Iterative Sequences for a Finite Number of Resolvent Operators on Complete Geodesic Spaces

Abstract

We consider Halpern’s and Mann’s types of iterative schemes to find a common minimizer of a finite number of proper lower semicontinuous convex functions defined on a complete geodesic space with curvature bounded above.

1. Introduction

We consider finding a common fixed point of a finite number of resolvents operators for proper lower semicontinuous convex functions on a geodesic space. To find this point, we often use iterative schemes. We focus on Mann’s [1] and Halpern’s [2] iterative schemes. We know many authors have considered these schemes by using nonexpansive mappings. In a Banach space, Reich [3] proved weak convergence of Mann-type iteration, and Takahashi and Tamura [4] proved that by using two nonexpansive mappings. In a Hilbert space, Wittmann [5] proved strong convergence of the Halpern-type iteration.

We also know many researchers have proved iterative schemes on geodesic spaces. In a CAT(0) space, Dhompongsa and Panyanak [6] proved $\Delta$-convergence of Mann’s iterative scheme, and Saejung [7] also proved convergence of Halpern’s iterative scheme. We know a large number of results by using Mann’s and Halpern’s iterative schemes in a CAT(1) space. Pia̧tek [8] considered Halpern’s iterative scheme by using a nonexpansive mapping in CAT(1) space. Kimura and Satô [9] proved that by using a strongly quasi-nonexpansive and $\Delta$-demiclosed mapping in a complete CAT(1) space. Kimura, Saejung, and Yotkaew [10] also proved convergence of Halpern’s iterative schemes under the same setting. Kimura and Kohsaka [11] proved convergence of Mann and Halpern types of iterative schemes with a sequence of resolvent operators for a single proper lower semicontinuous convex function. We are particularly interested in these results [9,10,11], and obtain Theorems 1 and 2 with a finite number of resolvent operators in a complete CAT(1) space.

In a Hilbert space, the resolvent operator $J_f$ is defined as follows. Let f be a proper lower semicontinuous convex function from a Hilbert space H to $\left]-\infty ,+\infty \right]$. Then, $J_f$ is defined by

$$J_{f}x=\underset{y\in H}{\mathrm{argmin}}\{f\left(y\right)+\frac{1}{2}{∥y-x∥}^2\}$$

for all $x\in H$. We know the resolvent $J_f$ is a single-valued mapping from H to H and it is nonexpansive. For a proper lower semicontinuous convex function f from a complete CAT(0) space X into $\left]-\infty ,+\infty \right]$, Jost [12] and Mayer [13] defined the resolvent $R_f$ of f by

$$R_{f}x=\underset{y\in X}{\mathrm{argmin}}\{f\left(y\right)+\frac{1}{2}d{(y,x)}^2\}$$

for all $x\in X$. We also know the resolvent $R_f$ is a single-valued mapping from X to X and it is nonexpansive. In this paper, we use the resolvent in a complete CAT(1) space defined by Kimura and Kohsaka [11,14].

2. Preliminaries

Let $(X,d)$ be a metric space. For $x,y\in X$, a geodesic between x and y is an isometric mapping $c:[0,d(x,y\left)\right]\to X$ with $c\left(0\right)=x$ and $c\left(d\right(x,y\left)\right)=y$. We say X is an r-geodesic space for $r>0$ if a geodesic exists for every pair of points in X satisfying $d(x,y)<r$. Further, a metric space X is said to be r-uniquely geodesic if such a geodesic is unique for each pair of points satisfying $d(x,y)<r$. The image of a unique geodesic between x and y is denoted by $[x,y]$.

For an r-uniquely geodesic space X, the convex combination between $x,y\in X$ with $d(x,y)<r$ is naturally defined. That is, for $\alpha \in [0,1]$, we denote by $\alpha x\oplus (1-\alpha )y$ the point $c\left(\right(1-\alpha \left)d\right(x,y\left)\right)$, where c is a geodesic between x and y. It follows that

$$d(\alpha x\oplus (1-\alpha )y,x)=(1-\alpha )d(x,y)\mathrm{and}d(\alpha x\oplus (1-\alpha )y,y)=\alpha d(x,y).$$

A subset C of X is said to be r-convex if $\alpha x\oplus (1-\alpha )y\in C$ for every $x,y\in C$ with $d(x,y)<r$ and $\alpha \in [0,1]$.

If X is r-geodesic for any $r>0$, then X is simply called a geodesic space. A uniquely geodesic space and a convex subset are also defined in the same way.

Let X be a uniquely geodesic space and $x,y,z\in X$. For a triangle $▵(x,y,z)=[y,z]\cup [z,x]\cup [x,y]\subset X$ satisfying $d(y,z)+d(z,x)+d(x,y)<2\pi$, we define its comparison triangle $▵(\overline{x},\overline{y},\overline{z})$ in the two-dimensional unit sphere ${\mathbb{S}}^2$ by the triangle such that each corresponding edge has the same length as that of the original triangle. Using this notion, we call X a CAT$\left(1\right)$ space if for every $x,y,z\in X$, $p,q\in ▵(x,y,z)$, and their corresponding points $\overline{p},\overline{q}\in {\mathbb{S}}^2$, the following relation is satisfied,

$$d(p,q)\le d_{{\mathbb{S}}^2}(\overline{x},\overline{y}),$$

where $d_{{\mathbb{S}}^2}$ is the spherical metric on ${\mathbb{S}}^2$.

The following results are fundamental and important for our work.

Lemma 1

(Kimura-Satô [15]). Let X be a CAT$\left(1\right)$ space. Then, for every $x,y,z\in X$ with $d(x,y)+d(y,z)+d(z,x)<2\pi$ and $\alpha \in \left[0,1\right]$, the following inequality holds,

$$cosd(x,w)sind(y,z)\ge cosd(x,y)sin\left(\alpha d\right(y,z\left)\right)+cosd(x,z)sin\left(\right(1-\alpha \left)d\right(y,z\left)\right),$$

where $w=\alpha y\oplus (1-\alpha )z$.

Lemma 2

(Kimura-Satô [9]). Let X be a CAT$\left(1\right)$ space. Then, for every $x,y,z\in X$ with $d(x,y)+d(y,z)+d(z,x)<2\pi$ and $\alpha \in \left[0,1\right]$, the following inequality holds,

$$cosd(x,w)\ge \alpha cosd(x,y)+(1-\alpha )cosd(x,z),$$

where $w=\alpha y\oplus (1-\alpha )z$.

Lemma 3

(Kimura-Satô [9]). Let X be a CAT$\left(1\right)$ space such that $d(v,v^{\prime })<\pi$ for every $v,v^{\prime }\in X$. Let $\alpha \in \left[0,1\right]$ and $u,y,z\in X$. Then,

$$\begin{array}{c}1-cosd(\alpha u\oplus (1-\alpha )y,z)\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\le (1-\beta )(1-cosd(y,z))+\beta \left(1-\frac{cosd(u,z)}{sind(u,y)tan\left(\frac{\alpha }{2}d(u,y)\right)+cosd(u,y)}\right),\end{array}$$

where

$$\beta =\left\{\begin{array}{cc}1-{\displaystyle \frac{sin\left(\right(1-\alpha \left)d\right(u,y\left)\right)}{sind(u,y)}}\hfill & (u\ne y),\hfill \\ \alpha \hfill & (u=y).\hfill \end{array}\right.$$

Let $\left\{x_n\right\}\subset X$ be a bounded sequence. We say a point $z\in X$ is an asymptotic center of $\left\{x_n\right\}$ if it is a minimizer of the function ${lim\; sup}_{n\to \infty }d(x_n,·)$, that is,

$$\underset{n\to \infty }{lim\; sup}d(x_n,z)\le \underset{n\to \infty }{lim\; sup}d(x_n,y)$$

for every $y\in X$. If $z\in X$ is the unique asymptotic center of all subsequences of $\left\{x_n\right\}$, then we say $\left\{x_n\right\}$ is $\Delta$-convergent to a $\Delta$-limit z. We know that in a CAT$\left(1\right)$ space, every sequence $\left\{x_n\right\}$ satisfying ${inf}_{y\in X}{lim\; sup}_{n\to \infty }d(x_n,y)<\pi /2$ has a unique asymptotic center and a $\Delta$-convergent subsequence.

Let X be a CAT$\left(1\right)$ space and $T:X\to X$. The set of all fixed points of T is denoted by $F\left(T\right)$. Namely, $F\left(T\right)=\{z\in X:z=Tz\}$. T is said to be quasi-nonexpansive if $F\left(T\right)\ne Ø$ and $d(Tx,z)\le d(x,z)$ for every $x\in X$ and $z\in F\left(T\right)$. A quasi-nonexpansive mapping T is said to be strongly quasi-nonexpansive if ${lim}_{n\to \infty }d(x_n,T{x}_n)=0$ whenever $\left\{x_n\right\}\subset X$ satisfies ${sup}_{n\in \mathbb{N}}d(x_n,p)<\pi /2$ and ${lim}_{n\to \infty }(cosd(x_n,p)/cosd(T{x}_n,p))=1$ for every $p\in F\left(T\right)$.

A mapping T is said to be $\Delta$-demiclosed if $z\in F\left(T\right)$ whenever $\left\{x_n\right\}$ is $\Delta$-convergent to z and ${lim}_{n\to \infty }d(x_n,T{x}_n)=0$.

Following [16], we define the notions of a strongly quasi-nonexpansive sequence and a $\Delta$-demiclosed sequence on CAT$\left(1\right)$ spaces as follows. Let $\left\{T_n\right\}$ be a sequence of mappings from X to X. $\left\{T_n\right\}$ is said to be a strongly quasi-nonexpansive sequence if each $T_n$ is quasi-nonexpansive and ${lim}_{n\to \infty }d(x_n,T_n{x}_n)=0$ whenever ${sup}_{n\in \mathbb{N}}d(x_n,p)<\pi /2$ and ${lim}_{n\to \infty }(cosd(x_n,p)/cosd(T_n{x}_n,p))=1$ for every $p\in {\bigcap }_{n=1}^{\infty }F\left(T_n\right)$. $\left\{T_n\right\}$ is said to be a $\Delta$-demiclosed sequence if $z\in {\bigcap }_{n=1}^{\infty }F\left(T_n\right)$ whenever $\left\{x_n\right\}$ is $\Delta$-convergent to z and ${lim}_{n\to \infty }d(x_n,T_n{x}_n)=0$.

Let X be a complete CAT$\left(1\right)$ space and $C\subset X$ a nonempty closed $\pi$-convex subset such that $d(x,C)={inf}_{y\in C}d(x,y)<\pi /2$ for every $x\in X$. Then, for each $x\in X$, there exists a unique point $y_x\in C$ satisfying $d(x,y_x)={inf}_{y\in C}d(x,y)$. Using this point, we define a metric projection $P_C:X\to C$ by $P_{C}x=y_x$ for $x\in X$.

Let X be a complete CAT$\left(1\right)$ space such that $d(v,v^{\prime })<\pi /2$ for every $v,v^{\prime }\in X$. Let $f:X\to \left]-\infty ,+\infty \right]$ be a proper lower semicontinuous convex function. The resolvent $R_f$ of f is defined by

$$R_{f}x=\underset{y\in X}{\mathrm{argmin}}(f\left(y\right)+tand(y,x)sind(y,x))$$

for all $x\in X$; (see in [14]). We know that $R_f$ is a single-valued mapping from X to X. We also know that the resolvent $R_f$ is strongly quasi-nonexpansive and $\Delta$-demiclosed such that $F\left(R_f\right)={\mathrm{argmin}}_{x\in X}f$ (see [11,14]).

We recall some lemmas useful for our results.

Lemma 4

(Kimura-Satô [17]). Let X be a complete CAT$\left(1\right)$ space such that $d(u,v)<\pi /2$ for all $u,v\in X$. Let $S,T$ be quasi-nonexpansive mappings from X to X with $F\left(S\right)\cap F\left(T\right)\ne Ø$. Then, for every $\alpha \in \left]0,1\right[$, $F\left(S\right)\cap F\left(T\right)=F(\alpha S\oplus (1-\alpha \left)T\right)$ and the mapping $\alpha S\oplus (1-\alpha )T$ is quasi-nonexpansive.

Lemma 5

(He-Fang-López-Li [18]). Let X be a complete CAT$\left(1\right)$ space and $p\in X$. If a sequence $\left\{x_n\right\}$ in X satisfies that ${lim\; sup}_{n\to \infty }d(x_n,p)<\pi /2$ and that $\left\{x_n\right\}$ is Δ-convergent to $x\in X$, then $d(x,p)\le {lim\; inf}_{n\to \infty }d(x_n,p)$.

Lemma 6

(Saejung-Yotkaew [19], Aoyama-Kimura-Kohsaka [20]). Let $\left\{s_n\right\}$ and $\left\{t_n\right\}$ be sequences of real numbers such that $s_n\ge 0$ for every $n\in \mathbb{N}$. Let $\left\{{\beta }_n\right\}$ be a sequence in $\left]0,1\right[$ such that ${\sum }_{n=0}^{\infty }{\beta }_n=\infty$. Suppose that $s_{n+1}\le (1-{\beta }_n)s_n+{\beta }_n{t}_n$ for every $n\in \mathbb{N}$. If ${lim\; sup}_{k\to \infty }{t}_{n_k}\le 0$ for every nondecreasing sequence $\left\{n_k\right\}$ of $\mathbb{N}$ satisfying ${lim\; inf}_{k\to \infty }(s_{n_k+1}-s_{n_k})\ge 0$, then ${lim}_{n\to \infty }{s}_n=0$.

3. Lemmas for a Finite Number of Resolvent Operators

In this section, we prove some lemmas by using a finite number of resolvent operators for iterative schemes. Throughout this section, let X be a CAT$\left(1\right)$ space such that $d(v,v^{\prime })<\pi /2$ for every $v,v^{\prime }\in X$.

Lemma 7.

For a given real number $a\in \left]0,\frac{1}{2}\right]$, let $\sigma \in \left[a,1-a\right]$. For given points $y,y^0,y^1\in X$, define $w\in X$ by

$$w=\sigma y^0\oplus (1-\sigma )y^1.$$

Then,

$$cosd(w,y)cos\left(ad(y^0,y^1)\right)\ge min\{cosd(y^0,y),cosd(y^1,y)\}.$$

Proof. 

If $y^0=y^1$, it is obvious. Otherwise, by Lemma 1, we have

$$\begin{array}{cc}\hfill cosd& (w,y)sind(y^0,y^1)\hfill \\ \hfill & \ge cosd(y^0,y)sin\left(\sigma d(y^0,y^1)\right)+cosd(y^1,y)sin\left((1-\sigma )d(y^0,y^1)\right)\hfill \\ \hfill & \ge min\{cosd(y^0,y),cosd(y^1,y)\}(sin\left(\sigma d(y^0,y^1)\right)+sin\left((1-\sigma )d(y^0,y^1)\right))\hfill \\ \hfill & =2min\{cosd(y^0,y),cosd(y^1,y)\}sin\frac{d(y^0,y^1)}{2}cos\frac{(2\sigma -1)d(y^0,y^1)}{2}.\hfill \end{array}$$

Dividing above by $2sin(d(y^0,y^1)/2)$, we have

$$\begin{array}{cc}\hfill cosd(w,y)cos& \frac{d(y^0,y^1)}{2}\hfill \\ \hfill & \ge min\{cosd(y^0,y),cosd(y^1,y)\}cos\frac{(2\sigma -1)d(y^0,y^1)}{2}\hfill \\ \hfill & \ge min\{cosd(y^0,y),cosd(y^1,y)\}cos\frac{(1-2a)d(y^0,y^1)}{2}.\hfill \end{array}$$

Moreover, dividing above by $cos((1-2a)d(y^0,y^1)/2)$, we have

$$\begin{array}{cc}\hfill & min\{cosd(y^0,y),cosd(y^1,y)\}\hfill \\ \hfill & \le cosd(w,y)\frac{cos{\displaystyle \frac{(1-2a)d(y^0,y^1)}{2}}cos\left(ad(y^0,y^1)\right)-sin{\displaystyle \frac{(1-2a)d(y^0,y^1)}{2}}sin\left(ad(y^0,y^1)\right)}{cos{\displaystyle \frac{(1-2a)d(y^0,y^1)}{2}}}\hfill \\ \hfill & \le cosd(w,y)cos\left(ad(y^0,y^1)\right).\hfill \end{array}$$

This completes the proof. □

Lemma 8.

For a given real number $a\in \left]0,\frac{1}{2}\right]$, let ${\sigma }^l\in \left[a,1-a\right]$ for every $l=0,1,\dots ,N-1$. For given points $y,y^k\in X$ for every $k=0,1,\dots ,N$, define $w^l\in X$ by

$$w^N=y^N\mathrm{and}{w}^l={\sigma }^l{y}^l\oplus (1-{\sigma }^l)w^{l+1}$$

for every $l=0,1,\dots ,N-1$. Then,

$$cosd(w^0,y)cos\left(ad(y^0,w^1)\right)\ge \underset{k\in \{0,1,\dots ,N\}}{min}cosd(y^k,y).$$

Proof. 

By Lemma 7,

$$cosd(w^0,y)cos\left(ad(y^0,w^1)\right)\ge min\{cosd(y^0,y),cosd(w^1,y)\}.$$

We also have

$$\begin{array}{cc}\hfill cosd(w^l,y)& \ge cosd(w^l,y)cos\left(ad(y^l,w^{l+1})\right)\hfill \\ \hfill & \ge min\{cosd(y^l,y),cosd(w^{l+1},y)\}\hfill \end{array}$$

for $l=1,2,\dots ,N-1$. Therefore, $cosd(w^0,y)cos\left(ad(y^0,w^1)\right)\ge {min}_{k\in \{0,1,\dots ,N\}}cosd(y^k,y)$. This completes the proof. □

Corollary 1.

Let $T^k$ be a quasi-nonexpansive mapping from X to X for every $k=0,1,\dots ,N$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let ${\sigma }^l\in \left[a,1-a\right]$ for every $l=0,1,\dots ,N-1$. Define $U^l:X\to X$ by

$$U^N=T^N\mathrm{and}{U}^l={\sigma }^l{T}^l\oplus (1-{\sigma }^l)U^{l+1}$$

for every $l=0,1,\dots ,N-1$. Let $x\in X$ and $p\in {\bigcap }_{k=0}^{N}F\left(T^k\right)$. Then,

$$cosd(U^{0}x,p)cos\left(ad(T^{0}x,U^{1}x)\right)\ge cosd(x,p).$$

Next, we show several properties of a sequence of resolvents. Let f be a proper lower semicontinuous convex function from X into $\left]-\infty ,+\infty \right]$ such that ${\mathrm{argmin}}_{X}f\ne Ø$ and let $\left\{{\lambda }_n\right\}$ be a real sequence such that $inf{\lambda }_n>0$. Then we know that $\left\{R_{{\lambda }_{n}f}\right\}$ is a strongly quasi-nonexpansive sequence and $\Delta$-demiclosed sequence (see [11]). Therefore, we obtain the following results, using Lemma 4.

Lemma 9.

Let $f^k$ be a proper lower semicontinuous convex function from X into $\left]-\infty ,+\infty \right]$ for every $k=0,1,\dots ,N$ such that ${\bigcap }_{k=0}^N{\mathrm{argmin}}_X{f}^k\ne Ø$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let ${\sigma }^l\in \left[a,1-a\right]$ for every $l=0,1,\dots ,N-1$ and ${\lambda }^k\in \left[a,+\infty \right[$ for every $k=0,1,\dots ,N$. Let $R_{{\lambda }^k{f}^k}$ be the resolvent of ${\lambda }^k{f}^k$ for every $k=0,1,\dots ,N$. Define $U^l:X\to X$ by

$$U^N=R_{{\lambda }^N{f}^N}\mathrm{and}{U}^l={\sigma }^l{R}_{{\lambda }^l{f}^l}\oplus (1-{\sigma }^l)U^{l+1}$$

for every $l=0,1,\dots ,N-1$. Then

$$F\left(U^0\right)=\bigcap _{k=0}^N\underset{X}{\mathrm{argmin}}{f}^k.$$

Lemma 10.

Let $\left\{T_n\right\}$ be a strongly quasi-nonexpansive sequence. Let f be a proper lower semicontinuous convex function from X into $\left]-\infty ,+\infty \right]$ such that ${\bigcap }_{n=1}^{\infty }F\left(T_n\right)\cap {\mathrm{argmin}}_{X}f\ne Ø$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let $\left\{{\sigma }_n\right\}\subset \left[a,1-a\right]$ and $\left\{{\lambda }_n\right\}\subset \left[a,+\infty \right[$. Let $R_{{\lambda }_{n}f}$ be the resolvent of ${\lambda }_{n}f$ for every $n\in \mathbb{N}$. Then $\{{\sigma }_n{R}_{{\lambda }_{n}f}\oplus (1-{\sigma }_n)T_n\}$ is a strongly quasi-nonexpansive sequence.

Proof. 

Let $V_n={\sigma }_n{R}_{{\lambda }_{n}f}\oplus (1-{\sigma }_n)T_n$ for every $n\in \mathbb{N}$. From Lemma 4, $V_n$ is a quasi-nonexpansive mapping for every $n\in \mathbb{N}$. From Corollary 1, for $\left\{x_n\right\}\subset X$ and $p\in {\bigcap }_{n=1}^{\infty }F\left(T_n\right)\cap {\mathrm{argmin}}_{X}f$ such that ${lim}_{n\to \infty }cosd(x_n,p)/cosd(V_n{x}_n,p)=1$ and ${sup}_{n\in \mathbb{N}}d(x_n,p)<\pi /2$, we have

$$cosd(V_n{x}_n,p)cos\left(ad(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\right)\ge cosd(x_n,p)$$

and thus

$$cos\left(ad(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\right)\ge \frac{cosd(x_n,p)}{cosd(V_n{x}_n,p)}\to 1.$$

That is, ${lim}_{n\to \infty }d(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)=0$. Therefore, we have

$$\underset{n\to \infty }{lim}d(T_n{x}_n,V_n{x}_n)=\underset{n\to \infty }{lim}{\sigma }_{n}d(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)=0.$$

As $1={lim}_{n\to \infty }cosd(x_n,p)/cosd(V_n{x}_n,p)={lim}_{n\to \infty }cosd(x_n,p)/cosd(T_n{x}_n,p)$, we have

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

Thus, we obtain

$$d(V_n{x}_n,x_n)\le d(V_n{x}_n,T_n{x}_n)+d(T_n{x}_n,x_n)\to 0.$$

This completes the proof. □

Corollary 2.

Let $f^k$ be the same as in Lemma 9 for $k=0,1,\dots ,N$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let $\left\{{\sigma }_n^l\right\}\subset \left[a,1-a\right]$ for every $l=0,1,\dots ,N-1$ and $\left\{{\lambda }_n^k\right\}\subset \left[a,+\infty \right[$ for every $k=0,1,\dots ,N$. Let $R_{{\lambda }_n^k{f}^k}$ be the resolvent of ${\lambda }_n^k{f}^k$ for every $k=0,1,\dots ,N$ and $n\in \mathbb{N}$. Define $U_n^l:X\to X$ by

$$U_n^N=R_{{\lambda }_n^N{f}^N}\mathrm{and}{U}_n^l={\sigma }_n^l{R}_{{\lambda }_n^l{f}^l}\oplus (1-{\sigma }_n^l)U_n^{l+1}$$

for every $l=0,1,\dots ,N-1$ and $n\in \mathbb{N}$. Then, $\left\{U_n^0\right\}$ is a strongly quasi-nonexpansive sequence.

Lemma 11.

Let $\left\{T_n\right\}$ be a quasi-nonexpansive and Δ-demiclosed sequence. Let f be a proper lower semicontinuous convex function from X into $\left]-\infty ,+\infty \right]$ such that ${\bigcap }_{n=1}^{\infty }F\left(T_n\right)\cap {\mathrm{argmin}}_{X}f\ne Ø$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let $\left\{{\sigma }_n\right\}\subset \left[a,1-a\right]$ and $\left\{{\lambda }_n\right\}\subset \left[a,+\infty \right[$. Let $R_{{\lambda }_{n}f}$ be the resolvent of ${\lambda }_{n}f$ for every $n\in \mathbb{N}$. Then $\{{\sigma }_n{R}_{{\lambda }_{n}f}\oplus (1-{\sigma }_n)T_n\}$ is a Δ-demiclosed sequence.

Proof. 

Let $V_n={\sigma }_n{R}_{{\lambda }_{n}f}\oplus (1-{\sigma }_n)T_n$ for every $n\in \mathbb{N}$. Let $p\in {\bigcap }_{n=1}^{\infty }F\left(T_n\right)\cap {\mathrm{argmin}}_{X}f$, $\left\{x_n\right\}\subset X$, and $z\in X$ such that ${lim}_{n\to \infty }d(V_n{x}_n,x_n)=0$ and suppose that $\left\{x_n\right\}$ is $\Delta$-convergent to z. Then,

$$cosd(V_n{x}_n,p)cos\left(ad(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\right)\ge cosd(x_n,p)$$

and thus

$$\begin{array}{cc}\hfill 1\ge cos\left(ad(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\right)& \ge \frac{cosd(x_n,p)}{cosd(V_n{x}_n,p)}\hfill \\ \hfill & \ge \frac{cos(d(x_n,V_n{x}_n)+d(V_n{x}_n,p))}{cosd(V_n{x}_n,p)}\to 1.\hfill \end{array}$$

Therefore, ${lim}_{n\to \infty }d(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)=0$. Thus, we have

$$\begin{array}{cc}\hfill d(R_{{\lambda }_{n}f}{x}_n,V_n{x}_n)& =(1-{\sigma }_n)d(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\hfill \\ \hfill & \le (1-a)d(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\to 0.\hfill \end{array}$$

Since $R_{{\lambda }_{n}f}$ is a $\Delta$-demiclosed sequence, we have $R_{{\lambda }_{n}f}z=z$. Similarly,

$$\begin{array}{cc}\hfill d(T_n{x}_n,V_n{x}_n)& ={\sigma }_{n}d(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\hfill \\ \hfill & \le (1-a)d(R_{{\lambda }_{n}f}{x}_n,T_n{x}_n)\to 0.\hfill \end{array}$$

Since $\left\{T_n\right\}$ is a $\Delta$-demiclosed sequence, we have $T_{n}z=z$. Thus, $V_{n}z=z$. This completes the proof. □

Corollary 3.

Let $f^k$, $\left\{{\sigma }_n^l\right\}$, $\left\{{\lambda }_n^k\right\}$ and $\left\{U_n^l\right\}$ be the same as in Corollary 2 for $k=0,1,\dots ,N$ and $l=0,1,\dots ,N-1$. Then $\left\{U_n^0\right\}$ is a Δ-demiclosed sequence.

4. Iterative Schemes for a Finite Resolvents Operators

We prove convergence of Mann and Halpern types of iterative sequences for finitely many convex functions by using the properties of a sequence of the resolvents in CAT$\left(1\right)$ space.

Theorem 1.

Let X be a complete CAT$\left(1\right)$ space such that $d(v,v^{\prime })<\pi /2$ for every $v,v^{\prime }\in X$. Let $f^k$ be a proper lower semicontinuous convex function from X into $\left]-\infty ,+\infty \right]$ for every $k=0,1,\dots ,N$ such that $F={\bigcap }_{k=0}^N{\mathrm{argmin}}_X{f}^k\ne Ø$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let $\left\{{\sigma }_n^l\right\}\subset \left[a,1-a\right]$ for every $l=0,1,\dots ,N-1$ and $\left\{{\lambda }_n^k\right\}\subset \left[a,+\infty \right[$ for every $k=0,1,\dots ,N$. Let $R_{{\lambda }_n^k{f}^k}$ be the resolvent of ${\lambda }_n^k{f}^k$ for every $k=0,1,\dots ,N$ and $n\in \mathbb{N}$. Define $U_n^l:X\to X$ by

$$U_n^N=R_{{\lambda }_n^N{f}^N}\mathrm{and}{U}_n^l={\sigma }_n^l{R}_{{\lambda }_n^l{f}^l}\oplus (1-{\sigma }_n^l)U_n^{l+1}$$

for every $l=0,1,\dots ,N-1$ and $n\in \mathbb{N}$. Let $\left\{{\alpha }_n\right\}$ be a real sequence in $\left[a,1-a\right]$. For a given point $x_1\in X$, let $\left\{x_n\right\}$ be the sequence in X generated by

$$x_{n+1}={\alpha }_n{x}_n\oplus (1-{\alpha }_n)U_n^0{x}_n$$

for $n\in \mathbb{N}$. Then, $\left\{x_n\right\}$ Δ-converges to a point of F.

Proof. 

Let $z\in F$. As $U_n^0$ is a quasi-nonexpansive mapping, it follows from Lemma 2 that

$$\begin{array}{cc}\hfill cosd(x_{n+1},z)& \ge {\alpha }_{n}cosd(x_n,z)+(1-{\alpha }_n)cosd(U_n^0{x}_n,z)\hfill \\ \hfill & \ge cosd(x_n,z).\hfill \end{array}$$

Thus we have $d(x_{n+1},z)\le d(x_n,z)$ for $n\in \mathbb{N}$. There exists $D={lim}_{n\to \infty }d(x_n,z)\le d(x_1,z)<\pi /2$. From Lemma 1, we get

$$\begin{array}{cc}\hfill & cosd(x_{n+1},z)sind(x_n,U_n^0{x}_n)\hfill \\ & \ge cosd(x_n,z)sin{\alpha }_{n}d(x_n,U_n^0{x}_n)+cosd(U_n^0{x}_n,z)sin(1-{\alpha }_n)d(x_n,U_n^0{x}_n)\hfill \\ \hfill & \ge 2cosd(x_n,z)sin\frac{d(x_n,U_n^0{x}_n)}{2}cos\frac{(2{\alpha }_n-1)d(x_n,U_n^0{x}_n)}{2}.\hfill \end{array}$$

If $d(x_n,U_n^0{x}_n)\ne 0$, we obtain

$$\begin{array}{c}\hfill cosd(x_{n+1},z)cos\frac{d(x_n,U_n^0{x}_n)}{2}\ge cosd(x_n,z)cos\frac{(2{\alpha }_n-1)d(x_n,U_n^0{x}_n)}{2}.\end{array}$$

As $\left\{{\alpha }_n\right\}\subset \left[a,1-a\right]$, we get

$$\begin{array}{c}\hfill 1>\frac{cos\frac{d(x_n,U_n^0{x}_n)}{2}}{cos\frac{(1-2a)d(x_n,U_n^0{x}_n)}{2}}\ge \frac{cos\frac{d(x_n,U_n^0{x}_n)}{2}}{cos\frac{(2{\alpha }_n-1)d(x_n,U_n^0{x}_n)}{2}}\ge \frac{cosd(x_n,z)}{cosd(x_{n+1},z)}.\end{array}$$

As $D={lim}_{n\to \infty }d(x_n,z)\le d(x_1,z)<\pi /2$, we have

$$\underset{n\to \infty }{lim}\frac{cos\frac{d(x_n,U_n^0{x}_n)}{2}}{cos\frac{(1-2a)d(x_n,U_n^0{x}_n)}{2}}=1$$

and thus ${lim}_{n\to \infty }d(x_n,U_n^0{x}_n)=0$. Let $x_0$ be an asymptotic center of $\left\{x_n\right\}$ and y an asymptotic center of any subsequence $\left\{x_{n_k}\right\}\subset \left\{x_n\right\}$. There exists $\{x_{n_{k_l}}\}\subset \left\{x_{n_k}\right\}$ such that $\{x_{n_{k_l}}\}$ $\Delta$-converges to w. As $\{U_{n_{k_l}}^0\}$ is a $\Delta$-demiclosed sequence and ${lim}_{n\to \infty }d(U_{n_{k_l}}^0{x}_{n_{k_l}},x_{n_{k_l}})=0$, we obtain $w\in F$. Since there exists ${lim}_{n\to \infty }d(x_{n_k},w)$, we have

$$\begin{array}{cc}\hfill \underset{k\to \infty }{lim\; sup}d(x_{n_k},w)=\underset{k\to \infty }{lim}d(x_{n_k},w)& =\underset{l\to \infty }{lim}d(x_{n_{k_l}},w)\hfill \\ & \le \underset{l\to \infty }{lim\; sup}d(x_{n_{k_l}},y)\le \underset{k\to \infty }{lim\; sup}d(x_{n_k},y).\hfill \end{array}$$

Therefore, we obtain $y=w\in F$. Similarly, we get $x_0=y$. Therefore, $\left\{x_n\right\}$ $\Delta$-converges to $x_0\in F$. □

Theorem 2.

Let X, $f^k$, $\left\{{\sigma }_n^l\right\}$, $\left\{{\lambda }_n^k\right\}$ and $\left\{U_n^l\right\}$ be the same as in Theorem 1 for $k=0,1,\dots ,N$ and $l=0,1,\dots ,N-1$. Let $\left\{{\alpha }_n\right\}$ be a real sequence in $\left]0,1\right[$ such that ${lim}_{n\to \infty }{\alpha }_n=0$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$. For given points $u,x_1\in X$, let $\left\{x_n\right\}$ be the sequence in X generated by

$$x_{n+1}={\alpha }_{n}u\oplus (1-{\alpha }_n)U_n^0{x}_n$$

for $n\in \mathbb{N}$. Suppose that one of the following conditions holds:

(a)

${sup}_{v,v^{\prime }\in X}d(v,v^{\prime })<\pi /2;$

(b)

$d(u,P_{F}u)<\pi /4$ and $d(u,P_{F}u)+d(x_0,P_{F}u)<\pi /2;$

(c)

${\sum }_{n=0}^{\infty }{\alpha }_n^2=\infty .$

Then, $\left\{x_n\right\}$ converges to $P_{F}u$.

To prove this theorem, we also employ the technique proposed in [9]. Note that $F={\bigcap }_{k=0}^N{\mathrm{argmin}}_X{f}^k$.

Proof. 

Let $p=P_{F}u$ and let

$$\begin{array}{cc}\hfill & s_n=1-cosd(x_n,p),\hfill \\ \hfill & t_n=1-\frac{cosd(u,p)}{sind(u,U_n^0{x}_n)tan\left(\frac{{\alpha }_n}{2}d(u,U_n^0{x}_n)\right)+cosd(u,U_n^0{x}_n)},\hfill \\ \hfill & {\beta }_n=\left\{\begin{array}{cc}1-{\displaystyle \frac{sin\left((1-{\alpha }_n)d(u,U_n^0{x}_n)\right)}{sind(u,U_n^0{x}_n)}}\hfill & (u\ne U_n^0{x}_n),\hfill \\ {\alpha }_n\hfill & (u=U_n^0{x}_n)\hfill \end{array}\right.\hfill \end{array}$$

for $n\in \mathbb{N}$. Since $U_n^0$ is a quasi-nonexpansive mapping, it follows from Lemma 3 that

$$s_{n+1}\le (1-{\beta }_n)(1-cosd(U_n^0{x}_n,p))+{\beta }_n{t}_n\le (1-{\beta }_n)s_n+{\beta }_n{t}_n$$

for $n\in \mathbb{N}$. By Lemma 2, we have

$$\begin{array}{cc}\hfill cosd(x_{n+1},p)& =cosd({\alpha }_{n}u\oplus (1-{\alpha }_n)U_n^0{x}_n,p)\hfill \\ \hfill & \ge {\alpha }_{n}cosd(u,p)+(1-{\alpha }_n)cosd(U_n^0{x}_n,p)\hfill \\ \hfill & \ge {\alpha }_{n}cosd(u,p)+(1-{\alpha }_n)cosd(x_n,p)\hfill \\ \hfill & \ge min\{cosd(u,p),cosd(x_n,p)\}\hfill \end{array}$$

for $n\in \mathbb{N}$. So we have

$$cosd(x_n,p)\ge min\{cosd(u,p),cosd(x_0,p)\}=cosmax\{d(u,p),d(x_0,p)\}>0$$

for $n\in \mathbb{N}$. Hence ${sup}_{n\in \mathbb{N}}d(x_n,p)\le max\{d(u,p),d(x_0,p)\}<\pi /2$. Next, we will show for each of the conditions (a–c) imply that ${\sum }_{n=0}^{\infty }{\beta }_n=\infty$. For the conditions (a) and (b), let $M={sup}_{n\in \mathbb{N}}d(u,U_n^0{x}_n)$. Thus, we will show $M<\pi /2$. In case (a), it is obvious. In case (b), as ${sup}_{n\in \mathbb{N}}d(x_n,p)\le max\{d(u,p),d(x_0,p)\}$, we have

$$\begin{array}{cc}\hfill M& \le \underset{n\in \mathbb{N}}{sup}(d(u,p)+d(U_n^0{x}_n,p))\hfill \\ \hfill & \le \underset{n\in \mathbb{N}}{sup}(d(u,p)+d(x_n,p))\hfill \\ \hfill & \le max\{2d(u,p),d(u,p)+d(x_0,p)\}<\pi /2.\hfill \end{array}$$

Thus, for cases (a) and (b), we have

$$\begin{array}{cc}\hfill {\beta }_n& \ge 1-\frac{sin\left((1-{\alpha }_n)M\right)}{sinM}\hfill \\ \hfill & =\frac{2}{sinM}sin\left(\frac{{\alpha }_n}{2}M\right)cos\left(\left(1-\frac{{\alpha }_n}{2}\right)M\right)\hfill \\ \hfill & \ge {\alpha }_{n}cosM\hfill \end{array}$$

for $n\in \mathbb{N}$. As ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, each of the conditions (a) and (b) implies that ${\sum }_{n=0}^{\infty }{\beta }_n=\infty$. In the case (c), we have

$${\beta }_n\ge 1-sin\frac{(1-{\alpha }_n)\pi }{2}=1-cos\frac{{\alpha }_n}{2}\ge \frac{{\alpha }_n^2{\pi }^2}{16}$$

for $n\in \mathbb{N}$. Hence the condition (c) also implies that ${\sum }_{n=0}^{\infty }{\beta }_n=\infty$. For $\left\{s_{n_i}\right\}\subset \left\{s_n\right\}$ with a nondecreasing real sequence $\left\{n_i\right\}\subset \mathbb{N}$ such that ${lim\; inf}_{i\to \infty }(s_{n_i+1}-s_{n_i})\ge 0$, we have

$$\begin{array}{cc}\hfill 0& \le \underset{i\to \infty }{lim\; inf}(s_{n_i+1}-s_{n_i})\hfill \\ \hfill & =\underset{i\to \infty }{lim\; inf}(cosd(x_{n_i},p)-cosd(x_{n_i+1},p))\hfill \\ \hfill & \le \underset{i\to \infty }{lim\; inf}(cosd(x_{n_i},p)-({\alpha }_{n_i}cosd(u,p)+(1-{\alpha }_{n_i})cosd(U_{n_i}^0{x}_{n_i},p)))\hfill \\ \hfill & =\underset{i\to \infty }{lim\; inf}(cosd(x_{n_i},p)-cosd(U_{n_i}^0{x}_{n_i},p))\hfill \\ \hfill & \le \underset{i\to \infty }{lim\; sup}(cosd(x_{n_i},p)-cosd(U_{n_i}^0{x}_{n_i},p))\le 0.\hfill \end{array}$$

Hence ${lim}_{i\to \infty }(cosd(x_{n_i},p)-cosd(U_{n_i}^0{x}_{n_i},p))=0$. Since ${sup}_{n\in \mathbb{N}}d(U_n^0{x}_n,p)<\pi /2$, we have ${lim}_{i\to \infty }(cosd(x_{n_i},p)/cosd(U_{n_i}^0{x}_{n_i},p))=1$. As $\left\{U_{n_i}^0\right\}$ is a strongly quasi-nonexpansive sequence, it follows that ${lim}_{i\to \infty }d(x_{n_i},U_{n_i}^0{x}_{n_i})=0$. Let $\left\{x_{n_j}\right\}\subset \left\{x_{n_i}\right\}$ be a $\Delta$-convergent subsequence such that ${lim}_{j\to \infty }d(u,x_{n_j})={lim\; inf}_{i\to \infty }d(u,x_{n_i})$. Since $\left\{U_n^0\right\}$ is a $\Delta$-demiclosed sequence and ${lim}_{j\to \infty }d(x_{n_j},U_{n_j}^0{x}_{n_j})=0$, the $\Delta$-limit $z\in \left\{x_{n_j}\right\}$ belongs to F. By Lemma 5, we have

$$\underset{i\to \infty }{lim\; inf}d(u,U_{n_i}^0{x}_{n_i})=\underset{i\to \infty }{lim\; inf}d(u,x_{n_i})=\underset{j\to \infty }{lim}d(u,x_{n_j})\ge d(u,z)\ge d(u,p).$$

Hence

$$\begin{array}{cc}\hfill \underset{i\to \infty }{lim\; sup}{t}_{n_i}& =\underset{i\to \infty }{lim\; sup}\left(1-\frac{cosd(u,p)}{sind(u,U_{n_i}^0{x}_{n_i})tan(\frac{{\alpha }_{n_i}}{2}d(u,U_{n_i}^0{x}_{n_i})+cosd(u,U_{n_i}^0{x}_{n_i})}\right)\hfill \\ \hfill & =\underset{i\to \infty }{lim\; sup}\left(1-\frac{cosd(u,p)}{cosd(u,U_{n_i}^0{x}_{n_i})}\right)\le 0.\hfill \end{array}$$

From Lemma 6, we have ${lim}_{n\to \infty }{s}_n=0$. Therefore, $\left\{x_n\right\}$ converges to p. This completes the proof. □

5. Applications to the Image Recovery Problem

At the end of this work, we apply our results to the problem of finding a point of the intersection of a finite family of closed convex subsets. This problem is also known as the image recovery problem. See the works in [21,22] and references therein.

Let C be a nonempty closed convex subset of a complete CAT$\left(1\right)$ space such that $d(v,v^{\prime })<\pi /2$ for every $v,v^{\prime }\in X$. Then, the indicator function $i_C:C\to X$ of C defined by

$$i_C\left(x\right)=\left\{\begin{array}{cc}0\hfill & (x\in C),\hfill \\ \infty \hfill & (x\notin C)\hfill \end{array}\right.$$

is proper, lower semicontinuous, and convex. As is mentioned in [14], the resolvent $R_{i_C}$ of this function coincides with the metric projection $P_C$. Using this fact, we obtain the following results for the image recovery problem. The first result can be proved by using Theorem 1.

Theorem 3.

Let X be a complete CAT$\left(1\right)$ space such that $d(v,v^{\prime })<\pi /2$ for every $v,v^{\prime }\in X$. Let $\{C_0,C_1,\dots ,C_N\}$ be a finite family of nonempty closed convex subsets of X such that $C={\bigcap }_{k=0}^N{C}_K\ne Ø$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let $\left\{{\sigma }_n^l\right\}\subset \left[a,1-a\right]$ for $l=0,1,\dots ,N-1$ and $n\in \mathbb{N}$. Let $P_{C_k}$ be the metric projection onto $C_k$ for $k=0,1,\dots ,N$. Define $U_n^l:X\to X$ by

$$U_n^N=P_{C_N}\mathrm{and}{U}_n^l={\sigma }_n^l{P}_{C_l}\oplus (1-{\sigma }_n^l)U_n^{l+1}$$

for every $l=0,1,\dots ,N-1$ and $n\in \mathbb{N}$. Let $\left\{{\alpha }_n\right\}$ be a real sequence in $\left[a,1-a\right]$. For a given point $x_1\in X$, let $\left\{x_n\right\}$ be the sequence in X generated by

$$x_{n+1}={\alpha }_n{x}_n\oplus (1-{\alpha }_n)U_n^0{x}_n$$

for $n\in \mathbb{N}$. Then, $\left\{x_n\right\}$ Δ-converges to a point of C.

Note that this theorem is a generalization of the result by [21] in the setting of Hilbert spaces, to complete CAT$\left(1\right)$ spaces.

On the other hand, by using Thoerem 2, we can also prove the following theorem which was obtained by the authors of [23].

Theorem 4

(Kasahara-Kimura [23]). Let X be a complete CAT$\left(1\right)$ space such that $d(v,v^{\prime })<\pi /2$ for every $v,v^{\prime }\in X$. Let $\{C_0,C_1,\dots ,C_N\}$ be a finite family of nonempty closed convex subsets of X such that $C={\bigcap }_{k=0}^N{C}_K\ne Ø$. For a given real number $a\in \left]0,\frac{1}{2}\right]$, let $\left\{{\sigma }_n^l\right\}\subset \left[a,1-a\right]$ for $l=0,1,\dots ,N-1$ and $n\in \mathbb{N}$. Let $P_{C_k}$ be the metric projection onto $C_k$ for $k=0,1,\dots ,N$. Define $U_n^l:X\to X$ by

$$U_n^N=P_{C_N}\mathrm{and}{U}_n^l={\sigma }_n^l{P}_{C_l}\oplus (1-{\sigma }_n^l)U_n^{l+1}$$

for every $l=0,1,\dots ,N-1$ and $n\in \mathbb{N}$. Let $\left\{{\alpha }_n\right\}$ be a real sequence in $\left]0,1\right[$ such that ${lim}_{n\to \infty }{\alpha }_n=0$ and ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$. For given points $u,x_1\in X$, let $\left\{x_n\right\}$ be the sequence in X generated by

$$x_{n+1}={\alpha }_{n}u\oplus (1-{\alpha }_n)U_n^0{x}_n$$

for $n\in \mathbb{N}$. Suppose that one of the following conditions holds:

(a)

${sup}_{v,v^{\prime }\in X}d(v,v^{\prime })<\pi /2;$

(b)

$d(u,P_{C}u)<\pi /4$ and $d(u,P_{C}u)+d(x_0,P_{C}u)<\pi /2;$

(c)

${\sum }_{n=0}^{\infty }{\alpha }_n^2=\infty .$

Then $\left\{x_n\right\}$ converges to $P_{C}u$.

6. Conclusions

We proposed a new type of iterative scheme for the problem of finding a common minimizer of finitely many convex functions defined on a complete CAT(1) space. We considered the resolvent operators for proper lower semicontinuous convex functions defined on a complete CAT(1) space and their convex combination. As the convex combination on a CAT(1) space is defined only between two points, we need to take it repeatedly for three or more points.

In the first result (Theorem 1), we adopted a Mann-type sequence defined by the following iterative formula: $x_1\in X$ is given and

$$x_{n+1}={\alpha }_n{x}_n\oplus (1-{\alpha }_n)U_n^0{x}_n$$

for $n\in \mathbb{N}$, where a mapping $U_n^0$ is defined by the convex combination of finitely many resolvents. Then, $\left\{x_n\right\}$ is $\Delta$-convergent to a solution to our problem.

In the second result (Theorem 2), we used a Halpern-type sequence defined as follows: $u,x_1\in X$ is given and

$$x_{n+1}={\alpha }_{n}u\oplus (1-{\alpha }_n)U_n^0{x}_n$$

for $n\in \mathbb{N}$. Then, it converges to $P_{F}u$, the nearest point of the solution set F to u.

Further, we showed that these results can be applied to the image recovery problem.