Triple Mann Iteration Method for Variational Inclusions, Equilibria, and Common Fixed Points of Finitely Many Quasi-Nonexpansive Mappings on Hadamard Manifolds

Abstract

In this paper, we introduce a triple Mann iteration method for approximating an element in the set of common solutions of a system of quasivariational inclusion issues, which is an equilibrium problem and a common fixed point problem (CFPP) of finitely many quasi-nonexpansive operators on a Hadamard manifold. Through some suitable assumptions, we prove that the sequence constructed in the suggested algorithm is convergent to an element in the set of common solutions. Finally, making use of the main result, we deal with the minimizing problem with a CFPP constraint and saddle point problem with a CFPP constraint on a Hadamard manifold, respectively.

1. Introduction

It is well known that the theory of variational inequalities, introduced by Hartman and Stampacchia [1] in the early 1960s, has played an important role in the study of physics, engineering, and economics, and it encompasses as special cases complementarity problems, optimization problems, the problem of finding zeros of operators, etc. Within the period of the past 20 years, a great deal of effort has gone into the existence and algorithms for variational inequality problems, related optimization problems, and related fixed-point problems; see, e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and the references therein.

Let $\mathcal{M}$ be a Hadamard manifold and C be a nonempty subset of $\mathcal{M}$. In 2012, Calao et al. [19] first studied the following equilibrium problem of finding $y^{*}\in C$ such that

$$\Phi (y^{*},x)\ge 0$$

(1)

for all $x\in C$, in which the function $\Phi$ is of equilibrium. It was proven in [19] that there holds the existence of solutions to the issue (1) for $\Phi$. Moreover, the outcomes were employed to settle mixed variational inequalities, fixed point issues, and Nash equilibrium issues in $\mathcal{M}$. Using Picard’s iterative method, the authors introduced an iterative algorithm for finding a solution of issue (1). Subsequently, it was pointed out in [20] that there hold certain gaps involving the existence argument for mixed variational inequality problems and the domain for the resolvent implicating the equilibrium issue of [19].

Recently, the authors [21,22] investigated the problem of finding an element $x^{*}\in \mathrm{Fix}\left(T\right)\cap {(A+B)}^{-1}0$, with T being the nonexpansive operator, B being the maximal monotone vector field, and A being the continuous monotone vector field in a Hadamard manifold $\mathcal{M}$. They proposed some Halpern-type and Mann-type iterative methods. Under some mild conditions, they proved that the iterative sequences fabricated in the suggested methods are strongly convergent to an element in the set of common solutions of the fixed-point problem of T and the variational inclusion problem for A and B.

Assume $\varnothing \ne C\subset \mathcal{M}$, where the bounded C is of both closedness and geodesic convexity in $\mathcal{M}$; let $T_r^{\Phi }:\mathcal{M}\to C$ be the resolvent of equilibrium function $\Phi$ for $r>0$, and let ${\exp }_x^{-1}$ be the inverse of the exponential map ${\exp }_x:T_x\mathcal{M}\to \mathcal{M}$ at $x\in \mathcal{M}$. Very recently, Chang et al. [23] considered the problem of finding an element

$$x^{*}\in \Omega :=\mathrm{Fix}\left(S\right)\cap \mathrm{EP}\left(\Phi \right)\cap (\bigcap _{i=1}^M{(A_i+B)}^{-1}0),$$

(2)

in which $\mathrm{Fix}\left(S\right)$ is the fixed point set of the quasi-nonexpansive operator $S:C\to C$, $\mathrm{EP}\left(\Phi \right)$ is the equilibrium point set of $\Phi$, and $B:C\to 2^{T\mathcal{M}}$ is the maximal monotone vector field; for $i=1,\dots ,N$, each continuous $A_i:C\to T\mathcal{M}$ is a monotone vector field, and ${\bigcap }_{m=1}^M{(A_m+B)}^{-1}0$ is the common singularity set of a quasivariational inclusion system. They proposed the following splitting iterative algorithm, that is, for any initial $x_0\in C$, the sequence $\left\{x_n\right\}$ is defined as follows:

$$\left\{\begin{array}{cc}& u_n^i=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n),i=1,2,\dots ,M,\hfill \\ & y_n=S{u}_n^{i_n}\mathrm{with}{i}_n\in \{1,2,\dots ,M\}\mathrm{s}.\mathrm{t}.d(u_n^{i_n},x_n)={\displaystyle \underset{1\le i\le M}{max}}d(u_n^i,x_n),\hfill \\ & x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right)\forall n\ge 0,\hfill \end{array}\right.$$

(3)

where $\left\{{\alpha }_n\right\}\subset (0,1),{\alpha }_n\to 1,{\sum }_{n=0}^{\infty }{\alpha }_n=\infty$, and $J_{\lambda }^B{\exp }_I(-\lambda A_i):C\to \mathcal{M}$ is the mapping defined by $B,A_i$ and $\lambda >0$ for $i=1,2,\dots ,M$. Under some mild assumptions, it was proven that the $\left\{x_n\right\}$ constructed in (3) is strongly convergent to an element of $\Omega$.

Inspired and aroused by the works above, this article considers the issue of seeking an element

$$y^{*}\in \Omega :=(\bigcap _{i=1}^N\mathrm{Fix}\left(S_i\right))\cap \mathrm{EP}\left(\Phi \right)\cap (\bigcap _{i=1}^M{(A_i+B)}^{-1}0),$$

(4)

in Hadamard manifold $\mathcal{M}$, where ${\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)$ is the set of common fixed points of finitely many quasi-nonexpansive mappings ${\left\{S_i\right\}}_{i=1}^N$, and $\Phi ,B,A_i,i=1,2,\dots ,M$ are the same as above. We propose a triple Mann iteration method and show that the sequence fabricated in suggested algorithm is convergent to an element in the set of common solutions for issue (4). Finally, making use of the main result, we deal with the minimizing issue with a CFPP constraint and saddle-point issue with a CFPP constraint on Hadamard manifolds, respectively. Our main result improves, extends, and develops Chang et al. [23], Theorem 3.1, in some aspects.

2. Basic Notions and Tools

To address the problem (4), one first releases certain preliminaries on geometric theory of manifolds, including some concepts and basic tools. It is worth mentioning that these can be found in many introductory books on Riemannian and differential geometry (see, e.g., [24]).

Assume $\mathcal{M}$ is a differentiable manifold of finite dimension. Then, the tangent space of $\mathcal{M}$ at $x\in \mathcal{M}$ is denoted as $T_x\mathcal{M}$. The tangent bundle of $\mathcal{M}$ is formulated as $T\mathcal{M}={\bigcup }_{x\in \mathcal{M}}{T}_x\mathcal{M}$. Naturally, it is a manifold. The inner product ${\mathcal{R}}_x(·,·)$ in $T_x\mathcal{M}$ is termed as a Riemannian metric in $T_x\mathcal{M}$. The tensor field $\mathcal{R}(·,·)$ is termed as a Riemannian metric on $\mathcal{M}$ if and only if, for each $x\in \mathcal{M}$, the tensor ${\mathcal{R}}_x(·,·)$ is a Riemannian metric on $T_x\mathcal{M}$. The associated norm with ${\mathcal{R}}_x(·,·)$ on $T_x\mathcal{M}$ is written as ${\parallel ·\parallel }_x$. The differentiable manifold $\mathcal{M}$ equipped with Riemannian metric $\mathcal{R}(·,·)$ is termed as Riemannian manifold. Let the curve $\gamma :[a,b]\to \mathcal{M}$ joining u to v (i.e., $\gamma \left(a\right)=u$ and $\gamma \left(b\right)=v$) be piecewise smooth. Then, the length of $\gamma$ is defined as $l\left(\gamma \right)={\int }_a^b\parallel {\gamma }^{\prime }\left(t\right)\parallel dt$. Moreover, the Riemannian distance $d(v,y)$, inducing the original topology on $\mathcal{M}$, is formulated as minimizing this length over the set of all such curves joining v to y.

The Riemannian manifold $\mathcal{M}$ is of completeness if and only if, for each $y\in \mathcal{M}$, any geodesic emanating from y is definable with respect to (w.r.t.) t in $\mathbf{R}:=(-\infty ,\infty )$. The geodesic joining v to y in $\mathcal{M}$ is termed as the minimal geodesic if and only if the length of it equals to $d(v,y)$. Riemannian manifold $\mathcal{M}$ endowed by Riemannian distance d is the metric space $(\mathcal{M},d)$. From the Hopf–Rinow result [24], it is easily known that, in case $\mathcal{M}$ is of completeness, each pair of elements in $\mathcal{M}$ is joined with the minimal geodesic. Meanwhile, the metric space $(\mathcal{M},d)$ is of completeness, and each closed bounded subset is of compactness.

Suppose that Riemannian manifold $\mathcal{M}$ is of completeness. The map ${\exp }_y:T_y\mathcal{M}\to \mathcal{M}$ at y is the exponential one formulated as ${\exp }_{y}v={\gamma }_v(1,y)$ for all $v\in T_y\mathcal{M}$, in which $\gamma (·)={\gamma }_v(·,y)$ denotes the geodesic initiating from y with velocity v, that is, ${\gamma }_v(0,y)=y$ and ${\gamma }_v^{\prime }(0,y)=v$. Then, ${\exp }_{y}tv={\gamma }_v(t,y)$ for any $t\in \mathbf{R}$. It is clear to check ${\exp }_{y}0={\gamma }_v(0,y)=y$, in which 0 denotes the zero tangent vector. It is noteworthy that map ${\exp }_y$ is the differentiable exponential one on $T_y\mathcal{M}$ w.r.t. $y\in \mathcal{M}$.

A complete simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard Manifold. In the rest of this paper, we always assume that $\mathcal{M}$ is a Hadamard Manifold of finite dimension without causing confusion.

Proposition 1 

(see [24]). For any two points $x,y\in \mathcal{M}$, there exists a unique normalized geodesic $\gamma :[0,1]\to \mathcal{M}$ joining $x=\gamma \left(0\right)$ to $y=\gamma \left(1\right)$, which is actually a minimal geodesic denoted by

$$\gamma \left(t\right)={\exp }_{x}t{\exp }_x^{-1}y\forall t\in [0,1].$$

(5)

Moreover, for each $\left\{x_n\right\}$ in $\mathcal{M}$ satisfying $x_n\to x_0$ in $\mathcal{M}$, there hold the following:

$${\exp }_{x_n}^{-1}y\to {\exp }_{x_0}^{-1}y\mathrm{and}{\exp }_y^{-1}{x}_n\to {\exp }_y^{-1}{x}_0\forall y\in \mathcal{M}.$$

(6)

It is easy to check that the following lemma is valid.

Lemma 1. 

(i) If $\gamma :[0,1]\to \mathcal{M}$ is a geodesic joining x to y, then we have

$$d(\gamma \left(t_1\right),\gamma \left(t_2\right))=|t_1-t_2|d(x,y)\forall t_1,t_2\in [0,1].$$

(7)

(From now on, $d(x,y)$ indicates the Riemannian distance).

(ii) For each $y,z,u,v,w\in \mathcal{M}$ and $t\in [0,1]$, there hold the inequalities below:

$$d({\exp }_{y}t{\exp }_y^{-1}v,z)\le (1-t)d(y,z)+td(v,z);$$

(8)

$$d^2({\exp }_{y}t{\exp }_y^{-1}v,z)\le (1-t)d^2(y,z)+t{d}^2(v,z)-t(1-t)d^2(y,v);$$

(9)

$$d({\exp }_{y}t{\exp }_y^{-1}v,{\exp }_{u}t{\exp }_u^{-1}w)\le (1-t)d(y,u)+td(v,w).$$

(10)

The set C in $\mathcal{M}$ is of geodesic convexity if and only if, for each $y,v\in C$, the geodesic joining y to v belongs to C. The geodesic convex hull of set $D\subset \mathcal{M}$ is the smallest geodesic convex set containing D in $\mathcal{M}$, and it is written as $\mathrm{co}\left(D\right)$.

In what follows, unless otherwise specified, we always assume that $\varnothing \ne C\subset \mathcal{M}$, where C is of both closedness and geodesic convexity in Hadamard manifold $\mathcal{M}$, and $\mathrm{Fix}\left(S\right)$ denotes the set of fixed points of an operator S.

The function $h:C\to \mathbf{R}\cup \{+\infty \}=(-\infty ,\infty ]$ is termed as being of geodesic convexity if and only if, for each geodesic $\gamma \left(t\right)$ ($\forall t\in [0,1]$) joining $y,v\in C$, the $h\circ \gamma$ is of convexity, that is,

$$h\left(\gamma \right(t\left)\right)\le th\left(\gamma \right(0\left)\right)+(1-t)h\left(\gamma \right(1\left)\right)=th\left(y\right)+(1-t)h\left(v\right).$$

Assume the metric space X is complete, and let $\varnothing \ne D\subset X$. The sequence $\left\{x_n\right\}$ in X is of Fejér monotonicity w.r.t. D if and only if, for each $v\in D$ and each non-negative n, one has $d(x_n,v)\ge d(x_{n+1},v)$.

Lemma 2 

(see [25,26]). Assume the metric space X is complete, and let $\varnothing D\subset X$. In the case that $\left\{y_n\right\}\subset X$ is of Fejér monotonicity w.r.t. D, $\left\{y_n\right\}$ is of boundedness. In addition, in the case that $\left\{y_n\right\}$ has a cluster point y lying in D, $\left\{y_n\right\}$ is convergent to y.

An operator S from C to itself is termed as being the following:

(a) Contractive if and only if $\exists \ell \in (0,1)$ such that

$$d(Sy,Sv)\le \ell d(y,v)\forall y,v\in C$$

(in particular, in the case of $\ell =1$, S is termed as being nonexpansive);

(b) Quasi-nonexpansive if and only if $\mathrm{Fix}\left(S\right)$ is not equal to ∅, and

$$d(Sy,v)\le d(y,v)\forall y\in C,v\in \mathrm{Fix}\left(S\right);$$

(c) Firmly nonexpansive [27] if and only if, for each $y,v$ in C, the map $\psi :[0,1]\to [0,\infty ]$ formulated by

$$\psi \left(t\right):=d({\exp }_{y}t{\exp }_y^{-1}Sy,{\exp }_{v}t{\exp }_v^{-1}Sv)\forall t\in [0,1]$$

(11)

is nonincreasing.

Proposition 2 

(see [27]). Let $S:C\to C$ be a mapping. Then, the following statements are equivalent:

(i) S is firmly nonexpansive;

(ii) For all $x,y\in C$ and $t\in [0,1]$,

$$d(Sx,Sy)\le d({\exp }_{x}t{\exp }_x^{-1}Sx,{\exp }_{y}t{\exp }_y^{-1}Sy);$$

(12)

(iii) for all $x,y\in C$,

$$\mathcal{R}({\exp }_{Sx}^{-1}Sy,{\exp }_{Sx}^{-1}x)+\mathcal{R}({\exp }_{Sy}^{-1}Sx,{\exp }_{Sy}^{-1}y)\le 0.$$

(13)

Lemma 3 

(see [23]). Let $S:C\to C$ be firmly nonexpansive, and assume $\mathrm{Fix}\left(S\right)$ is not equal to ∅. Then, for each $y\in C$ and $v\in \mathrm{Fix}\left(S\right)$, there holds the inequality below:

$$d^2(Sy,v)\le d^2(y,v)-d^2(Sy,y).$$

(14)

It is remarkable that, when $\mathrm{Fix}\left(S\right)\ne \varnothing$, by Lemma 3, one obtains the relations below:

The firm nonexpansivity of $S\Rightarrow$ the nonexpansivity of $S\Rightarrow$ the quasinonexpansivity of S.

However, the converse relations do not hold.

An operator S from C to itself is termed as being demiclosed at zero if and only if, for each $\left\{y_n\right\}\subset C$ satisfying $y_n\to y^{*}\in C$ and $d(y_n,S{y}_n)\to 0$, one has $y^{*}\in \mathrm{Fix}\left(S\right)$.

In what follows, one uses $\mathrm{\Omega }\left(\mathcal{M}\right)$ to denote the collection of all single-valued vector fields $A:\mathcal{M}\to T\mathcal{M}$ such that $Ay\in T_y\mathcal{M}\forall y\in \mathcal{M}$, and $D\left(A\right)$ the domain of A defined by

$$D\left(A\right)=\{y\in \mathcal{M}:Ay\in T_y\mathcal{M}\}.$$

Also, one uses $\mathcal{X}\left(\mathcal{M}\right)$ to indicate the collection of all set-valued vector fields $B:\mathcal{M}\to 2^{T\mathcal{M}}$ such that $By\subset T_y\mathcal{M}\forall y\in \mathcal{M}$, and $\mathcal{D}\left(B\right)$ is the domain of B formulated by $\mathcal{D}\left(B\right)=\{y\in \mathcal{M}:By\ne \varnothing \}$.

Next, one furnishes some vital notions (see [28]) that will be helpful to demonstrate the major outcome:

(i) A single-valued vector field $A\in \mathrm{\Omega }\left(\mathcal{M}\right)$ is termed as being monotone if and only if

$$\mathcal{R}(Ay,{\exp }_y^{-1}v)\le \mathcal{R}(Av,-{\exp }_v^{-1}y)\forall y,v\in \mathcal{M}.$$

(ii) A set-valued vector field $B\in \mathcal{X}\left(\mathcal{M}\right)$ is termed as being the following:

(a) Monotone if and only if, for each $y,z\in \mathcal{D}\left(B\right)$

$$\mathcal{R}(v,{\exp }_y^{-1}z)\le \mathcal{R}(w,-{\exp }_z^{-1}y)\forall v\in By,w\in Bz;$$

(b) Maximal monotone if and only if it is monotone and, for the given $y\in \mathcal{D}\left(B\right)$ and $v\in T_y\mathcal{M}$, the relation

$$\mathcal{R}(v,{\exp }_y^{-1}z)\le \mathcal{R}(w,-{\exp }_z^{-1}y)\forall z\in \mathcal{D}\left(B\right),w\in Bz$$

implies $v\in By$.

(iii) For given $\lambda >0$, the resolvent of B for $\lambda >0$ is a set-valued mapping $J_{\lambda }^B:\mathcal{M}\to 2^{T\mathcal{M}}$ formulated by

$$J_{\lambda }^B\left(y\right):=\{x\in \mathcal{M}:y\in {\exp }_x\lambda Bx\}\forall y\in \mathcal{M}.$$

(15)

Lemma 4 

(see [21]). Given a single-valued vector field $A\in \mathrm{\Omega }\left(\mathcal{M}\right)$ of monotonicity and a set-valued vector field $B\in \mathcal{X}\left(\mathcal{M}\right)$ of maximal monotonicity, one then has that for each $y\in \mathcal{M}$, there holds the equivalence of the assertions below:

(i) $y\in {(A+B)}^{-1}0$;

(ii) $y=J_{\lambda }^B\left({\exp }_y(-\lambda Ay)\right)\forall \lambda >0$.

In addition, we recall the resolvent of a bifunction on Hadamard manifold $\mathcal{M}$ introduced in Calao et al. [19], as outlined below:

Assume $\varnothing \ne C\subset \mathcal{M}$, where C is of both closedness and geodesic convexity in $\mathcal{M}$. Given a bifunction $\Phi :C\times C\to \mathbf{R}$, the resolvent of $\Phi$ is set-valued mapping $T_r^{\Phi }:\mathcal{M}\to 2^C$ such that

$$T_r^{\Phi }\left(y\right)=\{x\in C:\Phi (x,z)-{\displaystyle \frac{1}{r}}\langle {\exp }_x^{-1}y,{\exp }_x^{-1}z\rangle \ge 0\forall z\in C\}\forall y\in \mathcal{M}.$$

(16)

Lemma 5 

(see [19,20]). Let $\Phi :C\times C\to \mathbf{R}$ be a bifunction satisfying the following conditions:

(A1) $\Phi (x,x)=0\forall x\in C$;

(A2) $\Phi (x,y)+\Phi (y,x)\le 0\forall x,y\in C$, i.e., Φ is monotone;

(A3) $x\mapsto \Phi (x,y)$ is upper semicontinuous for each $y\in C$;

(A4) $y\mapsto \Phi (x,y)$ is geodesic convex and lower semicontinuous for each $x\in C$;

(A5) ∃ (compact set) D in $\mathcal{M}$ such that

$$y\in C\setminus D\Rightarrow \exists z\in C\cap D\mathrm{s}.\mathrm{t}.\Phi (y,z)<0.$$

One then has that for each $r>0$, the assertions listed below hold:

(i) The resolvent $T_r^{\Phi }$ is nonempty and single-valued;

(ii) The resolvent $T_r^{\Phi }$ is firmly nonexpansive;

(iii) $\mathrm{Fix}\left(T_r^{\Phi }\right)=\mathrm{EP}\left(\Phi \right)$, i.e., the fixed point set of $T_r^{\Phi }$ is the equilibrium point set of Φ;

(iv) $\mathrm{EP}\left(\Phi \right)$ is of both closedness and geodesic convexity.

3. Major Outcomes

Let $\mathcal{M}$ be a Hadamard manifold of finite dimension and the nonempty $C\subset \mathcal{M}$ be of both closedness and geodesic convexity in $\mathcal{M}$. Assume throughout that the following conditions hold:

• $B:C\to 2^{T\mathcal{M}}$ is a multi-valued vector field of maximal monotonicity, $A_i:C\to T\mathcal{M}$ is a single-valued vector field of both continuity and monotonicity for $i=1,2,\dots ,M$, and $J_{\lambda }^B\left({\exp }_I(-\lambda A_i)\right):C\to \mathcal{M}$ is the mapping defined by $B,A_i$ and $\lambda >0$ for $i=1,2,\dots ,M$;
• $\Phi :C\times C\to (-\infty ,\infty )$ is the bifunction fulfilling the hypotheses (A1)–(A5) of Lemma 5 and $T_r^{\Phi }:\mathcal{M}\to 2^C$ is the resolvent of $\Phi$ for $r>0$ and $D\left(T_r^{\Phi }\right)$ is of both closedness and geodesic convexity;
• ${\left\{S_i\right\}}_{i=1}^N$ is a finite family of L-Lipschitzian and demiclosed at zero quasi-nonexpansive self-mappings on C, where $L\ge 1$;
• $\Omega :=\left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)\cap \left({\bigcap }_{i=1}^M{(A_i+B)}^{-1}0\right)\ne \varnothing$;
• For given $\lambda >0$ and $\rho \in (0,1)$, the inequality holds

  $$d({\exp }_x(-\lambda A_{i}x),{\exp }_y(-\lambda A_{i}y))\le (1-\rho )d(x,y)\forall x,y\in C,i\in \{1,2,\dots ,M\};$$

  (17)
• $\left\{{\alpha }_n\right\}\subset (0,1)$ and $\left\{{\beta }_n\right\},\left\{{\gamma }_n\right\}\subset (0,1]$ such that the following are true:

  (i)

  ${lim}_{n\to \infty }{\alpha }_n=1$, and ${\sum }_{n=1}^{\infty }(1-{\alpha }_n)=\infty$;

  (ii)

  $0<{lim\; inf}_{n\to \infty }{\beta }_n$, and $0<{lim\; inf}_{n\to \infty }{\gamma }_n$.

In terms of Xu and Kim [18], we write $S_n:=S_{n\mathrm{mod}N}$ for integer $n\ge 1$ with the mod function taking values in the set $\{1,2,\dots ,N\}$, that is, if $n=jN+q$ for some integers $j\ge 0$ and $0\le q<N$, then $S_n=S_N$ if $q=0$ and $S_n=S_q$ if $0<q<N$.

Theorem 1. 

Let $\mathcal{M},C,{\left\{A_i\right\}}_{i=1}^M,B,{\left\{J_{\lambda }^B{\exp }_I(-\lambda A_i)\right\}}_{i=1}^M,\Phi ,T_r^{\Phi }$, and ${\left\{S_i\right\}}_{i=1}^N$ be chosen as in the above conditions. Given a starting $x_0\in C$ arbitrarily. Let $\left\{x_n\right\},{\left\{u_n^i\right\}}_{i=1}^M$ and $\left\{y_n\right\}$ be the sequences defined as follows:

$$\left\{\begin{array}{cc}& u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right),i=1,2,\dots ,M,\hfill \\ & y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)\mathrm{with}{i}_n\in \{1,2,\dots ,M\}\mathrm{s}.\mathrm{t}.d(u_n^{i_n},x_n)={\displaystyle \underset{1\le i\le M}{max}}d(u_n^i,x_n),\hfill \\ & x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right)\forall n\ge 0.\hfill \end{array}\right.$$

(18)

Then, we have the following: (i) The sequence $\left\{x_n\right\}$ converges to $x^{*}\in \Omega$ a solution of problem (4);

(ii) In addition, if ${\beta }_n={\gamma }_n=1\forall n\ge 0$, and $S_i=S,i=1,2,\dots ,N$, then the sequence $\left\{x_n\right\}$ defined by

$$\left\{\begin{array}{cc}& u_n^i=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n),i=1,2,\dots ,M,\hfill \\ & y_n=S{u}_n^{i_n}\mathrm{with}{i}_n\in \{1,2,\dots ,M\}\mathrm{s}.\mathrm{t}.d(u_n^{i_n},x_n)={\displaystyle \underset{1\le i\le M}{max}}d(u_n^i,x_n),\hfill \\ & x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right)\forall n\ge 0,\hfill \end{array}\right.$$

converges to a solution of problem (2);

(iii) In addition, if ${\beta }_n={\gamma }_n=1\forall n\ge 0$, and $A_j=B=0$ (the null operator), $j=1,2,\dots ,M$, then the sequence $\left\{x_n\right\}$ defined by

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{S}_n{x}_n\right)\forall n\ge 0,$$

converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)$;

(iv) In addition, if for $\Phi =0$, ${\beta }_n={\gamma }_n=1\forall n\ge 0$, and $S_i=I\forall i\in \{1,2,\dots ,N\}$, $\left\{x_n\right\}$ is the sequence constructed as

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_{x_n}(-\lambda A_{i_n}{x}_n)\right)\forall n\ge 0,$$

then $\left\{x_n\right\}$ converges to an element $x^{*}\in {\bigcap }_{i=1}^M{(A_i+B)}^{-1}0$;

(v) In addition, if for $\Phi =0$, ${\beta }_n={\gamma }_n=1\forall n\ge 0$, and $A_j=B=0\forall j\in \{1,2,\dots ,M\}$, $\left\{x_n\right\}$ is the sequence constructed as

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(S_n{x}_n\right)\forall n\ge 0,$$

then $\left\{x_n\right\}$ converges to $x^{*}\in {\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)$ a common fixed point of ${\left\{S_i\right\}}_{i=1}^N$;

(vi) In addition, if ${\beta }_n={\gamma }_n=1\forall n\ge 0$, and $A_i=A,i=1,2,\dots ,M$, then the sequence $\left\{x_n\right\}$ defined by

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }\left(S_n\left(J_{\lambda }^B{\exp }_{x_n}(-\lambda A{x}_n)\right)\right)\right)\forall n\ge 0,$$

converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)\cap {(A+B)}^{-1}0$;

(vii) In addition, if ${\beta }_n={\gamma }_n=1\forall n\ge 0$, and $A_i=0,i=1,2,\dots ,M$, then the sequence $\left\{x_n\right\}$ defined by

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }\left(S_n\left(J_{\lambda }^B{x}_n\right)\right)\right)\forall n\ge 0,$$

converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)\cap B^{-1}0$.

Proof. 

First of all, using the conditions in Theorem 1 and Lemmas 4 and 5, one knows the following:

(i) The resolvent $T_{\gamma }^{\Phi }$ of function $\Phi$ is the single-valued mapping of firm nonexpansivity such that $\mathrm{Fix}\left(T_{\gamma }^{\Phi }\right)=\mathrm{EP}\left(\Phi \right)\forall \gamma \in (0,\infty )$;

(ii) If $z\in \Omega$, then one has that $\forall \lambda >0$ and $\gamma >0$,

$$\begin{array}{cc}z\hfill & \in ({\displaystyle \bigcap _{i=1}^N}\mathrm{Fix}\left(S_i\right))\cap \mathrm{EP}\left(\Phi \right)\cap ({\displaystyle \bigcap _{i=1}^M}{(A_i+B)}^{-1}0)\hfill \\ & =({\displaystyle \bigcap _{i=1}^N}\mathrm{Fix}\left(S_i\right))\cap \mathrm{Fix}\left(T_{\gamma }^{\Phi }\right)\cap ({\displaystyle \bigcap _{i=1}^M}\mathrm{Fix}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)\right)).\hfill \end{array}$$

(19)

We now divide the rest of the proof into several steps.

Step 1. We claim that the sequences $\left\{x_n\right\},\left\{y_n\right\},\left\{u_n^i\right\},i=1,2,\dots ,M,\left\{S_n{u}_n^{i_n}\right\}$ and $\left\{T_r^{\Phi }{y}_n\right\}(r>0)$ all are bounded in C.

Indeed, by condition (17), for each $i=1,2,\dots ,M$ and $\lambda >0$, $J_{\lambda }^B{\exp }_I(-\lambda A_i)$ is a mapping of quasi-nonexpansivity. Take a fixed $p\in \Omega$ arbitrarily. Then, for $i=1,2,\dots ,M$, by (17) and Lemma 1 (ii), we have

$$\begin{array}{cc}d(u_n^i,p)\hfill & =d({\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right),{\exp }_p{\gamma }_n{\exp }_p^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)p\right))\hfill \\ & \le (1-{\gamma }_n)d(x_n,p)+{\gamma }_{n}d(J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n),J_{\lambda }^B{\exp }_p(-\lambda A_{i}p))\hfill \\ & \le (1-{\gamma }_n)d(x_n,p)+{\gamma }_n(1-\rho )d(x_n,p)\hfill \\ & =(1-{\gamma }_n\rho )d(x_n,p)\le d(x_n,p).\hfill \end{array}$$

(20)

Since $S_i$ is quasi-nonexpansive for $i=1,2,\dots ,N$, from Lemma 1 (ii), (18), and (20) we obtain

$$\begin{array}{cc}d(y_n,p)\hfill & =d({\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right),{\exp }_p{\beta }_n{\exp }_p^{-1}\left(S_{n}p\right))\hfill \\ & \le (1-{\beta }_n)d(x_n,p)+{\beta }_{n}d(S_n{u}_n^{i_n},S_{n}p)\hfill \\ & \le (1-{\beta }_n)d(x_n,p)+{\beta }_{n}d(u_n^{i_n},p)\hfill \\ & \le (1-{\beta }_n)d(x_n,p)+{\beta }_{n}d(x_n,p)=d(x_n,p).\hfill \end{array}$$

(21)

Thus, from Lemma 1 (ii), Lemma 5, (18), and (21), we have

$$\begin{array}{cc}{d}^2(x_{n+1},p)\hfill & =d^2({\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right),{\exp }_p{\alpha }_n{\exp }_p^{-1}\left(T_r^{\Phi }p\right))\hfill \\ & \le (1-{\alpha }_n)d^2(x_n,p)+{\alpha }_n{d}^2(T_r^{\Phi }{y}_n,T_r^{\Phi }p)-{\alpha }_n(1-{\alpha }_n)d^2(x_n,T_r^{\Phi }{y}_n)\hfill \\ & \le (1-{\alpha }_n)d^2(x_n,p)+{\alpha }_n{d}^2(y_n,p)-{\alpha }_n(1-{\alpha }_n)d^2(x_n,T_r^{\Phi }{y}_n)\hfill \\ & \le (1-{\alpha }_n)d^2(x_n,p)+{\alpha }_n{d}^2(x_n,p)-{\alpha }_n(1-{\alpha }_n)d^2(x_n,T_r^{\Phi }{y}_n)\hfill \\ & =d^2(x_n,p)-{\alpha }_n(1-{\alpha }_n)d^2(x_n,T_r^{\Phi }{y}_n).\hfill \end{array}$$

(22)

This implies that, for all $n\ge 0$ and $p\in \Omega$,

$$d(x_{n+1},p)\le d(x_n,p),$$

(23)

that is, $\left\{x_n\right\}$ is of Fejér monotonicity w.r.t. $\Omega$. Via Lemma 2, $\left\{x_n\right\}$ is of boundedness and so are the sequences $\left\{u_n^j\right\},j=1,2,\dots ,M,\left\{y_n\right\},\left\{T_r^{\Phi }{y}_n\right\}$. As a result, the limit ${lim}_{n\to \infty }d(x_n,p)$ exists for each $p\in \Omega$.

Step 2. We claim that for $i=1,2,\dots ,M$,

$$d(x_n,u_n^i)\to 0,d(x_n,y_n)\to 0\mathrm{and}d(x_{n+1},x_n)\to 0\mathrm{as}n\to \infty .$$

(24)

Indeed, from (20), (21), and (23) we obtain that

$$\left\{\begin{array}{cc}& d(x_n,u_n^i)\le d(x_n,p)+d(u_n^i,p)\le 2d(x_n,p)\mathrm{for}i=1,2,\dots ,M,\hfill \\ & d(x_n,y_n)\le d(x_n,p)+d(y_n,p)\le 2d(x_n,p),\hfill \\ & d(x_{n+1},x_n)\le d(x_n,p)+d(x_{n+1},p)\le 2d(x_n,p).\hfill \end{array}\right.$$

(25)

Taking $p\in \Omega$ and setting $\xi ={sup}_{n\ge 0}d(x_n,p)$, by Lemma 1 (ii), we deduce that, for each $i=1,2,\dots ,M$,

$$\begin{array}{cc}d(x_n,u_n^i)\hfill & \le 2d(x_n,p)=2d({\exp }_{x_{n-1}}{\alpha }_{n-1}{\exp }_{x_{n-1}}^{-1}\left(T_r^{\Phi }{y}_{n-1}\right),{\exp }_p{\alpha }_{n-1}{\exp }_p^{-1}\left(T_r^{\Phi }p\right))\hfill \\ & \le 2\{(1-{\alpha }_{n-1})d(x_{n-1},p)+{\alpha }_{n-1}d(T_r^{\Phi }{y}_{n-1},T_r^{\Phi }p)\}\hfill \\ & \le 2\{(1-{\alpha }_{n-1})d(x_{n-1},p)+{\alpha }_{n-1}d(y_{n-1},p)\}\hfill \\ & \le 2\{(1-{\alpha }_{n-1})\xi +{\alpha }_{n-1}d(x_{n-1},p)\}.\hfill \end{array}$$

By induction, one deduces that, for $1\le m\le n-1$,

$$d(x_n,u_n^i)\le 2\xi \sum _{j=m}^{n-1}\{(1-{\alpha }_j)\prod _{i=j+1}^{n-1}{\alpha }_i\}+2\xi \prod _{j=m}^{n-1}{\alpha }_j,i=1,2,\dots ,M.$$

(26)

Taking the limit in (26) as $n\to \infty$, we obtain

$$\underset{n\to \infty }{lim}d(x_n,u_n^i)\le 2\xi \sum _{j=m}^{\infty }\{(1-{\alpha }_j)\prod _{i=j+1}^{\infty }{\alpha }_i\}+2\xi \prod _{j=m}^{\infty }{\alpha }_j.$$

Since $\left\{{\alpha }_n\right\}\subset (0,1),{\alpha }_n\to 1(n\to \infty )$ and ${\sum }_{n=0}^{\infty }(1-{\alpha }_n)=\infty$, we obtain that

$$\begin{array}{cc}& {\displaystyle \underset{m\to \infty }{lim}\sum _{j=m}^{\infty }}\{(1-{\alpha }_j){\displaystyle \prod _{i=j+1}^{\infty }}{\alpha }_i\}=0,\hfill \\ & {\displaystyle \underset{m\to \infty }{lim}\prod _{j=m}^{\infty }}{\alpha }_j=0.\hfill \end{array}$$

Consequently,

$$\underset{n\to \infty }{lim}d(x_n,u_n^i)=0,i=1,2,\dots ,M.$$

In a similar way, from (25), we can also show that

$$d(x_n,y_n)\to 0\mathrm{and}d(x_n,x_{n+1})\to 0\mathrm{as}n\to \infty .$$

Thus, the assertions in (24) are valid.

Step 3. We claim that, for $i=1,2,\dots ,M$,

$$d(x_n,J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n)\to 0,d(x_n,S_n{x}_n)\to 0\mathrm{and}d(x_n,T_r^{\Phi }{x}_n)\to 0\mathrm{as}n\to \infty .$$

Indeed, by (18) and Lemma 1 (ii), we obatain

$$\begin{array}{cc}& d(x_n,J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n)\le d(x_n,u_n^i)+d(u_n^i,J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n)\hfill \\ & =d(x_n,u_n^i)+d({\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right),J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n)\hfill \\ & \le d(x_n,u_n^i)+(1-{\gamma }_n)d(x_n,J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n),\hfill \end{array}$$

which hence yields

$$d(x_n,J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n)\le {\displaystyle \frac{1}{{\gamma }_n}}d(x_n,u_n^i).$$

Noticing $0<{lim\; inf}_{n\to \infty }{\gamma }_n$, from (24) we have

$$\underset{n\to \infty }{lim}d(x_n,J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n)=0,i=1,2,\dots ,M.$$

(27)

Since each $S_n$ is L-Lipschitzian, using Lemma 1 (ii), one has

$$\begin{array}{cc}d(S_n{x}_n,x_n)\hfill & \le d(S_n{x}_n,y_n)+d(y_n,x_n)\hfill \\ & =d(y_n,x_n)+d({\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right),S_n{x}_n)\hfill \\ & \le d(x_n,y_n)+(1-{\beta }_n)d(x_n,S_n{x}_n)+{\beta }_{n}d(S_n{u}_n^{i_n},S_n{x}_n)\hfill \\ & \le d(x_n,y_n)+(1-{\beta }_n)d(x_n,S_n{x}_n)+{\beta }_{n}Ld(u_n^{i_n},x_n),\hfill \end{array}$$

which hence leads to

$$d(x_n,S_n{x}_n)\le {\displaystyle \frac{1}{{\beta }_n}}[d(x_n,y_n)+{\beta }_{n}Ld(u_n^{i_n},x_n)]\le {\displaystyle \frac{1}{{\beta }_n}}[d(x_n,y_n)+Ld(u_n^{i_n},x_n)].$$

Noticing $0<{lim\; inf}_{n\to \infty }{\beta }_n$, from (24), we have

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

(28)

Also, since $T_r^{\Phi }$ is firmly nonexpansive, by Lemma 1 (ii), we obatain

$$\begin{array}{cc}d(x_n,T_r^{\Phi }{x}_n)\hfill & \le d(x_n,x_{n+1})+d(x_{n+1},T_r^{\Phi }{x}_n)\hfill \\ & =d(x_n,x_{n+1})+d({\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right),T_r^{\Phi }{x}_n)\hfill \\ & \le d(x_n,x_{n+1})+(1-{\alpha }_n)d(x_n,T_r^{\Phi }{x}_n)+{\alpha }_{n}d(T_r^{\Phi }{y}_n,T_r^{\Phi }{x}_n)\hfill \\ & \le d(x_n,x_{n+1})+(1-{\alpha }_n)d(x_n,T_r^{\Phi }{x}_n)+{\alpha }_{n}d(y_n,x_n),\hfill \end{array}$$

which immediately implies that

$$\begin{array}{cc}d(x_n,T_r^{\Phi }{x}_n)\hfill & \le {\displaystyle \frac{1}{{\alpha }_n}}[d(x_n,x_{n+1})+{\alpha }_{n}d(y_n,x_n)]\hfill \\ & \le {\displaystyle \frac{1}{{\alpha }_n}}[d(x_n,x_{n+1})+d(y_n,x_n)].\hfill \end{array}$$

Noticing ${lim}_{n\to \infty }{\alpha }_n=1$, from (24), we obtain

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

(29)

Step 4. We claim that $\left\{x_n\right\}$ converges to some point $x^{*}\in \Omega$.

Indeed, since C is closed and geodesic convex in finite dimensional $\mathcal{M}$, and $\left\{x_n\right\}$ is of boundedness in C, one has that $\exists \left\{x_{n_k}\right\}\subset \left\{x_n\right\}$ such that $\left\{x_{n_k}\right\}$ is convergent to an element $x^{*}$ in C. We first claim $x^{*}\in {\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)$. Without loss of generality, we may assume $l=n_k\mathrm{mod}N$ for all k. Since combining $d(x_n,x_{n+1})\to 0$ (due to (24)) and $x_{n_k}\to x^{*}$ guarantees $x_{n_k+j}\to x^{*}$ for all $j\ge 1$, we deduce from (28) that $d(x_{n_k+j},T_{l+j}{x}_{n_k+j})=d(x_{n_k+j},T_{n_k+j}{x}_{n_k+j})\to 0$. Then, the demiclosedness assumption of $S_i,i=1,2,\dots ,N$ implies that $x^{*}\in \in \mathrm{Fix}\left(T_{l+j}\right)$ for all j. This ensures that $x^{*}\in {\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)$. Again, note that $J_{\lambda }^B{\exp }_I(-\lambda A_i),i=1,2,\dots ,M$ and $T_r^{\Phi }$ are all nonexpansive mappings, which are also demiclosed at zero. So, it follows from (27) and (29) that $x^{*}\in \left({\bigcap }_{i=1}^M\mathrm{Fix}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)\right)\right)\cap \mathrm{Fix}\left(T_r^{\Phi }\right)$. Since $\mathrm{Fix}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)\right)={(A_i+B)}^{-1}0$ for $i=1,2,\dots ,M$ and $\mathrm{Fix}\left(T_r^{\Phi }\right)=\mathrm{EP}\left(\Phi \right)$, we have $x^{*}\in \left({\bigcap }_{i=1}^M\mathrm{Fix}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)\right)\right)\cap \mathrm{Fix}\left(T_r^{\Phi }\right)=\left({\bigcap }_{i=1}^M{(A_i+B)}^{-1}0\right)\cap \mathrm{EP}\left(\Phi \right)$. Consequently, $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)\cap \left({\bigcap }_{i=1}^M{(A_i+B)}^{-1}0\right)=\Omega$. Since $\left\{x_n\right\}$ is Fejér monotone with respect to $\Omega$, in terms of Lemma 2, we obtain ${lim}_{n\to \infty }{x}_n=x^{*}$. The conclusion (i) of Theorem 1 is proved.

Step 5. We claim that there hold the conclusions (ii), (iii), (iv), (v), (vi), and (vii) in Theorem 1:

(1) Indeed, (a) if ${\gamma }_n=1\forall n\ge 0$, then we obtain that, for $i=1,2,\dots ,M$ and $\lambda >0$,

$$u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right)=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n);$$

(b) If ${\beta }_n=1\forall n\ge 0$ and $S_i=S,i=1,2,\dots ,N$, then we obtain

$$y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)=S{u}_n^{i_n}\mathrm{with}{i}_n\in \{1,2,\dots ,M\}\mathrm{s}.\mathrm{t}.d(u_n^{i_n},x_n)=\underset{1\le i\le M}{max}d(u_n^i,x_n).$$

Therefore, (18) reduces to the following iterative algorithm:

$$\left\{\begin{array}{cc}& u_n^i=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n),i=1,2,\dots ,M,\hfill \\ & y_n=S{u}_n^{i_n}\mathrm{with}{i}_n\in \{1,2,\dots ,M\}\mathrm{s}.\mathrm{t}.d(u_n^{i_n},x_n)={\displaystyle \underset{1\le i\le M}{max}}d(u_n^i,x_n),\hfill \\ & x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right)\forall n\ge 0,\hfill \end{array}\right.$$

which converges to $x^{*}\in \mathrm{Fix}\left(S\right)\cap \mathrm{EP}\left(\Phi \right)\cap ({\bigcap }_{i=1}^M{(A_i+B)}^{-1}0)$ as a solution of problem (2). The conclusion (ii) of Theorem 1 is proved.

(2) Indeed, (a) if ${\gamma }_n=1\forall n\ge 0$ and $A_i=B=0,i=1,2,\dots ,M$, then we obtain that, for $i=1,2,\dots ,M$ and $\lambda >0$,

$$u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right)=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n)=x_n;$$

(b) If ${\beta }_n=1\forall n\ge 0$, we obtain

$$y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)=S_n{u}_n^{i_n}=S_n{x}_n.$$

Therefore, (18) reduces to the following iterative algorithm:

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{S}_n{x}_n\right)\forall n\ge 0,$$

which converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)$. The conclusion (iii) is proved.

(3) Indeed, (a) if ${\beta }_n={\gamma }_n=1\forall n\ge 0$ and $S_i=I,i=1,2,\dots ,N$, then we obtain that, for $i=1,2,\dots ,M$ and $\lambda >0$,

$$u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right)=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n),$$

and hence

$$y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)=u_n^{i_n}=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_{i_n}{x}_n);$$

(b) If $\Phi =0$, then we have $T_r^{\Phi }=I\forall r>0$, and hence

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right)={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}{y}_n.$$

Therefore, (18) reduces to the following iterative algorithm:

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_{x_n}(-\lambda A_{i_n}{x}_n)\right)\forall n\ge 0,$$

which converges to an element $x^{*}\in {\bigcap }_{i=1}^M{(A_i+B)}^{-1}0$. The conclusion (iv) is proved.

(4) Indeed, if ${\beta }_n={\gamma }_n=1\forall n\ge 0$ and $A_i=B=0,i=1,2,\dots ,M$, then we obtain that, for $i=1,2,\dots ,M$ and $\lambda >0$,

$$u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right)=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n)=x_n,$$

and hence

$$y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)=S_n{u}_n^{i_n}=S_n{x}_n;$$

(b) If $\Phi =0$, then we have $T_r^{\Phi }=I\forall r>0$, and hence

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }{y}_n\right)={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}{y}_n.$$

Therefore, (18) reduces to the following iterative algorithm:

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(S_n{x}_n\right)\forall n\ge 0,$$

which converges to $x^{*}\in {\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)$ a common fixed point of ${\left\{S_i\right\}}_{i=1}^N$. The conclusion (v) is proved.

(5) Indeed, (a) if ${\gamma }_n=1\forall n\ge 0$, and $A_i=A,i=1,2,\dots ,M$, then we obtain that, for $i=1,2,\dots ,M$ and $\lambda >0$,

$$u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right)=J_{\lambda }^B{\exp }_{x_n}(-\lambda A{x}_n);$$

(b) If ${\beta }_n=1\forall n\ge 0$, then we obtain

$$y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)=S_n{u}_n^{i_n}=S_n(J_{\lambda }^B{\exp }_{x_n}(-\lambda A{x}_n)).$$

Therefore, (18) reduces to the following iterative algorithm:

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }\left(S_n\left(J_{\lambda }^B{\exp }_{x_n}(-\lambda A{x}_n)\right)\right)\right)\forall n\ge 0,$$

which converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)\cap {(A+B)}^{-1}0$. The conclusion (vi) is proved.

(6) Indeed, (a) if ${\gamma }_n=1\forall n\ge 0$, and $A_i=0,i=1,2,\dots ,M$, then we obtain that, for $i=1,2,\dots ,M$ and $\lambda >0$,

$$u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right)=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n)=J_{\lambda }^B{x}_n;$$

(b) If ${\beta }_n=1\forall n\ge 0$, then we obtain

$$y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)=S_n{u}_n^{i_n}=S_n\left(J_{\lambda }^B{x}_n\right).$$

Therefore, (18) reduces to the following iterative algorithm:

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }\left(S_n\left(J_{\lambda }^B{x}_n\right)\right)\right)\forall n\ge 0,$$

which converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)\cap B^{-1}0$. The conclusion (vii) is proved. □

Remark 1. 

Compared with Theorem 3.1 in Chang et al. [23], our Theorem 1 improves, extends, and develops it in the following aspects:

(i) The problem of finding an element of $\mathrm{Fix}\left(S\right)\cap \mathrm{EP}\left(\Phi \right)\cap \left({\bigcap }_{i=1}^M{(A_i+B)}^{-1}0\right)$ in [23] is extended to develop our problem of finding an element of $\left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap \mathrm{EP}\left(\Phi \right)\cap \left({\bigcap }_{i=1}^M{(A_i+B)}^{-1}0\right)$, where ${\left\{S_i\right\}}_{i=1}^N$ is a finite family of L-Lipschitzian ($L\ge 1$) demiclosed at zero and which features quasi-nonexpansive self-mappings on C.

(ii) The boundedness assumption of the nonempty closed and geodesic convex subset $C\subset \mathcal{M}$ in [23], Theorem 3.1, is dropped by our Theorem 1, and there is only the assumption of the nonempty closed and geodesic convex subset $C\subset \mathcal{M}$ in our Theorem 1.

(iii) Because $\mathcal{M}$ is diffeomorphic to an Euclidean space ${\mathbf{R}}^m$, $\mathcal{M}$ has the same topology and differential structure as ${\mathbf{R}}^m$. Moreover, Hadamard manifolds and Euclidean spaces have some similar geometrical properties. Therefore, the convergence statement of the sequence $\left\{x_n\right\}$ in our Theorem 1 is more general.

(iv) The splitting iterative algorithm of Theorem 3.1 in [23] is developed into the triple Mann-type iteration method of this paper, i.e., the iteration steps $u_n^i=J_{\lambda }^B{\exp }_{x_n}(-\lambda A_i{x}_n)$, $i=1,2,\dots ,M$, and $y_n=S{u}_n^{i_n}\mathrm{with}{i}_n\in \{1,2,\dots ,M\}\mathrm{s}.\mathrm{t}.d(u_n^{i_n},x_n)={max}_{1\le i\le M}d(u_n^i,x_n)$ in [23], Theorem 3.1, are extended to develop the ones $u_n^i={\exp }_{x_n}{\gamma }_n{\exp }_{x_n}^{-1}\left(J_{\lambda }^B{\exp }_I(-\lambda A_i)x_n\right)$, $i=1,2,\dots ,M$, and $y_n={\exp }_{x_n}{\beta }_n{\exp }_{x_n}^{-1}\left(S_n{u}_n^{i_n}\right)\mathrm{with}{i}_n\in \{1,2,\dots ,M\}\mathrm{s}.\mathrm{t}.d(u_n^{i_n},x_n)={max}_{1\le i\le M}d(u_n^i,x_n)$ in our Theorem 1, respectively.

4. Applicable Examples

Assume throughout that $\mathcal{M}$ is a Hadamard manifold of finite dimension, and the nonempty $C\subset \mathcal{M}$ is of both closedness and geodesic convexity in $\mathcal{M}$.

4.1. The Minimizing Problem with CFPP Constraint

Suppose that $f,g:C\to \mathbf{R}\cup \{+\infty \}$ are two proper functions of both lower semicontinuity and geodesic convexity. One first considers the minimizing problem of seeking an element $y^{*}\in C$ such that

$$f\left(y^{*}\right)+g\left(y^{*}\right)=\underset{y\in C}{min}\{f\left(y\right)+g\left(y\right)\},$$

(30)

where ${\Omega }_1$ denotes the set of solutions of the minimization issue (30), i.e.,

$${\Omega }_1:=\{x\in C:f\left(x\right)+g\left(x\right)=\underset{y\in C}{min}\{f\left(y\right)+g\left(y\right)\}\}.$$

(31)

This matter possesses a specific scenario for linear inverse issues, and some authors have furnished its applications in compressed sensing, image reconstruction, data retrieve and machine learning; refer to [29,30,31,32].

Next, one writes

$$\Phi (y,v):=\Theta (y,v)+g\left(v\right)-g\left(y\right)\forall y,v\in C,$$

in which $\Theta (y,v)=f\left(v\right)-f\left(y\right)$. It is easily known that function $\Phi (y,v):C\times C\to \mathbf{R}\cup \{+\infty \}$ satisfies the conditions below:

(A1) $\Phi (x,x)=0\forall x\in C$;

(A2) $\Phi (x,y)+\Phi (y,x)\le 0\forall x,y\in C$, i.e., $\Phi$ is monotone;

(A3) $x\mapsto \Phi (x,y)$ is upper semicontinuous for each $y\in C$;

(A4) $y\mapsto \Phi (x,y)$ is geodesic convex and lower semicontinuous for each $x\in C$;

Assume that ${\left\{S_i\right\}}_{i=1}^N$ is a finite family of L-Lipschitzian ($L\ge 1$) demiclosed at zero and with quasi-nonexpansive self-mappings on C. In Theorem 1, we put ${\beta }_n={\gamma }_n=1\forall n\ge 0$, and $A_i=B=0,i=1,2\dots .,M$. Then, using Theorem 1 (iii), we derive the following result.

Theorem 2. 

Assume $C,\mathcal{M},g,f,\Phi ,\Theta$ and ${\left\{S_i\right\}}_{i=1}^N$ are chosen as in the above conditions, and $T_r^{\Phi }:\mathcal{M}\to 2^C(r>0)$ is the resolvent of Φ formulated in (16), where $D\left(T_r^{\Phi }\right)$ is of both closedness and geodesic convexity. Suppose there hold the relations below:

(i) ∃ (compact set) $D\subset \mathcal{M}$ such that

$$y\in C\setminus D\Rightarrow \exists v\in C\cap D\mathrm{s}.\mathrm{t}.\Phi (y,v)<0;$$

(ii) $\left\{{\alpha }_n\right\}\subset (0,1),{\alpha }_n\to 1,{\sum }_{n=0}^{\infty }(1-{\alpha }_n)=\infty$, and $\left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap {\Omega }_1\ne \varnothing$.

Then the sequence $\left\{x_n\right\}$ defined by

$$x_{n+1}={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}\left(T_r^{\Phi }\left(S_n{x}_n\right)\right)\forall n\ge 0,$$

(32)

converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap {\Omega }_1$.

4.2. The Saddle Point Problem (SPP) with CFPP Constraint

Given Hadamard manifolds ${\mathcal{M}}_1,{\mathcal{M}}_2$, we consider geodesic convex subsets $C_1,C_2$ of ${\mathcal{M}}_1,{\mathcal{M}}_2$, respectively. The mapping $H:C_1\times C_2\to \mathbf{R}$ is termed as a saddle function iff the following hold:

(i) $v\mapsto H(y,v)$ is of geodesic convexity on $C_2$ w.r.t. $y\in C_1$;

(ii) $y\mapsto H(y,v)$ is of geodesic concavity, that is, $y\mapsto -H(y,v)$ is of geodesic convexity on $C_1$ w.r.t. $v\in C_2$.

An element ${\mathbf{z}}^{*}=(y^{*},v^{*})$ is termed as a saddle point of H if and only if

$$H(y,v^{*})\le H(y^{*},v^{*})\le H(y^{*},v)\forall \mathbf{z}=(y,v)\in C_1\times C_2,$$

(33)

where ${\Omega }_2$ denotes the saddle point set of problem (33).

Let $V_H:C_1\times C_2\to 2^{T{\mathcal{M}}_1}\times 2^{T{\mathcal{M}}_2}$ be an associated multi-valued vector field with saddle function H formulated below

$$V_H(y,v)=\partial (-H(·,v))\left(y\right)\times \partial \left(H(y,·)\right)\left(v\right)\forall (y,v)\in C_1\times C_2.$$

(34)

Let $\mathcal{M}={\mathcal{M}}_1\times {\mathcal{M}}_2$ denote a product space. Then, it is a Hadamard manifold, and the tangent space of it at $\mathbf{z}=(x,y)$ is $T_{\mathbf{z}}\mathcal{M}=T_y{\mathcal{M}}_1\times T_v{\mathcal{M}}_2$ (for more information, refer to Page 239 of [24]). The relevant metric is specified below

$${\mathcal{R}}_{\mathbf{z}}\langle \mathbf{u},{\mathbf{u}}^{\prime }\rangle ={\mathcal{R}}_y\langle x,x^{\prime }\rangle +{\mathcal{R}}_v\langle t,t^{\prime }\rangle \forall \mathbf{u}=(x,t),{\mathbf{u}}^{\prime }=(x^{\prime },t^{\prime })\in T_{\mathbf{z}}\mathcal{M}.$$

The product of two geodesics in ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ denotes a geodesic in the product manifold $\mathcal{M}$. So, for arbitrary two elements $\mathbf{z}=(y,v)$ and ${\mathbf{z}}^{\prime }=(y^{\prime },v^{\prime })$ in $\mathcal{M}$, one has

$${\exp }_{\mathbf{z}}^{-1}{\mathbf{z}}^{\prime }={\exp }_{(y,v)}^{-1}(y^{\prime },v^{\prime })=({\exp }_y^{-1}{y}^{\prime },{\exp }_v^{-1}{v}^{\prime }).$$

The vector field $V:{\mathcal{M}}_1\times {\mathcal{M}}_2\to 2^{T{\mathcal{M}}_1}\times 2^{T{\mathcal{M}}_2}$ is termed as being monotone if and only if, for each $\mathbf{z}=(y,v),{\mathbf{z}}^{\prime }=(y^{\prime },v^{\prime }),\mathbf{u}=(x,t)\in V\left(\mathbf{z}\right)$ and ${\mathbf{u}}^{\prime }=(x^{\prime },t^{\prime })\in V\left({\mathbf{z}}^{\prime }\right)$, one has

$$\langle x,{\exp }_y^{-1}{y}^{\prime }\rangle +\langle t,{\exp }_v^{-1}{v}^{\prime }\rangle \le \langle x^{\prime },-{\exp }_{y^{\prime }}^{-1}y\rangle +\langle t^{\prime },-{\exp }_{v^{\prime }}^{-1}v\rangle .$$

Lemma 6 

(see [33]). Suppose that H is a saddle function on $C=C_1\times C_2$, and $V_H$ is the multi-valued vector field formulated in (34). Then, $V_H$ is of maximal monotonicity.

It can be readily seen that an element ${\mathbf{z}}^{*}=(y^{*},v^{*})\in C=C_1\times C_2$ is a saddle point of H if and only if ${\mathbf{z}}^{*}$ is a singularity of $V_H$. Next, for $i=1,2,\dots ,N$, let the mapping $S_i:C\to C$ be L-Lipschitzian ($L\ge 1$), demiclosed at zero, and quasi-nonexpansive.

In Theorem 1, we put ${\beta }_n={\gamma }_n=1\forall n\ge 0$, $B=V_H,A_i=0,i=1,2,\dots ,M$ and $\Phi =0$. By Theorem 1 (vi), we immediately obtain the assertion below.

Theorem 3. 

Suppose that $H:C=C_1\times C_2\to (-\infty ,\infty )$ is a saddle function, and $V_H:C_1\times C_2\to 2^{T{\mathcal{M}}_1}\times 2^{T{\mathcal{M}}_2}$ is the corresponding vector field of maximal monotonicity. Assume ${\left\{S_i\right\}}_{i=1}^N$ is a finite family of L-Lipschitzian ($L\ge 1$), demiclosed at zero, and with quasi-nonexpansive self-mappings on C such that $\left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap {\Omega }_2\ne \varnothing$. Choose the initial point $x_0\in C$, and define $\left\{x_n\right\}$ as follows:

$$x_{n+1}:={\exp }_{x_n}{\alpha }_n{\exp }_{x_n}^{-1}{S}_n\left(J_{\lambda }^{V_H}{x}_n\right)\forall n,$$

for $\lambda >0$, with $\left\{{\alpha }_n\right\}\subset (0,1)$ satisfying ${\alpha }_n\to 1$ and ${\sum }_{n=0}^{\infty }(1-{\alpha }_n)=\infty$. Then, $\left\{x_n\right\}$ converges to an element $x^{*}\in \left({\bigcap }_{i=1}^N\mathrm{Fix}\left(S_i\right)\right)\cap {\Omega }_2$.

5. Conclusions

In this study, we have devised and analyzed a triple Mann iteration method for approximating a common solution of a system of quasi-variational inclusion problems, an equilibrium problem and a common fixed point problem (CFPP) of finitely many quasi-nonexpansive mappings on Hadamard manifolds. With the help of some suitable assumptions, we have shown that the iterative sequence constructed by the proposed algorithm converges to a common solution. In addition, we have furnished the applications of the major outcomes in some specific matters such as the minimization problem with CFPP constraint and saddle point problem with CFPP constraint on Hadamard manifolds. Our major outcomes are the improvement and development of those in Chang et al. [23]. Finally, it is worth mentioning that a section of our subsequent investigation is focused on presenting the new convergence outcome for the modification of our devised method with countably many quasi-nonexpansive mappings on Hadamard manifolds.