(1.5)

where P is a nonexpansive retraction of X onto C.

Remark 1.2 From the definitions, it is to know that each nonexpansive nonself mapping is an asymptotically nonexpansive nonself mapping with a sequence $\{k_n=1\}$, and each asymptotically nonexpansive mapping is a $(\{{\mu }_n\},\{{\nu }_n\},\zeta )$-total asymptotically nonexpansive mapping with ${\mu }_n=0$, ${\nu }_n=k_n-1$, $\mathrm{\forall }n\ge 1$ and $\zeta (t)=t$, $t\ge 0$.

Definition 1.3 A nonself mapping $T:C\to X$ is said to be uniformly L-Lipschitzian if there exists a constant $L>0$ such that

$$d(T{(PT)}^{n-1}x,T{(PT)}^{n-1}y)\le Ld(x,y),\mathrm{\forall }n\ge 1,x,y\in C.$$

(1.6)

Recently, Chang et al. [26] introduced the following Krasnoselskii-Mann type iteration for finding a fixed point of a total asymptotically nonexpansive mappings in $CAT(0)$ spaces.

$$x_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n{x}_n,n\ge 1.$$

(1.7)

Under some limit conditions, they proved that the sequence $\{x_n\}$ Δ-converges to a fixed point of T.

Inspired and motivated by the recent work of Chang et al. [26], Tang et al. [27], Laowang et al. [19] and so on, the purpose of this paper is to introduce the concept of total asymptotically nonexpansive nonself mappings and prove the demiclosed principle for this kind of mappings in $CAT(0)$ spaces. As a consequence, we obtain a Δ-convergence theorem of total asymptotically nonexpansive nonself mappings in $CAT(0)$ spaces. The results presented in this paper improve and extend the corresponding recent results in [19, 26, 27].

2 Preliminaries

The following lemma plays an important role in our paper.

In this paper, we write $(1-t)x\oplus ty$ for the unique point z in the geodesic segment joining from x to y such that

$$d(z,x)=td(x,y),d(z,y)=(1-t)d(x,y).$$

(2.1)

We also denote by $[x,y]$ the geodesic segment joining from x to y, that is, $[x,y]=\{(1-t)x\oplus ty:t\in [0,1]\}$.

A subset C of a $CAT(0)$ space is convex if $[x,y]\subset C$ for all $x,y\in C$.

Lemma 2.1 [18]

A geodesic space X is a $CAT(0)$ space if and only if the following inequality holds:

$$d^2((1-t)x\oplus ty,z)\le (1-t)d^2(x,z)+t{d}^2(y,z)-t(1-t)d^2(x,y)$$

(2.2)

for all $x,y,z\in X$ and all $t\in [0,1]$. In particular, if x, y, z are points in a $CAT(0)$ space and $t\in [0,1]$, then

$$d((1-t)x\oplus ty,z)\le (1-t)d(x,z)+td(y,z).$$

(2.3)

Let $\{x_n\}$ be a bounded sequence in a $CAT(0)$ space X. For $x\in X$, we set

$$r(x,\{x_n\})=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x,x_n).$$

The asymptotic radius $r(\{x_n\})$ of $\{x_n\}$ is given by

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

(2.4)

The asymptotic radius $r_C(\{x_n\})$ of $\{x_n\}$ with respect to $C\subset X$ is given by

$$r_C(\{x_n\})=inf\{r(x,\{x_n\}):x\in C\}.$$

(2.5)

The asymptotic center $A(\{x_n\})$ of $\{x_n\}$ is the set

$$A(\{x_n\})=\{x\in X:r(x,\{x_n\})=r(\{x_n\})\}.$$

(2.6)

And the asymptotic center $A_C(\{x_n\})$ of $\{x_n\}$ with respect to $C\subset X$ is the set

$$A_C(\{x_n\})=\{x\in C:r(x,\{x_n\})=r_C(\{x_n\})\}.$$

(2.7)

Recall that a bounded sequence $\{x_n\}$ in X is said to be regular if $r(\{x_n\})=r(\{u_n\})$ for every subsequence $\{u_n\}$ of $\{x_n\}$.

Proposition 2.2 [14]

Let X be a complete $CAT(0)$ space, let $\{x_n\}$ be a bounded sequence in X, and let C be a closed convex subset of X. Then

• (1) there exists a unique point $u\in C$ such that

  $$r(u,\{x_n\})=\underset{x\in C}{inf}r(x,\{x_n\});$$
• (2)
  $$A(\{x_n\})$$

  and $A_C(\{x_n\})$ both are singleton.

Definition 2.3 [6, 9]

Let X be a $CAT(0)$ space. A sequence $\{x_n\}$ in X is said to Δ-converge to $p\in X$ if p is the unique asymptotic center of $\{u_n\}$ for each subsequence $\{u_n\}$ of $\{x_n\}$. In this case we write $\mathrm{\Delta }\text{-}{lim}_{n\to \mathrm{\infty }}{x}_n=p$ and call p the Δ-limit of $\{x_n\}$.

Lemma 2.4 [9]

Every bounded sequence in a complete $CAT(0)$ space always has a Δ-convergent subsequence.

Lemma 2.5 [11]

Let X be a complete $CAT(0)$ space, and let C be a closed convex subset of X. If $\{x_n\}$ is a bounded sequence in C, then the asymptotic center of $\{x_n\}$ is in C.

Remark 2.6 Let X be a $CAT(0)$ space, and let C be a closed convex subset of X. Let $\{x_n\}$ be a bounded sequence in C. In what follows, we denote

$$\{x_n\}\rightharpoonup w\iff \mathrm{\Phi }(w)=\underset{x\in C}{inf}\mathrm{\Phi }(x),$$

where $\mathrm{\Phi }(x):={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x_n,x)$.

Now we give a connection between the ‘⇀’ convergence and Δ-convergence.

Proposition 2.7 [26]

Let X be a $CAT(0)$ space, let C be a closed convex subset of X, and let $\{x_n\}$ be a bounded sequence in C. Then $\mathrm{\Delta }\text{-}{lim}_{n\to \mathrm{\infty }}{x}_n=p$ implies that $\{x_n\}\rightharpoonup p$.

Lemma 2.8 Let C be a closed and convex subset of a complete $CAT(0)$ space X, and let $T:C\to X$ be a uniformly L-Lipschitzian and $(\{{\mu }_n\},\{{\nu }_n\},\zeta )$-total asymptotically nonexpansive nonself mapping. Let $\{x_n\}$ be a bounded sequence in C such that $\{x_n\}\rightharpoonup p$ and ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. Then $Tp=p$.

Proof By the definition, $\{x_n\}\rightharpoonup p$ if and only if $A_C(\{x_n\})=\{p\}$. By Lemma 2.5, we have $A(\{x_n\})=\{p\}$.

Since ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$, by induction we can prove that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{(PT)}^m{x}_n)=0\text{for each }m\ge 0.$$

(2.8)

In fact, it is obvious that the conclusion is true for $m=0$. Suppose that the conclusion holds for $m\ge 1$, now we prove that the conclusion is also true for $m+1$.

Indeed, since T is uniformly L-Lipschitzian, we have

$$\begin{array}{rcl}d(x_n,T{(PT)}^m{x}_n)& \le & d(x_n,T{(PT)}^{m-1}{x}_n)+d(T{(PT)}^{m-1}{x}_n,T{(PT)}^m{x}_n)\\ \le & d(x_n,T{(PT)}^{m-1}{x}_n)+Ld(x_n,PT{x}_n)\\ =& d(x_n,T{(PT)}^{m-1}{x}_n)+Ld(P{x}_n,PT{x}_n)\\ \le & d(x_n,T{(PT)}^{m-1}{x}_n)+Ld(x_n,T{x}_n)\to 0(\text{as }n\to \mathrm{\infty }).\end{array}$$

(2.8) is proved. Hence for each $x\in C$ and $m\ge 1$, from (2.8), we have

$$\mathrm{\Phi }(x):=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,x)=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(T{(PT)}^{m-1}(x_n),x).$$

(2.9)

In (2.9) taking $x=T{(PT)}^{m-1}p$, $m\ge 1$, we have

$$\begin{array}{rcl}\mathrm{\Phi }\left(T{(PT)}^{m-1}p\right)& =& \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(T{(PT)}^{m-1}(x_n),T{(PT)}^{m-1}p)\\ \le & \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\{d(x_n,p)+{\nu }_m\zeta (d(x_n,p))+{\mu }_m\}.\end{array}$$

Letting $m\to \mathrm{\infty }$ and taking the superior limit on the both sides, we get that

$$\underset{m\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\mathrm{\Phi }\left(T{(PT)}^{m-1}p\right)\le \mathrm{\Phi }(p).$$

(2.10)

Furthermore, for any $n,m\ge 1$, it follows from inequality (2.2) with $t=\frac{1}{2}$ that

$$\begin{array}{c}{d}^2(x_n,\frac{p\oplus T{(PT)}^{m-1}(p)}{2})\hfill \\ \le \frac{1}{2}{d}^2(x_n,p)+\frac{1}{2}{d}^2(x_n,T{(PT)}^{m-1}(p))-\frac{1}{4}{d}^2(p,T{(PT)}^{m-1}(p)).\hfill \end{array}$$

(2.11)

Letting $n\to \mathrm{\infty }$ and taking the superior limit on the both sides of the above inequality, for any $m\ge 1$, we get

$$\begin{array}{c}\mathrm{\Phi }{\left(\frac{p\oplus T{(PT)}^{m-1}(p)}{2}\right)}^2\hfill \\ \le \frac{1}{2}\mathrm{\Phi }{(p)}^2+\frac{1}{2}\mathrm{\Phi }{(T{(PT)}^{m-1}(p))}^2-\frac{1}{4}{d}^2(p,T{(PT)}^{m-1}(p)).\hfill \end{array}$$

(2.12)

Since $A(\{x_n\})=\{p\}$, for any $m\ge 1$, we have

$$\begin{array}{rcl}\mathrm{\Phi }{(p)}^2& \le & \mathrm{\Phi }{\left(\frac{p\oplus T{(PT)}^{m-1}(p)}{2}\right)}^2\\ \le & \frac{1}{2}\mathrm{\Phi }{(p)}^2+\frac{1}{2}\mathrm{\Phi }{(T{(PT)}^{m-1}(p))}^2-\frac{1}{4}{d}^2(p,T{(PT)}^{m-1}(p)),\end{array}$$

(2.13)

which implies that

$$d^2(p,T{(PT)}^{m-1}(p))\le 2\mathrm{\Phi }{(T{(PT)}^{m-1}(p))}^2-2\mathrm{\Phi }{(p)}^2.$$

(2.14)

By (2.10) and (2.14), we have ${lim}_{m\to \mathrm{\infty }}d(p,T{(PT)}^{m-1}p)=0$. Hence we have

$$\begin{array}{rcl}d(Tp,p)& \le & d(Tp,T{(PT)}^{m}p)+d(T{(PT)}^{m}p,p)\\ \le & Ld(p,T{(PT)}^{m-1}p)+d(T{(PT)}^{m}p,p)\to 0(\text{as }m\to \mathrm{\infty }),\end{array}$$

i.e., $p=Tp$, as desired. □

The following result can be obtained from Lemma 2.8 immediately.

Lemma 2.9 Let C be a closed and convex subset of a complete $CAT(0)$ space X, and let $T:C\to X$ be an asymptotically nonexpansive nonself mapping with a sequence $\{k_n\}\subset [1,\mathrm{\infty })$, $k_n\to 1$. Let $\{x_n\}$ be a bounded sequence in C such that ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$ and $\mathrm{\Delta }\text{-}{lim}_{n\to \mathrm{\infty }}{x}_n=p$. Then $Tp=p$.

Lemma 2.10 [26]

Let X be a $CAT(0)$ space, let $x\in X$ be a given point, and let $\{t_n\}$ be a sequence in $[b,c]$ with $b,c\in (0,1)$ and $0<b(1-c)\le \frac{1}{2}$. Let $\{x_n\}$ and $\{y_n\}$ be any sequences in X such that

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,x)\le r,\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(y_n,x)\le r\mathit{\text{and}}\hfill \\ \underset{n\to \mathrm{\infty }}{lim}d((1-t_n)x_n\oplus t_n{y}_n,x)=r,\hfill \end{array}$$

for some $r\ge 0$. Then

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,y_n)=0.$$

(2.15)

Lemma 2.11 [26]

Let $\{a_n\}$, $\{{\lambda }_n\}$ and $\{c_n\}$ be the sequences of nonnegative numbers such that

$$a_{n+1}\le (1+{\lambda }_n)a_n+c_n,\mathrm{\forall }n\ge 1.$$

If ${\sum }_{n=1}^{\mathrm{\infty }}{\lambda }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{c}_n<\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists. If there exists a subsequence of $\{a_n\}$ which converges to zero, then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Lemma 2.12 [18]

Let X be a complete $CAT(0)$ space, and let $\{x_n\}$ be a bounded sequence in X with $A(\{x_n\})=\{p\}$; $\{u_n\}$ is a subsequence of $\{x_n\}$ with $A(\{u_n\})=\{u\}$, and the sequence $\{d(x_n,u)\}$ converges, then $p=u$.

3 Main results

Theorem 3.1 Let C be a nonempty, closed and convex subset of a complete $CAT(0)$ space E. Let $T_i:C\to E$ be a uniformly L-Lipschitzian and total asymptotically nonexpansive nonself mapping with sequences $\{{\mu }_n^{(i)}\}$ and $\{{\upsilon }_n^{(i)}\}$ satisfying ${lim}_{n\to \mathrm{\infty }}{\mu }_n^{(i)}=0$ and ${lim}_{n\to \mathrm{\infty }}{\upsilon }_n^{(i)}=0$, and strictly increasing function ${\xi }^{(i)}:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with ${\xi }^{(i)}(0)=0$, $i=1,2$. For arbitrarily chosen $x_1\in K$, $\{x_n\}$ is defined as follows:

$$\{\begin{array}{c}{y}_n=P((1-{\beta }_n)x_n\oplus {\beta }_n{T}_2{(P{T}_2)}^{n-1}{x}_n),\hfill \\ x_{n+1}=P((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}_1{(P{T}_1)}^{n-1}{y}_n),\hfill \end{array}$$

(3.1)

where $\{{\mu }_n^{(1)}\}$, $\{{\mu }_n^{(2)}\}$, $\{{\upsilon }_n^{(1)}\}$, $\{{\upsilon }_n^{(2)}\}$, ${\xi }^{(1)}$, ${\xi }^{(2)}$, $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ satisfy the following conditions:

• (1)
  $${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n^{(i)}<\mathrm{\infty }$$

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

• (2) there exist constants $a,b\in (0,1)$ with $0<b(1-a)\le \frac{1}{2}$ such that $\{{\alpha }_n\}\subset [a,b]$ and $\{{\beta }_n\}\subset [a,b]$;

• (3) there exists a constant $M^{\ast }>0$ such that ${\xi }^{(i)}(r)\le M^{\ast }r$, $r\ge 0$, $i=1,2$.

Then the sequence $\{x_n\}$ defined in (3.1) Δ-converges to a common fixed point of $T_1$ and $T_2$.

Proof We divide the proof into three steps.

Step 1. We first show that ${lim}_{n\to \mathrm{\infty }}d(x_n,q)$ exists for each $q\in F(T_1)\cap F(T_2)$.

Set ${\mu }_n=max\{{\mu }_n^{(1)},{\mu }_n^{(2)}\}$ and ${\upsilon }_n=\{{\upsilon }_n^{(1)},{\upsilon }_n^{(2)}\}$, $n=1,2,\dots ,\mathrm{\infty }$. Since ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n^{(i)}<\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}{\upsilon }_n^{(i)}<\mathrm{\infty }$, $i=1,2$, we know that ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\upsilon }_n<\mathrm{\infty }$. For any $q\in F(T_1)\cap F(T_2)$, we have

$$\begin{array}{rcl}d(x_{n+1},q)& =& d(P((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}_1{(P{T}_1)}^{n-1}{y}_n),q)\\ \le & d((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}_1{(P{T}_1)}^{n-1}{y}_n,q)\\ \le & (1-{\alpha }_n)d(x_n,q)+{\alpha }_{n}d(T_1{(P{T}_1)}^{n-1}{y}_n,q)\\ \le & (1-{\alpha }_n)d(x_n,q)+{\alpha }_n[d(y_n,q)+{\upsilon }_n{\xi }^{(1)}(d(y_n,q))+{\mu }_n]\\ \le & (1-{\alpha }_n)d(x_n,q)+{\alpha }_n[(1+{\upsilon }_n{M}^{\ast })d(y_n,q)+{\mu }_n],\end{array}$$

(3.2)

where

$$\begin{array}{rcl}d(y_n,q)& =& d(P((1-{\beta }_n)x_n\oplus {\beta }_n{T}_2{(P{T}_2)}^{n-1}{x}_n),q)\\ \le & d((1-{\beta }_n)x_n\oplus {\beta }_n{T}_2{(P{T}_2)}^{n-1}{x}_n,q)\\ \le & (1-{\beta }_n)d(x_n,q)+{\beta }_{n}d(T_2{(P{T}_2)}^{n-1}{x}_n,q)\\ \le & (1-{\beta }_n)d(x_n,q)+{\beta }_n[d(x_n,q)+{\upsilon }_n{\xi }^{(2)}(d(x_n,q))+{\mu }_n]\\ \le & (1+{\beta }_n{\upsilon }_n{M}^{\ast })d(x_n,q)+{\beta }_n{\mu }_n.\end{array}$$

(3.3)

Substituting (3.3) into (3.2), we have

$$\begin{array}{rcl}d(x_{n+1},q)& \le & (1-{\alpha }_n)d(x_n,q)+{\alpha }_n[(1+{\upsilon }_n{M}^{\ast })((1+{\beta }_n{\upsilon }_n{M}^{\ast })d(x_n,q)+{\beta }_n{\mu }_n)+{\mu }_n]\\ \le & [1+(1+{\beta }_n+{\beta }_n{\upsilon }_n{M}^{\ast }){\alpha }_n{M}^{\ast }{\upsilon }_n]d(x_n,q)+(1+{\beta }_n)(1+{\upsilon }_n{M}^{\ast }){\alpha }_n{\mu }_n.\end{array}$$

(3.4)

Since ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\upsilon }_n<\mathrm{\infty }$, it follows from Lemma 2.11 that ${lim}_{n\to \mathrm{\infty }}d(x_n,q)$ exists for each $q\in F(T_1)\cap F(T_2)$.

Step 2. We show that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_1{x}_n)={lim}_{n\to \mathrm{\infty }}d(x_n,T_2{x}_n)=0$.

For each $q\in F(T_1)\cap F(T_2)$, from the proof of Step 1, we know that ${lim}_{n\to \mathrm{\infty }}d(x_n,q)$ exists. We may assume that ${lim}_{n\to \mathrm{\infty }}d(x_n,q)=k$. From (3.3), we have

$$d(y_n,q)\le (1+{\beta }_n{\upsilon }_n{M}^{\ast })d(x_n,q)+{\beta }_n{\mu }_n.$$

(3.5)

Taking lim sup on both sides in (3.5), we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(y_n,q)\le k.$$

(3.6)

In addition, since

$$\begin{array}{rcl}d(T_1{(P{T}_1)}^{n-1}{y}_n,q)& \le & d(y_n,q)+{\upsilon }_n{\xi }^{(2)}(d(y_n,q))+{\mu }_n\\ \le & (1+{\upsilon }_n{M}^{\ast })d(y_n,q)+{\mu }_n,\end{array}$$

we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(T_1{(P{T}_1)}^{(n-1)}{y}_n,q)\le k.$$

(3.7)

Since ${lim}_{n\to \mathrm{\infty }}d(x_{n+1},q)=k$, it is easy to prove that

$$\underset{n\to \mathrm{\infty }}{lim}d((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}_1{(P{T}_1)}^{(n-1)}{y}_n,q)=k.$$

(3.8)

It follows from Lemma 2.10 that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T_1{(P{T}_1)}^{n-1}{y}_n)=0.$$

(3.9)

On the other hand, since

$$\begin{array}{rcl}d(x_n,q)& \le & d(x_n,T_1{(P{T}_1)}^{n-1}{y}_n)+d(T_1{(P{T}_1)}^{n-1}{y}_n,q)\\ \le & d(x_n,T_1{(P{T}_1)}^{n-1}{y}_n)+d(y_n,q)+{\upsilon }_n{M}^{\ast }d(y_n,q)+{\mu }_n\\ =& d(x_n,T_1{(P{T}_1)}^{n-1}{y}_n)+(1+{\upsilon }_n{M}^{\ast })d(y_n,q)+{\mu }_n,\end{array}$$

we have ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}d(y_n,q)\ge k$. Combined with (3.6), it yields that

$$\underset{n\to \mathrm{\infty }}{lim}d(y_n,q)=k.$$

(3.10)

This implies that

$$\underset{n\to \mathrm{\infty }}{lim}d((1-{\beta }_n)x_n\oplus {\beta }_n{T}_2{(P{T}_2)}^{n-1}{x}_n,q)=k.$$

(3.11)

It is easy to show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(T_2{(P{T}_2)}^{n-1}{x}_n,q)\le k.$$

(3.12)

So, it follows from (3.12) and Lemma 2.10 that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T_2{(P{T}_2)}^{n-1}{x}_n)=0.$$

(3.13)

Observe that

$$\begin{array}{rcl}d(x_n,T_1{(P{T}_1)}^{n-1}{x}_n)& \le & d(x_n,T_1{(P{T}_1)}^{n-1}{y}_n)+d(T_1{(P{T}_1)}^{n-1}{y}_n,T_1{(P{T}_1)}^{n-1}{x}_n)\\ \le & d(x_n,T_1{(P{T}_1)}^{n-1}{y}_n)+d(x_n,y_n)+{\upsilon }_n{\xi }^{(1)}(d(x_n,y_n))+{\mu }_n\\ =& d(x_n,T_1{(P{T}_1)}^{n-1}{y}_n)+(1+{\upsilon }_n{M}^{\ast })d(x_n,y_n)+{\mu }_n,\end{array}$$

(3.14)

where

$$\begin{array}{rcl}d(x_n,y_n)& =& d(P((1-{\beta }_n)x_n\oplus {\beta }_n{T}_2{(P{T}_2)}^{n-1}{x}_n),x_n)\\ \le & {\beta }_{n}d(x_n,T_2{(P{T}_2)}^{n-1}{x}_n).\end{array}$$

(3.15)

It follows from (3.13) that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,y_n)=0.$$

(3.16)

Thus, from (3.9), (3.14) and (3.16), we have

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T_1{(P{T}_1)}^{n-1}{x}_n)=0.$$

(3.17)

In addition, since

$$\begin{array}{rcl}d(x_{n+1},x_n)& =& d(P((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}_1{(P{T}_1)}^{n-1}{y}_n),x_n)\\ \le & d((1-{\alpha }_n)x_n\oplus {\alpha }_n{T}_1{(P{T}_1)}^{n-1}{y}_n,x_n)\\ \le & {\alpha }_{n}d(T_1{(P{T}_1)}^{n-1}{y}_n,x_n),\end{array}$$

from (3.9), we have

$$\underset{n\to \mathrm{\infty }}{lim}d(x_{n+1},x_n)=0.$$

(3.18)

Finally, since

$$\begin{array}{rcl}d(x_n,T_1{x}_n)& \le & d(x_n,x_{n+1})+d(x_{n+1},T_1{(P{T}_1)}^n{x}_{n+1})\\ +d(T_1{(P{T}_1)}^n{x}_{n+1},T_1{(P{T}_1)}^n{x}_n)+d(T_1{(P{T}_1)}^n{x}_n,T_1{x}_n)\\ \le & (1+L)d(x_n,x_{n+1})+d(x_{n+1},T_1{(P{T}_1)}^n{x}_{n+1})+Ld(T_1{(P{T}_1)}^{n-1}{x}_n,x_n),\end{array}$$

it follows from (3.17) and (3.18) that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_1{x}_n)=0$. Similarly, we also can show that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_2{x}_n)=0$.

Step 3. We show that $\{x_n\}$ Δ-converges to a common fixed point of $T_1$ and $T_2$.

Let $W_{\omega }(x_n)={\bigcup }_{\{u_n\}\subset \{x_n\}}A(\{u_n\})$. Firstly, we show that $W_{\omega }\subset F(T_1)\cap F(T_2)$. Let $u\in W_{\omega }$, then there exists a subsequence $\{u_n\}$ of $\{x_n\}$ such that $A(\{u_n\})=\{u\}$. By Lemma 2.4 and Lemma 2.5, there exists a subsequence $\{u_{n_i}\}$ of $\{u_n\}$ such that $\mathrm{\Delta }\text{-}{lim}_{i\to \mathrm{\infty }}{u}_{n_i}=p\in K$. Since ${lim}_{n\to \mathrm{\infty }}d(x_n,T_1{x}_n)={lim}_{n\to \mathrm{\infty }}d(x_n,T_2{x}_n)=0$, it follows from Lemma 2.8 that $p\in F(T_1)\cap F(T_2)$. So, ${lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists. By Lemma 2.12, we know that $p=u\in F(T_1)\cap F(T_2)$. This implies that $W_{\omega }(x_n)\subset F(T_1)\cap F(T_2)$. Next, let $\{u_n\}$ be a subsequence of $\{x_n\}$ with $A(\{u_n\})=\{u\}$ and $A\{x_n\}=\{v\}$. Since $u\in W_{\omega }(x_n)\subset F(T_1)\cap F(T_2)$, ${lim}_{n\to \mathrm{\infty }}d(x_n,u)$ exists. By Lemma 2.12, we know that $v=u$. This implies that $W_{\omega }(x_n)$ contains only one point. Thus, since $W_{\omega }(x_n)\subset F(T_1)\cap F(T_2)$, $W_{\omega }(x_n)$ contains only one point and ${lim}_{n\to \mathrm{\infty }}d(x_n,q)$ exists for each $q\in F(T_1)\cap F(T_2)$, we know that $\{x_n\}$ Δ-converges to a common fixed point of $T_1$ and $T_2$. The proof is completed. □