Convergence Result for Solving the Split Fixed Point Problem with Multiple Output Sets in Nonlinear Spaces

Abstract

We study the split fixed point problem with multiple output sets in nonlinear spaces, particularly in CAT(0) spaces. We modify the existing self-adaptive algorithm for solving the split common fixed point problem with multiple output sets in the settings of generalized structures. We also present the consequences of our main theorem in terms of the split feasibility problem and the split common fixed point problem.

1. Introduction

Let A and B be nonempty, closed, and convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively, and $T:H_1\to H_2$ be a bounded operator. Let $T^{*}:H_2\to H_1$ be its adjoint. The split convex feasibility problem is to:

$$\begin{array}{c}\hfill \mathrm{find}\mathrm{an}\mathrm{element}{x}^{*}\in A\mathrm{such}\mathrm{that}T{x}^{*}\in B.\end{array}$$

The split convex feasibility problem (SCFP) was initially developed by Censor and Elfving [1] in 1994 for the modeling of inverse problems. In the real world, the SCFP appears in various domains, such as signal processing [2], medical care, and image reconstruction. One of the key uses of SCFP can be perceived in intensity-modulated radiation therapy (IMRT) [3]. Many mathematicians generalize the above problem in various directions. Some of these include the multiple-sets split feasibility problem [4,5], the split common fixed point problem [1,6], the split variational inequality/inclusion problem [7,8,9,10], and the split common null point problem [11,12,13]. Byrne’s $CQ$ algorithm [2] is a popular technique for finding the solution of the SCFP. This algorithm has been extended for the multiple-sets split convex feasibility problem by many authors, like [1,14,15]. In the year 2020, Reich and Tuyen [13] introduced and examined the split feasibility problem with multiple output sets in Hilbert Spaces.

In 2022 [16], Reich et al. considered a more general problem (the split common fixed point problem with multiple output sets) in Hilbert spaces in accordance with a new self-adaptive algorithm, defined as follows:

Assume that H and $H_i,i=1,2,...,N,$ are the real Hilbert spaces and $T_i:H\to H_i,i=1,2,\dots ,N,$ are the bounded operators. Let $S_j:H\to H,j=1,2,\dots ,M,{\Xi }_k^i:H_i\to H_i,i=1,2,\dots ,N,k=1,2,\dots ,M_i$ be nonexpansive mappings. Find $x\in H$, such that

$$\begin{array}{c}\hfill x\in \Omega =\left({\cap }_{j=1}^{M}Fix\left(S_j\right)\right)\cap \left({\cap }_{i=1}^N{T}_i^{-1}\left({\cap }_{k=1}^{M_i}Fix\left({\Xi }_k^i\right)\right)\right)\ne \varphi .\end{array}$$

(1)

Motivation

Motivated by all the above-mentioned advancements regarding feasibility problems, we proposed a split fixed point problem with multiple output sets in nonlinear spaces, particularly in CAT(0) spaces. We also extended the self-adaptive algorithm for this space as follows:

Let ${\rm Y}$ and ${{\rm Y}}_1$ be complete $CAT\left(0\right)$ spaces and $T:{\rm Y}\to {{\rm Y}}_1$ be a bounded operator. Let $S_j:{\rm Y}\to {\rm Y},j=1,2,\dots ,M,{\Xi }_k:{{\rm Y}}_1\to {{\rm Y}}_1,k=1,2,\dots ,K$ be nonexpansive mappings. The split common fixed point problem (SCFPP) is to:

$$\begin{array}{c}\hfill \mathrm{find}x\in C=\left({\cap }_{j=1}^{M}Fix\left(S_j\right)\right)\mathrm{in}\mathrm{the}\mathrm{way}\mathrm{that}Tx\in Q=\left({\cap }_{k=1}^{K}Fix\left({\Xi }_k\right)\right).\end{array}$$

This is a generalization of Problem (1.3) given in [16].

2. Preliminaries

Let $({\rm Y},\zeta )$ be a metric space. A geodesic path [17] connecting r and s, for the pair $(r,s)\in {\rm Y}\times {\rm Y}$, is considered to be map $\phi$ from $[0,\mathcal{L}]\subset \mathbb{R}$ to ${\rm Y}$ for $\mathcal{L}=\zeta (r,s)$, with conditions $\phi \left(0\right)=r$, $\phi \left(\mathcal{L}\right)=s$, and $\zeta (\phi \left(\kappa \right),\phi \left({\kappa }^{\prime }\right))=|\kappa -{\kappa }^{\prime }|,$ for each $\kappa ,{\kappa }^{\prime }\in [0,\mathcal{L}]$. The term geodesic segment is referred as the image $\vartheta$ of $\phi$, having endpoints r and s. The metric space $({\rm Y},\zeta )$ is known as a geodesic metric space if all possible two points of ${\rm Y}$ are connected by a geodesic segment. The metric space $({\rm Y},\zeta )$ is termed as uniquely geodesic if r and s are connected by precisely one geodesic for each $r,s\in {\rm Y}$. Assume that $({\rm Y},\zeta )$ is a geodesic metric space. A geodesic triangle consists of three elements, ${\varrho }_1,{\varrho }_2,{\varrho }_3\in {\rm Y}$, and three geodesics, $[{\varrho }_1,{\varrho }_2],[{\varrho }_2,{\varrho }_3],[{\varrho }_3,{\varrho }_1]$, denoted by $▵([{\varrho }_1,{\varrho }_2],[{\varrho }_2,{\varrho }_3],[{\varrho }_3,{\varrho }_1])$. A comparison triangle of a geodesic triangle $▵([{\varrho }_1,{\varrho }_2],[{\varrho }_2,{\varrho }_3],[{\varrho }_3,{\varrho }_1])$ in Euclidean space ${\mathbb{R}}^2$ is the triangle $\overline{▵}({\varrho }_1,{\varrho }_2,{\varrho }_3)=▵(\overline{{\varrho }_1},\overline{{\varrho }_2},\overline{{\varrho }_3})$, such that $\zeta ({\varrho }_i,{\varrho }_j)={\zeta }_{{\mathbb{R}}^2}(\overline{{\varrho }_i},\overline{{\varrho }_j})$ for all $i,j=1,2,3.$ A comparison triangle exists for a geodesic triangle without exception.

A geodesic metric space $({\rm Y},\zeta )$ is termed as a CAT(0) space if for every geodesic triangle in $▵$ in ${\rm Y}$ and a comparison triangle $\overline{▵}$ of $▵$, with $r,s\in ▵$ and $\overline{r},\overline{s}\in \overline{▵}$, the $CAT\left(0\right)$ inequality

$$\begin{array}{c}\hfill \zeta (r,s)\le {\zeta }_{{\mathbb{R}}^2}(\overline{r},\overline{s}),\end{array}$$

is established. Pre-Hilbert spaces, $\mathbb{R}$-trees, Euclidean buildings [18], and the complex Hilbert ball with a hyperbolic metric [19] are some examples of CAT(0) spaces.

Let $p,{\varrho }_1,{\varrho }_2$ be the elements in a $CAT\left(0\right)$ space, and ${\varrho }_0$ is a midpoint of $[{\varrho }_1,{\varrho }_2]$, symbolized by $\frac{{\varrho }_1\oplus {\varrho }_2}{2}$, after that the $CAT\left(0\right)$ inequality refers

$$\begin{array}{c}\hfill {\zeta }^2(p,\frac{{\varrho }_1\oplus {\varrho }_2}{2})={\zeta }^2(p,{\varrho }_0)\le \frac{1}{2}{\zeta }^2(p,{\varrho }_1)+\frac{1}{2}{\zeta }^2(p,{\varrho }_2)-\frac{1}{4}{\zeta }^2({\varrho }_1,{\varrho }_2).\end{array}$$

(2)

(2) is termed as (CN)-inequality [20]. A metric space that is geodesically connected is regarded as a $CAT\left(0\right)$ space if and only if the (CN) inequality is met.

Complete $CAT\left(0\right)$ spaces are frequently termed as Hadamard spaces (in honor to Jacques Hadamard) [19].

In 1976, Lim [21] originated the idea of $\Delta$-Convergence in $CAT\left(0\right)$ spaces that resembles weak convergence a great deal in a Banach space. Kirk and Panyanak [22] further extended the idea of $\Delta$-Convergence in $CAT\left(0\right)$ spaces. Assume that ${\eta }_n$ is a bounded sequence in ${\rm Y}$, $\upsilon (.,\left\{{\eta }_n\right\}):{\rm Y}\to [0,\infty ]$ is continuous, given as

$$\begin{array}{c}\hfill \upsilon (\eta ,\left\{{\eta }_n\right\})=\underset{n\to \infty }{lim\; sup}\zeta (\eta ,\left\{{\eta }_n\right\}).\end{array}$$

The asymptotic radius of $\left\{{\eta }_n\right\}$ is specified as

$$\begin{array}{c}\hfill \upsilon \left(\left\{{\eta }_n\right\}\right)=inf\{\upsilon (\eta ,\left\{{\eta }_n\right\}):\eta \in {\rm Y}\}.\end{array}$$

The asymptotic center $A_c\left({\eta }_n\right)$ of $\left\{{\eta }_n\right\}$ is

$$\begin{array}{c}\hfill A_c\left({\eta }_n\right)=\{\eta \in {\rm Y}:\upsilon (\eta ,\left\{{\eta }_n\right\})=\upsilon \left(\left\{{\eta }_n\right\}\right)\}.\end{array}$$

It is well known from Proposition 7 of [23] that $A_c\left({\eta }_n\right)$ contains one single point in a complete $CAT\left(0\right)$ space.

The sequence $\left\{{\eta }_n\right\}$ in a complete $CAT\left(0\right)$ space $({\rm Y},\zeta )$ is referred as $\Delta$-converges to $\eta \in {\rm Y}$, if $A\left({\eta }_{n_k}\right)=\eta$ for every subsequence $\left\{{\eta }_{n_k}\right\}$ of $\left\{{\eta }_n\right\}$ [22].

The initial notion of quasilinearization for a $CAT\left(0\right)$ space ${\rm Y}$ was introduced by Berg and Nikolaev in their work [24]. They indicated a pair $(c,w)\in {\rm Y}\times {\rm Y}$ by $\overrightarrow{vw}$ and name it a vector. The map $\langle .,.\rangle :({\rm Y}\times {\rm Y})\times ({\rm Y}\times {\rm Y})\to \mathbb{R}$ is a quasilinearization defined as $\langle \overrightarrow{cw},\overrightarrow{ez}\rangle =\frac{1}{2}[{\zeta }^2(c,w)+{\zeta }^2(e,z)-{\zeta }^2(c,e)-{\zeta }^2(w,z)]$ for every $c,w,e,z\in {\rm Y}$.

It is simple to demonstrate that $\langle \overrightarrow{cw},\overrightarrow{ez}\rangle =\langle \overrightarrow{ez},\overrightarrow{cw}\rangle$, $\langle \overrightarrow{cw},\overrightarrow{ez}\rangle =-\langle \overrightarrow{wc},\overrightarrow{ez}\rangle$, $\langle \overrightarrow{cw},\overrightarrow{cw}\rangle ={\zeta }^2(c,w)$ and $\langle \overrightarrow{cw},\overrightarrow{ez}\rangle =\langle \overrightarrow{cx},\overrightarrow{ez}\rangle +\langle \overrightarrow{xw},\overrightarrow{ez}\rangle$ for each $c,w,e,z,x\in {\rm Y}$. A metric space that is geodesically connected is a $CAT\left(0\right)$ space if and only if it meets the Cauchy–Schwarz inequality.

Theorem 1 

([25]). Assume that Φ is a nonempty convex subset of a complete $CAT\left(0\right)space$Υ, $\mu \in {\rm Y}$, and $\nu \in \mathcal{C}$. Then the metric projection $P_{\mathcal{C}}$ satisfies

$$\begin{array}{c}\hfill \nu =P_{\mathcal{C}}\mu \iff \langle \overrightarrow{y\nu },\overrightarrow{\nu \mu }\rangle \ge 0,\mathrm{for}\mathrm{each}y\in \mathsf{\Phi }.\end{array}$$

Assume that $({\rm Y},\zeta )$ is a metric space. A mapping $\omega :{\rm Y}\to {\rm Y}$ is termed as nonexpansive if

$$\begin{array}{c}\hfill \zeta \left(\omega \right(\gamma ),\omega (\xi \left)\right)\le \zeta (\gamma ,\xi ),\end{array}$$

for all $\gamma ,\xi \in {\rm Y}$. We denote the set of fixed points of $\omega$ by $F\left(\omega \right)$, i.e.,

$$\begin{array}{c}\hfill F\left(\omega \right)=\{x\in {\rm Y}:\omega x=x\}.\end{array}$$

Suppose that $({\rm Y},\zeta )$ is a $CAT\left(0\right)$ space. A mapping $T^{*}:{\rm Y}\to {\rm Y}$ is an adjoint operator [26] of T if, for all $\xi ,\eta ,\mu ,\nu \in {\rm Y}$, we have

$$\begin{array}{c}\hfill \langle \overrightarrow{T\xi T\mu },\overrightarrow{\eta \nu }\rangle =\langle T\overrightarrow{\xi \mu },\overrightarrow{\eta \nu }\rangle =\langle \overrightarrow{\xi \mu },T^{*}\overrightarrow{\eta \nu }\rangle =\langle \overrightarrow{\xi \mu },\overrightarrow{T^{*}\eta T^{*}\nu }\rangle .\end{array}$$

Whenever T is a linear operator, $T^{*}$ is also linear. As in a Hilbert space, we have

$$\begin{array}{c}\hfill d^2(T^{*}\eta ,T^{*}\nu )=d^2(T\xi ,T\mu )\le M{d}^2(\xi ,\nu ),\end{array}$$

and so, $T^{*}$ is bounded in ${\rm Y}$.

Lemma 1 

([27]). Let $p_1,p_2,p\in {\rm Y}$ and $\mu \in [0,1]$. Then

(i) 

$d(\mu p_1\oplus (1-\mu )p_2,p)\le \mu d(p_1,p)+(1-\mu )d(p_2,p)$,

(ii) 

$d^2(\mu p_1\oplus (1-\mu )p_2,p)\le \mu d^2(p_1,p)+(1-\mu )d^2(p_2,p)-\mu (1-\mu )d^2(p_1,p_2)$.

Lemma 2 

([28]). Let $p_1,p_2,p\in {\rm Y}$ and $\mu \in [0,1]$. Then

(i) 

$d(\mu p_1\oplus (1-\mu )p_2,\gamma p_1\oplus (1-\gamma )p_2)=|\mu -\gamma |d(p_1,p_2)$,

(ii) 

$d(\mu p_1\oplus (1-\mu )p_2,\mu p_1\oplus (1-\mu )p)\le (1-\mu )d(p_2,p)$.

Lemma 3 

([25]). Let $p_1,p_2,p\in {\rm Y}$ and $\mu \in [0,1]$. Then

$$\begin{array}{c}\hfill d^2(\mu p_1\oplus (1-\mu )p_2,p)\le {\mu }^{2}d(p_1,p)+{(1-\mu )}^{2}d(p_2,p)+2\mu (1-\mu )\langle \overrightarrow{p_{1}p},\overrightarrow{p_{2}p}\rangle .\end{array}$$

Lemma 4 

([29]). For $({\rm Y},d)$, the inequality stated below holds

$$d^2(p_1,p_3)\le d^2(p_2,p_3)+2\langle \overrightarrow{p_1{p}_2},\overrightarrow{p_1{p}_3}\rangle ,forall{p}_1,p_2,p_3\in {\rm Y}.$$

Lemma 5 

([22]). In $({\rm Y},d)$ space, every bounded sequence has a Δ-convergent subsequence.

Lemma 6. 

Suppose that $\Phi \ne \varphi$ is convex and a closed subset of the $CAT\left(0\right)$ space Υ and $\Pi :\Phi \to \Phi$ is a nonexpansive mapping, such that ${\iota }_n\Delta$-converges to $\iota \in \Phi$ and $d({\iota }_n,\Pi {\iota }_n)\to 0$. Then $\Pi \iota =\iota$.

Proof. 

Since

$$\begin{array}{ccc}\hfill \underset{n\to \infty }{lim\; sup}d(\Pi \left(\iota \right),{\iota }_n)& \le & \underset{n\to \infty }{lim\; sup}[d(\Pi \left(\iota \right),\Pi \left({\iota }_n\right)),d({\iota }_n,\Pi {\iota }_n)]\hfill \\ & \le & \underset{n\to \infty }{lim\; sup}[d(\iota ,{\iota }_n),d({\iota }_n,\Pi {\iota }_n)]\hfill \\ & =& \underset{n\to \infty }{lim\; sup}d(\iota ,{\iota }_n)\hfill \end{array}$$

By Opial’s condition in $CAT\left(0\right)$ spaces, $\Pi \left(\iota \right)=\iota$.    □

Lemma 7 

([30]). Assume that $\left\{{\xi }_n\right\}$ is a sequence of real numbers, such that there exists a subsequence $\left\{{\xi }_{n_j}\right\}$ of $\left\{{\xi }_n\right\}$ with ${\xi }_n<{\xi }_{n_j+1}$ for each $j\ge 1$. Then there exists a nondecreasing sequence $\left\{b_k\right\}$ of positive integers in the way that the given two inequalities:

$$\begin{array}{c}\hfill {\xi }_{b_k}\le {\xi }_{b_k+1}\mathit{and}{\xi }_k\le {\xi }_{b_k+1},\end{array}$$

satisfy for all (sufficiently large) numbers, k. Indeed, $b_k$ is the largest number, n, of the set $\{1,2,...,k\}$, in the way that the condition ${\xi }_n\le {\xi }_{n+1}$ satisfies.

Lemma 8 

([31]). Assume that the sequence $\left\{{\gamma }_n\right\}$ is of nonnegative real numbers, $\left\{{\rho }_n\right\}$ is a sequence of real numbers with ${lim\; sup}_{k\to \infty }{\rho }_n\le 0$, and $\left\{{\beta }_n\right\}$ is a sequence in (0,1) with ${\Sigma }_{n=0}^{\infty }{\beta }_n=\infty$. Suppose

$$\begin{array}{c}\hfill {\gamma }_{n+1}\le (1-{\beta }_n){\gamma }_n+{\alpha }_n{\rho }_n\mathit{for}\mathit{each}n\ge 1.\end{array}$$

Then ${lim}_{n\to \infty }{\gamma }_n=0$.

Lemma 9. 

Suppose that $\left\{{\gamma }_n\right\}$ is a sequence of positive real numbers, $\left\{{\rho }_n\right\}$ is a real numbers sequence, and $\left\{{\beta }_n\right\}$ is a sequence in the way that $0<{\beta }_n<1$ and ${\Sigma }_{n=0}^{\infty }{\beta }_n=\infty$. Let

$$\begin{array}{c}\hfill {\gamma }_{n+1}\le (1-{\beta }_n){\gamma }_n+{\beta }_n{\rho }_n\forall n\ge 1.\end{array}$$

If ${lim\; sup}_{k\to \infty }{\rho }_{n_k}\le 0$ for each subsequence $\left\{{\gamma }_{n_k}\right\}$ of $\left\{{\gamma }_n\right\}$, which satisfies ${lim\; inf}_{k\to \infty }({\gamma }_{n_k+1}-{\gamma }_{n_k})\ge 0$, then ${lim}_{n\to \infty }{\gamma }_n=0$.

Proof. 

There are two cases in the proof.

Case 1: There exists an $n_0\in \mathbb{N}$, in the way that, for every $n\ge n_0$, ${\gamma }_{n+1}\le {\gamma }_n$. It refers that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; inf}({\gamma }_{n+1}-{\gamma }_n)=0.\end{array}$$

Therefore, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}{\rho }_n\le 0.\end{array}$$

Lemma 8 leads to the conclusion.

Case 2: there is a subsequence $\left\{{\gamma }_{m_j}\right\}$ of $\left\{{\gamma }_n\right\}$, such that ${\gamma }_{m_j}<{\gamma }_{m_j+1}$ for each $j\in \mathbb{N}$. Lemma 7 can be used in this case to obtain a nondecreasing sequence $\left\{n_k\right\}$ of $\left\{n\right\}$, in the way that $n_k\to \infty$ and that the aforementioned two inequalities:

$$\begin{array}{c}\hfill {\gamma }_{n_k}\le {\gamma }_{n_k+1}\mathrm{and}{\gamma }_k\le {\gamma }_{n_k+1},\end{array}$$

satisfy for all (sufficiently large) numbers, k. The first inequality implies that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; inf}({\gamma }_{n_k+1}-{\gamma }_{n_k})\ge 0.\end{array}$$

It follows that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}{\rho }_{n_k}\le 0.\end{array}$$

Additionally, by repeating that first inequality, we obtain

$$\begin{array}{ccc}\hfill {\gamma }_{n_k+1}& \le & (1-{\beta }_{n_k}){\gamma }_{n_k}+{\beta }_{n_k}{\rho }_{n_k}\hfill \\ & \le & (1-{\beta }_{n_k}){\gamma }_{n_k+1}+{\beta }_{n_k}{\rho }_{n_k}.\hfill \end{array}$$

In particular, since each ${\beta }_{n_k}>0$, we have

$$\begin{array}{c}\hfill {\gamma }_{n_k+1}\le {\rho }_{n_k}.\end{array}$$

Finally, due to the second inequality, we obtain

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}{\gamma }_k\le \underset{k\to \infty }{lim\; sup}{\gamma }_{n_k+1}\le \underset{k\to \infty }{lim\; sup}{\rho }_{n_k}\le 0.\end{array}$$

Thus

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}{\gamma }_k=0.\end{array}$$

   □

3. Problem Formulation

Let ${\rm Y}$ and ${{\rm Y}}_i,i=1,2,\dots N,$ be the complete $CAT\left(0\right)$ spaces and $T_i:{\rm Y}\to {{\rm Y}}_i,i=1,2,\dots ,N,$ be bounded operators. Let $S_j:{\rm Y}\to {\rm Y},j=1,2,\dots ,M,{\Xi }_k^i:{{\rm Y}}_i\to {{\rm Y}}_i,i=1,2,\dots ,N,k=1,2,\dots ,M_i$ be nonexpansive mappings. Now we denote the solution set of SCFPP with multiple output sets by

$$\begin{array}{c}\hfill \Omega =\left({\cap }_{j=1}^{M}Fix\left(S_j\right)\right)\cap \left({\cap }_{i=1}^N{T}_i^{-1}\left({\cap }_{k=1}^{M_i}Fix\left({\Xi }_k^i\right)\right)\right)\ne \varphi .\end{array}$$

(3)

Assume that $f:{\rm Y}\to {\rm Y}$ is a strict contraction with a contraction coefficient $c\in [0,1)$. Our variational inequality problem (VIP) over the split fixed point problem with multiple output sets is to find an element $x\in \Omega$,

$$\begin{array}{c}\hfill \langle \overrightarrow{fxx},\overrightarrow{xy}\rangle \ge 0forally\in \Omega .\end{array}$$

(4)

4. Results

The following algorithm will be introduced first for solving the problem in (1).    

Algorithm 1: For $x_0\in {\rm Y}$ arbitrarily chosen, define the sequence $\left\{x_n\right\}$ as given below.

Step 1 Calculate $$\begin{array}{c}\hfill y_{j,n}=S_j{x}_n,\end{array}$$ for each $j=1,2,\dots ,M$ and suppose $$\begin{array}{ccc}\hfill d_n& =& \underset{j=1,2,\dots ,M}{max}\left\{d(y_{j,n},x_n)\right\}.\hfill \end{array}$$ Step 2 Compute $$\begin{array}{c}\hfill z_{k,n}^i={\Xi }_k^i\left(T_i{x}_n\right),\end{array}$$ for each $i=1,2,\dots ,N$ and $k=1,2,\dots ,M_i$, and suppose $$\begin{array}{ccc}\hfill d_{i,n}& =& \underset{k=1,2,\dots ,M_i}{max}\left\{d(z_{k,n}^i,T_i{x}_n)\right\},i=1,2,\dots ,N.\hfill \end{array}$$ Step 3 Let $$\begin{array}{c}\hfill {\Gamma }_n=max\{d_n,\underset{i=1,2,\dots ,N}{max}\left\{d_{i,n}\right\}\}.\end{array}$$ If $d_n={\Gamma }_n$, let $t_n=y_{j_n,n},j=1,2,\dots ,M$, and let $\Theta =I$. Else, $d_{i_n,n}={\Gamma }_n$, then let $t_n=z_{k_n,n}^{i_n},k=1,2,\dots ,M_i$ and $\Theta =T_{i_n}$. Step 4 Compute $$\begin{array}{c}\hfill u_n=\lambda x_n\oplus (1-\lambda ){\Theta }^{*}(\mu \Theta x_n\oplus (1-\mu )t_n).\end{array}$$ Step 5 Compute $$\begin{array}{c}\hfill x_{n+1}={\beta }_{n}f\left(x_n\right)\oplus (1-{\beta }_n)u_n,n\ge 0,\end{array}$$ where $\left\{{\beta }_n\right\}\subset (0,1)$ and $h:\Gamma \to \Gamma$ is a strict contraction having a contraction coefficient $l\in [0,1)$.

We start the analysis of our aforementioned algorithm with the underlying proposition.

Proposition 1. 

Let ${{\rm Y}}_1$ and ${{\rm Y}}_2$ be two CAT(0) spaces. Let $A:{{\rm Y}}_1\to {{\rm Y}}_2$ be a bounded linear operator and let $\Xi :{{\rm Y}}_2\to {{\rm Y}}_2$ be a nonexpansive mapping. Suppose that $\Omega =\{q\in {{\rm Y}}_1:Aq\in Fix(\Xi )\}\ne \varphi$. Then, for any $q\in \Omega$ and $x\in {{\rm Y}}_1$, we have

$$\begin{array}{ccc}\hfill d^2(\lambda x\oplus (1-\lambda )A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)& \le & \lambda d^2(x,q)+2\mu (1-\lambda )d^2(I\left(Ax\right),r)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2(\Xi \left(Ax\right),r)\hfill \\ & & -2\mu (1-\mu )(1-\lambda )d^2(I\left(Ax\right),\Xi \left(Ax\right))\hfill \\ & & +(1-\lambda )d^2(r,Aq)+(1-\lambda )d^2(A^{*}r,q).\hfill \end{array}$$

Proof. 

Consider

$$\begin{array}{c}\hfill \upsilon =\lambda x\oplus (1-\lambda )A^{*}(\mu I\oplus (1-\mu )\Xi )Ax.\end{array}$$

Then, we have

$$\begin{array}{ccc}\hfill d^2(\upsilon ,q)& =& d^2(\lambda x\oplus (1-\lambda )A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)\hfill \\ & =& {\lambda }^2{d}^2(x,q)+{(1-\lambda )}^2{d}^2(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)\hfill \\ & & +2\lambda (1-\lambda )\langle \overrightarrow{xq},\overrightarrow{A^{*}(\mu I\oplus (1-\mu )\Xi )Axq\rangle }\hfill \\ & \le & {\lambda }^2{d}^2(x,q)+{(1-\lambda )}^2{d}^2(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)\hfill \\ & & +\lambda (1-\lambda )2d(x,q)d(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)\hfill \\ & \le & {\lambda }^2{d}^2(x,q)+{(1-\lambda )}^2{d}^2(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)\hfill \\ & & +\lambda (1-\lambda )[d^2(x,q)+d^2(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)]\hfill \\ & =& {\lambda }^2{d}^2(x,q)+(1+{\lambda }^2-2\lambda )d^2(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)\hfill \\ & & +\lambda d^2(x,q)-{\lambda }^2{d}^2(x,q)\hfill \\ & & +(\lambda -{\lambda }^2)d^2(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)\hfill \\ & =& \lambda d^2(x,q)+(1-\lambda )d^2(A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q).\hfill \end{array}$$

Using the properties of quasilinearization and adjoint operator, we have

$$\begin{array}{ccc}\hfill d^2(\upsilon ,q)& \le & \lambda d^2(x,q)+(1-\lambda )\langle \overrightarrow{A^{*}(\mu I\oplus (1-\mu )\Xi )Axq},\overrightarrow{A^{*}(\mu I\oplus (1-\mu )\Xi )Axq}\rangle \hfill \\ & =& \lambda d^2(x,q)+(1-\lambda )\hfill \\ & & \{\langle \overrightarrow{A^{*}(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))A^{*}r},\overrightarrow{A^{*}(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))r}\rangle \hfill \\ & & +\langle \overrightarrow{A^{*}rq},\overrightarrow{A^{*}(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))q}\rangle \}\hfill \\ & =& \lambda d^2(x,q)+(1-\lambda )\hfill \\ & & \{\langle A^{*}\overrightarrow{\left[\right(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))r}],\overrightarrow{A^{*}(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))A^{*}r}\rangle \hfill \\ & & +\langle A^{*}\overrightarrow{\left[\right(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))r}],\overrightarrow{A^{*}rq}\rangle \hfill \\ & & +\langle \overrightarrow{A^{*}rq},\overrightarrow{A^{*}(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))A^{*}r}\rangle +\langle \overrightarrow{A^{*}rq},\overrightarrow{A^{*}rq}\rangle \}\hfill \\ & =& \lambda d^2(x,q)+(1-\lambda )\hfill \\ & & \{\langle \overrightarrow{\left(\mu I\right(Ax)\oplus (1-\mu )\Xi (Ax\left)\right)r},A\overrightarrow{(A^{*}\left[(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right))r\right]}\rangle \hfill \\ & & +\langle \overrightarrow{\left(\mu I\right(Ax)\oplus (1-\mu )\Xi (Ax\left)\right)r},A\overrightarrow{\left(A^{*}rq\right)}\rangle \hfill \\ & & +\langle \overrightarrow{A^{*}rq},A^{*}\overrightarrow{\left[\right(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right)\left)r\right]}\rangle +\langle \overrightarrow{A^{*}rq},\overrightarrow{A^{*}rq}\rangle \}\hfill \\ & =& \lambda d^2(x,q)+(1-\lambda )\hfill \\ & & \langle \overrightarrow{\left(\mu I\right(Ax)\oplus (1-\mu )\Xi (Ax\left)\right)r},\overrightarrow{\left(\mu I\right(Ax)\oplus (1-\mu )\Xi (Ax\left)\right)r}\rangle \hfill \\ & & +(1-\lambda )\langle \overrightarrow{\left(\mu I\right(Ax)\oplus (1-\mu )\Xi (Ax\left)\right)r},\overrightarrow{rAq}\rangle \hfill \\ & & +(1-\lambda )\langle A\overrightarrow{\left(A^{*}rq\right)},\overrightarrow{\left(\mu I\right(Ax)\oplus (1-\mu )\Xi (Ax\left)\right)r}\rangle +(1-\lambda )\langle \overrightarrow{A^{*}rq},\overrightarrow{A^{*}rq}\rangle \hfill \end{array}$$

$$\begin{array}{ccc}& =& \lambda d^2(x,q)+(1-\lambda )d^2(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right),r)\hfill \\ & & +2(1-\lambda )\langle \overrightarrow{\left(\mu I\right(Ax)\oplus (1-\mu )\Xi (Ax\left)\right)r},\overrightarrow{rAq}\rangle +(1-\lambda )d^2(A^{*}r,q)\hfill \\ & \le & \lambda d^2(x,q)+(1-\lambda )d^2(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right),r)\hfill \\ & & +2(1-\lambda )d(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right),r)d(r,Aq)+(1-\lambda )d^2(A^{*}r,q)\hfill \\ & \le & \lambda d^2(x,q)+(1-\lambda )d^2(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right),r)\hfill \\ & & +(1-\lambda )d^2(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right),r)+(1-\lambda )d^2(r,Aq)\hfill \\ & & +(1-\lambda )d^2(A^{*}r,q)\hfill \\ & =& \lambda d^2(x,q)+2(1-\lambda )d^2(\mu I\left(Ax\right)\oplus (1-\mu )\Xi \left(Ax\right),r)\hfill \\ & & +(1-\lambda )d^2(r,Aq)+(1-\lambda )d^2(A^{*}r,q)\hfill \\ & \le & \lambda d^2(x,q)+2(1-\lambda )\{\mu d^2(I\left(Ax\right),r)+(1-\mu )d^2(\Xi \left(Ax\right),r)\hfill \\ & & -\mu (1-\mu )d^2(I\left(Ax\right),\Xi \left(Ax\right))\}+(1-\lambda )d^2(r,Aq)\hfill \\ & & +(1-\lambda )d^2(A^{*}r,q)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill d^2(\upsilon ,q)& \le & \lambda d^2(x,q)+2\mu (1-\lambda )d^2(I\left(Ax\right),r)+2(1-\mu )(1-\lambda )d^2(\Xi \left(Ax\right),r)\hfill \\ & & -2\mu (1-\mu )(1-\lambda )d^2(I\left(Ax\right),\Xi \left(Ax\right))+(1-\lambda )d^2(r,Aq)\hfill \\ & & +(1-\lambda )d^2(A^{*}r,q).\hfill \end{array}$$

Thus, we proved that

$$\begin{array}{ccc}\hfill d^2(\lambda x\oplus (1-\lambda )A^{*}(\mu I\oplus (1-\mu )\Xi )Ax,q)& \le & \lambda d^2(x,q)\hfill \\ & & +2\mu (1-\lambda )d^2(I\left(Ax\right),r)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2(\Xi \left(Ax\right),r)\hfill \\ & & -2\mu (1-\mu )(1-\lambda )d^2(I\left(Ax\right),\Xi \left(Ax\right))\hfill \\ & & +(1-\lambda )d^2(r,Aq)\hfill \\ & & +(1-\lambda )d^2(A^{*}r,q).\hfill \end{array}$$

Our proof is now complete.    □

Now, we will prove a proposition consisting of two important properties that will be used to prove our convergence result.

Proposition 2. 

Assume that $\left\{x_n\right\}$ is a sequence produced using Algorithm 1. Consequently, the two statements that are given below hold true:

(i) For $\sigma \in \Omega$,

when $g_n=max\{g_n,{max}_{i=1,2,\dots ,N}\left\{g_{i,n}\right\}\}$

$$\begin{array}{ccc}\hfill {\Gamma }_n^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\lambda s_n-s_{n+1}+{\alpha }_n{d}^2(h\left(x_n\right),\sigma )\hfill \\ & & +2\mu (1-\lambda )d^2(x_n,r_n)+2(1-\mu )(1-\lambda )d^2(y_{j,n},r_n)\hfill \\ & & +2(1-\lambda )d^2\left(r_n\sigma \right)],\hfill \end{array}$$

(5)

and when $g_{i_n,n}=max\{g_n,{max}_{i=1,2,...,N}\left\{g_{i,n}\right\}\}$

$$\begin{array}{ccc}\hfill {\Gamma }_n^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\lambda s_n-s_{n+1}+{\alpha }_n{d}^2(h\left(x_n\right),\sigma )\hfill \\ & & +2\mu (1-\lambda )d^2({\Pi }_{i_n}{x}_n,\sigma )+2(1-\mu )(1-\lambda )d^2(E_{k_n}\left({\Pi }_{i_n}{x}_n\right),r_n)\hfill \\ & & +(1-\lambda )d^2(r_n,{\Pi }_{i_n}\sigma )+(1-\lambda )d^2({\Pi }_{i_n}^{*}{r}_n,\sigma ),\hfill \end{array}$$

(6)

where $s_n=d^2(x_n,q)$ and $\left\{r_n\right\}$ is a bounded sequence in ${{\rm Y}}_2$.

(ii) The inequality stated below holds true:

$$\begin{array}{cc}\hfill s_{n+1}\le & (1-{\beta }_n^{\prime })s_n+{\beta }_n^{\prime }{b}_n,\end{array}$$

(7)

where

$$\begin{array}{c}\hfill {\beta }_n^{\prime }=\frac{{\beta }_n(1-2l)}{1-{\beta }_{n}l}\mathrm{and}{b}_n=\frac{1}{1-2l}\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle .\end{array}$$

Proof. 

We take into consideration the given two cases.

Case I: $g_n=max\{g_n,{max}_{i=1,2,\dots ,N}\left\{g_{i,n}\right\}\}$

Using Proposition 1 to $A=\Theta =I$, and $\Xi =S_{j_n}$ in this case, we have, for any $\sigma \in \Omega$,

$$\begin{array}{ccc}\hfill d^2(u_n,\sigma )& =& d^2(\lambda x_n\oplus (1-\lambda ){\Theta }^{*}(\mu \Theta x_n\oplus (1-\mu )t_n,\sigma )\hfill \\ & =& d^2(\lambda x_n\oplus (1-\lambda ){\Theta }^{*}(\mu \Theta x_n\oplus (1-\mu )S_{j_n}{x}_n,\sigma )\hfill \\ & =& d^2(\lambda x_n\oplus (1-\lambda ){\Theta }^{*}(\mu I\oplus (1-\mu )S_{j_n})\Theta x_n,\sigma ).\hfill \end{array}$$

Proposition 1 implies

$$\begin{array}{ccc}\hfill d^2(u_n,\sigma )& \le & \lambda d^2(x_n,\sigma )+2\mu (1-\lambda )d^2(\Theta x_n,r_n)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2(S_{j_n}(\Theta x_n),r_n)\hfill \\ & & -2\mu (1-\mu )(1-\lambda )d^2(\Theta x_n,S_{j_n}(\Theta x_n))+(1-\lambda )d^2(r_n,\Theta \sigma )\hfill \\ & & +(1-\lambda )d^2({\Theta }^{*}{r}_n,\sigma )\hfill \\ & =& \lambda d^2(x_n,\sigma )+2\mu (1-\lambda )d^2(x_n,r_n)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2(S_{j_n}\left(x_n\right),r_n)\hfill \\ & & -2\mu (1-\mu )(1-\lambda )d^2(x_n,S_{j_n}{x}_n)+(1-\lambda )d^2(r_n,\sigma )\hfill \\ & & +(1-\lambda )d^2(r_n,\sigma )\hfill \\ & =& \lambda d^2(x_n,\sigma )+2\mu (1-\lambda )d^2(x_n,r_n)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2(y_{j,n},r_n)\hfill \\ & & -2\mu (1-\mu )(1-\lambda )d^2(x_n,y_{j,n})+2(1-\lambda )d^2(r_n,\sigma ).\hfill \end{array}$$

(8)

Case II: $g_{i_n,n}=max\{g_n,{max}_{i=1,2,\dots ,N}\left\{g_{i,n}\right\}\}$

Using Proposition 1 to $A=\Theta ={\Pi }_{i_n}$, and $\Xi ={\Xi }_{k_n}$, we have

$$\begin{array}{ccc}\hfill d^2(u_n,\sigma )& \le & \lambda d^2(x_n,\sigma )+2\mu (1-\lambda )d^2({\Pi }_{i_n}{x}_n,r_n)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2({\Xi }_{k_n}\left({\Pi }_{i_n}{x}_n\right),r_n)\hfill \\ & & -2\mu (1-\mu )(1-\lambda )d^2({\Pi }_{i_n}{x}_n,{\Xi }_{k_n}\left({\Pi }_{i_n}{x}_n\right))+(1-\lambda )d^2(r_n,{\Pi }_{i_n}\sigma )\hfill \\ & & +(1-\lambda )d^2({\Pi }_{i_n}^{*}{r}_n,\sigma ).\hfill \end{array}$$

(9)

We now demonstrate the boundedness of the sequence $\left\{x_n\right\}$.

$$\begin{array}{ccc}\hfill d(x_{n+1},\sigma )& =& d({\beta }_{n}h\left(x_n\right)\oplus (1-{\beta }_n)u_n,\sigma )\hfill \\ & \le & {\beta }_{n}d(h\left(x_n\right),\sigma )+(1-{\beta }_n)d(u_n,\sigma )\hfill \\ & \le & {\beta }_n[d(h\left(x_n\right),h\left(\sigma \right))+d(h\left(\sigma \right),\sigma )]+(1-{\beta }_n)d(u_n,\sigma )\hfill \\ & \le & l{\beta }_{n}d(x_n,\sigma )+{\beta }_{n}d(h\left(\sigma \right),\sigma )+(1-{\beta }_n)d(u_n,\sigma )\hfill \\ & \le & l{\beta }_{n}d(x_n,\sigma )+{\beta }_{n}d(h\left(\sigma \right),\sigma )+(1-{\beta }_n)d(x_n,\sigma )\hfill \\ & =& [l{\beta }_n+(1-{\beta }_n)]d(x_n,\sigma )+{\beta }_{n}d(h\left(\sigma \right),\sigma )\hfill \\ & =& [1-(1-l){\beta }_n]d(x_n,\sigma )+{\beta }_n(1-l)\frac{d\left(h\right(\sigma ),\sigma )}{(1-l)}\hfill \end{array}$$

$$\begin{array}{ccc}& \le & max\{d(x_n,\sigma ),\frac{d\left(h\right(\sigma ),\sigma )}{(1-l)}\}\hfill \\ & \le & max\{d(x_o,\sigma ),\frac{d\left(h\right(\sigma ),\sigma )}{(1-l)}\}.\hfill \end{array}$$

It follows that $\left\{x_n\right\}$ is a bounded sequence.

$$\begin{array}{ccc}\hfill d^2(x_{n+1},\sigma )& =& d^2({\beta }_{n}h\left(x_n\right)\oplus (1-{\beta }_n)u_n,\sigma )\hfill \\ & \le & {\beta }_n{d}^2(h\left(x_n\right),\sigma )+(1-{\beta }_n)d^2(u_n,\sigma )\hfill \\ & \le & {\beta }_n{d}^2(h\left(x_n\right),\sigma )+d^2(u_n,\sigma ).\hfill \end{array}$$

(10)

In Case I

$$\begin{array}{c}\hfill {\Gamma }_n=g_n=\underset{j=1,2,\dots ,M}{max}\left\{d(x_n,y_{j,n})\right\},\end{array}$$

combining (8) and (10)

$$\begin{array}{ccc}\hfill d^2(x_{n+1},\sigma )& \le & {\beta }_n{d}^2(h\left(x_n\right),\sigma )+\lambda d^2(x_n,\sigma )+2\mu (1-\lambda )d^2(x_n,r_n)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2(y_{j,n},r_n)-2\mu (1-\mu )(1-\lambda ){\Gamma }_n^2\hfill \\ & & +2(1-\lambda )d^2(r_n,\sigma )\hfill \\ \hfill 2\mu (1-\mu )(1-\lambda ){\Gamma }_n^2& \le & {\beta }_n{d}^2(h\left(x_n\right),\sigma )+\lambda d^2(x_n,\sigma )-d^2(x_{n+1},\sigma )\hfill \\ & & +2\mu (1-\lambda )d^2(x_n,r_n)+2(1-\mu )(1-\lambda )d^2(y_{j,n},r_n)\hfill \\ & & +2(1-\lambda )d^2(r_n,\sigma )\hfill \\ \hfill {\Gamma }_n^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\lambda s_n-s_{n+1}+{\beta }_n{d}^2(h\left(x_n\right),\sigma )\hfill \\ & & +2\mu (1-\lambda )d^2(x_n,r_n)+2(1-\mu )(1-\lambda )d^2(y_{j,n},r_n)\hfill \\ & & +2(1-\lambda )d^2(r_n,\sigma )].\hfill \end{array}$$

In Case II

$$\begin{array}{c}\hfill {\Gamma }_n=g_{i,n}=\underset{k=1,2,...,M_i}{max}\left\{d(z_{k,n}^i,{\Pi }_i{x}_n)\right\},i=1,2,...,N.\end{array}$$

By joining (9) and (10), we obtain

$$\begin{array}{ccc}\hfill d^2(x_{n+1},\sigma )& \le & {\beta }_n{d}^2(h\left(x_n\right),\sigma )+\lambda d^2(x_n,\sigma )+2\mu (1-\lambda )d^2({\Pi }_{i_n}{x}_n,r_n)\hfill \\ & & +2(1-\mu )(1-\lambda )d^2({\Xi }_{k_n}\left({\Pi }_{i_n}{x}_n\right),r_n)-2\mu (1-\mu )(1-\lambda ){\Gamma }_n^2\hfill \\ & & +(1-\lambda )d^2(r_n,{\Pi }_{i_n}\sigma )+(1-\lambda )d^2({\Pi }_{i_n}^{*}{r}_n,\sigma )\hfill \\ \hfill 2\mu (1-\mu )(1-\lambda ){\Gamma }_n^2& \le & {\beta }_n{d}^2(h\left(x_n\right),\sigma )+\lambda d^2(x_n,\sigma )-d^2(x_{n+1},\sigma )\hfill \\ & & +2\mu (1-\lambda )d^2({\Pi }_{i_n}{x}_n,r_n)+2(1-\mu )(1-\lambda )d^2({\Xi }_{k_n}\left({\Pi }_{i_n}{x}_n\right),r_n)\hfill \\ & & +(1-\lambda )d^2(r_n,{\Pi }_{i_n}\sigma )+(1-\lambda )d^2({\Pi }_{i_n}^{*}{r}_n,\sigma )\hfill \\ \hfill {\Gamma }_n^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\lambda s_n-s_{n+1}+{\beta }_n{d}^2(h\left(x_n\right),\sigma )\hfill \\ & & +2\mu (1-\lambda )d^2({\Pi }_{i_n}{x}_n,r_n)+2(1-\mu )(1-\lambda )d^2({\Xi }_{k_n}\left({\Pi }_{i_n}{x}_n\right),r_n)\hfill \\ & & +(1-\lambda )d^2(r_n,{\Pi }_{i_n}\sigma )+(1-\lambda )d^2({\Pi }_{i_n}^{*}{r}_n,\sigma )].\hfill \end{array}$$

Let us set

$$\begin{array}{c}\hfill z_n={\beta }_n\sigma \oplus (1-{\beta }_n)u_n.\end{array}$$

It follows

$$\begin{array}{ccc}\hfill d^2(x_{n+1},\sigma )& =& d^2({\beta }_{n}h\left(x_n\right)\oplus (1-{\beta }_n)u_n,\sigma )\hfill \\ & \le & d^2(z_n,\sigma )+2\langle \overrightarrow{x_{n+1}{z}_n},\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & =& d{({\beta }_n\sigma \oplus (1-{\beta }_n)u_n,\sigma )}^2+2\langle \overrightarrow{({\beta }_{n}h\left(x_n\right)\oplus (1-{\beta }_n)u_n)z_n},\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & \le & {[{\beta }_{n}d(\sigma ,\sigma )+(1-{\beta }_n)d(u_n,\sigma )]}^2+2[{\beta }_n\langle \overrightarrow{h\left(x_n\right)z_n},\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & & +(1-{\beta }_n)\langle \overrightarrow{u_n{z}_n},\overrightarrow{x_{n+1}\sigma }\rangle ]\hfill \\ & =& {(1-{\beta }_n)}^2{d}^2(u_n,\sigma )+2[{\beta }_n\langle \overrightarrow{h\left(x_n\right)({\beta }_n\sigma \oplus (1-{\beta }_n)u_n)},\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & & +(1-{\beta }_n)\langle \overrightarrow{u_n({\beta }_n\sigma \oplus (1-{\beta }_n)u_n)},\overrightarrow{x_{n+1}\sigma }\rangle ]\hfill \\ & \le & {(1-{\beta }_n)}^2{d}^2(u_n,\sigma )+2[{\beta }_n^2\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle +\{{\beta }_n(1-{\beta }_n)\hfill \\ & & \times \langle \overrightarrow{h\left(x_n\right)u_n},\overrightarrow{x_{n+1}\sigma }\rangle \}+{\beta }_n(1-{\beta }_n)\langle \overrightarrow{u_n\sigma },\overrightarrow{x_{n+1}\sigma }\rangle +\{{(1-{\beta }_n)}^2\hfill \\ & & \times \langle \overrightarrow{u_n{u}_n},\overrightarrow{x_{n+1}\sigma }\rangle \left\}\right]\hfill \\ & =& {(1-{\beta }_n)}^2{d}^2(u_n,\sigma )+2{\beta }_n^2\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & & +2{\beta }_n(1-{\beta }_n)\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & =& {(1-{\beta }_n)}^2{d}^2(u_n,\sigma )+2{\beta }_n^2\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & & +2{\beta }_n\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle -2{\beta }_n^2\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & =& {(1-{\beta }_n)}^2{d}^2(u_n,\sigma )+2{\beta }_n\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle .\hfill \\ \hfill d^2(x_{n+1},\sigma )& \le & \left(1-{\beta }_n\right)d^2(u_n,\sigma )+2{\beta }_n\langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle .\hfill \end{array}$$

(11)

Consider

$$\begin{array}{ccc}\hfill \langle \overrightarrow{h\left(x_n\right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle & =& \langle \overrightarrow{h\left(x_n\right)h\left(\sigma \right)},\overrightarrow{x_{n+1}\sigma }\rangle +\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & \le & d(h\left(x_n\right),h\left(\sigma \right))d(x_{n+1},\sigma )+\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & \le & ld(x_n,\sigma )d(x_{n+1},\sigma )+\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ & \le & \frac{l}{2}[d^2(x_n,\sigma )+d^2(x_{n+1},\sigma )]+\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle .\hfill \end{array}$$

(12)

Now, by putting (12) in (11), we obtain

$$\begin{array}{cc}\hfill d^2(x_{n+1},\sigma )\le & (1-{\beta }_n)d^2(u_n,\sigma )+{\beta }_{n}l{d}^2(x_n,\sigma )\hfill \\ \hfill & +{\beta }_{n}l{d}^2(x_{n+1},\sigma )+2{\beta }_n\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ \hfill d^2(x_{n+1},\sigma )-{\beta }_{n}l{d}^2(x_{n+1},\sigma )\le & (1-{\beta }_n+{\beta }_{n}l)d^2(x_n,\sigma )\hfill \\ \hfill & +2{\beta }_n\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ \hfill (1-{\beta }_{n}l)d^2(x_{n+1},\sigma )\le & (1-(1-l){\beta }_n)d^2(x_n,\sigma )+2{\beta }_n\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle \hfill \\ \hfill d^2(x_{n+1},\sigma )\le & \frac{(1-(1-l){\beta }_n)}{1-{\beta }_{n}l}{d}^2(x_n,\sigma )\hfill \\ \hfill & +\frac{2{\beta }_n}{1-{\beta }_{n}l}\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle .\hfill \\ \hfill s_{n+1}\le & (1-{\beta }_n^{\prime })s_n+{\beta }_n^{\prime }{b}_n,\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\beta }_n^{\prime }=\frac{{\beta }_n(1-2l)}{1-{\beta }_{n}l}\mathrm{and}{b}_n=\frac{1}{1-2l}\langle \overrightarrow{h\left(\sigma \right)\sigma },\overrightarrow{x_{n+1}\sigma }\rangle .\end{array}$$

   □

We now present the strong convergence of the sequence produced by Algorithm 1.

Theorem 2. 

If $\left\{x_n\right\}$ is bounded sequence and the sequence $\left\{{\beta }_n\right\}$ satisfies that

$$\begin{array}{c}\hfill {\beta }_n\to 0,\sum _{n=0}^{\infty }{\beta }_n=\infty ,\end{array}$$

then the sequence $\left\{x_n\right\}$ iteratively produced by Algorithm 1 strongly converges to an element $x\in \Omega$, which is the only solution of the variational inequality V.I

$$\begin{array}{c}\hfill \langle \overrightarrow{hxx},\overrightarrow{xy}\rangle \ge 0for ally\in \Omega .\end{array}$$

Proof. 

Let us suppose that the solution of V.I is x.

$$\begin{array}{c}\hfill \langle \overrightarrow{hxx},\overrightarrow{xy}\rangle \ge 0for ally\in \Omega ,\end{array}$$

i.e., $x=P_{\Omega }h\left(x\right)$

Assume that $s_n=d^2(x_n,x)$. By applying Proposition 2, we obtain

$$\begin{array}{cc}\hfill s_{n+1}\le & (1-{\beta }_n^{\prime })s_n+{\beta }_n^{\prime }{b}_n,\end{array}$$

where

$$\begin{array}{c}\hfill {\beta }_n^{\prime }=\frac{{\beta }_n(1-2l)}{1-{\beta }_{n}l}\mathrm{and}{b}_n=\frac{1}{1-2l}\langle \overrightarrow{h\left(x\right)x},\overrightarrow{x_{n+1}x}\rangle .\end{array}$$

We now use the Lemma 9 to demonstrate that $s_n\to 0$.

Assume that $\left\{s_{n_k}\right\}$ is an arbitrarily chosen subsequence of $\left\{s_n\right\}$ that satisfies ${lim\; inf}_{k\to \infty }$$(s_{n_k+1}-s_{n_k})\ge 0$.

By (5), we have

$$\begin{array}{ccc}\hfill {\Gamma }_{n_k}^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\lambda s_{n_k}-s_{n_k+1}+{\beta }_{n_k}{d}^2(h\left(x_{n_k}\right),x)\hfill \\ & & +2\mu (1-\lambda )d^2(x_{n_k},r_{n_k})+2(1-\mu )(1-\lambda )d^2(y_{j,n},r_{n_k})\hfill \\ & & +2(1-\lambda )d^2(r_{n_k},x)].\hfill \end{array}$$

Since $\left\{x_n\right\}$ and $\left\{r_n\right\}$ are bounded, and so are $\left\{f\left(x_n\right)\right\}$ and $\left\{y_{j,n}\right\}$, we have

$$\begin{array}{ccc}\hfill d^2(x_{n_k},r_{n_k})& \le & {\beta }_{n_k}M,\hfill \\ \hfill d^2(y_{j,n},r_{n_k})& \le & {\beta }_{n_k}N,\hfill \\ \hfill d^2(r_{n_k},x)& \le & {\beta }_{n_k}R.\hfill \end{array}$$

Also, ${\beta }_n\to 0$, which implies that

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim\; sup}{\Gamma }_{n_k}^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\underset{k\to \infty }{lim\; sup}(\lambda s_{n_k}-s_{n_k+1})+\underset{k\to \infty }{lim\; sup}{\beta }_{n_k}{d}^2(h\left(x_{n_k}\right),x)\hfill \\ & & +2\underset{k\to \infty }{lim\; sup}\mu (1-\lambda )d^2(x_{n_k},r_{n_k})+2\underset{k\to \infty }{lim\; sup}(1-\mu )(1-\lambda )d^2(y_{j,n},r_{n_k})\hfill \\ & & +2\underset{k\to \infty }{lim\; sup}(1-\lambda )d^2(r_{n_k},x)]\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\underset{k\to \infty }{lim\; sup}(\lambda s_{n_k}-s_{n_k+1})+\underset{k\to \infty }{lim\; sup}{\beta }_{n_k}{d}^2(h\left(x_{n_k}\right),x)\hfill \\ & & +2\underset{k\to \infty }{lim\; sup}\mu (1-\lambda ){\beta }_{n_k}M+2\underset{k\to \infty }{lim\; sup}(1-\mu )(1-\lambda ){\beta }_{n_k}N\hfill \\ & & +2\underset{k\to \infty }{lim\; sup}(1-\lambda ){\beta }_{n_k}R]\hfill \\ & \le & 0.\hfill \end{array}$$

Now, by (6)

$$\begin{array}{ccc}\hfill {\Gamma }_{n_k}^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\lambda s_{n_k}-s_{n_k+1}+{\beta }_n{d}^2(h\left(x_{n_k}\right),x)\hfill \\ & & +2\mu (1-\lambda )d^2({\Pi }_{i_n}{x}_{n_k},r_{n_k})+2(1-\mu )(1-\lambda )d^2({\Xi }_{k_n}\left({\Pi }_{i_n}{x}_{n_k}\right),r_{n_k})\hfill \\ & & +(1-\lambda )d^2(r_{n_k},{\Pi }_{i_n}x)+(1-\lambda )d^2({\Pi }_{i_n}^{*}{r}_{n_k},x)].\hfill \end{array}$$

We have

$$\begin{array}{ccc}\hfill d^2({\Pi }_{i_n}{x}_{n_k},r_{n_k})& \le & {\beta }_{n_k}H,\hfill \\ \hfill d^2({\Xi }_{k_n}\left({\Pi }_{i_n}{x}_{n_k}\right),r_{n_k})& \le & {\beta }_{n_k}I,\hfill \\ \hfill d^2(r_{n_k},{\Pi }_{i_n}x)& \le & {\beta }_{n_k}J,\hfill \\ \hfill d^2({\Pi }_{i_n}^{*}{r}_{n_k},x)& \le & {\beta }_{n_k}K.\hfill \end{array}$$

It follows that

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim\; sup}{\Gamma }_{n_k}^2& \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\underset{k\to \infty }{lim\; sup}(\lambda s_{n_k}-s_{n_k+1})+\underset{k\to \infty }{lim\; sup}{\beta }_n{d}^2(h\left(x_{n_k}\right),x)\hfill \\ & & +2\underset{k\to \infty }{lim\; sup}\mu (1-\lambda )d^2({\Pi }_{i_n}{x}_{n_k},r_{n_k})+2\underset{k\to \infty }{lim\; sup}(1\hfill \\ & & -\mu )(1-\lambda )d^2({\Xi }_{k_n}\left({\Pi }_{i_n}{x}_{n_k}\right),r_{n_k})\hfill \\ & & +\underset{k\to \infty }{lim\; sup}(1-\lambda )d^2(r_{n_k},{\Pi }_{i_n}x)+\underset{k\to \infty }{lim\; sup}(1-\lambda )d^2({\Pi }_{i_n}^{*}{r}_{n_k},x)]\hfill \\ & \le & \frac{1}{2\mu (1-\mu )(1-\lambda )}[\underset{k\to \infty }{lim\; sup}(\lambda s_{n_k}-s_{n_k+1})+\underset{k\to \infty }{lim\; sup}{\beta }_n{d}^2(h\left(x_{n_k}\right),x)\hfill \\ & & +2\underset{k\to \infty }{lim\; sup}\mu (1-\lambda ){\beta }_{n_k}H+2\underset{k\to \infty }{lim\; sup}(1-\mu )(1-\lambda ){\beta }_{n_k}I\hfill \\ & & +\underset{k\to \infty }{lim\; sup}(1-\lambda ){\beta }_{n_k}J+\underset{k\to \infty }{lim\; sup}(1-\lambda ){\beta }_{n_k}K]\hfill \\ & \le & 0.\hfill \end{array}$$

Hence, we can see ${\Gamma }_{n_k}\to 0$ as $k\to \infty$ in both cases. The sequences $\left\{g_{n_k}\right\}$ and $\left\{g_{i,n_k}\right\}$ also converge to 0, according to the definition of ${\Gamma }_n$ in Algorithm 1. It follows that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}d(x_{n_k},S_j{x}_{n_k})=0,j=1,2,...,M,\end{array}$$

(13)

and

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}d({\Pi }_i{x}_{n_k},E_k^i\left({\Pi }_i{x}_{n_k}\right))=0,i=1,2,...,N,k=1,2,...,M_i.\end{array}$$

(14)

Now, we assume that ${lim\; sup}_{k\to \infty }{b}_{n_k}\le 0$. Assume that $\left\{x_{n_{k_l}}\right\}$ is a subsequence of $\left\{x_{n_k}\right\}$, such that

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}\langle \overrightarrow{hxx},\overrightarrow{x_{n_k}x}\rangle =\underset{k\to \infty }{lim}\langle \overrightarrow{hxx},\overrightarrow{x_{n_{k_l}}x}\rangle .\end{array}$$

A subsequence of $\left\{x_{n_{k_l}}\right\}$$\Delta$-converges to $x^{*}$ because $\left\{x_{n_{k_l}}\right\}$ is bounded. Let us assume that $\left\{x_{n_{k_l}}\right\}\Delta$-converges to $x^{*}$, without losing any generality. Let us suppose that $x^{*}\in \Omega$.

Every ${\Pi }_i$ is a bounded linear operator, hence it is evident that ${\Pi }_i{x}_{n_{k_l}}\Delta$-converges to ${\Pi }_i{x}^{*}$ for each $i=1,2,...,N$. Hence, by Lemma 6 and $d({\Pi }_i{x}_{n_k},E_k^i\left({\Pi }_i{x}_{n_k}\right))\to 0$, which holds for each $i=1,2,...,N$ and $k=1,2,...,M_i$, we obtain

$$\begin{array}{ccc}\hfill E_k^i\left({\Pi }_i{x}^{*}\right)& =& {\Pi }_i{x}^{*}.\hfill \end{array}$$

This implies

$$\begin{array}{c}\hfill {\Pi }_i{x}^{*}\in {\cap }_{k=1}^{M_i}Fix\left(E_k^i\right)\\ \hfill x^{*}\in ({\cap }_{i=1}^N{\Pi }_i^{-1}\left({\cap }_{k=1}^{M_i}Fix\left(E_k^i\right)\right).\end{array}$$

Thus, $x^{*}\in \Omega$, as we assumed.

In view of $x=P_{\Omega }h\left(x\right)$ and Lemma 1

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim\; sup}\langle \overrightarrow{hxx},\overrightarrow{x_{n_k}x}\rangle =\langle \overrightarrow{hxx},\overrightarrow{x^{*}x}\rangle \le 0.\end{array}$$

(15)

We demonstrate that $d(x_{n_k+1},x_{n_k})\to 0$ as $k\to \infty$. In fact, as the sequence $\left\{x_n\right\},\left\{u_n\right\}$ and $\left\{f\left(x_n\right)\right\}$ are bounded, ${\beta }_n\to 0$ and the inequality

$$\begin{array}{ccc}\hfill d(x_{n_k+1},u_{n_k})& =& d({\beta }_{n_k}h\left(x_{n_k}\right)\oplus (1-{\beta }_{n_k})u_{n_k},u_{n_k})\hfill \\ & \le & {\beta }_{n_k}d(h\left(x_{n_k}\right),u_{n_k})+(1-{\beta }_{n_k})d(u_{n_k},u_{n_k})\hfill \\ \hfill \underset{k\to \infty }{lim}d(x_{n_k+1},u_{n_k})& \le & \underset{k\to \infty }{lim}{\beta }_{n_k}d(h\left(x_{n_k}\right),u_{n_k}),\hfill \end{array}$$

implies that

$$\begin{array}{c}\hfill d(x_{n_k+1},u_{n_k})\to 0.\end{array}$$

(16)

So, from Step 4 of Algorithm 1, we obtain

$$\begin{array}{ccc}\hfill d(u_{n_k},x_{n_k})& =& d(\lambda x_{n_k}\oplus (1-\lambda ){\Theta }^{*}(\mu \Theta x_{n_k}\oplus (1-\mu )t_{n_k}),x_{n_k})\hfill \\ & \le & \lambda d(x_{n_k},x_{n_k})+(1-\lambda )d({\Theta }^{*}(\mu \Theta x_{n_k}\oplus (1-\mu )t_{n_k}),x_{n_k})\hfill \\ & \le & \mu (1-\lambda )d(x_{n_k},x_{n_k})+(1-\lambda )(1-\mu )d({\Theta }^{*}{t}_{n_k},x_{n_k}).\hfill \end{array}$$

In case 1, $\Theta =I$ and $t_{n_k}=S_{j_n}{x}_{n_k}$, so

$$\begin{array}{ccc}\hfill d(u_{n_k},x_{n_k})& \le & (1-\lambda )(1-\mu )d(t_{n_k},x_{n_k}).\hfill \end{array}$$

So from (13), we have

$$\begin{array}{c}\hfill d(u_{n_k},x_{n_k})\to 0,\end{array}$$

which, when combined with (16), gives

$$\begin{array}{c}\hfill d(x_{n_k+1},x_{n_k})\to 0.\end{array}$$

(17)

From (13) and (15), the implication is that ${lim\; sup}_{k\to \infty }{b}_{n_k}\le 0$, as we supposed. Thus, all the suppositions of Lemma 9 are fulfilled. Hence, we have $s_n\to 0$, that is, $x_n\to x=P_{\Omega }h\left(x\right)$.

Our proof is now complete.    □

5. Some Consequences

In this section, some corollaries are stated related to the solution of the variational inequality problem (VIP) over the split feasibility problem with multiple output sets and the split common fixed point problem for nonexpansive mappings.

5.1. Split Feasibility Problem with Multiple Output Sets

Assume that ${\rm Y}$ and ${{\rm Y}}_i,i=1,2,\dots ,N,$ are the complete $CAT\left(0\right)$ spaces and ${\Pi }_i:{\rm Y}\to {{\rm Y}}_i,i=1,2,\dots ,N,$ are the bounded operators. Suppose ${\Phi }_j,j=1,2,\dots ,M$ and ${\Psi }_k^i,i=1,2,\dots ,N$ and $k=1,2,\dots ,M_i$ are nonempty, convex and closed subsets of ${\rm Y}$ and ${{\rm Y}}_i$, respectively.

We have to find $x^{†}\in {\rm Y}$, in the way that

$$\begin{array}{c}\hfill x^{†}\in {\Omega }^{SFPMOS}=\left({\cap }_{j=1}^M{\Phi }_j\right)\cap ({\cap }_{i=1}^N{\Pi }_i^{-1}\left({\cap }_{k=1}^{M_i}\left({\Psi }_k^i\right)\right).\end{array}$$

(18)

By utilizing Theorem 2 to ${\Pi }_j=P_{{\Phi }_j}$ for all $j=1,2,\dots ,M$ and ${\Xi }_k^i=P_{{\Psi }_k^i}$ for all $i=1,2,\dots ,N,k=1,2,\dots ,M_i$, we obtain the next corollary to find the solution of Problem (18).

Corollary 1. 

Let $\left\{x_n\right\}$ be a sequence produced by Algorithm 1 with ${\Pi }_j=P_{{\Phi }_j}$ for all $j=1,2,...,M$ and ${\Xi }_k^i=P_{{\Psi }_k^i}$ for all $i=1,2,...,N,k=1,2,...,M_i$, respectively. If the sequence $\left\{{\beta }_n\right\}$ satisfies the condition ${\beta }_n\to 0$ and ${\sum }_{n=0}^{\infty }{\beta }_n=\infty$, then the sequence $\left\{x_n\right\}$ converges strongly to $x^{†}\in {\Omega }^{SFPMOS}$, which is the unique solution of the variational inequality

$$\begin{array}{c}\hfill \langle \overrightarrow{h{x}^{†}{x}^{†}},\overrightarrow{x^{†}y}\rangle \ge 0\forall y\in {\Omega }^{SFPMOS}.\end{array}$$

5.2. The Split Common Fixed Point Problem for Nonexpansive Mappings

Suppose that ${\rm Y}$ and ${{\rm Y}}_1$ are the complete $CAT\left(0\right)$ spaces and $\Pi :{\rm Y}\to {{\rm Y}}_1$ are bounded operators. Suppose ${\mathcal{S}}_j:{\rm Y}\to {\rm Y},j=1,2,...,M$, and ${\Xi }_k:{{\rm Y}}_1\to {{\rm Y}}_1,k=1,2,...,K,$ are nonexpansive mappings. We have to find $x^{†}\in {\rm Y}$, in the way that

$$\begin{array}{c}\hfill x^{†}\in {\Omega }^{SCFPP}=\left({\cap }_{j=1}^{M}Fix\left({\mathcal{S}}_j\right)\right)\cap T^{-1}\left({\cap }_{k=1}^{K}Fix\left({\Xi }_k\right)\right)\ne \varphi .\end{array}$$

(19)

We approach the following algorithm, which is based on Algorithm 1, to obtain a solution to the split common fixed point problem for nonexpansive mappings (19).

Theorem 2 gives the following corollary for the strong convergence of Algorithm 2.

Algorithm 2: Let $x_0\in {\rm Y}$, $\left\{x_n\right\}$ is iteratively generated as follows.

Step 1 Calculate $$\begin{array}{c}\hfill y_{j,n}={\mathcal{S}}_j{x}_n,\end{array}$$ for each $j=1,2,\dots ,M$ and suppose $$\begin{array}{ccc}\hfill g_{1,n}& =& \underset{j=1,2,\dots ,M}{max}\left\{d(y_{j,n},x_n)\right\}.\hfill \end{array}$$ Step 2  Compute $$\begin{array}{c}\hfill z_{k,n}={\Xi }_k(\Pi x_n),\end{array}$$ for each $k=1,2,\dots ,K$, and suppose $$\begin{array}{ccc}\hfill g_{2,n}& =& \underset{k=1,2,\dots ,K}{max}\left\{d(z_{k,n},\Pi x_n)\right\}.\hfill \end{array}$$ Step 3  Let $$\begin{array}{c}\hfill {\Gamma }_n=max\{g_{1,n},g_{2,n}\}.\end{array}$$ If $g_{1,n}={\Gamma }_n$, let $t_n=y_{j_n,n},j=1,2,\dots ,M$, and let $\Theta =I$. Else, if $g_{2,n}={\Gamma }_n$, then let $t_n=z_{k_n,n},k=1,2,\dots ,K$ and $\Theta =\Pi$. Step 4 Compute $$\begin{array}{c}\hfill u_n=\lambda x_n\oplus (1-\lambda ){\Theta }^{*}(\mu \Theta x_n\oplus (1-\mu )t_n).\end{array}$$ Step 5   Compute $$\begin{array}{c}\hfill x_{n+1}={\beta }_{n}h\left(x_n\right)\oplus (1-{\beta }_n)u_n,n\ge 0.\end{array}$$ where $\left\{{\beta }_n\right\}\subset (0,1)$ and $h:{\rm Y}\to {\rm Y}$ is a strict contraction having a contraction coefficient $l\in [0,1)$.

Corollary 2. 

If $\left\{{\beta }_n\right\}$ is the sequence with conditions ${\beta }_n\to 0$ and ${\sum }_{n=0}^{\infty }{\beta }_n=\infty$, then the sequence $\left\{x_n\right\}$ produced by Algorithm 2 strongly converges to $x^{†}\in {\Omega }^{SCFPP}$, which is the only solution of the variational inequality (V.I)

$$\begin{array}{c}\hfill \langle \overrightarrow{h{x}^{†}{x}^{†}},\overrightarrow{x^{†}y}\rangle \ge 0\forall y\in {\Omega }^{SCFPP}.\end{array}$$

6. Numerical Illustration

In this section, the numerical implementation of Algorithm 1 is provided. We also checked the convergence of our main result under our proposed algorithm.

Example 1. 

Consider the following problem: find an element $x\in {\mathbb{R}}^4$, such that

$$\begin{array}{c}\hfill x\in \Omega =\left({\cap }_{j=1}^{M}Fix\left(S_j\right)\right)\cap \left({\cap }_{i=1}^N{T}_i^{-1}\left({\cap }_{k=1}^{M_i}Fix\left({\Xi }_k^i\right)\right)\right)\ne \varphi ,\end{array}$$

where $S_j:H\to H,j=1,2,...,M$ and ${\Xi }_i^k:H_i\to H_i,i=1,2,...,N,k=1,2,...,M_i$ are the nonexpansive mappings defined as

$$\begin{array}{ccc}\hfill S_j& :& {\mathbb{R}}^4\to {\mathbb{R}}^4\hfill \\ \hfill S_j(a_1,a_2,a_3,a_4)& =& \left(j(\frac{-1}{2}{a}_1,\frac{-1}{2}{a}_2,\frac{1}{2}{a}_3,\frac{1}{2}{a}_4)\right)\hfill \\ \hfill \Xi i^k& :& {\mathbb{R}}^2\to {\mathbb{R}}^2\hfill \\ \hfill \Xi i^k(a_1,a_2)& =& \left(ik(2a_1,0)\right).\hfill \end{array}$$

and $T_i:H\to H_i,i=1,2,...,m$ is the bounded linear operators, defined as

$$\begin{array}{ccc}\hfill T_i& :& {\mathbb{R}}^4\to {\mathbb{R}}^2\hfill \\ \hfill T_i(a_1,a_2,a_3,a_4)& =& \left(i(a_1,a_1+a_3)\right).\hfill \end{array}$$

We defined $h:H\to H$ a strict contraction having a contraction coefficient $l\in [0,1)$ by

$$\begin{array}{ccc}\hfill h& :& {\mathbb{R}}^4\to {\mathbb{R}}^4\hfill \\ \hfill h(a_1,a_2,a_3,a_4)& =& (0.4a_1,0.4a_2,0.4a_3,0.4a_4).\hfill \end{array}$$

Now, we will examine the convergence of the sequence $\left\{x_n\right\}$, which is generated by Algorithm 1.

Let $x_0$ = (0.1,0,0.2,0)

Step1: for $j=1,n=0$

$$\begin{array}{ccc}\hfill y_{1,0}& =& S_1{x}_0=S_1(0.1,0,0.2,0)=(\frac{-1}{2}\left(0.1\right),\frac{-1}{2}\left(0\right),\frac{1}{2}\left(0.2\right),\frac{1}{2}\left(0\right))=(-0.05,0,0.1,0)\hfill \\ \hfill d_0& =& max\left\{d(y_{1,0},x_0)\right\}=d((-0.05,0,0.1,0),(0.1,0,0.2,0))=0.180278\hfill \end{array}$$

Step2: for $i=1,k=1$

$$\begin{array}{ccc}\hfill z_{1,0}^1& =& {\Xi }_1^1(T_1,x_0)={\Xi }_1^1\left(T_1(0.1,0,0.2,0)\right)=(0.2,0)\hfill \\ \hfill d_{1,0}& =& max\left\{d(z_{1,0}^1,\left(T_1{x}_0\right))\right\}=\sqrt{{(0.1-0.2)}^2+{(0.3-0)}^2}=\sqrt{0.01+0.9}=0.316228\hfill \end{array}$$

Step3:

$$\begin{array}{ccc}\hfill {\Gamma }_0& =& max\left(\{d_0,d_{1,0}\}\right)=d_0=0.180278\hfill \\ \hfill t_0& =& y_{(}{1}_0,0)\&\theta =I\hfill \\ \hfill t_0& =& (-0.05,0,0.1,0)\hfill \end{array}$$

Step4: for $\lambda =0.3,\mu =0.5$,

$$\begin{array}{c}\hfill u_0=\lambda x_0\oplus (1-\lambda ){\theta }^{*}(\mu \theta x_0\oplus (1-\mu )t_0)=(0.0475,0,0.48,0)\end{array}$$

Step5: let ${\alpha }_0=0.6$

$$\begin{array}{c}\hfill x_1={\alpha }_{0}h\left(x_0\right)\oplus (1-\alpha )u_0=(0.043,0,0.24,0).\end{array}$$

2nd Iteration:

Step1: for $j=2,n=1$

$$\begin{array}{ccc}\hfill y_{2,1}& =& S_2{x}_1=S_2(2\left(0.043\right),2\left(0\right),2\left(0.24\right),2\left(0\right)))=(-0.0432,0,0.24,0)\hfill \\ \hfill d_1& =& max\left(d(y_{(2,1)},x_1)\right)=d((-0.0432,0,0.24,0),(0.043,0,0.24,0))=0.086197\hfill \end{array}$$

Step2: for $i=2,k=2$

$$\begin{array}{c}\hfill z_{2,1}^2={\Xi }_2^2\left(T_2{x}_1\right)={\Xi }_2^2\left(T_2(0.043,0,0.24,0)\right)={\Xi }_2^2(2\left(0.043\right),2(0.043+0.24))=(0.664,0)\end{array}$$

$$\begin{array}{c}\hfill d_{2,1}=max\{d(z_{2,1}^2,\left(T_2{x}_1\right))\}=d((0.664,0),(0.086,1.34))=1.45931\end{array}$$

Step3:

$$\begin{array}{ccc}\hfill {\Gamma }_1& =& max(d_1,d_{2,1})=d_{2,1}=1.45931\hfill \\ \hfill t_1& =& z_{2_1,1}^{2_1}\&\theta =T_{2_0}\hfill \\ \hfill t_1& =& (0.664,0)\hfill \end{array}$$

Step4:

$$\begin{array}{ccc}\hfill u_1& =& \lambda x_1\oplus (1-\lambda ){\theta }^{*}(\mu \theta x_1\oplus (1-\mu )t_1)\hfill \\ & =& 0.3(0.043,0,0.24,0)+0.7T_{2_0}^{*}[(0.5T_{2_0}(0.043,0,0.24,0)+0.5(0.664,0)]\hfill \\ & =& (0.0129,0,0.072,0)+0.7T_{2_0}^{*}[(0.5(0.086,1.34)+(0.332,0)]\hfill \\ & =& (0.0129,0,0.072,0)+0.7T_{2_0}^{*}(0.375,0.67)\hfill \end{array}$$

To find adjoint of $T_{2_0}$

$$\begin{array}{ccc}\hfill <(a_1,a_2,a_3,a_4),T_{2_0}^{*}(b_1,b_2)>& =& <(a_1,a_1+a_3),(b_1,b_2)>\hfill \\ & =& b_1\left(a_1\right)+b_2(a_1+a_3)\hfill \\ & =& (b_1+b_2)a_1+b_2{a}_3\hfill \\ & =& <(a_1,a_2,a_3,a_4),(b_1+b_2,0,b_2,0)>\hfill \\ & =& <(a_1,a_2,a_3,a_4),T_{2_0}^{*}(b_1,b_2)\hfill \\ \hfill T_{2_0}^{*}(b_1,b_2)& =& (b_1+b_2,0,b_2,0)\hfill \\ \hfill T_{2_0}^{*}(0.375,0.67)& =& (0.375+0.67,0,0.67,0)\hfill \\ & =& (1.045,0,0.67,0)\hfill \end{array}$$

$$\begin{array}{c}\hfill u_1=(0.0129,0,0.072,0)+0.7(1.045,0,0.67,0)=(0.7444,0,0.541,0)\end{array}$$

Step5:

$$\begin{array}{ccc}\hfill x_2& =& {\alpha }_{1}h\left(x_1\right)\oplus (1-{\alpha }_1)u_1\hfill \\ & =& 0.7h(0.043,0,0.24,0)+0.3(0.7444,0,0.541,0)\hfill \\ & =& 0.7\left(0.4\right(0.043),0.4(0),0.4(0.24),0.4(0\left)\right)+(0.22332,0,0.1623,0)\hfill \\ & =& (0.23536,0,0.2295,0).\hfill \end{array}$$

Continuing in a similar way, we can construct the required converging sequence, which satisfies the condition of Algorithm 1.

7. Conclusions

We first introduced the split fixed point problems with multiple output sets and some other generalizations in $CAT\left(0\right)$ spaces. We presented two algorithms to solve the variational inequality problem over the split common fixed point problem with multiple output sets and the split common fixed point problem for nonexpansive mappings in $CAT\left(0\right)$ spaces, and then we proved the strong convergence of the sequence generated by the first algorithm and also stated some corollaries related to the solution of our variational inequality problem. We also proved some Lemmas and Propositions, which are essential for our main result.