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An intermixed iteration for constrained convex minimization problem and split feasibility problem

  • Kanyanee Saechou
  • Atid KangtunyakarnEmail author
Open Access
Research
  • 197 Downloads

Abstract

In this paper, we first introduce the two-step intermixed iteration for finding the common solution of a constrained convex minimization problem, and also we prove a strong convergence theorem for the intermixed algorithm. By using our main theorem, we prove a strong convergence theorem for the split feasibility problem. Finally, we apply our main theorem for the numerical example.

Keywords

Constrained convex minimization problem Split feasibility problem Variational inequality 

MSC

46N10 47H09 74G60 

1 Introduction

Let H be a real Hilbert space with inner product \(\langle \cdot ,\cdot \rangle \) and norm \(\|\cdot \|\). Let C be a nonempty, closed, and convex subset of a real Hilbert space H.

We denote the fixed point set of a mapping T by \(F(T)\). Fixed point theory can be applied to variational inequality problems, equilibrium problems, split feasibility problems, optimization problems, etc. These problems are encountered in various fields such as engineering, physics, game theory, and economics.

A mapping T of C into itself is called nonexpansive if
$$ \Vert Tx-Ty \Vert \leq \Vert x-y \Vert ,\quad \forall x,y\in C. $$

In mathematics, conventional optimization problems arise in the process of making a trading system more effective and are usually stated in terms of minimization problems. In this paper, we give a new iteration for solving two constrained convex minimization problems.

Convex constrained minimization problem is popular and very important to various branches in physics, engineering and economics, e.g., to find the minimum travel distance or to find the lowest cost. Consider the constrained convex minimization problem as follows:
$$ \mbox{minimize } \bigl\{ f(x):x\in C\bigr\} , $$
(1)
where \(f:C\rightarrow \mathbb{R}\) is a real-valued convex function. If f is (Fréchet) differentiable, then the gradient-projection algorithm (GPA) generates a sequence \(\{x_{n}\}\) using the following recursive formula:
$$ x_{n+1}=P_{C}\bigl(x_{n}-\lambda \nabla f(x_{n})\bigr),\quad \forall n\geq 0, $$
(2)
or more generally,
$$ x_{n+1}=P_{C}\bigl(x_{n}-\lambda _{n}\nabla f(x_{n})\bigr),\quad \forall n\geq 0, $$
(3)
where both in (2) and (3) the initial guess \(x_{0}\) is taken from C arbitrarily, and the parameters, λ or \(\lambda _{n}\), are positive real numbers satisfying certain conditions. The convergence of the algorithms (2) and (3) depends on the behavior of the gradient ∇f. In fact, it is known that if ∇f is α-strongly monotone and L-Lipschitz with constants \(\alpha ,L \geq 0\), then the operator
$$ T := P_{C}(I-\lambda \nabla f) $$
(4)
is a contraction; hence, the sequence \(\{x_{n}\}\) defined by the algorithm (2) converges in norm to the unique minimizer of (1). However, if the gradient ∇f fails to be strongly monotone, the operator T defined by (4) could fail to be contractive; consequently, the sequence \(\{x_{n}\}\) generated by the algorithm (2) may fail to converge strongly [1]. If ∇f is Lipschitz, then the algorithms (2) and (3) can still converge in the weak topology under certain conditions [2, 3, 4].
The variational inequality problem is to find a point \(u\in C\) such that
$$ \langle v-u,Au\rangle \geq 0,\quad \forall v\in C. $$
(5)
We denote the set of solutions of the variational inequality by \(\operatorname{VI}(C,A)\). Many models of variational inequalities are used in practice, including a mathematical theory, some interesting connections to numerous disciplines and a wide range of important applications in engineering, physics, optimization, minimax problems, game theory, and economics; for more details, see [5, 6].

Su and Xu [3] introduced the relation of a solution to the minimization problem (1) and solutions of the variational inequality (5) as stated in the following Lemma 1, and this lemma helps to prove the theorem about the minimization problem more effectively; for more details, see [7, 8, 9].

Lemma 1

(Optimality condition, [3])

A necessary condition for a point\(x^{*}\in C\)to be a solution of the minimization problem (1) is that\(x^{*}\)solves the variational inequality
$$ \bigl\langle \nabla f\bigl(x^{*}\bigr),x-x^{*} \bigr\rangle \geq 0,\quad \forall x\in C. $$
(6)
Equivalently, \(x^{*}\in C\)solves the fixed point equation
$$ x^{*}=P_{C}\bigl(x^{*}-\lambda \nabla f \bigl(x^{*}\bigr)\bigr), $$
for every constant\(\lambda >0\). If, in addition, fis convex, then the optimality condition (6) is also sufficient.

By \(U_{f}\) we denote the set of solutions of (1).

In 2011, Ceng et al. [10] introduced the following iterative scheme that generates a sequence \(\{x_{n}\}\) in an explicit way:
$$ x_{n+1}=P_{C}\bigl[s_{n}\gamma Vx_{n}+(I-s_{n}\mu F)T_{n}x_{n} \bigr],\quad \forall n\geq 0, $$
where \(s_{n}=\frac{2-\lambda _{n}L}{4}\) and \(P_{C}(I-\lambda _{n}\nabla f)=s_{n}I+(1-s_{n})T_{n}\) for each \(n\geq 0\). He proved that the sequence \(\{x_{n}\}\) converges strongly to a minimizer \(x^{*}\in S\) of (1).

In 2014, Ming and Lei [11] introduced an explicit composite iterative method for finding the common element of the set of solutions to an equilibrium problem and the solution set to a constrained convex minimization problem, as well as proved a strong convergence theorem, as follows:

Algorithm 1

Given \(x_{1}\in C\), let the sequences \(\{u_{n}\}\) and \(\{x_{n}\}\) be generated iteratively by
$$ \textstyle\begin{cases} \phi (u_{n},y)+\frac{1}{\beta _{n}}\langle y-u_{n},u_{n}-x_{n}\rangle \geq 0,& \forall y\in C, \\ x_{n+1}=\alpha _{n}\gamma Vu_{n}+(I-\alpha _{n}A)T_{n}u_{n}, & \forall n\in \mathbb{N}, \end{cases} $$
where \(T_{n}\) is a nonexpansive mapping from \(P_{C}(I-\lambda _{n} \nabla f)=s_{n}I+(1-s_{n})T_{n}\), which is \(\frac{2+\lambda _{n}L}{4}\)-averaged with \(s_{n}= \frac{2-\lambda _{n}L}{4}\), and ∇f is an L-Lipschitz mapping, for all \(L\geq 0\), \(V:C\rightarrow C\) is an l-Lipschitz mapping with constant \(l \geq 0\), \(A:C\rightarrow C\) is a strongly positive bounded linear operator with coefficient \(\bar{\gamma } \geq 0\) and \(0<\gamma <\frac{\bar{\gamma }}{l}\), \(u_{n}=Q_{\beta _{n}}x_{n}\), \(\{\lambda _{n}\}\subset (0,\frac{2}{L})\), \(\{\alpha _{n}\}\subset (0,1)\), \(\{\beta _{n}\}\subset (0,\infty )\) and \(\{s_{n}\}\subset (0, \frac{1}{2})\).

In 2015, Yao et al. [12] introduced the intermixed algorithm for two strict pseudocontractions S and T as follows:

Algorithm 2

For arbitrarily given \(x_{0}\in C\), \(y_{0}\in C\), let the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) be generated iteratively by
$$ \textstyle\begin{cases} x_{n+1}=(1-\beta _{n})x_{n}+\beta _{n}P_{C}[\alpha _{n}f(y_{n})+(1-k- \alpha _{n})x_{n}+kTx_{n}], & n\geq 0, \\ y_{n+1}=(1-\beta _{n})y_{n}+\beta _{n}P_{C}[\alpha _{n}g(x_{n})+(1-k- \alpha _{n})y_{n}+kSy_{n}], & n\geq 0, \end{cases} $$
(7)
where \(S,T:C\rightarrow C\) are λ-strictly pseudocontractions, \(f:C\rightarrow H\) is a \(\rho _{1}\)-contraction, and \(g:C\rightarrow H\) is a \(\rho _{2}\)-contraction, \(k\in (0,1-\lambda )\) is a constant, and \(\{\alpha _{n}\}\), \(\{\beta _{n}\}\) are two real number sequences in \((0,1)\).

Furthermore, under some control conditions, they proved that the iterative sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) defined by (7) converge independently to \(P_{F(T)}f(y^{*})\) and \(P_{F(S)}g(x^{*})\), respectively, where \(x^{*}\in F(T)=\{z\in C:Tz=z \}\) and \(y^{*}\in F(S)=\{z^{*}\in C:Tz^{*}=z^{*}\}\).

Motivated by Yao et al. [12] and Ming et al. [11], we introduce the new iterative method as follows:

Algorithm 3

Given \(x_{1},y_{1}\in C\), let the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) be defined by
$$ \textstyle\begin{cases} x_{n+1}=(1-\mu _{n})x_{n}+\mu _{n}P_{C}(\alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}), \\ y_{n+1}=(1-\mu _{n})y_{n}+\mu _{n}P_{C}(\alpha _{n}g(x_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{2}}y_{n}), \end{cases} $$
(8)
where \(f,g:H\rightarrow H\) are \(a_{f}\)- and \(a_{g}\)-contraction mappings with \(a_{f},a_{g}\in (0,1)\) and \(a=\max \{a_{f},a_{g}\}\), \(\nabla \widetilde{f_{i}}\) is an \(\frac{1}{L_{i}}\)-inverse strongly monotone with \(L_{i}\geq 0\), for all \(i=1,2\), \(\{\mu _{n}\}\), \(\{\alpha _{n}\} \subseteq [0,1]\), \(P_{C}(I-\lambda _{n}^{i}\nabla \widetilde{f}_{i})=s _{n}^{i}I+(1-s_{n}^{i})T_{n}^{\widetilde{f}_{i}}\), \(\forall i=1,2\) and \(s_{n}^{i}=\frac{2-\lambda _{n}^{i}L_{i}}{4}\), \(\{\lambda _{n}^{i}\} \subset (0,\frac{2}{L_{i}})\) and \(0<\overline{\theta }\leq \mu _{n} \leq \theta \) for all \(n\in \mathbb{N}\) and for some \(\overline{ \theta },\theta >0\).

The purpose of this article is to combine the GPA and averaged mapping approach to design a two-step intermixed iteration for finding the common solution of a constrained convex minimization problem, and also prove a strong convergence theorem for the intermixed algorithm generated by (8). Applying our main result, we prove a strong convergence theorem for the split feasibility problem. Moreover, we utilize our main theorem in the numerical example.

2 Preliminaries

Throughout this article, we always assume that C is a nonempty, closed, and convex subset of a real Hilbert space H. We use “⇀” for weak convergence and “→” for strong convergence. For every \(x\in H\), there is a unique nearest point \(P_{C}x\) in C such that
$$ \Vert x-P_{C}x \Vert \leq \Vert x-y \Vert , \quad \forall y \in C. $$
Such an operator \(P_{C}\) is called the metric projection of H onto C.
Assume that C is a nonempty closed and convex subset of H. A mapping \(V:C \rightarrow C\) is said to be an l-Lipschitz if there exists a constant \(l \geq 0\) such that
$$ \Vert Vx-Vy \Vert \leq l \Vert x-y \Vert , \quad \forall x,y \in C. $$
If \(l \in [0,1)\), then V is called a contraction. Obviously, if \(l=1\), V is a nonexpansive mapping.

Definition 1

A mapping \(T:H \rightarrow H\) is said to be firmly nonexpansive if and only if \(2T-I\) is nonexpansive, or equivalently,
$$ \langle x-y,Tx-Ty\rangle \geq \Vert Tx-Ty \Vert ^{2}, \quad x,y \in H. $$
Alternatively, T is firmly nonexpansive if and only if T can be expressed as
$$ T=\frac{1}{2}(I+S), $$
where \(S:H \rightarrow H\) is nonexpansive.

Definition 2

(Positive operator)

An operator A is called positive if it is self-adjoint and \(\langle Ax,x\rangle \geq 0\) for all \(x \in H\).

An operator A on H is strongly positive if there exists a constant \(\overline{\gamma } > 0\) with the property
$$ \langle Ax,x\rangle \geq \overline{\gamma } \Vert x \Vert ^{2}, \quad \forall x \in H. $$

Lemma 2

([13])

For a given\(z\in H\)and\(u\in C\),
$$ u=P_{C}z\quad \Longleftrightarrow\quad \langle u-z,v-u\rangle \geq 0, \quad \forall v\in C. $$
Furthermore, \(P_{C}\)is a firmly nonexpansive mapping ofHontoC.

Lemma 3

([14])

LetHbe a real Hilbert space. Then the following results hold:
  1. (i)
    For all\(x,y\in H\)and\(\alpha \in [0,1]\),
    $$ \bigl\Vert \alpha x+(1-\alpha )y \bigr\Vert ^{2} = \alpha \Vert x \Vert ^{2}+(1-\alpha ) \Vert y \Vert ^{2}- \alpha (1- \alpha ) \Vert x-y \Vert ^{2}, $$
     
  2. (ii)

    \(\|x+y\|^{2} \leq \|x\|^{2}+2\langle y,x+y\rangle \), for each\(x,y\in H\).

     

Lemma 4

([4])

Let\(\{s_{n}\}\)be a sequence of nonnegative real numbers satisfying
$$ s_{n+1} = (1-\alpha _{n})s_{n}+\delta _{n}, \quad \forall n\geq 0, $$
where\(\{\alpha _{n}\}\)is a sequence in\((0,1)\)and\(\{\delta _{n}\}\)is a sequence such that
  1. (1)

    \(\sum_{n=1}^{\infty }\alpha _{n}=\infty \),

     
  2. (2)

    \(\limsup_{n\rightarrow \infty }\frac{\delta _{n}}{\alpha _{n}}\leq 0\)or\(\sum_{n=1}^{\infty }|\delta _{n}|<\infty \).

     
Then\(\lim_{n\rightarrow \infty }s_{n}=0\).

Definition 3

A mapping \(T:H \rightarrow H\) is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,
$$ T=(1-\alpha )I+\alpha S, $$
(9)
where α is a number in \((0,1)\) and \(S: H \rightarrow H\) is nonexpansive. More precisely, when (9) holds, we say that T is α-averaged.

Clearly, a firmly nonexpansive mapping is a \(\frac{1}{2}\)-averaged mapping.

Proposition 1

For given operators\(S, T, V:H \rightarrow H\):
  1. (i)

    If\(T=(1-\alpha )S+\alpha V\)for some\(\alpha \in (0,1)\)and ifSis averaged andVis nonexpansive, thenTis averaged.

     
  2. (ii)

    Tis firmly nonexpansive if and only if the complement\(I-T\)is firmly nonexpansive.

     
  3. (iii)

    If\(T=(1-\alpha )S+\alpha V\)for some\(\alpha \in (0,1)\), Sis firmly nonexpansive andVis nonexpansive, thenTis averaged.

     
  4. (iv)

    The composition of finitely many averaged mappings is averaged. That is, if each of the mappings\(\{T_{i}\}_{i=1}^{N}\)is averaged, then so is the composition\(T_{1}\circ T_{2} \circ \cdots \circ T_{N}\). In particular, if\(T_{1}\)is\(\alpha _{1}\)-averaged, and\(T_{2}\)is\(\alpha _{2}\)-averaged, where\(\alpha _{1},\alpha _{2} \in (0,1)\), then the composition\(T_{1}\circ T_{2}\)isα-averaged, where\(\alpha =\alpha _{1}+\alpha _{2}-\alpha _{1}\alpha _{2}\).

     

Lemma 5

([11])

For given\(x \in H\)and let\(P_{C}:H \rightarrow C\)be a metric projection. Then
  1. (a)

    \(z=P_{C}x\)if and only if\(\langle x-z,y-z\rangle \leq 0\), \(\forall y\in C\).

     
  2. (b)

    \(z=P_{C}x\)if and only if\(\| x-z\|^{2} \leq \| x-y\|^{2}- \| y-z\|^{2}\), \(\forall y \in C\).

     
  3. (c)

    \(\langle P_{C}x-P_{C}y,x-y\rangle \geq \| P_{C}x-P_{C}y\| ^{2}\), \(\forall x,y\in H\).

     
Consequently, \(P_{C}\)is nonexpansive and monotone.

Lemma 6

([15])

Each Hilbert spaceHsatisfies Opial’s condition, i.e., for any sequence\(\{ u_{n} \} \subset H\)with\(u_{n} \rightharpoonup u\), the inequality
$$ \liminf_{n \rightarrow \infty } \Vert u_{n}-u \Vert < \liminf _{n \rightarrow \infty } \Vert u_{n}-v \Vert $$
holds for every\(v \in H\)with\(v \neq u\).

Definition 4

A nonlinear operator T whose domain \(D(T)\subseteq H\) and range \(R(T)\subseteq H\) is said to be:
  1. (a)
    monotone if
    $$ \langle x-y,Tx-Ty \rangle \geq 0, \quad \forall x,y \in D(T); $$
     
  2. (b)
    β-strongly monotone if there exists \(\beta > 0\) such that
    $$ \langle x-y,Tx-Ty \rangle \geq \beta \Vert x-y \Vert ^{2}, \quad \forall x,y \in D(T); $$
     
  3. (c)
    v-inverse strongly monotone (for short, v-ism) if there exists \(v>0\) such that
    $$ \langle x-y,Tx-Ty \rangle \geq v \Vert Tx-Ty \Vert ^{2}, \quad \forall x,y \in D(T). $$
     

Proposition 2

LetTbe an operator fromHto itself. Then
  1. (a)

    Tis nonexpansive if and only if the complement\(I-T\)is\(\frac{1}{2}\)-ism;

     
  2. (b)

    IfTisv-ism, then for\(\gamma >0\), γTis\(\frac{v}{\gamma }\)-ism;

     
  3. (c)

    Tis averaged if and only if the complement\(I-T\)isv-ism for some\(v > \frac{1}{2}\). Indeed, for\(\alpha \in (0,1)\), Tisα-averaged if and only if\(I-T\)is\(\frac{1}{2\alpha }\)-ism.

     

Lemma 7

([16])

Assume\(A:H \rightarrow H\)is a strongly positive bounded linear operator with coefficient\(\overline{\gamma }>0\)and\(0< t\leq \|A\| ^{-1}\). Then\(\|I-tA\|\leq 1-t\overline{\gamma }\).

3 Main results

Let \(V:C\rightarrow C\) be l-Lipschitz with coefficient \(l \geq 0\), and \(A:C\rightarrow C\) a strongly positive bounded linear operator with coefficient γ̅ and \(0<\gamma <\frac{\overline{ \gamma }}{l}\). Let \(f:C\rightarrow \mathbb{R}\) be a real-valued convex function and assume that ∇f is an L-Lipschitz mapping with \(L \geq 0\). From Xu [1], we have that \(P_{C}(I-\lambda \nabla f)\) is \(\frac{2+\lambda L}{4}\)-averaged for \(0<\lambda < \frac{2}{L}\) and for each \(n\in \mathbb{N}\), that is, we can write
$$ P_{C}(I-\lambda _{n}\nabla f)=(1-s_{n})I+s_{n}T_{n}^{f}, $$
where \(T_{n}^{f}\) is nonexpansive and \(s_{n}=\frac{2+\lambda _{n}L}{4}\).

Theorem 1

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. For every\(i=1,2\), \(\widetilde{f}_{i}:C \rightarrow \mathbb{R}\)be a real-valued convex function and assume that\(\nabla \widetilde{f} _{i}\)is an\(\frac{1}{L_{i}}\)-inverse strongly monotone with\(L_{i}> 0\)and\(U_{\widetilde{f}_{i}}\neq \emptyset \). Let\(f,g:H \rightarrow H\)be\(a_{f}\)- and\(a_{g}\)-contraction mappings, respectively, with\(a_{f},a_{g}\in (0,1)\)and\(a=\max \{a_{f},a_{g}\}\). Let the sequences\(\{x_{n}\}\), \(\{y_{n}\}\)be generated by\(x_{1},y _{1}\in C\)and
$$ \textstyle\begin{cases} x_{n+1}=(1-\mu _{n})x_{n}+\mu _{n}P_{C}(\alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}), \\ y_{n+1}=(1-\mu _{n})y_{n}+\mu _{n}P_{C}(\alpha _{n}g(x_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{2}}y_{n}), \end{cases} $$
(10)
where\(\{\mu _{n}\},\{\alpha _{n}\}\subseteq [0,1]\), \(P_{C}(I-\lambda _{n}^{i}\nabla \widetilde{f}_{i})=s_{n}^{i}I+(1-s_{n}^{i})T_{n}^{ \widetilde{f}_{i}}\), \(s_{n}^{i}=\frac{2-\lambda _{n}^{i}L_{i}}{4}\)and\(\{\lambda _{n}^{i}\}\subset (0,\frac{2}{L_{i}})\)for all\(i=1,2\). Assume that the following conditions hold:
  1. (i)

    \(\lim_{n\rightarrow \infty }\alpha _{n}=0\)and\(\sum_{n=1}^{\infty }\alpha _{n}=\infty \),

     
  2. (ii)

    \(0<\overline{\theta }\leq \mu _{n} \leq \theta \)for all\(n\in \mathbb{N}\)and for some\(\overline{\theta },\theta >0\),

     
  3. (iii)

    \(\sum_{n=1}^{\infty }|\alpha _{n+1}-\alpha _{n}|<\infty \), \(\sum_{n=1}^{\infty }|\mu _{n+1}-\mu _{n}|<\infty \).

     
Then\(\{x_{n}\}\)and\(\{y_{n}\}\)converge strongly as\(s_{n}^{i} \rightarrow 0\) (\(\Longleftrightarrow \lambda _{n}^{i}\rightarrow \frac{2}{L _{i}}\)) \(\forall i=1,2\), to\(x^{*}=P_{U_{\widetilde{f}_{1}}}f(y^{*})\)and\(y^{*}=P_{U_{\widetilde{f}_{2}}}g(x^{*})\), respectively.

Proof

First, we show that \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded. Assume that \(\widetilde{x}\in U_{\widetilde{f}_{1}}\) and \(\widetilde{y}\in U_{ \widetilde{f}_{2}}\). Then we have
$$\begin{aligned} \Vert x_{n+1}-\widetilde{x} \Vert =& \bigl\Vert (1-\mu _{n})x_{n}+\mu _{n}P_{C}\bigl( \alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n} \bigr)- \widetilde{x} \bigr\Vert \\ =& \bigl\Vert (1-\mu _{n}) (x_{n}-\widetilde{x})+\mu _{n}\bigl(P_{C}\bigl(\alpha _{n}f(y_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}\bigr)- \widetilde{x}\bigr) \bigr\Vert \\ \leq& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert + \mu _{n} \bigl\Vert \alpha _{n}f(y_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert \\ \leq& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert + \mu _{n}\bigl(\alpha _{n} \bigl\Vert f(y_{n})- \widetilde{x} \bigr\Vert +(1-\alpha _{n}) \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert \bigr) \\ \leq& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert + \mu _{n}\bigl(\alpha _{n} \bigl\Vert f(y_{n})- \widetilde{x} \bigr\Vert +(1-\alpha _{n}) \Vert x_{n}- \widetilde{x} \Vert \bigr) \\ =& (1-\alpha _{n}\mu _{n}) \Vert x_{n}- \widetilde{x} \Vert +\alpha _{n}\mu _{n} \bigl\Vert f(y _{n})-\widetilde{x} \bigr\Vert \\ \leq& (1-\alpha _{n}\mu _{n}) \Vert x_{n}- \widetilde{x} \Vert +\alpha _{n}\mu _{n}\bigl( \bigl\Vert f(y_{n})-f(\widetilde{y}) \bigr\Vert + \bigl\Vert f( \widetilde{y})-\widetilde{x} \bigr\Vert \bigr) \\ \leq& (1-\alpha _{n}\mu _{n}) \Vert x_{n}- \widetilde{x} \Vert +\alpha _{n}\mu _{n}a \Vert y_{n}-\widetilde{y} \Vert +\alpha _{n}\mu _{n} \bigl\Vert f(\widetilde{y})- \widetilde{x} \bigr\Vert . \end{aligned}$$
(11)
Similarly, we get
$$ \Vert y_{n+1}-\widetilde{y} \Vert \leq (1-\alpha _{n}\mu _{n}) \Vert y_{n}- \widetilde{y} \Vert +\alpha _{n}\mu _{n}a \Vert x_{n}- \widetilde{x} \Vert +\alpha _{n} \mu _{n} \bigl\Vert g( \widetilde{x})-\widetilde{y} \bigr\Vert . $$
(12)
Combining (11) and (12), we have
$$\begin{aligned} \Vert x_{n+1}-\widetilde{x} \Vert + \Vert y_{n+1}- \widetilde{y} \Vert \leq{} & \bigl(1-\alpha _{n}\mu _{n}(1-a)\bigr) \bigl( \Vert x_{n}-\widetilde{x} \Vert + \Vert y_{n}-\widetilde{y} \Vert \bigr) \\ &{}+\alpha _{n}\mu _{n}\bigl( \bigl\Vert f(\widetilde{y})- \widetilde{x} \bigr\Vert + \bigl\Vert g( \widetilde{x})-\widetilde{y} \bigr\Vert \bigr). \end{aligned}$$
By induction, we can derive that
$$ \Vert x_{n}-\widetilde{x} \Vert + \Vert y_{n}- \widetilde{y} \Vert \leq \max \bigl\{ \Vert x_{1}- \widetilde{x} \Vert + \Vert y_{1}-\widetilde{y} \Vert , \bigl\Vert f( \widetilde{y})- \widetilde{x} \bigr\Vert + \bigl\Vert g(\widetilde{x})- \widetilde{y} \bigr\Vert \bigr\} , $$
for every \(n \in \mathbb{N}\). This implies that \(\{x_{n}\}\) and \(\{y_{n}\}\) are bounded.
Next, we show that \(\|x_{n+1}-x_{n}\|\rightarrow 0\) and \(\|y_{n+1}-y _{n}\|\rightarrow 0\). Observe that
$$\begin{aligned}& \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}-T_{n-1}^{\widetilde{f}_{1}}x_{n-1} \bigr\Vert \\& \quad \leq \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}-T_{n}^{\widetilde{f}_{1}}x _{n-1} \bigr\Vert + \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n-1}-T_{n-1}^{\widetilde{f} _{1}}x_{n-1} \bigr\Vert \\ & \quad \leq \Vert x_{n}-x_{n-1} \Vert + \biggl\Vert \biggl( \frac{4P_{C}(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1})-(2-\lambda _{n}^{1}L_{1})}{2+\lambda _{n}^{1}L _{1}} \biggr)x_{n-1} \\ & \qquad {} - \biggl(\frac{4P_{C}(I-\lambda _{n-1}^{1}\nabla \widetilde{f}_{1})-(2- \lambda _{n-1}^{1}L_{1})}{2+\lambda _{n-1}^{1}L_{1}} \biggr)x_{n-1} \biggr\Vert \\ & \quad \leq \Vert x_{n}-x_{n-1} \Vert + \biggl\Vert \biggl( \frac{4P_{C}(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1})}{2+\lambda _{n}^{1}L_{1}} \biggr)x_{n-1}- \biggl(\frac{4P _{C}(I-\lambda _{n-1}^{1}\nabla \widetilde{f}_{1})}{2+\lambda _{n-1} ^{1}L_{1}} \biggr)x_{n-1} \biggr\Vert \\ & \qquad {} + \biggl\Vert \biggl(\frac{2-\lambda _{n-1}^{1}L_{1}}{2+\lambda _{n-1}^{1}L _{1}} \biggr)x_{n-1}- \biggl( \frac{2-\lambda _{n}^{1}L_{1}}{2+\lambda _{n} ^{1}L_{1}} \biggr)x_{n-1} \biggr\Vert \\ & \quad = \Vert x_{n}-x_{n-1} \Vert \\ & \qquad {} + \biggl\Vert \frac{4(2+\lambda _{n-1}^{1}L_{1})P_{C}(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1})x_{n-1}-4(2+\lambda _{n}^{1}L_{1})P_{C}(I- \lambda _{n-1}^{1}\nabla \widetilde{f}_{1})x_{n-1}}{(2+\lambda _{n}^{1}L _{1})(2+\lambda _{n-1}^{1}L_{1})} \biggr\Vert \\ & \qquad {} + \biggl\Vert \frac{(2-\lambda _{n-1}^{1}L_{1})(2+\lambda _{n}^{1}L_{1})x _{n-1}-(2-\lambda _{n}^{1}L_{1})(2+\lambda _{n-1}^{1}L_{1})x_{n-1}}{(2+ \lambda _{n}^{1}L_{1})(2+\lambda _{n-1}^{1}L_{1})} \biggr\Vert \\ & \quad = \Vert x_{n}-x_{n-1} \Vert \\ & \qquad {} + \biggl\Vert \frac{4(2+\lambda _{n-1}^{1}L_{1})P_{C}(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1})x_{n-1}-4(2+\lambda _{n}^{1}L_{1})P_{C}(I- \lambda _{n-1}^{1}\nabla \widetilde{f}_{1})x_{n-1}}{(2+\lambda _{n}^{1}L _{1})(2+\lambda _{n-1}^{1}L_{1})} \biggr\Vert \\ & \qquad {} + \biggl(\frac{4L_{1} \vert \lambda _{n}^{1}-\lambda _{n-1}^{1} \vert }{(2+\lambda _{n}^{1}L_{1})(2+\lambda _{n-1}^{1}L_{1})} \biggr) \Vert x_{n-1} \Vert \\ & \quad = \Vert x_{n}-x_{n-1} \Vert \\ & \qquad {} + \biggl\Vert \frac{4L_{1}(\lambda _{n-1}^{1}-\lambda _{n}^{1})P_{C}(I- \lambda _{n}^{1}\nabla \widetilde{f}_{1})x_{n-1}}{(2+\lambda _{n}^{1}L _{1})(2+\lambda _{n-1}^{1}L_{1})} \\ & \qquad {} + \frac{4(2+\lambda _{n}^{1}L_{1})(P_{C}(I-\lambda _{n}^{1}\nabla \widetilde{f}_{1})x_{n-1}-P_{C}(I-\lambda _{n-1}^{1}\nabla \widetilde{f}_{1})x_{n-1})}{(2+\lambda _{n}^{1}L_{1})(2+\lambda _{n-1} ^{1}L_{1})} \biggr\Vert \\ & \qquad {} + \biggl(\frac{4L_{1} \vert \lambda _{n}^{1}-\lambda _{n-1}^{1} \vert }{(2+\lambda _{n}^{1}L_{1})(2+\lambda _{n-1}^{1}L_{1})} \biggr) \Vert x_{n-1} \Vert \\ & \quad \leq \Vert x_{n}-x_{n-1} \Vert \\ & \qquad {} + \frac{4L_{1} \vert \lambda _{n-1}^{1}-\lambda _{n}^{1} \vert \Vert P_{C}(I-\lambda _{n}^{1}\nabla \widetilde{f}_{1})x_{n-1} \Vert }{(2+\lambda _{n}^{1}L_{1})(2+ \lambda _{n-1}^{1}L_{1})}+\frac{4 \Vert \lambda _{n-1}^{1}\nabla \widetilde{f}_{1}x_{n-1}-\lambda _{n}^{1}\nabla \widetilde{f}_{1}x_{n-1} \Vert }{2+\lambda _{n-1}^{1}L_{1}} \\ & \qquad {} + \biggl(\frac{4L_{1} \vert \lambda _{n}^{1}-\lambda _{n-1}^{1} \vert }{(2+\lambda _{n}^{1}L_{1})(2+\lambda _{n-1}^{1}L_{1})} \biggr) \Vert x_{n-1} \Vert \\ & \quad \leq \Vert x_{n}-x_{n-1} \Vert +L_{1} \bigl\vert \lambda _{n-1}^{1}-\lambda _{n}^{1} \bigr\vert \bigl\Vert P _{C}\bigl(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1}\bigr)x_{n-1} \bigr\Vert \\ & \qquad {} +4 \bigl\vert \lambda _{n-1}^{1}-\lambda _{n}^{1} \bigr\vert \Vert \nabla \widetilde{f}_{1}x _{n-1} \Vert +L_{1} \bigl\vert \lambda _{n}^{1}-\lambda _{n-1}^{1} \bigr\vert \Vert x_{n-1} \Vert \\ & \quad \leq \Vert x_{n}-x_{n-1} \Vert \\ & \qquad {} + \bigl\vert \lambda _{n-1}^{1}-\lambda _{n}^{1} \bigr\vert \bigl(L_{1} \bigl\Vert P_{C}\bigl(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1}\bigr)x_{n-1} \bigr\Vert +4 \Vert \nabla \widetilde{f}_{1}x_{n-1} \Vert +L_{1} \Vert x_{n-1} \Vert \bigr) \\ & \quad \leq \Vert x_{n}-x_{n-1} \Vert +M_{1} \bigl\vert \lambda _{n-1}^{1}-\lambda _{n}^{1} \bigr\vert , \end{aligned}$$
(13)
for some \(M_{1}>0\) such that \(M_{1}\geq L_{1}\|P_{C}(I-\lambda _{n} ^{1}\nabla \widetilde{f}_{1})x_{n-1}\|+4\|\nabla \widetilde{f}_{1}x _{n-1}\|+L_{1}\|x_{n-1}\|\), \(\forall n\geq 1 \).
From the definition of \(x_{n}\) and (13), we have
$$\begin{aligned}& \Vert x_{n+1}-x_{n} \Vert \\& \quad = \bigl\Vert (1-\mu _{n})x_{n}+\mu _{n}P_{C}\bigl(\alpha _{n}f(y_{n})+(1- \alpha _{n})T _{n}^{\widetilde{f}_{1}}x_{n}\bigr) \\& \qquad {} -\bigl((1-\mu _{n-1})x_{n-1}+\mu _{n-1}P_{C} \bigl(\alpha _{n-1}f(y_{n-1})+(1- \alpha _{n-1})T_{n-1}^{\widetilde{f}_{1}}x_{n-1} \bigr)\bigr) \bigr\Vert \\& \quad \leq (1-\mu _{n}) \Vert x_{n}-x_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert x_{n-1} \Vert \\& \qquad {} + \mu _{n} \bigl\Vert P_{C}\bigl(\alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{ \widetilde{f}_{1}}x_{n} \bigr) \\& \qquad {} -P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T_{n-1}^{ \widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \quad \leq (1-\mu _{n}) \Vert x_{n}-x_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert x_{n-1} \Vert \\& \qquad {} + \mu _{n} \bigl\Vert \alpha _{n}f(y_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f} _{1}}x_{n}-\bigl( \alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T_{n-1}^{ \widetilde{f}_{1}}x_{n-1} \bigr) \bigr\Vert \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \quad \leq (1-\mu _{n}) \Vert x_{n}-x_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert x_{n-1} \Vert \\& \qquad {} + \mu _{n}\bigl( \bigl\Vert \alpha _{n}f(y_{n})- \alpha _{n-1}f(y_{n-1}) \bigr\Vert + \bigl\Vert (1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}-(1-\alpha _{n-1})T_{n-1}^{ \widetilde{f}_{1}}x_{n-1} \bigr\Vert \bigr) \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \quad \leq (1-\mu _{n}) \Vert x_{n}-x_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert x_{n-1} \Vert \\& \qquad {} + \mu _{n}\bigl(\alpha _{n} \bigl\Vert f(y_{n})-f(y_{n-1}) \bigr\Vert + \vert \alpha _{n}-\alpha _{n-1} \vert \bigl\Vert f(y_{n-1}) \bigr\Vert \\& \qquad {} + (1-\alpha _{n}) \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}-T_{n-1}^{ \widetilde{f}_{1}}x_{n-1} \bigr\Vert + \vert \alpha _{n-1}-\alpha _{n} \vert \bigl\Vert T_{n-1}^{ \widetilde{f}_{1}}x_{n-1} \bigr\Vert \bigr) \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \quad \leq (1-\mu _{n}) \Vert x_{n}-x_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert x_{n-1} \Vert \\& \qquad {} + \mu _{n}\bigl(\alpha _{n} \bigl\Vert f(y_{n})-f(y_{n-1}) \bigr\Vert + \vert \alpha _{n}-\alpha _{n-1} \vert \bigl\Vert f(y_{n-1}) \bigr\Vert \\& \qquad {} + (1-\alpha _{n}) \bigl( \Vert x_{n}-x_{n-1} \Vert +M_{1} \bigl\vert \lambda _{n-1}^{1}- \lambda _{n}^{1} \bigr\vert \bigr)+ \vert \alpha _{n-1}-\alpha _{n} \vert \bigl\Vert T_{n-1}^{\widetilde{f}_{1}}x _{n-1} \bigr\Vert \bigr) \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \quad \leq (1-\mu _{n}) \Vert x_{n}-x_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert x_{n-1} \Vert \\& \qquad {} + \mu _{n}\biggl(\alpha _{n} \bigl\Vert f(y_{n})-f(y_{n-1}) \bigr\Vert + \vert \alpha _{n}-\alpha _{n-1} \vert \bigl\Vert f(y_{n-1}) \bigr\Vert \\& \qquad {} + (1-\alpha _{n}) \Vert x_{n}-x_{n-1} \Vert +(1-\alpha _{n}) \frac{4M_{1}}{L_{1}} \bigl\vert s_{n}^{1}-s_{n-1}^{1} \bigr\vert \\& \qquad {} + \vert \alpha _{n-1}-\alpha _{n} \vert \bigl\Vert T_{n-1}^{\widetilde{f}_{1}}x_{n-1} \bigr\Vert \biggr) \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \quad \leq (1-\mu _{n}\alpha _{n}) \Vert x_{n}-x_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert x _{n-1} \Vert \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \qquad {} + \mu _{n}\biggl(\alpha _{n}a \Vert y_{n}-y_{n-1} \Vert + \vert \alpha _{n}-\alpha _{n-1} \vert \bigl\Vert f(y _{n-1}) \bigr\Vert \\& \qquad {} +(1-\alpha _{n})\frac{4M_{1}}{L_{1}} \bigl\vert s_{n}^{1}-s_{n-1}^{1} \bigr\vert + \vert \alpha _{n-1}-\alpha _{n} \vert \bigl\Vert T_{n-1}^{\widetilde{f}_{1}}x_{n-1} \bigr\Vert \biggr). \end{aligned}$$
(14)
Using the same method as derived in (14), we have
$$\begin{aligned}& \Vert y_{n+1}- y_{n} \Vert \\& \quad \leq (1-\mu _{n}\alpha _{n}) \Vert y_{n}-y_{n-1} \Vert + \vert \mu _{n-1}-\mu _{n} \vert \Vert y _{n-1} \Vert \\& \qquad {} + \vert \mu _{n}-\mu _{n-1} \vert \bigl\Vert P_{C}\bigl(\alpha _{n-1}g(x_{n-1})+(1-\alpha _{n-1})T _{n-1}^{\widetilde{f}_{2}}y_{n-1}\bigr) \bigr\Vert \\& \qquad {} + \mu _{n}\biggl(\alpha _{n}a \Vert x_{n}-x_{n-1} \Vert + \vert \alpha _{n}-\alpha _{n-1} \vert \bigl\Vert g(x _{n-1}) \bigr\Vert \\& \qquad {} +(1-\alpha _{n})\frac{4M_{2}}{L_{2}} \bigl\vert s_{n}^{2}-s_{n-1}^{2} \bigr\vert + \vert \alpha _{n-1}-\alpha _{n} \vert \bigl\Vert T_{n-1}^{\widetilde{f}_{2}}y_{n-1} \bigr\Vert \biggr), \end{aligned}$$
(15)
for some \(M_{2}>0\) such that \(M_{2}\geq L_{2}\|P_{C}(I-\lambda _{n} ^{2}\nabla \widetilde{f}_{2})y_{n-1}\|+4\|\nabla \widetilde{f}_{2}y _{n-1}\|+L_{2}\|y_{n-1}\|\), \(\forall n\geq 1\).
From (14) and (15), we have
$$\begin{aligned}& \Vert x _{n+1}-x_{n} \Vert + \Vert y_{n+1}-y_{n} \Vert \\& \quad \leq \bigl(1-(1-a)\mu _{n}\alpha _{n}\bigr) \bigl( \Vert x_{n}-x_{n-1} \Vert + \Vert y_{n}-y_{n-1} \Vert \bigr) \\& \qquad {} + \vert \mu _{n-1}-\mu _{n} \vert \bigl( \Vert x_{n-1} \Vert + \Vert y_{n-1} \Vert \\& \qquad {} + \bigl\Vert P_{C}\bigl(\alpha _{n-1}f(y_{n-1})+(1- \alpha _{n-1})T_{n-1}^{ \widetilde{f}_{1}}x_{n-1}\bigr) \bigr\Vert \\& \qquad {} + \bigl\Vert P_{C}\bigl(\alpha _{n-1}g(x_{n-1})+(1- \alpha _{n-1})T_{n-1}^{ \widetilde{f}_{2}}y_{n-1}\bigr) \bigr\Vert \bigr) \\& \qquad {} + \vert \alpha _{n}-\alpha _{n-1} \vert \bigl( \bigl\Vert f(y_{n-1}) \bigr\Vert + \bigl\Vert g(x_{n-1}) \bigr\Vert + \bigl\Vert T_{n-1} ^{\widetilde{f}_{1}}x_{n-1} \bigr\Vert + \bigl\Vert T_{n-1}^{\widetilde{f}_{2}}y_{n-1} \bigr\Vert \bigr) \\& \qquad {} +(1-\alpha _{n}) \biggl(\frac{4M_{1}}{L_{1}} \bigl\vert s_{n}^{1}-s_{n-1}^{1} \bigr\vert + \frac{4M _{2}}{L_{2}} \bigl\vert s_{n}^{2}-s_{n-1}^{2} \bigr\vert \biggr). \end{aligned}$$
Applying Lemma 4 and condition (iii), we can conclude that
$$ \Vert x_{n+1}-x_{n} \Vert \rightarrow 0\quad \mbox{and}\quad \Vert y_{n+1}-y_{n} \Vert \rightarrow 0\quad \mbox{as } n\rightarrow \infty . $$
(16)
Next, we show that \(\|x_{n}-W_{n}\|\rightarrow 0\) where \(W_{n}=\alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}\) and \(\|y_{n}-V_{n}\|\rightarrow 0\) where \(V_{n}=\alpha _{n}g(x_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f}_{2}}y_{n}\). Let \(\widetilde{x}\in U _{\widetilde{f}_{1}}\) and \(\widetilde{y}\in U_{\widetilde{f}_{2}}\). Then we derive that
$$\begin{aligned} \Vert x_{n+1}-\widetilde{x} \Vert ^{2} =& \bigl\Vert (1-\mu _{n})x_{n}+\mu _{n}P_{C}W_{n}- \widetilde{x} \bigr\Vert ^{2} \\ =& \bigl\Vert (1-\mu _{n}) (x_{n}-\widetilde{x})+\mu _{n}(P_{C}W_{n}- \widetilde{x}) \bigr\Vert ^{2} \\ =& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+\mu _{n} \Vert P_{C}W_{n}- \widetilde{x} \Vert ^{2} \\ &{} - (1-\mu _{n})\mu _{n} \Vert x_{n}-P_{C}W_{n} \Vert ^{2} \\ \leq& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+\mu _{n} \bigl\Vert \alpha _{n}f(y _{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert ^{2} \\ &{} - (1-\mu _{n})\mu _{n} \Vert x_{n}-P_{C}W_{n} \Vert ^{2} \\ =& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+\mu _{n} \bigl\Vert \alpha _{n}\bigl(f(y _{n})-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr)+T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert ^{2} \\ &{} - (1-\mu _{n})\mu _{n} \Vert x_{n}-P_{C}W_{n} \Vert ^{2} \\ \leq & (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+ \mu _{n}\bigl( \bigl\Vert T_{n}^{ \widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert ^{2} \\ &{} +2\alpha _{n}\bigl\langle f(y_{n})-T_{n}^{\widetilde{f}_{1}}x_{n}, \alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\rangle \bigr) \\ &{} - (1-\mu _{n})\mu _{n} \Vert x_{n}-P_{C}W_{n} \Vert ^{2} \\ \leq& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+ \mu _{n}\bigl( \bigl\Vert T_{n}^{ \widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert ^{2} \\ &{} +2\alpha _{n} \bigl\Vert f(y_{n})-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr\Vert \bigl\Vert \alpha _{n}f(y _{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}-\widetilde{x} \bigr\Vert \bigr) \\ &{} - (1-\mu _{n})\mu _{n} \Vert x_{n}-P_{C}W_{n} \Vert ^{2} \\ \leq& (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+\mu _{n} \Vert x_{n}- \widetilde{x} \Vert ^{2} \\ &{} +2\alpha _{n}\mu _{n} \bigl\Vert f(y_{n})-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr\Vert \bigl\Vert \alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert \\ &{} - (1-\mu _{n})\mu _{n} \Vert x_{n}-P_{C}W_{n} \Vert ^{2} \\ =& \Vert x_{n}-\widetilde{x} \Vert ^{2} \\ &{} +2\alpha _{n}\mu _{n} \bigl\Vert f(y_{n})-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr\Vert \bigl\Vert \alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert \\ &{} - (1-\mu _{n})\mu _{n} \Vert x_{n}-P_{C}W_{n} \Vert ^{2}, \end{aligned}$$
which implies that
$$\begin{aligned}& (1-\mu _{n})\mu _{n}\|x_{n}-P_{C}W_{n}\|^{2} \\& \quad \leq \Vert x_{n}-\widetilde{x} \Vert ^{2}- \Vert x_{n+1}-\widetilde{x} \Vert ^{2} \\& \qquad {} +2\alpha _{n}\mu _{n} \bigl\Vert f(y_{n})-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr\Vert \bigl\Vert \alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert \\& \quad \leq \Vert x_{n}-x_{n+1} \Vert \bigl( \Vert x_{n}-\widetilde{x} \Vert + \Vert x_{n+1}- \widetilde{x} \Vert \bigr) \\& \qquad {} +2\alpha _{n}\mu _{n} \bigl\Vert f(y_{n})-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr\Vert \bigl\Vert \alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x} \bigr\Vert . \end{aligned}$$
By (16), as well as conditions (i) and (ii), we get
$$ \Vert P_{C}W_{n}-x_{n} \Vert \rightarrow 0 \quad \mbox{as } n\rightarrow \infty . $$
(17)
From definition of \(x_{n}\) and applying the same method as (17), we have
$$ \Vert P_{C}V_{n}-y_{n} \Vert \rightarrow 0\quad \mbox{as } n\rightarrow \infty . $$
(18)
Considering
$$\begin{aligned} \Vert P_{C}W_{n}-\widetilde{x} \Vert ^{2} &= \Vert P_{C}W_{n}-P_{C}\widetilde{x} \Vert ^{2} \\ &\leq \langle W_{n}-\widetilde{x},P_{C}W_{n}- \widetilde{x} \rangle \\ &= \frac{1}{2}\bigl( \Vert W_{n}-\widetilde{x} \Vert ^{2}+ \Vert P_{C}W_{n}- \widetilde{x} \Vert ^{2}- \Vert W_{n}-P_{C}W_{n} \Vert ^{2}\bigr) \end{aligned}$$
implies that
$$ \Vert P_{C}W_{n}-\widetilde{x} \Vert \leq \Vert W_{n}-\widetilde{x} \Vert ^{2}- \Vert W_{n}-P _{C}W_{n} \Vert ^{2}. $$
(19)
Observe that
$$\begin{aligned} \Vert W_{n}-\widetilde{x} \Vert ^{2} &= \bigl\Vert \alpha _{n}\bigl(f(y_{n})-\widetilde{x} \bigr)+(1- \alpha _{n}) \bigl(T_{n}^{\widetilde{f}_{1}}x_{n}- \widetilde{x}\bigr) \bigr\Vert ^{2} \\ &\leq \alpha _{n} \bigl\Vert f(y_{n})-\widetilde{x} \bigr\Vert ^{2}+(1-\alpha _{n}) \bigl\Vert T _{n}^{\widetilde{f}_{1}}x_{n}-\widetilde{x} \bigr\Vert ^{2} \\ &\leq \alpha _{n} \bigl\Vert f(y_{n})-\widetilde{x} \bigr\Vert ^{2}+(1-\alpha _{n}) \Vert x _{n}- \widetilde{x} \Vert ^{2}. \end{aligned}$$
(20)
From (19) and (20), we obtain
$$\begin{aligned} \Vert x_{n+1}-\widetilde{x} \Vert ^{2} = & \bigl\Vert (1-\mu _{n})x_{n}+\mu _{n}P_{C}\bigl( \alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n} \bigr)- \widetilde{x} \bigr\Vert ^{2} \\ = & \bigl\Vert (1-\mu _{n}) (x_{n}-\widetilde{x})+\mu _{n}(P_{C}W_{n}- \widetilde{x}) \bigr\Vert ^{2} \\ \leq & (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+\mu _{n} \Vert P_{C}W_{n}- \widetilde{x} \Vert ^{2} \\ \leq & (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2}+\mu _{n}\bigl( \Vert W_{n}- \widetilde{x} \Vert ^{2}- \Vert W_{n}-P_{C}W_{n} \Vert ^{2}\bigr) \\ \leq & (1-\mu _{n}) \Vert x_{n}-\widetilde{x} \Vert ^{2} \\ &{}+\mu _{n}\bigl(\alpha _{n} \bigl\Vert f(y_{n})-\widetilde{x} \bigr\Vert ^{2}+(1-\alpha _{n}) \Vert x _{n}-\widetilde{x} \Vert ^{2}- \Vert W_{n}-P_{C}W_{n} \Vert ^{2} \bigr), \end{aligned}$$
implying that
$$\begin{aligned} \mu _{n} \Vert W_{n}-P_{C}W_{n} \Vert ^{2}&\leq (1-\alpha _{n}\mu _{n}) \Vert x_{n}- \widetilde{x} \Vert ^{2}- \Vert x_{n+1}- \widetilde{x} \Vert ^{2}+\alpha _{n}\mu _{n} \bigl\Vert f(y _{n})-\widetilde{x} \bigr\Vert ^{2} \\ &\leq \Vert x_{n}-\widetilde{x} \Vert ^{2}- \Vert x_{n+1}-\widetilde{x} \Vert ^{2}+ \alpha _{n}\mu _{n} \bigl\Vert f(y_{n})-\widetilde{x} \bigr\Vert ^{2} \\ &\leq \Vert x_{n}-x_{n+1} \Vert \bigl( \Vert x_{n}-\widetilde{x} \Vert + \Vert x_{n+1}- \widetilde{x} \Vert \bigr)+\alpha _{n}\mu _{n} \bigl\Vert f(y_{n})-\widetilde{x} \bigr\Vert ^{2}. \end{aligned}$$
From \(\|x_{n+1}-x_{n}\|\rightarrow 0\) as \(n\rightarrow \infty \) and condition (i), we have
$$ \Vert W_{n}-P_{C}W_{n} \Vert \rightarrow 0 \quad \mbox{as } n\rightarrow \infty . $$
(21)
From definition of \(V_{n}\) and applying the same argument as (21), we also obtain
$$ \Vert V_{n}-P_{C}V_{n} \Vert \rightarrow 0 \quad \mbox{as } n\rightarrow \infty . $$
(22)
Since
$$\begin{aligned} \Vert x_{n}-W_{n} \Vert &= \Vert x_{n}-P_{C}W_{n}+P_{C}W_{n}-W_{n} \Vert \\ &\leq \Vert x_{n}-P_{C}W_{n} \Vert + \Vert P_{C}W_{n}-W_{n} \Vert . \end{aligned}$$
From (17) and (21), we have
$$ \Vert x_{n}-W_{n} \Vert \rightarrow 0\quad \mbox{as } n\rightarrow \infty . $$
(23)
From definition of \(y_{n}\) and applying the same method as in (23), we also have
$$ \Vert y_{n}-V_{n} \Vert \rightarrow 0\quad \mbox{as } n\rightarrow \infty . $$
(24)
Next, we show that \(\|W_{n}-P_{C}(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1})W_{n}\|\rightarrow 0\) as \(n\rightarrow \infty \) and \(\|V_{n}-P_{C}(I-\frac{2}{L_{2}}\nabla \widetilde{f}_{2})V_{n}\| \rightarrow 0\) as \(n\rightarrow \infty \). Observe that
$$ W_{n}-x_{n} = \alpha _{n}\bigl(f(y_{n})-x_{n} \bigr)+(1-\alpha _{n}) \bigl(T_{n}^{ \widetilde{f}_{1}}x_{n}-x_{n} \bigr), $$
which yields
$$ (1-\alpha _{n}) \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}-x_{n} \bigr\Vert \leq \Vert W_{n}-x _{n} \Vert +\alpha _{n} \bigl\Vert f(y_{n})-x_{n} \bigr\Vert . $$
From (23) and condition (i), we have
$$ \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}-x_{n} \bigr\Vert \rightarrow 0 \quad \mbox{as } n \rightarrow \infty . $$
(25)
Since
$$\begin{aligned} \bigl\Vert W_{n}-T_{n}^{\widetilde{f}_{1}}W_{n} \bigr\Vert &= \bigl\Vert W_{n}-x_{n}+x_{n}-T_{n} ^{\widetilde{f}_{1}}x_{n}+T_{n}^{\widetilde{f}_{1}}x_{n}-T_{n}^{ \widetilde{f}_{1}}W_{n} \bigr\Vert \\ &\leq \Vert W_{n}-x_{n} \Vert + \bigl\Vert x_{n}-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr\Vert + \bigl\Vert T _{n}^{\widetilde{f}_{1}}x_{n}-T_{n}^{\widetilde{f}_{1}}W_{n} \bigr\Vert \\ &\leq \Vert W_{n}-x_{n} \Vert + \bigl\Vert x_{n}-T_{n}^{\widetilde{f}_{1}}x_{n} \bigr\Vert + \Vert x _{n}-W_{n} \Vert \\ &= 2 \Vert x_{n}-W_{n} \Vert + \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}-x_{n} \bigr\Vert . \end{aligned}$$
From (23) and (25), we get
$$ \bigl\Vert T_{n}^{\widetilde{f}_{1}}W_{n}-W_{n} \bigr\Vert \rightarrow 0 \quad \mbox{as } n \rightarrow \infty . $$
(26)
Observe that
$$\begin{aligned} \bigl\Vert P_{C}\bigl(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1}\bigr)W_{n}-W_{n} \bigr\Vert &= \bigl\Vert s _{n}^{1}W_{n}+ \bigl(1-s_{n}^{1}\bigr)T_{n}^{\widetilde{f}_{1}}W_{n}-W_{n} \bigr\Vert \\ &= \bigl(1-s_{n}^{1}\bigr) \bigl\Vert T_{n}^{\widetilde{f}_{1}}W_{n}-W_{n} \bigr\Vert \\ &\leq \bigl\Vert T_{n}^{\widetilde{f}_{1}}W_{n}-W_{n} \bigr\Vert , \end{aligned}$$
(27)
where \(s_{n}^{1}=\frac{2-\lambda _{n}^{1}L_{1}}{4}\in (0,\frac{1}{2})\).
From (27), we have
$$\begin{aligned}& \biggl\Vert P _{C}\biggl(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1}\biggr)W_{n}-W_{n} \biggr\Vert \\& \quad \leq \biggl\Vert P_{C}\biggl(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1}\biggr)W_{n}-P_{C}\bigl(I- \lambda _{n}^{1}\nabla \widetilde{f}_{1} \bigr)W_{n} \biggr\Vert + \bigl\Vert P_{C}\bigl(I- \lambda _{n} ^{1}\nabla \widetilde{f}_{1} \bigr)W_{n}-W_{n} \bigr\Vert \\& \quad \leq \biggl\Vert \biggl(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1} \biggr)W_{n}-\bigl(I-\lambda _{n}^{1}\nabla \widetilde{f}_{1}\bigr)W_{n} \biggr\Vert + \bigl\Vert P_{C}\bigl(I-\lambda _{n}^{1} \nabla \widetilde{f}_{1}\bigr)W_{n}-W_{n} \bigr\Vert \\& \quad \leq \biggl(\frac{2}{L_{1}}-\lambda _{n}^{1}\biggr) \bigl\Vert \nabla \widetilde{f}_{1}(W _{n}) \bigr\Vert + \bigl\Vert T_{n}^{\widetilde{f}_{1}}W_{n}-W_{n} \bigr\Vert . \end{aligned}$$
From the boundedness of \(\{W_{n}\}\), \(s_{n}^{1}\rightarrow 0\) (\(\Longleftrightarrow \lambda _{n}^{1}\rightarrow \frac{2}{L_{1}}\)) and (26), we conclude that
$$ \lim_{n\rightarrow \infty } \biggl\Vert W_{n}-P_{C}\biggl(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1}\biggr)W_{n} \biggr\Vert =0. $$
(28)
Applying the same method as for (28), we also have
$$ \lim_{n\rightarrow \infty } \biggl\Vert V_{n}-P_{C}\biggl(I-\frac{2}{L_{2}}\nabla \widetilde{f}_{2}\biggr)V_{n} \biggr\Vert =0. $$
(29)

Next, we show that \(\limsup_{n\rightarrow \infty }\langle f(y^{*})-x^{*},W_{n}-x^{*} \rangle \leq 0\), where \(x^{*}=P_{U_{\widetilde{f}_{1}}}f(y^{*})\) and \(\limsup_{n\rightarrow \infty }\langle g(x^{*})-y^{*},V_{n}-y^{*} \rangle \leq 0\), where \(y^{*}=P_{U_{\widetilde{f}_{2}}}g(x^{*})\).

Indeed, take a subsequence \(\{W_{n_{k}}\}\) of \(\{W_{n}\}\) such that
$$ \limsup_{n\rightarrow \infty }\bigl\langle f\bigl(y^{*} \bigr)-x^{*},W_{n}-x^{*} \bigr\rangle = \limsup_{k\rightarrow \infty }\bigl\langle f\bigl(y^{*} \bigr)-x^{*},W_{n_{k}}-x^{*} \bigr\rangle . $$
Since \(\{x_{n}\}\) is bounded, without loss of generality, we may assume that \(x_{n_{k}}\rightharpoonup \widehat{x}\) as \(k\rightarrow \infty \). From (23), we obtain \(W_{n_{k}}\rightharpoonup \widehat{x}\) as \(k\rightarrow \infty \). Assume that \(\widehat{x} \neq P_{C}(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1}) \widehat{x}\). By nonexpansiveness of \(P_{C}(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1})\), (28) and Opial’s property, we have
$$\begin{aligned} \liminf_{k\rightarrow \infty } \Vert W_{n_{k}}- \widehat{x} \Vert < {}& \liminf_{k\rightarrow \infty } \biggl\Vert W_{n_{k}}-P_{C}\biggl(I-\frac{2}{L_{1}} \nabla \widetilde{f}_{1}\biggr)\widehat{x} \biggr\Vert \\ \leq {}& \liminf_{k\rightarrow \infty }\biggl( \biggl\Vert W_{n_{k}}-P_{C}\biggl(I-\frac{2}{L_{1}} \nabla \widetilde{f}_{1}\biggr)W_{n_{k}} \biggr\Vert \\ &{} + \biggl\Vert P_{C}\biggl(I-\frac{2}{L_{1}}\nabla \widetilde{f}_{1}\biggr)W_{n_{k}}-P_{C}\biggl(I- \frac{2}{L _{1}}\nabla \widetilde{f}_{1}\biggr)\widehat{x} \biggr\Vert \biggr) \\ \leq{} & \liminf_{k\rightarrow \infty } \Vert W_{n_{k}}- \widehat{x} \Vert . \end{aligned}$$
This is a contradiction, thus we have
$$ \widehat{x}\in F \biggl(P_{C}\biggl(I- \frac{2}{L_{1}}\nabla \widetilde{f}_{1}\biggr) \biggr)=U_{\widetilde{f}_{1}}. $$
(30)
Since \(W_{n_{k}}\rightharpoonup \widehat{x}\) as \(k\rightarrow \infty \), due to (30) and Lemma 2, we can derive that
$$\begin{aligned} \limsup_{n\rightarrow \infty }\bigl\langle f \bigl(y^{*}\bigr)-x^{*},W_{n}-x^{*} \bigr\rangle &= \limsup_{k\rightarrow \infty }\bigl\langle f \bigl(y^{*}\bigr)-x^{*},W_{n_{k}}-x^{*} \bigr\rangle \\ &= \bigl\langle f\bigl(y^{*}\bigr)-x^{*}, \widehat{x}-x^{*}\bigr\rangle \\ &\leq 0. \end{aligned}$$
(31)
Similarly, take a subsequence \(\{V_{n_{k}}\}\) of \(\{V_{n}\}\) such that
$$ \limsup_{n\rightarrow \infty }\bigl\langle g\bigl(x^{*} \bigr)-y^{*},V_{n}-y^{*} \bigr\rangle = \limsup_{k\rightarrow \infty }\bigl\langle g\bigl(x^{*} \bigr)-y^{*},V_{n_{k}}-y^{*} \bigr\rangle . $$
Since \(\{y_{n}\}\) is bounded, without loss of generality, we may assume that \(y_{n_{k}}\rightharpoonup \widehat{y}\) as \(k\rightarrow \infty \). From (24), we obtain \(V_{n_{k}}\rightharpoonup \widehat{y}\) as \(k\rightarrow \infty \). Following the same method as for (31), we easily obtain that
$$ \limsup_{n\rightarrow \infty }\bigl\langle g \bigl(x^{*}\bigr)-y^{*},V_{n}-y^{*} \bigr\rangle \leq 0. $$
(32)

Finally, we show that \(\{x_{n}\}\) converges strongly to \(x^{*}\), where \(x^{*}=P_{U_{\widetilde{f}_{1}}}f(y^{*})\) and \(\{y_{n}\}\) converges strongly to \(y^{*}\), where \(y^{*}=P_{U_{\widetilde{f}_{2}}}g(x^{*})\).

Let \(W_{n}=\alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{\widetilde{f}_{1}}x _{n}\) and \(V_{n}=\alpha _{n}g(x_{n})+(1-\alpha _{n})T_{n}^{ \widetilde{f}_{2}}y_{n}\). From the definition of \(x_{n}\), we get
$$\begin{aligned}& \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \\& \quad = \bigl\Vert (1-\mu _{n})x_{n}+\mu _{n}P_{C}\bigl(\alpha _{n}f(y_{n})+(1- \alpha _{n})T _{n}^{\widetilde{f}_{1}}x_{n} \bigr)-x^{*} \bigr\Vert ^{2} \\& \quad = \bigl\Vert (1-\mu _{n}) \bigl(x_{n}-x^{*} \bigr)+\mu _{n}\bigl(P_{C}\bigl(\alpha _{n}f(y_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n} \bigr)-x^{*}\bigr) \bigr\Vert ^{2} \\& \quad = (1-\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\mu _{n} \bigl\Vert P_{C}\bigl( \alpha _{n}f(y_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n} \bigr)-x^{*} \bigr\Vert ^{2} \\& \quad \leq (1-\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\mu _{n} \bigl\Vert \alpha _{n}f(y_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f}_{1}}x_{n}-x^{*} \bigr\Vert ^{2} \\& \quad = (1-\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +\mu _{n} \bigl\Vert \alpha _{n}\bigl(f(y_{n})-x^{*} \bigr)+(1-\alpha _{n}) \bigl(T_{n}^{ \widetilde{f}_{1}}x_{n}-x^{*} \bigr) \bigr\Vert ^{2} \\& \quad \leq (1-\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +\mu _{n}\bigl((1-\alpha _{n}) \bigl\Vert T_{n}^{\widetilde{f}_{1}}x_{n}-x^{*} \bigr\Vert ^{2}+2 \alpha _{n}\bigl\langle f(y_{n})-x^{*},W_{n}-x^{*} \bigr\rangle \bigr) \\& \quad \leq (1-\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +\mu _{n}(1-\alpha _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+2\alpha _{n}\mu _{n}\bigl\langle f(y_{n})-x^{*},W_{n}-x^{*} \bigr\rangle \\& \quad = (1-\alpha _{n}\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+2\alpha _{n}\mu _{n} \bigl\langle f(y_{n})-x^{*},W_{n}-x^{*} \bigr\rangle \\& \quad = (1-\alpha _{n}\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +2\alpha _{n}\mu _{n}\bigl(\bigl\langle f(y_{n})-f\bigl(y^{*}\bigr),W_{n}-x^{*} \bigr\rangle + \bigl\langle f\bigl(y^{*}\bigr)-x^{*},W_{n}-x^{*} \bigr\rangle \bigr) \\& \quad \leq (1-\alpha _{n}\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +2\alpha _{n}\mu _{n}\bigl( \bigl\Vert f(y_{n})-f\bigl(y^{*}\bigr) \bigr\Vert \bigl\Vert W_{n}-x^{*} \bigr\Vert +\bigl\langle f\bigl(y ^{*}\bigr)-x^{*},W_{n}-x^{*}\bigr\rangle \bigr) \\& \quad \leq (1-\alpha _{n}\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +2\alpha _{n}\mu _{n} \bigl\Vert f(y_{n})-f \bigl(y^{*}\bigr) \bigr\Vert \bigl( \Vert W_{n}-x_{n+1} \Vert + \bigl\Vert x_{n+1}-x ^{*} \bigr\Vert \bigr) \\& \qquad {} +2\alpha _{n}\mu _{n}\bigl\langle f\bigl(y^{*} \bigr)-x^{*},W_{n}-x^{*}\bigr\rangle \\& \quad \leq (1-\alpha _{n}\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +2\alpha _{n}\mu _{n}a \bigl\Vert y_{n}-y^{*} \bigr\Vert \Vert W_{n}-x_{n+1} \Vert +2\alpha _{n}\mu _{n}a \bigl\Vert y_{n}-y^{*} \bigr\Vert \bigl\Vert x_{n+1}-x^{*} \bigr\Vert \\& \qquad {} +2\alpha _{n}\mu _{n}\bigl\langle f\bigl(y^{*} \bigr)-x^{*},W_{n}-x^{*}\bigr\rangle \\& \quad \leq (1-\alpha _{n}\mu _{n}) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +2\alpha _{n}\mu _{n}a \bigl\Vert y_{n}-y^{*} \bigr\Vert \Vert W_{n}-x_{n+1} \Vert +\alpha _{n}\mu _{n}a\big(\bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}+ \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2}\big) \\& \qquad {} +2\alpha _{n}\mu _{n}\bigl\langle f\bigl(y^{*} \bigr)-x^{*},W_{n}-x^{*}\bigr\rangle , \end{aligned}$$
which yields
$$\begin{aligned}& \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \\& \quad \leq \frac{1-\alpha _{n}\mu _{n}}{1-\alpha _{n}\mu _{n}a} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert y_{n}-y^{*} \bigr\Vert \Vert W_{n}-x_{n+1} \Vert \\& \qquad {} + \frac{\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}}{1-\alpha _{n}\mu _{n}a}\bigl\langle f\bigl(y^{*}\bigr)-x ^{*},W_{n}-x^{*}\bigr\rangle \\& \quad = \biggl(1-\frac{\alpha _{n}\mu _{n}-\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \biggr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert y_{n}-y^{*} \bigr\Vert \Vert W_{n}-x_{n+1} \Vert \\& \qquad {} + \frac{\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}}{1-\alpha _{n}\mu _{n}a}\bigl\langle f\bigl(y^{*}\bigr)-x ^{*},W_{n}-x^{*}\bigr\rangle \\& \quad = \biggl(1-\frac{\alpha _{n}\mu _{n}(1-a)}{1-\alpha _{n}\mu _{n}a} \biggr) \bigl\Vert x _{n}-x^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert y _{n}-y^{*} \bigr\Vert \Vert W_{n}-x_{n+1} \Vert \\& \qquad {} + \frac{\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}}{1-\alpha _{n}\mu _{n}a}\bigl\langle f\bigl(y^{*}\bigr)-x ^{*},W_{n}-x^{*}\bigr\rangle . \end{aligned}$$
(33)
Similarly, as derived above, we also have
$$\begin{aligned}& \bigl\Vert y_{n+1}-y^{*} \bigr\Vert ^{2} \\& \quad \leq \biggl(1-\frac{\alpha _{n}\mu _{n}(1-a)}{1-\alpha _{n}\mu _{n}a} \biggr) \bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert x_{n}-x^{*} \bigr\Vert \Vert V_{n}-y_{n+1} \Vert \\& \qquad {} + \frac{\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\frac{2\alpha _{n}\mu _{n}}{1-\alpha _{n}\mu _{n}a}\bigl\langle g\bigl(x^{*}\bigr)-y ^{*},V_{n}-y^{*}\bigr\rangle . \end{aligned}$$
(34)
From (33) and (34), we deduce that
$$\begin{aligned}& \bigl\Vert x _{n+1}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n+1}-y^{*} \bigr\Vert ^{2} \\& \quad \leq \biggl( 1-\frac{\alpha _{n}\mu _{n}(1-a)}{1-\alpha _{n}\mu _{n}a} \biggr) \bigl( \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}\bigr) \\& \qquad {} + \frac{2\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a}\bigl( \bigl\Vert y_{n}-y^{*} \bigr\Vert \Vert W_{n}-x_{n+1} \Vert + \bigl\Vert x_{n}-x^{*} \bigr\Vert \Vert V_{n}-y_{n+1} \Vert \bigr) \\& \qquad {} + \frac{\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a}\bigl( \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}\bigr) \\& \qquad {} + \frac{2\alpha _{n}\mu _{n}}{1-\alpha _{n}\mu _{n}a}\bigl(\bigl\langle f\bigl(y^{*}\bigr)-x ^{*},W_{n}-x^{*}\bigr\rangle +\bigl\langle g \bigl(x^{*}\bigr)-y^{*},V_{n}-y^{*}\bigr\rangle \bigr) \\& \quad = \biggl(1-\frac{\alpha _{n}\mu _{n}(1-2a)}{1-\alpha _{n}\mu _{n}a} \biggr) \bigl( \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+ \bigl\Vert y_{n}-y^{*} \bigr\Vert ^{2}\bigr) \\& \qquad {} + \frac{2\alpha _{n}\mu _{n}a}{1-\alpha _{n}\mu _{n}a}\bigl( \bigl\Vert y_{n}-y^{*} \bigr\Vert \Vert W_{n}-x_{n+1} \Vert + \bigl\Vert x_{n}-x^{*} \bigr\Vert \Vert V_{n}-y_{n+1} \Vert \bigr) \\& \qquad {} + \frac{2\alpha _{n}\mu _{n}}{1-\alpha _{n}\mu _{n}a}\bigl(\bigl\langle f\bigl(y^{*}\bigr)-x ^{*},W_{n}-x^{*}\bigr\rangle +\bigl\langle g \bigl(x^{*}\bigr)-y^{*},V_{n}-y^{*}\bigr\rangle \bigr). \end{aligned}$$
(35)
By (16), (23), (24), (31), (32), condition (i) and Lemma 4, we have \(\lim_{n\rightarrow \infty }(\|x_{n}-x^{*}\|+\|y_{n}-y^{*}\|)=0\). It implies that the sequences \(\{x_{n}\}\), \(\{y_{n}\}\) converge to \(x^{*}=P_{U_{\widetilde{f}_{1}}}f(y^{*})\), \(y^{*}=P_{U_{\widetilde{f} _{2}}}g(x^{*})\), respectively. This completes the proof. □

Corollary 1

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. Let\(\widetilde{f}:C \rightarrow \mathbb{R}\)be a real-valued convex function and assume thatis\(\frac{1}{L}\)-inverse strongly monotone with\(L> 0\)and\(U_{ \widetilde{f}}\neq \emptyset \). Let\(f:H \rightarrow H\)be ana-contraction mapping with\(a\in (0,1)\). Let the sequence\(\{x_{n}\}\)be generated by\(x_{1}\in C\)and
$$ x_{n+1}=(1-\mu _{n})x_{n}+\mu _{n}P_{C}\bigl(\alpha _{n}f(x_{n})+(1- \alpha _{n})T_{n}^{\widetilde{f}}x_{n}\bigr), $$
(36)
where\(\{\mu _{n}\},\{\alpha _{n}\}\subseteq [0,1]\), \(P_{C}(I-\lambda _{n}^{i}\nabla \widetilde{f})=s_{n}I+(1-s_{n})T_{n}^{\widetilde{f}}\)and\(s_{n}=\frac{2-\lambda _{n}L}{4}\), \(\{\lambda _{n}\}\subset (0, \frac{2}{L})\). Assume that the following conditions hold:
  1. (i)

    \(\lim_{n\rightarrow \infty }\alpha _{n}=0\)and\(\sum_{n=1}^{\infty }\alpha _{n}=\infty \),

     
  2. (ii)

    \(0<\overline{\theta }\leq \mu _{n} \leq \theta \)for all\(n\in \mathbb{N}\)and for some\(\overline{\theta },\theta >0\),

     
  3. (iii)

    \(\sum_{n=1}^{\infty }|\alpha _{n+1}-\alpha _{n}|<\infty \), \(\sum_{n=1}^{\infty }|\mu _{n+1}-\mu _{n}|<\infty \).

     
Then\(\{x_{n}\}\)converges strongly, as\(s_{n}\rightarrow 0\) (\(\Longleftrightarrow \lambda _{n}\rightarrow \frac{2}{L}\)), to\(x^{*}=P_{U_{\widetilde{f}}}f(x ^{*})\).

Proof

If we put \(f\equiv g\), \(x_{n}=y_{n}\), in Theorem 1, we obtain the desired conclusion. □

4 Application

Let \(H_{1}\), \(H_{2}\) be two real Hilbert spaces. Let C, Q be nonempty closed convex subsets of \(H_{1}\) and \(H_{2}\), respectively.

In 1994, Censor and Elfving [17] introduced the split feasibility problem (SFP), which is to find a point x such that
$$ x \in C \quad \text{and} \quad Dx\in Q, $$
where \(D:H_{1}\rightarrow H_{2}\) is a bounded linear operator.
Throughout this paper, we assume that the SFP is consistent, that is, the solution set Γ of the SFP is nonempty. Let \(f:\mathcal{H}_{1} \rightarrow \mathbb{R}\) be a continuous differentiable function. The minimization problem
$$ \min_{x\in C}f(x) := \min_{x\in C} \frac{1}{2} \Vert Ax-P_{Q}Ax \Vert ^{2} $$
(37)
is ill-posed.

Before proving Theorem 2, we need the following:

Proposition 3

([18])

Given\(x^{*}\in \mathcal{H}_{1}\), the following statements are equivalent:
  1. (i)

    \(x^{*}\)solves the SFP;

     
  2. (ii)

    \(P_{C}(I-\lambda \nabla f)x^{*}=P_{C}(I-\lambda A^{*}(I-P_{Q})A)x^{*}=x ^{*}\);

     
  3. (iii)
    \(x^{*}\)solves the variational inequality problem of finding\(x^{*}\in C\)such that
    $$ \bigl\langle \nabla f\bigl(x^{*}\bigr), x-x^{*}\bigr\rangle \geq 0, \quad \forall x\in C, $$
    (38)
    where\(\nabla f=A^{*}(I-P_{Q})A\)and\(A^{*}\)is the adjoint ofA.
     

Theorem 2

LetCandQbe nonempty, closed, and convex subsets of\(H_{1}\)and\(H_{2}\), respectively, and let\(A_{i}:H_{1}\rightarrow H _{2}\)be bounded linear operators for all\(i=1,2\)with\(L_{i}\)being the spectral radius of\(A_{i}^{*}A_{i}\)for all\(i=1,2\)with\(\varGamma _{i} \neq \emptyset\). Let\(f,g:H\rightarrow H\)be\(a_{f}\)- and\(a_{g}\)-contraction mappings with\(a_{f},a_{g}\in (0,1)\)and\(a=\max \{a_{f},a_{g}\}\). Let the sequences\(\{x_{n}\}\), \(\{y_{n}\}\)be generated by\(x_{1},y_{1} \in C\)and
$$ \textstyle\begin{cases} x_{n+1}=(1-\mu _{n})x_{n}+\mu _{n}P_{C}(\alpha _{n}f(y_{n})+(1-\alpha _{n})T_{n}^{a_{1}}x_{n}), \\ y_{n+1}=(1-\mu _{n})y_{n}+\mu _{n}P_{C}(\alpha _{n}g(x_{n})+(1-\alpha _{n})T_{n}^{a_{2}}y_{n}), \end{cases} $$
(39)
where\(\{\mu _{n}\},\{\alpha _{n}\}\subseteq [0,1]\), \(P_{C}(I-\lambda _{n}^{i}(A_{i}^{*}(I-P_{Q})A_{i}))=s_{n}^{i}I+(1-s_{n}^{i})T_{n}^{a _{i}}\), \(\forall i=1,2\)and\(s_{n}^{i}= \frac{2-\lambda _{n}^{i}L_{i}}{4}\), \(\{\lambda _{n}^{i}\}\subset (0,\frac{2}{L _{i}})\). Assume that the following conditions hold:
  1. (i)

    \(\lim_{n\rightarrow \infty }\alpha _{n}=0\)and\(\sum_{n=1}^{\infty }\alpha _{n}=\infty \),

     
  2. (ii)

    \(0<\overline{\theta }\leq \mu _{n} \leq \theta \)for all\(n\in \mathbb{N}\)and for some\(\overline{\theta },\theta >0\),

     
  3. (iii)

    \(\sum_{n=1}^{\infty }|\alpha _{n+1}-\alpha _{n}|<\infty \), \(\sum_{n=1}^{\infty }|\mu _{n+1}-\mu _{n}|<\infty \).

     
Then\(\{x_{n}\}\)and\(\{y_{n}\}\)converge strongly, as\(s_{n}^{i} \rightarrow 0\) (\(\Longleftrightarrow \lambda _{n}^{i}\rightarrow \frac{2}{L _{i}}\)) \(\forall i=1,2\), to\(x^{*}=P_{\varGamma _{1}}f(y^{*})\)with\(\varGamma _{1}=\{x\in C;A_{1}x\in Q\}\)and\(y^{*}=P_{\varGamma _{2}}g(x^{*})\)with\(\varGamma _{2}=\{\bar{x}\in C;A_{2}\bar{x}\in Q\}\), respectively.

Proof

Letting \(x,y\in C\) and \(\nabla f_{i}=A_{i}^{*}(I-P_{Q})A_{i}\) for all \(i=1,2\), we have
$$\begin{aligned} \bigl\Vert \nabla f_{i}(x)-\nabla f_{i}(y) \bigr\Vert ^{2} &= \bigl\Vert A_{i}^{*}(I-P_{Q})A_{i}x-A _{i}^{*}(I-P_{Q})A_{i}y \bigr\Vert ^{2} \\ &\leq L_{i} \bigl\Vert (I-P_{Q})A_{i}x-(I-P_{Q})A_{i}y \bigr\Vert ^{2}. \end{aligned}$$
(40)
From the property of \(P_{C}\), we have
$$\begin{aligned}& \bigl\Vert (I-P_{Q}) A_{i}x -(I-P_{Q})A_{i}y \bigr\Vert ^{2} \\& \quad = \bigl\langle (I-P_{Q})A_{i}x-(I-P_{Q})A_{i}y,(I-P_{Q})A_{i}x-(I-P_{Q})A _{i}y\bigr\rangle \\& \quad = \bigl\langle (I-P_{Q})A_{i}x-(I-P_{Q})A_{i}y,A_{i}x-A_{i}y \bigr\rangle \\& \qquad {} -\bigl\langle (I-P_{Q})A_{i}x-(I-P_{Q})A_{i}y,P_{Q}A_{i}x-P_{Q}A_{i}y \bigr\rangle \\& \quad = \bigl\langle A_{i}^{*}(I-P_{Q})A_{i}x-A_{i}^{*}(I-P_{Q})A_{i}y,x-y \bigr\rangle \\& \qquad {} -\bigl\langle (I-P_{Q})A_{i}x-(I-P_{Q})A_{i}y,P_{Q}A_{i}x-P_{Q}A_{i}y \bigr\rangle \\& \quad = \bigl\langle A_{i}^{*}(I-P_{Q})A_{i}x-A_{i}^{*}(I-P_{Q})A_{i}y,x-y \bigr\rangle \\& \qquad {} -\bigl\langle (I-P_{Q})A_{i}x,P_{Q}A_{i}x-P_{Q}A_{i}y \bigr\rangle \\& \qquad {} +\bigl\langle (I-P_{Q})A_{i}y,P_{Q}A_{i}x-P_{Q}A_{i}y \bigr\rangle \\& \quad \leq \bigl\langle A_{i}^{*}(I-P_{Q})A_{i}x-A_{i}^{*}(I-P_{Q})A_{i}y,x-y \bigr\rangle . \end{aligned}$$
(41)
Substituting (41) into (40), we have
$$\begin{aligned} \bigl\Vert \nabla f_{i}(x)-\nabla f_{i}(y) \bigr\Vert ^{2} &\leq L_{i}\bigl\langle A_{i}^{*}(I-P _{Q})A_{i}x-A_{i}^{*}(I-P_{Q})A_{i}y,x-y \bigr\rangle \\ &= L_{i}\bigl\langle \nabla f_{i}(x)-\nabla f_{i}(y),x-y\bigr\rangle . \end{aligned}$$
It follows that
$$ \bigl\langle \nabla f_{i}(x)-\nabla f_{i}(y),x-y\bigr\rangle \geq \frac{1}{L _{i}} \bigl\Vert \nabla f_{i}(x)-\nabla f_{i}(y) \bigr\Vert ^{2}. $$
Then \(\nabla f_{i}\) is \(\frac{1}{L_{i}}\)-inverse strongly monotone, for all \(i=1,2\).

From Proposition 3 and Theorem 1, we can conclude that Theorem 2 is true. □

Corollary 2

LetCandQbe nonempty, closed, and convex subsets of\(H_{1}\)and\(H_{2}\), respectively, and let\(A:H_{1}\rightarrow H_{2}\)be bounded linear operator withLbeing the spectral radius of\(A^{*}A\)with\(\varGamma \neq \emptyset\). Let\(f:H\rightarrow H\)be ana-contraction mapping with\(a\in (0,1)\). Let the sequence\(\{x_{n}\}\)be generated by\(x_{1}\in C\)and
$$ x_{n+1}=(1-\mu _{n})x_{n}+\mu _{n}P_{C}\bigl(\alpha _{n}f(x_{n})+(1- \alpha _{n})T_{n}^{a_{1}}x_{n}\bigr), $$
(42)
where\(\{\mu _{n}\},\{\alpha _{n}\}\subseteq [0,1]\), \(P_{C}(I-\lambda _{n}(A^{*}(I-P_{Q})A))=s_{n}I+(1-s_{n})T_{n}^{a_{1}}\)and\(s_{n}=\frac{2- \lambda _{n}L}{4}\), \(\{\lambda _{n}\}\subset (0,\frac{2}{L})\). Assume that the following conditions hold:
  1. (i)

    \(\lim_{n\rightarrow \infty }\alpha _{n}=0\)and\(\sum_{n=1}^{\infty }\alpha _{n}=\infty \),

     
  2. (ii)

    \(0<\overline{\theta }\leq \mu _{n} \leq \theta \)for all\(n\in \mathbb{N}\)and for some\(\overline{\theta },\theta >0\),

     
  3. (iii)

    \(\sum_{n=1}^{\infty }|\alpha _{n+1}-\alpha _{n}|<\infty \), \(\sum_{n=1}^{\infty }|\mu _{n+1}-\mu _{n}|<\infty \).

     
Then\(\{x_{n}\}\)converges strongly, as\(s_{n}\rightarrow 0\) (\(\Longleftrightarrow \lambda _{n}\rightarrow \frac{2}{L}\)), to\(x^{*}=P_{\varGamma }f(x^{*})\)with\(\varGamma =\{x\in C;Ax\in Q\}\).

Proof

If we put \(f \equiv g\), \(x_{n}=y_{n}\) in Theorem 2, then the conclusion follows. □

5 Numerical examples

Example 1

Let \(C=[-10,10]\times [-10,10]\) and let \(\langle \cdot , \cdot \rangle : \mathbb{R}^{2} \times \mathbb{R}^{2}\rightarrow \mathbb{R}\) be an inner product defined by \(\langle \mathbf{x},\mathbf{y} \rangle = \mathbf{x}\cdot \mathbf{y}=x_{1}y_{1}+x_{2}y_{2}\), for all \(\mathbf{x}=(x_{1},x_{2})\in \mathbb{R}^{2}\) and \(\mathbf{y}=(y_{1},y _{2})\in \mathbb{R}^{2}\). For every \(i=1,2\), let \(\widetilde{f_{i}}:C \rightarrow \mathbb{R}\) be defined by \(\widetilde{f_{1}}(x_{1},x_{2})=2x _{1}^{2}+x_{2}\) and \(\widetilde{f_{2}}(x_{1},x_{2})=(x_{1}-1)+x_{2} ^{2}\), \(\forall x_{1},x_{2}\in \mathbb{R}\). Let \(f,g:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\), defined by \(f(x_{1},x_{2})=( \frac{x_{1}}{3},\frac{x_{2}}{3})\) and \(g(x_{1},x_{2})=( \frac{x_{1}}{4},\frac{x_{2}}{4})\), be \(\frac{1}{2}\)- and \(\frac{1}{3}\)-contraction mappings and \(a=\max \{\frac{1}{2}, \frac{1}{3}\}=\frac{1}{2}\). Let the sequences \(\{x_{n}\}\), \(\{y_{n}\}\) be generated by \(x_{1},y_{1}\in C\). Putting \(\alpha _{n}=\frac{1}{4n}\) and \(\mu _{n}=\frac{3n+1}{7n}\), we can rewrite (10) as follows:
$$ \textstyle\begin{cases} x_{n+1}=(\frac{4n-1}{7n})x_{n}+(\frac{3n+1}{7n})P_{C}(\frac{1}{4n}f(y _{n})+(\frac{4n-1}{4n})T_{n}^{\widetilde{f}_{1}}x_{n}), \\ y_{n+1}=(\frac{4n-1}{7n})y_{n}+(\frac{3n+1}{7n})P_{C}(\frac{1}{4}g(x _{n})+(\frac{4n-1}{4n})T_{n}^{\widetilde{f}_{2}}y_{n}), \end{cases} $$
(43)
where \(P_{C}(x_{1},x_{2})=(\max \{\min \{x_{1},10\},-10\},\max \{ \min \{x_{2},10\},-10\})\) and also \(P_{C}(I-\lambda _{n}^{i}\nabla \widetilde{f_{i}})=s_{n}^{i}I+(1-s_{n}^{i})T_{n}^{\widetilde{f_{i}}}\) and \(s_{n}^{i}=\frac{2-\lambda _{n}^{i}(16)}{4}\), where \(\lambda _{n} ^{i}=\frac{n^{2}}{8n^{2}+1}\)\(\forall i=1,2\).
Then, since \(\widetilde{f_{1}}(x_{1},x_{2})=2x_{1}^{2}+x_{2}\) and \(\widetilde{f_{2}}(x_{1},x_{2})=(x_{1}-1)+x_{2}^{2}\), we have
$$ \nabla \widetilde{f_{1}}(x_{1},x_{2})=(4x_{1},1)\quad \mbox{and} \quad \nabla \widetilde{f_{2}}(x_{1},x_{2})=(1,2x_{2}). $$
It is obvious that \(\nabla \widetilde{f_{i}}\) is a \(\frac{1}{16}\)-inverse strongly monotone, \(\forall i=1,2\).
Consider \((0,-10),(-10,0)\in [-10,10]\times [-10,10]\) for which
$$\begin{aligned} P_{C}\bigl(I-\lambda _{n}^{1}\nabla \widetilde{f_{1}}\bigr) (0,-10) &= P_{[-10,10] \times [-10,10]}\biggl(I- \frac{1}{16}\nabla \widetilde{f_{1}}\biggr) (0,-10) \\ &= P_{[-10,10]\times [-10,10]}\biggl(0,\frac{-161}{16}\biggr) \\ &= \biggl(P_{[-10,10]}(0),P_{[-10,10]}\biggl(\frac{-161}{16}\biggr) \biggr) \\ &= \biggl(\max \bigl\{ \min \{0,10\},-10\bigr\} ,\max \biggl\{ \min \biggl\{ \frac{-161}{16},10\biggr\} ,-10 \biggr\} \biggr) \\ &= (0,-10), \end{aligned}$$
thus \((0,-10)\in U_{\widetilde{f_{1}}}\).
Similarly,
$$\begin{aligned} P_{C}\bigl(I-\lambda _{n}^{2}\nabla \widetilde{f_{2}}\bigr) (-10,0) &= P_{[-10,10] \times [-10,10]}\biggl(I- \frac{1}{16}\nabla \widetilde{f_{2}}\biggr) (-10,0) \\ &= P_{[-10,10]\times [-10,10]}\biggl(\frac{-161}{16},0\biggr) \\ &= \biggl(P_{[-10,10]}\biggl(\frac{-161}{16}\biggr),P_{[-10,10]}(0) \biggr) \\ &= \biggl(\max \biggl\{ \min \biggl\{ \frac{-161}{16},10\biggr\} ,-10\biggr\} , \max \bigl\{ \min \{0,10\},-10 \bigr\} \biggr) \\ &= (-10,0), \end{aligned}$$
thus \((-10,0)\in U_{\widetilde{f_{2}}}\).
It is clear that the sequences \(\{\alpha _{n}\}\), \(\{\mu _{n} \}\) satisfy all the conditions of Theorem 1, so we can conclude that the sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) converge strongly to \((0,-10)\) and \((-10,0)\), respectively. Table 1 shows the values of \(\{x_{n}\}\) and \(\{y_{n}\}\) with \(x_{n}^{1}=-10\), \(x_{n}^{2}=10\), \(y_{n}^{1}=10\), \(y_{n}^{2}=-10\), and \(n=N=400\).
Table 1

The values of \(\{x_{n}\}\) and \(\{y_{n}\}\) with \(x_{n}^{1}=-10\), \(x_{n}^{2}=10\), \(y_{n}^{1}=10\), \(y_{n}^{2}=-10\), and \(n=N=400\)

n

\(x_{n}=(x_{n}^{1},x_{n}^{2})\)

\(y_{n}=(y_{n}^{1},y_{n}^{2})\)

1

(−10.0000,10.0000)

(10.0000,−10.0000)

2

(−6.0784,8.0448)

(8.1639,−7.2059)

3

(−4.2192,7.3380)

(7.5048,−5.8538)

4

(−3.0380,6.9152)

(7.1109,−4.9143)

250

(−0.0050,−7.7785)

(−7.5696,−0.0076)

396

(−0.0043,−9.9937)

(−9.9937,−0.0065)

397

(−0.0042,−9.9937)

(−9.9937,−0.0064)

398

(−0.0042,−9.9937)

(−9.9937,−0.0064)

399

(−0.0042,−9.9937)

(−9.9937,−0.0064)

400

(−0.0042,−9.9937)

(−9.9937,−0.0064)

Figure 1

The convergence of \(\{x_{n}\}\) and \(\{y_{n}\}\) with \(x_{n}^{1}=-10\), \(x_{n}^{2}=10\), \(y_{n}^{1}=10\), \(y_{n}^{2}=-10\), and \(n=N=400\)

Figure 2

The convergence of \(\{x_{n}\}\) with \(x_{n}^{1}=-10\), \(x_{n}^{2}=10\), and \(n=N=400\)

Remark 1

If we choose \(f\equiv g\) and \(x_{n}=y_{n}\) in Example 1, we can rewrite (36) as follows:
$$ x_{n+1}=\biggl(\frac{4n-1}{7n}\biggr)x_{n}+\biggl( \frac{3n+1}{7n}\biggr)P_{C}\biggl(\frac{1}{4n}f(x _{n})+\biggl(\frac{4n-1}{4n}\biggr)T_{n}^{\widetilde{f}}x_{n} \biggr), $$
where \(P_{C}(x_{1},x_{2})=(\max \{\min \{x_{1},10\},-10\},\max \{ \min \{x_{2},10\},-10\})\) and also \(P_{C}(I-\lambda _{n}\nabla \widetilde{f})=s_{n}I+(1-s_{n})T_{n}^{\widetilde{f}}\) and \(s_{n}=\frac{2- \lambda _{n}(16)}{4}\), where \(\lambda _{n}=\frac{n^{2}}{8n^{2}+1}\). From Corollary 1, we can conclude that the sequence \(\{x_{n}\}\) converges strongly to \((0,-10)\). Table 2 shows the values of \(\{x_{n}\}\) with \(x_{n}^{1}=-10\), \(x_{n}^{2}=10\), and \(n=N=400\).
Table 2

The values of \(\{x_{n}\}\) with \(x_{n}^{1}=-10\), \(x_{n}^{2}=10\), and \(n=N=400\)

n

\(x_{n}=(x_{n}^{1},x_{n}^{2})\)

1

(−10.0000,10.0000)

2

(−7.0308,8.9972)

3

(−5.2235,8.5685)

4

(−3.9531,8.2876)

250

(0.0000,-6.2974)

396

(0.0000,−9.9958)

397

(0.0000,−9.9958)

398

(0.0000,−9.9958)

399

(0.0000,−9.9958)

400

(0.0000,−9.9958)

Conclusion

  1. 1.

    Theorem 1 guarantees the convergence of \(\{x_{n}\}\) and \(\{y_{n}\}\) in Example 1.

     
  2. 2.

    Corollary 1 guarantees the convergence of \(\{x_{n}\}\) in Remark 1.

     
  3. 3.

    By using the concepts of an intermixed algorithm and gradient-projection algorithm (GPA), we give a new iteration for solving two constrained convex minimization problems.

     

Notes

Acknowledgements

This work is supported by King Mongkut’s Institute of Technology Ladkrabang.

Availability of data and materials

Not applicable.

Authors’ contributions

The two authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Funding

Not applicable.

Competing interests

The authors declare that they have no competing interests.

References

  1. 1.
    Xu, H.K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bretarkas, D.P., Gafin, E.M.: Projection methods for variational inequalities with applications to the traffic assignment problem. Math. Program. Stud. 17, 139–159 (1982) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Su, M., Xu, H.K.: Remarks on the gradient-projection algorithm. J. Nonlinear Anal. Optim. 1, 35–43 (2010) MathSciNetzbMATHGoogle Scholar
  4. 4.
    Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–517 (1967) CrossRefGoogle Scholar
  6. 6.
    Kangtunyakarn, A.: A new iterative algorithm for the set of fixed-point problems of nonexpansive mappings and the set of equilibrium problem and variational inequalities problem. Abstr. Appl. Anal. 2011, Article ID 562689 (2011).  https://doi.org/10.1155/2011/562689 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ke, Y., Ma, C.: Iterative algorithm of common solutions for a constrained convex minimization problem, a quasi-variational inclusion problem and the fixed point problem of a strictly pseudo-contractive mapping. Fixed Point Theory Appl. 2014, 54 (2014) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chahn, Y.-J., Nazeer, W., Naqvi, S.-A., Shin, M.-K.: An implicit viscosity technique of nonexpansive mappings in Hilbert spaces. Int. J. Pure Appl. Math. 108(3), 635–650 (2016) Google Scholar
  9. 9.
    Nazeer, W., Munir, M.: Strong convergence of new viscosity rules of nonexpansive mappings. J. Appl. Math. 35(5–6), 423–438 (2017) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ceng, L.-C., Ansari, Q.H., Yao, J.-C.: Some iterative methods for finding fixed points and for solving constrained convex minimization problems. Nonlinear Anal. 74, 5286–5302 (2011) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ming, T., Liu, L.: General iterative methods for equilibrium and constrained convex minimization problem. J. Optim. Theory Appl. 63, 1367–1385 (2014) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yao, Z., Kang, S.M., Li, H.J.: An intermixed algorithm for strict pseudo-contractions in Hilbert spaces. Fixed Point Theory Appl. 2015, 206 (2015) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000) zbMATHGoogle Scholar
  14. 14.
    Osilike, M.O., Isiogugu, F.O.: Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. Nonlinear Anal. 74, 1814–1822 (2011) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Opial, Z.: Weak convergence of the sequence of successive approximation of nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Marino, G., Xu, H.-K.: A general method for nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 318, 43–52 (2016) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ceng, L.-C., Ansari, Q.H., Yao, J.-C.: An extragradient method for solving split feasibility and fixed point problems. Comput. Math. Appl. 64, 633–642 (2012) MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s Institute of Technology LadkrabangBangkokThailand

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