# Another type of Mann iterative scheme for two mappings in a complete geodesic space

## Abstract

In this paper, we show a Δ-convergence theorem for a Mann iteration procedure in a complete geodesic space with two quasinonexpansive and Δ-demiclosed mappings. The proposed method is different from known procedures with respect to the order of taking the convex combination.

## 1 Introduction

The fixed point approximation has been studied in a variety of ways and its results are useful for the other studies. In 1953, Mann [1] introduced an iteration procedure for approximating fixed points of a nonexpansive mapping T in a Hilbert space. Later, Reich [2] discussed this iteration procedure in a uniformly convex Banach space whose norm is Fréchet differentiable. In 1998, Takahashi and Tamura [3] considered an iteration procedure with two nonexpansive mappings and obtained weak convergence theorems for this procedure in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. On the other hand, in 2008, Dhompongsa and Panyanak [4] proved the following theorem.

Theorem 1.1 Let C be a bounded closed convex subset of a complete $CAT\left(0\right)$ space and $T:C\to C$ a nonexpansive mapping. For any initial point ${x}_{0}$ in C, define the Mann iterative sequence $\left\{{x}_{n}\right\}$ by

${x}_{n+1}=\left(1-{t}_{n}\right){x}_{n}\oplus {t}_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n=0,1,2,\dots ,$

where $\left\{{t}_{n}\right\}$ is a sequence in $\left[0,1\right]$, with the restrictions that ${\sum }_{n=0}^{\mathrm{\infty }}{t}_{n}$ diverges and ${lim sup}_{n\to \mathrm{\infty }}{t}_{n}<1$. Then $\left\{{x}_{n}\right\}$ Δ-converges to a fixed point of T.

Further, in a $CAT\left(1\right)$ space, Kimura et al. [5] proved the Δ-convergence theorem for a family of nonexpansive mappings including the following scheme:

${x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}\oplus {\alpha }_{n}\left(\left(1-{\beta }_{n}\right)S{x}_{n}\oplus {\beta }_{n}T{x}_{n}\right).$

In a Hilbert space H, the following equality holds for any $x,y,z\in H$:

$\alpha x+\left(1-\alpha \right)\left(\beta y+\left(1-\beta \right)z\right)=\gamma \left(\delta x+\left(1-\delta \right)y\right)+\left(1-\gamma \right)z,$

where $\alpha ,\beta ,\gamma ,\delta \in \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$ such that $\alpha =\gamma \delta$ and $\beta =\gamma \left(1-\delta \right)/\left(1-\gamma \delta \right)$. However, in $CAT\left(\kappa \right)$ spaces with $\kappa >0$, it does not hold in general, that is, the value of the convex combination taken twice depends on their order. Thus, the following formulas are different in general:

$\begin{array}{r}{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}\oplus {\alpha }_{n}\left(\left(1-{\beta }_{n}\right)S{x}_{n}\oplus {\beta }_{n}T{x}_{n}\right),\\ {x}_{n+1}=\left(1-{\alpha }_{n}\right)\left({\beta }_{n}{x}_{n}\oplus \left(1-{\beta }_{n}\right)S{x}_{n}\right)\oplus {\alpha }_{n}\left(\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}T{x}_{n}\right).\end{array}$
(1)

In this paper, we show an analogous result to Theorem 1.1 using the iterative scheme (1) in a complete $CAT\left(1\right)$ space with two quasinonexpansive and Δ-demiclosed mappings. We also deal with the image recovery problem for two closed convex sets.

## 2 Preliminaries

Let X be a metric space. For $x,y\in X$, a mapping $c:\left[0,l\right]\to X$ is said to be a geodesic if c satisfies $c\left(0\right)=x$, $c\left(l\right)=y$ and $d\left(c\left(s\right),c\left(t\right)\right)=|s-t|$ for all $s,t\in \left[0,l\right]$. An image $\left[x,y\right]$ of c is called a geodesic segment joining x and y. For $r>0$, X is said to be an r-geodesic metric space if, for any $x,y\in X$ with $d\left(x,y\right), there exists a geodesic segment $\left[x,y\right]$. In particular, if a segment $\left[x,y\right]$ is unique for any $x,y\in X$ with $d\left(x,y\right), then X is said to be a uniquely r-geodesic metric space. In what follows, we always assume $d\left(x,y\right)<\pi /2$ for any $x,y\in X$. Thus, we say X is a geodesic metric space instead of a $\pi /2$-geodesic metric space. For the more general case, see [6].

Let X be a uniquely geodesic metric space. A geodesic triangle is defined by $\mathrm{△}\left(x,y,z\right)=\left[x,y\right]\cup \left[y,z\right]\cup \left[z,x\right]$. Let M be the two-dimensional unit sphere in ${\mathbb{R}}^{3}$. For $\overline{x},\overline{y},\overline{z}\in M$, a triangle $\mathrm{△}\left(\overline{x},\overline{y},\overline{z}\right)\subset M$ is called a comparison triangle of $\mathrm{△}\left(x,y,z\right)$ if $d\left(x,y\right)={d}_{M}\left(\overline{x},\overline{y}\right)$, $d\left(y,z\right)={d}_{M}\left(\overline{y},\overline{z}\right)$, $d\left(z,x\right)={d}_{M}\left(\overline{z},\overline{x}\right)$. Further, for any $x,y\in X$ and $t\in \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$, if $z\in \left[x,y\right]$ satisfies $d\left(x,z\right)=\left(1-t\right)d\left(x,y\right)$ and $d\left(z,y\right)=td\left(x,y\right)$, then z is denoted by $z=tx\oplus \left(1-t\right)y$. A point $\overline{z}\in \left[\overline{x},\overline{y}\right]$ is called a comparison point of $z\in \left[x,y\right]$ if $d\left(x,z\right)={d}_{M}\left(\overline{x},\overline{z}\right)$. X is said to be a $CAT\left(1\right)$ space if, for any $p,q\in \mathrm{△}\left(x,y,z\right)\subset X$ and its comparison points $\overline{p},\overline{q}\in \mathrm{△}\left(\overline{x},\overline{y},\overline{z}\right)\subset M$, the inequality $d\left(p,q\right)\le {d}_{M}\left(\overline{p},\overline{q}\right)$ holds.

Let X be a geodesic metric space and $\left\{{x}_{n}\right\}$ a bounded sequence of X. For $x\in X$, we put $r\left(x,\left\{{x}_{n}\right\}\right)={lim sup}_{n\to \mathrm{\infty }}d\left(x,{x}_{n}\right)$. The asymptotic radius of $\left\{{x}_{n}\right\}$ is defined by $r\left(\left\{{x}_{n}\right\}\right)={inf}_{x\in X}r\left(x,\left\{{x}_{n}\right\}\right)$. Further, the asymptotic center of $\left\{{x}_{n}\right\}$ is defined by $AC\left(\left\{{x}_{n}\right\}\right)=\left\{x\in X:r\left(x,\left\{{x}_{n}\right\}\right)=r\left(\left\{{x}_{n}\right\}\right)\right\}$. If, for any subsequences $\left\{{x}_{{n}_{k}}\right\}$ of $\left\{{x}_{n}\right\}$, $AC\left(\left\{{x}_{{n}_{k}}\right\}\right)=\left\{{x}_{0}\right\}$, i.e., their asymptotic center consists of the unique element ${x}_{0}$, then we say $\left\{{x}_{n}\right\}$ Δ-converges to ${x}_{0}$ and we denote it by ${x}_{n}\stackrel{\mathrm{\Delta }}{⇀}{x}_{0}$.

Let X be a metric space. A mapping $T:X\to X$ is said to be a nonexpansive if T satisfies $d\left(Tx,Ty\right)\le d\left(x,y\right)$ for any $x,y\in X$. The set of fixed points of T is denoted by $F\left(T\right)=\left\{z\in X:Tz=z\right\}$. Further, a mapping $T:X\to X$ with $F\left(T\right)\ne \mathrm{\varnothing }$ is said to be a quasinonexpansive if T satisfies $d\left(Tx,z\right)\le d\left(x,z\right)$ for any $x\in X$ and $z\in F\left(T\right)$. Moreover, T is said to be Δ-demiclosed if, for any bounded sequence $\left\{{x}_{n}\right\}\subset X$ and ${x}_{0}\in X$ satisfying $d\left({x}_{n},T{x}_{n}\right)\to 0$ and ${x}_{n}\stackrel{\mathrm{\Delta }}{⇀}{x}_{0}$, we have ${x}_{0}\in F\left(T\right)$.

## 3 Tools for the main results

In this section, we introduce some tools for using the main theorem.

Theorem 3.1 (Kimura and Satô [7])

Let $\mathrm{△}\left(x,y,z\right)$ be a geodesic triangle in a $CAT\left(1\right)$ space such that $d\left(x,y\right)+d\left(y,z\right)+d\left(z,x\right)<2\pi$. Let $u=tx\oplus \left(1-t\right)y$ for some $t\in \left[0,1\right]$. Then

$cosd\left(u,z\right)sind\left(x,y\right)\ge cosd\left(x,z\right)sintd\left(x,y\right)+cosd\left(y,z\right)sin\left(1-t\right)d\left(x,y\right).$

Corollary 3.2 (Kimura and Satô [8])

Let $\mathrm{△}\left(x,y,z\right)$ be a geodesic triangle in a $CAT\left(1\right)$ space such that $d\left(x,y\right)+d\left(y,z\right)+d\left(z,x\right)<2\pi$. Let $u=tx\oplus \left(1-t\right)y$ for some $t\in \left[0,1\right]$. Then

$cosd\left(u,z\right)\ge tcosd\left(x,z\right)+\left(1-t\right)cosd\left(y,z\right).$

Theorem 3.3 (Espínola and Fernández-León [9])

Let X be a complete $CAT\left(1\right)$ space and $\left\{{x}_{n}\right\}$ a sequence in X. If $r\left(\left\{{x}_{n}\right\}\right)<\pi /2$, then the following hold.

1. (i)

$AC\left(\left\{{x}_{n}\right\}\right)$ consists of exactly one point;

2. (ii)

$\left\{{x}_{n}\right\}$ has a Δ-convergent subsequence.

Theorem 3.4 (Kimura and Satô [8])

Let X be a metric space and T a mapping from X into itself. If T is a nonexpansive with $F\left(T\right)\ne \mathrm{\varnothing }$, then T is quasinonexpansive and Δ-demiclosed.

The following lemmas are important properties of real numbers and they are easy to show. Thus, we omit the proofs.

Lemma 3.5 Let δ be a real number such that $-1<\delta <0$ and $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$ real sequences satisfying $\delta \le {b}_{n}\le 1$, $\delta \le {c}_{n}\le 1$ and ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}{c}_{n}\ge 1$. Then ${lim}_{n\to \mathrm{\infty }}{b}_{n}={lim}_{n\to \mathrm{\infty }}{c}_{n}=1$.

Lemma 3.6 Let $s\in \phantom{\rule{0.2em}{0ex}}\right]0,\mathrm{\infty }\left[$ and $\left\{{b}_{n}\right\}$, $\left\{{c}_{n}\right\}$ bounded real sequences satisfying ${b}_{n}\le 0$, $s<{c}_{n}$ and ${lim}_{n\to \mathrm{\infty }}{b}_{n}/{c}_{n}=0$. Then ${lim}_{n\to \mathrm{\infty }}{b}_{n}=0$.

Lemma 3.7 Let $\left\{{b}_{n}\right\}$ and $\left\{{c}_{n}\right\}$ be bounded real sequences satisfying ${lim}_{n\to \mathrm{\infty }}\left({b}_{n}-{c}_{n}\right)=0$. Then ${lim inf}_{n\to \mathrm{\infty }}{b}_{n}={lim inf}_{n\to \mathrm{\infty }}{c}_{n}$.

## 4 The main result

In this section, we show the main result.

Theorem 4.1 Let X be a complete $CAT\left(1\right)$ space such that for any $u,v\in X$, $d\left(u,v\right)<\pi /2$. Let S and T be quasinonexpansive and Δ-demiclosed mappings from X into itself with $F\left(S\right)\cap F\left(T\right)\ne \mathrm{\varnothing }$. Let $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$ and $\left\{{\gamma }_{n}\right\}$ be sequences of $\left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$. Define a sequence $\left\{{x}_{n}\right\}\subset X$ by the following recurrence formula: ${x}_{1}\in X$ and

$\left\{\begin{array}{l}{u}_{n}=\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}S{x}_{n},\\ {v}_{n}=\left(1-{\gamma }_{n}\right){x}_{n}\oplus {\gamma }_{n}T{x}_{n},\\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){u}_{n}\oplus {\alpha }_{n}{v}_{n}\end{array}$

for $n\in \mathbb{N}$. Then $\left\{{x}_{n}\right\}$ Δ-converges to a common fixed point of S and T.

Proof Let $z\in F\left(S\right)\cap F\left(T\right)$. By Corollary 3.2, we have

$\begin{array}{c}\begin{array}{rl}cosd\left({u}_{n},z\right)& \ge \left(1-{\beta }_{n}\right)cosd\left({x}_{n},z\right)+{\beta }_{n}cosd\left(S{x}_{n},z\right)\\ \ge \left(1-{\beta }_{n}\right)cosd\left({x}_{n},z\right)+{\beta }_{n}cosd\left({x}_{n},z\right)\\ =cosd\left({x}_{n},z\right),\end{array}\hfill \\ \begin{array}{rl}cosd\left({v}_{n},z\right)& \ge \left(1-{\gamma }_{n}\right)cosd\left({x}_{n},z\right)+{\gamma }_{n}cosd\left(T{x}_{n},z\right)\\ \ge \left(1-{\gamma }_{n}\right)cosd\left({x}_{n},z\right)+{\gamma }_{n}cosd\left({x}_{n},z\right)\\ =cosd\left({x}_{n},z\right).\end{array}\hfill \end{array}$

Then, by Corollary 3.2 again, we have

$\begin{array}{rl}cosd\left({x}_{n+1},z\right)& \ge \left(1-{\alpha }_{n}\right)cosd\left({u}_{n},z\right)+{\alpha }_{n}cosd\left({v}_{n},z\right)\\ \ge \left(1-{\alpha }_{n}\right)cosd\left({x}_{n},z\right)+{\alpha }_{n}cosd\left({x}_{n},z\right)\\ \ge cosd\left({x}_{n},z\right).\end{array}$

So, we get $d\left({x}_{n+1},z\right)\le d\left({x}_{n},z\right)$ for all $n\in \mathbb{N}$ and there exists ${d}_{0}={lim}_{n\to \mathrm{\infty }}d\left({x}_{n},z\right)\le d\left({x}_{1},z\right)<\pi /2$.

Furthermore, by Theorem 3.1, we have

$\begin{array}{r}cosd\left({u}_{n},z\right)sind\left({x}_{n},S{x}_{n}\right)\\ \phantom{\rule{1em}{0ex}}\ge cosd\left({x}_{n},z\right)sin\left(1-{\beta }_{n}\right)d\left({x}_{n},S{x}_{n}\right)+cosd\left(S{x}_{n},z\right)sin{\beta }_{n}d\left({x}_{n},S{x}_{n}\right)\\ \phantom{\rule{1em}{0ex}}\ge 2cosd\left({x}_{n},z\right)sin\frac{d\left({x}_{n},S{x}_{n}\right)}{2}cos\frac{\left(1-2{\beta }_{n}\right)d\left({x}_{n},S{x}_{n}\right)}{2}\end{array}$
(2)

and

$\begin{array}{r}cosd\left({v}_{n},z\right)sind\left({x}_{n},T{x}_{n}\right)\\ \phantom{\rule{1em}{0ex}}\ge cosd\left({x}_{n},z\right)sin\left(1-{\gamma }_{n}\right)d\left({x}_{n},T{x}_{n}\right)+cosd\left(T{x}_{n},z\right)sin{\gamma }_{n}d\left({x}_{n},T{x}_{n}\right)\\ \phantom{\rule{1em}{0ex}}\ge 2cosd\left({x}_{n},z\right)sin\frac{d\left({x}_{n},T{x}_{n}\right)}{2}cos\frac{\left(1-2{\gamma }_{n}\right)d\left({x}_{n},T{x}_{n}\right)}{2}.\end{array}$
(3)

Let ${d}_{n}=d\left({x}_{n},z\right)$, ${s}_{n}=d\left({x}_{n},S{x}_{n}\right)/2$ and ${t}_{n}=d\left({x}_{n},T{x}_{n}\right)/2$ for $n\in \mathbb{N}$. If there exists ${n}_{0}\in \mathbb{N}$ such that ${s}_{{n}_{0}}={t}_{{n}_{0}}=0$, then we have ${x}_{{n}_{0}}\in F\left(S\right)\cap F\left(T\right)$ and since

$\begin{array}{rl}{x}_{{n}_{0}+1}& =\left(1-{\alpha }_{{n}_{0}}\right)\left(\left(1-{\beta }_{{n}_{0}}\right){x}_{{n}_{0}}\oplus {\beta }_{{n}_{0}}S{x}_{{n}_{0}}\right)\oplus {\alpha }_{{n}_{0}}\left(\left(1-{\gamma }_{{n}_{0}}\right){x}_{{n}_{0}}\oplus {\gamma }_{{n}_{0}}T{x}_{{n}_{0}}\right)\\ =\left(1-{\alpha }_{{n}_{0}}\right){x}_{{n}_{0}}\oplus {\alpha }_{{n}_{0}}{x}_{{n}_{0}}\\ ={x}_{{n}_{0}},\end{array}$

and the proof is finished. So, we may assume ${s}_{n}\ne 0$ or ${t}_{n}\ne 0$ for all $n\in \mathbb{N}$.

If ${s}_{n}=0$ and ${t}_{n}\ne 0$, then we have ${u}_{n}={x}_{n}$. From (2), (3), and Corollary 3.2, we get

$\begin{array}{r}2cos{d}_{n+1}sin{t}_{n}cos{t}_{n}\\ \phantom{\rule{1em}{0ex}}=cos{d}_{n+1}sin2{t}_{n}\\ \phantom{\rule{1em}{0ex}}\ge \left(1-{\alpha }_{n}\right)cosd\left({u}_{n},z\right)sin2{t}_{n}+{\alpha }_{n}cosd\left({v}_{n},z\right)sin2{t}_{n}\\ \phantom{\rule{1em}{0ex}}\ge 2\left(1-{\alpha }_{n}\right)cos{d}_{n}sin{t}_{n}cos{t}_{n}+2{\alpha }_{n}cos{d}_{n}sin{t}_{n}cos\left(1-2{\gamma }_{n}\right){t}_{n}.\end{array}$

Dividing by $2sin{t}_{n}>0$, we get

$cos{d}_{n+1}cos{t}_{n}\ge \left(1-{\alpha }_{n}\right)cos{d}_{n}cos{t}_{n}+{\alpha }_{n}cos{d}_{n}cos\left(1-2{\gamma }_{n}\right){t}_{n}.$
(4)

If ${t}_{n}=0$ and ${s}_{n}\ne 0$, then we have ${v}_{n}={x}_{n}$. In a similar way as above, we get

$cos{d}_{n+1}cos{s}_{n}\ge \left(1-{\alpha }_{n}\right)cos{d}_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}+{\alpha }_{n}cos{d}_{n}cos{s}_{n}.$
(5)

If ${s}_{n}\ne 0$ and ${t}_{n}\ne 0$, then from (2), (3), and Corollary 3.2, we get

$\begin{array}{r}cos{d}_{n+1}sin2{s}_{n}sin2{t}_{n}\\ \phantom{\rule{1em}{0ex}}\ge \left(1-{\alpha }_{n}\right)cosd\left({u}_{n},z\right)sin2{s}_{n}sin2{t}_{n}+{\alpha }_{n}cosd\left({v}_{n},z\right)sin2{s}_{n}sin2{t}_{n}\\ \phantom{\rule{1em}{0ex}}\ge 4cos{d}_{n}sin{s}_{n}sin{t}_{n}\left(\left(1-{\alpha }_{n}\right)cos{t}_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}+{\alpha }_{n}cos{s}_{n}cos\left(1-2{\gamma }_{n}\right){t}_{n}\right).\end{array}$

Dividing by $4sin{s}_{n}sin{t}_{n}>0$, we get

$\begin{array}{r}cos{d}_{n+1}cos{s}_{n}cos{t}_{n}\\ \phantom{\rule{1em}{0ex}}\ge \left(1-{\alpha }_{n}\right)cos{d}_{n}cos{t}_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}+{\alpha }_{n}cos{d}_{n}cos{s}_{n}cos\left(1-2{\gamma }_{n}\right){t}_{n}.\end{array}$
(6)

Therefore, (4) and (5) can be reduced to the inequality (6) and it is equivalent to

$\left(\frac{{\epsilon }_{n}cos{s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}-\frac{1-{\alpha }_{n}}{{\alpha }_{n}}\right)\left(\frac{{\epsilon }_{n}cos{t}_{n}}{\left(1-{\alpha }_{n}\right)cos\left(1-2{\gamma }_{n}\right){t}_{n}}-\frac{{\alpha }_{n}}{1-{\alpha }_{n}}\right)\ge 1,$

where ${\epsilon }_{n}=cos{d}_{n+1}/cos{d}_{n}$ for $n\in \mathbb{N}$. It follows that ${lim}_{n\to \mathrm{\infty }}{\epsilon }_{n}=cos{d}_{0}/cos{d}_{0}=1$. Since $\left\{{\alpha }_{n}\right\}\subset \left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$ for $n\in \mathbb{N}$, we get

$\underset{n\to \mathrm{\infty }}{lim inf}\left(\frac{cos{s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}-\frac{1-{\alpha }_{n}}{{\alpha }_{n}}\right)\left(\frac{cos{t}_{n}}{\left(1-{\alpha }_{n}\right)cos\left(1-2{\gamma }_{n}\right){t}_{n}}-\frac{{\alpha }_{n}}{1-{\alpha }_{n}}\right)\ge 1.$
(7)

Then we show that there exists ${n}_{0}\in \mathbb{N}$ such that for all $n\ge {n}_{0}$, the following hold:

$-\frac{1}{2}\le \frac{cos{s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}-\frac{1-{\alpha }_{n}}{{\alpha }_{n}}\le 1$
(8)

and

$-\frac{1}{2}\le \frac{cos{t}_{n}}{\left(1-{\alpha }_{n}\right)cos\left(1-2{\gamma }_{n}\right){t}_{n}}-\frac{{\alpha }_{n}}{1-{\alpha }_{n}}\le 1.$
(9)

First, we show the right inequality of (8). Since $\left\{{\beta }_{n}\right\}\subset \left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$ for $n\in \mathbb{N}$, we get $cos{s}_{n}\le cos|1-2{\beta }_{n}|{s}_{n}=cos\left(1-2{\beta }_{n}\right){s}_{n}$. Hence we get

$\frac{cos{s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}-\frac{1-{\alpha }_{n}}{{\alpha }_{n}}\le \frac{1}{{\alpha }_{n}}-\frac{1-{\alpha }_{n}}{{\alpha }_{n}}=1.$

By the same method as above, the right inequality of (9) also holds. Next, let us show the left inequality of (8). If it does not hold, then letting

${\sigma }_{n}=\frac{cos{s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}-\frac{1-{\alpha }_{n}}{{\alpha }_{n}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\tau }_{n}=\frac{cos{t}_{n}}{\left(1-{\alpha }_{n}\right)cos\left(1-2{\gamma }_{n}\right){t}_{n}}-\frac{{\alpha }_{n}}{1-{\alpha }_{n}},$

we can find a subsequence $\left\{{\sigma }_{{n}_{i}}\right\}\subset \left\{{\sigma }_{n}\right\}$ such that ${\sigma }_{{n}_{i}}<-1/2$ for $i\in \mathbb{N}$ and ${lim}_{i\to \mathrm{\infty }}{\sigma }_{{n}_{i}}=\sigma \le -1/2$. Since $\left\{{\alpha }_{n}\right\},\left\{{\gamma }_{n}\right\}\subset \left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$ and $\left\{{t}_{n}\right\}\subset \left[0,\pi /4\left[\phantom{\rule{0.2em}{0ex}}\subset \left[0,\pi /2\left[$, we have $\left\{{\tau }_{n}\right\}$ is bounded. Therefore, by taking a subsequence again if necessary, we may assume that $\left\{{\tau }_{{n}_{i}}\right\}$ converges to $\tau \in \mathbb{R}$. Then, by (7), we get $\sigma \tau ={lim}_{i\to \mathrm{\infty }}{\sigma }_{{n}_{i}}{\tau }_{{n}_{i}}\ge {lim inf}_{n\to \mathrm{\infty }}{\sigma }_{n}{\tau }_{n}\ge 1$. Hence we may assume that ${\tau }_{{n}_{i}}<0$ for all $i\in \mathbb{N}$. Since $\sqrt{2}/2, $\sqrt{2}/2, $0, $0 and $\left\{{\alpha }_{n}\right\}\subset \left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$, we also have

$0<\frac{\sqrt{2}}{2b}\le \frac{cos{s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}0<\frac{\sqrt{2}}{2\left(1-a\right)}\le \frac{cos{t}_{n}}{\left(1-{\alpha }_{n}\right)cos\left(1-2{\gamma }_{n}\right){t}_{n}}.$
(10)

Let ρ be a real number such that

$0<\rho
(11)

Then, by (10), we get

$\rho -\frac{1-{\alpha }_{{n}_{i}}}{{\alpha }_{{n}_{i}}}\le {\sigma }_{{n}_{i}}<0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\rho -\frac{{\alpha }_{{n}_{i}}}{1-{\alpha }_{{n}_{i}}}\le {\tau }_{{n}_{i}}<0.$
(12)

Then, by (11) and (12), we have

$\begin{array}{rl}{\sigma }_{{n}_{i}}{\tau }_{{n}_{i}}& \le \left(\rho -\frac{1-{\alpha }_{{n}_{i}}}{{\alpha }_{{n}_{i}}}\right)\left(\rho -\frac{{\alpha }_{{n}_{i}}}{1-{\alpha }_{{n}_{i}}}\right)\\ ={\rho }^{2}-\left(\frac{1-{\alpha }_{{n}_{i}}}{{\alpha }_{{n}_{i}}}+\frac{{\alpha }_{{n}_{i}}}{1-{\alpha }_{{n}_{i}}}\right)\rho +1\\ \le {\rho }^{2}-\left(\frac{1-b}{b}+\frac{a}{1-a}\right)\rho +1\\ =\rho \left(\rho -\left(\frac{1-b}{b}+\frac{a}{1-a}\right)\right)+1.\end{array}$

Thus, as $i\to \mathrm{\infty }$, we have

$1\le \sigma \tau \le \rho \left(\rho -\left(\frac{1-b}{b}+\frac{a}{1-a}\right)\right)+1<1.$
(13)

This is a contradiction. We also obtain the left inequality of (9) in a similar way. Hence we get

$\underset{n\to \mathrm{\infty }}{lim}\left(\frac{cos{s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}-\frac{1-{\alpha }_{n}}{{\alpha }_{n}}\right)=\underset{n\to \mathrm{\infty }}{lim}\left(\frac{cos{t}_{n}}{\left(1-{\alpha }_{n}\right)cos\left(1-2{\gamma }_{n}\right){t}_{n}}-\frac{{\alpha }_{n}}{1-{\alpha }_{n}}\right)=1$
(14)

by Lemma 3.5, (8), and (9). Furthermore, from (14), we get

$\underset{n\to \mathrm{\infty }}{lim}\frac{cos{s}_{n}-cos\left(1-2{\beta }_{n}\right){s}_{n}}{{\alpha }_{n}cos\left(1-2{\beta }_{n}\right){s}_{n}}=0.$
(15)

By Lemma 3.6 and (15), we get

$\underset{n\to \mathrm{\infty }}{lim}\left(cos{s}_{n}-cos\left(1-2{\beta }_{n}\right){s}_{n}\right)=0.$
(16)

Moreover, by Lemma 3.7 and (16), we get

$\underset{n\to \mathrm{\infty }}{lim inf}cos{s}_{n}=\underset{n\to \mathrm{\infty }}{lim inf}cos\left(1-2{\beta }_{n}\right){s}_{n}=\underset{n\to \mathrm{\infty }}{lim inf}cos|1-2{\beta }_{n}|{s}_{n}.$

Hence we get

$\underset{n\to \mathrm{\infty }}{lim sup}{s}_{n}=\underset{n\to \mathrm{\infty }}{lim sup}\left(|1-2{\beta }_{n}|{s}_{n}\right)\le \underset{n\to \mathrm{\infty }}{lim sup}|1-2{\beta }_{n}|\underset{n\to \mathrm{\infty }}{lim sup}{s}_{n},$

and we have

$0\ge \left(1-\underset{n\to \mathrm{\infty }}{lim sup}|1-2{\beta }_{n}|\right)\underset{n\to \mathrm{\infty }}{lim sup}{s}_{n}=\underset{n\to \mathrm{\infty }}{lim inf}\left(1-|1-2{\beta }_{n}|\right)\underset{n\to \mathrm{\infty }}{lim sup}{s}_{n}.$

Since $\left\{{\beta }_{n}\right\}\subset \left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$ for $n\in \mathbb{N}$, we have ${lim inf}_{n\to \mathrm{\infty }}\left(1-|1-2{\beta }_{n}|\right)>0$ and thus, ${lim sup}_{n\to \mathrm{\infty }}{s}_{n}\le 0$. Therefore, we get ${lim sup}_{n\to \mathrm{\infty }}{s}_{n}=0$ and we also get ${lim sup}_{n\to \mathrm{\infty }}{t}_{n}=0$. It implies $d\left({x}_{n},S{x}_{n}\right)\to 0$ and $d\left({x}_{n},T{x}_{n}\right)\to 0$.

Next, let $\left\{{y}_{k}\right\}$ be a subsequence of $\left\{{x}_{n}\right\}$. Since $r\left(\left\{{x}_{n}\right\}\right)\le {d}_{0}<\pi /2$, by Theorem 3.3(i), there exists a unique asymptotic center ${x}_{0}$ of $\left\{{y}_{k}\right\}$. Moreover, since $r\left(\left\{{y}_{k}\right\}\right)<\pi /2$, by Theorem 3.3(ii), there exists a subsequence $\left\{{z}_{l}\right\}$ of $\left\{{y}_{k}\right\}$ such that ${z}_{l}\stackrel{\mathrm{\Delta }}{⇀}{z}_{0}\in X$. Further, since $d\left({z}_{l},S{z}_{l}\right)\to 0$, $d\left({z}_{l},T{z}_{l}\right)\to 0$ and S, T are Δ-demiclosed, we have ${z}_{0}\in F\left(S\right)\cap F\left(T\right)$. Then we can show that ${x}_{0}={z}_{0}$, i.e., ${x}_{0}\in F\left(S\right)\cap F\left(T\right)$. If not, from the uniqueness of the asymptotic centers ${x}_{0}$, ${z}_{0}$ of $\left\{{y}_{k}\right\}$, $\left\{{z}_{l}\right\}$, respectively, due to Theorem 3.3(i), we have

$\begin{array}{rl}\underset{k\to \mathrm{\infty }}{lim sup}d\left({y}_{k},{x}_{0}\right)& <\underset{k\to \mathrm{\infty }}{lim sup}d\left({y}_{k},{z}_{0}\right)\\ =\underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n},{z}_{0}\right)\\ =\underset{l\to \mathrm{\infty }}{lim sup}d\left({z}_{l},{z}_{0}\right)\\ <\underset{l\to \mathrm{\infty }}{lim sup}d\left({z}_{l},{x}_{0}\right)\\ \le \underset{k\to \mathrm{\infty }}{lim sup}d\left({y}_{k},{x}_{0}\right).\end{array}$

This is a contradiction. Hence we get ${x}_{0}\in F\left(S\right)\cap F\left(T\right)$. Next, we show that for any subsequences of $\left\{{x}_{n}\right\}$, their asymptotic center consists of the unique element. Let $\left\{{u}_{k}\right\}$, $\left\{{v}_{k}\right\}$ be subsequences of $\left\{{x}_{n}\right\}$, ${x}_{0}\in AC\left(\left\{{u}_{k}\right\}\right)$ and ${x}_{0}^{\prime }\in AC\left(\left\{{v}_{k}\right\}\right)$. We show ${x}_{0}={x}_{0}^{\prime }$ by using contradiction. Assume ${x}_{0}\ne {x}_{0}^{\prime }$. Then ${x}_{0}^{\prime }\notin AC\left(\left\{{u}_{k}\right\}\right)$ and ${x}_{0}\notin AC\left(\left\{{v}_{k}\right\}\right)$ by Theorem 3.3(i). It follows that

$\begin{array}{rl}\underset{k\to \mathrm{\infty }}{lim sup}d\left({u}_{k},{x}_{0}\right)& <\underset{k\to \mathrm{\infty }}{lim sup}d\left({u}_{k},{x}_{0}^{\prime }\right)\\ =\underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n},{x}_{0}^{\prime }\right)\\ =\underset{k\to \mathrm{\infty }}{lim sup}d\left({v}_{k},{x}_{0}^{\prime }\right)\\ <\underset{k\to \mathrm{\infty }}{lim sup}d\left({v}_{k},{x}_{0}\right)\\ =\underset{n\to \mathrm{\infty }}{lim}d\left({x}_{n},{x}_{0}\right)\\ =\underset{k\to \mathrm{\infty }}{lim sup}d\left({u}_{k},{x}_{0}\right).\end{array}$

This is a contradiction. Hence we get ${x}_{0}={x}_{0}^{\prime }$. Therefore, we have $\left\{{x}_{n}\right\}$ Δ-converges to a common fixed point of S and T. □

By Theorem 3.4, we know that a nonexpansive mapping having a fixed point satisfies the assumptions in Theorem 4.1. Thus, we get the following result.

Corollary 4.2 Let X be a complete $CAT\left(1\right)$ space such that for any $u,v\in X$, $d\left(u,v\right)<\pi /2$. Let S and T be nonexpansive mappings of X into itself such that $F\left(S\right)\cap F\left(T\right)\ne \mathrm{\varnothing }$. Let $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$ and $\left\{{\gamma }_{n}\right\}$ be sequences in $\left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$. Define a sequence $\left\{{x}_{n}\right\}\subset X$ as the following recurrence formula: ${x}_{1}\in X$ and

$\left\{\begin{array}{l}{u}_{n}=\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}S{x}_{n},\\ {v}_{n}=\left(1-{\gamma }_{n}\right){x}_{n}\oplus {\gamma }_{n}T{x}_{n},\\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){u}_{n}\oplus {\alpha }_{n}{v}_{n}\end{array}$

for $n\in \mathbb{N}$. Then $\left\{{x}_{n}\right\}$ Δ-converges to a common fixed point of S and T.

## 5 An application to the image recovery

The image recovery problem is formulated as to find the nearest point in the intersection of family of closed convex subsets from a given point by using corresponding metric projection of each subset. In this section, we consider this problem for two subsets of a complete $CAT\left(1\right)$ space.

Theorem 5.1 Let X be a complete $CAT\left(1\right)$ space such that for any $u,v\in X$, $d\left(u,v\right)<\pi /2$. Let ${C}_{1}$ and ${C}_{2}$ be nonempty closed convex subsets of X such that ${C}_{1}\cap {C}_{2}\ne \mathrm{\varnothing }$. Let ${P}_{1}$ and ${P}_{2}$ be metric projections onto ${C}_{1}$ and ${C}_{2}$, respectively. Let $\left\{{\alpha }_{n}\right\}$, $\left\{{\beta }_{n}\right\}$ and $\left\{{\gamma }_{n}\right\}$ be real sequences in $\left[a,b\right]\subset \phantom{\rule{0.2em}{0ex}}\right]0,1\left[$. Define a sequence $\left\{{x}_{n}\right\}\subset X$ by the following recurrence formula: ${x}_{1}\in X$ and

$\left\{\begin{array}{l}{u}_{n}=\left(1-{\beta }_{n}\right){x}_{n}\oplus {\beta }_{n}{P}_{1}{x}_{n},\\ {v}_{n}=\left(1-{\gamma }_{n}\right){x}_{n}\oplus {\gamma }_{n}{P}_{2}{x}_{n},\\ {x}_{n+1}=\left(1-{\alpha }_{n}\right){u}_{n}\oplus {\alpha }_{n}{v}_{n}\end{array}$

for $n\in \mathbb{N}$. Then $\left\{{x}_{n}\right\}$ Δ-converges to a fixed point of the intersection of ${C}_{1}$ and ${C}_{2}$.

Proof We see that ${P}_{1}$ and ${P}_{2}$ are quasinonexpansive [9] and Δ-demiclosed [8]. Further, we also get $F\left({P}_{1}\right)={C}_{1}$ and $F\left({P}_{2}\right)={C}_{2}$. Thus, letting $S={P}_{1}$ and $T={P}_{2}$ in Theorem 4.1, we obtain the desired result. □

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## Acknowledgements

The authors thank the anonymous referees for their valuable comments and suggestions. The first author is supported by Grant-in-Aid for Scientific Research No. 22540175 from the Japan Society for the Promotion of Science.

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Correspondence to Koichi Nakagawa.

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The authors declare that they have no competing interests.

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The authors have contributed in this work on an equal basis. All authors read and approved the final manuscript.

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Kimura, Y., Nakagawa, K. Another type of Mann iterative scheme for two mappings in a complete geodesic space. J Inequal Appl 2014, 72 (2014). https://doi.org/10.1186/1029-242X-2014-72

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• DOI: https://doi.org/10.1186/1029-242X-2014-72

### Keywords

• Nonexpansive Mapping
• Common Fixed Point
• Nonempty Closed Convex Subset
• Iteration Procedure
• Real Sequence