Approximation of the Solution of Split Equality Fixed Point Problem for Family of Multivalued Demicontractive Operators with Application

Abstract

In this paper, a new viscosity type iterative algorithm is used for obtaining a strong convergence result of split equality fixed point solutions for infinite families of multivalued demicontractive mappings in real Hilbert spaces. Our iterative scheme is based on choosing the step-sizes without calculating or estimating the operator norms and the condition of hemicompactness was relaxed to prove the strong convergence result. As an application, the solution of split convex minimization problem was approximated. The result presented herein unifies and extends several comparable results in the literature.

1. Introduction and Preliminaries

Let H be a Hilbert space, C a non-empty closed and convex set in H and T a self mapping on C. The set $\{p^{*}\in C:p^{*}=T{p}^{*}\}$ of all fixed points of T is denoted by $F\left(T\right).$

A mapping T is called Lipschitzian if there exists a constant $L>0$, such that

$$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel$$

holds for all $x,y\in C.$ If in the above inequality, we restrict L to vary only in the interval $(0,1),$ then T is called a contraction. The mapping T is called non-expansive, if we set $L=1$ in the above inequality. Non-expansive mappings not only generalize contraction mappings but also appear in several applications of a practical nature, see for example [1].

Let us recall the following definitions.

Definition 1.

Let T be self mapping on H with $F\left(T\right)\ne \varnothing$. Then for all $x\in H$ and $p^{*}\in F\left(T\right)$, the mapping T is called

• Quasi-non-expansive if

  $$\parallel Tx-p^{*}\parallel \le \parallel x-p^{*}\parallel ,$$
• Firmly quasi-non-expansive if

  $$\parallel Tx-p^{*}{\parallel }^2\le \parallel x-p^{*}{\parallel }^2-{\parallel x-Tx\parallel }^2,$$
• Quasi-pseudo contraction if

  $$\parallel Tx-p^{*}{\parallel }^2\le \parallel x-p^{*}{\parallel }^2+{\parallel x-Tx\parallel }^2,$$
• k-demicontraction if there is a constant $k\in (-\infty ,1)$ such that

  $$\parallel Tx-p^{*}{\parallel }^2\le \parallel x-p^{*}{\parallel }^2+k{\parallel x-Tx\parallel }^2.$$

Note that the class of k-demicontraction is more general class of mappings than that of quasi-non-expansive mappings (see Example 1.1 of [2]). Moreover, if we put $k=-1$ we obtain firmly quasi-non-expansive mapping.

The following concepts for multivalued notations are also needed in the sequel.

The set $CB\left(H\right)$ denotes the collection of all non-empty closed and bounded subsets of H. Note that, the mapping $T:H\to CB\left(H\right)$ can be identified by its graph given by $\left\{\right(x,y):$$x\in H$ and $y\in Tx\}$. An element $x\in H$ is called a fixed point (strict fixed point) of T if $x\in T\left(x\right)$ ($T\left(x\right)=\left\{x\right\}$). Let $\Pi$ be the Hausdorff metric on $CB\left(H\right),$ given by

$$\Pi (C,D)=max\{\underset{x\in C}{sup}d(x,D),\underset{y\in D}{sup}d(y,C)\}$$

where, $C,D\in CB\left(H\right)$ and $d(x,C)={inf}_{y\in H}\parallel x-y\parallel$.

A multivalued mapping $T:C\to CB\left(H\right)$ is said to be L-Lipschitzian if there exists a constant $L>0$, such that for all $x,y\in C,$ we have

$$\Pi \left(Tx,Ty\right)\le L\parallel x-y\parallel .$$

If in the above inequality, we take L in $(0,1),$ then T is called multivalued contraction while T is called multivalued non-expansive if $L=1$.

A multivalued mapping $T:C\to CB\left(H\right)$ is said to be quasi-non-expansive if $F\left(T\right)\ne \varnothing$, and for $p^{*}\in F\left(T\right)$ and $x\in C,$ we have

$$\Pi \left(Tx,T{p}^{*}\right)\le \parallel x-p^{*}\parallel .$$

The mapping T is said to be k-strictly pseudo contractive in the sense of [3], if there exists $k\in (0,1)$, such that $x,y\in C$ and $u\in Tx,v\in Ty$ we have

$${\Pi }^2\left(Tx,Ty\right)\le {\parallel x-y\parallel }^2+k{\parallel (x-y)-(u-v)\parallel }^2.$$

and T is said to be demicontractive in the sense of [4], if $F\left(T\right)\ne \varnothing$, and for $p^{*}\in F\left(T\right)$, $x\in C$ there exists $k\in (-\infty ,1)$ such that the following holds:

$${\Pi }^2\left(Tx,T{p}^{*}\right)\le {\parallel x-p^{*}\parallel }^2+k{d}^2(x,Tx).$$

The mapping T is said to be demiclosed at 0, if for any sequence $\left\{x_n\right\}$ in H with $x_n\rightharpoonup x^{*}$ and $d(x_n,T{x}_n)\to 0$, we have $x^{*}\in T{x}^{*}$.

Every multivalued non-expansive mapping T with $F\left(T\right)\ne \varnothing$ is obviously a quasi-non-expansive. However, the converse does not hold in general, see [5]. Additionally, a multivalued quasi-non-expansive mapping is a multivalued demicontractive type mapping but every demicontractive mapping may not multivalued quasi-non-expansive mapping, (for details see [2]).

The metric projection $P_C:H\to C$ is defined as follows:

For each $x\in H,$ there is a unique element $x\in C$, such that

$$\parallel x-P_{C}x\parallel =inf\{\parallel x-q\parallel :q\in C\}.$$

Note that, for any $x\in H,$

$$\langle x-P_{C}x,P_{C}x-z\rangle \ge 0,\forall z\in C.$$

(1)

It is also known that the metric projection $P_C$ is firmly non-expansive mapping, that is, for all $x,y\in H$, we have

$$\parallel P_{C}x-P_C{y\parallel }^2\le {\parallel x-y\parallel }^2+{\parallel (I-P_C)x-(I-P_C)y\parallel }^2.$$

(2)

For more details about metric projection we refer to [6].

Throughout the paper, we denote the strong and weak converge of a sequence $\left\{x_n\right\}$ to a point $x^{*}$ by $x_n\to x^{*}$ and $x_n\rightharpoonup x^{*}$, respectively.

Now we define the split feasibility problem (SFP) [7] as follows;

Let $H_1$ and $H_2$ be two Hilbert spaces and C and Q two closed and convex subsets of $H_1$ and $H_2,$ respectively, and $A:H_1\to H_2$ be a bounded linear operator.

$$\mathrm{Find}\mathrm{a}\mathrm{point}{x}^{*}\in C\mathrm{such}\mathrm{that}A\left(x^{*}\right)\in Q.$$

That is, the split feasibility problem is the problem of finding a point of a closed and convex subset of a Hilbert space, such that the image of the point under bounded linear operator is in a closed convex subset of another Hilbert space. The SFP has attracted the attention of many researchers due to its applications in real-world problems, such as image recognizing, signal processing, intensity-modulated radiation therapy, and many others. For further details about the results dealing with SPF, we refer to [8,9,10,11,12,13,14,15,16]. Later on, different variants of split feasibility problems were studied and one of the interesting generalizations of SFP is the split fixed point problem (SFPP) which was first studied by Censor and Segal [17] in 2009 for the class of quasi-firmly non-expansive mappings.

Let us recall that the SFPP is the problem of finding a common fixed point of mapping on a space $H_1$ whose image under a bounded linear operator is a fixed point of another family of mappings on the image space $H_2$.

The split fixed point problem is defined as follows

$$\mathrm{Find}{p}^{*}\in F\left(S\right)\mathrm{such}\mathrm{that}A\left(p^{*}\right)\in F\left(T\right)$$

(3)

where $A:H_1\to H_2$ is a bounded linear operator, $H_1$ and $H_2$ are real Hilbert spaces and $S:H_1\to H_1$ and $T:H_2\to H_2$. Afterward, many researchers studied the SFPP for different classes of mappings (see [18,19,20,21,22,23,24,25,26,27,28]).

Moudafi and Al-Shemas [29] introduced the split equality fixed point problem (SEFPP) which generalizes the SFFP. The SEFPP is defined as follows:

$$\mathrm{Find}{p}^{*}\in F\left(S\right)\mathrm{and}{q}^{*}\in F\left(T\right)\mathrm{such}\mathrm{that}A{p}^{*}=B{q}^{*},$$

(4)

where $A:H_1\to H_3$ and $B:H_2\to H_3$ are bounded linear operators, $H_1$, $H_2$ and $H_3$ are real Hilbert spaces and $S:H_1\to H_1$ and $T:H_2\to H_2$.

Note that if $H_2=H_3$ and $B=I$ (where I is the identity map on $H_2$) in (4), then problem (4) reduces to SFP and (3).

To approximate the solution of SEFPP (4), Moudafi and Al-Shemas [29] proposed the following numerical scheme:

$$\begin{array}{cc}\hfill \mathrm{Take}{x}_0& \in H_1,\mathrm{and}{y}_0\in H_2.\mathrm{Define}\hfill \\ \hfill & x_{n+1}=S\left(x_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)\right)\mathrm{and}\hfill \\ \hfill & y_{n+1}=T\left(y_n-{\gamma }_n{B}^{*}(A{x}_n-B{y}_n)\right),n\in N,\hfill \end{array}$$

where $S:H_1\to H_1$ and $T:H_2\to H_2$ are firmly quasi-non-expansive mappings and $\left\{{\gamma }_n\right\}$ is the sequence of parameters which depends on the spectral radius of $A^{*}A$ and $B^{*}B$.

Shehu [5] introduced the multiple set split equality fixed point problem (MSSEFPP) for infinite families of multivalued quasi-non-expansive mappings as follows

$$\mathrm{Find}{p}^{*}\in \bigcap _{i=1}^{\infty }F\left(S_i\right)\mathrm{and}{q}^{*}\in \bigcap _{i=1}^{\infty }F\left(T_i\right)\mathrm{such}\mathrm{that}A{p}^{*}=B{q}^{*},$$

(5)

where $S_i:H_1\to CB\left(H_1\right)$ and $T_i:H_2\to CB\left(H_2\right)$ are two infinite families of multivalued quasi-non-expansive mappings. The simultaneous numerical method to approximate the solution of (MSSEFPP) was proposed by Shehu which is given by

$$\begin{array}{cc}\hfill \mathrm{Take}{x}_0& \in H_1,\mathrm{and}{y}_0\in H_2.\mathrm{Define}\hfill \\ \hfill u_n=& x_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)\hfill \\ \hfill x_{n+1}=& {\eta }_{n}u+({\alpha }_{0,n}-{\eta }_n)u_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{z}_{i,n},\hfill \\ \hfill v_n=& y_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n)\hfill \\ \hfill y_{n+1}=& {\eta }_{n}v+({\alpha }_{0,n}-{\eta }_n)v_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{w}_{i,n},\hfill \end{array}$$

(6)

where $z_{i,n}\in S_i{u}_n$ and $w_{i,n}\in T_i{u}_n$, ${\eta }_n\in (0,1)$ and ${\gamma }_n$ is the step size and ${\alpha }_{i,n}$ is a sequence of non-negative real numbers satisfying ${\sum }_{i=0}^{\infty }{\alpha }_{i,n}$.

The generalized split common fixed point problem is defined as follows

$$\mathrm{Find}{p}^{*}\in \bigcap _{i=1}^{\infty }F\left(S_i\right)\mathrm{such}\mathrm{that}A{p}^{*}\in \bigcap _{i=1}^{\infty }F\left(T_i\right).$$

(7)

The solution set of problem (7) denoted by ${\Omega }_{GSCFPP}$. If we take B as identity mapping in (5) then the problem (5) reduces to problem (7). For $i=1$ the problem (7) reduces to the split common fixed point problem (SCFPP), which is defined as follows

$$\mathrm{Find}{p}^{*}\in F\left(S\right)\mathrm{such}\mathrm{that}A{p}^{*}\in F\left(T\right).$$

(8)

The solution set of SCFPP is denoted by ${\Omega }_{SCFPP}$.

As the fixed points problem is directly linked with minimization problem, split feasibility problem, variational inequality problem, equilibrium problem and many others. So, it has interesting applications in different disciplines of mathematical sciences dealing with these problems, for details we refer [30,31,32].

Different iterative processes help in approximating the solution of functional equations which fail to have a closed form solution. Recently, many authors have proposed the fixed point iterative schemes to approximate the solution certain non-linear problems. For instance, in [33], an inertial iterative algorithm was studied with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a non-expansive mapping. In 2018, Toscano and Vetro [34] applied the fixed point iterative schemes to approximate the solution of variational inequality problems, via admissible perturbations of projection operators in real Hilbert spaces. Khunpanuk et al. [35] discussed certain proximal-type algorithms for solving equilibrium problems in real Hilbert space. The algorithms were analogous to the well-known two-step extragradient algorithm for solving the variational inequality problem in Hilbert spaces. The proposed iterative algorithms in [35] used the step size rule based on local bifunction information instead of the line search technique.

Recently, Wang et al. [36] studied the solution of the split equality fixed point problem for the class of single valued demicontractive mapping. The proposed method is given as follows

$$\begin{array}{cc}\hfill & \mathrm{Take}{x}_0\in H_1,\mathrm{and}{y}_0\in H_2.\mathrm{Define}\hfill \\ \hfill x_{n+1}& ={\alpha }_n{f}_1\left(x_n\right)+(1-{\alpha }_n)[x_n-{\gamma }_n(x_n-S{x}_n+A^{*}(A{x}_n-B{y}_n)])\hfill \\ \hfill & y_{n+1}={\alpha }_n{f}_2\left(y_n\right)+(1-{\alpha }_n)[y_n-{\gamma }_n(y_n-T{y}_n+A^{*}(B{y}_n-A{x}_n)]),n\in N,\hfill \end{array}$$

where $S:H_1\to H_1$ and $T:H_2\to H_2$ are demicontractive, $f_1:H_1\to H_1$ and $f_2:H_2\to H_2$ are contraction mappings and $\left\{{\gamma }_n\right\}$ is the sequence of step sizes.

Motivated by the recent work, we in this paper propose the viscosity type iterative algorithm for infinite family of multivalued demicontractive mappings to approximate the solution of the problem (5). We used the iterative scheme based on choosing the step-sizes without calculating or estimating the operator norms because to find the step-size which depends upon the operator (matrix) norms $\parallel A\parallel$ and $\parallel B\parallel$, first one has to compute or at least estimate the operator norms of A and B which is, in general, practically not easy. To overcome this difficulty many authors prefers and use the iterative scheme which did not require the operator norm. Continuing in this direction, we have also used such a technique in our iterative scheme. We denote the solution set of the problem (5) by $\Omega$. The following results are needed in the sequel.

Lemma 1

([37]). For any $x,y\in H,$ we have

$$2\langle x,y\rangle ={\parallel x\parallel }^2+{\parallel y\parallel }^2{-\parallel x-y\parallel }^2={\parallel x+y\parallel }^2-{\parallel x\parallel }^2-{\parallel y\parallel }^2.$$

Lemma 2

([37]). For any $x,y\in H,$ we have

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2\langle y,x+y\rangle .$$

Lemma 3

([38]). Suppose that $\left\{x_n\right\}\subset R^{+},$$\left\{y_n\right\}\subset R$,(where $R^{+}$ and R denote the set of non-negative real numbers and the set of real numbers, respectively)$\left\{{\sigma }_n\right\}\subset (0,1)$, such that ${\sum }_{n=0}^{\infty }{\sigma }_n=\infty$ and following condition holds:

$$x_{n+1}\le (1-{\sigma }_n)x_n+{\sigma }_n{y}_n,\forall ,n\in N.$$

If $\underset{n\to \infty }{lim\; sup}{y}_n\le 0,$ then $\underset{n\to \infty }{lim}{x}_n=0.$

Lemma 4

([39]). If X is a uniformly convex Banach space and $c>0,$ then there exists a continuous strictly increasing function $h:[0,\infty )\to [0,\infty )$ with $h\left(0\right)=0$ such that

$$\parallel \sum _{n=1}^{\infty }{\alpha }_n{x}_n{\parallel }^2\le \sum _{n=1}^{\infty }{\alpha }_n\parallel x_n{\parallel }^2-{\alpha }_m{\alpha }_{n}h(\parallel x_n-x_n\parallel )$$

holds, for any $m,n\in N$ with $m<n$ where $x_n\in H$ with $\parallel x_n\parallel \le c$, and ${\sum }_{n=1}^{\infty }{\alpha }_n=1.$

2. Main Results

Theorem 1.

Assume that $S_i:H_1\to CB\left(H_1\right)$ and $T_i:H_2\to CB\left(H_2\right)$ are two infinite families of multivalued demicontractive-type mappings and for each $i\ge 1,$ $S_i$ and $T_i$ are demiclosed at 0 and ${\bigcap }_{i=1}^{\infty }F\left(S_i\right)\ne \varnothing$, ${\bigcap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing$ and Ω: is the solution set of (5), which is non-empty. Suppose that $f_1:H_1\to H_1$ and $f_2:H_2\to H_2$ are two contraction mappings with contraction constants $c_1$ and $c_2,$ respectively. Let $\left\{{\alpha }_{i,n}\right\}$ and $\left\{{\eta }_n\right\}$ be sequences in $(0,1)$, for each $i\ge 1$, $ϵ\ge 0$, and γ a fixed non-negative real number. Let $x_1\in H_1$ and $y_1\in H_2$ and define the sequences $\left\{x_n\right\}$ and $\left\{y_n\right\}$ by

$$\begin{array}{cc}\hfill u_n=& P_C\left[x_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)\right]\hfill \\ \hfill x_{n+1}=& {\eta }_n{f}_1\left(u_n\right)+({\alpha }_{0,n}-{\eta }_n)u_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{z}_{i,n},\hfill \\ \hfill v_n=& P_Q\left[y_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n)\right]\hfill \\ \hfill y_{n+1}=& {\eta }_n{f}_2\left(v_n\right)+({\alpha }_{0,n}-{\eta }_n)v_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{w}_{i,n},\hfill \end{array}$$

(9)

where A and B are bounded linear operators as described above $z_{i,n}\in S_i{u}_n$, $w_{i,n}\in T_i{u}_n$ and

$$I_n=\left(ϵ,\frac{2\parallel A{x}_n-B{y}_n{\parallel }^2}{\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2+{\parallel B^{*}(A{x}_n-B{y}_n)\parallel }^2}-ϵ\right).$$

The step size ${\gamma }_n$ is chosen from $I_n$ provided that $n\in \Gamma =\{n:A{x}_n-B{y}_n\ne 0\}$ otherwise take ${\gamma }_n$ equal to $\gamma .$ If

$\left(i\right)$

${\sum }_{i=0}^{\infty }{\alpha }_{i,n}=1$ for each $n\ge 1$,

$\left(ii\right)$

${lim}_{n\to \infty }{\eta }_n=0$ and ${\sum }_{i=1}^{\infty }{\eta }_n=\infty$,

$\left(iii\right)$

for each $i\ge 1$${lim\; inf}_{n\to \infty }{\alpha }_{0,n}{\alpha }_{i,n}\ge 0$,

$\left(iv\right)$

${\eta }_n\le {\alpha }_{0,n}$ for each $n\ge 1$,

$\left(v\right)$

for each $p\in {\bigcap }_{i=1}^{\infty }F\left(S_i\right),S_i\left(p\right)=\left\{p\right\}$ and for each $q\in {\bigcap }_{i=1}^{\infty }F\left(T_i\right),S_i\left(q\right)=\left\{q\right\}$, for each $i\ge 1$.

Then $\left\{(x_n,y_n)\right\}$ converges strongly to some point in Ω.

Proof. 

We divide the proof into two steps.

Step 1. First we prove that the sequence $\left\{(x_n,y_n)\right\}$ is bounded in $H_1\times H_2$. Let $(p^{*},q^{*})\in$$\Omega .$ Using the convexity of ${\parallel .\parallel }^2$ and the Cauchy–Schwarz inequality, we have

$$\begin{array}{cc}\hfill \parallel x_{n+1}-p^{*}{\parallel }^2=& \parallel {\eta }_n{f}_1\left(u_n\right)+({\alpha }_{0,n}-{\eta }_n)u_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{z}_{i,n}-p^{*}{\parallel }^2\hfill \\ \hfill =& \parallel ({\alpha }_{0,n}-{\eta }_n)(u_n-p^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*})+{\eta }_n(f_1\left(u_n\right)-p^{*}){\parallel }^2\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n)\parallel u_n-p^{*}{\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}\parallel z_{i,n}-p^{*}{\parallel }^2+{\eta }_n{\parallel f_1\left(u_n\right)-p^{*}\parallel }^2\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n){\parallel u_n-p^{*}\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}{\left(d(z_{i,n},S_i\left(p^{*}\right))\right)}^2\hfill \\ \hfill & +{\eta }_n{\parallel f_1\left(u_n\right)-f_1\left(p^{*}\right)+f_1\left(p^{*}\right)-p^{*}\parallel }^2\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n){\parallel u_n-p^{*}\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}{\Pi }^2(S_i\left(u_n\right),S_i\left(p^{*}\right))\hfill \\ \hfill & +{\eta }_n[\parallel f_1\left(u_n\right)-f_1\left(p^{*}\right){\parallel }^2+\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+2\langle f_1\left(u_n\right)-f_1\left(p^{*}\right),f_1\left(p^{*}\right)-p^{*}\rangle ]\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n)\parallel u_n-p^{*}{\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}{\parallel u_n-p^{*}\parallel }^2\hfill \\ \hfill & +{\eta }_n[\parallel f_1\left(u_n\right)-f_1\left(p^{*}\right){\parallel }^2+\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+2\parallel f_1\left(u_n\right)-f_1\left(p^{*}\right)\parallel \parallel f_1\left(p^{*}\right)-p^{*}\parallel ]\hfill \\ \hfill \le & (1-{\eta }_n)\parallel u_n-p^{*}{\parallel }^2+2{\eta }_n[c_1^2\parallel u_n-p^{*}{\parallel }^2+\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2]\hfill \\ \hfill =& (1-(1-2c_1^2){\eta }_n)\parallel u_n-p^{*}{\parallel }^2+2{\eta }_n{\parallel f_1\left(p^{*}\right)-p^{*}\parallel }^2.\hfill \end{array}$$

(10)

Now, by (9) and firmly non-expansive property of the projection mapping, we obtain that

$$\begin{array}{cc}\hfill \parallel u_n-p^{*}{\parallel }^2\le & \parallel x_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)-p^{*}{\parallel }^2-{\parallel x_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)-u_n\parallel }^2\hfill \\ \hfill =& \parallel x_n-p^{*}{\parallel }^2+{\gamma }_n^2{\parallel A^{*}(A{x}_n-B{y}_n)\parallel }^2-2{\gamma }_n\langle x_n-p^{*},A^{*}(A{x}_n-B{y}_n)\rangle \hfill \\ \hfill & -\parallel x_n-u_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n){\parallel }^2\hfill \\ \hfill =& \parallel x_n-p^{*}{\parallel }^2+{\gamma }_n^2{\parallel A^{*}(A{x}_n-B{y}_n)\parallel }^2-2{\gamma }_n\langle A{x}_n-A{p}^{*},(A{x}_n-B{y}_n)\rangle \hfill \\ \hfill & -\parallel x_n-u_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n){\parallel }^2\hfill \\ \hfill =& \parallel x_n-p^{*}{\parallel }^2+{\gamma }_n^2\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2-{\gamma }_n\parallel A{x}_n-A{p}^{*}{\parallel }^2-{\gamma }_n{\parallel A{x}_n-B{y}_n\parallel }^2\hfill \\ \hfill & +{\gamma }_n\parallel B{y}_n-A{p}^{*}{\parallel }^2-{\parallel x_n-u_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)\parallel }^2.\hfill \end{array}$$

(11)

Inserting (11) in (10) we have

$$\begin{array}{cc}\hfill \parallel x_{n+1}-p^{*}{\parallel }^2\le & (1-(1-2c_1^2){\eta }_n)[\parallel x_n-p^{*}{\parallel }^2+{\gamma }_n^2\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2-{\gamma }_n{\parallel A{x}_n-A{p}^{*}\parallel }^2\hfill \\ \hfill & -{\gamma }_n\parallel A{x}_n-B{y}_n{\parallel }^2+{\gamma }_n\parallel B{y}_n-A{p}^{*}{\parallel }^2-\parallel x_n-u_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n){\parallel }^2]\hfill \\ \hfill & +2{\eta }_n{\parallel f_1\left(p^{*}\right)-p^{*}\parallel }^2.\hfill \end{array}$$

(12)

Similarly, by (9) we have

$$\begin{array}{cc}\hfill \parallel y_{n+1}-q^{*}{\parallel }^2=& \parallel {\eta }_n{f}_2\left(v_n\right)+({\alpha }_{0,n}-{\eta }_n)v_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{w}_{i,n}-q^{*}{\parallel }^2\hfill \\ \hfill =& \parallel ({\alpha }_{0,n}-{\eta }_n)(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*})+{\eta }_n(f_2\left(v_n\right)-q^{*}){\parallel }^2\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n)\parallel v_n-q^{*}{\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}\parallel w_{i,n}-q^{*}{\parallel }^2+{\eta }_n{\parallel f_2\left(v_n\right)-q^{*}\parallel }^2\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n){\parallel v_n-q^{*}\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}{\left(d(w_{i,n},T_i\left(q^{*}\right))\right)}^2\hfill \\ \hfill & +{\eta }_n{\parallel f_2\left(v_n\right)-f_2\left(q^{*}\right)+f_2\left(q^{*}\right)-q^{*}\parallel }^2\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n){\parallel v_n-q^{*}\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}{\Pi }^2(T_i\left(v_n\right),T_i\left(q^{*}\right))\hfill \\ \hfill & +{\eta }_n[\parallel f_2\left(v_n\right)-f_2\left(q^{*}\right){\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2+2\langle f_2\left(v_n\right)-f_2\left(q^{*}\right),f_2\left(q^{*}\right)-q^{*}\rangle ]\hfill \\ \hfill \le & ({\alpha }_{0,n}-{\eta }_n)\parallel v_n-q^{*}{\parallel }^2+\sum _{i=1}^{\infty }{\alpha }_{i,n}{\parallel v_n-q^{*}\parallel }^2\hfill \\ \hfill & +{\eta }_n[\parallel f_2\left(v_n\right)-f_2\left(q^{*}\right){\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2+2\parallel f_2\left(v_n\right)-f_2\left(q^{*}\right)\parallel \parallel f_2\left(q^{*}\right)-q^{*}\parallel ]\hfill \\ \hfill \le & (1-{\eta }_n)\parallel v_n-q^{*}{\parallel }^2+2{\eta }_n[c_2^2\parallel v_n-q^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2]\hfill \\ \hfill =& (1-(1-2c_2^2){\eta }_n)\parallel v_n-q^{*}{\parallel }^2+2{\eta }_n{\parallel f_2\left(q^{*}\right)-q^{*}\parallel }^2.\hfill \end{array}$$

(13)

Now, by (9) and firmly non-expansive property of the projection mapping, we have

$$\begin{array}{cc}\hfill \parallel v_n-q^{*}{\parallel }^2\le & \parallel y_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n)-q^{*}{\parallel }^2-{\parallel y_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n)-v_n\parallel }^2\hfill \\ \hfill =& \parallel y_n-q^{*}{\parallel }^2+{\gamma }_n^2{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2-2{\gamma }_n\langle y_n-q^{*},B^{*}(B{y}_n-A{x}_n)\rangle \hfill \\ \hfill & -\parallel y_n-v_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n){\parallel }^2\hfill \\ \hfill =& \parallel y_n-q^{*}{\parallel }^2+{\gamma }_n^2{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2-2{\gamma }_n\langle B{y}_n-B{q}^{*},(B{y}_n-A{x}_n)\rangle \hfill \\ \hfill & -\parallel y_n-v_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n){\parallel }^2\hfill \\ \hfill =& \parallel y_n-q^{*}{\parallel }^2+{\gamma }_n^2\parallel B^{*}(B{y}_n-A{x}_n){\parallel }^2-{\gamma }_n\parallel B{y}_n-B{q}^{*}{\parallel }^2-{\gamma }_n{\parallel A{x}_n-B{y}_n\parallel }^2\hfill \\ \hfill & +{\gamma }_n\parallel A{x}_n-B{q}^{*}{\parallel }^2-{\parallel y_n-v_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n)\parallel }^2.\hfill \end{array}$$

(14)

Inserting (14) in (13) we obtain that

$$\begin{array}{cc}\hfill \parallel y_{n+1}-q^{*}{\parallel }^2\le & (1-(1-2c_2^2){\eta }_n)[\parallel y_n-q^{*}{\parallel }^2+{\gamma }_n^2\parallel B^{*}(B{y}_n-A{x}_n){\parallel }^2-{\gamma }_n{\parallel B{y}_n-B{q}^{*}\parallel }^2\hfill \\ \hfill & -{\gamma }_n\parallel A{x}_n-B{y}_n{\parallel }^2+{\gamma }_n\parallel A{x}_n-B{q}^{*}{\parallel }^2-\parallel y_n-v_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n){\parallel }^2]\hfill \\ \hfill & +2{\eta }_n{\parallel f_2\left(q^{*}\right)-q^{*}\parallel }^2.\hfill \end{array}$$

(15)

Adding (12) and (15), taking $c=max\{c_1,c_2\}$ and using $A{p}^{*}=B{q}^{*}$, we obtain

$$\begin{array}{cc}\hfill \parallel x_{n+1}-p^{*}{\parallel }^2+{\parallel y_{n+1}-q^{*}\parallel }^2\le & (1-(1-2c^2){\eta }_n)[\parallel x_n-p^{*}{\parallel }^2+\parallel y_n-q^{*}{\parallel }^2-{\gamma }_n[2\parallel A{x}_n-B{y}_n{\parallel }^2\hfill \\ \hfill & -({\gamma }_n\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2+\parallel B^{*}(B{y}_n-A{x}_n){\parallel }^2)\left]\right]\hfill \\ \hfill & +2{\eta }_n(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2).\hfill \end{array}$$

(16)

Hence

$$\begin{array}{cc}\hfill \parallel x_{n+1}-p^{*}{\parallel }^2+{\parallel y_{n+1}-q^{*}\parallel }^2\le & (1-(1-2c^2){\eta }_n)\parallel x_n-p^{*}{\parallel }^2+{\parallel y_n-q^{*}\parallel }^2\hfill \\ \hfill & +2{\eta }_n(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2).\hfill \end{array}$$

(17)

If we set ${\mathsf{\Theta }}_n=\parallel x_n-p^{*}{\parallel }^2+{\parallel y_n-q^{*}\parallel }^2$, then by the above expression we have

$$\begin{array}{cc}\hfill {\mathsf{\Theta }}_{n+1}\le & (1-(1-2c^2){\eta }_n){\mathsf{\Theta }}_n+2{\eta }_n(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2)\hfill \\ \hfill =& (1-(1-2c^2){\eta }_n){\mathsf{\Theta }}_n+(1-2c^2){\eta }_n\frac{2(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2)}{1-2c^2}\hfill \\ \hfill \le & max\left\{{\mathsf{\Theta }}_n,\frac{2(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2)}{1-2c^2}\right\}\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill \le & max\left\{{\mathsf{\Theta }}_1,\frac{2(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2)}{1-2c^2}\right\}.\hfill \end{array}$$

So, $\left\{{\mathsf{\Theta }}_n\right\}$ is bounded, and $\left\{(x_n,y_n)\right\}$ is bounded. Thus, $\left\{x_n\right\}$, $\left\{y_n\right\}$ are bounded sequences. Consequently, $\left\{u_n\right\}$, $\left\{v_n\right\}$, $\left\{w_{i,n}\right\}$ and $\left\{z_{i,n}\right\}$ are bounded.

Step 2. We now prove the strong convergence of $\left\{(x_n,y_n)\right\}$ to $(p^{*},q^{*})$. We divide the proof further into two cases.

Case 1. Assume that $\left\{{\mathsf{\Theta }}_n\right\}$ is monotonically decreasing.

Then by (16) we have

$$\begin{gathered} 1-(1-2c^2){\eta }_n{\gamma }_n^2\left(\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2+{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2\right) \\ \le (1-(1-2c^2){\eta }_n)[\parallel x_n-p^{*}{\parallel }^2+\parallel y_n-q^{*}{\parallel }^2]-(\parallel x_{n+1}-p^{*}{\parallel }^2 \\ +\parallel y_{n+1}-q^{*}{\parallel }^2)+2{\eta }_n(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2). \end{gathered}$$

So,

$$\begin{array}{cc}\hfill {\gamma }_n^2(\parallel A^{*}(A{x}_n-& B{y}_n{)\parallel }^2+{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2)\hfill \\ \hfill \le & \parallel x_n-p^{*}{\parallel }^2+\parallel y_n-q^{*}{\parallel }^2-\frac{1}{1-(1-2c^2){\eta }_n}({\parallel x_{n+1}-p^{*}\parallel }^2\hfill \\ \hfill & +\parallel y_{n+1}-q^{*}{\parallel }^2)+\frac{2{\eta }_n}{1-(1-2c^2){\eta }_n}(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2).\hfill \end{array}$$

(18)

Since ${\eta }_n\to 0$ as $n\to \infty$, we have

$${\gamma }_n^2\left(\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2+{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2\right)\to 0,\mathrm{as}n\to \infty .$$

As, ${\gamma }_n\in \left(ϵ,\frac{2\parallel A{x}_n-B{y}_n{\parallel }^2}{\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2+{\parallel B^{*}(A{x}_n-B{y}_n)\parallel }^2}-ϵ\right)$, where $n\in \Gamma$ we obtain

$$\underset{n\to \infty }{lim}\left(\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2+{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2\right)=0.$$

(19)

Additionally, $A{x}_n-B{y}_n=0$ in case $n\notin \Gamma$. Hence

$$\underset{n\to \infty }{lim}\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2=\underset{n\to \infty }{lim}{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2=0.$$

(20)

Now by Lemmas 1 and 4, we have

$$\begin{array}{cc}\hfill \parallel x_{n+1}-p^{*}{\parallel }^2=& \parallel {\eta }_n{f}_1\left(u_n\right)+({\alpha }_{0,n}-{\eta }_n)u_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{z}_{i,n}-p^{*}{\parallel }^2\hfill \\ \hfill =& \parallel {\alpha }_{0,n}(u_n-p^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*})+{\eta }_n(f_1\left(u_n\right)-u_n){\parallel }^2\hfill \\ \hfill =& \parallel {\alpha }_{0,n}(u_n-p^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*}){\parallel }^2+{\eta }_n^2{\parallel (f_1\left(u_n\right)-u_n)\parallel }^2\hfill \\ \hfill +& 2{\eta }_n\langle f_1\left(u_n\right)-u_n,{\alpha }_{0,n}(u_n-p^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*})\rangle \hfill \\ \hfill \le & \parallel u_n-p^{*}{\parallel }^2-{\alpha }_{0,n}{\alpha }_{i,n}h(\parallel u_n-z_{i,n}\parallel )+{\eta }_n^2{\parallel (f_1\left(u_n\right)-u_n)\parallel }^2\hfill \\ \hfill +& 2{\eta }_n\langle f_1\left(u_n\right)-u_n,{\alpha }_{0,n}(u_n-p^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*})\rangle .\hfill \end{array}$$

(21)

Similarly,

$$\begin{array}{cc}\hfill \parallel y_{n+1}-q^{*}{\parallel }^2=& \parallel {\eta }_n{f}_2\left(v_n\right)+({\alpha }_{0,n}-{\eta }_n)v_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{w}_{i,n}-q^{*}{\parallel }^2\hfill \\ \hfill =& \parallel {\alpha }_{0,n}(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*})+{\eta }_n(f_2\left(v_n\right)-v_n){\parallel }^2\hfill \\ \hfill =& \parallel {\alpha }_{0,n}(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*}){\parallel }^2+{\eta }_n^2{\parallel (f_2\left(v_n\right)-v_n)\parallel }^2\hfill \\ \hfill +& 2{\eta }_n\langle f_2\left(v_n\right)-v_n,{\alpha }_{0,n}(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*})\rangle \hfill \\ \hfill \le & \parallel v_n-q^{*}{\parallel }^2-{\alpha }_{0,n}{\alpha }_{i,n}h(\parallel v_n-w_{i,n}\parallel )+{\eta }_n^2{\parallel (f_2\left(v_n\right)-v_n)\parallel }^2\hfill \\ \hfill +& 2{\eta }_n\langle f_2\left(v_n\right)-v_n,{\alpha }_{0,n}(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*})\rangle .\hfill \end{array}$$

(22)

Adding (21) and (22) we have

$$\begin{array}{cc}\hfill \parallel x_{n+1}-p^{*}{\parallel }^2& +\parallel y_{n+1}-q^{*}{\parallel }^2\hfill \\ \hfill \le & \parallel u_n-p^{*}{\parallel }^2+{\parallel v_n-q^{*}\parallel }^2-{\alpha }_{0,n}{\alpha }_{i,n}\left(h(\parallel u_n-z_{i,n}\parallel )+h(\parallel v_n-w_{i,n}\parallel )\right)\hfill \\ \hfill +& {\eta }_n^2\left(\parallel (f_1\left(u_n\right)-u_n){\parallel }^2+{\parallel (f_2\left(v_n\right)-v_n)\parallel }^2\right)\hfill \\ \hfill +& 2{\eta }_n(\langle f_1\left(u_n\right)-u_n,{\alpha }_{0,n}(u_n-p^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*})\rangle \hfill \\ \hfill +& \langle f_2\left(v_n\right)-v_n,{\alpha }_{0,n}(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*})\rangle ).\hfill \end{array}$$

(23)

By adding the inequalities (11) and (14), using $A{p}^{*}=B{q}^{*}$ and the assumption on ${\gamma }_n$, we obtain

$$\begin{array}{cc}\hfill \parallel u_n-p^{*}{\parallel }^2+{\parallel v_n-q^{*}\parallel }^2\le & \parallel x_n-p^{*}{\parallel }^2+\parallel y_n-q^{*}{\parallel }^2-{\gamma }_n[2{\parallel (A{x}_n-B{y}_n)\parallel }^2\hfill \\ \hfill & -{\gamma }_n^2\left(\parallel A^{*}(A{x}_n-B{y}_n){\parallel }^2+{\parallel B^{*}(B{y}_n-A{x}_n)\parallel }^2\right)]\hfill \\ \hfill \le & \parallel x_n-p^{*}{\parallel }^2+{\parallel y_n-q^{*}\parallel }^2.\hfill \end{array}$$

(24)

Using (24) in (23), we obtain

$$\begin{array}{cc}\hfill \parallel x_{n+1}-p^{*}{\parallel }^2& +\parallel y_{n+1}-q^{*}{\parallel }^2\hfill \\ \hfill \le & \parallel x_n-p^{*}{\parallel }^2+{\parallel y_n-q^{*}\parallel }^2-{\alpha }_{0,n}{\alpha }_{i,n}\left(h(\parallel u_n-z_{i,n}\parallel )+h(\parallel v_n-w_{i,n}\parallel )\right)\hfill \\ \hfill +& {\eta }_n^2\left(\parallel (f_1\left(u_n\right)-u_n){\parallel }^2+{\parallel (f_2\left(v_n\right)-v_n)\parallel }^2\right)\hfill \\ \hfill +& 2{\eta }_n(\langle f_1\left(u_n\right)-u_n,{\alpha }_{0,n}(u_n-p^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*})\rangle \hfill \\ \hfill +& \langle f_2\left(v_n\right)-v_n,{\alpha }_{0,n}(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*})\rangle ).\hfill \end{array}$$

(25)

As $\left\{u_n\right\}$, $\left\{v_n\right\}$, $\left\{w_{i,n}\right\}$ and $\left\{z_{i,n}\right\}$ are bounded, there exist $K>0$, such that

$$\begin{array}{cc}\hfill [{\eta }_n& \left(\parallel (f_1\left(u_n\right)-u_n){\parallel }^2+{\parallel (f_2\left(v_n\right)-v_n)\parallel }^2\right)+2(\langle f_1\left(u_n\right)-u_n,{\alpha }_{0,n}(u_n-p^{*})\hfill \\ \hfill +& \sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-p^{*})\rangle +\langle f_2\left(v_n\right)-v_n,{\alpha }_{0,n}(v_n-q^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(w_{i,n}-q^{*})\rangle \left)\right]\le K.\hfill \end{array}$$

Now by (25) we obtain

$$\begin{array}{cccc}\hfill {\alpha }_{0,n}{\alpha }_{i,n}& \left(h(\parallel u_n-z_{i,n}\parallel )+h(\parallel v_n-w_{i,n}\parallel )\right)\hfill & \hfill & \hfill \\ \hfill \le & \parallel x_n-p^{*}{\parallel }^2+\parallel y_n-q^{*}{\parallel }^2-({\parallel x_{n+1}-p^{*}\parallel }^2\hfill & \hfill +\parallel y_{n+1}-q^{*}{\parallel }^2)+{\eta }_{n}K& \to 0,\hfill \end{array}$$

(26)

as $n\to \infty$. Thus by condition $\left(iii\right),$ we obtain

$$h(\parallel u_n-z_{i,n}\parallel )+h(\parallel v_n-w_{i,n}\parallel )\to 0,\mathrm{as}n\to \infty .$$

Hence

$$\underset{n\to \infty }{lim}h(\parallel u_n-z_{i,n}\parallel )=\underset{n\to \infty }{lim}h(\parallel v_n-w_{i,n}\parallel )=0.$$

(27)

Since h is strictly increasing, continuous and $h\left(0\right)=0$, we have

$$\underset{n\to \infty }{lim}\parallel u_n-z_{i,n}\parallel =0$$

and

$$\underset{n\to \infty }{lim}\parallel v_n-w_{i,n}\parallel =0.$$

Thus,

$$\underset{n\to \infty }{lim}d(u_n,S_i{u}_n)\le \underset{n\to \infty }{lim}\parallel u_n-z_{i,n}\parallel =0,$$

(28)

and

$$\underset{n\to \infty }{lim}d(v_n,T_i{v}_n)\le \underset{n\to \infty }{lim}\parallel v_n-w_{i,n}\parallel =0.$$

(29)

By (11), (14) together with (17) we obtain

$$\underset{n\to \infty }{lim}\parallel x_n-u_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)\parallel =\underset{n\to \infty }{lim}\parallel y_n-v_n-{\gamma }_n{B}^{*}(B{y}_n-A{x}_n)\parallel =0.$$

(30)

By (20) and (30) we have

$$\parallel x_n-u_n\parallel \le \parallel x_n-u_n-{\gamma }_n{A}^{*}(A{x}_n-B{y}_n)\parallel +\parallel {\gamma }_n{A}^{*}(A{x}_n-B{y}_n)\parallel .$$

On taking limit as $n\to \infty$ we obtain that

$$\underset{n\to \infty }{lim}\parallel u_n-x_n\parallel =0.$$

(31)

Similarly,

$$\underset{n\to \infty }{lim}\parallel v_n-y_n\parallel =0.$$

(32)

Using the condition for ${\gamma }_n$ and (20) in (16), we obtain

$$\underset{n\to \infty }{lim}\parallel (A{x}_n-B{y}_n)\parallel =0.$$

As $\left\{x_n\right\}$ is a bounded sequence, there exists $x^{*}\in H_1$, such that $x_n\rightharpoonup x^{*}$. So by (31), we have $u_n\rightharpoonup x^{*}$. Using the demi-closedness of $S_i$ and (28) we have $x^{*}\in F\left(S_i\right)$ and, hence, $x^{*}\in {\bigcap }_{i=1}^{\infty }F\left(S_i\right)$. Similarly, $\left\{y_n\right\}$ is a bounded sequence, there exists $y^{*}\in H_2$, such that $y_n\rightharpoonup y^{*}$ and hence by (32), we have $v_n\rightharpoonup y^{*}$. Using the demi-closedness of $T_i$ and (29) we have $y^{*}\in F\left(T_i\right)$, and, hence, $y^{*}\in {\bigcap }_{i=1}^{\infty }F\left(T_i\right)$. Since A and B are bounded linear operators, we have $A{x}_n\rightharpoonup A{x}^{*}$ and $B{y}_n\rightharpoonup B{y}^{*}$, and then by the weak continuity of the norm, we have

$$\parallel (A{x}^{*}-B{y}^{*})\parallel \le \underset{n\to \infty }{lim\; inf}\parallel (A{x}_n-B{y}_n)\parallel =0.$$

Thus, we have $(x^{*},y^{*})\in \Omega$.

We now prove that $x_n\to x^{*}$ and $y_n\to y^{*}$. Using Lemma 2 in (9) we have

$$\begin{array}{cc}\hfill \parallel x_{n+1}-x^{*}{\parallel }^2=& \parallel {\eta }_n{f}_1\left(u_n\right)+({\alpha }_{0,n}-{\eta }_n)u_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{z}_{i,n}-x^{*}{\parallel }^2\hfill \\ \hfill =& \parallel ({\alpha }_{0,n}-{\eta }_n)(u_n-x^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-x^{*})+{\eta }_n(f_1\left(u_n\right)-x^{*}){\parallel }^2\hfill \\ \hfill \le & \parallel ({\alpha }_{0,n}-{\eta }_n)(u_n-x^{*})+\sum _{i=1}^{\infty }{\alpha }_{i,n}(z_{i,n}-x^{*}){\parallel }^2+{\eta }_n\langle x_{n+1}-x^{*},f_1\left(u_n\right)-x^{*}\rangle \hfill \\ \hfill \le & {\left[({\alpha }_{0,n}-{\eta }_n)\parallel u_n-x^{*}\parallel +\sum _{i=1}^{\infty }{\alpha }_{i,n}\parallel z_{i,n}-x^{*}\parallel \right]}^2+2{\eta }_n\langle x_{n+1}-x^{*},f_1\left(u_n\right)-x^{*}\rangle \hfill \\ \hfill \le & (1-{\eta }_n){\parallel u_n-x^{*}\parallel }^2+2{\eta }_n\langle x_{n+1}-x^{*},f_1\left(u_n\right)-x^{*}\rangle .\hfill \end{array}$$

(33)

Similarly,

$$\parallel y_{n+1}-y^{*}{\parallel }^2\le (1-{\eta }_n){\parallel v_n-y^{*}\parallel }^2+2{\eta }_n\langle y_{n+1}-y^{*},f_2\left(v_n\right)-y^{*}\rangle .$$

(34)

Now adding the inequalities (33) and (34) we obtain

$$\begin{array}{cc}\hfill \parallel x_{n+1}-x^{*}{\parallel }^2+{\parallel y_{n+1}-y^{*}\parallel }^2\le & (1-{\eta }_n)\left[\parallel u_n-x^{*}{\parallel }^2+{\parallel v_n-y^{*}\parallel }^2\right]\hfill \\ \hfill & +2{\eta }_n\left(\langle x_{n+1}-y^{*},f_1\left(u_n\right)-x^{*}\rangle +\langle y_{n+1}-y^{*},f_2\left(v_n\right)-y^{*}\rangle \right)\hfill \\ \hfill \le & (1-{\eta }_n)\left[\parallel x_n-x^{*}{\parallel }^2+{\parallel y_n-y^{*}\parallel }^2\right]\hfill \\ \hfill & +2{\eta }_n\left(\langle x_{n+1}-x^{*},f_1\left(u_n\right)-x^{*}\rangle +\langle y_{n+1}-y^{*},f_2\left(v_n\right)-y^{*}\rangle \right).\hfill \end{array}$$

(35)

Since $x_n\rightharpoonup x^{*}$ and $y_n\rightharpoonup y^{*}$ then we have

$$\langle x_{n+1}-x^{*},f_1\left(u_n\right)-x^{*}\rangle +\langle y_{n+1}-y^{*},f_2\left(v_n\right)-y^{*}\rangle \to 0\mathrm{as}n\to \infty .$$

Using Lemma 3 in (35) we obtain

$$\parallel x_n-x^{*}\parallel +\parallel y_n-y^{*}\parallel \to 0,\mathrm{as}n\to \infty .$$

Hence,

$$\underset{n\to \infty }{lim}\parallel x_n-x^{*}\parallel =\underset{n\to \infty }{lim}\parallel y_n-y^{*}\parallel =0.$$

Thus $\left\{(x_n,y_n)\right\}$ converges strongly to $(x^{*},y^{*})$ in the solution set $\Omega$.

Case 2. Suppose that $\left\{{\mathsf{\Theta }}_n\right\}$ is not monotonically decreasing. For all $n\ge n_0$ for some large $n_0$, we define the mapping $\varphi$ by

$$\varphi \left(n\right)=max\{k\in N:k\le n,{\mathsf{\Theta }}_n\le {\mathsf{\Theta }}_{n+1}\}.$$

As $\left\{\varphi \right(n\left)\right\}$ is a non-decreasing sequence, such that $\varphi \left(n\right)\to \infty$ as $n\to \infty$ and

$${\mathsf{\Theta }}_{\varphi \left(n\right)}\le {\mathsf{\Theta }}_{\varphi \left(n\right)+1},\mathrm{for}n\ge n_0.$$

It follows from (18) that

$$\begin{array}{cc}\hfill {\gamma }_{\varphi \left(n\right)}^2(\parallel A^{*}(A{x}_{\varphi \left(n\right)}-& B{y}_{\varphi \left(n\right)}{)\parallel }^2+{\parallel B^{*}(B{y}_{\varphi \left(n\right)}-A{x}_{\varphi \left(n\right)})\parallel }^2)\hfill \\ \hfill \le & \parallel x_{\varphi \left(n\right)}-p^{*}{\parallel }^2+\parallel y_{\varphi \left(n\right)}-q^{*}{\parallel }^2-\frac{1}{1-(1-2c^2){\eta }_{\varphi \left(n\right)}}({\parallel x_{\varphi \left(n\right)+1}-p^{*}\parallel }^2\hfill \\ \hfill & +\parallel y_{\varphi \left(n\right)+1}-q^{*}{\parallel }^2)+\frac{2{\eta }_{\varphi \left(n\right)}}{1-(1-2c^2){\eta }_{\varphi \left(n\right)}}(\parallel f_1\left(p^{*}\right)-p^{*}{\parallel }^2+\parallel f_2\left(q^{*}\right)-q^{*}{\parallel }^2).\hfill \end{array}$$

Hence, we have

$${\gamma }_{\varphi \left(n\right)}^2\left(\parallel A^{*}(A{x}_{\varphi \left(n\right)}-B{y}_{\varphi \left(n\right)}){\parallel }^2+{\parallel B^{*}(B{y}_{\varphi \left(n\right)}-A{x}_{\varphi \left(n\right)})\parallel }^2\right)\to 0.$$

By the condition ${\gamma }_{\varphi \left(n\right)}\in \left(ϵ,\frac{2\parallel A{x}_{\varphi \left(n\right)}-B{y}_{\varphi \left(n\right)}{\parallel }^2}{\parallel A^{*}(A{x}_{\varphi \left(n\right)}-B{y}_{\varphi \left(n\right)}){\parallel }^2+{\parallel B^{*}(A{x}_{\varphi \left(n\right)}-B{y}_{\varphi \left(n\right)})\parallel }^2}-ϵ\right)$, where $\varphi \left(n\right)\in \Gamma$ we obtain

$$\underset{n\to \infty }{lim}\left(\parallel A^{*}(A{x}_{\varphi \left(n\right)}-B{y}_{\varphi \left(n\right)}){\parallel }^2+{\parallel B^{*}(B{y}_{\varphi \left(n\right)}-A{x}_{\varphi \left(n\right)})\parallel }^2\right)=0.$$

Additionally, $A{x}_{\varphi \left(n\right)}-B{y}_{\varphi \left(n\right)}=0$ if $\varphi \left(n\right)\notin \Gamma$. Hence

$$\underset{n\to \infty }{lim}\parallel A^{*}(A{x}_{\varphi \left(n\right)}-B{y}_{\varphi \left(n\right)}){\parallel }^2=\underset{n\to \infty }{lim}{\parallel B^{*}(B{y}_{\varphi \left(n\right)}-A{x}_{\varphi \left(n\right)})\parallel }^2=0.$$

Following arguments similar to those given in the Case 1, we have

$$\underset{n\to \infty }{lim}d(u_{\varphi \left(n\right)},S_i{u}_{\varphi \left(n\right)})=\underset{n\to \infty }{lim}d(v_{\varphi \left(n\right)},T_i{v}_{\varphi \left(n\right)})=0,$$

and, hence, $\left\{(x_{\varphi \left(n\right)},y_{\varphi \left(n\right)})\right\}$ converges weakly to $(x^{*},y^{*})\in \Omega$. For all $n\ge n_0$, we obtain from (35) that

$$\begin{array}{cc}\hfill 0\le & \parallel x_{\varphi \left(n\right)+1}-x^{*}{\parallel }^2{\parallel y_{\varphi \left(n\right)+1}-q^{*}\parallel }^2-\left(\parallel x_{\varphi \left(n\right)}-x^{*}{\parallel }^2+{\parallel y_{\varphi \left(n\right)}-y^{*}\parallel }^2\right)\hfill \\ \hfill \le & (1-{\eta }_{\varphi \left(n\right)})\left(\parallel u_{\varphi \left(n\right)}-x^{*}{\parallel }^2+{\parallel v_{\varphi \left(n\right)}-y^{*}\parallel }^2\right)\hfill \\ \hfill & +2{\eta }_{\varphi \left(n\right)}(\langle x_{\varphi \left(n\right)+1}-x^{*},f_1\left(u_{\varphi \left(n\right)}\right)-x^{*}\rangle +\langle y_{\varphi \left(n\right)+1}-y^{*},f_2\left(v_{\varphi \left(n\right)}\right)-y^{*}\rangle \hfill \\ \hfill -& \left(\parallel x_{\varphi \left(n\right)}-x^{*}{\parallel }^2+{\parallel y_{\varphi \left(n\right)}-q^{*}\parallel }^2\right)).\hfill \\ \hfill \le & (1-{\eta }_{\varphi \left(n\right)})\left(\parallel x_{\varphi \left(n\right)}-x^{*}{\parallel }^2+{\parallel y_{\varphi \left(n\right)}-y^{*}\parallel }^2\right)\hfill \\ \hfill & +2{\eta }_{\varphi \left(n\right)}(\langle x_{\varphi \left(n\right)+1}-x^{*},f_1\left(u_{\varphi \left(n\right)}\right)-x^{*}\rangle +\langle y_{\varphi \left(n\right)+1}-y^{*},f_2\left(v_{\varphi \left(n\right)}\right)-y^{*}\rangle \hfill \\ \hfill -& \left(\parallel x_{\varphi \left(n\right)}-x^{*}{\parallel }^2+{\parallel y_{\varphi \left(n\right)}-q^{*}\parallel }^2\right)).\hfill \\ \hfill =& 2\left[{\eta }_{\varphi \left(n\right)}\right(\langle x_{\varphi \left(n\right)+1}-x^{*},f_1\left(u_{\varphi \left(n\right)}\right)-x^{*}\rangle +\langle y_{\varphi \left(n\right)+1}-y^{*},f_2\left(v_{\varphi \left(n\right)}\right)-y^{*}\rangle \hfill \\ \hfill & -\frac{1}{2}\left(\parallel x_{\varphi \left(n\right)}-x^{*}{\parallel }^2+{\parallel y_{\varphi \left(n\right)}-y^{*}\parallel }^2\right)].\hfill \end{array}$$

Thus

$$\begin{array}{cc}\hfill \parallel x_{\varphi \left(n\right)}-x^{*}{\parallel }^2+{\parallel y_{\varphi \left(n\right)}-y^{*}\parallel }^2\le & \langle x_{\varphi \left(n\right)+1}-x^{*},f_1\left(u_{\varphi \left(n\right)}\right)-p^{*}\rangle \hfill \\ \hfill & +\langle y_{\varphi \left(n\right)+1}-y^{*},f_2\left(v_{\varphi \left(n\right)}\right)-y^{*}\rangle \to 0.\hfill \end{array}$$

Hence, we have

$$\underset{n\to \infty }{lim}\parallel (x_{\varphi \left(n\right)}-x^{*})\parallel =\underset{n\to \infty }{lim}\parallel (y_{\varphi \left(n\right)}-y^{*})\parallel =0.$$

Therefore,

$$\underset{n\to \infty }{lim}{\mathsf{\Theta }}_{\varphi \left(n\right)}=\underset{n\to \infty }{lim}{\mathsf{\Theta }}_{\varphi \left(n\right)+1}=0.$$

Furthermore, for $n\ge n_0$ observe that ${\mathsf{\Theta }}_{\varphi \left(n\right)}\le {\mathsf{\Theta }}_{\varphi \left(n\right)+1}$ if $n\ne \varphi \left(n\right)$ because ${\mathsf{\Theta }}_j\le {\mathsf{\Theta }}_{j+1}$ for $\varphi \left(n\right)+1\le j\le n.$ Consequently, for all $n\ge n_0,$ we have

$${\mathsf{\Theta }}_n\le max\{{\mathsf{\Theta }}_{\varphi \left(n\right)},{\mathsf{\Theta }}_{\varphi \left(n\right)+1}\}={\mathsf{\Theta }}_{\varphi \left(n\right)+1}.$$

So, ${lim}_{n\to \infty }{\mathsf{\Theta }}_{\varphi \left(n\right)}=0.$ Hence, $\left\{(x_n,y_n)\right\}$ converges strongly to $(x^{*},y^{*})\in \Omega$. □

Corollary 1.

Assume that ${\Omega }_{GSCFPP}\ne \varnothing$ and $\left\{{\alpha }_{i,n}\right\}$ and $\left\{{\eta }_n\right\}$ are sequences in $(0,1)$, for each $i\ge 1$, $ϵ\ge 0$, and γ a fixed non-negative real number. Let $x_1\in H_1$ and $y_1\in H_2$. Define the sequences $\left\{x_n\right\}$ and $\left\{y_n\right\}$ by

$$\begin{array}{cc}\hfill u_n=& P_C\left[x_n-{\gamma }_n{A}^{*}(A{x}_n-y_n)\right]\hfill \\ \hfill x_{n+1}=& {\eta }_n{f}_1\left(u_n\right)+({\alpha }_{0,n}-{\eta }_n)u_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{z}_{i,n},\hfill \\ \hfill v_n=& P_Q\left[y_n-{\gamma }_n(y_n-A{x}_n)\right]\hfill \\ \hfill y_{n+1}=& {\eta }_n{f}_2\left(v_n\right)+({\alpha }_{0,n}-{\eta }_n)v_n+\sum _{i=1}^{\infty }{\alpha }_{i,n}{w}_{i,n},\hfill \end{array}$$

(36)

where $z_{i,n}\in S_i{u}_n$ and $w_{i,n}\in T_i{u}_n$ and

$$I_n=\left(ϵ,\frac{2\parallel A{x}_n-y_n{\parallel }^2}{\parallel A^{*}(A{x}_n-y_n){\parallel }^2+{\parallel (A{x}_n-y_n)\parallel }^2}-ϵ\right).$$

The step size ${\gamma }_n$ is chosen from $I_n$ provided that $n\in \Gamma =\{n:A{x}_n-y_n\ne 0\}$ otherwise take ${\gamma }_n$ equal to $\gamma .$ If

$\left(i\right)$

${\sum }_{i=0}^{\infty }{\alpha }_{i,n}=1$ for each $n\ge 1$,

$\left(ii\right)$

${lim}_{n\to \infty }{\eta }_n=0$ and ${\sum }_{i=1}^{\infty }{\eta }_n=\infty$,

$\left(iii\right)$

for each $i\ge 1$${lim\; inf}_{n\to \infty }{\alpha }_{0,n}{\alpha }_{i,n}\ge 0$,

$\left(iv\right)$

${\eta }_n\le {\alpha }_{0,n}$ for each $n\ge 1$,

$\left(v\right)$

for each $p\in {\bigcap }_{i=1}^{\infty }F\left(S_i\right),S_i\left(p\right)=\left\{p\right\}$ and for each $q\in {\bigcap }_{i=1}^{\infty }F\left(T_i\right),S_i\left(q\right)=\left\{q\right\}$, for each $i\ge 1$.

Then, $\left\{(x_n,y_n)\right\}$ converges strongly to some point in the solution set of (7).

Proof. 

Take $B=I$ in Theorem 1 then $H_2=H_3$. By Theorem 1 we obtain the required result. □

Remark 1.

If we take $i=1$, and $S,T:H\to H$ in Theorem 1 then Theorem 1 gives the solution of SPPF defined in (8).

3. Application

Split Convex Minimization Problem

Assume that $f:C\to C$ and $g:Q\to Q$ are convex continuously differentiable operators. The split convex minimization problem (SCMP) is defined as follows: For $q^{*}=A{p}^{*}$

$$\begin{array}{cc}\hfill & \mathrm{Find}{p}^{*}\in \bigcap _{i=1}^{\infty }F\left(S_i\right)\mathrm{and}{q}^{*}\in \bigcap _{i=1}^{\infty }F\left(T_i\right)\mathrm{such}\mathrm{that}\hfill \\ \hfill & p^{*}=arg\underset{x\in C}{min}f\left(x\right),q^{*}=arg\underset{y\in Q}{min}g\left(y\right).\hfill \end{array}$$

(37)

We denote the solution set of problem (37) by ${\Omega }_{SCMP}$. Let us recall the following result from [40].

Theorem 2.

Let C be a closed and convex subset of a Hilbert space H, $\tau \ge 0$ and $f:C\to R$ be differentiable and convex. Then the following are equivalent

(1) 

A point $p^{*}\in C$ is a minimizer of the function $f.$

(2) 

$p^{*}$ solves the variational inequality $\langle \nabla f\left(p^{*}\right),x-p^{*}\rangle \ge 0$, for all $x\in C$.

(3) 

$p^{*}$ is the solution of fixed point equation: $P_C(I-\tau \nabla f)p^{*}=p^{*}$.

If the multivalued mapping $S:H\to 2^H$ is a maximal monotone mapping, then the resolvent operator ${(I+\lambda S)}^{-1}=J_{\lambda }^S$ is indeed a non-expansive and single valued mapping and the set of its fixed points $F\left(J_{\lambda }^S\right)=S^{-1}\left(0\right).$

Recall that, an operator $g:H\to H$ is called $v-$inverse strongly monotone if for all $x,y\in H$ we have $\langle g\left(x\right)-g\left(y\right),x-y\rangle \ge {v\parallel g\left(x\right)-g\left(y\right)\parallel }^2$, where v is a positive real number.

Recall that, the mapping $f:H\to R$ is said to be lower semi-continuous at a point $x_0\in H$ if $lim{inf}_{x\to x_0}f\left(x\right)\ge f\left(x_0\right)$.

Let $f:C\to R$ be convex and lower semi-continuous mapping and $\nabla f$ the subgradient of f with $\nabla f$ a $v-$inverse strongly monotone operator. Additionally, $P_C(I-\tau \nabla f)$ on a closed and convex subset C of H is a non-expansive mapping for $\tau \in [0,2v]$. Now for non-expansive mapping, the hierarchical variational inequality problem is defined as follow:

$$\mathrm{Find}{p}^{*}\in S^{-1}\left(0\right)\mathrm{such}\mathrm{that}\langle \nabla f\left(p^{*}\right),x-p^{*}\rangle \ge 0\mathrm{for}\mathrm{all}x\in S^{-1}\left(0\right).$$

(38)

By definition of subgradient, we have $f\left(x\right)-f\left(p^{*}\right)\ge \langle \nabla f\left(p^{*}\right),x-p^{*}\rangle \ge 0$. Thus, for all $x\in S^{-1}\left(0\right)$, we have $f\left(x\right)-f\left(p^{*}\right)\ge 0.$ Hence, the Equation (38) is equivalent to the minimization problem which is given as

$$\mathrm{Find}{p}^{*}\in S^{-1}\left(0\right)\mathrm{such}\mathrm{that}f\left(p^{*}\right)=\underset{x\in S^{-1}\left(0\right)}{min}f\left(x\right).$$

(39)

Taking $C=S^{-1}\left(0\right)$, then (39) becomes the split fixed point problem

$$\mathrm{Find}{p}^{*}\in C\mathrm{such}\mathrm{that}{p}^{*}\in F\left(P_C(I-\tau \nabla f)\right).$$

(40)

Using the similar arguments as above, if $T:H\to 2^H$ is a maximal monotone mapping and $g:Q\to R$ is convex and lower semi-continuous mapping and $\nabla g$ is $v-$inverse strongly monotone operator, then by taking $Q=T^{-1}\left(0\right)$ and $A{p}^{*}=q^{*}$ we have

$$\mathrm{Find}{q}^{*}\in Q\mathrm{such}\mathrm{that}{q}^{*}\in F\left(P_Q(I-\tau \nabla g)\right).$$

(41)

Using the Remark 1, (by taking $S=P_C(I-\tau \nabla f)$ and $T=P_Q(I-\tau \nabla g)$ ) we obtain the solution of (37).

4. Conclusions

In this paper, we studied the split equality fixed point problem which is a generalization of split fixed point problem. A new viscosity type iterative technique for approximation of the solution of SEFPP for multivalued demicontractive type operators in Hilbert spaces was proposed and a strong convergence result was proved. As the class of multivalued demicontractive mappings is larger than the class of multivalued quasi-non-expansive mappings [5]. So our Theorem 1 contributes in the following ways:

(1)

The class of mappings we considered in this paper is more general than the mappings in [5,29,36,41,42].

(2)

The problems in [43,44] are special cases of the problem presented in this paper.

Chidume et al. [45] employed the concept of multivalued demicompact mappings. In our paper, we have relaxed the condition of hemicompactness using metric projections. In this way, Theorem 1 has extended Theorem 3.2 in [46].

Open problem: can we prove the Theorem 1 by replacing projection on closed and bounded subset by a half space using more general class of mappings than demicontractive mapping.