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Shrinking Cesáro means method for the split equilibrium and fixed point problems in Hilbert spaces

Abstract

We propose a modified version of the classical Cesáro means method endowed with the hybrid shrinking projection method to solve the split equilibrium and fixed point problems (SEFPP) in Hilbert spaces. One of the main reasons to equip the classical Cesáro means method with the shrinking projection method is to establish strong convergence results which are often required in infinite-dimensional functional spaces. As a consequence, the convergence analysis is carried out under mild conditions on the underlying shrinking Cesáro means method. We emphasize that the results accounted in this manuscript can be considered as an improvement and generalization of various existing exciting results in this field of study.

Introduction

Throughout the introduction, we first fix some necessary notions and concepts which will be required in the sequel. The inner product and the induced norm associated with a real Hilbert space H are denoted by \(\langle \cdot ,\cdot \rangle \) and \(\Vert \cdot \Vert :=\sqrt{ \langle \cdot ,\cdot \rangle }\), respectively. For a sequence \(\{x_{n}\}_{n=1}^{\infty }\) in H, the strong convergence characteristics (resp. weak convergence characteristics) of \(\{x_{n}\}_{n=1}^{\infty }\) is denoted as \(x_{n}\rightarrow x\) (resp. \(x_{n}\rightharpoonup x\)). For a self-mapping T over a nonempty subset C of H, the set of all fixed points of the mapping T is denoted by \(F(T)\). Recall that the self-mapping T is said to be total asymptotically nonexpansive mapping [1] if, for all \(x,y\in C\), we have

$$ \bigl\Vert T^{n}x-T^{n}y \bigr\Vert \leq \Vert x-y \Vert +\lambda _{n}\xi \bigl( \Vert x-y \Vert \bigr)+\mu _{n}\quad \text{for all }n\geq 1, $$
(1)

where \(\{ u_{n} \} \) and \(\{ v_{n} \} \) are nonnegative sequences satisfying \(\lambda _{n}\), \(\mu _{n} \overset{n\rightarrow \infty }{\longrightarrow }0\) and \(\xi :[0,\infty )\rightarrow {}[ 0,\infty )\) satisfying \(\xi (0)=0\) and \(\xi (x)<\xi (y)\) for \(x< y\).

It is remarked that the fixed points of certain nonlinear mappings can be constructed with the effective iterative algorithms by employing a suitable set of control conditions. On the other hand, computational investigation associated with the fixed points of generalized nonexpansive mappings is the only main tool to establish the consequent fixed point property in various framework of spaces. We remark that the convergence analysis of effective iterative algorithms associated with the class of total asymptotically nonexpansive mappings, as defined in (1), contributes significantly in metric fixed point theory. The introduction of the aforementioned class of mappings includes, inter alia, the generalization of nonlinear mappings as well as unification of different notions associated with the class of asymptotically nonexpansive mappings. Some useful results concerning the iterative construction of fixed points can be found in [18, 19, 22] and the references cited therein.

In 1975, Baillon [2] established the nonlinear version of a classical ergodic theorem involving a nonexpansive self-mapping T defined over a closed bounded convex subset C of a Hilbert space H. In fact, he proved that, for every \(x\in C\), the Cesáro (-arithmetical) means method

$$ \frac{x+Tx+\cdots +T^{n}x}{n+1} $$

exhibits weak convergence towards a fixed point of the mapping provided that \(F(T)\) is nonempty. This nonlinear ergodic theorem was then generalized to Banach spaces in [4, 15, 26]. Moreover, Hirano and Takahashi [16] extended the Baillon’s result to asymptotically nonexpansive mappings. Since then, the Cesáro means method has been employed for the construction of fixed points of (asymptotically) nonexpansive mappings; see, for instance, [2729] and the references cited therein. We remark that the so-called ergodic average \(\frac{1}{n+1}\sum_{i=0}^{n}T^{i}x\) converges weakly, whereas a strongly convergent iterative algorithm involving a nonlinear mapping is much more desirable than the weakly convergent iterative algorithm in infinite dimensional functional spaces. In 1967, Halpern [13] suggested the following strongly convergent iterative algorithm:

$$ P_{0}^{\alpha }x=x, \qquad P_{n+1}^{\alpha }x= \alpha _{n+1}x+ ( 1- \alpha _{n+1} ) TP_{n}^{\alpha }x, $$

where \(\alpha = \{ \alpha _{n} \} _{n=1}^{\infty }\) is a sequence in \([ 0,1 ] \). Note that for a particularly important choice for T being positively homogeneous (i.e., \(T(tx)=tTx\) for any \(t\geq 0\) and \(x\in C\)) and \(\{ \alpha _{n} \} _{n=1}^{\infty }\) to be \(\{ \frac{1}{n+1} \} _{n=1}^{\infty }\), the so-called Halpern iterative algorithm is a nonlinear generalization of the Cesáro means method such that \(P_{n}^{\alpha }x=\frac{1}{n+1}S_{n}x\), where \(S_{0}x:=x\) and \(S_{n+1}x:=x+T(S_{n}x)\). In 1968, Haugazeau [14] proposed and analyzed a hybrid projection algorithm or outer-approximation methods in Hilbert spaces. This method has been modified in different ways to ensure the strong convergence characteristics of an iterative algorithm. In 2008, Takahashi et al. [32] firstly proposed a strongly convergent hybrid algorithm, based on the shrinking effect of the half-space, for nonexpansive mappings in Hilbert spaces. We are aiming to employ a modified version of the classical Cesáro means method endowed the hybrid shrinking projection method for the construction of common fixed points of a finite family of total asymptotically nonexpansive mappings.

In 2012, Censor et al. [8] coined the concept of split variational inequality problem (SVIP) in Hilbert spaces. The concept of SVIP was then generalized to split monotone variational inclusions (SMVI) in [25]. The concept of split equilibrium problem (SEP) is considered as a special case of SMVI which aims to solve a pair of equilibrium problems in such a way that the equilibrium points of an equilibrium problem solve another equilibrium problem under the action of a given bounded linear operator. The above stated theoretical problems have been successfully implemented to real world applications, for example, medical image reconstruction, see [6, 7]. Moreover, SEP can also be employed for problems in phase retrieval, data compression, sensor networks, inverse problems, and computerized tomography; see, for example, [5, 10]. We now introduce the concept of SEP:

Let \(F:C\times C\rightarrow \mathbb{R}\) and \(G:Q\times Q\rightarrow \mathbb{R}\) be two bifunctions, where \(\emptyset \neq C\subseteq H_{1}\) and \(\emptyset \neq Q\subseteq H_{2}\), respectively. An SEP is as follows:

$$ \text{find a point }x^{\ast }\in C\quad \text{which solves }F \bigl( x^{ \ast },x \bigr) \geq 0\text{ for all }x\in C $$
(2)

and

$$ \text{find the image }y^{\ast }=Ax^{\ast }\in Q \quad \text{which solves }G \bigl( y^{\ast },y \bigr) \geq 0\text{ for all }y\in Q, $$
(3)

where A is a bounded linear operator from \(H_{1}\) onto \(H_{2}\).

The set \(\varOmega =\{z\in \operatorname{EP}(F):Az\in \operatorname{EP}(G)\}\) denotes the equilibrium points of SEP (2) and (3). Some important implications of SEP are as follows: If \(H_{1}=H_{2}\), \(C=Q\) and \(A:=I\) (the identity mapping), then inequalities (2) and (3) coincide with the classical equilibrium problem whose solution set is denoted as \(\operatorname{EP}(F)\). Moreover, if \(F ( x^{\ast },x ) = \langle f ( x^{\ast } ) ,x-x^{\ast } \rangle \) and \(G ( x^{\ast },x ) = \langle g ( x^{\ast } ) ,x-x^{\ast } \rangle \) in (2) and (3), then we have the concept of SVIP. As a consequence, the existence and approximation results for these problems can easily be derived from the ones established for SEFFP. Hence, it shows the significance and range of applicability of SEFFP. Some interesting methods have been proposed and analyzed to find a feasible solution of SEFPP associated with different classes of nonlinear mappings in the framework of Hilbert spaces [9, 12, 17, 20, 21, 23, 24, 30, 31, 33].

Inspired and motivated by the aforementioned results, we aim to establish convergence results of the shrinking Cesáro means method satisfying an appropriate set of conditions devised for the control sequences of parameters. As a consequence, the proposed algorithm strongly converges to an element in the set of solutions of SEFPP associated with a finite family of total asymptotically nonexpansive mappings. We emphasize that the results accounted in this manuscript can be considered as an improvement and generalization of various existing exciting results in this field of study.

The remainder of the manuscript is furnished in the following manner. In Sect. 2, we recall some definitions, mathematical tools, and important results in the form of lemmas required in the sequel. In Sect. 3, we establish results concerning the convergence characteristics of shrinking Cesáro means method in Hilbert spaces. Section 4 deals with the results deduced from the main results of Sect. 3.

Preliminaries

This section is devoted to recalling some useful definitions, entailing mathematical tools, and helpful results in the form of lemmas required in the sequel.

We assume that C is a nonempty closed convex subset of a Hilbert space \(H_{1}\). For each \(x\in H_{1}\), we can find a projection denoted by \(P_{C}x\), which is the unique nearest point in C such that

$$ \Vert x-P_{C}x \Vert :=\inf \bigl\{ \Vert x-y \Vert : \text{for all }y\in C \bigr\} . $$

Such a mapping \(P_{C}\) is known as the nearest point projector or the metric projection of \(H_{1}\) onto C. Now, for all \(x,y\in C\), we concluded that the metric projection \(P_{C}\):

  1. (i)

    satisfies nonexpansiveness in Hilbert spaces;

  2. (ii)

    satisfies firm nonexpansiveness in Hilbert spaces, that is,

    $$ \Vert P_{C}x-P_{C}y \Vert ^{2}\leq \langle x-y,P_{C}x-P_{C}y \rangle . $$

Moreover, if \(P_{C}x\in C\), then for all \(x\in H_{1}\) and for all \(y\in C\), we have

$$ \langle x-P_{C}x,P_{C}x-y \rangle \geq 0. $$
(4)

It is remarked that (4) is equivalent to

$$ \Vert x-y \Vert ^{2}\geq \Vert x-P_{C}x \Vert ^{2}+ \Vert y-P_{C}x \Vert ^{2}. $$
(5)

We now recall some important classes of monotone operators required in the sequel. A nonlinear mapping \(A:C\rightarrow H_{1}\) is: (i) monotone if \(\langle Ax-Ay,x-y \rangle \geq 0\) for all \(x,y\in C\) and (ii) λ-inverse strongly monotone if for \(\lambda >0\) we have that \(\langle Ax-Ay,x-y \rangle \geq \lambda \Vert Ax-Ay \Vert ^{2}\) for all \(x,y\in C\). Note that a λ-inverse strongly monotone operator satisfies monotonicity defined in (i) as well as Lipschitz continuity with the Lipschitz constant being \(( \frac{1}{\lambda } ) \). Moreover, the relation between the concepts of metric projection operator and variational inequality problem can be expressed as follows:

$$ x^{\ast }\in VI(C,f)\quad \Longleftrightarrow\quad x^{\ast }=P_{C} \bigl( x^{ \ast }-\lambda Ax^{\ast } \bigr)\quad \text{for all } \lambda >0. $$

It is remarked that the linear operator defined as \(A=I-T\) is \(( \frac{1}{2} ) \)-inverse strongly monotone provided that T satisfies nonexpansiveness.

The following lemma collects some useful equations and inequalities in the context of a real Hilbert space.

Lemma 2.1

Let\(H_{1}\)be a real Hilbert space, then:

  1. (i)

    \(\Vert x-y \Vert ^{2}= \Vert x \Vert ^{2}- \Vert y \Vert ^{2}-2 \langle x-y,y \rangle \)for all\(x,y\in H_{1}\);

  2. (ii)

    \(2 \langle y,x+y \rangle \geq \Vert x+y \Vert ^{2}- \Vert x \Vert ^{2}\)for all\(x,y\in H_{1}\);

  3. (iii)

    \(2 \langle x-y,u-v \rangle = \Vert x-v \Vert ^{2}+ \Vert y-u \Vert ^{2}- \Vert x-u \Vert ^{2}- \Vert y-v \Vert ^{2}\)for all\(x,y,u,v\in H_{1}\);

  4. (iv)

    \(\Vert \lambda x+(1-\lambda )y \Vert ^{2}+\lambda (1- \lambda ) \Vert x-y \Vert ^{2}=\lambda \Vert x \Vert ^{2}+(1-\lambda ) \Vert y \Vert ^{2}\)for all\(x,y\in H_{1}\)and\(\lambda \in \mathbb{R} \).

An important tool in metric fixed point theory is the well-known Opial condition which is defined as follows: let \(\{x_{n}\}\) be a sequence in a Hilbert space \(H_{1}\) satisfying \(x_{n}\overset{n\rightarrow \infty }{\rightharpoonup }x\), then the following inequality

$$ \liminf_{n\rightarrow \infty } \Vert x_{n}-x \Vert < \liminf _{n\rightarrow \infty } \Vert x_{n}-y \Vert $$

holds for all \(y\in H_{1}\) with \(x\neq y\).

Another important tool in metric fixed point theory is the demiclosedness principal which states that a mapping \(T:H_{1}\rightarrow H_{1}\) is demiclosed at the origin provided that a sequence \(\{x_{n}\}\) in \(H_{1}\) satisfies \(x_{n}\overset{n\rightarrow \infty }{\rightharpoonup }x\) as well as \(\Vert x_{n}-Tx_{n} \Vert \overset{n\rightarrow \infty }{\longrightarrow }0\), we have \(x=Tx\).

The class of equilibrium problems, involving a bifunction \(F:C\times C\rightarrow \mathbb{R}\), can be solved assuming the following essential conditions (cf. [3] and [11]):

  1. (A1)

    \(F(x,x)=0 \) for all \(x\in C\);

  2. (A2)

    \(F(x,y)+F(y,x)\leq 0 \) for all \(x,y\in C\) implies that F is monotone;

  3. (A3)

    for each \(x,z\in C\), the following relation

    $$ \lim_{t\rightarrow 0}F\bigl(tz+(1-t)x,y\bigr)\leq F(x,y) $$

    implies that the function \(x\mapsto F(x,y)\) is upper hemicontinuous for all \(y\in C\);

  4. (A4)

    the function \(y\mapsto F(x,y)\) is convex and lower semicontinuous for all \(x\in C\).

Lemma 2.2

([11])

LetCbe a closed convex subset of a real Hilbert space\(H_{1}\), and let\(F:C\times C\rightarrow \mathbb{R}\)be a bifunction satisfying conditions (A1)(A4). For\(r>0\)and\(x\in H_{1}\), there exists\(z\in C\)such that

$$ F(z,y)+\frac{1}{r}\langle y-z,z-x\rangle \geq 0 \quad \textit{for all }y\in C. $$

Moreover, define a mapping\(T_{r}^{F}:H_{1}\rightarrow C\)by

$$ T_{r}^{F}(x)= \biggl\{ z\in C:F(z,y)+ \frac{1}{r}\langle y-z,z-x \rangle \geq 0 \textit{ for all }y\in C \biggr\} $$

for all\(x\in H_{1}\). Then the following hold:

  1. (i)

    \(T_{r}^{F}\)is single-valued;

  2. (ii)

    \(T_{r}^{F}\)is firmly nonexpansive, i.e., for every\(x,y\in H\),

    $$ \bigl\Vert T_{r}^{F}x-T_{r}^{F}y \bigr\Vert ^{2}\leq \bigl\langle T_{r}^{F}x-T_{r}^{F}y,x-y \bigr\rangle ; $$
  3. (iii)

    \(F(T_{r}^{F})=\operatorname{EP}(F)\);

  4. (iv)

    \(\operatorname{EP}(F)\)is closed and convex.

It is remarked that if \(G:Q\times Q\rightarrow \mathbb{R}\) is a bifunction satisfying conditions (A1)–(A4), then for \(s>0\) and \(w\in H_{2}\) we can define a mapping

$$ T_{s}^{G}(w)= \biggl\{ d\in C:G(d,e)+ \frac{1}{s}\langle e-d,d-w \rangle \geq 0 \text{ for all }e\in Q \biggr\} , $$

which is nonempty, single-valued, and firmly nonexpansive. Moreover, \(\operatorname{EP}(G)\) is closed and convex, and \(F(T_{s}^{G})=\operatorname{EP}(G)\).

Main results

We now prove our main result of this section.

Theorem 3.1

LetC, Qbe two nonempty closed convex subsets of two real Hilbert spaces\(H_{1}\)and\(H_{2}\), respectively. Let\(\{ f_{i} \} _{i=1}^{N}:C\times C\rightarrow \mathbb{R}\)and\(\{ g_{i} \} _{i=1}^{N}:Q\times Q\rightarrow \mathbb{R}\)be two finite families of bifunctions satisfying conditions (A1)(A4) such that each\(g_{i} \)is upper semicontinuous for each\(i\in \{1,2,3,\ldots ,N\}\). Let\(\{ A_{i} \} _{i=1}^{N}:H_{1}\rightarrow H_{2}\)be a finite family of bounded linear operators, and let\(\{ S_{i} \} _{i=1}^{N}:C\rightarrow C\)be a finite family of uniformly continuous total asymptotically nonexpansive mappings satisfying the condition that

$$ \lim_{n\rightarrow \infty }\sup_{x\in K} \bigl\Vert S_{i}^{n+1}x-S_{i}^{n}x \bigr\Vert =0,\quad 1\leq i\leq N, $$
(6)

for any bounded subsetKofC. Assume that the solution set\(\mathbb{F}:= [ \bigcap_{i=1}^{N}F(S_{i}) ] \cap \varOmega \neq \emptyset \), where\(\varOmega = \{ z\in C:z\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\textit{ and }A_{i}z \in \bigcap_{i=1}^{N}\operatorname{EP}(g_{i})\textit{ for }1\leq i\leq N \} \). Let\(\{x_{n}\} \)be a sequence generated by

$$ \begin{aligned} &x_{1}\in C_{1}=C, \\ &u_{n}=T_{r_{n}}^{f_{n ( \operatorname {mod}N ) }} \bigl( x_{n}- \gamma A_{n ( \operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }} \bigr) A_{n ( \operatorname {mod}N ) }x_{n} \bigr) , \\ &y_{n}=\alpha _{n}u_{n}+(1-\alpha _{n})\frac{1}{N}\sum_{i=1}^{N}S_{i}^{n}u_{n}, \\ &C_{n+1}= \bigl\{ {z\in C:} \Vert {y_{n}-z} \Vert {^{2} \leq \Vert {x_{n}-z} \Vert ^{2}+ \theta _{n}} \bigr\} , \\ &x_{n+1}=P_{C_{n+1}}x_{1},\quad n\geq 1,\end{aligned} $$
(7)

where\(\theta _{n}=(1-\alpha _{n}) \{ \lambda _{n}\xi _{n}(M_{n})+ \lambda _{n}M_{n}^{\ast }D_{n}+\mu _{n} \} \)with\(D_{n}=\sup \{ \Vert x_{n}-p \Vert :p \in \mathbb{F} \} \). Let\(\{r_{n}\}\), \(\{s_{n}\}\)be two positive real sequences, and let\(\{\alpha _{n}\}\)be in\((0,1)\)such that\(\alpha _{n}\leq a\). Assume that if the following set of conditions holds:

  1. (C1)

    \(\gamma \in ( 0,\frac{1}{L} ) \)where\(L=\max \{ L_{1},L_{2},\ldots ,L_{N} \} \)and\(L_{i}\)is the spectral radius of the operator\(A_{i}^{\ast }A_{i}\)where\(A_{i}^{\ast }\)is the adjoint of\(A_{i}\)for each\(i\in \{1,2,3,\ldots ,N\}\);

  2. (C2)

    \(\liminf_{n\rightarrow \infty }r_{n}>0\)and\(\liminf_{n\rightarrow \infty }s_{n}>0\);

  3. (C3)

    \(\sum_{n=1}^{\infty }\lambda _{n}<\infty \)and\(\sum_{n=1}^{\infty }\mu _{n}<\infty \);

  4. (C4)

    there exist constants\(M_{i}, M_{i}^{\ast }>0\)such that\(\xi _{i} ( \lambda _{i} ) \leq M_{i}^{ \ast }\lambda _{i}\)for all\(\lambda _{i}\geq M_{i}\), \(i=1,2,3,\ldots ,N\), then the sequence\(\{x_{n}\}\)generated by (7) converges strongly to\(P_{\mathbb{F}}x_{1}\).

Proof

For the sake of simplicity, we divide the proof into five steps.

Step 1. The sequence \(\{x_{n}\}\) is well defined.

Proof of Step 1. We first show by mathematical induction that \(\mathbb{F}\subset C_{n}\) for all \(n\geq 1\). It is obvious from the assumption that \(\mathbb{F}\subset C_{1}=C\). Let \(\mathbb{F}\subset C_{k}\) for some \(k\geq 1\). We show that \(\mathbb{F}\subset C_{k+1}\) for some \(k\geq 1\). It follows from (7) that

$$\begin{aligned} \Vert u_{k}-p \Vert ^{2}={}& \bigl\Vert T_{r_{k}}^{f_{k (\operatorname {mod}N ) }} \bigl( x_{k}-\gamma A_{k (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }} \bigr) A_{k (\operatorname {mod}N ) }x_{k} \bigr) -T_{r_{k}}^{f_{k (\operatorname {mod}N ) }}p \bigr\Vert ^{2} \\ \leq{}& \bigl\Vert x_{k}-\gamma A_{k (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }} \bigr) A_{k ( \operatorname {mod}N ) }x_{k}-p \bigr\Vert ^{2} \\ \leq{} & \Vert x_{k}-p \Vert ^{2}+\gamma ^{2} \bigl\Vert A_{k ( \operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k ( \operatorname {mod}N ) }} \bigr) A_{k (\operatorname {mod}N ) }x_{k} \bigr\Vert ^{2} \\ &{}+2\gamma \bigl\langle p-x_{k},A_{k (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }} \bigr) A_{k ( \operatorname {mod}N ) }x_{k} \bigr\rangle \\ \leq{}& \Vert x_{k}-p \Vert ^{2} +\gamma ^{2} \bigl\langle A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k}, \\ &{}A_{k (\operatorname {mod}N ) }A_{k (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }} \bigr) A_{k (\operatorname {mod}N ) }x_{k} \bigr\rangle \\ &{}+2\gamma \bigl\langle p-x_{k},A_{k (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }} \bigr) A_{k ( \operatorname {mod}N ) }x_{k} \bigr\rangle \\ \leq{}& \Vert x_{k}-p \Vert ^{2} \\ &{}+L\gamma ^{2} \bigl\langle A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k},A_{k ( \operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k ( \operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k} \bigr\rangle \\ &{}+2\gamma \bigl\langle p-x_{k},A_{k (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }} \bigr) A_{k ( \operatorname {mod}N ) }x_{k} \bigr\rangle \\ ={}& \Vert x_{k}-p \Vert ^{2}+L\gamma ^{2} \bigl\Vert A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k ( \operatorname {mod}N ) }x_{k} \bigr\Vert ^{2} \\ &{}+2\gamma \bigl\langle p-x_{k},A_{k (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }} \bigr) A_{k (\operatorname {mod}N ) }x_{k} \bigr\rangle . \end{aligned}$$
(8)

Denote \(\varLambda =2\gamma \langle p-x_{k},A_{k ( \operatorname {mod}N ) }^{ \ast } ( I-T_{s_{k}}^{g_{k ( \operatorname {mod}N ) }} ) A_{k ( \operatorname {mod}N ) }x_{k} \rangle \), we have

$$\begin{aligned} \varLambda ={}&2\gamma \bigl\langle p-x_{k},A_{k (\operatorname {mod}N ) }^{ \ast } \bigl( I-T_{s_{k}}^{g_{k (\operatorname {mod}N )}} \bigr) A_{k (\operatorname {mod}N ) }x_{k} \bigr\rangle \\ ={}&2\gamma \bigl\langle A_{k (\operatorname {mod}N ) } (p-x_{k} ) ,A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k ( \operatorname {mod}N ) }}A_{k (\operatorname {mod}N )}x_{k} \bigr\rangle \\ ={}&2\gamma \bigl\langle A_{k (\operatorname {mod}N ) } ( p-x_{k} ) + \bigl( A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k} \bigr) \\ & {}- \bigl( A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k} \bigr),A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k} \bigr\rangle \\ ={}&2\gamma \bigl\{ \bigl\langle A_{k (\operatorname {mod}N ) }p-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k},A_{k ( \operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k} \bigr\rangle \\ & {}- \bigl\Vert A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k ( \operatorname {mod}N ) }x_{k} \bigr\Vert ^{2} \bigr\} \\ \leq{}& 2\gamma \biggl\{ \frac{1}{2} \bigl\Vert A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k ( \operatorname {mod}N ) }}A_{k ( \operatorname {mod}N ) }x_{k} \bigr\Vert ^{2} \\ &{}- \bigl\Vert A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k (\operatorname {mod}N ) }}A_{k ( \operatorname {mod}N ) }x_{k} \bigr\Vert ^{2} \biggr\} \\ ={}&-\gamma \bigl\Vert A_{k (\operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k ( \operatorname {mod}N ) }}A_{k (\operatorname {mod}N ) }x_{k} \bigr\Vert ^{2}. \end{aligned}$$

Substituting the above simplified value of Λ in (8), we have

$$ \Vert u_{k}-p \Vert ^{2}\leq \Vert x_{k}-p \Vert ^{2}+\gamma ( L\gamma -1 ) \bigl\Vert A_{k ( \operatorname {mod}N ) }x_{k}-T_{s_{k}}^{g_{k ( \operatorname {mod}N ) }}A_{k ( \operatorname {mod}N ) }x_{k} \bigr\Vert ^{2}. $$
(9)

From the definition of γ and condition (C1), we obtain

$$ \Vert u_{k}-p \Vert ^{2}\leq \Vert x_{k}-p \Vert ^{2}. $$
(10)

Let \(S^{k}=\frac{1}{N}\sum_{i=1}^{N}S_{i}^{k}\), it then follows that

$$\begin{aligned} \bigl\Vert S^{k}x-S^{k}y \bigr\Vert =& \Biggl\Vert \frac{1}{N}\sum_{i=1}^{N}S_{i}^{k}x- \frac{1}{N}\sum_{i=1}^{N}S_{i}^{k}y \Biggr\Vert \\ \leq &\frac{1}{N}\sum_{i=1}^{N} \bigl( \Vert x-y \Vert +\lambda _{k}\xi _{k}\bigl( \Vert x-y \Vert \bigr)+\mu _{k}\bigr) \\ \leq & \Vert x-y \Vert +\lambda _{k}\xi _{k} \bigl( \Vert x-y \Vert \bigr)+\mu _{k}\quad \text{for all }x,y \in C. \end{aligned}$$
(11)

Now, for any \(p\in \mathbb{F}\), we have \(S^{k}p=\frac{1}{N}\sum_{i=1}^{N}S_{i}^{k}p=p\). It follows from (8) and (11) that

$$\begin{aligned} \Vert y_{k}-p \Vert =& \bigl\Vert \alpha _{k}u_{k}+ ( 1-\alpha _{k} ) S^{k}u_{k}-p \bigr\Vert \\ =&\alpha _{k} \Vert u_{k}-p \Vert + ( 1-\alpha _{k} ) \bigl\Vert S^{k}u_{k}-p \bigr\Vert \\ \leq &\alpha _{k} \Vert u_{k}-p \Vert + ( 1- \alpha _{k} ) \bigl\{ \Vert u_{k}-p \Vert +\lambda _{k}\xi _{k}\bigl( \Vert u_{k}-p \Vert \bigr)+\mu _{k} \bigr\} \\ \leq & \Vert u_{k}-p \Vert + ( 1-\alpha _{k} ) \bigl\{ \lambda _{k}\xi _{k} ( M_{k} ) + \lambda _{k}M_{k}^{ \ast } \Vert u_{k}-p \Vert )+\mu _{k} \bigr\} \\ \leq & \Vert x_{k}-p \Vert + ( 1-\alpha _{k} ) \bigl\{ \lambda _{k}\xi _{k} ( M_{k} ) + \lambda _{k}M_{k}^{ \ast } \Vert x_{k}-p \Vert ^{2}+\mu _{k} \bigr\} \\ \leq & \Vert x_{k}-p \Vert +\theta _{k}, \end{aligned}$$
(12)

where \(\theta _{k}=(1-\alpha _{k})[(\lambda _{k}\xi _{k} ( M_{k} ) +\lambda _{k}D_{k}M_{k}^{\ast }+\mu _{k})]\) with \(D_{k}=\sup \{ \Vert x_{k}-p \Vert :p\in \mathbb{F} \} \). Estimate (12) implies that \(p\in C_{k+1}\) and hence \(\mathbb{F}\subset C_{n}\) for all \(n\geq 1\). Since

$$ \bigl\{ {z\in C:} \Vert {y_{n}-z} \Vert {^{2}\leq \Vert {x_{n}-z} \Vert ^{2}+\theta _{n}} \bigr\} = \bigl\{ {z\in C:} \Vert {y_{n}} \Vert {^{2}- \Vert {x_{n}} \Vert {^{2}} \leq 2} \langle {y_{n}}-x_{n},z \rangle +{ \theta _{n}} \bigr\} , $$

it is closed and convex; therefore the sequence \(\{ x_{n} \} \) is well defined.

Step 2. The sequence \(\{ \Vert x_{n}-x_{1}\Vert \} \) is Cauchy.

Proof of Step 2. Note that \(x_{n}=P_{C_{n}}x_{1}\), therefore we have

$$ 0\leq \bigl\langle x_{n}-x_{1},x^{\ast }-x_{n} \bigr\rangle \quad \text{for each }x^{\ast }\in C_{n}. $$

In particular,

$$ 0\leq \langle x_{n}-x_{1},p-x_{n} \rangle \quad \text{for each }p\in \mathbb{F}. $$

This further implies that

$$\begin{aligned} 0 \leq & \langle x_{n}-x_{1},p-x_{n} \rangle \\ =& \langle x_{n}-x_{1},p+x_{1}-x_{1}-x_{n} \rangle \\ =& \langle x_{n}-x_{1},x_{1}-x_{n} \rangle + \langle x_{n}-x_{1},p-x_{1} \rangle \\ =&- \Vert x_{n}-x_{1} \Vert ^{2}+ \Vert x_{n}-x_{1} \Vert \Vert p-x_{1} \Vert . \end{aligned}$$

That is,

$$ \Vert x_{n}-x_{1} \Vert \leq \Vert p-x_{1} \Vert \quad \text{for all }p\in \mathbb{F}\text{ and }n \geq 1. $$

Moreover, from \(x_{n}=P_{C_{n}}x_{1}\) and \(x_{n+1}=P_{C_{n+1}}x_{1}\in C_{n+1}\subset C_{n}\), we have

$$ 0\leq \langle x_{n}-x_{1},x_{n+1}-x_{n} \rangle $$

and

$$\begin{aligned} 0 \leq & \langle x_{n}-x_{1},x_{n+1}-x_{n} \rangle \\ =& \langle x_{n}-x_{1},x_{n+1}+x_{1}-x_{1}-x_{n} \rangle \\ =& \langle x_{n}-x_{1},x_{1}-x_{n} \rangle + \langle x_{n}-x_{1},x_{n+1}-x_{1} \rangle \\ =&- \Vert x_{n}-x_{1} \Vert ^{2}+ \Vert x_{n}-x_{1} \Vert \Vert x_{n+1}-x_{1} \Vert . \end{aligned}$$

This implies that

$$ \Vert x_{n}-x_{1} \Vert \leq \Vert x_{n+1}-x_{1} \Vert \quad \text{for all }n\geq 1. $$

Hence, the sequence \(\{ \Vert x_{n}-x_{1}\Vert \} \) is bounded and nondecreasing; therefore we have

$$ \lim_{n\rightarrow \infty } \Vert x_{n}-x_{1} \Vert \quad \text{exists}. $$
(13)

Note that

$$\begin{aligned} \Vert x_{n+1}-x_{n} \Vert ^{2} =& \Vert x_{n+1}-x_{1}+x_{1}-x_{n} \Vert ^{2} \\ =& \Vert x_{n+1}-x_{1} \Vert ^{2}+ \Vert x_{n}-x_{1} \Vert ^{2}-2 \langle x_{n}-x_{1},x_{n+1}-x_{1} \rangle \\ =& \Vert x_{n+1}-x_{1} \Vert ^{2}+ \Vert x_{n}-x_{1} \Vert ^{2}-2 \langle x_{n}-x_{1},x_{n+1}-x_{n}+x_{n}-x_{1} \rangle \\ =& \Vert x_{n+1}-x_{1} \Vert ^{2}- \Vert x_{n}-x_{1} \Vert ^{2}-2 \langle x_{n}-x_{1},x_{n+1}-x_{n} \rangle \\ \leq & \Vert x_{n+1}-x_{1} \Vert ^{2}- \Vert x_{n}-x_{1} \Vert ^{2}. \end{aligned}$$

It now follows from estimate (13) that

$$ \lim_{n\rightarrow \infty } \Vert x_{n+1}-x_{n} \Vert =0. $$
(14)

Step 3. Show that:

  1. (i)

    \(\lim_{n\rightarrow \infty } \Vert y_{n}-x_{n+1} \Vert = \lim_{n\rightarrow \infty } \Vert y_{n}-x_{n} \Vert =0\),

  2. (ii)

    \(\lim_{n\rightarrow \infty } \Vert u_{n}-x_{n} \Vert =0\),

  3. (iii)

    \(\lim_{n\rightarrow \infty } \Vert y_{n}-u_{n} \Vert =0\),

  4. (iv)

    \(\lim_{n\rightarrow \infty } \Vert S_{n}u_{n}-u_{n} \Vert =0\).

Proof of Step 3. By \(x_{n+1}\in C_{n+1}\), we have \(\Vert y_{n}-x_{n+1} \Vert \leq \Vert x_{n}-x_{n+1} \Vert +\theta _{n}\). Using (14), we have

$$ \lim_{n\rightarrow \infty } \Vert y_{n}-x_{n+1} \Vert =0 \quad \text{for all }n\geq 1. $$
(15)

Since \(\Vert y_{n}-x_{n} \Vert \leq \Vert y_{n}-x_{n+1} \Vert + \Vert x_{n+1}-x_{n} \Vert \), therefore using (14)–(15) we obtain

$$ \lim_{n\rightarrow \infty } \Vert y_{n}-x_{n} \Vert =0 \quad \text{for all }n\geq 1 $$
(16)

as \(n\rightarrow \infty \).

Altogether, we deduce from (7), (9), and (12) that

$$\begin{aligned}& \gamma ( 1-\gamma L ) \bigl\Vert A_{n ( \operatorname {mod}N ) }x_{n}-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }}A_{n ( \operatorname {mod}N ) }x_{n} \bigr\Vert ^{2} \\& \quad \leq \Vert x_{n}-p \Vert ^{2}- \Vert u_{n}-p \Vert ^{2} \\& \quad \leq \Vert x_{n}-p \Vert ^{2}- \Vert y_{n}-p \Vert ^{2}+\theta _{n} \\& \quad \leq \bigl( \Vert x_{n}-p \Vert + \Vert y_{n}-p \Vert \bigr) \Vert x_{n}-y_{n} \Vert +\theta _{n}. \end{aligned}$$

From \(\gamma (1-Ly)>0\) and (16), we obtain

$$ \lim_{n\rightarrow \infty } \bigl\Vert A_{n ( \operatorname {mod}N ) }x_{n}-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }}A_{n ( \operatorname {mod}N ) }x_{n} \bigr\Vert ^{2}=0\quad \text{for all }n\geq 1. $$
(17)

Next, we show that \(\Vert u_{n}-x_{n}\Vert \rightarrow 0\) as \(n\rightarrow \infty \). Since \(p\in \mathbb{F}\), we have

$$\begin{aligned} \Vert u_{n}-p \Vert ^{2} =& \bigl\Vert T_{r_{n}}^{f_{n (\operatorname {mod}N ) }} \bigl( x_{n}-\gamma A_{n (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }} \bigr) A_{n ( \operatorname {mod}N ) }x_{n} \bigr) -T_{r_{n}}^{f_{n (\operatorname {mod}N ) }}p \bigr\Vert ^{2} \\ \leq & \bigl\langle u_{n}-p,x_{n}-\gamma A_{n (\operatorname {mod}N ) }^{ \ast } \bigl( I-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n}-p \bigr\rangle \\ =&\frac{1}{2}\bigl\{ \Vert u_{n}-p \Vert ^{2}+ \bigl\Vert x_{n}- \gamma A_{n (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n}-p \bigr\Vert ^{2} \\ &{}- \bigl\Vert u_{n}-x_{n}+\gamma A_{n (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }} \bigr) A_{n ( \operatorname {mod}N ) }x_{n} \bigr\Vert ^{2}\bigr\} \\ \leq &\frac{1}{2} \bigl\{ \Vert u_{n}-p \Vert ^{2}+ \Vert x_{n}-p \Vert ^{2} \\ &{}- \bigl\Vert u_{n}-x_{n}+\gamma A_{n (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n} \bigr\Vert ^{2} \bigr\} \\ =&\frac{1}{2}\bigl\{ \Vert u_{n}-p \Vert ^{2}+ \Vert x_{n}-p \Vert ^{2} \\ &-\bigl( \Vert u_{n}-x_{n} \Vert ^{2}+\gamma ^{2} \bigl\Vert A_{n (\operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n} \bigr\Vert ^{2} \\ &{}-2\gamma \bigl\langle u_{n}-x_{n},A_{n (\operatorname {mod}N ) }^{ \ast } \bigl( I-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n} \bigr\rangle \bigr)\bigr\} \\ \leq & \Vert x_{n}-p \Vert ^{2}- \Vert u_{n}-x_{n} \Vert ^{2} \\ &{}+2\gamma \Vert u_{n}-x_{n} \Vert \bigl\Vert A_{n (\operatorname {mod}N ) }x_{n}-T_{s_{n}}^{g_{n (\operatorname {mod}N ) }}A_{n ( \operatorname {mod}N ) }x_{n} \bigr\Vert . \end{aligned}$$
(18)

Consider the following variant of (12) together with (10):

$$ \Vert y_{n}-p \Vert ^{2}\leq \alpha _{n} \Vert x_{n}-p \Vert ^{2}+ ( 1-\alpha _{n} ) \Vert u_{n}-p \Vert ^{2}+\theta _{n}. $$
(19)

Altogether, it follows from (18) and (19) that

$$\begin{aligned} ( 1-\alpha _{n} ) \Vert u_{n}-x_{n} \Vert ^{2} \leq & \bigl( \Vert x_{n}-p \Vert + \Vert y_{n}-p \Vert \bigr) \Vert x_{n}-y_{n} \Vert \\ &{}+2\gamma \Vert u_{n}-x_{n} \Vert \bigl\Vert A_{n ( \operatorname {mod}N ) }x_{n}-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }}A_{n ( \operatorname {mod}N ) }x_{n} \bigr\Vert +\theta _{n}. \end{aligned}$$

From (16) and (17), the above estimate implies that

$$ \lim_{n\rightarrow \infty } \Vert u_{n}-x_{n} \Vert =0 \quad \text{for all }n\geq 1. $$
(20)

Utilizing (16) and (20), we get

$$\begin{aligned} \Vert y_{n}-u_{n} \Vert \leq & \Vert y_{n}-x_{n} \Vert + \Vert x_{n}-u_{n} \Vert \\ \rightarrow &0\quad \text{as }n\rightarrow \infty . \end{aligned}$$
(21)

Since \(\Vert y_{n}-u_{n} \Vert = ( 1-\alpha _{n} ) \Vert S^{n}u_{n}-u_{n} \Vert \) and \(\alpha _{n}\leq a<1\), then from (21) it follows that

$$ \lim_{n\rightarrow \infty } \bigl\Vert S^{n}u_{n}-u_{n} \bigr\Vert =0 \quad \text{for all }n\geq 1. $$
(22)

From (20) and (22), we obtain

$$\begin{aligned} \bigl\Vert S^{n}u_{n}-x_{n} \bigr\Vert \leq & \bigl\Vert S^{n}u_{n}-u_{n} \bigr\Vert + \Vert x_{n}-u_{n} \Vert \\ \rightarrow &0\quad \text{as }n\rightarrow \infty . \end{aligned}$$
(23)

Reasoning as above, we obtain

$$ \lim_{n\rightarrow \infty } \bigl\Vert S^{n}u_{n}-y_{n} \bigr\Vert =0 \quad \text{for all }n\geq 1. $$
(24)

It is evident from (20) and (23) that the following estimate implies that

$$\begin{aligned} \bigl\Vert S^{n}x_{n}-x_{n} \bigr\Vert \leq & \bigl\Vert S^{n}x_{n}-S^{n}u_{n} \bigr\Vert + \bigl\Vert S^{n}u_{n}-x_{n} \bigr\Vert \\ \leq &\bigl(1+{\lambda _{n}} {M_{n}^{\ast }} \bigr) \Vert x_{n}-u_{n} \Vert +\lambda _{n}\xi _{n}({M_{n}})+\mu _{n}+ \bigl\Vert S^{n}u_{n}-x_{n} \bigr\Vert \\ \rightarrow &0\quad \text{as }n\rightarrow \infty . \end{aligned}$$
(25)

Note that \(\frac{1}{N} \Vert S_{i}^{n}u_{n}-u_{n} \Vert ^{2}\leq \frac{1}{N}\sum_{i=1}^{N} \Vert S_{i}^{n}u_{n}-u_{n} \Vert ^{2}\), therefore using (22) we have

$$ \lim_{n\rightarrow \infty } \bigl\Vert S_{i}^{n}u_{n}-u_{n} \bigr\Vert =0\quad \text{for each }i=1,2,\ldots ,N. $$

Similarly, we also have that

$$ \lim_{n\rightarrow \infty } \bigl\Vert S_{i}^{n}x_{n}-x_{n} \bigr\Vert =0\quad \text{for each }i=1,2,\ldots ,N. $$
(26)

Moreover, utilizing the uniform continuity of \(S_{i}\) and (26), we get

$$\begin{aligned} \Vert x_{n}-S_{i}x_{n} \Vert \leq & \bigl\Vert x_{n}-S_{i}^{n}x_{n} \bigr\Vert + \bigl\Vert S_{i}^{n}x_{n}-S_{i}x_{n} \bigr\Vert \\ \rightarrow &0\quad \text{as }n\rightarrow \infty \text{ for each }i=1,2, \ldots ,N. \end{aligned}$$

Similarly, we also have that

$$ \lim_{n\rightarrow \infty } \Vert u_{n}-S_{i}u_{n} \Vert =0 \quad \text{for each }i=1,2,\ldots ,N. $$

Now, we show that \(\omega (x_{n})\subset \mathbb{F}\), where \(\omega (x_{n})\) is the set of all weak ω-limits of \(\{x_{n}\}\). Since \(\{x_{n}\}\) is bounded, therefore \(\omega (x_{n})\neq \emptyset \). Let \(q\in \omega (x_{n}) \), then there exists a subsequence \(\{x_{Nn+i}\}\) of \(\{x_{n}\}\) such that \(x_{Nn+i}\rightharpoonup q\). Using the fact that \(S_{Nn+i}=S_{i}\) for all \(n\geq 1\) and the demiclosedness principle for each \(S_{i}\), we have that \(x\in F(S_{i}) \) for each \(1\leq i\leq N\). Next, we show that \(q\in \varOmega \), i.e., \(q\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\) and \(A_{i}q\in \operatorname{EP}(g_{i})\) for each \(1\leq i\leq N\). In order to show that \(q\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\), that is, \(q\in \operatorname{EP}(f_{i})\) for each \(1\leq i\leq N\), we define a subsequence \(\{ n_{j} \} \) of index \(\{ n \} \) such that \(n_{j}=Nj+i\) for all \(n\geq 1\). As a consequence, we can write \(f_{n_{j}}=f_{i}\) for \(1\leq i\leq N\).

From \(u_{n_{j}}=T_{r_{n_{j}}}^{f_{i ( \operatorname {mod}N ) }} ( I- \gamma A_{n_{j ( \operatorname {mod}N ) }}^{\ast } ( I-T_{s_{n_{j}}}^{g_{n_{j} ( \operatorname {mod}N ) }} ) A_{n_{j ( \operatorname {mod}N ) }} ) x_{n_{j}}\), for all \(n\geq 1\), we have

$$\begin{aligned} &f_{i (\operatorname {mod}N ) }(u_{n_{j}},y) \\ &\quad {}+\frac{1}{r_{n_{j}}} \bigl\langle y-u_{n_{j}},u_{n_{j}}-x_{n_{j}}- \gamma A_{n_{j (\operatorname {mod}N ) }}^{\ast } \bigl( I-T_{s_{n_{j}}}^{g_{n_{j (\operatorname {mod}N ) }}} \bigr) A_{n_{j (\operatorname {mod}N ) }}x_{n_{j}} \bigr\rangle \geq 0 \end{aligned}$$

for all \(y\in C\).

This implies that

$$\begin{aligned} &f_{i (\operatorname {mod}N ) }(u_{n_{j}},y)+\frac{1}{r_{n_{j}}} \langle y-u_{n_{j}},u_{n_{j}}-x_{n_{j}} \rangle \\ & \quad {}-\frac{1}{r_{n_{j}}} \bigl\langle y-u_{n_{j}}, \gamma A_{n_{j (\operatorname {mod}N ) }}^{ \ast } \bigl( I-T_{s_{n_{j}}}^{g_{n_{j (\operatorname {mod}N ) }}} \bigr) A_{n_{j (\operatorname {mod}N ) }}x_{n_{j}} \bigr\rangle \geq 0. \end{aligned}$$

From condition (A2), we have

$$\begin{aligned} &\frac{1}{r_{n_{j}}} \langle y-u_{n_{j}},u_{n_{j}}-x_{n_{j}} \rangle \\ &\quad {}-\frac{1}{r_{n_{j}}} \bigl\langle y-u_{n_{j}},\gamma A_{n_{j ( \operatorname {mod}N ) }}^{\ast } \bigl( I-T_{s_{n_{j}}}^{g_{n_{j (\operatorname {mod}N ) }}} \bigr) A_{n_{j (\operatorname {mod}N ) }}x_{n_{j}} \bigr\rangle \geq f_{i (\operatorname {mod}N ) }(y,u_{n_{j}}) \end{aligned}$$

for all \(y\in C\). Since \(\liminf_{j\rightarrow \infty }r_{n_{i}}>0\) (by (C2)), it follows from (17) and (20) that

$$ f_{i (\operatorname {mod}N ) }(y,q)\leq 0 \quad \text{for all }y\in C \text{ and for }1 \leq i\leq N. $$

Let \(y_{t}=ty+(1-t)q\) for some \(0< t<1\) and \(y\in C\). Since \(q\in C\), this implies that \(y_{t}\in C\). Using conditions (A1) and (A4), the following estimate

$$ 0=f_{i (\operatorname {mod}N ) }(y_{t},y_{t})\leq tf_{i (\operatorname {mod}N ) }(y_{t},y)+(1-t)f_{i (\operatorname {mod}N ) }(y_{t},q) \leq tf_{i (\operatorname {mod}N ) }(y_{t},y) $$

implies that

$$ f_{i (\operatorname {mod}N ) }(y_{t},y)\geq 0\quad \text{for }1\leq i \leq N. $$

Letting \(t\rightarrow 0\), we have \(f_{i (\operatorname {mod}N ) }(q,y)\geq 0\) for all \(y\in C\). Thus, \(q\in \operatorname{EP}(f_{i})\) for \(1\leq i\leq N\). That is, \(q\in \bigcap_{i=1}^{N}\operatorname{EP}(F_{i})\). Reasoning as above, we show that \(A_{i ( \operatorname {mod}N ) }q\in \operatorname{EP}(g_{i})\) for each \(1\leq i\leq N\). Since \(x_{n_{l}}\longrightarrow q\) and \(A_{n_{l} (\operatorname {mod}N ) }\) is a bounded linear operator, therefore \(A_{n_{l (\operatorname {mod}N ) }}x_{n_{l}}\longrightarrow A_{n_{l (\operatorname {mod}N ) }}q\). Hence, it follows from (17) that

$$ T_{s_{n_{l}}}^{g_{n_{l} (\operatorname {mod}N ) }}A_{n_{l (\operatorname {mod}N ) }}x_{n_{l}} \longrightarrow A_{n_{l (\operatorname {mod}N ) }}q \quad \text{as }l\rightarrow \infty . $$

Now, from Lemma 2.2, we have

$$\begin{aligned} &g_{i (\operatorname {mod}N ) } \bigl( T_{s_{n_{l}}}^{g_{n_{l ( \operatorname {mod}N ) }}}A_{n_{l} (\operatorname {mod}N ) }x_{n_{l}},z \bigr) \\ &\quad {}+\frac{1}{s_{n_{l}}} \bigl\langle z-T_{s_{n_{l}}}^{g_{n_{l (\operatorname {mod}N ) }}}A_{n_{l} ( \operatorname {mod}N ) }x_{n_{l}},T_{s_{n_{l}}}^{g_{n_{l ( \operatorname {mod}N ) }}}A_{n_{l (\operatorname {mod}N ) }}x_{n_{l}}-A_{n_{l (\operatorname {mod}N ) }}x_{n_{l}} \bigr\rangle \geq 0 \end{aligned}$$

for all \(z\in Q\).

Since \(g_{i}\) is upper hemicontinuous in the first argument for each \(1\leq i\leq N\), taking lim sup on both sides of the above estimate as \(l\rightarrow \infty \) and utilizing (C2) and (17), we get

$$ g_{i ( \operatorname {mod}N ) } ( A_{n_{l} ( \operatorname {mod}N ) }x,z ) \geq 0 \quad \text{for all }z\in Q\text{ and for each }1\leq i \leq N. $$

Hence \(A_{i (\operatorname {mod}N ) }q\in \operatorname{EP}(g_{i})\) for each \(1\leq i\leq N\) and consequently \(q\in \mathbb{F}\). It remains to show that \(x_{n}\rightarrow q=P_{\mathbb{F}}x_{1}\). Let \(x=P_{\mathbb{F}}x_{1}\), then from \(\Vert x_{n}-x_{1} \Vert \leq \Vert x-x_{1} \Vert \) we have

$$\begin{aligned} \Vert x-x_{1} \Vert \leq & \Vert q-x_{1} \Vert \\ \leq &\liminf_{j\rightarrow \infty } \Vert x_{n_{j}}-x_{1} \Vert \\ \leq &\limsup_{j\rightarrow \infty } \Vert x_{n_{j}}-x_{1} \Vert \\ \leq & \Vert x-x_{1} \Vert . \end{aligned}$$

This implies that

$$ \lim_{j\rightarrow \infty } \Vert x_{n_{j}}-x_{1} \Vert = \Vert q-x_{1} \Vert . $$

Hence \(x_{n_{j}}\rightarrow q=P_{\mathbb{F}}x_{1}\). From the arbitrariness of the subsequence \(\{ x_{n_{j}} \} \) of \(\{ x_{n} \} \), we conclude that \(x_{n}\rightarrow x\) as \(n\rightarrow \infty \). It is easy to see that \(y_{n,i}\rightarrow x\) and \(u_{n,i}\rightarrow x\). This completes the proof. □

We now give an example to justify the main result of this section.

Example 3.2

Let \(H_{1}=H_{2}=\mathbb{R}\), \(C=Q=[0,10]\). Let \(S_{i}:C\rightarrow C\) be defined by \(S_{i}x=\frac{x}{i+1}\) for each \(i=1,2,\ldots ,N\) with strictly increasing function \(\xi :[0,\infty )\rightarrow {}[ 0,\infty )\) satisfying \(\xi (0)=0\) and \(\lambda _{n}=\mu _{n}=\frac{1}{n^{2}} \overset{n\rightarrow \infty }{\longrightarrow }0\). Then observe that, for each fixed \(i=1,2,\ldots ,N\),

$$\begin{aligned}& \bigl\Vert S_{i}^{n}x-S_{i}^{n}y \bigr\Vert - \Vert x-y \Vert -\lambda _{n}\xi _{n} \bigl( \Vert x-y \Vert \bigr)-\mu _{n} \\& \quad \leq \frac{1}{ ( i+1 ) ^{n}} \Vert x-y \Vert - \Vert x-y \Vert - \lambda _{n}\xi _{n}\bigl( \Vert x-y \Vert \bigr)-\mu _{n} \\& \quad \leq \Vert x-y \Vert - \Vert x-y \Vert - \lambda _{n} \xi _{n}\bigl( \Vert x-y \Vert \bigr)-\mu _{n}\leq 0. \end{aligned}$$

This shows that, for each fixed \(i=1,2,\ldots ,N\), the mapping \(S_{i}\) is total asymptotically nonexpansive with \(\bigcap_{i=1}^{N}F(S_{i})= \{ 0 \} \). Let \(A:\mathbb{R}\rightarrow \mathbb{R}\) be defined by \(Ax=x\) for all \(x\in H_{1}=\mathbb{R}\). This implies that \(A^{\ast }y=y\) for all \(y\in H_{2}=\mathbb{R}\). The two bifunctions f and g are defined by \(f_{i}(u,v)=f(u,v)=2u(v-u)\) for all \(u,v\in C\) and \(g_{i}(x,y)=g(x,y)=x(y-x)\) for all \(x,y\in Q\), respectively. It is easy to check that f and g satisfy all the conditions in Theorem 3.1 (Main Result) with \(\varOmega = \{ 0 \} \), and hence \(\mathbb{F}= \{ 0 \} \). Set \(\beta _{n}=r_{n}=\frac{n}{100n+1}\) and \(\gamma =\frac{1}{100}\). For each \(r>0\) and \(x\in C\), we compute our iteration as follows.

Step 1. Find \(z\in Q\) such that \(g(z,y)+\frac{1}{r}\langle y-z,z-Ax\rangle \geq 0\) for all \(y\in Q\). Since \(Ax=x\), we have

$$\begin{aligned} g(z,y)+\frac{1}{r}\langle y-z,z-Ax\rangle \geq 0 \quad \Leftrightarrow \quad &z(y-z)+\frac{1}{r}\langle y-z,z-x \rangle \geq 0, \\ \quad \Leftrightarrow \quad &rz(y-z)+(y-z) (z-x)\geq 0, \\ \quad \Leftrightarrow \quad &(y-z) \bigl((1+r)z-x\bigr)\geq 0. \end{aligned}$$

It follows from Lemma 2.2(i) that \(T_{r}^{g}Ax\) is single-valued; therefore we get \(z=\frac{x}{1+r}\). This implies that \(T_{r}^{g}Ax=\frac{x}{1+r}\).

Step 2. Find \(s\in C\) such that \(s=w-\gamma A^{\ast }(I-T_{r}^{G})Aw\). It follows from Step 1 that

$$\begin{aligned} s =&x-\gamma A^{\ast }\bigl(I-T_{r}^{g} \bigr)Ax \\ =&x-\frac{1}{100}A^{\ast } \biggl( x-\frac{x}{1+r} \biggr) \\ =&x-\frac{1}{100} \biggl( x-\frac{x}{1+r} \biggr) \\ =& \biggl( 1-\frac{1}{100} \biggr) x+\frac{1}{100} \biggl( \frac{x}{1+r} \biggr) . \end{aligned}$$

Step 3. Find \(u\in C\) such that \(f(u,v)+\frac{1}{r}\langle v-u,u-s\rangle \geq 0\) for all \(v\in C\). It follows from Step 2 that

$$\begin{aligned} f(u,v)+\frac{1}{r}\langle v-u,u-s\rangle \geq 0 \quad \Leftrightarrow \quad &2u(v-u)+\frac{1}{r}\langle v-u,u-s\rangle \geq 0, \\ \quad \Leftrightarrow\quad &2ru(v-u)+(v-u) (u-s)\geq 0, \\ \quad \Leftrightarrow \quad &(v-u) \bigl((1+2r)u-s\bigr)\geq 0. \end{aligned}$$

Similarly, from Lemma 2.2(i) we obtain that \(u=\frac{s}{1+2r}= ( 1-\frac{1}{100} ) \frac{x}{1+2r}+\frac{1}{100} ( \frac{x}{ ( 1+r ) ( 1+2r ) } ) \).

Step 4. Find \(y_{n}=\alpha _{n}u_{n}+(1-\alpha _{n})\frac{1}{N}\sum_{i=1}^{N}S_{i}^{n}u_{n}\) where \(u_{n}= ( 1-\frac{1}{100} ) \frac{x_{n}}{1+2r_{n}}+\frac{1}{100} ( \frac{x_{n}}{ ( 1+r_{n} ) ( 1+2r_{n} ) } ) \).

Step 5. Find \(C_{n+1}= \{ {z\in C:\Vert y_{n}-z\Vert \leq \Vert w_{n}-z\Vert } \} \) where \(C_{1}=[0,10]\). Since \(0\leq y_{1}\leq x_{1}\leq 10\), therefore \(C_{2}= \{ {z\in C_{1}:\Vert y_{1}-z\Vert \leq \Vert x_{1}-z \Vert } \} = [ 0,\frac{y_{1}+x_{1}}{2} ] \). Since \(\frac{y_{1}+x_{1}}{2}\leq x_{1}\) and in particular \(\frac{y_{1}+x_{1}}{2}\leq x_{1}\), therefore \(x_{2}=P_{C_{2}}x_{1}=\frac{y_{1}+x_{1}}{2}\). In a similar fashion, we have \(C_{n+1}= [ 0,\frac{y_{n}+x_{n}}{2} ] \) and \(x_{n+1}=P_{C_{n+1}}x_{1}=\frac{y_{n}+x_{n}}{2}\).

Step 6. Compute the numerical results of \(x_{n+1}=P_{C_{n+1}}x_{1}\).

Table 1 exhibits the performance of sequence \(\{x_{n}\}\) defined in Theorem 3.1.

Table 1 Numerical results of Example 3.2 with \(\alpha _{n}=0.5\) and initial guess \(x_{1}=10\).

Applications

In this section, we deduce some results from Theorem 3.1. An immediate consequence of Theorem 3.1 is to establish the same result for a class of nonexpansive mappings.

Corollary 4.1

Let\(H_{1}\)and\(H_{2}\)be two real Hilbert spaces, and let\(C\subseteq H_{1}\)and\(Q\subseteq H_{2}\)be nonempty closed convex subsets of Hilbert spaces\(H_{1}\)and\(H_{2}\), respectively. Let\(f_{i}:C\times C\rightarrow \mathbb{R}\)and\(g_{i}:Q\times Q\rightarrow \mathbb{R}\)be two finite families of bifunctions satisfying conditions (A1)(A4) such that each\(g_{i}\)is upper semicontinuous for each\(i\in \{1,2,3,\ldots ,N\}\). Let\(S_{i}:C\rightarrow C\)be a finite family of nonexpansive mappings, and let\(A_{i}:H_{1}\rightarrow H_{2}\)be a finite family of bounded linear operators for each\(i\in \{1,2,3,\ldots ,N\}\). Suppose that\(\mathbb{F}:=F(S)\cap \varOmega \neq \emptyset \), where\(\varOmega = \{ z\in C:z\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\textit{ and }A_{i}z \in \bigcap_{i=1}^{N}\operatorname{EP}(g_{i})\textit{ for }1\leq i\leq N \} \). Let\(\{x_{n}\} \)be a sequence generated by

$$ \begin{aligned} &x_{1}\in C_{1}=C, \\ &u_{n}=T_{r_{n}}^{f_{n (\operatorname {mod}N ) }} \bigl( x_{n}- \gamma A_{n ( \operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n} \bigr) , \\ &y_{n}=\alpha _{n}u_{n}+(1-\alpha _{n})\frac{1}{N}\sum_{i=1}^{N}S^{i}u_{n}, \\ &C_{n+1}= \bigl\{ {z\in C:} \Vert {y_{n}-z} \Vert {^{2} \leq \Vert {x_{n}-z} \Vert ^{2}+ \theta _{n}} \bigr\} , \\ &x_{n+1}=P_{C_{n+1}}x_{1}, \quad n\geq 1,\end{aligned} $$
(27)

where\(\theta _{n}=(1-\alpha _{n}) \{ \lambda _{n}\xi _{n}(M_{n})+ \lambda _{n}M_{n}^{\ast }D_{n}+\mu _{n} \} \)with\(D_{n}=\sup \{ \Vert x_{n}-p \Vert :p \in \mathbb{F} \} \). Let\(\{r_{n}\}\), \(\{s_{n}\}\)be two positive real sequences, and let\(\{\alpha _{n}\}\)be in\((0,1)\). Assume that if the following set of conditions holds:

  1. (C1)

    \(0\leq k< a\leq \alpha _{n}\leq b<1\)and\(\gamma \in ( 0,\frac{1}{L} ) \)where\(L=\max \{ L_{1},L_{2},\ldots ,L_{N} \} \)and\(L_{i}\)is the spectral radius of the operator\(A_{i}^{\ast }A_{i}\)and\(A_{i}^{\ast }\)is the adjoint of\(A_{i}\)for each\(i\in \{1,2,3,\ldots ,N\}\);

  2. (C2)

    \(\liminf_{n\rightarrow \infty }r_{n}>0\)and\(\liminf_{n\rightarrow \infty }s_{n}>0\);

  3. (C3)

    \(\sum_{n=1}^{\infty }\lambda _{n}<\infty \)and\(\sum_{n=1}^{\infty }\mu _{n}<\infty \);

  4. (C4)

    there exist constants\(M_{i}, M_{i}^{\ast }>0\)such that\(\xi _{i} ( \lambda _{i} ) \leq M_{i}^{ \ast }\lambda _{i}\)for all\(\lambda _{i}\geq M_{i}\), \(i=1,2,3,\ldots ,N\), then the sequence\(\{x_{n}\}\)generated by (27) converges strongly to\(P_{\mathbb{F}}x_{1}\).

Corollary 4.2

Let\(H_{1}\)and\(H_{2}\)be two real Hilbert spaces, and let\(C\subseteq H_{1}\)and\(Q\subseteq H_{2}\)be nonempty closed convex subsets of Hilbert spaces\(H_{1}\)and\(H_{2}\), respectively. Let\(f_{i}:C\times C\rightarrow \mathbb{R}\)and\(g_{i}:Q\times Q\rightarrow \mathbb{R}\)be two finite families of bifunctions satisfying conditions (A1)(A4) such that each\(g_{i}\)is upper semicontinuous for each\(i\in \{1,2,3,\ldots ,N\}\). Let\(S:C\rightarrow C\)be a nonexpansive mapping, and let\(A_{i}:H_{1}\rightarrow H_{2}\)be a finite family of bounded linear operators for each\(i\in \{1,2,3,\ldots ,N\}\). Suppose that\(\mathbb{F}:=F(S)\cap \varOmega \neq \emptyset \), where\(\varOmega = \{ z\in C:z\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\textit{ and }A_{i}z \in \bigcap_{i=1}^{N}\operatorname{EP}(g_{i}) \textit{ for }1\leq i\leq N \} \). Let\(\{x_{n}\}\)be a sequence generated by

$$ \begin{aligned} &x_{1}\in C_{1}=C, \\ &u_{n}=T_{r_{n}}^{f_{n (\operatorname {mod}N ) }} \bigl( x_{n}- \gamma A_{n ( \operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }} \bigr) A_{n ( \operatorname {mod}N ) }x_{n} \bigr) , \\ &y_{n}=\alpha _{n}u_{n}+(1-\alpha _{n})Su_{n}, \\ &C_{n+1}= \bigl\{ {z\in C:} \Vert {y_{n}-z} \Vert {^{2} \leq \Vert {x_{n}-z} \Vert ^{2}+ \theta _{n}} \bigr\} , \\ &x_{n+1}=P_{C_{n+1}}x_{1},\quad n\geq 1,\end{aligned} $$
(28)

where\(\theta _{n}=(1-\alpha _{n}) \{ \lambda _{n}\xi _{n}(M_{n})+ \lambda _{n}M_{n}^{\ast }D_{n}+\mu _{n} \} \)with\(D_{n}=\sup \{ \Vert x_{n}-p \Vert :p \in \mathbb{F} \} \). Let\(\{r_{n}\}\), \(\{s_{n}\}\)be two positive real sequences, and let\(\{\alpha _{n}\}\)be in\((0,1)\). Assume that if the following set of conditions holds:

  1. (C1)

    \(0\leq k< a\leq \alpha _{n}\leq b<1\)and\(\gamma \in ( 0,\frac{1}{L} ) \)where\(L=\max \{ L_{1},L_{2},\ldots ,L_{N} \} \)and\(L_{i}\)is the spectral radius of the operator\(A_{i}^{\ast }A_{i}\)and\(A_{i}^{\ast }\)is the adjoint of\(A_{i}\)for each\(i\in \{1,2,3,\ldots ,N\}\);

  2. (C2)

    \(\liminf_{n\rightarrow \infty }r_{n}>0\)and\(\liminf_{n\rightarrow \infty }s_{n}>0\);

  3. (C3)

    \(\sum_{n=1}^{\infty }\lambda _{n}<\infty \)and\(\sum_{n=1}^{\infty }\mu _{n}<\infty \);

  4. (C4)

    there exist constants\(M_{i}, M_{i}^{\ast }>0\)such that\(\xi _{i} ( \lambda _{i} ) \leq M_{i}^{ \ast }\lambda _{i}\)for all\(\lambda _{i}\geq M_{i}\), \(i=1,2,3,\ldots ,N\), then the sequence\(\{x_{n}\}\)generated by (28) converges strongly to\(P_{\mathbb{F}}x_{1}\).

Proof

Set \(S^{i}=S\) for \(i\in \{1,2,3,\ldots ,N\}\), then the desired result follows from Corollary 4.1 immediately. □

The following results suggest an iterative construction for a common solution of the classical equilibrium problem together with the fixed point problem.

Corollary 4.3

Let\(H_{1}\)and\(H_{2}\)be two real Hilbert spaces, and let\(C\subseteq H_{1}\)and\(Q\subseteq H_{2}\)be nonempty closed convex subsets of Hilbert spaces\(H_{1}\)and\(H_{2}\), respectively. Let\(f_{i}:C\times C\rightarrow \mathbb{R}\)and\(g_{i}:Q\times Q\rightarrow \mathbb{R}\)be two finite families of bifunctions satisfying conditions (A1)(A4) such that each\(g_{i}\)is upper semicontinuous for each\(i\in \{1,2,3,\ldots ,N\}\). Let\(S_{i}:C\rightarrow C\)be a finite family of uniformly continuous total asymptotically nonexpansive mappings, and let\(A_{i}:H_{1}\rightarrow H_{2}\)be a finite family of bounded linear operators for each\(i\in \{1,2,3,\ldots ,N\}\). Suppose that\(\mathbb{F}:= [ \bigcap_{i=1}^{N}F(S_{i}) ] \cap \varOmega \neq \emptyset \), where\(\varOmega = \{ z\in C:z\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\textit{ and }A_{i}z \in \bigcap_{i=1}^{N}\operatorname{EP}(g_{i})\textit{ for }1\leq i\leq N \} \). Let\(\{x_{n}\} \)be a sequence generated by

$$ \begin{aligned} &x_{1}\in C_{1}=C, \\ &u_{n}=T_{r_{n}}^{f_{n (\operatorname {mod}N ) }} \bigl( x_{n}- \gamma A_{n ( \operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n} \bigr) , \\ &y_{n}=\alpha _{n}u_{n}+(1-\alpha _{n})\frac{1}{N}\sum_{i=1}^{N}S_{i}^{n}u_{n}, \\ &C_{n+1}= \bigl\{ {z\in C:} \Vert {y_{n}-z} \Vert {^{2} \leq \Vert {x_{n}-z} \Vert ^{2}+ \theta _{n}} \bigr\} , \\ &x_{n+1}=P_{C_{n+1}}x_{1},\quad n\geq 1,\end{aligned} $$
(29)

where\(\theta _{n}=(1-\alpha _{n}) \{ \lambda _{n}\xi _{n}(M_{n})+ \lambda _{n}M_{n}^{\ast }D_{n}+\mu _{n} \} \)with\(D_{n}=\sup \{ \Vert x_{n}-p \Vert :p \in \mathbb{F} \} \). Let\(\{r_{n}\}\), \(\{s_{n}\}\)be two positive real sequences, and let\(\{\alpha _{n}\}\)be in\((0,1)\). Assume that if the following set of conditions holds:

  1. (C1)

    \(0\leq k< a\leq \alpha _{n}\leq b<1\)and\(\gamma \in ( 0,\frac{1}{L} ) \)where\(L=\max \{ L_{1},L_{2},\ldots ,L_{N} \} \)and\(L_{i}\)is the spectral radius of the operator\(A_{i}^{\ast }A_{i}\)and\(A_{i}^{\ast }\)is the adjoint of\(A_{i}\)for each\(i\in \{1,2,3,\ldots ,N\}\);

  2. (C2)

    \(\liminf_{n\rightarrow \infty }r_{n}>0\)and\(\liminf_{n\rightarrow \infty }s_{n}>0\);

  3. (C3)

    \(\sum_{n=1}^{\infty }\lambda _{n}<\infty \)and\(\sum_{n=1}^{\infty }\mu _{n}<\infty \);

  4. (C4)

    there exist constants\(M_{i}, M_{i}^{\ast }>0\)such that\(\xi _{i} ( \lambda _{i} ) \leq M_{i}^{ \ast }\lambda _{i}\)for all\(\lambda _{i}\geq M_{i}\), \(i=1,2,3,\ldots ,N\), then the sequence\(\{x_{n}\}\)generated by (29) converges strongly to\(P_{\mathbb{F}}x_{1}\).

Proof

Set \(H_{1}=H_{2}\), \(C=Q\) and \(A_{i}=I\) (the identity mapping) for \(i=1,2,3,\ldots ,N\), then the desired result follows from Theorem 3.1 immediately. □

Corollary 4.4

Let\(H_{1}\)and\(H_{2}\)be two real Hilbert spaces, and let\(C\subseteq H_{1}\)and\(Q\subseteq H_{2}\)be nonempty closed convex subsets of Hilbert spaces\(H_{1}\)and\(H_{2}\), respectively. Let\(f_{i}:C\times C\rightarrow \mathbb{R}\)and\(g_{i}:Q\times Q\rightarrow \mathbb{R}\)be two finite families of bifunctions satisfying conditions (A1)(A4) such that each\(g_{i}\)is upper semicontinuous for each\(i\in \{1,2,3,\ldots ,N\}\). Let\(S_{i}:C\rightarrow C\)be a finite family of nonexpansive mappings, and let\(A_{i}:H_{1}\rightarrow H_{2}\)be a finite family of bounded linear operators for each\(i\in \{1,2,3,\ldots ,N\}\). Suppose that\(\mathbb{F}:= [ \bigcap_{i=1}^{N}F(S_{i}) ] \cap \varOmega \neq \emptyset \), where\(\varOmega = \{ z\in C:z\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\textit{ and }A_{i}z \in \bigcap_{i=1}^{N}\operatorname{EP}(g_{i}) \textit{ for }1\leq i\leq N \} \). Let\(\{x_{n}\}\)be a sequence generated by

$$ \begin{aligned} &x_{1}\in C_{1}=C, \\ &u_{n}=T_{r_{n}}^{f_{n (\operatorname {mod}N ) }} \bigl( x_{n}- \gamma A_{n ( \operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }} \bigr) A_{n (\operatorname {mod}N ) }x_{n} \bigr) , \\ &y_{n}=\alpha _{n}u_{n}+(1-\alpha _{n})\frac{1}{N}\sum_{i=1}^{N}S^{i}u_{n}, \\ &C_{n+1}= \bigl\{ {z\in C:} \Vert {y_{n}-z} \Vert {^{2} \leq \Vert {x_{n}-z} \Vert ^{2}+ \theta _{n}} \bigr\} , \\ &x_{n+1}=P_{C_{n+1}}x_{1}, \quad n\geq 1,\end{aligned} $$
(30)

where\(\theta _{n}=(1-\alpha _{n}) \{ \lambda _{n}\xi _{n}(M_{n})+ \lambda _{n}M_{n}^{\ast }D_{n}+\mu _{n} \} \)with\(D_{n}=\sup \{ \Vert x_{n}-p \Vert :p \in \mathbb{F} \} \). Let\(\{r_{n}\}\), \(\{s_{n}\}\)be two positive real sequences, and let\(\{\alpha _{n}\}\)be in\((0,1)\). Assume that if the following set of conditions holds:

  1. (C1)

    \(0\leq k< a\leq \alpha _{n}\leq b<1\)and\(\gamma \in ( 0,\frac{1}{L} ) \)where\(L=\max \{ L_{1},L_{2},\ldots ,L_{N} \} \)and\(L_{i}\)is the spectral radius of the operator\(A_{i}^{\ast }A_{i}\)and\(A_{i}^{\ast }\)is the adjoint of\(A_{i}\)for each\(i\in \{1,2,3,\ldots ,N\}\);

  2. (C2)

    \(\liminf_{n\rightarrow \infty }r_{n}>0\)and\(\liminf_{n\rightarrow \infty }s_{n}>0\);

  3. (C3)

    \(\sum_{n=1}^{\infty }\lambda _{n}<\infty \)and\(\sum_{n=1}^{\infty }\mu _{n}<\infty \);

  4. (C4)

    there exist constants\(M_{i}, M_{i}^{\ast }>0\)such that\(\xi _{i} ( \lambda _{i} ) \leq M_{i}^{ \ast }\lambda _{i}\)for all\(\lambda _{i}\geq M_{i}\), \(i=1,2,3,\ldots ,N\), then the sequence\(\{x_{n}\}\)generated by (27) converges strongly to\(P_{\mathbb{F}}x_{1}\).

Proof

Set \(H_{1}=H_{2}\), \(C=Q\), and \(A_{i}=I\) (the identity mapping) for \(i=1,2,3,\ldots ,N\), then the desired result follows from Corollary 4.1 immediately. □

Corollary 4.5

Let\(H_{1}\)and\(H_{2}\)be two real Hilbert spaces, and let\(C\subseteq H_{1}\)and\(Q\subseteq H_{2}\)be nonempty closed convex subsets of Hilbert spaces\(H_{1}\)and\(H_{2}\), respectively. Let\(f_{i}:C\times C\rightarrow \mathbb{R}\)and\(g_{i}:Q\times Q\rightarrow \mathbb{R}\)be two finite families of bifunctions satisfying conditions (A1)(A4) such that each\(g_{i}\)is upper semicontinuous for each\(i\in \{1,2,3,\ldots ,N\}\). Let\(S:C\rightarrow C\)be a nonexpansive mapping, and let\(A_{i}:H_{1}\rightarrow H_{2}\)be a finite family of bounded linear operators for each\(i\in \{1,2,3,\ldots ,N\}\). Suppose that\(\mathbb{F}:=F(S)\cap \varOmega \neq \emptyset \), where\(\varOmega = \{ z\in C:z\in \bigcap_{i=1}^{N}\operatorname{EP}(f_{i})\textit{ and }A_{i}z \in \bigcap_{i=1}^{N}\operatorname{EP}(g_{i}) \textit{ for }1\leq i\leq N \} \). Let\(\{x_{n}\}\)be a sequence generated by

$$ \begin{aligned} &x_{1}\in C_{1}=C, \\ &u_{n}=T_{r_{n}}^{f_{n ( \operatorname {mod}N ) }} \bigl( x_{n}- \gamma A_{n ( \operatorname {mod}N ) }^{\ast } \bigl( I-T_{s_{n}}^{g_{n ( \operatorname {mod}N ) }} \bigr) A_{n ( \operatorname {mod}N ) }x_{n} \bigr) , \\ &y_{n}=\alpha _{n}u_{n}+(1-\alpha _{n})Su_{n}, \\ &C_{n+1}= \bigl\{ {z\in C:} \Vert {y_{n}-z} \Vert {^{2} \leq \Vert {x_{n}-z} \Vert ^{2}+ \theta _{n}} \bigr\} , \\ &x_{n+1}=P_{C_{n+1}}x_{1},\quad n\geq 1,\end{aligned} $$
(31)

where\(\theta _{n}=(1-\alpha _{n}) \{ \lambda _{n}\xi _{n}(M_{n})+ \lambda _{n}M_{n}^{\ast }D_{n}+\mu _{n} \} \)with\(D_{n}=\sup \{ \Vert x_{n}-p \Vert :p \in \mathbb{F} \} \). Let\(\{r_{n}\}\), \(\{s_{n}\}\)be two positive real sequences, and let\(\{\alpha _{n}\}\)be in\((0,1)\). Assume that if the following set of conditions holds:

  1. (C1)

    \(0\leq k< a\leq \alpha _{n}\leq b<1\)and\(\gamma \in ( 0,\frac{1}{L} ) \)where\(L=\max \{ L_{1},L_{2},\ldots ,L_{N} \} \)and\(L_{i}\)is the spectral radius of the operator\(A_{i}^{\ast }A_{i}\)and\(A_{i}^{\ast }\)is the adjoint of\(A_{i}\)for each\(i\in \{1,2,3,\ldots ,N\}\);

  2. (C2)

    \(\liminf_{n\rightarrow \infty }r_{n}>0\)and\(\liminf_{n\rightarrow \infty }s_{n}>0\);

  3. (C3)

    \(\sum_{n=1}^{\infty }\lambda _{n}<\infty \)and\(\sum_{n=1}^{\infty }\mu _{n}<\infty \);

  4. (C4)

    there exist constants\(M_{i}, M_{i}^{\ast }>0\)such that\(\xi _{i} ( \lambda _{i} ) \leq M_{i}^{ \ast }\lambda _{i}\)for all\(\lambda _{i}\geq M_{i}\), \(i=1,2,3,\ldots ,N\), then the sequence\(\{x_{n}\}\)generated by (31) converges strongly to\(P_{\mathbb{F}}x_{1}\).

Proof

Set \(S^{i}=S\) for \(i\in \{1,2,3,\ldots ,N\}\), then the desired result follows from Corollary 4.4 immediately. □

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Acknowledgements

The authors express their sincere thanks to the referee for his/her careful reading and suggestions that helped to improve this paper.

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This project was supported by the deanship of scientific research at Prince Sattam Bin Abdulaziz University under the research project 2019/01/10960.

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The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Correspondence to Kottakkaran Sooppy Nisar.

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Harisa, S.A., Khan, M.A.A., Mumtaz, F. et al. Shrinking Cesáro means method for the split equilibrium and fixed point problems in Hilbert spaces. Adv Differ Equ 2020, 345 (2020). https://doi.org/10.1186/s13662-020-02800-z

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MSC

  • 47H05
  • 47H09
  • 47J05
  • 49H05

Keywords

  • Cesáro means method
  • Shrinking projection method
  • Split equilibrium fixed point problem
  • Total asymptotically nonexpansive mapping