Let C be a nonempty closed convex subset of a real Hilbert space H. The fixed point problem for the mapping $T:C\to C$ is to find x ∈ C such that

$$x=Tx.$$

We denote the fixed point set of a mapping T by Fix(T).

Definition 1.1. Let $T:C\to C$ be a mapping. Then

(i) a mapping T is called nonexpansive if

$$‖Tx-Ty‖\le ‖x-y‖,\forall x,y\in C;$$

(ii) T is said to be κ-strictly pseudo-contractive if there exists a constant $\kappa \in [0,1)$such that

$${‖Tx-Ty‖}^2\le {‖x-y‖}^2+\kappa {‖(I-T)x-(I-T)y‖}^2,\forall x,y\in C.$$

Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is, a nonexpansive mapping is a 0-strictly pseudo-contractive mapping.

In 2008, Kohsaka and Takahashi [6] introduced a nonspreading mapping T in Hilbert space H as follows:

(1.1)$$2\Vert Tx-Ty{\Vert }^2\le \Vert Tx-y{\Vert }^2+\Vert x-Ty{\Vert }^2,\forall x,y\in C.$$

In 2009, it is shown by Iemoto and Takahashi [2] that (1.1) is equivalent to the following equation.

$$\Vert Tx-Ty{\Vert }^2\le \Vert x-y{\Vert }^2+2⟨x-Tx,y-Ty⟩,\text{ for all }x,y\in C.$$

Later, in 2011, Osilike and Isiogugu [13] proposed a κ-strictly pseudononspreading mapping, that is, a mapping $T:C\to C$ is said to be a κ-strictly pseudononspreading mapping if there exists $\kappa \in [0,1)$ such that

$$\Vert Tx-Ty{\Vert }^2\le \Vert x-y{\Vert }^2+\kappa \Vert (I-T)x-(I-T)y{\Vert }^2+2⟨x-Tx,y-Ty⟩,\text{ for all }x,y\in C.$$

Obviously, every nonspreading mapping is a κ-strictly pseudononspreading mapping, that is, a nonspreading mapping is a 0-stricly pseudononspreading mapping.

Many mathematicians tried to proposed iterative algorithms and proved the strong convergence theorems for a nonspreading mapping and a strictly pseudononspreading mapping in Hilbert space to find their fixed points, see, for instance, [7, 13, 8, 1].

Over the past decades, many others have constructed various types of iterative methods to approximate fixed points. The first one is the Mann iteration introduced by Mann [9] in 1953 and is defined as follows:

(1.2)$$\left\{\begin{array}{l}\begin{array}{l}{x}_0\in H\text{ arbitrary chosen,}\hfill \\ x_{n+1}=\left(1-{\alpha }_n\right)x_n+{\alpha }_{n}T{x}_n,\forall n\ge 0,\hfill \end{array}\hfill \end{array}\right.$$

where C is a nonempty closed convex subset of a normed space, $T:C\to C$ is a mapping and the sequence $\left\{{\alpha }_n\right\}$ is in the interval (0,1). But this algorithm has only weak convergence. Thus, many mathematicians have been trying to modify Mann's iteration (1.1) and construct new iterative method to obtain the strong convergence theorem.

By modification of Mann's iteration (1.1), the next iteration process is referred to as Ishikawa's iteration process [3] which is defined recursively as follows:

(1.3)$$\left\{\begin{array}{l}\begin{array}{l}{x}_0\in H\text{ arbitrary chosen,}\hfill \\ y_n={\beta }_n{x}_n+\left(1-{\beta }_n\right)T{x}_n,\hfill \\ x_{n+1}={\alpha }_n{x}_n+\left(1-{\alpha }_n\right)T{y}_n,\forall n\ge 0,\hfill \end{array}\hfill \end{array}\right.$$

where $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ are real sequences in [0,1]. He also obtain the strong convergence theorem for the iterative method (1.3) converging to a fixed point of mapping T. Observe that if β_n=1, then the Ishikawa's iteration (1.3) reduces to the Mann's iteration (1.2).

In 2000, Moudafi [11] introduced the viscosity approximation method for nonexpansive mapping S as follows:

Let C be a closed convex subset of a real Hilbert space H and let $S:C\to C$ be a nonexpansive mapping such that Fix(S) is nonempty. Let $f:C\to C$ be a contraction, that is, there exists $\alpha \in (0,1)$ such that $‖fx-fy‖\le \alpha ‖x-y‖,\forall x,y\in C$, and let $\left\{x_n\right\}$ be a sequence defined by

(1.4)$$\left\{\begin{array}{l}\begin{array}{l}{x}_1\in C\text{ arbitrary chosen,}\hfill \\ x_{n+1}=\frac{1}{1+{ϵ}_n}S{x}_n+\frac{{ϵ}_n}{1+{ϵ}_n}f\left(x_n\right),\forall n\in \mathbb{N},\hfill \end{array}\hfill \end{array}\right.$$

where $\left\{{\epsilon }_n\right\}\subset (0,1)$ satisfies certain conditions. Then the sequence $\left\{x_n\right\}$ converges strongly to $z\in Fix(S)$, where $z=P_{Fix(S)}f(z)$ and $P_{Fix(S)}$ is the metric projection of H onto Fix(S).

In 2006, using the concept of the viscosity approximation method (1.4), Marino and Xu [10] introduced the general iterative method and obtained the strong convergence theorem.

Let $T:H\to H$ be a nonexpansive mapping with $Fix(T)\ne \varnothing$. Let $f:H\to H$ be a contractive mapping on H and let $\left\{x_n\right\}$ be generated by

(1.5)$$\left\{\begin{array}{l}\begin{array}{l}{x}_0\in H\text{ arbitrary chosen,}\hfill \\ x_{n+1}=\left(I-{\alpha }_{n}A\right)T{x}_n+{\alpha }_n\gamma f\left(x_n\right),n\ge 0,\hfill \end{array}\hfill \end{array}\right.$$

where $\left\{{\alpha }_n\right\}$ is a sequence in (0,1) satisfying the appropriate conditions. Then $\left\{x_n\right\}$ converges strongly to a fixed point $\tilde{x}$ of T which solves the variational inequality:

$$⟨(A-\gamma f)\tilde{x},\tilde{x}-z⟩\le 0,z\in Fix(T).$$

In 2015, Yao et al. [18] proposed the intermixed algorithm for two strict pseudocontractions S and T as follows:

Algorithm 1.2. For arbitrarily given $x_0\in C,y_0\in C$, let the sequences $\left\{x_n\right\}$ and $\left\{y_n\right\}$ be generated iteratively by

(1.6)$$\begin{array}{l}{x}_{n+1}=\left(1-{\beta }_n\right)x_n+{\beta }_n{P}_C\left[{\alpha }_{n}f\left(y_n\right)+\left(1-k-{\alpha }_n\right)x_n+kT{x}_n\right],n\ge 0,\\ y_{n+1}=\left(1-{\beta }_n\right)y_n+{\beta }_n{P}_C\left[{\alpha }_{n}g\left(x_n\right)+\left(1-k-{\alpha }_n\right)y_n+kS{y}_n\right],n\ge 0,\end{array}$$

where $T:C\to C$ is a λ-strict pseudo-contraction, $f:C\to H$ is a ρ₁-contraction and $g:C\to H$ is a ${\rho }_2$-contraction, $k\in (0,1-\lambda )$ is a constant and $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$ are two real number sequences in (0,1).

Furthermore, under some control conditions, they proved that the iterative sequences $\left\{x_n\right\}$ and $\left\{y_n\right\}$ defined by (1.6) converges independently to $P_{Fix(T)}f\left(y^{*}\right)$ and $P_{Fix(S)}g\left(x^{*}\right)$, respectively, where $x^{*}\in Fix(T)$ and $y^{*}\in Fix(S)$.

Motivated by Yao et al. [18], in 2018, Suwannaut [15] introduce the S-intermixed iteration for two finite families of nonlinear mappings without considering the constant k as in the following algorithm:

Algorithm 1.3. Starting with $x_1,y_1,z_1\in C$, let the sequences $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ be defined by

(1.7)$$\begin{array}{l}{x}_{n+1}=\left(1-{\beta }_n\right)x_n+{\beta }_n\left({\alpha }_n{f}_1\left(y_n\right)+\left(1-{\alpha }_n\right)S{x}_n\right),\\ y_{n+1}=\left(1-{\beta }_n\right)y_n+{\beta }_n\left({\alpha }_n{f}_2\left(x_n\right)+\left(1-{\alpha }_n\right)T{y}_n\right),n\ge 1,\end{array}$$

where $S,T:C\to C$, is a nonlinear mapping with $Fix(S)\cap Fix\left(T\right)\ne \varnothing$, $f_i:C\to C$ is a contractive mapping with coefficients ${\alpha }_i;i=1,2$ and $\left\{{\beta }_n\right\}$,$\left\{{\alpha }_n\right\}$ are real sequences in (0,1), $\forall n\ge 1$.

Under appropriate conditions, they prove a strong convergence theorem for finding a common solution of two finite families of equilibrium problems.

Inspired by the previous work, we introduce the new iterative method called the three-step intermixed iteration for two finite families of nonlinear mappings as the following algorithm:

Algorithm 1.4. Starting with $x_1,y_1,z_1\in C$, let the sequences $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ be defined by

(1.8)$$\begin{array}{l}{x}_{n+1}={\delta }_n{x}_n+{\eta }_n{S}_1{x}_n+{\mu }_n{P}_C\left[{\alpha }_n{\gamma }_1{f}_1(y_n)+(I-{\alpha }_n{A}_1)T_1{x}_n\right],\\ y_{n+1}={\delta }_n{y}_n+{\eta }_n{S}_2{y}_n+{\mu }_n{P}_C\left[{\alpha }_n{\gamma }_2{f}_2(z_n)+(I-{\alpha }_n{A}_2)T_2{y}_n\right],\\ z_{n+1}={\delta }_n{z}_n+{\eta }_n{S}_3{z}_n+{\mu }_n{P}_C\left[{\alpha }_n{\gamma }_3{f}_3(x_n)+(I-{\alpha }_n{A}_3)T_3{z}_n\right],n\ge 1,\end{array}$$

where $S_i,T_i:C\to C$, where i=1,2,3, is nonlinear mappings with $Fix(S_i)\cap Fix\left(T_i\right)\ne \varnothing ,\forall i=1,2,3$, f_i is a contractive mapping with coefficients ${\xi }_i,A_i:C\to C$ is a strongly positive linear bounded operator with coefficient ${\beta }_i>0$ and $0<\gamma <\frac{\beta }{\xi }$, where $\gamma ={\max }_{i\in \{1,2,3\}}{\gamma }_i$, $\xi ={\max }_{i\in \{1,2,3\}}{\xi }_i$ and $\beta ={\min }_{i\in \{1,2,3\}}{\beta }_i$, $\left\{{\delta }_n\right\}$, $\left\{{\eta }_n\right\}$, $\left\{{\mu }_n\right\}$ and $\left\{{\alpha }_n\right\}$ are real sequences in (0,1) and ${\delta }_n+{\eta }_n+{\mu }_n=1,\forall n\ge 1$.

Remark 1.5. From Algorithm 1.2 and 1.4, we observe that Algorithm 1.4 can be seen as a modification and extension of Algorithm 1.2 in senses that we choose to consider the three-step intermixed algorithm for approximating fixed points of two finite families of nonlinear mappings and we study the general iterative method without a constant k.

Remark 1.6. If we take $S_i\equiv I$, ${\gamma }_i=1$ and $A_i\equiv I$ for i=1,2,3, then the iterative method (1.8) reduces to

(1.9)$$\begin{array}{l}{x}_{n+1}=\left(1-{\mu }_n\right)x_n+{\mu }_n\left[{\alpha }_n{f}_1(y_n)+(1-{\alpha }_n)T_1{x}_n\right],\\ y_{n+1}=\left(1-{\mu }_n\right)y_n+{\mu }_n\left[{\alpha }_n{f}_2(z_n)+(1-{\alpha }_n)T_2{y}_n\right],\\ z_{n+1}=\left(1-{\mu }_n\right)z_n+{\mu }_n\left[{\alpha }_n{f}_3(x_n)+(1-{\alpha }_n)T_3{z}_n\right].\end{array}$$

The iteration (1.9) is a modification and improvement of iteration (1.7) in sense that it extends to three-step iteration for three nonlinear mappings.

Inspired by the previous research, we introduce the three-steps intermixed iteration for two finite families of nonlinear mappings. Under appropriate conditions, we prove a strong convergence theorem for finding a common fixed point of a strictly pseudo-contractive mapping and a strictly pseudononspreading mapping. Finally, we give a numerical example for the main theorem in a space of real numbers.

We denote weak convergence and strong convergence by notations ${}^{‵‵}{\rightharpoonup }^{\prime \prime }$ and ${}^{‵‵}{\to }^{\prime \prime }$, respectively. For every x ∈ H, there is a unique nearest point P_Cx in C such that

$$\Vert x-P_{C}x\Vert \le \Vert x-y\Vert ,\forall y\in C.$$

Such an operator P_C is called the metric projection of H onto C.

We now recall the following definition and well-known lemmas.

Lemma 2.1. ([16]) For a given z ∈ H and u ∈ C,

$$u=P_{C}z\iff ⟨u-z,v-u⟩\ge 0,\forall v\in C.$$

Furthermore, P_C is a firmly nonexpansive mapping of H onto C and satisfies

$${‖P_{C}x-P_{C}y‖}^2\le ⟨P_{C}x-P_{C}y,x-y⟩,\forall x,y\in H.$$

Lemma 2.2. ([12]) Each Hilbert space H satisfies Opial's condition, i.e., for any sequence $\left\{x_n\right\}\subset H$ with $x_n\rightharpoonup x$, the inequality

$$\underset{n\to \infty }{\lim \inf }‖x_n-x‖<\underset{n\to \infty }{\lim \inf }‖x_n-y‖$$

holds for every y ∈ H with y ≠ x.

Lemma 2.3. ([13]) Let H be a real Hilbert space. Then the following results hold:

(i) For all x,y ∈ H and $\alpha \in [0,1]$,

$${‖\alpha x+(1-\alpha )y‖}^2=\alpha {‖x‖}^2+(1-\alpha ){‖y‖}^2-\alpha (1-\alpha ){‖x-y‖}^2,$$

(ii) $\Vert x+y{\Vert }^2\le \Vert x{\Vert }^2+2⟨y,x+y⟩,$ for each $x,y\in H.$

Lemma 2.4. ([17]) Let $\left\{s_n\right\}$ be a sequence of nonnegative real numbers satisfying

$$s_{n+1}\le (1-{\alpha }_n)s_n+{\delta }_n,\forall n\ge 0,$$

where α_n is a sequence in (0,1) and $\left\{{\delta }_n\right\}$ is a sequence such that

(1) ${\displaystyle \sum _{n=1}^{\infty }{\alpha }_n}=\infty$,

(2) $\underset{n\to \infty }{\lim \sup }\frac{{\delta }_n}{{\alpha }_n}\le 0$ or ${\displaystyle \sum _{n=1}^{\infty }|}{\delta }_n|<\infty$.

Then, $\underset{n\to \infty }{\lim }{s}_n=0$.

Lemma 2.5. ([10]) Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient $\beta >0$ and $0<\delta <\Vert A{\Vert }^{-1}$. Then $\Vert I-\delta A\Vert \le 1-\delta \beta$.

Lemma 2.6. ([4, 14]) Let C be a nonempty closed convex subset of a real Hilbert space H and let $T:C\to C$ be a κ-strictly pseudo-contractive mapping with $Fix(T)\ne \varnothing$. Then, we there hold the following statement:

(i) $Fix(T)=VI(C,I-T)$;

(ii) For every u ∈ C and $v\in Fix(T)$,

$$\Vert P_C(I-\lambda (I-T))u-v\Vert \le \Vert u-v\Vert ,\text{ for }u\in C,v\in Fix(T)\text{ and }\lambda \in (0,1-\kappa ).$$

By applying Remark 2.10 in [5], we easily obtain the following result:

Lemma 2.7. Let $S:C\to C$ be a κ-strictly pseudo nonspreading mapping with $Fix(S)\ne \varnothing$. Define $T:C\to C$ by $Tx:=(1-\lambda )x+\lambda Sx$, where $\lambda \in (0,1-\kappa )$. Then there hold the following statement:

(i) Fix(S) = Fix(T)

(ii) T is a quasi-nonexpansive mapping, that is,

$$\Vert Tx-y\Vert \le \Vert x-y\Vert ,\text{ for every }x\in C\text{ and }y\in Fix(S).$$

Proof. It is clear to prove that (i) holds.

(ii) Let x ∈ H and $y\in Fix(S)$. Then we derive

$$\begin{array}{l}\Vert Tx-y{\Vert }^2=\Vert (1-\lambda )(x-y)+\lambda (Sx-y){\Vert }^2\\ =(1-\lambda )\Vert x-y{\Vert }^2+\lambda \Vert Sx-y{\Vert }^2-\lambda (1-\lambda )\Vert Sx-x{\Vert }^2\\ \le (1-\lambda )\Vert x-y{\Vert }^2+\lambda \left(\Vert x-y{\Vert }^2+\kappa \Vert x-Sx{\Vert }^2\right)-\lambda (1-\lambda )\Vert Sx-x{\Vert }^2\\ =\Vert x-y{\Vert }^2+\kappa \lambda \Vert x-Sx{\Vert }^2-\lambda (1-\lambda )\Vert Sx-x{\Vert }^2\\ =\Vert x-y{\Vert }^2-\lambda ((1-\kappa )-\lambda )\Vert x-Sx{\Vert }^2\\ \le \Vert x-y{\Vert }^2.\end{array}$$

This implies that T is a quasi-nonexpansive mapping.

Let C be a nonempty closed convex subset of a real Hilbert space H. For i=1,2,3, let $f_i:C\to C$ be a contractive mappings with a coefficient ξ_i and $\xi ={\max }_{i\in \{1,2,3\}}{\xi }_i$, let $S_i:C\to C$ be a κ_i-strictly pseudo-contractive mapping and $W_i:C\to C$ be ρ_i-strictly pseudo-nonspreading mapping with ${\Omega }_i=Fix(S_i)\cap Fix(W_i)\ne \varnothing$. For each i=1,2,3, define a mapping $T_n^i:C\to C$ by $T_n^{i}x=\left(1-{\omega }_n\right)x+{\omega }_n{W}_{i}x,$ for all x ∈ C, and let $A_i:C\to C$ be a strongly positive linear bounded operator with a coefficient ${\beta }_i>0$ and $0<\gamma <\frac{\beta }{\xi }$, where $\gamma ={\max }_{i\in \{1,2,3,\}}{\gamma }_i$ and $\beta ={\min }_{i\in \{1,2,3\}}{\beta }_i$. Let $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ be sequences generated by $x_1,y_1,z_1\in C$ and

$$\left\{\begin{array}{l}\begin{array}{c}{x}_{n+1}={\delta }_n{x}_n+{\eta }_n{P}_C\left(I-{\lambda }_n\left(I-S_1\right)\right)x_n\\ +{\mu }_n{P}_C\left[{\alpha }_n{\gamma }_1{f}_1(y_n)+(I-{\alpha }_n{A}_1)T_n^1{x}_n\right],\\ y_{n+1}={\delta }_n{y}_n+{\eta }_n{P}_C\left(I-{\lambda }_n\left(I-S_2\right)\right)y_n\\ +{\mu }_n{P}_C\left[{\alpha }_n{\gamma }_2{f}_2(z_n)+(I-{\alpha }_n{A}_2)T_n^2{y}_n\right],\\ z_{n+1}={\delta }_n{z}_n+{\eta }_n{P}_C\left(I-{\lambda }_n\left(I-S_3\right)\right)z_n\\ +{\mu }_n{P}_C\left[{\alpha }_n{\gamma }_3{f}_3(x_n)+(I-{\alpha }_n{A}_3)T_n^3{z}_n\right],\end{array}\hfill \end{array}\right.$$

for n ≥ 1, where $\left\{{\alpha }_n\right\}$, $\left\{{\delta }_n\right\}$, $\left\{{\eta }_n\right\}$, $\left\{{\mu }_n\right\}\subset (0,1)$, $\left\{{\lambda }_n\right\}\subset (0,1-\kappa )$, $\kappa ={\min }_{i\in \{1,2,3\}}$, $\left\{{\omega }_n\right\}\subset (0,1-\rho )$, where $\rho ={\min }_{i\in \{1,2,3\}}$ and ${\delta }_n+{\mu }_n+{\eta }_n=1$ satisfying the following conditions:

(i) $\underset{n\to \infty }{\lim }{\alpha }_n=0$ and ${\displaystyle \sum _{n=1}^{\infty }{\alpha }_n}=\infty$;

(ii) $0<\tau \le {\delta }_n,{\eta }_n,{\mu }_n,\le \upsilon <1$, for some $\tau ,\upsilon >0$;

(iii) ${\displaystyle \sum _{n=1}^{\infty }{\lambda }_n}<\infty$, ${\displaystyle \sum _{n=1}^{\infty }{\omega }_n}<\infty$;

(iv) ${\displaystyle \sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|}<\infty$, ${\displaystyle \sum _{n=1}^{\infty }\left|{\delta }_{n+1}-{\delta }_n\right|}<\infty$, ${\displaystyle \sum _{n=1}^{\infty }\left|{\mu }_{n+1}-{\mu }_n\right|}<\infty$,

${\displaystyle \sum _{n=1}^{\infty }\left|{\eta }_{n+1}-{\eta }_n\right|}<\infty$, ${\displaystyle \sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_n\right|}<\infty$, ${\displaystyle \sum _{n=1}^{\infty }\left|{\omega }_{n+1}-{\omega }_n\right|}<\infty$.

Then the sequences $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ converge strongly to $\tilde{x}=P_{{\Omega }_1}\left((I-A_1)\tilde{x}+\gamma f_1\left(\tilde{y}\right)\right)$, $\tilde{y}=P_{{\Omega }_2}\left((I-A_2)\tilde{y}+\gamma f_2\left(\tilde{z}\right)\right)$ and $\tilde{z}=P_{{\Omega }_3}\left((I-A_3)\tilde{z}+\gamma f_3\left(\tilde{x}\right)\right)$, respectively.

Proof. The proof of this theorem will be divided into five steps.

Step 1. We show that $\left\{x_n\right\}$ is bounded.

Since ${\alpha }_n\to 0$ as $n\to \infty$, without loss of generality, we may assume that ${\alpha }_n<\frac{1}{‖A_i‖},$ for all i=1,2,3 and $n\in \mathbb{N}$.

Let $x^{*}\in {\Omega }_1$, $y^{*}\in {\Omega }_2$, $z^{*}\in {\Omega }_3$, $\beta ={\min }_{i\in \{1,2,3\}}{\beta }_i$, $\xi ={\max }_{i\in \{1,2,3\}}{\xi }_i$ and $\gamma ={\max }_{i\in \{1,2,3\}}{\gamma }_i$. Then we have

(3.1)$$\begin{array}{l}‖x_{n+1}-x^{*}‖\\ \le \Vert {\delta }_n\left(x_n-x^{*}\right)+{\eta }_n\left(P_C\left(I-{\lambda }_n\left(I-S_1\right)\right)x_n-x^{*}\right)\\ +{\mu }_n\left(P_C\left[{\alpha }_n{\gamma }_1{f}_1(y_n)+(I-{\alpha }_n{A}_1)T_n^1{x}_n\right]-x^{*}\right)\Vert \\ \le {\delta }_n‖x_n-x^{*}‖+{\eta }_n‖P_C\left(I-{\lambda }_n\left(I-S_1\right)\right)x_n-x^{*}‖\\ +{\mu }_n‖P_C\left[{\alpha }_n{\gamma }_1{f}_1(y_n)+(I-{\alpha }_n{A}_1)T_n^1{x}_n\right]-x^{*}‖\\ \le \left(1-{\mu }_n\right)‖x_n-x^{*}‖+{\mu }_n\left[{\alpha }_n‖{\gamma }_1{f}_1\left(y_n\right)-A_1{x}^{*}‖+‖I-{\alpha }_n{A}_1‖‖T_n^1{x}_n-x^{*}‖\right]\\ \le \left(1-{\mu }_n\right)‖x_n-x^{*}‖\\ +{\mu }_n\left[{\alpha }_n{\gamma }_1{\xi }_1‖y_n-y^{*}‖+{\alpha }_n‖{\gamma }_1{f}_1\left(y^{*}\right)-A_1{x}^{*}‖+\left(1-{\alpha }_n\beta \right)‖x_n-x^{*}‖\right]\\ \le \left(1-{\mu }_n{\alpha }_n\beta \right)‖x_n-x^{*}‖+{\mu }_n{\alpha }_n\gamma \xi ‖y_n-y^{*}‖+{\mu }_n{\alpha }_n‖{\gamma }_1{f}_1\left(y^{*}\right)-A_1{x}^{*}‖.\end{array}$$

Similarly, we get

(3.2)$$\begin{array}{l}‖y_{n+1}-y^{*}‖\\ \le \left(1-{\mu }_n{\alpha }_n\beta \right)‖y_n-y^{*}‖+{\mu }_n{\alpha }_n\gamma \xi ‖z_n-z^{*}‖+{\mu }_n{\alpha }_n‖{\gamma }_2{f}_2\left(z^{*}\right)-A_2{y}^{*}‖\end{array}$$

and

(3.3)$$\begin{array}{l}‖z_{n+1}-z^{*}‖\\ \le \left(1-{\mu }_n{\alpha }_n\beta \right)‖z_n-z^{*}‖+{\mu }_n{\alpha }_n\gamma \xi ‖x_n-x^{*}‖+{\mu }_n{\alpha }_n‖{\gamma }_3{f}_3\left(x^{*}\right)-A_3{z}^{*}‖.\end{array}$$

Combining (3.1), (3.2) and (3.3), we have

$$\begin{array}{l}‖x_{n+1}-x^{*}‖+‖y_{n+1}-y^{*}‖+‖z_{n+1}-z^{*}‖\\ \le \left(1-{\mu }_n{\alpha }_n\left(\beta -\gamma \xi \right)\right)\left(‖x_n-x^{*}‖+‖y_n-y^{*}‖+‖z_n-z^{*}‖\right)\\ +{\mu }_n{\alpha }_n\left(‖{\gamma }_1{f}_1\left(y^{*}\right)-A_1{x}^{*}‖+‖{\gamma }_2{f}_2\left(z^{*}\right)-A_2{y}^{*}‖+‖{\gamma }_3{f}_3\left(x^{*}\right)-A_3{z}^{*}‖\right).\end{array}$$

By induction, we can derive that

$$\begin{array}{l}‖x_n-x^{*}‖+‖y_n-y^{*}‖+‖z_n-z^{*}‖\\ \le \max \{‖x_1-x^{*}‖+‖y_1-y^{*}‖+‖z_1-z^{*}‖,\\ \frac{‖{\gamma }_1{f}_1\left(y^{*}\right)-A_1{x}^{*}‖+‖{\gamma }_2{f}_2\left(z^{*}\right)-A_2{y}^{*}‖+‖{\gamma }_3{f}_3\left(x^{*}\right)-A_3{z}^{*}‖}{\beta -\gamma \xi }\},\end{array}$$

for every $n\in \mathbb{N}$. This implies that $\left\{x_n\right\},\left\{y_n\right\}$ and $\left\{z_n\right\}$ are bounded.

Step 2. Claim that $\underset{n\to \infty }{\lim }‖x_{n+1}-x_n‖=0$.

First, we let

$$u_n=P_C\left[{\alpha }_n{\gamma }_1{f}_1(y_n)+(I-{\alpha }_n{A}_1)T_n^1{x}_n\right],$$ $$v_n=P_C\left[{\alpha }_n{\gamma }_2{f}_2(z_n)+(I-{\alpha }_n{A}_2)T_n^2{y}_n\right]$$

and

$$w_n=P_C\left[{\alpha }_n{\gamma }_3{f}_3(x_n)+(I-{\alpha }_n{A}_3)T_n^3{z}_n\right].$$

Then, observe that

(3.4)$$\begin{array}{l}‖u_n-u_{n-1}‖\\ =\parallel P_C\left[{\alpha }_n{\gamma }_1{f}_1\left(y_n\right)+\left(I-{\alpha }_n{A}_1\right)T_n^1{x}_n\right]\\ -P_C\left[{\alpha }_{n-1}{\gamma }_1{f}_1\left(y_{n-1}\right)+\left(I-{\alpha }_{n-1}{A}_1\right)T_{n-1}^1{x}_{n-1}\right]\parallel \\ \le {\alpha }_n{\gamma }_1‖f_1\left(y_n\right)-f_1\left(y_{n-1}\right)‖+{\gamma }_1\left|{\alpha }_n-{\alpha }_{n-1}\right|‖f_1\left(y_{n-1}\right)‖\\ +‖I-{\alpha }_n{A}_1‖‖T_n^1{x}_n-T_{n-1}^1{x}_{n-1}‖\\ +‖\left(I-{\alpha }_n{A}_1\right)T_{n-1}^1{x}_{n-1}-\left(I-{\alpha }_{n-1}{A}_1\right)T_{n-1}^1{x}_{n-1}‖\\ \le {\alpha }_n{\gamma }_1{\xi }_1‖y_n-y_{n-1}‖+{\gamma }_1\left|{\alpha }_n-{\alpha }_{n-1}\right|‖f_1\left(y_{n-1}\right)‖\\ +\left(1-{\alpha }_n{\beta }_1\right)(\left(1-{\omega }_n\right)‖x_n-x_{n-1}‖\\ +\left|{\omega }_n-{\omega }_{n-1}\right|‖x_{n-1}‖+{\omega }_n‖W_1{x}_n-W_1{x}_{n-1}‖+\left|{\omega }_n-{\omega }_{n-1}\right|‖W_1{x}_{n-1}‖)\\ +\left|{\alpha }_n-{\alpha }_{n-1}\right|‖A_1{T}_n^1{x}_{n-1}‖\\ \le {\alpha }_n\gamma \xi ‖y_n-y_{n-1}‖+\left|{\alpha }_n-{\alpha }_{n-1}\right|\left(\gamma ‖f_1\left(y_{n-1}\right)‖+‖A_1{T}_n^1{x}_{n-1}‖\right)\\ +\left(1-{\alpha }_n\beta \right)(\left(1-{\omega }_n\right)‖x_n-x_{n-1}‖+\left|{\omega }_n-{\omega }_{n-1}\right|\left(‖x_{n-1}‖+‖W_1{x}_{n-1}‖\right)\\ +{\omega }_n‖W_1{x}_n-W_1{x}_{n-1}‖).\end{array}$$

By the definition of x_n, we obtain

$$\begin{array}{l}‖x_{n+1}-x_n‖\\ \le {\delta }_n‖x_n-x_{n-1}‖+\left|{\delta }_n-{\delta }_{n-1}\right|‖x_{n-1}‖\\ +{\eta }_n‖P_C\left(I-{\lambda }_n(I-S_1)\right)x_n-P_C\left(I-{\lambda }_{n-1}(I-S_1)\right)x_{n-1}‖\\ +\left|{\eta }_n-{\eta }_{n-1}\right|‖P_C\left(I-{\lambda }_{n-1}(I-S_1)\right)x_{n-1}‖+{\mu }_n‖u_n-u_{n-1}‖\\ +\left|{\mu }_n-{\mu }_{n-1}\right|‖u_{n-1}‖\\ \le {\delta }_n‖x_n-x_{n-1}‖+\left|{\delta }_n-{\delta }_{n-1}\right|‖x_{n-1}‖+{\eta }_n‖x_n-x_{n-1}‖\\ +{\eta }_n‖{\lambda }_n(I-S_1)x_n-{\lambda }_{n-1}(I-S_1)x_{n-1}‖\\ +\left|{\eta }_n-{\eta }_{n-1}\right|‖P_C\left(I-{\lambda }_{n-1}(I-S_1)\right)x_{n-1}‖\\ +{\mu }_n[{\alpha }_n\gamma \xi ‖y_n-y_{n-1}‖+\left|{\alpha }_n-{\alpha }_{n-1}\right|\left(\gamma ‖f_1\left(y_{n-1}\right)‖+‖A_1{T}_n^1{x}_{n-1}‖\right)\\ +\left(1-{\alpha }_n\beta \right)(\left(1-{\omega }_n\right)‖x_n-x_{n-1}‖+\left|{\omega }_n-{\omega }_{n-1}\right|\left(‖x_{n-1}‖+‖W_1{x}_{n-1}‖\right)\\ +{\omega }_n‖W_1{x}_n-W_1{x}_{n-1}‖)]+\left|{\mu }_n-{\mu }_{n-1}\right|‖u_{n-1}‖\end{array}$$ $$\begin{array}{l}\le \left(1-{\mu }_n\right)‖x_n-x_{n-1}‖+\left|{\delta }_n-{\delta }_{n-1}\right|‖x_{n-1}‖\\ +{\eta }_n{\lambda }_n‖(I-S_1)x_n-(I-S_1)x_{n-1}‖\\ +{\eta }_n\left|{\lambda }_n-{\lambda }_{n-1}\right|‖(I-S_1)x_{n-1}‖\\ +\left|{\eta }_n-{\eta }_{n-1}\right|‖P_C\left(I-{\lambda }_{n-1}(I-S_1)\right)x_{n-1}‖\\ +{\mu }_n{\alpha }_n\gamma \xi ‖y_n-y_{n-1}‖+\left|{\alpha }_n-{\alpha }_{n-1}\right|\left(\gamma ‖f_1\left(y_{n-1}\right)‖+‖A_1{T}_n^1{x}_{n-1}‖\right)\\ +{\mu }_n\left(1-{\alpha }_n\beta \right)‖x_n-x_{n-1}‖+\left|{\omega }_n-{\omega }_{n-1}\right|\left(‖x_{n-1}‖+‖W_1{x}_{n-1}‖\right)\\ +{\omega }_n‖W_1{x}_n-W_1{x}_{n-1}‖+\left|{\mu }_n-{\mu }_{n-1}\right|‖u_{n-1}‖\end{array}$$ (3.5)$$\begin{array}{l}\le \left(1-{\mu }_n{\alpha }_n\beta \right)‖x_n-x_{n-1}‖+{\mu }_n{\alpha }_n\gamma \xi ‖y_n-y_{n-1}‖+\left|{\delta }_n-{\delta }_{n-1}\right|‖x_{n-1}‖\\ +{\lambda }_n‖(I-S_1)x_n-(I-S_1)x_{n-1}‖+\left|{\lambda }_n-{\lambda }_{n-1}\right|‖(I-S_1)x_{n-1}‖\\ +\left|{\eta }_n-{\eta }_{n-1}\right|‖P_C\left(I-{\lambda }_{n-1}(I-S_1)\right)x_{n-1}‖\\ +\left|{\alpha }_n-{\alpha }_{n-1}\right|\left(\gamma ‖f_1\left(y_{n-1}\right)‖+‖A_1{T}_n^1{x}_{n-1}‖\right)\\ +\left|{\omega }_n-{\omega }_{n-1}\right|\left(‖x_{n-1}‖+‖W_1{x}_{n-1}‖\right)+{\omega }_n‖W_1{x}_n-W_1{x}_{n-1}‖\\ +\left|{\mu }_n-{\mu }_{n-1}\right|‖u_{n-1}‖.\end{array}$$

Using the same method as derived in (3.5), we have

(3.6)$$\begin{array}{l}‖y_{n+1}-y_n‖\\ \le \left(1-{\mu }_n{\alpha }_n\beta \right)‖y_n-y_{n-1}‖+{\mu }_n{\alpha }_n\gamma \xi ‖z_n-z_{n-1}‖+\left|{\delta }_n-{\delta }_{n-1}\right|‖y_{n-1}‖\\ +{\lambda }_n‖(I-S_2)y_n-(I-S_2)y_{n-1}‖+\left|{\lambda }_n-{\lambda }_{n-1}\right|‖(I-S_2)y_{n-1}‖\\ +\left|{\eta }_n-{\eta }_{n-1}\right|‖P_C\left(I-{\lambda }_{n-1}(I-S_2)\right)y_{n-1}‖\\ +\left|{\alpha }_n-{\alpha }_{n-1}\right|\left(\gamma ‖f_2\left(z_{n-1}\right)‖+‖A_2{T}_n^2{y}_{n-1}‖\right)\\ +\left|{\omega }_n-{\omega }_{n-1}\right|\left(‖y_{n-1}‖+‖W_2{y}_{n-1}‖\right)+{\omega }_n‖W_2{y}_n-W_2{y}_{n-1}‖\\ +\left|{\mu }_n-{\mu }_{n-1}\right|‖v_{n-1}‖\end{array}$$

and

(3.7)$$\begin{array}{l}‖z_{n+1}-z_n‖\\ \le \left(1-{\mu }_n{\alpha }_n\beta \right)‖z_n-z_{n-1}‖+{\mu }_n{\alpha }_n\gamma \xi ‖x_n-x_{n-1}‖+\left|{\delta }_n-{\delta }_{n-1}\right|‖z_{n-1}‖\\ +{\lambda }_n‖(I-S_3)z_n-(I-S_3)z_{n-1}‖+\left|{\lambda }_n-{\lambda }_{n-1}\right|‖(I-S_3)z_{n-1}‖\\ +\left|{\eta }_n-{\eta }_{n-1}\right|‖P_C\left(I-{\lambda }_{n-1}(I-S_3)\right)z_{n-1}‖\\ +\left|{\alpha }_n-{\alpha }_{n-1}\right|\left(\gamma ‖f_3\left(x_{n-1}\right)‖+‖A_3{T}_n^3{z}_{n-1}‖\right)\\ +\left|{\omega }_n-{\omega }_{n-1}\right|\left(‖z_{n-1}‖+‖W_3{z}_{n-1}‖\right)+{\omega }_n‖W_3{z}_n-W_3{z}_{n-1}‖\\ +\left|{\mu }_n-{\mu }_{n-1}\right|‖w_{n-1}‖.\end{array}$$

From (3.5), (3.6) and (3.7), then we get

$$\begin{array}{l}‖x_{n+1}-x_n‖+‖y_{n+1}-y_n‖+‖z_{n+1}-z_n‖\\ \le \left(1-{\mu }_n{\alpha }_n\left(\beta -\gamma \xi \right)\right)\left[‖x_n-x_{n-1}‖+‖y_n-y_{n-1}‖+‖z_n-z_{n-1}‖\right]\\ +\left|{\delta }_n-{\delta }_{n-1}\right|\left(‖x_{n-1}‖+‖y_{n-1}‖+‖z_{n-1}‖\right)+{\lambda }_n(‖(I-S_1)x_n-(I-S_1)x_{n-1}‖\\ +‖(I-S_2)y_n-(I-S_2)y_{n-1}‖+‖(I-S_3)z_n-(I-S_3)z_{n-1}‖)\\ +\left|{\lambda }_n-{\lambda }_{n-1}\right|\left(‖(I-S_1)x_{n-1}‖+‖(I-S_2)y_{n-1}‖+‖(I-S_3)z_{n-1}‖\right)\\ +\left|{\eta }_n-{\eta }_{n-1}\right|(‖P_C\left(I-{\lambda }_{n-1}(I-S_1)\right)x_{n-1}‖+‖P_C\left(I-{\lambda }_{n-1}(I-S_2)\right)y_{n-1}‖\end{array}$$ $$\begin{array}{l}+‖P_C\left(I-{\lambda }_{n-1}(I-S_3)\right)z_{n-1}‖)+\left|{\alpha }_n-{\alpha }_{n-1}\right|(\gamma (‖f_1\left(x_{n-1}\right)‖+‖f_2\left(y_{n-1}\right)‖\\ +‖f_3\left(z_{n-1}\right)‖)+\left(‖A_1{T}_n^1{x}_{n-1}‖+‖A_2{T}_n^2{y}_{n-1}‖+‖A_3{T}_n^3{z}_{n-1}‖\right))\\ +\left|{\omega }_n-{\omega }_{n-1}\right|(\left(‖x_{n-1}‖+‖y_{n-1}‖+‖z_{n-1}‖\right)\\ +\left(‖W_1{x}_{n-1}‖+‖W_2{y}_{n-1}‖+‖W_3{z}_{n-1}‖\right))\\ +{\omega }_n\left(‖W_1{x}_n-W_1{x}_{n-1}‖+‖W_2{y}_n-W_2{y}_{n-1}‖+‖W_3{z}_n-W_3{z}_{n-1}‖\right)\\ +\left|{\mu }_n-{\mu }_{n-1}\right|\left(‖u_{n-1}‖+‖v_{n-1}‖+‖w_{n-1}‖\right).\end{array}$$

Applying Lemma 2.4 and the condition(iii), (iv), we can conclude that

(3.8)$$‖x_{n+1}-x_n‖\to 0,‖y_{n+1}-y_n‖\to 0\text{ and }‖z_{n+1}-z_n‖\to 0\text{ as }n\to \infty .$$

Step 3. Prove that $\underset{n\to \infty }{\lim }‖u_n-P_C\left(I-{\lambda }_n(I-S_1)\right)u_n‖=\underset{n\to \infty }{\lim }‖u_n-T_n^1{u}_n‖=0$.

To show this, take ${\tilde{u}}_n={\alpha }_n{\gamma }_1{f}_1(y_n)+(I-{\alpha }_n{A}_1)T_n^1{x}_n$. Then we derive that

$$\begin{array}{l}{‖x_{n+1}-x^{*}‖}^2\\ ={‖{\delta }_n\left(x_n-x^{*}\right)+{\eta }_n\left(P_C\left(I-{\lambda }_n(I-S_1)\right)x_n-x^{*}\right)+{\mu }_n\left(u_n-x^{*}\right)‖}^2\\ \le {\delta }_n{‖x_n-x^{*}‖}^2+{\eta }_n{‖P_C\left(I-{\lambda }_n(I-S_1)\right)x_n-x^{*}‖}^2+{\mu }_n{‖u_n-x^{*}‖}^2\\ -{\delta }_n{\eta }_n{‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖}^2\\ \le \left(1-{\mu }_n\right){‖x_n-x^{*}‖}^2+{\mu }_n{‖{\alpha }_n\left({\gamma }_1{f}_1\left(y_n\right)-A_1{T}_n^1{x}_n\right)+\left(T_n^1{x}_n-x^{*}\right)‖}^2\\ -{\delta }_n{\eta }_n{‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖}^2\\ \le \left(1-{\mu }_n\right){‖x_n-x^{*}‖}^2\\ +{\mu }_n[{‖T_n^1{x}_n-x^{*}‖}^2+2{\alpha }_n⟨{\gamma }_1{f}_1\left(y_n\right)-A_1{T}_n^1{x}_n,{\tilde{u}}_n-x^{*}⟩]\\ -{\delta }_n{\eta }_n{‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖}^2\\ \le {‖x_n-x^{*}‖}^2+2{\mu }_n{\alpha }_n‖{\gamma }_1{f}_1\left(y_n\right)-A_1{T}_n^1{x}_n‖‖{\tilde{u}}_n-x^{*}‖\\ -{\delta }_n{\eta }_n{‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖}^2,\end{array}$$

which implies that

$$\begin{array}{l}{\delta }_n{\eta }_n{‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖}^2\\ \le {‖x_n-x^{*}‖}^2-{‖x_{n+1}-x^{*}‖}^2+2{\mu }_n{\alpha }_n‖{\gamma }_1{f}_1\left(y_n\right)-A_1{T}_n^1{x}_n‖‖{\tilde{u}}_n-x^{*}‖\\ \le ‖x_n-x_{n+1}‖\left(‖x_n-x^{*}‖+‖x_{n+1}-x^{*}‖\right)\\ +2{\mu }_n{\alpha }_n‖{\gamma }_1{f}_1\left(y_n\right)-A_1{T}_n^1{x}_n‖‖{\tilde{u}}_n-x^{*}‖.\end{array}$$

By (3.8), the condition (i) and (ii), thus we get

(3.9)$$‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖\to 0\text{ as }n\to \infty .$$

Observe that

$$x_{n+1}-x_n={\eta }_n\left(P_C\left(I-{\lambda }_n(I-S_1)\right)x_n-x_n\right)+{\mu }_n\left(u_n-x_n\right).$$

This follows that

$${\mu }_n‖u_n-x_n‖\le {\eta }_n‖P_C\left(I-{\lambda }_n(I-S_1)\right)x_n-x_n‖+‖x_{n+1}-x_n‖.$$

From (3.8) and (3.9), we obtain

(3.10)$$‖u_n-x_n‖\to 0\text{ as }n\to \infty .$$

Observe that

$$\begin{array}{l}‖u_n-P_C\left(I-{\lambda }_n(I-S_1)\right)u_n‖\\ \le ‖u_n-x_n‖+‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖\\ +‖P_C\left(I-{\lambda }_n(I-S_1)\right)x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)u_n‖\\ \le 2‖u_n-x_n‖+‖x_n-P_C\left(I-{\lambda }_n(I-S_1)\right)x_n‖+{\lambda }_n‖\left(I-S_1\right)x_n-\left(I-S_1\right)u_n‖.\end{array}$$

Hence, by (3.9), (3.10) and the condition (iii), we obtain

(3.11)$$‖u_n-P_C\left(I-{\lambda }_n(I-S_1)\right)u_n‖\to 0\text{ as }n\to \infty .$$

Applying the same argument as (3.11), we also obtain

(3.12)$$‖v_n-P_C\left(I-{\lambda }_n(I-S_2)\right)v_n‖\to 0\text{ and }‖w_n-P_C\left(I-{\lambda }_n(I-S_3)\right)w_n‖\to 0\text{ as }n\to \infty .$$

Consider

$$‖x_{n+1}-u_n‖\le ‖x_{n+1}-x_n‖+‖x_n-u_n‖,$$

by (3.8) and (3.10), we have

(3.13)$$‖x_{n+1}-u_n‖\to 0\text{ as }n\to \infty .$$

Since

$$\begin{array}{l}‖x_n-T_n^1{x}_n‖\le ‖x_n-x_{n+1}‖+‖x_{n+1}-u_n‖+‖u_n-T_n^1{x}_n‖\\ \le ‖x_n-x_{n+1}‖+‖x_{n+1}-u_n‖+‖{\tilde{u}}_n-T_n^1{x}_n‖\\ =‖x_n-x_{n+1}‖+‖x_{n+1}-u_n‖+{\alpha }_n‖{\gamma }_1{f}_1\left(y_n\right)-A_1{T}_n^1{x}_n‖,\end{array}$$

from (3.8), (3.13) and the condition (i), we get

(3.14)$$‖x_n-T_n^1{x}_n‖\to 0\text{ as }n\to \infty .$$

Consider

$$\begin{array}{l}‖u_n-T_n^1{u}_n‖\le ‖u_n-x_n‖+‖x_n-T_n^1{x}_n‖+‖T_n^1{x}_n-T_n^1{u}_n‖\\ \le 2‖u_n-x_n‖+‖x_n-T_n^1{x}_n‖+{\omega }_n‖W_1{x}_n-W_1{u}_n‖.\end{array}$$

Therefore, by (3.10), (3.14) and the condition (iii), we have

(3.15)$$‖u_n-T_n^1{u}_n‖\to 0\text{ as }n\to \infty .$$

Applying the same method as (3.15), we also have

(3.16)$$‖v_n-T_n^2{v}_n‖\to 0\text{ and }‖w_n-T_n^3{w}_n‖\to 0\text{ as }n\to \infty .$$

Step 4. Claim that

$$\begin{array}{l}\underset{n\to \infty }{\lim \sup }⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩\le 0,\text{ where }\\ \tilde{x}=P_{{\Omega }_1}\left((I-A_1)\tilde{x}+{\gamma }_1{f}_1\left(\tilde{y}\right)\right).\end{array}$$

First, take a subsequence $\left\{u_{n_k}\right\}$ of $\left\{u_n\right\}$ such that

(3.17)$$\underset{n\to \infty }{\lim \sup }⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩=\underset{k\to \infty }{\lim }⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_{n_k}-\tilde{x}⟩.$$

Since $\left\{x_n\right\}$ is bounded, then we can assume that $x_{n_k}\rightharpoonup \widehat{x}$ as $k\to \infty$. From (3.10), we obtain $u_{n_k}\rightharpoonup \widehat{x}$ as $k\to \infty$.

Next, assume $\widehat{x}\notin Fix\left(S_1\right)$. Since $Fix\left(S_1\right)=Fix\left(P_C\left(I-{\lambda }_{n_k}(I-S_1)\right)\right)$, then we get $\widehat{x}\ne P_C\left(I-{\lambda }_{n_k}(I-S_1)\right)\widehat{x}$.

By nonexpansiveness of P_C, (3.11), the condition (iii) and the Opial's condition, we obtain

$$\begin{array}{l}\underset{k\to \infty }{\underset{k\to \infty }{\lim \inf }‖u_{n_k}-\widehat{x}‖<\lim \inf }‖u_{n_k}-P_C\left(I-{\lambda }_{n_k}(I-S_1)\right)\widehat{x}‖\\ \underset{k\to \infty }{\le \lim \inf }[‖u_{n_k}-P_C\left(I-{\lambda }_{n_k}(I-S_1)\right)u_{n_k}‖\\ +‖P_C\left(I-{\lambda }_{n_k}(I-S_1)\right)u_{n_k}-P_C\left(I-{\lambda }_{n_k}(I-S_1)\right)\widehat{x}‖]\\ \underset{k\to \infty }{\le \lim \inf }[‖u_{n_k}-P_C\left(I-{\lambda }_{n_k}(I-S_1)\right)u_{n_k}‖+‖u_{n_k}-\widehat{x}‖\\ +{\lambda }_{n_k}‖(I-S_1)u_{n_k}-(I-S_1)\widehat{x}‖]\\ \underset{k\to \infty }{=\lim \inf }‖u_{n_k}-\widehat{x}‖.\end{array}$$

This is a contradiction. Therefore

(3.18)$$\widehat{x}\in Fix\left(S_1\right).$$

Assume that $\widehat{x}\notin Fix\left(W_1\right)$. Because $Fix\left(W_1\right)=Fix\left(T_{n_k}^1\right)$, then we have $\widehat{x}\ne T_{n_k}^1\widehat{x}$.

From (3.15) and the Opial's condition, we deduce that

$$\begin{array}{l}\underset{k\to \infty }{\underset{k\to \infty }{\lim \inf }‖u_{n_k}-\widehat{x}‖<\lim \inf }‖u_{n_k}-T_{n_k}^1\widehat{x}‖\\ \underset{k\to \infty }{\le \lim \inf }\left[‖u_{n_k}-T_{n_k}^1{u}_{n_k}‖+‖T_{n_k}^1{u}_{n_k}-T_{n_k}^1\widehat{x}‖\right]\\ \le \underset{k\to \infty }{\lim \inf }[‖u_{n_k}-T_{n_k}^1{u}_{n_k}‖+‖u_{n_k}-\widehat{x}‖\\ +{\omega }_{n_k}‖\left(I-W_1\right)u_{n_k}-\left(I-W_1\right)\widehat{x}‖]\\ \underset{k\to \infty }{=\lim \inf }‖u_{n_k}-\widehat{x}‖.\end{array}$$

This is a contradiction. Thus we obtain

(3.19)$$\widehat{x}\in Fix\left(W_1\right).$$

By (3.18) and (3.19), this yields that

(3.20)$$\widehat{x}\in {\Omega }_1=Fix\left(S_1\right)\cap Fix\left(W_1\right).$$

Since $x_{n_k}\rightharpoonup \widehat{x}$ as $k\to \infty$, (3.20) and Lemma 2.1, we can derive that

(3.21)$$\begin{array}{l}\underset{n\to \infty }{\lim \sup }⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩=\underset{k\to \infty }{\lim }⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_{n_k}-\tilde{x}⟩\\ =⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},\widehat{x}-\tilde{x}⟩\\ =⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x}+\tilde{x}-\tilde{x},\widehat{x}-\tilde{x}⟩\\ \le 0.\end{array}$$

Following the same method as (3.21), we easily obtain that

(3.22)$$\underset{n\to \infty }{\lim \sup }⟨{\gamma }_2{f}_2\left(\tilde{z}\right)-A_2\tilde{y},v_n-\tilde{y}⟩\le 0\text{ and }\underset{n\to \infty }{\lim \sup }⟨{\gamma }_3{f}_3\left(\tilde{x}\right)-A_3\tilde{z},w_n-\tilde{z}⟩\le 0.$$

Step 5. Finally, Prove that the sequence $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ converge strongly to $\tilde{x}=P_{{\Omega }_1}\left((I-A_1)\tilde{x}+{\gamma }_1{f}_1\left(\tilde{y}\right)\right)$, $\tilde{y}=P_{{\Omega }_2}\left((I-A_2)\tilde{y}+{\gamma }_2{f}_2\left(\tilde{z}\right)\right)$ and $\tilde{z}=P_{{\Omega }_3}\left((I-A_3)\tilde{z}+{\gamma }_3{f}_3\left(\tilde{x}\right)\right)$, respectively.

By firmly-nonexpansiveness of P_C, we derive that

$$\begin{array}{l}{‖u_n-\tilde{x}‖}^2={‖P_C\widetilde{u_n}-\tilde{x}‖}^2\\ \le ⟨\widetilde{u_n}-\tilde{x},u_n-\tilde{x}⟩\\ =⟨{\alpha }_n\left({\gamma }_1{f}_1\left(y_n\right)-A_1\tilde{x}\right)+\left(I-{\alpha }_n{A}_1\right)\left(T_n^1{x}_n-\tilde{x}\right),u_n-\tilde{x}⟩\\ ={\alpha }_n⟨{\gamma }_1{f}_1\left(y_n\right)-A_1\tilde{x},u_n-\tilde{x}⟩+⟨\left(I-{\alpha }_n{A}_1\right)\left(T_n^1{x}_n-\tilde{x}\right),u_n-\tilde{x}⟩\\ \le {\alpha }_n{\gamma }_1⟨f_1\left(y_n\right)-f_1\left(\tilde{y}\right),u_n-\tilde{x}⟩+{\alpha }_n⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩\\ +‖\left(I-{\alpha }_n{A}_1\right)\left(T_n^1{x}_n-\tilde{x}\right)‖‖u_n-\tilde{x}‖\\ \le {\alpha }_n{\gamma }_1{\xi }_1‖y_n-\tilde{y}‖‖u_n-\tilde{x}‖+{\alpha }_n⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩\\ +\left(1-{\alpha }_n{\beta }_1\right)‖x_n-\tilde{x}‖‖u_n-\tilde{x}‖\\ \le \frac{{\alpha }_n\gamma \xi }{2}\left({‖y_n-\tilde{y}‖}^2+{‖u_n-\tilde{x}‖}^2\right)+{\alpha }_n⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩\\ +\frac{1-{\alpha }_n\beta }{2}\left({‖x_n-\tilde{x}‖}^2+{‖u_n-\tilde{x}‖}^2\right)\\ =\frac{{\alpha }_n\gamma \xi }{2}{‖y_n-\tilde{y}‖}^2+\frac{1-{\alpha }_n\beta }{2}{‖x_n-\tilde{x}‖}^2+\frac{1-{\alpha }_n(\beta -\gamma \xi )}{2}{‖u_n-\tilde{x}‖}^2\\ +{\alpha }_n⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩,\end{array}$$

which yields that

(3.23)$$\begin{array}{l}{‖u_n-\tilde{x}‖}^2\le \frac{{\alpha }_n\gamma \xi }{1+{\alpha }_n(\beta -\gamma \xi )}{‖y_n-\tilde{y}‖}^2+\frac{1-{\alpha }_n\beta }{1+{\alpha }_n(\beta -\gamma \xi )}{‖x_n-\tilde{x}‖}^2\\ +\frac{2{\alpha }_n}{1+{\alpha }_n(\beta -\gamma \xi )}⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩.\end{array}$$

From the definition of x_n and (3.23), we get

(3.24)$$\begin{array}{l}{‖x_{n+1}-\tilde{x}‖}^2\\ \le {\delta }_n{‖x_n-\tilde{x}‖}^2+{\eta }_n{‖P_C\left(I-{\lambda }_n(I-S_1)\right)x_n-\tilde{x}‖}^2+{\mu }_n{‖u_n-\tilde{x}‖}^2\\ \le \left(1-{\mu }_n\right){‖x_n-\tilde{x}‖}^2+{\mu }_n[\frac{{\alpha }_n\gamma \xi }{1+{\alpha }_n(\beta -\gamma \xi )}{‖y_n-\tilde{y}‖}^2+\frac{1-{\alpha }_n\beta }{1+{\alpha }_n(\beta -\gamma \xi )}{‖x_n-\tilde{x}‖}^2\\ +\frac{2{\alpha }_n}{1+{\alpha }_n(\beta -\gamma \xi )}⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩]\\ =\left(1-{\mu }_n+\frac{{\mu }_n\left(1-{\alpha }_n\beta \right)}{1+{\alpha }_n(\beta -\gamma \xi )}\right){‖x_n-\tilde{x}‖}^2+\frac{{\mu }_n{\alpha }_n\gamma \xi }{1+{\alpha }_n(\beta -\gamma \xi )}{‖y_n-\tilde{y}‖}^2\\ +\frac{2{\mu }_n{\alpha }_n}{1+{\alpha }_n(\beta -\gamma \xi )}⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩\\ =\left(1-\frac{{\mu }_n\left(1+{\alpha }_n(\beta -\gamma \xi )\right)-{\mu }_n\left(1-{\alpha }_n\beta \right)}{1+{\alpha }_n(\beta -\gamma \xi )}\right){‖x_n-\tilde{x}‖}^2+\frac{{\mu }_n{\alpha }_n\gamma \xi }{1+{\alpha }_n(\beta -\gamma \xi )}{‖y_n-\tilde{y}‖}^2\\ =\left(1-\frac{{\mu }_n{\alpha }_n(2\beta -\gamma \xi )}{1+{\alpha }_n(\beta -\gamma \xi )}\right){‖x_n-\tilde{x}‖}^2+\frac{{\mu }_n{\alpha }_n\gamma \xi }{1+{\alpha }_n(\beta -\gamma \xi )}{‖y_n-\tilde{y}‖}^2\\ +\frac{2{\mu }_n{\alpha }_n}{1+{\alpha }_n(\beta -\gamma \xi )}⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩.\end{array}$$

Similarly, as derived above, we also have

(3.25)$$\begin{array}{l}{‖y_{n+1}-\tilde{y}‖}^2\le \left(1-\frac{{\mu }_n{\alpha }_n(2\beta -\gamma \xi )}{1+{\alpha }_n(\beta -\gamma \xi )}\right){‖y_n-\tilde{y}‖}^2+\frac{{\mu }_n{\alpha }_n\gamma \xi }{1+{\alpha }_n(\beta -\gamma \xi )}{‖z_n-\tilde{z}‖}^2\\ +\frac{2{\mu }_n{\alpha }_n}{1+{\alpha }_n(\beta -\gamma \xi )}⟨{\gamma }_2{f}_2\left(\tilde{z}\right)-A_2\tilde{y},v_n-\tilde{y}⟩\end{array}$$

and

(3.26)$$\begin{array}{l}{‖z_{n+1}-\tilde{z}‖}^2\le \left(1-\frac{{\mu }_n{\alpha }_n(2\beta -\gamma \xi )}{1+{\alpha }_n(\beta -\gamma \xi )}\right){‖z_n-\tilde{z}‖}^2+\frac{{\mu }_n{\alpha }_n\gamma \xi }{1+{\alpha }_n(\beta -\gamma \xi )}{‖x_n-\tilde{x}‖}^2\\ +\frac{2{\mu }_n{\alpha }_n}{1+{\alpha }_n(\beta -\gamma \xi )}⟨{\gamma }_3{f}_3\left(\tilde{x}\right)-A_3\tilde{z},w_n-\tilde{z}⟩.\end{array}$$

From (3.24), (3.25) and (3.26), we deduces that

(3.27)$$\begin{array}{l}{‖x_{n+1}-\tilde{x}‖}^2+{‖y_{n+1}-\tilde{y}‖}^2+{‖z_{n+1}-\tilde{z}‖}^2\\ \le \left(1-\frac{2{\mu }_n{\alpha }_n(\beta -\gamma \xi )}{1+{\alpha }_n(\beta -\gamma \xi )}\right)\left({‖x_n-\tilde{x}‖}^2+{‖y_n-\tilde{y}‖}^2+{‖z_n-\tilde{z}‖}^2\right)\\ +\frac{2{\mu }_n{\alpha }_n}{1+{\alpha }_n(\beta -\gamma \xi )}(⟨{\gamma }_1{f}_1\left(\tilde{y}\right)-A_1\tilde{x},u_n-\tilde{x}⟩+⟨{\gamma }_2{f}_2\left(\tilde{z}\right)-A_2\tilde{y},v_n-\tilde{y}⟩\\ +⟨{\gamma }_3{f}_3\left(\tilde{x}\right)-A_3\tilde{z},w_n-\tilde{z}⟩).\end{array}$$

By (3.21), (3.22), the condition (i) and Lemma 2.4, this implies by (3.27) that the sequences $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ converge strongly to $\tilde{x}=P_{{\Omega }_1}\left((I-A_1)\tilde{x}+{\gamma }_1{f}_1\left(\tilde{y}\right)\right)$, $\tilde{y}=P_{{\Omega }_2}\left((I-A_2)\tilde{y}+{\gamma }_2{f}_2\left(\tilde{z}\right)\right)$ and $\tilde{z}=P_{{\Omega }_3}\left((I-A_3)\tilde{z}+{\gamma }_3{f}_3\left(\tilde{x}\right)\right)$, respectively. This completes the proof.

The following Corollary is a direct consequence of Theorem 3.1.

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. For i=1,2,3, let $f_i:C\to C$ be a contractive mappings with a coefficient ξ_i and $\xi ={\max }_{i\in 1,2,3}{\xi }_i$ and let $W_i:C\to C$ be ρ_i-strictly pseudo-nonspreading mapping with $Fix(W_i)\ne \varnothing$. Define a mapping $T_n^i:C\to C$ by $T_n^{i}x=\left(1-{\omega }_n\right)x+{\omega }_n{W}_{i}x,$ for all x ∈ C and i=1,2,3. Let $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ be sequences generated by $x_1,y_1,z_1\in C$ and

(3.28)$$\left\{\begin{array}{l}\begin{array}{c}{x}_{n+1}=\left(1-{\mu }_n\right)x_n+{\mu }_n\left[{\alpha }_n{f}_1(y_n)+(1-{\alpha }_n)T_n^1{x}_n\right],\\ y_{n+1}=\left(1-{\mu }_n\right)y_n+{\mu }_n\left[{\alpha }_n{f}_2(z_n)+(1-{\alpha }_n)T_n^2{y}_n\right],\\ z_{n+1}=\left(1-{\mu }_n\right)z_n+{\mu }_n\left[{\alpha }_n{f}_3(x_n)+(1-{\alpha }_n)T_n^3{z}_n\right],\end{array}\hfill \end{array}\right.$$

for $n\ge 1$, where $\left\{{\alpha }_n\right\}$, $\left\{{\mu }_n\right\}\subset (0,1)$ and $\left\{{\omega }_n\right\}\subset (0,1-\rho )$, where $\rho ={\min }_{i\in \{1,2,3\}}{\rho }_i$, satisfying the following conditions:

• (i) $\underset{n\to \infty }{\lim }{\alpha }_n=0$ and ${\displaystyle \sum _{n=1}^{\infty }{\alpha }_n}=\infty$;

• (ii) $0<\tau \le {\mu }_n,\le \upsilon <1$, for some $\tau ,\upsilon >0$;

• (iii) ${\displaystyle \sum _{n=1}^{\infty }{\omega }_n}<\infty$;

• (iv) ${\displaystyle \sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_n\right|}<\infty$, ${\displaystyle \sum _{n=1}^{\infty }\left|{\mu }_{n+1}-{\mu }_n\right|}<\infty$, ${\displaystyle \sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_n\right|}<\infty$,${\displaystyle \sum _{n=1}^{\infty }\left|{\omega }_{n+1}-{\omega }_n\right|}<\infty$

Then the sequences $\left\{x_n\right\}$, $\left\{y_n\right\}$ and $\left\{z_n\right\}$ converge strongly to $\tilde{x}=P_{Fix(W_1)}{f}_1\left(\tilde{y}\right)$, $\tilde{y}=P_{Fix(W_2)}{f}_2\left(\tilde{z}\right)$ and $\tilde{z}=P_{Fix(W_3)}{f}_3\left(\tilde{x}\right)$, respectively.

Proof. For each i=1,2,3, put $S_i\equiv I$, ${\gamma }_i=1$ and $A_i\equiv I$. Then, by Theorem 3.1, we obtain the desired result.