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# Iterative methods for split variational inclusion and fixed point problem of nonexpansive semigroup in Hilbert spaces

Journal of Inequalities and Applications20152015:24

https://doi.org/10.1186/s13660-014-0528-9

• Received: 5 April 2014
• Accepted: 13 December 2014
• Published:

## Abstract

In this paper, we introduce a general iterative method for a split variational inclusion and nonexpansive semigroups in Hilbert spaces. We also prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups, which also solves a class of variational inequalities as an optimality condition for a minimization problem. Moreover, a numerical example is given, to illustrate our methods and results, which may be viewed as a refinement and improvement of the previously known results announced by many other authors.

## Keywords

• split variational inclusion
• nonexpansive semigroup
• fixed point
• averaged mapping
• general iterative method

• 47H09
• 47H10
• 47J22

## 1 Introduction

Let $$H_{1}$$ and $$H_{2}$$ be real Hilbert spaces with inner product $$\langle \cdot,\cdot\rangle$$ and norm $$\|\cdot\|$$, respectively. Recall that a mapping $$T: H_{1}\rightarrow H_{1}$$ is called nonexpansive if
$$\|Tx-Ty\|\leq\|x-y\|, \quad \forall x,y\in H_{1}.$$
A one-parameter family $\mathcal{T}:=\left\{T\left(s\right):0\le s<\mathrm{\infty }\right\}$ is said to be a nonexpansive semigroup on $$H_{1}$$ if the following conditions are satisfied:
1. (1)

$$T(0)x=x$$ for all $$x\in H_{1}$$;

2. (2)

$$T(s+t)=T(s)T(t)$$ for all $$s,t\geq0$$;

3. (3)

$$\|T(s)x-T(s)y\|\leq\|x-y\|$$, for all $$x,y\in H_{1}$$ and $$s>0$$;

4. (4)

for each $$x\in H_{1}$$, the mapping $$s\mapsto T(s)x$$ is continuous.

Denote by $Fix\left(\mathcal{T}\right)$ the common fixed point set of the semigroup $\mathcal{T}$, i.e., $Fix\left(\mathcal{T}\right):=\left\{x\in {H}_{1}:T\left(s\right)x=x,\mathrm{\forall }s>0\right\}$. It is well known that $Fix\left(\mathcal{T}\right)$ is closed and convex (see Lemma 1 in Browder ).
Recently, the fixed point problem of nonexpansive mappings and its iterative methods have become an attractive subject, and various algorithms have been developed for solving variational inequalities and equilibrium problems; see  and the references therein. In 2006, Marino and Xu  introduced the following general iterative methods to approximate a fixed point of a nonexpansive mapping:
$$x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I- \alpha_{n}B)Tx_{n},$$
(1.1)
where $$\alpha_{n}\in[0,1]$$ satisfies certain conditions, f is a contraction of $$H_{1}$$ into itself, and B is a strongly positive bounded linear operator on $$H_{1}$$. Moreover, they prove that $$\{x_{n}\}$$ converges strongly to $$x^{*}\in \operatorname{Fix}(T)$$, the unique solution of the following variational inequality:
$$\bigl\langle (B-\gamma f)x^{*}, x^{*}-w\bigr\rangle \leq0, \quad \forall w \in \operatorname{Fix}(T),$$
which is also the optimality condition of the minimization problem. Thereafter, Li et al.  and Cianciaruso et al.  modified the general iterative method (1.1) to the case of nonexpansive semigroups and equilibrium problems. To obtain a mean ergodic theorem of nonexpansive mappings, Shehu  proposed an iterative method for nonexpansive semigroups, variational inclusions, and generalized equilibrium problems.
Recall also that a multi-valued mapping $$M: H_{1}\rightarrow2^{H_{1}}$$ is called monotone if, for all $$x,y\in H_{1}$$, $$u\in Mx$$ and $$v\in My$$ such that
$$\langle x-y,u-v\rangle\geq0.$$
A monotone mapping M is maximal if the $$\operatorname{Graph}(M)$$ is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping M is maximal if and only if for $$(x,u)\in H_{1}\times H_{1}$$, $$\langle x-y,u-v\rangle\geq0$$ for every $$(y,v)\in \operatorname{Graph}(M)$$ implies that $$u\in Mx$$.
Let $$M: H_{1}\rightarrow2^{H_{1}}$$ be a multi-valued maximal monotone mapping. Then the resolvent mapping $$J_{\lambda}^{M}: H_{1}\rightarrow H_{1}$$ associated with M is defined by
$$J_{\lambda}^{M}(x):=(I+\lambda M)^{-1}(x), \quad \forall x\in H_{1},$$
for some $$\lambda>0$$, where I stands for the identity operator on $$H_{1}$$. Note that for all $$\lambda>0$$ the resolvent operator $$J_{\lambda}^{M}$$ is single-valued, nonexpansive, and firmly nonexpansive.
In 2011, Moudafi  introduced the following split monotone variational inclusion problem: Find $$x^{*}\in H_{1}$$ such that
$$\left \{ \begin{array}{l} 0\in f_{1}(x^{*})+B_{1}(x^{*}), \\ y^{*}=Ax^{*}\in H_{2}\mbox{:} \quad 0\in f_{2}(y^{*})+B_{2}(y^{*}), \end{array} \right .$$
(1.2)
where $$B_{1}: H_{1}\rightarrow2^{H_{1}}$$ and $$B_{2}: H_{2}\rightarrow 2^{H_{2}}$$ are multi-valued maximal monotone mappings. The split monotone variational inclusion problem (1.2) includes as special cases: the split common fixed point problem, the split variational inequality problem, the split zero problem, and the split feasibility problem, which have already been studied and used in practice as a model in intensity-modulated radiation therapy treatment planning, see e.g. . This formalism is also at the core of the modeling of many inverse problems arising for phase retrieval and other real-world problems; for instance, in sensor networks in computerized tomography and data compression; see [13, 14] and the references therein.
If $$f_{1}\equiv0$$ and $$f_{2}\equiv0$$, then problem (1.2) reduces to the following split variational inclusion problem: Find $$x^{*}\in H_{1}$$ such that
$$\left \{ \begin{array}{l} 0\in B_{1}(x^{*}), \\ y^{*}=Ax^{*}\in H_{2}\mbox{:}\quad 0\in B_{2}(y^{*}), \end{array} \right .$$
(1.3)
which constitutes a pair of variational inclusion problems connected with a bounded linear operator A in two different Hilbert spaces $$H_{1}$$ and $$H_{2}$$. The solution set of problem (1.3) is denoted by .
Very recently, Byrne et al.  studied the weak and strong convergence of the following iterative method for problem (1.3): For given $$x_{0}\in H_{1}$$ and $$\lambda>0$$, compute iterative sequence $$\{x_{n}\}$$ generated by the following scheme:
$$x_{n+1}=J_{\lambda}^{B_{1}} \bigl[x_{n}+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr].$$
(1.4)

In 2013, Kazmi and Rizvi  modified scheme (1.3) to the case of a split variational inclusion and the fixed point problem of a nonexpansive mapping. To be more precise, they proved the following strong convergence theorem.

### Theorem KR

Let $$H_{1}$$ and $$H_{2}$$ be two real Hilbert spaces and $$A: H_{1}\rightarrow H_{2}$$ be a bounded linear operator. Let $$f: H_{1}\rightarrow H_{1}$$ be a contraction mapping with constant $$\rho\in(0,1)$$ and $$T: H_{1}\rightarrow H_{1}$$ be a nonexpansive mapping such that $\mathrm{\Omega }=Fix\left(T\right)\cap \mathcal{Z}\ne \mathrm{\varnothing }$. For a given $$x_{0}\in H_{1}$$ arbitrarily, let the iterative sequences $$\{u_{n}\}$$ and $$\{x_{n}\}$$ be generated by
$$\left \{ \begin{array}{l} u_{n}=J_{\lambda}^{B_{1}}[x_{n}+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)Ax_{n}], \\ x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})Tu_{n}, \end{array} \right .$$
(1.5)
where $$\lambda>0$$ and $$\epsilon\in(0,1/L)$$, L is the spectral radius of the operator $$A^{*}A$$, and $$A^{*}$$ is the adjoint of A; $$\{\alpha_{n}\}$$ is a sequence in $$(0,1)$$ such that $$\lim_{n\rightarrow\infty}\alpha_{n}=0$$, $$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$, and $$\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty$$. Then the sequences $$\{u_{n}\}$$ and $$\{x_{n}\}$$ both converge strongly to $$z\in \Omega$$, where $$z=P_{\Omega} f(z)$$.
Inspired and motivated by research going on in this area, we introduce a modified general iterative method for a split variational inclusion and nonexpansive semigroups, which is defined in the following way:
$$x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I- \alpha_{n}B)\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)J_{\lambda}^{B_{1}} \bigl[x_{n}+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr]\, ds,$$
(1.6)
where $$\gamma\in[0,1]$$ and $$\alpha_{n}, \beta_{n}\in[0,1]$$, B is a strongly positive bounded linear operator on $$H_{1}$$. Note that, if $$\gamma=1$$, $$B=I$$, and $$T(s)=T$$, a nonexpansive mapping, scheme (1.6) reduces to the approximate method (1.5), which is mainly due to Byrne et al.  and has been applied by Kazmi and Rizvi .

Our purpose is not only to modify the general iterative method (1.5) to the case of a split variational inclusion and nonexpansive semigroups from a nonexpansive mapping, but also to prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups, which also solves a class of variational inequalities as an optimality condition for a minimization problem. Moreover, a numerical example is given to illustrate our algorithm and our results, which improve and extend the corresponding results of [2, 4, 5, 15, 16] and many others.

## 2 Preliminaries

Let C be a nonempty, closed, and convex subset of a real Hilbert space $$H_{1}$$. For every point $$x\in H_{1}$$, there exists a unique nearest point in C, denoted by $$P_{C}$$, such that
$$\|x-P_{C}x\|\leq\|x-y\|, \quad \forall y\in C.$$
Then $$P_{C}$$ is called the metric projection of $$H_{1}$$ onto C. It is well known that $$P_{C}$$ is a nonexpansive mapping and the following inequality holds:
$$\langle x-u, y-u\rangle\leq0, \quad \forall y\in C,$$
if and only if $$u=P_{C}x$$ for given $$x\in H_{1}$$ and $$u\in C$$.
Recall that a mapping $$f: H_{1}\rightarrow H_{1}$$ is a contraction, if there exists a constant $$\rho\in(0,1)$$ such that
$$\bigl\Vert f(x)-f(y)\bigr\Vert \leq\rho\|x-y\|, \quad \forall x,y\in H_{1}.$$
Throughout the rest of this paper, we always assume that B is strongly positive; that is, there is a constant $$\overline{\gamma}>0$$ such that
$$\langle Bx,x\rangle\geq\overline{\gamma}\|x\|^{2}, \quad \forall x\in H_{1}.$$

A mapping $$S: H_{1}\rightarrow H_{1}$$ is said to be averaged if and only if it can be written as the average of the identity mapping and a nonexpansive mapping, i.e., $$S:=(1-\alpha)I+\alpha T$$ where $$\alpha\in(0,1)$$ and $$T: H_{1}\rightarrow H_{1}$$ is nonexpansive and I is the identity operator on $$H_{1}$$. We note that averaged mappings are nonexpansive. Further, firmly nonexpansive mappings (in particular, projections on nonempty, closed, and convex subsets and resolvent operators of maximal monotone operators) are averaged.

In order to prove our main results, we need the following lemmas and propositions.

### Lemma 2.1

Let $$H_{1}$$ be a real Hilbert space. The following well-known results hold:
1. (i)

$$\|x+y\|^{2}\leq\|x\|^{2}+2\langle y,(x+y)\rangle$$, $$\forall x,y\in H_{1}$$;

2. (ii)

$$\|tx+(1-t)y\|^{2}=t\|x\|^{2}+(1-t)\|y\|^{2}-t(1-t)\|x-y\|^{2}$$, $$t \in[0,1]$$, $$\forall x,y\in H_{1}$$.

### Lemma 2.2

[4, 17]

Let D be a nonempty, bounded, closed, and convex subset of a real Hilbert space H and let $\mathcal{T}:=\left\{T\left(s\right):0\le s<\mathrm{\infty }\right\}$ a nonexpansive semigroup on D, then for any $$u\geq0$$,
$$\lim_{t\rightarrow\infty}\sup_{x\in D}\biggl\Vert \frac{1}{t}\int_{0}^{t} T(s)x\,ds-T(u) \frac{1}{t}\int_{0}^{t} T(s)x\, ds\biggr\Vert =0.$$

### Lemma 2.3

[5, 18]

Let $$S: H_{1}\rightarrow H_{1}$$ be averaged and $$T: H_{1}\rightarrow H_{1}$$ be nonexpansive; we have:
1. (i)

$$W=(1-\alpha)S+\alpha T$$ is averaged, where $$\alpha\in(0,1)$$.

2. (ii)

The composite of finitely many averaged mappings is averaged.

### Lemma 2.4



The split variational inclusion problem (1.5) is equivalent to finding $$x^{*}\in H_{1}$$ such that $$y^{*}=Ax^{*}\in H_{2}$$: $$x^{*}=J_{\lambda}^{B_{1}}(x^{*})$$ and $$y^{*}=J_{\lambda}^{B_{2}}(y^{*})$$ for some $$\lambda>0$$.

### Lemma 2.5



Let B be a strongly positive linear bounded operator on a Hilbert space $$H_{1}$$ with a coefficient $$\overline{\gamma}>0$$ and $$0<\varrho<\|B\|^{-1}$$. Then $$\|I-\varrho B\|\leq1-\varrho\overline{\gamma}$$.

### Lemma 2.6



Let C be a nonempty, closed, and convex subset of a Hilbert space $$H_{1}$$. Assume that $$f:C\rightarrow C$$ is a contraction with a coefficient $$\rho\in(0,1)$$ and B is a strongly positive linear bounded operator with a coefficient $$\overline{\gamma}>0$$. Then, for $$0<\gamma<\overline{\gamma}/\rho$$,
$$\bigl\langle x-y, (B-\gamma f)x-(B-\gamma f)y\bigr\rangle \geq(\overline{\gamma }-\gamma\rho)\|x-y\|^{2}, \quad \forall x,y\in H_{1}.$$
That is, $$B-\gamma f$$ is strongly monotone with coefficient $$\overline{\gamma}-\gamma\rho$$.

### Lemma 2.7



Let $$\{a_{n}\}_{n=1}^{\infty}$$ be a sequence of nonnegative real numbers such that
$$a_{n+1}\leq(1-\gamma_{n})a_{n}+ \gamma_{n}b_{n}+\sigma_{n},$$
where $$\{\gamma_{n}\}_{n=1}^{\infty}\subset(0,1)$$ and $$\{b_{n}\}_{n=1}^{\infty}$$, $$\{\sigma_{n}\}_{n=1}^{\infty}$$ are sequences in such that
1. (i)

$$\lim_{n\rightarrow\infty}\gamma_{n}=0$$ and $$\sum_{n=1}^{\infty}\gamma _{n}=\infty$$;

2. (ii)

$$\limsup_{n\rightarrow\infty}b_{n}\leq0$$;

3. (iii)

$$\sigma_{n}\geq0$$ and $$\sum_{n=1}^{\infty}\sigma_{n}<\infty$$.

Then $$\lim_{n\rightarrow\infty}a_{n}=0$$.

## 3 Main results

### Theorem 3.1

Let $$H_{1}$$ and $$H_{2}$$ be two real Hilbert spaces, let $$A: H_{1}\rightarrow H_{2}$$ be a bounded linear operator and B be a strongly positive bounded linear operator on $$H_{1}$$ with constant $$\overline{\gamma}>0$$. Let $$B_{1}: H_{1}\rightarrow2^{H_{1}}$$, $$B_{2}: H_{2}\rightarrow2^{H_{1}}$$ be maximal monotone mappings and $\mathcal{T}:=\left\{T\left(s\right):0\le s<\mathrm{\infty }\right\}$ be a one-parameter nonexpansive semigroup on $$H_{1}$$ such that $Fix\left(\mathcal{T}\right)\cap \mathcal{Z}\ne \mathrm{\varnothing }$. Assume that $$f: H_{1}\rightarrow H_{1}$$ is a contraction mapping with constant $$\rho\in (0,1)$$. For any $$\alpha\in(0,1)$$, define the mapping Φ on $$H_{1}$$ by
$$\Phi(x)=\alpha\gamma f(x)+(I-\alpha B)\frac{1}{t}\int_{0}^{t}T(s)J_{\lambda}^{B_{1}} \bigl[x+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax\bigr]\, ds,$$
where $$t>0$$, $$\gamma\in(0,\frac{\overline{\gamma}}{\rho})$$, and $$\epsilon\in(0,\frac{1}{L})$$, L is the spectral radius of the operator $$A^{*}A$$, and $$A^{*}$$ is the adjoint of A. Then the mapping Φ is a contraction and has a unique fixed point.

### Proof

Since $$J_{\lambda}^{B_{1}}$$ and $$J_{\lambda}^{B_{2}}$$ are firmly nonexpansive, they are averaged. For $$\epsilon\in(0,1/L)$$, the mapping $$I+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)A$$ is averaged; see e.g. . It follows from Lemma 2.3(ii) that the mapping $$J_{\lambda}^{B_{1}}(I+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)A)$$ is averaged and hence nonexpansive. By Lemma 2.5, for any $$x,y\in H_{1}$$, we have
\begin{aligned} \bigl\Vert \Phi(x)-\Phi(y)\bigr\Vert =&\biggl\Vert \alpha\gamma f(x)+(I- \alpha B)\frac {1}{t}\int_{0}^{t}T(s)J_{\lambda}^{B_{1}} \bigl[x+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax\bigr]\, ds \\ &{}-\alpha\gamma f(y)-(I-\alpha B)\frac{1}{t}\int_{0}^{t}T(s)J_{\lambda}^{B_{1}} \bigl[y+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I\bigr)Ay\bigr]\, ds \biggr\Vert \\ \leq&\alpha\gamma\bigl\Vert f(x)-f(y)\bigr\Vert +(1-\alpha\overline{ \gamma}) \bigl\Vert J_{\lambda}^{B_{1}}\bigl[x+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax\bigr] \\ &{}-J_{\lambda}^{B_{1}}\bigl[y+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ay\bigr] \bigr\Vert \\ \leq&\alpha\gamma\rho \Vert x-y\Vert +(1-\alpha\overline{\gamma})\Vert x-y \Vert \\ =&\bigl[1-\alpha(\overline{\gamma}-\gamma\rho)\bigr]\Vert x-y\Vert . \end{aligned}
It follows from $$\gamma\in(0,\frac{\overline{\gamma}}{\rho})$$ that Φ is a contraction mapping. Therefore, by the Banach contraction principle, $$\Phi(x)$$ has a unique fixed point $$x_{\alpha}$$, that is,
$$x_{\alpha}=\alpha\gamma f(x_{\alpha})+(I-\alpha B)\frac{1}{t} \int_{0}^{t}T(s)J_{\lambda}^{B_{1}} \bigl[x_{\alpha}+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{\alpha}\bigr]\, ds.$$
□

### Theorem 3.2

Let $$H_{1}$$ and $$H_{2}$$ be two real Hilbert spaces, let $$A: H_{1}\rightarrow H_{2}$$ be a bounded linear operator and B be a strongly positive bounded linear operator on $$H_{1}$$ with constant $$\overline{\gamma}>0$$. Let $$B_{1}: H_{1}\rightarrow2^{H_{1}}$$, $$B_{2}: H_{2}\rightarrow2^{H_{1}}$$ be maximal monotone mappings and $\mathcal{T}:=\left\{T\left(s\right):0\le s<\mathrm{\infty }\right\}$ be a one-parameter nonexpansive semigroup on $$H_{1}$$ such that $\mathrm{\Omega }=Fix\left(\mathcal{T}\right)\cap \mathcal{Z}\ne \mathrm{\varnothing }$. Assume that $$f: H_{1}\rightarrow H_{1}$$ is a contraction mapping with constant $$\rho\in(0,1)$$, L is the spectral radius of the operator $$A^{*}A$$, and $$A^{*}$$ is the adjoint of A. For given $$x_{1}\in H_{1}$$, $$\lambda>0$$, $$\gamma\in(0,\frac{\overline{\gamma}}{\rho})$$, and $$\epsilon\in(0,\frac{1}{L})$$, suppose that the sequences $$\{\alpha_{n}\}\subset(0,1)$$ and $$\{t_{n}\}\subset(0,\infty)$$ satisfy:
1. (i)

$$\lim_{n\rightarrow\infty}\alpha_{n}=0$$, $$\sum_{n=1}^{\infty}\alpha _{n}=\infty$$, and $$\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty$$;

2. (ii)

$$\lim_{n\rightarrow\infty}s_{n}=+\infty$$ and $$\lim_{n\rightarrow \infty}\frac{|s_{n}-s_{n-1}|}{s_{n}}\frac{1}{\alpha_{n}}=0$$.

Then the sequence $$\{x_{n}\}$$ generated by (1.6) converges strongly to $$q\in\Omega$$, which is the unique solution of the following variational inequality:
$$\bigl\langle (B-\gamma f)q, q-w\bigr\rangle \leq0,\quad \forall w\in \Omega.$$

### Proof

Taking $p\in \mathrm{\Omega }=Fix\left(\mathcal{T}\right)\cap \mathcal{Z}$, we have $$p=J_{\lambda}^{B_{1}}p$$, $$Ap=J_{\lambda}^{B_{2}}(Ap)$$, and $$T(s)p=p$$. From (1.6), $$u_{n}=J_{\lambda}^{B_{1}}[x_{n}+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)Ax_{n}]$$, and Lemma 2.4, we estimate
\begin{aligned} \Vert u_{n}-p\Vert ^{2}&=\bigl\Vert J_{\lambda}^{B_{1}}\bigl[x_{n}+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n} \bigr]-J_{\lambda}^{B_{1}}p\bigr\Vert ^{2} \\ &\leq\bigl\Vert x_{n}+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}-p\bigr\Vert ^{2} \\ &\leq \Vert x_{n}-p\Vert ^{2}+2\epsilon\bigl\langle x_{n}-p, A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\rangle +\epsilon^{2}\bigl\Vert A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2}. \end{aligned}
(3.1)
By the definition of A and $$A^{*}$$, we obtain
\begin{aligned} \epsilon^{2}\bigl\Vert A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert ^{2}&\leq\epsilon^{2}\bigl\langle \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}, AA^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\rangle \\ &\leq L\epsilon^{2}\bigl\langle \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}, \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\rangle \\ &= L\epsilon^{2}\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert ^{2}. \end{aligned}
(3.2)
Using a similar method to [20, Theorem 2.1] and [21, Theorem 3.1], we have
\begin{aligned} \Lambda&=2\epsilon\bigl\langle x_{n}-p, A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\rangle \\ &=2\epsilon\bigl\langle A(x_{n}-p), \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\rangle \\ &=2\epsilon\bigl\langle A(x_{n}-p)+\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}-\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}, \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\rangle \\ &=2\epsilon \bigl[\bigl\langle J_{\lambda}^{B_{2}}Ax_{n}-Ap, \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\rangle -\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2} \bigr] \\ &\leq2\epsilon \biggl[\frac{1}{2}\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert ^{2}-\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2} \biggr] \\ &\leq-\epsilon\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert ^{2}. \end{aligned}
Combining (3.1) and (3.2), we obtain
$$\|u_{n}-p\|^{2}\leq\|x_{n}-p \|^{2}+\epsilon(L\epsilon-1)\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2}.$$
(3.3)
Setting $$w_{n}=\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n}\, ds$$ for $$n\geq0$$, it follows from $$\epsilon\in(0,\frac{1}{L})$$ and (3.3) that
$$\|w_{n}-p\|=\biggl\Vert \frac{1}{s_{n}}\int _{0}^{s_{n}}\bigl[T(s)u_{n}-T(s)p\bigr]\, ds \biggr\Vert \leq\|u_{n}-p\|\leq\|x_{n}-p\|.$$
(3.4)
It follows from (1.6), (3.4), and Lemma 2.5 that
\begin{aligned} \Vert x_{n+1}-p\Vert &=\biggl\Vert \alpha_{n}\bigl(\gamma f(x_{n})-Bp\bigr)+(I- \alpha_{n}B)\frac {1}{s_{n}}\int_{0}^{s_{n}} \bigl[T(s)u_{n}-T(s)p\bigr]\, ds\biggr\Vert \\ &\leq\alpha_{n}\bigl\Vert \gamma f(x_{n})-Bp\bigr\Vert +(1-\alpha_{n}\overline{\gamma})\biggl\Vert \frac{1}{s_{n}}\int _{0}^{s_{n}}\bigl[T(s)u_{n}-T(s)p\bigr]\, ds \biggr\Vert \\ &\leq\alpha_{n}\gamma\bigl\Vert f(x_{n})-f(p)\bigr\Vert +\alpha_{n}\bigl\Vert \gamma f(p)-Bp\bigr\Vert +(1-\alpha _{n}\overline{\gamma})\Vert u_{n}-p\Vert \\ &\leq\bigl[1-\alpha_{n}(\overline{\gamma}-\gamma\rho)\bigr]\Vert x_{n}-p\Vert +\alpha_{n}\bigl\Vert \gamma f(p)-Bp\bigr\Vert . \end{aligned}
By a simple induction, we have
$$\|x_{n}-p\|\leq\max \biggl\{ \|x_{0}-p\|, \frac{1}{\overline{\gamma}-\gamma\rho }\bigl\Vert \gamma f(p)-Bp\bigr\Vert \biggr\} .$$
(3.5)
Therefore, $$\{x_{n}\}$$ is bounded, and so are $$\{u_{n}\}$$ and $$\{w_{n}\}$$.
Now, we show that $$\lim_{n\rightarrow\infty}\|x_{n+1}-x_{n}\|=0$$. From (1.6), we have
\begin{aligned} \Vert x_{n+1}-x_{n}\Vert =&\bigl\Vert \alpha_{n}\gamma\bigl[f(x_{n})-f(x_{n-1})\bigr]+( \alpha_{n}-\alpha _{n-1})\gamma f(x_{n-1}) \\ &{} +(I-\alpha_{n} B) (w_{n}-w_{n-1})-( \alpha_{n}-\alpha _{n-1})Bw_{n-1}\bigr\Vert \\ \leq&\alpha_{n}\gamma\rho \Vert x_{n}-x_{n-1} \Vert +(1-\alpha_{n}\overline{\gamma})\Vert w_{n}-w_{n-1} \Vert \\ &{}+|\alpha_{n}-\alpha_{n-1}|\bigl[\Vert Bw_{n-1}\Vert +\gamma\bigl\Vert f(x_{n-1})\bigr\Vert \bigr]. \end{aligned}
(3.6)
On the other hand, for $$p\in\Omega$$, we have
\begin{aligned} \|w_{n}-w_{n-1}\| =&\biggl\Vert \frac{1}{s_{n}}\int_{0}^{s_{n}}\bigl[T(s)u_{n}-T(s)u_{n-1} \bigr]\, ds \\ &{}+ \biggl(\frac{1}{s_{n}}-\frac{1}{s_{n-1}} \biggr)\int _{0}^{s_{n-1}}\bigl[T(s)u_{n-1}-T(s)p\bigr]\, ds \\ &{}+\frac{1}{s_{n}}\int_{s_{n-1}}^{s_{n}} \bigl[T(s)u_{n-1}-T(s)p\bigr]\, ds\biggr\Vert . \end{aligned}
(3.7)
Given that
$$\biggl(\frac{1}{v}-\frac{1}{w} \biggr)w=- \frac{v-w}{v}, \quad v,w\neq0.$$
It follows from (3.7) that
$$\|w_{n}-w_{n-1}\|\leq\|u_{n}-u_{n-1} \|+ \biggl(\frac{2|s_{n}-s_{n-1}|}{s_{n}} \biggr)\|u_{n-1}-p\|.$$
(3.8)
Moreover, for $$\epsilon\in(0,\frac{1}{L})$$, mapping $$J_{\lambda}^{B_{1}}[I+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)A]$$ is averaged and hence nonexpansive, then we have
\begin{aligned} \Vert u_{n}-u_{n-1}\Vert &=\bigl\Vert J_{\lambda}^{B_{1}}\bigl[x_{n}+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n} \bigr]-J_{\lambda}^{B_{1}}\bigl[x_{n-1}+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n-1}\bigr]\bigr\Vert \\ &\leq\bigl\Vert J_{\lambda}^{B_{1}}\bigl[I+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)A\bigr]x_{n}-J_{\lambda}^{B_{1}}\bigl[I+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I\bigr)A \bigr]x_{n-1}\bigr\Vert \\ &\leq \Vert x_{n}-x_{n-1}\Vert . \end{aligned}
(3.9)
Combining (3.6), (3.8), and (3.9), we obtain
\begin{aligned} \Vert x_{n+1}-x_{n}\Vert \leq& \alpha_{n}\gamma\rho \Vert x_{n}-x_{n-1}\Vert +(1-\alpha _{n}\overline{\gamma}) \biggl[\Vert x_{n}-x_{n-1} \Vert + \biggl(\frac {2|s_{n}-s_{n-1}|}{s_{n}} \biggr)\Vert u_{n-1}-p\Vert \biggr] \\ &{}+|\alpha_{n}-\alpha_{n-1}|\bigl[\Vert Bw_{n-1}\Vert +\gamma\bigl\Vert f(x_{n-1})\bigr\Vert \bigr] \\ \leq&\bigl[1-\alpha_{n}(\overline{\gamma}-\gamma\rho)\bigr]\Vert x_{n}-x_{n-1}\Vert + \biggl(|\alpha_{n}- \alpha_{n-1}|+\frac{2|s_{n}-s_{n-1}|}{s_{n}} \biggr)M_{1}, \end{aligned}
(3.10)
where $$M_{1}=\max \{\sup_{n\in N}[\|Bw_{n-1}\|+\gamma\|f(x_{n-1})\|], \sup_{n\in N}\|u_{n-1}-p\| \}$$. It follows from conditions (i)-(ii) and Lemma 2.7 that
$$\lim_{n\rightarrow\infty}\|x_{n+1}-x_{n} \|=0.$$
(3.11)
Next, we will show that $$\lim_{n\rightarrow\infty}\|x_{n}-u_{n}\|=0$$. Note that $$w_{n}=\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n}\, ds$$ and
\begin{aligned} \Vert x_{n}-w_{n}\Vert &\leq \Vert x_{n}-x_{n+1}\Vert +\Vert x_{n+1}-w_{n} \Vert \\ &=\Vert x_{n}-x_{n+1}\Vert +\bigl\Vert \alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}B)w_{n}-w_{n} \bigr\Vert \\ &\leq \Vert x_{n}-x_{n+1}\Vert +\alpha_{n} \bigl\Vert \gamma f(x_{n})-Bw_{n}\bigr\Vert . \end{aligned}
Together with condition (i) and (3.11), we obtain
$$\lim_{n\rightarrow\infty}\|x_{n}-w_{n} \|=\lim_{n\rightarrow\infty} \biggl\| x_{n}-\frac{1}{s_{n}}\int _{0}^{s_{n}}T(s)u_{n}\, ds \biggr\| =0.$$
(3.12)
Observe that
\begin{aligned} \bigl\Vert x_{n}-T(u)x_{n}\bigr\Vert \leq& \biggl\Vert x_{n}-\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n} \, ds\biggr\Vert \\ &{}+\biggl\Vert \frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n} \, ds-T(u)\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n} \, ds\biggr\Vert \\ &{}+\biggl\Vert T(u)\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n} \, ds-T(u)x_{n}\biggr\Vert \\ \leq&2\biggl\Vert x_{n}-\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n} \, ds\biggr\Vert \\ &{}+\biggl\Vert \frac {1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n} \, ds-T(u)\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n} \, ds\biggr\Vert . \end{aligned}
It follows from (3.12) and Lemma 2.2 that
$$\lim_{n\rightarrow\infty}\bigl\Vert x_{n}-T(u)x_{n} \bigr\Vert =0.$$
(3.13)
By (3.3), (3.4), and Lemma 2.1, we have
\begin{aligned} \Vert x_{n+1}-p\Vert ^{2}&=\bigl\Vert w_{n}-p+\alpha_{n}\bigl[\gamma f(x_{n})-Bw_{n} \bigr]\bigr\Vert ^{2} \\ &\leq \Vert w_{n}-p\Vert ^{2} + 2\alpha_{n} \bigl\langle \gamma f(x_{n})-Bw_{n},x_{n+1}-p\bigr\rangle \\ &\leq \Vert u_{n}-p\Vert ^{2} + 2\alpha_{n} \bigl\langle \gamma f(x_{n})-Bw_{n},x_{n+1}-p\bigr\rangle \\ &\leq \bigl[\Vert x_{n}-p\Vert ^{2}+\epsilon(L \epsilon-1)\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert ^{2} \bigr]+2\alpha_{n} \bigl\langle \gamma f(x_{n})-Bw_{n},x_{n+1}-p\bigr\rangle \\ &\leq \Vert x_{n}-p\Vert ^{2}-\epsilon(1-L\epsilon)\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2}+2\alpha_{n}M_{2}^{2}, \end{aligned}
(3.14)
where $$M_{2}=\max \{\sup_{n\in N}\|\gamma f(x_{n})-Bw_{n}\|, \sup_{n\in N}\|x_{n+1}-p\| \}$$ and $$\epsilon\in(0,\frac{1}{L})$$, which implies that
\begin{aligned} \epsilon(1-L\epsilon)\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert ^{2}&\leq \Vert x_{n}-p \Vert ^{2}-\Vert x_{n+1}-p\Vert ^{2}+2 \alpha_{n}M_{2}^{2} \\ &\leq \Vert x_{n+1}-x_{n}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)+2 \alpha_{n}M_{2}^{2}. \end{aligned}
(3.15)
It follows from condition (i) and (3.11) that
$$\lim_{n\rightarrow\infty}\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert =0.$$
(3.16)
Furthermore, using (3.1), (3.3), and $$\epsilon\in(0,\frac{1}{L})$$, we obtain
\begin{aligned} \Vert u_{n}-p\Vert ^{2} =& \bigl\Vert J_{\lambda}^{B_{1}}\bigl[x_{n}+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n} \bigr]-J_{\lambda}^{B_{1}}p \bigr\Vert ^{2} \\ \leq&\bigl\langle u_{n}-p, x_{n}+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}-p\bigr\rangle \\ =&\frac{1}{2} \bigl\{ \Vert u_{n}-p\Vert ^{2}+ \bigl\Vert x_{n}+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}-p\bigr\Vert ^{2} \\ &{}-\bigl\Vert u_{n}-p-\bigl[x_{n}+\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}-p\bigr]\bigr\Vert ^{2} \bigr\} \\ \leq&\frac{1}{2} \bigl\{ \Vert u_{n}-p\Vert ^{2}+ \Vert x_{n}-p\Vert ^{2}+\epsilon(L\epsilon-1)\bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2} \\ &{}-\bigl\Vert u_{n}-x_{n}-\epsilon A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2} \bigr\} \\ \leq&\frac{1}{2} \bigl\{ \Vert u_{n}-p\Vert ^{2}+ \Vert x_{n}-p\Vert ^{2}- \bigl[\Vert u_{n}-x_{n}\Vert ^{2}+\epsilon^{2} \bigl\Vert A^{*}\bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert ^{2} \\ &{}-2\epsilon\bigl\langle u_{n}-x_{n}, A^{*} \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\rangle \bigr] \bigr\} \\ \leq&\frac{1}{2} \bigl\{ \Vert u_{n}-p\Vert ^{2}+ \Vert x_{n}-p\Vert ^{2}-\Vert u_{n}-x_{n} \Vert ^{2}+2\epsilon\bigl\Vert A(u_{n}-x_{n}) \bigr\Vert \bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr\Vert \bigr\} , \end{aligned}
which implies that
$$\|u_{n}-p\|^{2}\leq\|x_{n}-p \|^{2}-\|u_{n}-x_{n}\|^{2}+2\epsilon \bigl\Vert A(u_{n}-x_{n})\bigr\Vert \bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert .$$
(3.17)
It follows from (3.14) and (3.17) that
\begin{aligned} \|x_{n+1}-p\|^{2}&\leq\|u_{n}-p \|^{2} + 2\alpha_{n}M_{2}^{2} \\ &\leq\|x_{n}-p\|^{2}-\|u_{n}-x_{n} \|^{2}+2\epsilon\bigl\Vert A(u_{n}-x_{n})\bigr\Vert \bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n} \bigr\Vert + 2\alpha_{n}M_{2}^{2}, \end{aligned}
that is,
\begin{aligned} \Vert u_{n}-x_{n}\Vert ^{2} \leq& \Vert x_{n}-p\Vert ^{2}-\Vert x_{n+1}-p\Vert ^{2}+2\epsilon\bigl\Vert A(u_{n}-x_{n})\bigr\Vert \bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n} \bigr\Vert + 2\alpha_{n}M_{2}^{2} \\ \leq&\Vert x_{n}-x_{n+1}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr) \\ &{}+2\epsilon \bigl\Vert A(u_{n}-x_{n})\bigr\Vert \bigl\Vert \bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr\Vert + 2 \alpha_{n}M_{2}^{2}. \end{aligned}
Combining condition (i), (3.11), and (3.16), we have
$$\lim_{n\rightarrow\infty}\|u_{n}-x_{n} \|=0.$$
(3.18)
Since $$\{x_{n}\}$$ and $$\{u_{n}\}$$ are bounded, we consider a weak cluster point w of $$\{x_{n}\}$$. Without loss of generality, we may assume that subsequence $$\{x_{n_{j}}\}$$ of $$\{x_{n}\}$$ converges weakly to w, i.e., $$x_{n_{j}}\rightharpoonup w$$ as $$j\rightarrow\infty$$. From (3.18), we have $$\{u_{n_{j}}\}$$ of $$\{u_{n}\}$$, which converges weakly to w. Moreover, $$u_{n_{j}}=J_{\lambda}^{B_{1}}[x_{n_{j}}+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)Ax_{n_{j}}]$$ can be rewritten as
$$\frac{(x_{n_{j}}-u_{n_{j}})+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)Ax_{n_{j}}}{\lambda}\in B_{1}u_{n_{j}}.$$
(3.19)
By passing to the limit $$j\rightarrow\infty$$ in (3.19) and by taking into account (3.16), (3.18), and the fact that the graph of a maximal monotone operator is weakly-strongly closed, we obtain $$0\in B_{1}(w)$$. Furthermore, since $$\{x_{n}\}$$ and $$\{u_{n}\}$$ have the same asymptotical behavior, $$\{Ax_{n_{j}}\}$$ weakly converges to Aw. From (3.16) and the fact that the resolvent $$J_{\lambda}^{B_{2}}$$ is nonexpansive, we obtain $$Aw\in B_{2}(Aw)$$. It follows from Lemma 2.4 that $w\in \mathcal{Z}$.
We now show that $$\limsup_{n\rightarrow\infty}\langle\gamma f(q)-Bq, x_{n}-q\rangle\leq0$$, where $$q=P_{\Omega}(I-B+\gamma f)q$$. Note that the subsequence $$\{x_{n_{j}}\}$$ of $$\{x_{n}\}$$ converges weakly to w and
$$\limsup_{n\rightarrow\infty}\bigl\langle \gamma f(q)-Bq, x_{n}-q\bigr\rangle =\lim_{j\rightarrow\infty}\bigl\langle \gamma f(q)-Bq, x_{n_{j}}-q\bigr\rangle .$$
(3.20)
Assume that $$w\neq T(u)w$$. By (3.13) and Opial’s property, we obtain
\begin{aligned} \liminf_{j\rightarrow\infty} \Vert x_{n_{j}}-w\Vert &< \liminf_{j\rightarrow\infty }\bigl\Vert x_{n_{j}}-T(u)w\bigr\Vert \\ &\leq\liminf_{j\rightarrow\infty}\bigl(\bigl\Vert x_{n_{j}}-T(u)x_{n_{j}} \bigr\Vert +\bigl\Vert T(u)x_{n_{j}}-T(u)w\bigr\Vert \bigr) \\ &\leq\liminf_{j\rightarrow\infty}\bigl(\bigl\Vert x_{n_{j}}-T(u)x_{n_{j}} \bigr\Vert +\Vert x_{n_{j}}-w\Vert \bigr) \\ &\leq\liminf_{j\rightarrow\infty} \Vert x_{n_{j}}-w\Vert . \end{aligned}
This is a contradiction. Then $w\in Fix\left(\mathcal{T}\right)$. Consequently, $w\in \mathrm{\Omega }=Fix\left(\mathcal{T}\right)\cap \mathcal{Z}$. It follows from (3.20) that
$$\limsup_{n\rightarrow\infty}\bigl\langle \gamma f(q)-Bq, x_{n}-q\bigr\rangle =\bigl\langle \gamma f(q)-Bq, w-q\bigr\rangle \leq0.$$
(3.21)
On the other hand, we shall show that the uniqueness of a solution of the variational inequality
$$\bigl\langle (B-\gamma f)x, x-w\bigr\rangle \leq0, \quad w\in \Omega.$$
(3.22)
Suppose $$q\in\Omega$$ and $$\hat{q}\in\Omega$$ both are solutions to (3.22), then
$$\bigl\langle (B-\gamma f)q, q-\hat{q}\bigr\rangle \leq0$$
(3.23)
and
$$\bigl\langle (B-\gamma f)\hat{q}, \hat{q}-q\bigr\rangle \leq0.$$
(3.24)
Adding up (3.23) and (3.24) one gets
$$\bigl\langle (B-\gamma f)q-(B-\gamma f)\hat{q}, q-\hat{q}\bigr\rangle \leq0.$$
(3.25)
By Lemma 2.6, the strong monotonicity of $$B-\gamma f$$, we obtain $$q=\hat{q}$$ and the uniqueness is proved.
Finally, we show that $$\{x_{n}\}$$ converges strongly to q as $$n\rightarrow\infty$$. From (1.6), (3.4), and Lemma 2.1, we have (note that $$w_{n}=\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)u_{n}\, ds$$)
\begin{aligned} \Vert x_{n+1}-q\Vert ^{2} =&\bigl\langle \alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}B)w_{n}-q, x_{n+1}-q \bigr\rangle \\ =&\alpha_{n}\bigl\langle \gamma f(x_{n})-Bq, x_{n+1}-q \bigr\rangle +\bigl\langle (I-\alpha _{n}B) (w_{n}-q), x_{n+1}-q \bigr\rangle \\ \leq&\alpha_{n}\gamma\bigl\langle f(x_{n})-f(q), x_{n+1}-q\bigr\rangle + \alpha _{n}\bigl\langle \gamma f(q)-Bq, x_{n+1}-q \bigr\rangle \\ &{}+(1-\alpha_{n}\overline{\gamma})\Vert w_{n}-q\Vert \Vert x_{n+1}-q\Vert \\ \leq&\alpha_{n}\gamma\rho \Vert x_{n}-q\Vert \Vert x_{n+1}-q\Vert + \alpha_{n}\bigl\langle \gamma f(q)-Bq, x_{n+1}-q\bigr\rangle \\ &{}+(1-\alpha_{n}\overline{\gamma})\Vert x_{n}-q\Vert \Vert x_{n+1}-q\Vert \\ =&\bigl[1-\alpha_{n}(\overline{\gamma}-\gamma\rho)\bigr]\Vert x_{n}-q\Vert \Vert x_{n+1}-q\Vert + \alpha_{n}\bigl\langle \gamma f(q)-Bq, x_{n+1}-q\bigr\rangle \\ \leq&\frac{1-\alpha_{n}(\overline{\gamma}-\gamma\rho)}{2}\bigl(\Vert x_{n}-q\Vert ^{2}+ \Vert x_{n+1}-q\Vert ^{2}\bigr)+ \alpha_{n}\bigl\langle \gamma f(q)-Bq, x_{n+1}-q\bigr\rangle \\ \leq&\frac{1-\alpha_{n}(\overline{\gamma}-\gamma\rho)}{2}\Vert x_{n}-q\Vert ^{2} + \frac{1}{2}\Vert x_{n+1}-q\Vert ^{2}+ \alpha_{n}\bigl\langle \gamma f(q)-Bq, x_{n+1}-q\bigr\rangle . \end{aligned}
It follows that
$$\|x_{n+1}-q\|^{2}\leq\bigl[1-(\overline{ \gamma}-\gamma\rho)\alpha_{n}\bigr]\|x_{n}-q\| ^{2} + 2\alpha_{n}\bigl\langle \gamma f(q)-Bq, x_{n+1}-q\bigr\rangle .$$
(3.26)
From $$0<\gamma<\frac{\overline{\gamma}}{\rho}$$, condition (i), and (3.21), we can arrive at the desired conclusion $$\lim_{n\rightarrow \infty}\|x_{n}- q\|=0$$ by applying Lemma 2.7 to (3.26). This completes the proof. □

### Theorem 3.3

Let $$H_{1}$$ and $$H_{2}$$ be two real Hilbert spaces and $$A: H_{1}\rightarrow H_{2}$$ be a bounded linear operator. Let $$B_{1}: H_{1}\rightarrow2^{H_{1}}$$, $$B_{2}: H_{2}\rightarrow2^{H_{1}}$$ be maximal monotone mappings and $\mathcal{T}:=\left\{T\left(s\right):0\le s<\mathrm{\infty }\right\}$ be a one-parameter nonexpansive semigroup on $$H_{1}$$ such that $\mathrm{\Omega }=Fix\left(\mathcal{T}\right)\cap \mathcal{Z}\ne \mathrm{\varnothing }$. Assume that $$f: H_{1}\rightarrow H_{1}$$ is a contraction mapping with constant $$\rho\in(0,1)$$, L is the spectral radius of the operator $$A^{*}A$$, and $$A^{*}$$ is the adjoint of A. For given $$x_{1}\in H_{1}$$, $$\lambda>0$$, and $$\epsilon\in(0,\frac{1}{L})$$, define $$\{x_{n}\}$$ in the following manner:
$$x_{n+1}=\alpha_{n}f(x_{n})+(1- \alpha_{n})\frac{1}{s_{n}}\int_{0}^{s_{n}}T(s)J_{\lambda}^{B_{1}} \bigl[x_{n}+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I \bigr)Ax_{n}\bigr]\, ds,$$
(3.27)
where the sequence $$\{\alpha_{n}\}\subset(0,1)$$ satisfies the following conditions:
1. (i)

$$\lim_{n\rightarrow\infty}\alpha_{n}=0$$, $$\sum_{n=1}^{\infty}\alpha _{n}=\infty$$, and $$\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty$$;

2. (ii)

$$\lim_{n\rightarrow\infty}s_{n}=+\infty$$ and $$\lim_{n\rightarrow \infty}\frac{|s_{n}-s_{n-1}|}{s_{n}}\frac{1}{\alpha_{n}}=0$$.

Then the sequence $$\{x_{n}\}$$ converges strongly to $$q=P_{\Omega}f(q)$$, which is equivalent to the unique solution of the following variational inequality:
$$\bigl\langle (I-f)q, q-w\bigr\rangle \leq0, \quad \forall w\in\Omega.$$

### Proof

Putting $$\gamma=1$$ and $$B=I$$, iterative scheme (1.6) reduces to viscosity iteration (3.27). The desired conclusion follows immediately from Theorem 3.2. This completes the proof. □

### Theorem 3.4

Let $$H_{1}$$ and $$H_{2}$$ be two real Hilbert spaces, let $$A: H_{1}\rightarrow H_{2}$$ be a bounded linear operator and B be a strongly positive bounded linear operator on $$H_{1}$$ with constant $$\overline{\gamma}>0$$. Let $$B_{1}: H_{1}\rightarrow2^{H_{1}}$$, $$B_{2}: H_{2}\rightarrow2^{H_{1}}$$ be maximal monotone mappings and $$T:H_{1}\rightarrow H_{1}$$ be a nonexpansive mapping such that $\mathrm{\Omega }=Fix\left(T\right)\cap \mathcal{Z}\ne \mathrm{\varnothing }$. Assume that $$f: H_{1}\rightarrow H_{1}$$ is a contraction mapping with constant $$\rho\in (0,1)$$, L is the spectral radius of the operator $$A^{*}A$$, and $$A^{*}$$ is the adjoint of A. For given $$x_{1}\in H_{1}$$, $$\lambda>0$$, $$\gamma\in(0,\frac{\overline{\gamma}}{\rho})$$, and $$\epsilon\in(0,\frac{1}{L})$$, define $$\{x_{n}\}$$ in the following manner:
$$\left \{ \begin{array}{l} u_{n}=J_{\lambda}^{B_{1}}[x_{n}+\epsilon A^{*}(J_{\lambda}^{B_{2}}-I)Ax_{n}], \\ x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}B)Tu_{n}, \end{array} \right .$$
(3.28)
where the sequence $$\{\alpha_{n}\}\subset(0,1)$$ satisfies $$\lim_{n\rightarrow\infty}\alpha_{n}=0$$, $$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$, and $$\sum_{n=1}^{\infty}|\alpha_{n}-\alpha_{n-1}|<\infty$$. Then the sequence $$\{x_{n}\}$$ converges strongly to q, which is the unique solution of the following variational inequality:
$$\bigl\langle (B-\gamma f)q, q-w\bigr\rangle \leq0, \quad \forall w\in \Omega.$$

### Proof

Clearly, Theorem 3.2 is valid for a nonexpansive mapping. Therefore, the desired conclusion follows immediately from Theorem 3.2. This completes the proof. □

### Remark 3.1

Theorems 3.2 and 3.3 extend the approximation scheme of Byrne et al.  and the viscosity results of Kazmi and Rizvi  to a general iterative method for a split variational inclusion and one-parameter nonexpansive semigroups, which includes the results of [15, 16] as special cases.

### Remark 3.2

Theorems 3.2 and 3.4 improve and extend the main results of Marino and Xu  for nonexpansive mappings, and  for nonexpansive semigroups in different directions.

## 4 Numerical results

In this section, we give an example and numerical results to illustrate our algorithm and the main result of this paper.

### Example 4.1

Let $$H_{1}=H_{2}=\mathbb{R}$$. Let $$B_{1}x=2x$$ and $$B_{2}x=3x$$. Let $\mathcal{T}:=\left\{T\left(s\right):0\le s<\mathrm{\infty }\right\}$, where $$T(s)x=\frac{1}{1+2s}x$$, $$\forall x\in\mathbb{R}$$. Then $$B_{1}$$, $$B_{2}$$, and $\mathcal{T}$ satisfy all conditions of Theorem 3.2 and $\mathrm{\Omega }=Fix\left(\mathcal{T}\right)\cap \mathcal{L}=\left\{0\right\}$. Let $$\{x_{n}\}$$ be the sequence generated by $$x_{0}$$ and
$$x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I- \alpha_{n}B)\frac{1}{s_{n}}\int_{0}^{s_{n}} \frac{1}{1+2s}J_{\lambda}^{B_{1}}\bigl[x_{n}+\epsilon A^{*}\bigl(J_{\lambda}^{B_{2}}-I\bigr)Ax_{n}\bigr]\, ds.$$
(4.1)
Let $$A=B=I$$, the identity operator, and $$f(x)=\frac{1}{2}x$$, $$\forall x\in\mathbb{R}$$. Putting $$\gamma=\lambda=1$$, $$\epsilon=\frac{1}{2}$$, and $$\alpha_{n}=\frac{1}{\sqrt{n}}$$, $$s_{n}=n$$, then scheme (4.1) reduces to
$$x_{n+1}=\frac{1}{2\sqrt{n}} \biggl[x_{n}+ \frac{5}{24n}(\sqrt{n}-1)\ln (1+2n)x_{n} \biggr].$$
(4.2)
Setting $$\|x_{n}-x^{*}\|\leq10^{-4}$$ as stop criterion, then we obtain the numerical results of scheme (4.2) with different initial points $$x_{1}$$ in Table 1.

The computations are performed by Matlab R2007a running on a PC Desktop Intel(R) Core(TM)i3-2330M, CPU @2.20 GHz, 790 MHz, 1.83 GB, 2 GB RAM.

## Declarations

### Acknowledgements

The authors are grateful to the anonymous editor and referees for valuable remarks suggestions, which helped them very much in improving this manuscript. This work was supported by the National Science Foundation of China (11471059), Basic and Advanced Research Project of Chongqing (cstc2014jcyjA00037), Science and Technology Research Project of Chongqing Municipal Education Commission (KJ1400614, KJ1400618).

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

## Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

## References 