New iterative methods for a common solution of fixed points for pseudo-contractive mappings and variational inequalities

Abstract

In this paper, we introduce three iterative methods for finding a common element of the set of fixed points for a continuous pseudo-contractive mapping and the solution set of a variational inequality problem governed by continuous monotone mappings. Strong convergence theorems for the proposed iterative methods are proved. Our results improve and extend some recent results in the literature.

MSC:47H05, 47H09, 47J25, 65J15.

1 Introduction

The theory of variational inequalities represents, in fact, a very natural generalization of the theory of boundary value problems and allows us to consider new problems arising from many fields of applied mathematics, such as mechanics, physics, engineering, the theory of convex programming, and the theory of control. While the variational theory of boundary value problems has its starting point in the method of orthogonal projection, the theory of variational inequalities has its starting point in the projection on a convex set.

Let C be a nonempty closed and convex subset of a real Hilbert space H. The classical variational inequality problem is to find $u\in C$ such that $〈v-u,Au〉\ge 0$ for all $v\in C$, where A is a nonlinear mapping. The set of solutions of the variational inequality is denoted by $\mathit{VI}(C,A)$. The variational inequality problem has been extensively studied in the literature; see [1–9] and the references therein. In the context of the variational inequality problem, this implies that $u\in \mathit{VI}(C,A)\iff u=P_C(u-\lambda Au)$, $\mathrm{\forall }\lambda >0$, where $P_C$ is a metric projection of H into C.

Let A be a mapping from C to H, then A is called monotone if and only if for each $x,y\in C$,

$$〈x-y,Ax-Ax〉\ge 0.$$

(1.1)

An operator A is said to be strongly positive on H if there exists a constant $\overline{\gamma }>0$ such that

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

A mapping A of C into itself is called L-Lipschitz continuous if there exits a positive number L such that

$$\parallel Ax-Ay\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

A mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that

$$〈x-y,Ax-Ay〉\ge \alpha {\parallel Ax-Ay\parallel }^2$$

for all $x,y\in C$; see [9–14]. If A is an α-inverse strongly monotone mapping of C into H, then it is obvious that A is $\frac{1}{\alpha }$-Lipschitz continuous, that is, $\parallel Ax-Ay\parallel \le \frac{1}{\alpha }\parallel x-y\parallel$ for all $x,y\in C$. Clearly, the class of monotone mappings includes the class of α-inverse strongly monotone mappings.

A mapping A of C into H is called $\overline{\gamma }$-strongly monotone if there exists a positive real number $\overline{\gamma }$ such that

$$〈x-y,Ax-Ay〉\ge \overline{\gamma }{\parallel x-y\parallel }^2$$

for all $x,y\in C$; see [15]. Clearly, the class of $\overline{\gamma }$-strongly monotone mappings includes the class of strongly positive mappings.

Recall that a mapping T of C into H is called pseudo-contractive if for each $x,y\in C$, we have

$$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2.$$

(1.2)

T is said to be a k-strict pseudo-contractive mapping if there exists a constant $0\le k\le 1$ such that

$$〈x-y,Tx-Ty〉\le {\parallel x-y\parallel }^2-k{\parallel (I-T)x-(I-T)y\parallel }^2\text{for all }x,y\in D(T).$$

A mapping T of C into itself is called nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in C$. We denote by $F(T)$ the set of fixed points of T. Clearly, the class of pseudo-contractive mappings includes the class of nonexpansive and strict pseudo-contractive mappings.

For a sequence $\{{\alpha }_n\}$ of real numbers in $(0,1)$ and arbitrary $u\in C$, let the sequence $\{x_n\}$ in C be iteratively defined by $x_0\in C$ and

$$x_{n+1}:={\alpha }_{n+1}u+(1-{\alpha }_{n+1})T{x}_n,n\ge 0,$$

(1.3)

where T is a nonexpansive mapping of C into itself. Halpern [16] was first to study the convergence of algorithm (1.3) in the framework of Hilbert spaces. Lions [17] and Wittmann [18] improved the result of Halpern by proving strong convergence of $\{x_n\}$ to a fixed point of T if the real sequence $\{{\alpha }_n\}$ satisfies certain conditions. Reich [19], Shioji and Takahashi [20], and Zegeye and Shahzad [21] extended the result of Wittmann [18] to the case of a Banach space.

In 2000, Moudafi [22] introduced a viscosity approximation method and proved that if H is a real Hilbert space, for given $x_0\in C$, the sequence $\{x_n\}$ generated by the algorithm

$$x_{n+1}:={\alpha }_{n}f(x_n)+(1-{\alpha }_n)T{x}_n,n\ge 0,$$

(1.4)

where $f:C\to C$ is a contraction mapping and $\{{\alpha }_n\}\subset (0,1)$ satisfies certain conditions, converges strongly to the unique solution $x^{\ast }$ in C of the variational inequality

$$〈(I-f)x^{\ast },x-x^{\ast }〉\ge 0,x\in C.$$

(1.5)

Moudafi [22] generalized Halpern’s theorems in the direction of viscosity approximations. In [23], Zegeye et al. extended Moudafi’s result to the case of Lipschitz pseudo-contractive mappings in Banach spaces more general that Hilbert spaces.

In 2006, Marino and Xu [24] introduced the following general iterative method:

$$x_{n+1}:={\alpha }_n\gamma f(x_n)+(1-{\alpha }_{n}A)T{x}_n,n\ge 0.$$

(1.6)

They proved that if the sequence $\{{\alpha }_n\}$ of parameters satisfies appropriate conditions, then the sequence $\{x_n\}$ generated by (1.6) converges strongly to the unique solution of the variational inequality

$$〈(A-\gamma f)x^{\ast },x-x^{\ast }〉\ge 0,x\in C,$$

(1.7)

which is the optimality condition for the minimization problem

$$\underset{x\in C}{min}\frac{1}{2}〈Ax,x〉-h(x),$$

where h is a potential function for γf (i.e., $h^{\prime }(x)=\gamma f(x)$ for $x\in H$).

Recently, Zegeye and Shahzad [25] introduced an iterative method and proved that if C is a nonempty subset of a real Hilbert space H, $T_1:C\to C$ is a pseudo-contractive mapping and $T_2:C\to H$ is a continuous monotone mapping such that $F:=F(T_1)\cap \mathit{VI}(C,T_2)\ne \mathrm{\varnothing }$. For $\{r_n\}\subset (0,\mathrm{\infty })$ defined $T_{r_n},F_{r_n}:H\to C$ by the following: for $x\in H$ and $\{r_n\}\subset (0,\mathrm{\infty })$, define

$$T_{r_n}x:=\{z\in C:〈y-z,T_{1}z〉-\frac{1}{r_n}〈y-z,(1+r_n)z-x〉\le 0,y\in C\},$$

(1.8)

$$F_{r_n}x:=\{z\in C:〈y-z,T_{2}z〉+\frac{1}{r_n}〈y-z,z-x〉\ge 0,y\in C\}.$$

(1.9)

Then the sequence $\{x_n\}$ generated by $x_1\in C$ and

$$x_{n+1}:={\alpha }_{n}f(x_n)+(1-{\alpha }_n)T_{r_n}{F}_{r_n}{x}_n,n\ge 1,$$

(1.10)

where $f:C\to C$ is a contraction mapping and $\{{\alpha }_n\}\subset [0,1]$ and $\{r_n\}$ satisfy certain conditions, converges strongly to $z\in F$, where $z=P_{F}f(z)$.

In this paper, motivated and inspired by the method of Marino and Xu [24] and the work of Zegeye and Shahzad [25], we introduce a viscosity approximation method for finding a common fixed point of a set of fixed points of continuous pseudo-contractive mappings more general than nonexpansive mappings and a solution set of the variational inequality problem for continuous monotone mappings more general than α-inverse strongly monotone mappings in a real Hilbert space. Our result extend and unify most of the results that have been proved for important classes of nonlinear operators.

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let $T_1,T_2:C\to H$ be a continuous pseudo-contractive mapping and a continuous monotone mapping, respectively. For $x\in H$ and $\{r_n\}\subset (0,\mathrm{\infty })$, let $T_{r_n},F_{r_n}:H\to C$ be defined by (1.8) and (1.9).

We consider the three iterative methods given as follows:

$$\begin{array}{r}{x}_1\in H,\\ x_{n+1}:={\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T_{r_n}{F}_{r_n}{x}_n,n\ge 1,\end{array}$$

(1.11)

$$\begin{array}{r}{y}_1\in H,\\ y_{n+1}:={\alpha }_n\gamma f(T_{r_n}{F}_{r_n}{y}_n)+(I-{\alpha }_{n}A)T_{r_n}{F}_{r_n}{y}_n,n\ge 1,\end{array}$$

(1.12)

$$\begin{array}{r}{z}_1\in H,\\ z_{n+1}:=T_{r_n}{F}_{r_n}({\alpha }_n\gamma f(z_n)+(I-{\alpha }_{n}A)z_n),n\ge 1,\end{array}$$

(1.13)

where A is a $\overline{\gamma }$-strongly monotone and L-Lipschitzian continuous operator and $f:H\to H$ is a contraction mapping. We prove in Section 3 that if $\{{\alpha }_n\}$ and $\{r_n\}$ of parameters satisfy appropriate conditions, then the sequences $\{x_n\}$, $\{y_n\}$ and $\{z_n\}$ converge strongly to $z=P_{\mathfrak{F}}(I-A+\gamma f)(z)$.

2 Preliminaries

Let C be a closed and convex subset of a real Hilbert space H. For every $x\in H$, there exists a unique nearest point in C, denoted by $P_{C}x$, such that

$$\parallel x-P_{C}x\parallel =\parallel x-y\parallel ,\mathrm{\forall }y\in C.$$

(2.1)

$P_C$ is called the metric projection of H onto C. We know that $P_C$ is a nonexpansive mapping of H onto C. In connection with metric projection, we have the following lemma.

Lemma 2.1 Let H be a real Hilbert space. The following identity holds:

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

Lemma 2.2 Let C be a nonempty closed convex subset of a Hilbert space H. Let $x\in H$ and $y\in C$. Then $y=P_{C}x$ if and only if

$$〈x-y,y-z〉\ge 0,\mathrm{\forall }z\in C.$$

(2.2)

Lemma 2.3 [26]

Let $\{a_n\}$ be a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\gamma }_n)a_n+{\sigma }_n,\mathrm{\forall }n\ge 0,$$

where

• (i) $\{{\gamma }_n\}\subset [0,1]$, ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }$ and

• (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\sigma }_n}{{\gamma }_n}\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\sigma }_n|<\mathrm{\infty }$.

Then $a_n\to 0$ as $n\to \mathrm{\infty }$.

Lemma 2.4 [27]

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let $A:C\to H$ be a continuous monotone mapping. Then, for $r>0$ and $x\in H$, there exists $z\in C$ such that

$$〈y-z,Az〉+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C.$$

(2.3)

Moreover, by a similar argument as in the proof of Lemmas 2.8 and 2.9 in [28], Zegeye [27]obtained the following lemmas.

Lemma 2.5 [27]

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let $A:C\to H$ be a continuous monotone mapping. For $r>0$ and $x\in H$, define a mapping $F_r:H\to C$ as follows:

$$F_{r}x:=\{z\in C:〈y-z,Az〉+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C\}$$

for all $x\in H$. Then the following hold:

• (1) $F_r$ is single-valued;

• (2) $F_r$ is a firmly nonexpansive type mapping, i.e., for all $x,y\in H$,

  $${\parallel F_{r}x-F_{r}y\parallel }^2\le 〈F_{r}x-F_{r}y,x-y〉;$$

• (3) $F(F_r)=\mathit{VI}(C,A)$;

• (4) $\mathit{VI}(C,A)$ is closed and convex.

In the sequel, we shall make use of the following lemmas.

Lemma 2.6 [27]

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let $T:C\to H$ be a continuous pseudo-contractive mapping. Then, for $r>0$ and $x\in H$, there exists $z\in C$ such that

$$〈y-z,Tz〉-\frac{1}{r}〈y-z,(1+r)z-x〉\le 0,\mathrm{\forall }y\in C.$$

(2.4)

Lemma 2.7 [27]

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let $T:C\to C$ be a continuous pseudo-contractive mapping. For $r>0$ and $x\in H$, define a mapping $T_r:H\to C$ as follows:

$$T_{r}x:=\{z\in C:〈y-z,Tz〉+\frac{1}{r}〈y-z,(1+r)z-x〉\le 0,\mathrm{\forall }y\in C\}$$

for all $x\in H$. Then the following hold:

• (1) $T_r$ is single-valued;

• (2) $T_r$ is a firmly nonexpansive type mapping, i.e., for all $x,y\in H$,

  $${\parallel T_{r}x-T_{r}y\parallel }^2\le 〈T_{r}x-T_{r}y,x-y〉;$$

• (3) $F(T_r)=F(T)$;

• (4) $F(T)$ is closed and convex.

Lemma 2.8 [15]

Let $0<\alpha <1$ and let f be an α-contraction of a real Hilbert space H into itself, and let A be a $\overline{\gamma }$-strongly monotone and L-Lipschitzian continuous operator of H into itself with $\overline{\gamma }>0$ and $L>0$. Take μ, γ to be real numbers as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2},0<\gamma <\frac{\overline{\gamma }-\frac{L^2\mu }{2}}{\alpha }.$$

If $\{{\alpha }_n\}\subseteq (0,1)$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and $\tau =\overline{\gamma }-\frac{L^2\mu }{2}$, then

$$\parallel (I-{\alpha }_{n}A)x-(I-{\alpha }_{n}A)y\parallel \le (1-{\alpha }_n\tau )\parallel x-y\parallel ,\mathrm{\forall }x,y\in H.$$

3 Main results

Now, we prove our main theorems.

Theorem 3.1 Let H be a real Hilbert space, $T_1:C\to C$ be a continuous pseudo-contractive mapping and $T_2:C\to H$ be a continuous monotone mapping such that $\mathfrak{F}:=F(T_1)\cap \mathit{VI}(C,T_2)\ne \mathrm{\varnothing }$. Let $0<\alpha <1$ and let f be an α-contraction of H into itself, and let A be a $\overline{\gamma }$-strongly monotone and L-Lipschitzian continuous operator of C into H with $\overline{\gamma }>0$ and $L>0$. Take μ, γ to be real numbers as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2},0<\gamma <\frac{\overline{\gamma }-\frac{L^2\mu }{2}}{\alpha }.$$

For $x_1\in H$, let $\{x_n\}$ be a sequence generated by (1.11), where $\{{\alpha }_n\}\subset [0,1]$ and $\{r_n\}\subset (0,\mathrm{\infty })$ are such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$. Then the sequence $\{x_n\}$ converges strongly to $z\in \mathfrak{F}$, where $z=P_{\mathfrak{F}}(I-A+\gamma f)(z)$.

Proof Since ${\alpha }_n\to 0$ as $n\to \mathrm{\infty }$, we may assume, without loss of generality, ${\alpha }_n<1$ for all n. For $Q=P_{\mathfrak{F}}$, it implies that $Q(I-A+\gamma f)$ is a contraction of H into itself. Since H is a real Hilbert space, there exists a unique element $z\in H$ such that $z=P_{\mathfrak{F}}(I-A+\gamma f)(z)$.

Let $v\in \mathfrak{F}$, and let $u_n:=T_{r_n}{w}_n$, where $w_n:=F_{r_n}{x}_n$. Then we have from Lemma (2.5) and (2.7) that

$$\parallel u_n-v\parallel =\parallel T_{r_n}{w}_n-T_{r_n}v\parallel \le \parallel w_n-v\parallel =\parallel F_{r_n}{x}_n-F_{r_n}v\parallel \le \parallel x_n-v\parallel .$$

(3.1)

Moreover, from (1.11) and (3.1), we get that

$$\begin{array}{rcl}\parallel x_{n+1}-v\parallel & =& \parallel {\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)u_n-v\parallel \\ =& \parallel {\alpha }_n(\gamma f(x_n)-A(v))+(I-{\alpha }_{n}A)u_n-(I-{\alpha }_{n}A)v\parallel \\ \le & (1-{\alpha }_n\tau )\parallel x_n-v\parallel +{\alpha }_n\gamma \alpha \parallel x_n-v\parallel +{\alpha }_n\parallel \gamma f(v)-A(v)\parallel \\ \le & (1-{\alpha }_n(\tau -\gamma \alpha ))\parallel x_n-v\parallel +{\alpha }_n\parallel \gamma f(v)-A(v)\parallel \\ =& (1-{\alpha }_n(\tau -\gamma \alpha ))\parallel x_n-v\parallel +{\alpha }_n(\tau -\gamma \alpha )\frac{\parallel \gamma f(v)-A(v)\parallel }{\tau -\gamma \alpha }.\end{array}$$

(3.2)

It follows from induction that

$$\parallel x_n-v\parallel \le max\{\parallel x_1-v\parallel ,\frac{\parallel \gamma f(v)-A(v)\parallel }{\tau -\gamma \alpha }\},n\ge 1.$$

(3.3)

Thus $\{x_n\}$ is bounded, and hence so are $\{u_n\}$, $\{w_n\}$ and $\{f(x_n)\}$. Next, to show that $\parallel x_{n+1}-x_n\parallel \to 0$, we have

$$\begin{array}{rcl}\parallel x_{n+1}-x_n\parallel & =& \parallel {\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)u_n-({\alpha }_{n-1}\gamma f(x_{n-1})+(I-{\alpha }_{n-1}A)u_{n-1})\parallel \\ =& \parallel {\alpha }_n\gamma f(x_n)-{\alpha }_n\gamma f(x_{n-1})+{\alpha }_n\gamma f(x_{n-1})-{\alpha }_{n-1}\gamma f(x_{n-1})\\ +(I-{\alpha }_{n}A)u_n-(I-{\alpha }_{n}A)u_{n-1}+(I-{\alpha }_{n}A)u_{n-1}-(I-{\alpha }_{n-1}A)u_{n-1}\parallel \\ \le & {\alpha }_n\gamma \alpha \parallel x_n-x_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\gamma \parallel f(x_{n-1})\parallel \\ +(1-{\alpha }_n\tau )\parallel u_n-u_{n-1}\parallel +|{\alpha }_n-{\alpha }_{n-1}|\parallel A{u}_{n-1}\parallel \\ \le & {\alpha }_n\gamma \alpha \parallel x_n-x_{n-1}\parallel +(1+\gamma )|{\alpha }_n-{\alpha }_{n-1}|K+(1-{\alpha }_n\tau )\parallel u_n-u_{n-1}\parallel \\ \le & {\alpha }_n\gamma \alpha \parallel x_n-x_{n-1}\parallel +(1+\gamma )|{\alpha }_n-{\alpha }_{n-1}|K+(1-{\alpha }_n\tau )\parallel w_n-w_{n-1}\parallel ,\end{array}$$

(3.4)

where .

Moreover, since $w_n=F_{r_n}{x}_n$, $w_{n+1}=F_{r_{n+1}}{x}_{n+1}$, we get that

$$〈y-w_n,T_2{w}_n〉+\frac{1}{r_n}〈y-w_n,w_n-x_n〉\ge 0\text{for all }y\in C,$$

(3.5)

$$〈y-w_{n+1},T_2{w}_{n+1}〉+\frac{1}{r_{n+1}}〈y-w_{n+1},w_{n+1}-x_{n+1}〉\ge 0\text{for all }y\in C.$$

(3.6)

Putting $y=w_{n+1}$ in (3.5) and $y=w_n$ in (3.6), we get that

$$〈w_{n+1}-w_n,T_2{w}_n〉+\frac{1}{r_n}〈w_{n+1}-w_n,w_n-x_n〉\ge 0,$$

(3.7)

$$〈w_n-w_{n+1},T_2{w}_{n+1}〉+\frac{1}{r_{n+1}}〈w_n-w_{n+1},w_{n+1}-x_{n+1}〉\ge 0.$$

(3.8)

Adding (3.7) and (3.8), we have

$$〈w_{n+1}-w_n,T_2{w}_n-T_2{w}_{n+1}〉+〈w_{n+1}-w_n,\frac{w_n-x_n}{r_n}-\frac{w_{n+1}-x_{n+1}}{r_{n+1}}〉\ge 0,$$

(3.9)

which implies that

$$-〈w_{n+1}-w_n,T_2{w}_{n+1}-T_2{w}_n〉+〈w_{n+1}-w_n,\frac{w_n-x_n}{r_n}-\frac{w_{n+1}-x_{n+1}}{r_{n+1}}〉\ge 0.$$

(3.10)

Now, using the fact that $T_2$ is monotone, we get that

$$〈w_{n+1}-w_n,\frac{w_n-x_n}{r_n}-\frac{w_{n+1}-x_{n+1}}{r_{n+1}}〉\ge 0,$$

(3.11)

and hence

$$〈w_{n+1}-w_n,w_n-w_{n+1}+w_{n+1}-x_n-\frac{r_n}{r_{n+1}}(w_{n+1}-x_{n+1})〉\ge 0.$$

(3.12)

Without loss of generality, let us assume that there exists a real number b such that $r_n>b>0$ for all . Then we have

$$\begin{array}{rcl}{\parallel w_{n+1}-w_n\parallel }^2& \le & 〈w_{n+1}-w_n,x_{n+1}-x_n+(1-\frac{r_n}{r_{n+1}})(w_{n+1}-x_{n+1})〉\\ \le & \parallel w_{n+1}-w_n\parallel \{\parallel x_{n+1}-x_n\parallel +\left|(1-\frac{r_n}{r_{n+1}})\right|\parallel w_{n+1}-x_{n+1}\parallel \},\end{array}$$

(3.13)

and hence from (3.13) we obtain that

$$\begin{array}{rcl}\parallel w_{n+1}-w_n\parallel & \le & \parallel x_{n+1}-x_n\parallel +\frac{1}{r_{n+1}}|r_{n+1}-r_n|\parallel w_{n+1}-x_{n+1}\parallel \\ \le & \parallel x_{n+1}-x_n\parallel +\frac{1}{b}|r_{n+1}-r_n|L,\end{array}$$

(3.14)

where .

Furthermore, from (3.4) and (3.14) we have that

$$\begin{array}{rcl}\parallel x_{n+1}-x_n\parallel & \le & {\alpha }_n\gamma \alpha \parallel x_n-x_{n-1}\parallel +(1+\gamma )|{\alpha }_n-{\alpha }_{n-1}|K\\ +(1-{\alpha }_n\tau )(\parallel x_n-x_{n-1}\parallel +\frac{1}{b}|r_n-r_{n-1}|L).\end{array}$$

(3.15)

Hence by Lemma 2.3, we have

$$\parallel x_{n+1}-x_n\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

(3.16)

Consequently, from (3.14) and (3.16), we have that

$$\parallel w_{n+1}-w_n\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

(3.17)

Moreover, since $u_n=T_{r_n}{w}_n$, $u_{n+1}=T_{r_{n+1}}{w}_{n+1}$, we get that

$$〈y-u_n,T_1{u}_n〉-\frac{1}{r_n}〈y-u_n,(1+r_n)u_n-w_n〉\le 0\text{for all }y\in C.$$

(3.18)

and

$$〈y-u_{n+1},T_1{u}_{n+1}〉-\frac{1}{r_{n+1}}〈y-u_{n+1},(1+r_{n+1})u_{n+1}-w_{n+1}〉\le 0\text{for all }y\in C.$$

(3.19)

Putting $y=u_{n+1}$ in (3.18) and $y=u_n$ in (3.19), we get that

$$〈u_{n+1}-u_n,T_2{u}_n〉-\frac{1}{r_n}〈u_{n+1}-u_n,(1+r_n)u_n-w_n〉\le 0$$

(3.20)

and

$$〈u_n-u_{n+1},T_2{u}_{n+1}〉-\frac{1}{r_{n+1}}〈u_n-u_{n+1},(1+r_{n+1})u_{n+1}-w_{n+1}〉\le 0.$$

(3.21)

Adding (3.20) and (3.21), we have

$$\begin{array}{r}〈u_{n+1}-u_n,T_2{u}_n-T_2{u}_{n+1}〉\\ -〈u_{n+1}-u_n,\frac{(1+r_n)u_n-w_n}{r_n}-\frac{(1+r_{n+1})u_{n+1}-w_{n+1}}{r_{n+1}}〉\le 0,\end{array}$$

(3.22)

which implies that

$$\begin{array}{r}〈u_{n+1}-u_n,(u_{n+1}-T_2{u}_{n+1})-(u_n-T_2{u}_n)〉\\ -〈u_{n+1}-u_n,\frac{u_n-w_n}{r_n}-\frac{u_{n+1}-w_{n+1}}{r_{n+1}}〉\le 0.\end{array}$$

(3.23)

Now, using the fact that $T_1$ is pseudo-contractive, we get that

$$〈u_{n+1}-u_n,\frac{u_n-w_n}{r_n}-\frac{u_{n+1}-w_{n+1}}{r_{n+1}}〉\ge 0,$$

(3.24)

and hence

$$〈u_{n+1}-u_n,u_n-u_{n+1}+u_{n+1}-w_n-\frac{r_n}{r_{n+1}}(u_{n+1}-w_{n+1})〉\ge 0.$$

(3.25)

Thus, using the method in (3.13) and (3.14), we have that

$$\begin{array}{rcl}\parallel u_{n+1}-u_n\parallel & \le & \parallel x_{n+1}-x_n\parallel +\frac{1}{r_{n+1}}|r_{n+1}-r_n|\parallel u_{n+1}-w_{n+1}\parallel \\ \le & \parallel x_{n+1}-x_n\parallel +\frac{1}{b}|r_{n+1}-r_n|L,\end{array}$$

(3.26)

where .

Therefore, from (3.16) and the property of $\{r_n\}$, we have that

$$\parallel u_{n+1}-u_n\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

(3.27)

Furthermore, since $x_n={\alpha }_{n-1}\gamma f(x_{n-1})+(I-{\alpha }_{n-1}A)u_{n-1}$, we have that

$$\begin{array}{rcl}\parallel x_n-u_n\parallel & \le & \parallel x_n-u_{n-1}\parallel +\parallel u_{n-1}-u_n\parallel \\ =& \parallel {\alpha }_{n-1}\gamma f(x_{n-1})+(I-{\alpha }_{n-1}A)u_{n-1}-u_{n-1}\parallel +\parallel u_{n-1}-u_n\parallel \\ =& {\alpha }_{n-1}\parallel \gamma f(x_{n-1})-A{u}_{n-1}\parallel +\parallel u_{n-1}-u_n\parallel .\end{array}$$

(3.28)

From ${\alpha }_n\to 0$, we have $\parallel x_n-u_n\parallel \to 0$.

Now, for $v\in \mathfrak{F}$, using Lemma 2.5, we get that

$$\begin{array}{rl}{\parallel w_n-v\parallel }^2& ={\parallel F_{r_n}{x}_n-F_{r_n}v\parallel }^2\\ \le 〈F_{r_n}{x}_n-F_{r_n}v,x_n-v〉\\ =〈w_n-v,x_n-v〉\\ =\frac{1}{2}({\parallel w_n-v\parallel }^2+{\parallel x_n-v\parallel }^2-{\parallel x_n-w_n\parallel }^2),\end{array}$$

(3.29)

and hence

$${\parallel w_n-v\parallel }^2\le {\parallel x_n-v\parallel }^2-{\parallel x_n-w_n\parallel }^2.$$

(3.30)

Therefore, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-v\parallel }^2& =& {\parallel {\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)u_n-v\parallel }^2\\ =& {\parallel {\alpha }_n(\gamma f(x_n)-Av)+(I-{\alpha }_{n}A)(u_n-v)\parallel }^2\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel u_n-v\parallel }^2+2{\alpha }_n〈\gamma f(x_n)-Av,x_{n+1}-v〉\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel w_n-v\parallel }^2+2{\alpha }_n〈\gamma f(x_n)-\gamma f(v)+\gamma f(v)-Av,x_{n+1}-v〉\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel w_n-v\parallel }^2+2{\alpha }_n\gamma 〈f(x_n)-f(v),x_{n+1}-v〉\\ +2{\alpha }_n〈\gamma f(v)-Av,x_{n+1}-v〉\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel w_n-v\parallel }^2+2{\alpha }_n\gamma \alpha \parallel x_n-v\parallel \parallel x_{n+1}-v\parallel \\ +2{\alpha }_n\parallel \gamma f(v)-Av\parallel \parallel x_{n+1}-v\parallel \\ \le & {(1-{\alpha }_n\tau )}^2({\parallel x_n-v\parallel }^2-{\parallel x_n-w_n\parallel }^2)+2{\alpha }_n\gamma \alpha \parallel x_n-v\parallel \parallel x_{n+1}-v\parallel \\ +2{\alpha }_n\parallel \gamma f(v)-Av\parallel \parallel x_{n+1}-v\parallel \\ =& (1-2{\alpha }_n\tau +{({\alpha }_n\tau )}^2){\parallel x_n-v\parallel }^2-{(1-{\alpha }_n\tau )}^2{\parallel x_n-w_n\parallel }^2\\ +2{\alpha }_n\gamma \alpha \parallel x_n-v\parallel \parallel x_{n+1}-v\parallel +2{\alpha }_n\parallel \gamma f(v)-Av\parallel \parallel x_{n+1}-v\parallel .\end{array}$$

(3.31)

Hence

$$\begin{array}{r}{(1-{\alpha }_n\tau )}^2{\parallel x_n-w_n\parallel }^2\\ \le {\parallel x_n-v\parallel }^2-{\parallel x_{n+1}-v\parallel }^2+{\alpha }_n{\overline{\gamma }}^2{\parallel x_n-v\parallel }^2\\ +2{\alpha }_n\gamma \alpha \parallel x_n-v\parallel \parallel x_{n+1}-v\parallel +2{\alpha }_n\parallel \gamma f(v)-Av\parallel \parallel x_{n+1}-v\parallel \\ \le \parallel x_n-x_{n+1}\parallel \{\parallel x_n-v\parallel +\parallel x_{n+1}-v\parallel \}+{\alpha }_n{\tau }^2{\parallel x_n-v\parallel }^2\\ +2{\alpha }_n\gamma \alpha \parallel x_n-v\parallel \parallel x_{n+1}-v\parallel +2{\alpha }_n\parallel \gamma f(v)-Av\parallel \parallel x_{n+1}-v\parallel .\end{array}$$

(3.32)

So, we have

$$\parallel x_n-w_n\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

(3.33)

Since $\parallel u_n-w_n\parallel \le \parallel u_n-x_n\parallel +\parallel x_n-w_n\parallel$, it follows that

$$\parallel u_n-w_n\parallel \to 0\text{as }n\to \mathrm{\infty }.$$

Next, we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(A-\gamma f)z,z-x_n〉\le 0,$$

where $z=P_{\mathfrak{F}}(I-A+\gamma f)(z)$.

To show this equality, we choose a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(A-\gamma f)z,z-x_{n_i}〉=\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(A-\gamma f)z,z-x_n〉.$$

Since $\{x_{n_i}\}$ is bounded, there exists a subsequence $\{x_{n_{i_j}}\}$ of $\{x_{n_i}\}$ and $w\in H$ such that $x_{n_{i_j}}\rightharpoonup w$. Without loss of generality, we may assume that $x_{n_i}\rightharpoonup w$. Since $\{x_{n_i}\}\subset H$ and H is closed and convex, we get that $w\in H$. Moreover, since $x_n-w_n\to 0$ as $n\to \mathrm{\infty }$, we have that $w_{n_i}\rightharpoonup w$.

Now, we show that $w\in \mathfrak{F}$. Note that from the definition of $w_n$, we have

$$〈y-w_{n_i},T_2{w}_{n_i}〉+〈y-w_{n_i},\frac{w_{n_i}-x_{n_i}}{r_{n_i}}〉\ge 0\text{for all }y\in C.$$

(3.34)

Put $z_t=tv+(1-t)w$ for all $t\in (0,1]$ and $v\in H$. Consequently, we get that $z_t\in H$. From (3.34) it follows that

$$\begin{array}{rcl}〈z_t-w_{n_i},T_2{z}_t〉& \ge & 〈z_t-w_{n_i},T_2{z}_t〉-〈z_t-w_{n_i},T_2{w}_{n_i}〉-〈z_t-w_{n_i},\frac{w_{n_i}-x_{n_i}}{r_{n_i}}〉\\ =& 〈z_t-w_{n_i},T_2{z}_t-T_2{w}_{n_i}〉-〈z_t-w_{n_i},\frac{w_{n_i}-x_{n_i}}{r_{n_i}}〉.\end{array}$$

From the fact that $w_{n_i}-x_{n_i}\to 0$ as $n\to \mathrm{\infty }$, we obtain that $\frac{w_{n_i}-x_{n_i}}{r_{n_i}}\to 0$ as $n\to \mathrm{\infty }$.

Since $T_2$ is monotone, we have that $〈z_t-w_{n_i},T_2{z}_t-T_2{w}_{n_i}〉\ge 0$. Thus, it follows that

$$0\le \underset{i\to \mathrm{\infty }}{lim}〈z_t-w_{n_i},T_2{z}_t〉=〈z_t-w,T_2{z}_t〉,$$

and hence

$$〈z-w,T_2{z}_t〉\ge 0\text{for all }z\in C.$$

(3.35)

Letting $t\to 0$ and using the fact that $T_2$ is continuous, we obtain that

$$〈z-w,T_{2}w〉\ge 0\text{for all }v\in C.$$

(3.36)

This implies that $w\in \mathit{VI}(C,T_2)$.

Furthermore, from the definition of $u_{n_i}$ we have that

$$〈y-u_{n_i},T_2{u}_{n_i}〉-\frac{1}{r_{n_i}}〈y-u_{n_i},(r_{n_i}+1)u_{n_i}-x_{n_i}〉\le 0\text{for all }y\in C.$$

(3.37)

Put $z_t=tv+(1-t)w$ for all $t\in (0,1]$ and $v\in H$. Consequently, we get that $z_t\in H$. From (3.33) and pseudo-contractivity of $T_1$, it follows that

$$\begin{array}{rcl}〈u_{n_i}-z_t,T_1{z}_t〉& \ge & 〈u_{n_i}-z_t,T_1{z}_t〉+〈z_t-u_{n_i},T_1{u}_{n_i}〉-\frac{1}{r_{n_i}}〈z_t-u_{n_i},(r_{n_i}+1)u_{n_i}-w_{n_i}〉\\ =& -〈z_t-u_{n_i},T_1{z}_t-T_1{u}_{n_i}〉-\frac{1}{r_{n_i}}〈z_t-u_{n_i},u_{n_i}-w_{n_i}〉-〈z_t-u_{n_i},u_{n_i}〉\\ \ge & -{\parallel z_t-u_{n_i}\parallel }^2-\frac{1}{r_{n_i}}〈z_t-u_{n_i},u_{n_i}-w_{n_i}〉-〈z_t-u_{n_i},u_{n_i}〉\\ =& -〈z_t-u_{n_i},z_t〉-〈z_t-u_{n_i},\frac{u_{n_i}-w_{n_i}}{r_{n_i}}〉.\end{array}$$

(3.38)

Then, since $w_n-x_n\to 0$ as $n\to \mathrm{\infty }$, we obtain that $\frac{w_{n_i}-x_{n_i}}{r_{n_i}}\to 0$ as $i\to \mathrm{\infty }$. Thus, as $i\to \mathrm{\infty }$, it follows that

$$〈w-z_t,T_1{z}_t〉\ge 〈w-z_t,z_t〉,$$

(3.39)

and hence

$$-〈v-w,T_1{z}_t〉\ge -〈v-w,z_t〉\text{for all }v\in C.$$

(3.40)

Letting $t\to 0$ and using the fact that $T_1$ is continuous, we obtain that

$$-〈v-w,T_{1}w〉\ge -〈v-w,w〉\text{for all }v\in C.$$

(3.41)

Now, let $v=T_{1}w$. Then we obtain that $w=T_{1}w$ and hence $w\in F(T_1)$.

Therefore, $w\in F(T_1)\cap \mathit{VI}(C,T_2)$ and since $z=P_{\mathfrak{F}}(I-A+\gamma f)(z)$, by Lemma 2.2, implies that

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\gamma f-A)(z),x_n-z〉& =\underset{i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(I-A+\gamma f)(z),x_{n_i}-z〉\\ =〈(\gamma f-A)(z),w-z〉\le 0.\end{array}$$

(3.42)

Now, we show that $x_n\to z$ as $n\to \mathrm{\infty }$. From $x_{n+1}-z={\alpha }_n(\gamma f(x_n)-Az)+(I-{\alpha }_{n}A)(u_n-z)$, we have that

$$\begin{array}{rcl}{\parallel x_{n+1}-z\parallel }^2& =& {\parallel {\alpha }_n(\gamma f(x_n)-Az)+(I-{\alpha }_{n}A)(u_n-z)\parallel }^2\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel u_n-z\parallel }^2+2{\alpha }_n〈\gamma f(x_n)-Az,x_{n+1}-z〉\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel x_n-z\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-\gamma f(z)+\gamma f(z)-Az,x_{n+1}-z〉\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel x_n-z\parallel }^2+2{\alpha }_n\gamma 〈f(x_n)-f(z),x_{n+1}-z〉\\ +2{\alpha }_n〈\gamma f(z)-Az,x_{n+1}-z〉\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel x_n-z\parallel }^2+2{\alpha }_n\gamma \alpha \parallel x_n-z\parallel \parallel x_{n+1}-z\parallel \\ +2{\alpha }_n〈\gamma f(z)-Az,x_{n+1}-z〉\\ \le & {(1-{\alpha }_n\tau )}^2{\parallel x_n-z\parallel }^2+{\alpha }_n\gamma \alpha \{{\parallel x_n-z\parallel }^2+{\parallel x_{n+1}-z\parallel }^2\}\\ +2{\alpha }_n〈\gamma f(z)-Az,x_{n+1}-z〉\\ =& ({(1-{\alpha }_n\tau )}^2+{\alpha }_n\gamma \alpha ){\parallel x_n-z\parallel }^2+{\alpha }_n\gamma \alpha {\parallel x_{n+1}-z\parallel }^2\\ +2{\alpha }_n〈\gamma f(z)-Az,x_{n+1}-z〉.\end{array}$$

(3.43)

This implies that

$$\begin{array}{rcl}{\parallel x_{n+1}-z\parallel }^2& \le & \frac{1-2{\alpha }_n\tau +{({\alpha }_n\tau )}^2+{\alpha }_n\gamma \alpha }{1-{\alpha }_n\gamma \alpha }{\parallel x_n-z\parallel }^2\\ +\frac{2{\alpha }_n}{1-{\alpha }_n\gamma \alpha }〈\gamma f(z)-Az,x_{n+1}-z〉\\ =& [1-\frac{2(\tau -\gamma \alpha ){\alpha }_n}{1-{\alpha }_n\gamma \alpha }]{\parallel x_n-z\parallel }^2+\frac{{({\alpha }_n\tau )}^2}{1-{\alpha }_n\gamma \alpha }{\parallel x_n-z\parallel }^2\\ +\frac{2{\alpha }_n}{1-{\alpha }_n\gamma \alpha }〈\gamma f(z)-Az,x_{n+1}-z〉\\ \le & [1-\frac{2(\tau -\gamma \alpha ){\alpha }_n}{1-{\alpha }_n\gamma \alpha }]{\parallel x_n-z\parallel }^2\\ +\frac{2(\tau -\gamma \alpha ){\alpha }_n}{1-{\alpha }_n\gamma \alpha }\{\frac{({\alpha }_n\tau )M}{2(\overline{\gamma }-\gamma \alpha )}+\frac{1}{\tau -\gamma \alpha }〈\gamma f(z)-Az,x_{n+1}-z〉\}\\ =& (1-{\delta }_n){\parallel x_n-z\parallel }^2+{\delta }_n{\beta }_n,\end{array}$$

(3.44)

where , ${\delta }_n=\frac{2(\tau -\gamma \alpha ){\alpha }_n}{1-{\alpha }_n\gamma \alpha }$ and ${\beta }_n=\frac{({\alpha }_n\tau )M}{2(\tau -\gamma \alpha )}+\frac{1}{\tau -\gamma \alpha }〈\gamma f(z)-Az,x_{n+1}-z〉$. We put ${\xi }_n={\delta }_n{\beta }_n$. It is easy to see that ${\delta }_n\to 0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\delta }_n=\mathrm{\infty }$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\frac{{\xi }_n}{{\delta }_n}\le 0$ by (3.42). Hence, by Lemma 2.3, the sequence $\{x_n\}$ converges strongly to z. This completes the proof. □

Theorem 3.2 Let H be a real Hilbert space, $T_1:C\to H$ be a continuous pseudo-contractive mapping and $T_2:C\to H$ be a continuous monotone mapping such that $\mathfrak{F}:=F(T_1)\cap \mathit{VI}(C,T_2)\ne \mathrm{\varnothing }$. Let $0<\alpha <1$ and let f be an α-contraction of H into itself, and let A be a $\overline{\gamma }$-strongly monotone and L-Lipschitzian continuous operator of H into itself H with $\overline{\gamma }>0$ and $L>0$. Take μ, γ to be real numbers as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2},0<\gamma <\frac{\overline{\gamma }-\frac{L^2\mu }{2}}{\alpha }.$$

For $y_1\in H$, let $\{y_n\}$ be a sequence generated by (1.12), where $\{{\alpha }_n\}\subset [0,1]$ and $\{r_n\}\subset (0,\mathrm{\infty })$ are such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$. The sequence $\{y_n\}$ converges strongly to $z\in \mathfrak{F}$, where $z=P_{\mathfrak{F}}(I-A+\gamma f)(z)$.

Proof Let $\{x_n\}$ be the sequence given by $x_1=y_1$ and

$$x_{n+1}={\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T_{r_n}{F}_{r_n}{x}_n,\mathrm{\forall }n\ge 1.$$

From Theorem 3.1, $x_n\to z$. We claim that $y_n\to z$. Indeed, we estimate

$$\begin{array}{rcl}\parallel x_{n+1}-y_{n+1}\parallel & \le & \parallel {\alpha }_n\gamma f(x_n)-{\alpha }_n\gamma f(T_{r_n}{F}_{r_n}{y}_n)\parallel \\ +\parallel (I-{\alpha }_{n}A)T_{r_n}{F}_{r_n}{x}_n-(I-{\alpha }_{n}A)T_{r_n}{F}_{r_n}{y}_n\parallel \\ \le & {\alpha }_n\gamma \alpha \parallel T_{r_n}{F}_{r_n}{y}_n-x_n\parallel +(1-{\alpha }_n\tau )\parallel x_n-y_n\parallel \\ \le & {\alpha }_n\gamma \alpha \parallel T_{r_n}{F}_{r_n}{y}_n-T_{r_n}{F}_{r_n}z\parallel +{\alpha }_n\gamma \alpha \parallel T_{r_n}{F}_{r_n}z-x_n\parallel +(1-{\alpha }_n\tau )\parallel x_n-y_n\parallel \\ \le & {\alpha }_n\gamma \alpha \parallel y_n-z\parallel +{\alpha }_n\gamma \alpha \parallel z-x_n\parallel +(1-{\alpha }_n\tau )\parallel x_n-y_n\parallel \\ \le & {\alpha }_n\gamma \alpha \parallel y_n-x_n\parallel +{\alpha }_n\gamma \alpha \parallel x_n-z\parallel +{\alpha }_n\gamma \alpha \parallel z-x_n\parallel +(1-{\alpha }_n\tau )\parallel x_n-y_n\parallel \\ =& (1-{\alpha }_n(\tau -\gamma \alpha ))\parallel x_n-y_n\parallel +{\alpha }_n(\tau -\gamma \alpha )\frac{2\gamma \alpha }{\tau -\gamma \alpha }\parallel x_n-z\parallel .\end{array}$$

(3.45)

It follows from ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-z\parallel =0$ and Lemma 2.3 that $\parallel x_n-y_n\parallel \to 0$.

Consequently, $y_n\to z$ as required. □

Theorem 3.3 Let H be a real Hilbert space, $T_1:C\to H$ be a continuous pseudo-contractive mapping and $T_2:C\to H$ be a continuous monotone mapping such that $\mathfrak{F}:=F(T_1)\cap \mathit{VI}(C,T_2)\ne \mathrm{\varnothing }$. Let $0<\alpha <1$ and let f be an α-contraction of H into itself, and let A be a $\overline{\gamma }$-strongly monotone and L-Lipschitzian continuous operator of H into itself H with $\overline{\gamma }>0$ and $L>0$. Take μ, γ to be real numbers as follows:

$$0<\mu <\frac{2\overline{\gamma }}{L^2},0<\gamma <\frac{\overline{\gamma }-\frac{L^2\mu }{2}}{\alpha }.$$

For $z_1\in H$, let $\{z_n\}$ be a sequence generated by (1.13), where $\{{\alpha }_n\}\subset [0,1]$ and $\{r_n\}\subset (0,\mathrm{\infty })$ are such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${\sum }_{n=1}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${\sum }_{n=1}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$. The sequence $\{z_n\}$ converges strongly to $z\in \mathfrak{F}$, where $z=P_{\mathfrak{F}}(I-A+\gamma f)(z)$.

Proof Define the sequences $\{y_n\}$ and $\{{\beta }_n\}$ by

$$y_n={\alpha }_n\gamma f(z_n)+(I-{\alpha }_{n}A)z_n\text{and}{\beta }_n={\alpha }_{n+1},\mathrm{\forall }n\ge 1.$$

Taking $p\in \mathfrak{F}$, we have

$$\begin{array}{rcl}\parallel z_{n+1}-p\parallel & =& \parallel T_{r_n}{F}_{r_n}{y}_n-T_{r_n}{F}_{r_n}p\parallel \le \parallel y_n-p\parallel \\ =& \parallel {\alpha }_n\gamma f(z_n)+(I-{\alpha }_{n}A)z_n-(I-{\alpha }_{n}A)p-{\alpha }_{n}A(p)\parallel \\ \le & (1-{\alpha }_n\tau )\parallel z_n-p\parallel +{\alpha }_n\parallel \gamma f(z_n)-A(p)\parallel \\ =& (1-{\alpha }_n\tau )\parallel z_n-p\parallel +{\alpha }_n\tau \frac{\parallel \gamma f(z_n)-A(p)\parallel }{\tau }.\end{array}$$

(3.46)

It follows from induction that

$$\parallel z_{n+1}-p\parallel \le max\{\parallel z_1-p\parallel ,\frac{\parallel \gamma f(z_1)-A(p)\parallel }{\tau }\},n\ge 1.$$

(3.47)

Thus both $\{z_n\}$ and $\{y_n\}$ are bounded. We observe that

$$y_{n+1}={\alpha }_{n+1}\gamma f(z_{n+1})+(I-{\alpha }_{n+1}A)z_{n+1}={\beta }_n\gamma f(T_{r_n}{F}_{r_n}{y}_n)+(I-{\beta }_{n}A)T_{r_n}{F}_{r_n}{y}_n.$$

Thus Theorem 3.2 implies that $\{y_n\}$ converges to some point z. In this case, we also have

$$\parallel z_n-z\parallel \le \parallel z_n-y_n\parallel +\parallel y_n-z\parallel ={\alpha }_n\parallel \gamma f(z_n)-A(z_n)\parallel +\parallel y_n-z\parallel \to 0.$$

Hence the sequence $\{z_n\}$ converges to some point z. This completes the proof. □

Setting $\gamma =1$, $A\equiv I$, where I is the identity mapping in Theorem 3.1, we have the following result.

Corollary 3.4 Let H be a real Hilbert space, $T_1:C\to C$ be a continuous pseudo-contractive mapping and $T_2:C\to H$ be a continuous monotone mapping such that $\mathfrak{F}:=F(T_1)\cap \mathit{VI}(C,T_2)\ne \mathrm{\varnothing }$. Let f be a contraction of H into itself and let $\{x_n\}$ be a sequence generated by $x_1\in H$ and

$$x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)F_{r_n}{T}_{r_n}{x}_n,$$

(3.48)

where $\{{\alpha }_n\}\subset [0,1]$ and $\{r_n\}\subset (0,\mathrm{\infty })$ are such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${\sum }_{n=0}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$. The sequence $\{x_n\}$ converges strongly to $z\in \mathfrak{F}$, where $z=P_{\mathfrak{F}}f(z)$.

In Theorem 3.1, $\gamma =1$, $A\equiv I$, $f:=u\in H$ is a constant mapping, then we get $z=P_{\mathfrak{F}}(u)$. In fact, we have the following corollary.

Corollary 3.5 Let H be a real Hilbert space, $T_1:C\to C$ be a continuous pseudo-contractive mapping and $T_2:C\to H$ be a continuous monotone mapping such that $\mathfrak{F}:=F(T_1)\cap \mathit{VI}(C,T_2)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be a sequence generated by $x_1,u\in H$ and

$$x_{n+1}={\alpha }_{n}u+(1-{\alpha }_n)F_{r_n}{T}_{r_n}{x}_n,$$

(3.49)

where $\{{\alpha }_n\}\subset [0,1]$ and $\{r_n\}\subset (0,\mathrm{\infty })$ are such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${\sum }_{n=0}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$. The sequence $\{x_n\}$ converges strongly to $z\in \mathfrak{F}$, where $z=P_{\mathfrak{F}}(u)$.

In Theorem 3.1, $\gamma =1$ and $A,T_1\equiv I$, where I is the identity mapping, then we have the following corollary.

Corollary 3.6 Let H be a real Hilbert space and $T_2:C\to H$ be a continuous monotone mapping such that $\mathit{VI}(C,T_2)\ne \mathrm{\varnothing }$. Let f be a contraction of H into itself, and let $\{x_n\}$ be a sequence generated by $x_1\in H$ and

$$x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)T_{r_n}{x}_n,$$

(3.50)

where $\{{\alpha }_n\}\subset [0,1]$ and $\{r_n\}\subset (0,\mathrm{\infty })$ are such that ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$, ${\sum }_{n=0}^{\mathrm{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\mathrm{\infty }$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{r}_n>0$ and ${\sum }_{n=0}^{\mathrm{\infty }}|r_{n+1}-r_n|<\mathrm{\infty }$. The sequence $\{x_n\}$ converges strongly to $z\in F(T_1)$, where $z=P_{F(T_1)}(z)$.

Remark 3.7 Our results extend and unify most of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 3.1 extends Theorem 3.1 of Iiduka and Takahashi [12] and Zegeye et al. [25], Corollary 3.2 of Su et al. [29] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings. Corollary 3.4 also extends Theorem 4.2 of Iiduka and Takahashi [12] in the sense that our convergence is for the more general class of continuous pseudo-contractive and continuous monotone mappings.