A new explicit iteration method for variational inequalities on the set of common fixed points for a finite family of nonexpansive mappings

Abstract

In this paper, we introduce a new explicit iteration method based on the steepest descent method and Krasnoselskii-Mann type method for finding a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common fixed points for a finite family of nonexpansive mappings in a real Hilbert space.

MSC:41A65, 47H17, 47H20.

1 Introduction and preliminaries

Let C be a nonempty closed and convex subset of a real Hilbert space H with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$, and let $F:H\to H$ be a nonlinear mapping. The variational inequality problem is to find a point $p^{\ast }\in C$ such that

$$〈F\left(p^{\ast }\right),p-p^{\ast }〉\ge 0,\mathrm{\forall }p\in C.$$

(1.1)

Variational inequalities were initially studied by Kinderlehrer and Stampacchia in [1], and since then have been widely investigated. They cover partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [1–3]).

It is well known that if F is an L-Lipschitz continuous and η-strongly monotone, i.e., F satisfies the following conditions:

$$\begin{array}{r}\parallel F(x)-F(y)\parallel \le L\parallel x-y\parallel ,\\ 〈F(x)-F(y),x-y〉\ge \eta {\parallel x-y\parallel }^2,\end{array}$$

where L and η are fixed positive numbers, then (1.1) has a unique solution. It is also known that (1.1) is equivalent to the fixed point equation

$$p=P_C(p-\mu F(p)),$$

(1.2)

where $P_C$ denotes the metric projection from $x\in H$ onto C and μ is an arbitrarily positive constant.

The fixed point formulation (1.2) involves the metric projection $P_C$. To overcome the complexity caused by $P_C$, Yamada [4] introduced a hybrid steepest descent method for solving (1.1). His idea is stated as follows. Assume that $C={\bigcap }_{i=1}^{N}Fix(T_i)$, the set of common fixed points of a finite family of nonexpansive mappings $T_i$ on H with an integer $N\ge 1$.

Recall that $T:H\to H$ is nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in H$$

and

$$Fix(T)=\{x\in H:x=Tx\}$$

denotes the fixed point set of T. Yamada proposed the following algorithm in [4]

$$u_{k+1}=T_{[k+1]}{u}_k-{\lambda }_{k+1}\mu F(T_{[k+1]}{u}_k),$$

(1.3)

where $T_{[n]}=T_{nmodN}$, for integer $n\ge 1$, with the mod-function taking values in the set $\{1,2,\dots ,N\}$, $\mu \in (0,2\eta /L^2)$ and $\{{\lambda }_k\}\subset (0,1)$, and proved that the sequence $\{u_k\}$ in (1.3) converges strongly to $p^{\ast }$ under the following conditions:

(L1) $lim{\lambda }_k=0$;

(L2) $\sum {\lambda }_k=\mathrm{\infty }$;

(L3) $\sum |{\lambda }_k-{\lambda }_{k+N}|<\mathrm{\infty }$.

Further, Zeng and Yao [5] proved the same result with (L3) replaced by

(L4) $lim({\lambda }_k-{\lambda }_{k+N})/{\lambda }_{k+N}=0$.

Theorem 1.1 [5]

Let H be a real Hilbert space, and let $F:H\to H$ be an L-Lipschitz continuous and η-strongly monotone mapping for some constants $L,\eta >0$. Let ${\{T_i\}}_{i=1}^N$ be N nonexpansive self-maps of H such that $C={\bigcap }_{i=1}^{N}Fix(T_i)\ne \mathrm{\varnothing }$, $\mu \in (0,2\eta /L^2)$, and let conditions (L1), (L2), (L4) be satisfied. Assume in addition that

$$\begin{array}{rl}C& =\bigcap _{i=1}^{N}Fix(T_i)=Fix(T_1{T}_2\cdot \cdot \cdot T_N)\\ =Fix(T_N{T}_1{T}_2\cdot \cdot \cdot T_{N-1})\\ =\cdots \\ =Fix(T_2{T}_3\cdot \cdot \cdot T_N{T}_1).\end{array}$$

(1.4)

Then the sequence $\{u_k\}$ defined by (1.3) converges strongly to the unique element $p^{\ast }$ in (1.1).

It is not difficult to show that (L3) implies (L4) if $lim{\lambda }_k/{\lambda }_{k+N}$ exists. However, in general, conditions (L3) and (L4) are not comparable, i.e., neither one of them implies the other (see [6] for details).

Recently, Zeng et al. [7] proposed the following iterative scheme:

$$u_{k+1}=T_{[k+1]}{u}_k-{\lambda }_{k+1}{\mu }_{k+1}F(T_{[k+1]}{u}_k),$$

(1.5)

and proved the following result.

Theorem 1.2 [7]

Let H be a real Hilbert space, and let $F:H\to H$ be an L-Lipschitz continuous and η-strongly monotone mapping for some constants $L,\eta >0$. Let ${\{T_i\}}_{i=1}^N$ be N nonexpansive self-maps of H such that

$$C=\bigcap _{i=1}^{N}Fix(T_i)\ne \mathrm{\varnothing },$$

and let ${\mu }_k\in (0,2\eta /L^2)$. Assume that the following conditions hold:

• (i) $\sum {\lambda }_k=\mathrm{\infty }$, where $\{{\lambda }_k\}\subset (0,1)$;

• (ii) $|{\mu }_k-\eta /L^2|\le \sqrt{{\eta }^2-c{L}^2}/L^2$ for some $c\in (0,{\eta }^2/L^2)$;

• (iii) $lim({\mu }_{k+N}-({\lambda }_k/{\lambda }_{k+N}){\mu }_k)=0$.

Assume in addition that (1.4) holds. If

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈T_{[k+N]}\cdot \cdot \cdot T_{[k+1]}{u}_k-u_{k+N},T_{[k+N]}\cdot \cdot \cdot T_{[k+1]}{u}_k-u_k〉\le 0,$$

(1.6)

then the sequence $\{u_k\}$ defined by (1.5) converges strongly to the unique element $p^{\ast }$ in (1.1).

They also showed that conditions (L1), (L2) and (L4) are sufficient for $\{u_k\}$ to be bounded and

$$\underset{k\to \mathrm{\infty }}{lim}\parallel u_k-T_{[k+N]}\cdot \cdot \cdot T_{[k+1]}{u}_k\parallel =0.$$

So, (1.6) is satisfied. They did not give another sufficient condition different from (L1), (L2) and (L4).

Let $Fx=Ax-u$, where A is a self-adjoint bounded linear mapping such that A is strongly positive, i.e.,

$$〈Ax,x〉\ge \eta {\parallel x\parallel }^2,\mathrm{\forall }x\in H$$

and u is some fixed element in H. Xu [6] introduced the following iteration process:

$$u_{k+1}=(I-{\lambda }_{k+1}A)T_{k+1}{u}_k+{\lambda }_{k+1}u,$$

(1.7)

where I is the identity mapping of H, and proved the following result.

Theorem 1.3 [6]

Let conditions (L1), (L2) and (L3) or (L4) be satisfied. Assume in addition that (1.4) holds. Then the sequence $\{u_k\}$ generated by algorithm (1.7) converges strongly to the unique solution of (1.1) with $Fx=Ax-u$.

Very recently, Liu and Cui [8] showed that the condition

$$C=\bigcap _{i=1}^{N}Fix(T_i)=Fix(T_1{T}_2\cdot \cdot \cdot T_N)$$

(1.8)

is sufficient for (1.4) if $C\ne \mathrm{\varnothing }$.

In this paper, we introduce a new algorithm based on a combination of the steepest descent method for variational inequalities with the Krasnoselskii-Mann method for fixed point problems to solve (1.1) with $C={\bigcap }_{i=1}^{N}Fix(T_i)$, where $T_i$ is a nonexpansive mapping on H for each i.

Given a starting point $x_1\in H$, the iteration is defined by

$$\{\begin{array}{c}{x}_1\in H,\hfill \\ y_k^0=(I-{\lambda }_k\mu F)x_k,\hfill \\ y_k^i=(1-{\beta }_k^i)x_k+{\beta }_k^i{T}_i{y}_k^{i-1},i=1,\dots ,N,\hfill \\ x_{k+1}=y_k^N,\hfill \end{array}$$

(1.9)

and the sequences of parameters $\{{\lambda }_k\}$ and $\{{\beta }_k^i\}\subset (0,1)$ satisfy the following conditions:

$$\{\begin{array}{c}{\lambda }_k\to 0\text{as }k\to \mathrm{\infty }\text{and}\sum {\lambda }_k=\mathrm{\infty };\hfill \\ {\beta }_k^i\to {\beta }^i,0<{\beta }^i<1,i=1,\dots ,N-1;\hfill \\ 0<{lim\hspace{0.17em}inf}_{k\to \mathrm{\infty }}{\beta }_k^N\le {lim\hspace{0.17em}sup}_{k\to \mathrm{\infty }}{\beta }_k^N<1.\hfill \end{array}$$

(1.10)

In Section 2, we prove the strong convergence theorem for (1.9)-(1.10) without conditions (L3), (L4) and (1.8). An application to the case that $T_i$ is a ${\gamma }_i$-strictly pseudocontractive mapping is given in Section 3.

2 Main results

We need the following lemmas for the proof of our main result.

Lemma 2.1 [9]

• (i) ${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉$, $\mathrm{\forall }x,y\in H$.

• (ii) ${\parallel (1-t)x+ty\parallel }^2=(1-t){\parallel x\parallel }^2+t{\parallel y\parallel }^2-(1-t)t{\parallel x-y\parallel }^2$, $\mathrm{\forall }x,y\in H$, and for any fixed $t\in [0,1]$.

From [4], we have the following lemma.

Lemma 2.2 [4]

$\parallel T^{\lambda }x-T^{\lambda }y\parallel \le (1-\lambda \tau )\parallel x-y\parallel$, $\mathrm{\forall }x,y\in H$ and for a fixed number $\mu \in (0,2\eta /L^2)$, $\lambda \in (0,1)$, where $\tau =1-\sqrt{1-\mu (2\eta -\mu L^2)}\in (0,1)$ and $T^{\lambda }x=(I-\lambda \mu F)x$ for $x\in H$.

Lemma 2.3 [10, 11]

Assume that T is a nonexpansive self-map of a closed convex subset K of a real Hilbert space H. If T has a fixed point, then $I-T$ is demiclosed; that is, whenever $\{x_k\}$ is a sequence in K weakly converging to some $x\in K$ and the sequence $\{(I-T)x_k\}$ strongly converges to some y, it follows that $(I-T)x=y$.

Lemma 2.4 [12]

Let $\{x_k\}$ and $\{z_k\}$ be bounded sequences in a Banach space E such that

$$x_{k+1}=(1-{\beta }_k)x_k+{\beta }_k{z}_k$$

for $k\ge 1$, where $\{{\beta }_k\}$ is in $[0,1]$ such that

$$0<\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}{\beta }_k\le \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}{\beta }_k<1.$$

Assume that

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel z_{k+1}-z_k\parallel -\parallel x_{k+1}-x_k\parallel \le 0.$$

Then ${lim}_{k\to \mathrm{\infty }}\parallel x_k-z_k\parallel =0$.

Lemma 2.5 [6]

Let $\{a_k\}$ be a sequence of nonnegative real numbers satisfying the condition

$$a_{k+1}\le (1-b_k)a_k+b_k{c}_k,$$

where $\{b_k\}$ and $\{c_k\}$ are sequences of real numbers such that

• (i) $b_k\in [0,1]$ and ${\sum }_{k=1}^{\mathrm{\infty }}{b}_k=\mathrm{\infty }$;

• (ii) ${lim\hspace{0.17em}sup}_{k\to \mathrm{\infty }}{c}_k\le 0$.

Then ${lim}_{k\to \mathrm{\infty }}{a}_k=0$.

Now, we are in a position to prove the following main result.

Theorem 2.6 Let H be a real Hilbert space, and let $F:H\to H$ be an L-Lipschitz continuous and η-strongly monotone mapping for some constants $L,\eta >0$. Let ${\{T_i\}}_{i=1}^N$ be N nonexpansive self-maps of H such that

$$C=\bigcap _{i=1}^{N}Fix(T_i)\ne \mathrm{\varnothing }.$$

Then the sequence $\{x_k\}$ defined by (1.9)-(1.10) converges strongly to the unique element $p^{\ast }$ in (1.1).

Proof First, we prove that $\{x_k\}$ is bounded. By Lemma 2.2, we have, for any $p\in C$, from (1.9) that

$$\begin{array}{rl}\parallel y_k^0-p\parallel & =\parallel (I-{\lambda }_k\mu F)x_k-p\parallel =\parallel (I-{\lambda }_k\mu F)x_k-(I-{\lambda }_k\mu F)p-{\lambda }_k\mu F(p)\parallel \\ \le (1-{\lambda }_k\tau )\parallel x_k-p\parallel +{\lambda }_k\mu \parallel F(p)\parallel .\end{array}$$

Put $M_p=max\{\parallel x_1-p\parallel ,\frac{\mu }{\tau }\parallel F(p)\parallel \}$. Then $\parallel x_1-p\parallel \le M_p$. So, if $\parallel x_k-p\parallel \le M_p$, then $\parallel y_k^0-p\parallel \le M_p$. This conclusion has a place for $\{y_k^i\}$ with $i=1,\dots ,N-1$. Indeed,

$$\begin{array}{rl}\parallel y_k^i-p\parallel & =\parallel (1-{\beta }_k^i)(x_k-p)+{\beta }_k^i(T_i{y}_k^{i-1}-T_{i}p)\parallel \\ \le (1-{\beta }_k^i)\parallel x_k-p\parallel +{\beta }_k^i\parallel y_k^{i-1}-p\parallel \\ \le (1-{\beta }_k^i)M_p+{\beta }_k^i{M}_p\\ =M_p.\end{array}$$

Then

$$\begin{array}{rl}\parallel x_{k+1}-p\parallel & =\parallel (1-{\beta }_k^N)(x_k-p)+{\beta }_k^N(T_i{y}_k^{i-1}-T_{i}p)\parallel \\ \le (1-{\beta }_k^N)\parallel x_k-p\parallel +{\beta }_k^{i}N\parallel y_k^{N-1}-p\parallel \\ \le (1-{\beta }_k^i)M_p+{\beta }_k^i{M}_p\\ =M_p.\end{array}$$

Therefore, the sequence $\{x_k\}$ is bounded. So, the sequences $\{F(x_k)\}$, $\{y_k^i\}$, and $\{T_i{y}_k^{i-1}\}$ ($i=1,2,\dots ,N$) are also bounded. Without loss of generality, we assume that they are bounded by a positive constant $M_1$.

Let $z_k=T_N{y}_k^{N-1}$. Then we have from (1.9) that

$$x_{k+1}=(1-{\beta }_k^N)x_k+{\beta }_k^N{z}_k$$

and

$$\begin{array}{rl}\parallel z_{k+1}-z_k\parallel =& \parallel T_N{y}_{k+1}^{N-1}-T_N{y}_k^{N-1}\parallel \le \parallel y_{k+1}^{N-1}-y_k^{N-1}\parallel \\ =& \parallel (1-{\beta }_{k+1}^{N-1})x_{k+1}+{\beta }_{k+1}^{N-1}{T}_{N-1}{y}_{k+1}^{N-2}\\ -[(1-{\beta }_k^{N-1})x_k+{\beta }_k^{N-1}{T}_{N-1}{y}_k^{N-2}]\parallel \\ \le & (1-{\beta }_{k+1}^{N-1})\parallel x_{k+1}-x_k\parallel +2M_1|{\beta }_{k+1}^{N-1}-{\beta }_k^{N-1}|\\ +{\beta }_{k+1}^{N-1}\parallel T_{N-1}{y}_{k+1}^{N-2}-T_{N-1}{y}_k^{N-2}\parallel \\ \le & (1-{\beta }_{k+1}^{N-1})\parallel x_{k+1}-x_k\parallel +2M_1|{\beta }_{k+1}^{N-1}-{\beta }_k^{N-1}|\\ +{\beta }_{k+1}^{N-1}\parallel y_{k+1}^{N-2}-y_k^{N-2}\parallel \\ \le & (1-{\beta }_{k+1}^{N-1})\parallel x_{k+1}-x_k\parallel +2M_1|{\beta }_{k+1}^{N-1}-{\beta }_k^{N-1}|\\ +{\beta }_{k+1}^{N-1}[(1-{\beta }_{k+1}^{N-2})\parallel x_{k+1}-x_k\parallel +2M_1|{\beta }_{k+1}^{N-2}-{\beta }_k^{N-2}|\\ +{\beta }_{k+1}^{N-2}\parallel y_{k+1}^{N-3}-y_k^{N-3}\parallel ]\\ \le & (1-{\beta }_{k+1}^{N-1}{\beta }_{k+1}^{N-2})\parallel x_{k+1}-x_k\parallel +2M_1\left(\right|{\beta }_{k+1}^{N-1}-{\beta }_k^{N-1}|\\ +|{\beta }_{k+1}^{N-2}-{\beta }_k^{N-2}|)+{\beta }_{k+1}^{N-1}{\beta }_{k+1}^{N-2}\parallel y_{k+1}^{N-3}-y_k^{N-3}\parallel \\ \le & \cdots \\ \le & (1-\prod _{i=1}^{N-1}{\beta }_{k+1}^i)\parallel x_{k+1}-x_k\parallel +\prod _{i=1}^{N-1}{\beta }_{k+1}^i\parallel y_{k+1}^0-y_k^0\parallel \\ +2M_1\sum _{i=1}^{N-1}|{\beta }_{k+1}^i-{\beta }_k^i|.\end{array}$$

On the other hand,

$$\begin{array}{rl}\parallel y_{k+1}^0-y_k^0\parallel & =\parallel (I-{\lambda }_{k+1}\mu F)x_{k+1}-(I-{\lambda }_k\mu F)x_k\parallel \\ \le \parallel x_{k+1}-x_k\parallel +M_1({\lambda }_{k+1}+{\lambda }_k).\end{array}$$

So, we obtain that

$$\parallel z_{k+1}-z_k\parallel -\parallel x_{k+1}-x_k\parallel \le M_1({\lambda }_{k+1}+{\lambda }_k)\prod _{i=1}^{N-1}{\beta }_{k+1}^i+2M_1\sum _{i=1}^{N-1}|{\beta }_{k+1}^i-{\beta }_k^i|.$$

Since ${\lambda }_k\to 0$ and ${\beta }_k^i\to {\beta }^i$, $i=1,\dots ,N-1$, we have

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel z_{k+1}-z_k\parallel -\parallel x_{k+1}-x_k\parallel \le 0.$$

By Lemma 2.4, $\parallel x_k-z_k\parallel \to 0$ as $k\to \mathrm{\infty }$, i.e., $\parallel x_k-T_N{y}_k^{N-1}\parallel \to 0$.

Now, we prove that $\parallel x_k-T_i{y}_k^{i-1}\parallel \to 0$ for $i=1,\dots ,N-1$. First, we prove $\parallel x_k-T_{N-1}{y}_k^{N-2}\parallel \to 0$. Let $\{x_{k_n}\}$ be a subsequence of $\{x_k\}$ such that

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_k-T_{N-1}{y}_k^{N-2}\parallel =\underset{n\to \mathrm{\infty }}{lim}\parallel x_{k_n}-T_{N-1}{y}_{k_n}^{N-2}\parallel$$

and let $\{x_{k_j}\}$ be a subsequence of $\{x_{k_n}\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_{k_n}-p\parallel =\underset{j\to \mathrm{\infty }}{lim}\parallel x_{k_j}-p\parallel .$$

Further,

$$\begin{array}{rl}\parallel x_{k_j}-p\parallel \le & \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +\parallel T_N{y}_{k_j}^{N-1}-T_{N}p\parallel \\ \le & \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +\parallel y_{k_j}^{N-1}-p\parallel \\ \le & \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +(1-{\beta }_k^{N-1})\parallel x_{k_j}-p\parallel \\ +{\beta }_{k_j}^{N-1}\parallel T_{N-1}{y}_{k_j}^{N-2}-T_{N-1}p\parallel \\ \le & \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +(1-{\beta }_k^{N-1})\parallel x_{k_j}-p\parallel +{\beta }_{k_j}^{N-1}\parallel y_{k_j}^{N-2}-p\parallel \\ \le & \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +(1-{\beta }_k^{N-1})\parallel x_{k_j}-p\parallel \\ +{\beta }_{k_j}^{N-1}[(1-{\beta }_k^{N-2})\parallel x_{k_j}-p\parallel +{\beta }_{k_j}^{N-2}\parallel T_{N-2}{y}_{k_j}^{N-3}-T_{N-2}p\parallel ]\\ \le & \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +(1-{\beta }_k^{N-1}{\beta }_{k_j}^{N-2})\parallel x_{k_j}-p\parallel \\ +{\beta }_{k_j}^{N-1}{\beta }_{k_j}^{N-2}\parallel y_{k_j}^{N-2}-p\parallel \\ \le & \cdots \\ \le & \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +(1-\prod _{i=1}^{N-1}{\beta }_{k_j}^i)\parallel x_{k_j}-p\parallel +\prod _{i=1}^{N-1}{\beta }_{k_j}^i\parallel y_{k_j}^0-p\parallel .\end{array}$$

Since

$$\parallel y_{k_j}^0-p\parallel \le (1-{\lambda }_{k_j}\tau )\parallel x_{k_j}-p\parallel +{\lambda }_{k_j}\mu \parallel F(p)\parallel ,$$

we have

$$\begin{array}{rl}\parallel x_{k_j}-p\parallel & \le \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +\parallel y_{k_j}^{N-1}-p\parallel \\ \le \parallel x_{k_j}-T_N{y}_{k_j}^{N-1}\parallel +\parallel x_{k_j}-p\parallel +\prod _{i=1}^{N-1}{\beta }_{k_j}^i{\lambda }_{k_j}\mu \parallel F(p)\parallel .\end{array}$$

Therefore,

$$\underset{j\to \mathrm{\infty }}{lim}\parallel x_{k_j}-p\parallel =\underset{j\to \mathrm{\infty }}{lim}\parallel y_{k_j}^{N-1}-p\parallel .$$

(2.1)

Next, by Lemma 2.1 we have

$$\begin{array}{rl}{\parallel y_{k_j}^{N-1}-p\parallel }^2=& (1-{\beta }_{k_j}^{N-1}){\parallel x_{k_j}-p\parallel }^2+{\beta }_{k_j}^{N-1}{\parallel T_{N-1}{y}_{k_j}^{N-2}-p\parallel }^2\\ -(1-{\beta }_{k_j}^{N-1}){\beta }_{k_j}^{N-1}{\parallel x_{k_j}-T_{N-1}{y}_{k_j}^{N-2}\parallel }^2\\ \le & (1-{\beta }_{k_j}^{N-1}){\parallel x_{k_j}-p\parallel }^2+{\beta }_{k_j}^{N-1}{\parallel y_{k_j}^{N-2}-p\parallel }^2\\ -(1-{\beta }_{k_j}^{N-1}){\beta }_{k_j}^{N-1}{\parallel x_{k_j}-T_{N-1}{y}_{k_j}^{N-2}\parallel }^2\\ \le & (1-{\beta }_{k_j}^{N-1}){\parallel x_{k_j}-p\parallel }^2\\ +{\beta }_{k_j}^{N-1}{\parallel (1-{\beta }_{k_j}^{N-2})(x_{k_j}-p)+{\beta }_{k_j}^{N-2}(T_{N-2}{y}_{k_j}^{N-3}-p)\parallel }^2\\ -(1-{\beta }_{k_j}^{N-1}){\beta }_{k_j}^{N-1}{\parallel x_{k_j}-T_{N-1}{y}_{k_j}^{N-2}\parallel }^2\\ \le & (1-{\beta }_{k_j}^{N-1}{\beta }_{k_j}^{N-2}){\parallel x_{k_j}-p\parallel }^2+{\beta }_{k_j}^{N-1}{\beta }_{k_j}^{N-2}{\parallel y_{k_j}^{N-3}-p\parallel }^2\\ -(1-{\beta }_{k_j}^{N-1}){\beta }_{k_j}^{N-1}{\parallel x_{k_j}-T_{N-1}{y}_{k_j}^{N-2}\parallel }^2\\ \le & \cdots \\ \le & (1-\prod _{i=1}^{N-1}{\beta }_{k_j}^i){\parallel x_{k_j}-p\parallel }^2+\prod _{i=1}^{N-1}{\beta }_{k_j}^i{\parallel y_{k_j}^0-p\parallel }^2\\ -(1-{\beta }_{k_j}^{N-1}){\beta }_{k_j}^{N-1}{\parallel x_{k_j}-T_{N-1}{y}_{k_j}^{N-2}\parallel }^2.\end{array}$$

On the other hand, by Lemma 2.1 we get

$$\begin{array}{rl}{\parallel y_k^0-p\parallel }^2& ={\parallel (I-{\lambda }_k\mu F)x_k-p\parallel }^2\\ =\parallel x_k-p-(I-{\lambda }_k\mu F)x_k\parallel \\ \le {\parallel x_k-p\parallel }^2-2{\lambda }_k\mu 〈F(x_k),x_k-p〉\\ \le {\parallel x_k-p\parallel }^2+2{\lambda }_k\mu M_1{M}_p.\end{array}$$

Without loss of generality, assume that $\alpha \le {\beta }_k^i\le \beta$, $k\ge 1$ for $i=1,\dots ,N-1$ and some $\alpha ,\beta \in (0,1)$. Then we obtain that

$$\alpha (1-\beta ){\parallel x_{k_j}-T_{N-1}{y}_{k_j}^{N-2}\parallel }^2\le {\parallel x_{k_j}-p\parallel }^2-{\parallel y_{k_j}^{N-1}-p\parallel }^2+2{\lambda }_{k_j}\mu M_1{M}_p\prod _{i=1}^{N-1}{\beta }_{k_j}^i,$$

which with ${\lambda }_k\to 0$ and (2.1) implies that $\parallel x_{k_j}-T_{N-1}{y}_{k_j}^{N-2}\parallel \to 0$ as $j\to \mathrm{\infty }$. So, $\parallel x_k-T_{N-1}{y}_k^{N-2}\parallel \to 0$ as $k\to \mathrm{\infty }$.

Similarly, we obtain that $\parallel x_k-T_{N-2}{y}_k^{N-3}\parallel \to 0$, …, $\parallel x_k-T_1{y}_k^0\parallel \to 0$ as $k\to \mathrm{\infty }$.

Further, we prove that $\parallel x_k-T_i{x}_k\parallel \to 0$ as $k\to \mathrm{\infty }$ for $i=1,\dots ,N$. First, note that $\parallel y_k^0-x_k\parallel ={\lambda }_k\parallel F(x_k)\parallel \to 0$ as $k\to \mathrm{\infty }$ because ${\lambda }_k\to 0$ and $\parallel F(x_k)\parallel \le M_1$, and $\parallel y_k^i-x_k\parallel ={\beta }_k^i\parallel x_k-T_i{y}_k^{i-1}\parallel \to 0$ because $\parallel x_k-T_i{y}_k^{i-1}\parallel \to 0$, for $i=1,\dots ,N$. Now, from

$$\begin{array}{rl}\parallel x_k-T_i{x}_k\parallel & \le \parallel x_k-T_i{y}_k^{i-1}\parallel +\parallel T_i{y}_k^{i-1}-T_i{x}_k\parallel \\ \le \parallel x_k-T_i{y}_k^{i-1}\parallel +\parallel y_k^{i-1}-x_k\parallel \end{array}$$

and $\parallel x_k-T_i{y}_k^{i-1}\parallel ,\parallel y_k^{i-1}-x_k\parallel \to 0$, it follows that $\parallel x_k-T_i{x}_k\parallel \to 0$ for $i=1,\dots ,N$.

Further, we have

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈F\left(p^{\ast }\right),p^{\ast }-x_k〉\le 0.$$

(2.2)

Indeed, let $\{x_{k_j}\}$ be a subsequence of $\{x_k\}$ that converges weakly to $\tilde{p}$ such that

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈F\left(p^{\ast }\right),p^{\ast }-x_k〉=\underset{j\to \mathrm{\infty }}{lim}〈F\left(p^{\ast }\right),p^{\ast }-x_{k_j}〉.$$

Then $\parallel x_{k_j}-T_i{x}_{k_j}\parallel \to 0$. So, by Lemma 2.3, $\tilde{p}\in C$. Therefore, from (1.1) it implies (2.2).

Finally, we estimate the value

$$\begin{array}{rl}{\parallel x_{k+1}-p^{\ast }\parallel }^2\le & (1-{\beta }_k^N){\parallel x_k-p^{\ast }\parallel }^2+{\beta }_k^N{\parallel T_N{y}_k^{N-1}-T_N{p}^{\ast }\parallel }^2\\ \le & (1-{\beta }_k^N){\parallel x_k-p^{\ast }\parallel }^2+{\beta }_k^N{\parallel y_k^{N-1}-T_N{p}^{\ast }\parallel }^2\\ \le & (1-{\beta }_k^N){\parallel x_k-p^{\ast }\parallel }^2\\ +{\beta }_k^N[(1-{\beta }_k^{N-1}){\parallel x_k-p^{\ast }\parallel }^2+{\beta }_k^{N-1}{\parallel T_{N-1}{y}_k^{N-2}-T_{N-1}{p}^{\ast }\parallel }^2]\\ \le & (1-{\beta }_k^N{\beta }_k^{N-1}){\parallel x_k-p^{\ast }\parallel }^2+{\beta }_k^N{\beta }_k^{N-1}{\parallel y_k^{N-1}-T_N{p}^{\ast }\parallel }^2\\ \le & (1-{\beta }_k^N{\beta }_k^{N-1}){\parallel x_k-p^{\ast }\parallel }^2+{\beta }_k^N{\beta }_k^{N-1}[(1-{\beta }_k^{N-2}){\parallel x_k-p^{\ast }\parallel }^2\\ +{\beta }_k^{N-2}{\parallel T_{N-2}{y}_k^{N-3}-T_{N-2}{p}^{\ast }\parallel }^2]\\ \le \cdots \\ \le & (1-\prod _{i=1}^N{\beta }_k^i){\parallel x_k-p^{\ast }\parallel }^2+\prod _{i=1}^N{\beta }_k^i{\parallel y_k^0-p^{\ast }\parallel }^2.\end{array}$$

On the other hand, since

$$\begin{array}{rl}{\parallel y_k^0-p^{\ast }\parallel }^2& ={\parallel (I-{\lambda }_k\mu F)x_k-p^{\ast }\parallel }^2\\ ={\parallel (I-{\lambda }_k\mu F)(x_k-p^{\ast })-{\lambda }_k\mu F\left(p^{\ast }\right)\parallel }^2\\ \le (1-{\lambda }_k\tau ){\parallel x_k-p^{\ast }\parallel }^2-2{\lambda }_k\mu 〈F(x_k),y_k^0-p^{\ast }〉,\end{array}$$

we have

$$\begin{array}{rl}{\parallel x_{k+1}-p^{\ast }\parallel }^2\le & (1-\prod _{i=1}^N{\beta }_k^i){\parallel x_k-p^{\ast }\parallel }^2+\prod _{i=1}^N{\beta }_k^i(1-{\lambda }_k\tau ){\parallel x_k-p^{\ast }\parallel }^2\\ +[2{\lambda }_k\mu 〈F\left(p^{\ast }\right),p^{\ast }-x_k〉+2{\lambda }_k\mu 〈F\left(p^{\ast }\right),x_k-y_k^0〉]\prod _{i=1}^N{\beta }_k^i\\ \le & (1-{\lambda }_k\tau \prod _{i=1}^N{\beta }_k^i){\parallel x_k-p^{\ast }\parallel }^2\\ +{\lambda }_k\tau \prod _{i=1}^N{\beta }_k^i[\frac{2\mu }{\tau }〈F\left(p^{\ast }\right),p^{\ast }-x_k〉+\frac{2\mu }{\tau }〈F\left(p^{\ast }\right),x_k-y_k^0〉].\end{array}$$

Using Lemma 2.5 with

$$\begin{array}{r}{a}_k=\parallel x_k-p^{\ast }\parallel ,\\ b_k={\lambda }_k\tau \prod _{i=1}^N{\beta }_k^i,\\ c_k=\frac{2\mu }{\tau }〈F\left(p^{\ast }\right),p^{\ast }-x_k〉+\frac{2\mu }{\tau }〈F\left(p^{\ast }\right),x_k-y_k^0〉,\end{array}$$

$\parallel x_k-y_k^0\parallel \to 0$ and (2.2), we have that $\parallel x_k-p^{\ast }\parallel \to 0$. This completes the proof. □

3 Application

Recall that a mapping $S:H\to H$ is called γ-strictly pseudocontractive if there exists a constant $\gamma \in [0,1)$ such that

$${\parallel Sx-Sy\parallel }^2\le {\parallel x-y\parallel }^2+\gamma {\parallel (I-S)x-(I-S)y\parallel }^2,\mathrm{\forall }x,y\in H.$$

It is well known [9] that a mapping $T:H\to H$ by $Tx=\alpha x+(1-\alpha )Sx$ with a fixed $\alpha \in [\gamma ,1)$ for all $x\in H$ is a nonexpansive mapping and $Fix(T)=Fix(S)$.

Using this fact, we can extend our result to the case $C={\bigcap }_{i=1}^{N}Fix(S_i)$, where $S_i$ is ${\gamma }_i$-strictly pseudocontractive as follows.

Let ${\alpha }_i\in [{\gamma }_i,1)$ be fixed numbers. Then $C={\bigcap }_{i=1}^{N}Fix({\tilde{T}}_i)$ with

$${\tilde{T}}_{i}y={\alpha }_{i}y+(1-{\alpha }_i)S_{i}y,$$

(3.1)

a nonexpansive mapping, for each $i=1,\dots ,N$. So, we have the following result.

Theorem 3.1 Let H be a real Hilbert space, and let $F:H\to H$ be an L-Lipschitzian and η-strongly monotone mapping for some constants $L,\eta >0$. Let ${\{S_i\}}_{i=1}^N$ be N ${\gamma }_i$-strictly pseudocontractive self-maps of H such that

$$C=\bigcap _{i=1}^{N}Fix(S_i)\ne \mathrm{\varnothing }.$$

Let ${\alpha }_i\in [{\gamma }_i,1)$, $\mu \in (0,2\eta /L^2)$. Assume that $\{{\lambda }_k\},\{{\beta }_k^i\}\subset (0,1)$ satisfy (1.10). Then the sequence $\{x_k\}$ defined by (1.9) with $T_i$ replaced by ${\tilde{T}}_i$ of (3.1) converges strongly to the unique element $p^{\ast }$ of (1.1).

4 Numerical example

Consider the following optimization problem: find an element

$$p^{\ast }\in C:\phi \left(p^{\ast }\right)=\underset{C}{min}\phi (x),$$

where $\phi (x)={\parallel x\parallel }^2/2$, $x=(x_1,x_2)\in {\mathbf{E}}^2$, Euclid space, and $C=C_1\cap C_2$, defined by

$$\begin{array}{r}{C}_1=\{(x_1,x_2)\in {\mathbf{E}}^2:x_1-2x_2+1\le 0\},\\ C_2=\{(x_1,x_2)\in {\mathbf{E}}^2:4x_1-x_2-3\ge 0\}.\end{array}$$

Clearly, the above problem possesses a unique solution $p^{\ast }=(1;1)$ and F, the Fréchet derivative of φ, is 1-Lipschitz continuous and $(1/2)$-strongly monotone. Starting with the point $x^0=(x_1^0;x_2^0)=(4;15)$, $\mu =1/10\in (0;2\eta /L^2)$ and ${\lambda }_k={(k+1)}^{1/2}$, we obtained the result in Table 1.

Table 1 Iterations of scheme ( 1.9 ), where starting point ${\mathit{x}}^{\mathbf{0}}\mathbf{=}\mathbf{(}\mathbf{4}\mathbf{;}\mathbf{15}\mathbf{)}$