Strong convergence theorem for total quasi-ϕ-asymptotically nonexpansive mappings in a Banach space

Abstract

In this paper, we prove strong convergence theorems to a point which is a fixed point of multi-valued mappings, a zero of an α-inverse-strongly monotone operator and a solution of the equilibrium problem. Next, we obtain strong convergence theorems to a solution of the variational inequality problem, a fixed point of multi-valued mappings and a solution of the equilibrium problem. The results presented in this paper are improvement and generalization of the previously known results.

1 Introduction

Let E be a real Banach space with dual $E^{\ast }$, and let C be a nonempty closed convex subset of E. Let $A:C\to E^{\ast }$ be an operator. A is called monotone if

$$〈Ax-Ay,x-y〉\ge 0,\mathrm{\forall }x,y\in C;$$

α-inverse-strongly monotone if there exists a constant $\alpha >0$ such that

$$〈Ax-Ay,x-y〉\ge \alpha {\parallel Ax-Ay\parallel }^2,\mathrm{\forall }x,y\in C;$$

L-Lipschitz continuous if there exists a constant $L>0$ such that

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

If A is α-inverse strongly monotone, then it is $\frac{1}{\alpha }$-Lipschitz continuous, i.e.,

$$\parallel Ax-Ay\parallel \le \frac{1}{\alpha }\parallel x-y\parallel ,\mathrm{\forall }x,y\in C.$$

A monotone operator A is said to be maximal if its graph $G(A)=\{(x,x^{\ast }):x^{\ast }\in Ax\}$ is not properly contained in the graph of any other monotone operator.

Let A be a monotone operator. We consider the problem of finding $x\in E$ such that

$$0\in Ax,$$

(1.1)

a point $x\in E$ is called a zero point of A. Denote by $A^{-1}0$ the set of all points $x\in E$ such that $0\in Ax$. This problem is very important in optimization theory and related fields.

Let A be a monotone operator. The classical variational inequality problem for an operator A is to find $\hat{z}\in C$ such that

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

(1.2)

The set of solutions of (1.2) is denoted by $\mathit{VI}(A,C)$. This problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point $x\in E$ satisfying $Ax=0$.

The value of $x^{\ast }\in E^{\ast }$ at $x\in E$ will be denoted by $〈x,x^{\ast }〉$ or $x^{\ast }(x)$. For each $p>1$, the generalized duality mapping $J_p:E\to 2^{E^{\ast }}$ is defined by

$$J_p(x)=\{x^{\ast }\in E^{\ast }:〈x,x^{\ast }〉={\parallel x\parallel }^p,\parallel x^{\ast }\parallel ={\parallel x\parallel }^{p-1}\}$$

for all $x\in E$. In particular, $J=J_2$ is called the normalized duality mapping. If E is a Hilbert space, then $J=I$, where I is the identity mapping.

Consider the functional defined by

$$\varphi (y,x)={\parallel y\parallel }^2-2〈y,Jx〉+{\parallel x\parallel }^2\text{for }x,y\in E,$$

(1.3)

where J is the normalized duality mapping. It is obvious from the definition of ϕ that

$${(\parallel y\parallel -\parallel x\parallel )}^2\le \varphi (y,x)\le {(\parallel y\parallel +\parallel x\parallel )}^2,\mathrm{\forall }x,y\in E.$$

(1.4)

Alber [1] introduced that the generalized projection ${\mathrm{\Pi }}_C:E\to C$ is a map that assigns to an arbitrary point $x\in E$ the minimum point of the functional $\varphi (x,y)$, that is, ${\mathrm{\Pi }}_{C}x=\overline{x}$, where $\overline{x}$ is the solution of the minimization problem

$$\varphi (\overline{x},x)=\underset{y\in C}{inf}\varphi (y,x),$$

(1.5)

existence and uniqueness of the operator ${\mathrm{\Pi }}_C$ follows from the properties of the functional $\varphi (x,y)$ and strict monotonicity of the mapping J.

Iiduka and Takahashi [2] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E: $x_1=x\in C$ and

$$x_{n+1}={\mathrm{\Pi }}_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\mathrm{\forall }n\ge 1,$$

(1.6)

where ${\mathrm{\Pi }}_C$ is the generalized projection from E onto C, J is the duality mapping from E into $E^{\ast }$ and $\{{\lambda }_n\}$ is a sequence of positive real numbers. They proved that the sequence $\{x_n\}$ generated by (1.6) converges weakly to some element of $\mathit{VI}(A,C)$. In connection, Iiduka and Takahashi [3] studied the following iterative scheme for finding a zero point of a monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E:

$$\{\begin{array}{c}{x}_1=x\in E\text{chosen arbitrarily},\hfill \\ y_n=J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ X_n=\{z\in E:\varphi (z,y_n)\le \varphi (z,x_n)\},\hfill \\ Y_{n+1}=\{z\in E:〈x_n-z,Jx-J{x}_n〉\ge 0\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{X_n\cap Y_n}(x),\hfill \end{array}$$

(1.7)

where ${\mathrm{\Pi }}_{X_n\cap Y_n}$ is the generalized projection from E onto $X_n\cap Y_n$, J is the duality mapping from E into $E^{\ast }$ and $\{{\lambda }_n\}$ is a sequence of positive real numbers. They proved that the sequence $\{x_n\}$ converges strongly to an element of $A^{-1}0$. Moreover, under the additional suitable assumption they proved that the sequence $\{x_n\}$ converges strongly to some element of $\mathit{VI}(A,C)$. Some solution methods have been proposed to solve the variational inequality problem; see, for instance, [4–6].

A mapping $T:C\to C$ is said to be ϕ-nonexpansive [7, 8] if

$$\varphi (Tx,Ty)\le \varphi (x,y),\mathrm{\forall }x,y\in C.$$

T is said to be quasi-ϕ-nonexpansive [7, 8] if $F(T)\ne \mathrm{\varnothing }$ and

$$\varphi (p,Tx)\le \varphi (p,x),\mathrm{\forall }x\in C,p\in F(T).$$

T is said to be total quasi-ϕ-asymptotically nonexpansive, if $F(T)\ne \mathrm{\varnothing }$ and there exist nonnegative real sequences ${\nu }_n$, ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\phi (0)=0$ such that

$$\varphi (p,T^{n}x)\le \varphi (p,x)+{\nu }_n\phi (\varphi (p,x))+{\mu }_n,\mathrm{\forall }n\ge 1,\mathrm{\forall }x\in C,p\in F(T).$$

Let $2^C$ be the family of all nonempty subsets of C, and let $S:C\to 2^C$ be a multi-valued mapping. For a point $q\in C$, $n\ge 1$ define an iterative sequence as follows:

$$\begin{array}{c}Sq:=\{q_1:q_1\in Sq\},\hfill \\ S^{2}q=SSq:=\bigcup _{q_1\in Sq}S{q}_1,\hfill \\ S^{3}q=S{S}^{2}q:=\bigcup _{q_2\in T^{2}q}S{q}_2,\hfill \\ ⋮\hfill \\ S^{n}q=S{S}^{n-1}q:=\bigcup _{q_{n-1}\in S^{n-1}q}S{q}_{n-1}.\hfill \end{array}$$

A point $p\in C$ is said to be an asymptotic fixed point of S if there exists a sequence $\{x_n\}$ in C such that $\{x_n\}$ converges weakly to p and

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S{x}_n):=\underset{n\to \mathrm{\infty }}{lim}\underset{x\in S{x}_n}{inf}\parallel x_n-x\parallel =0.$$

The asymptotic fixed point set of S is denoted by $\hat{F}(S)$.

A multi-valued mapping S is said to be total quasi-ϕ-asymptotically nonexpansive if $F(S)\ne \mathrm{\varnothing }$ and there exist nonnegative real sequences ${\nu }_n$, ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\phi (0)=0$ such that for all $x\in C$, $p\in F(S)$,

$$\varphi (p,w_n)\le \varphi (p,x)+{\nu }_n\phi (\varphi (p,x))+{\mu }_n,\mathrm{\forall }n\ge 1,w_n\in S^{n}x.$$

S is said to be closed if for any sequence $\{x_n\}$ and $\{w_n\}$ in C with $w_n\in S{x}_n$ if $x_n\to x$ and $w_n\to w$, then $w\in Sx$.

A multi-valued mapping S is said to be uniformly asymptotically regular on C if

$$\underset{n\to \mathrm{\infty }}{lim}(\underset{x\in C}{sup}\parallel s_{n+1}-s_n\parallel )=0,s_n\in S^{n}x.$$

Every quasi-ϕ-asymptotically nonexpansive multi-valued mapping implies a quasi-ϕ-asymptotically nonexpansive mapping but the converse is not true.

In 2012, Chang et al. [9] introduced the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and then proved some strong convergence theorem by using the hybrid shrinking projection method.

Let $f:C\times C\to \mathbb{R}$ be a bifunction, the equilibrium problem is to find $x\in C$ such that

$$f(x,y)\ge 0,\mathrm{\forall }y\in C.$$

(1.8)

The set of solutions of (1.8) is denoted by $\mathit{EP}(f)$. The equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, min-max problems, saddle point problem, fixed point problem, Nash EP. In 2008, Takahashi and Zembayashi [10, 11] introduced iterative sequences for finding a common solution of an equilibrium problem and a fixed point problem. Some solution methods have been proposed to solve the equilibrium problem; see, for instance, [12–21].

For a mapping $A:C\to E^{\ast }$, let $f(x,y)=〈Ax,y-x〉$ for all $x,y\in C$. Then $x\in \mathit{EP}(f)$ if and only if $〈Tx,y-x〉\ge 0$ for all $y\in C$; i.e., x is a solution of the variational inequality.

Motivated and inspired by the work mentioned above, in this paper, we introduce and prove strong convergence of a new hybrid projection algorithm for a fixed point of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings, the solution of the equilibrium problem, a zero point of monotone operators. Moreover, we prove strong convergence to the solution of the variation inequality in a uniformly smooth and 2-uniformly convex Banach space.

2 Preliminaries

A Banach space E with the norm $\parallel \cdot \parallel$ is called strictly convex if $\parallel \frac{x+y}{2}\parallel <1$ for all $x,y\in E$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. Let $U=\{x\in E:\parallel x\parallel =1\}$ be the unit sphere of E. A Banach space E is called smooth if the limit ${lim}_{t\to 0}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$ exists for each $x,y\in U$. It is also called uniformly smooth if the limit exists uniformly for all $x,y\in U$. The modulus of convexity of E is the function $\delta :[0,2]\to [0,1]$ defined by

$$\delta (\epsilon )=inf\{1-\parallel \frac{x+y}{2}\parallel :x,y\in E,\parallel x\parallel =\parallel y\parallel =1,\parallel x-y\parallel \ge \epsilon \}.$$

A Banach space E is uniformly convex if and only if $\delta (\epsilon )>0$ for all $\epsilon \in (0,2]$. Let p be a fixed real number with $p\ge 2$. A Banach space E is said to be p-uniformly convex if there exists a constant $c>0$ such that $\delta (\epsilon )\ge c{\epsilon }^p$ for all $\epsilon \in [0,2]$. Observe that every p-uniform convex is uniformly convex. Every uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence $\{x_n\}\subset E$, if $x_n\rightharpoonup x\in E$ and $\parallel x_n\parallel \to \parallel x\parallel$, then $x_n\to x$.

Let E be a real Banach space with dual $E^{\ast }$, E is uniformly smooth if and only if $E^{\ast }$ is a uniformly convex Banach space. If E is a uniformly smooth Banach space, then E is a smooth and reflexive Banach space.

Remark 2.1

• If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

• If E is reflexive smooth and strictly convex, then the normalized duality mapping J is single-valued, one-to-one and onto.

• If E is a reflexive strictly convex and smooth Banach space and J is the duality mapping from E into $E^{\ast }$, then $J^{-1}$ is also single-valued, bijective and is also the duality mapping from $E^{\ast }$ into E and thus $J{J}^{-1}=I_{E^{\ast }}$ and $J^{-1}J=I_E$.

See [22] for more details.

Remark 2.2 If E is a reflexive, strictly convex and smooth Banach space, then $\varphi (x,y)=0$ if and only if $x=y$. It is sufficient to show that if $\varphi (x,y)=0$, then $x=y$. From (1.3) we have $\parallel x\parallel =\parallel y\parallel$. This implies that $〈x,Jy〉={\parallel x\parallel }^2={\parallel Jy\parallel }^2$. From the definition of J, one has $Jx=Jy$. Therefore, we have $x=y$ (see [22, 23] for more details).

Lemma 2.3 (Beauzamy [24] and Xu [25])

If E is a 2-uniformly convex Banach space, then, for all $x,y\in E$, we have

$$\parallel x-y\parallel \le \frac{2}{c^2}\parallel Jx-Jy\parallel ,$$

where J is the normalized duality mapping of E and $0<c\le 1$.

The best constant $\frac{1}{c}$ in the lemma is called the p-uniformly convex constant of E.

Lemma 2.4 (Beauzamy [24] and Zalinescu [26])

If E is a p-uniformly convex Banach space, and let p be a given real number with $p\ge 2$, then, for all $x,y\in E$, $J_x\in J_p(x)$ and $J_y\in J_p(y)$,

$$〈x-y,J_x-J_y〉\ge \frac{c^p}{2^{p-2}p}{\parallel x-y\parallel }^p,$$

where $J_p$ is the generalized duality mapping of E and $\frac{1}{c}$ is the p-uniformly convex constant of E.

Lemma 2.5 (Kamimura and Takahashi [27])

Let E be a uniformly convex and smooth Banach space, and let $\{x_n\}$, $\{y_n\}$ be two sequences of E. If $\varphi (x_n,y_n)\to 0$ and either $\{x_n\}$ or $\{y_n\}$ is bounded, then $\parallel x_n-y_n\parallel \to 0$.

Lemma 2.6 (Alber [1])

Let C be a nonempty closed convex subset of a smooth Banach space E, and let $x\in E$. Then $x_0={\mathrm{\Pi }}_{C}x$ if and only if

$$〈x_0-y,Jx-J{x}_0〉\ge 0,\mathrm{\forall }y\in C.$$

Lemma 2.7 (Alber [1])

Let E be a reflexive strictly convex and smooth Banach space, C be a nonempty closed convex subset of E, and let $x\in E$. Then

$$\varphi (y,{\mathrm{\Pi }}_{C}x)+\varphi ({\mathrm{\Pi }}_{C}x,x)\le \varphi (y,x),\mathrm{\forall }y\in C.$$

Lemma 2.8 (Chang et al. [9])

Let C be a nonempty, closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let $S:C\to 2^C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequence ${\nu }_n$ and ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\phi (0)=0$. If ${\mu }_1=0$, then the fixed point set $F(S)$ is a closed convex subset of C.

For solving the equilibrium problem for a bifunction $f:C\times C\to \mathbb{R}$, let us assume that f satisfies the following conditions:

• (A1) $f(x,x)=0$ for all $x\in C$;

• (A2) f is monotone, i.e., $f(x,y)+f(y,x)\le 0$ for all $x,y\in C$;

• (A3) for each $x,y,z\in C$,

  $$\underset{t\downarrow 0}{lim}f(tz+(1-t)x,y)\le f(x,y);$$

• (A4) for each $x\in C$, $y\mapsto f(x,y)$ is convex and lower semi-continuous.

Lemma 2.9 (Blum and Oettli [28])

Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4), and let $r>0$ and $x\in E$. Then there exists $z\in C$ such that

$$f(z,y)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\mathrm{\forall }y\in C.$$

Lemma 2.10 (Takahashi and Zembayashi [11])

Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E, and let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)-(A4). For all $r>0$ and $x\in E$, define a mapping $T_r:E\to C$ as follows:

$$T_{r}x=\{z\in C:f(z,y)+\frac{1}{r}〈y-z,Jz-Jx〉\ge 0,\mathrm{\forall }y\in C\}.$$

Then the following hold:

• (1) $T_r$ is single-valued;

• (2) $T_r$ is a firmly nonexpansive-type mapping [29], that is, for all $x,y\in E$,

  $$〈T_{r}x-T_{r}y,J{T}_{r}x-J{T}_{r}y〉\le 〈T_{r}x-T_{r}y,Jx-Jy〉;$$

• (3) $F(T_r)=\mathit{EP}(f)$;

• (4) $\mathit{EP}(f)$ is closed and convex.

Lemma 2.11 (Takahashi and Zembayashi [11])

Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from $C\times C$ to ℝ satisfying (A1)-(A4), and let $r>0$. Then, for $x\in E$ and $q\in F(T_r)$,

$$\varphi (q,T_{r}x)+\varphi (T_{r}x,x)\le \varphi (q,x).$$

Let A be an inverse-strongly monotone mapping of C into $E^{\ast }$ which is said to be hemicontinuous if for all $x,y\in C$, the mapping h of $[0,1]$ into $E^{\ast }$, defined by $h(t)=A(tx+(1-t)y)$, is continuous with respect to the weak^∗ topology of $E^{\ast }$. We define by $N_C(v)$ the normal cone for C at a point $v\in C$, that is,

$$N_C(v)=\{x^{\ast }\in E^{\ast }:〈v-y,x^{\ast }〉\ge 0,\mathrm{\forall }y\in C\}.$$

(2.1)

Theorem 2.12 (Rockafellar [30])

Let C be a nonempty, closed convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into $E^{\ast }$. Let $B\subset E\times E^{\ast }$ be an operator defined as follows:

$$Bv=\{\begin{array}{cc}Av+N_C(v),\hfill & v\in C;\hfill \\ \mathrm{\varnothing },\hfill & \mathit{\text{otherwise}}.\hfill \end{array}$$

(2.2)

Then B is maximal monotone and $B^{-1}0=\mathit{VI}(A,C)$.

Theorem 2.13 (Takahashi [31])

Let C be a nonempty subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into $E^{\ast }$ with $C=D(A)$. Then

$$\mathit{VI}(A,C)=\{u\in C:〈v-u,Av〉\ge 0,\mathrm{\forall }v\in C\}.$$

(2.3)

It is obvious that the set $\mathit{VI}(A,C)$ is a closed and convex subset of C and the set $A^{-1}0=\mathit{VI}(A,E)$ is a closed and convex subset of E.

Theorem 2.14 (Takahashi [31])

Let C be a nonempty compact convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into $E^{\ast }$ with $C=D(A)$. Then $\mathit{VI}(A,C)$ is nonempty.

We make use of the following mapping V studied in Alber [1]:

$$V(x,x^{\ast })={\parallel x\parallel }^2-2〈x,x^{\ast }〉+{\parallel x^{\ast }\parallel }^2,\mathrm{\forall }x\in E,x^{\ast }\in E^{\ast },$$

(2.4)

that is, $V(x,x^{\ast })=\varphi (x,J^{-1}(x^{\ast }))$.

Lemma 2.15 (Alber [1])

Let E be a reflexive strictly convex smooth Banach space, and let V be as in (2.4). Then we have

$$V(x,x^{\ast })+2〈J^{-1}\left(x^{\ast }\right)-x,y^{\ast }〉\le V(x,x^{\ast }+y^{\ast }),\mathrm{\forall }x\in E,x^{\ast },y^{\ast }\in E^{\ast }.$$

Lemma 2.16 (Beauzamy [24] and Xu [25])

If E is a 2-uniformly convex Banach space, then, for all $x,y\in E$, we have

$$\parallel x-y\parallel \le \frac{2}{c^2}\parallel Jx-Jy\parallel ,$$

where J is the normalized duality mapping of E and $0<c\le 1$.

Lemma 2.17 (Cho et al. [32])

Let E be a uniformly convex Banach space, and let $B_r(0)=\{x\in E:\parallel x\parallel \le r\}$ be a closed ball of E. Then there exists a continuous strictly increasing convex function $g:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $g(0)=0$ such that

$${\parallel \lambda x+\mu y+\gamma z\parallel }^2\le {\parallel \lambda x\parallel }^2+{\parallel \mu y\parallel }^2+{\parallel \gamma z\parallel }^2-\lambda \mu g(\parallel x-y\parallel )$$

for all $x,y,z\in B_r(0)$ and $\lambda ,\mu ,\gamma \in [0,1]$ with $\lambda +\mu +\gamma =1$.

Lemma 2.18 (Pascali and Sburlan [33])

Let E be a real smooth Banach space, and let $A:E\to 2^{E^{\ast }}$ be a maximal monotone mapping. Then $A^{-1}0$ is a closed and convex subset of E and the graph $G(A)$ of A is demiclosed in the following sense: if $\{x_n\}\subset D(A)$ with $x_n\rightharpoonup x\in E$ and $y_n\in A{x}_n$ with $y_n\to y\in E^{\ast }$, then $x\in D(A)$ and $y\in Ax$.

3 Main results

Theorem 3.1 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of E into $E^{\ast }$. Let $S:C\to 2^C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ${\nu }_n$, ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\psi (0)=0$. Assume that S is uniformly asymptotically regular on C with ${\mu }_1=0$ and $F:=F(S)\cap \mathit{EP}(f)\cap A^{-1}0\ne \mathrm{\varnothing }$. For arbitrary $x_1\in C$, $C_1=C$, generate a sequence $\{x_n\}$ by

$$\{\begin{array}{c}{z}_n=J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ u_n=T_{r_n}{z}_n,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n),w_n\in S^n{x}_n,\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,z_n)\le \varphi (v,x_n)+K_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,n\in \mathbb{N},\hfill \end{array}$$

(3.1)

where $K_n={\nu }_n{sup}_{q\in F}\psi (\varphi (q,x_n))+{\mu }_n$. Assume that the control sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{{\lambda }_n\}$ and $\{r_n\}$ satisfy the following conditions:

• 1. $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\gamma }_n\}$ are sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$,

• 2. $\{{\lambda }_n\}\subset [a,b]$ for some a, b with $0<a<b<\frac{c^2\alpha }{2}$ and $\frac{1}{c}$ is the 2-uniformly convex constant of E,

• 3. $\{r_n\}\subset [d,\mathrm{\infty })$ for some $d>0$,

then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_F{x}_1$.

Proof We will show that $C_n$ is closed and convex for all $n\in \mathbb{N}$. Since $C_1=C$ is closed and convex. Suppose that $C_n$ is closed and convex for all $n\in \mathbb{N}$. For any $v\in C_n$, we know that $\varphi (v,y_n)\le \varphi (v,x_n)+K_n$ is equivalent to

$$2〈v,J{x}_n-J{y}_n〉\le {\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2+K_n.$$

That is, $C_{n+1}$ is closed and convex, hence $C_n$ is closed and convex for all $n\in \mathbb{N}$.

We show by induction that $F\subset C_n$ for all $n\in \mathbb{N}$. It is obvious that $F\subset C=C_1$. Suppose that $F\subset C_n$ where $n\in \mathbb{N}$. Let $q\in F$, we have

$$\begin{array}{rcl}\varphi (q,z_n)& =& \varphi (q,J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n))\\ =& V(q,J{x}_n-{\lambda }_{n}A{x}_n)\\ \le & V(q,(J{x}_n-{\lambda }_{n}A{x}_n)+{\lambda }_{n}A{x}_n)-2〈J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)-q,{\lambda }_{n}A{x}_n〉\\ =& V(q,J{x}_n)-2{\lambda }_n〈J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)-q,A{x}_n〉\\ =& \varphi (q,x_n)-2{\lambda }_n〈x_n-q,A{x}_n〉+2〈J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)-x_n,-{\lambda }_{n}A{x}_n〉.\end{array}$$

(3.2)

Since A is an α-inverse-strongly monotone mapping, we get

$$\begin{array}{rcl}-2{\lambda }_n〈x_n-q,A{x}_n〉& =& -2{\lambda }_n〈x_n-q,A{x}_n-Aq〉-2{\lambda }_n〈x_n-q,Aq〉\\ \le & -2{\lambda }_n〈x_n-q,A{x}_n-Aq〉\\ =& -2\alpha {\lambda }_n{\parallel A{x}_n-Aq\parallel }^2.\end{array}$$

(3.3)

It follows from Lemma 2.17 that

$$\begin{array}{rcl}2〈J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)-x_n,-{\lambda }_{n}A{x}_n〉& =& 2〈J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)-J^{-1}(J{x}_n),-{\lambda }_{n}A{x}_n〉\\ \le & 2\parallel J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)-J^{-1}(J{x}_n)\parallel \parallel {\lambda }_{n}A{x}_n\parallel \\ \le & \frac{4}{c^2}\parallel J{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)-J{J}^{-1}(J{x}_n)\parallel \parallel {\lambda }_{n}A{x}_n\parallel \\ =& \frac{4}{c^2}\parallel J{x}_n-{\lambda }_{n}A{x}_n-J{x}_n\parallel \parallel {\lambda }_{n}A{x}_n\parallel \\ =& \frac{4}{c^2}{\parallel {\lambda }_{n}A{x}_n\parallel }^2\\ =& \frac{4}{c^2}{\lambda }_n^2{\parallel A{x}_n\parallel }^2\\ \le & \frac{4}{c^2}{\lambda }_n^2{\parallel A{x}_n-Aq\parallel }^2.\end{array}$$

(3.4)

Replacing (3.2) by (3.3) and (3.4), we get

$$\begin{array}{rcl}\varphi (q,z_n)& \le & \varphi (q,x_n)-2\alpha {\lambda }_n{\parallel A{x}_n-Aq\parallel }^2+\frac{4}{c^2}{\lambda }_n^2{\parallel A{x}_n-Aq\parallel }^2\\ =& \varphi (q,x_n)+2{\lambda }_n(\frac{2}{c^2}{\lambda }_n-\alpha ){\parallel A{x}_n-Aq\parallel }^2\\ \le & \varphi (q,x_n).\end{array}$$

(3.5)

From Lemma 2.11, we know that

$$\varphi (q,u_n)=\varphi (q,T_{r_n}{z}_n)\le \varphi (q,z_n)\le \varphi (q,x_n).$$

(3.6)

Since S is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and $w_n\in S^n{x}_n$, it follows that

$$\begin{array}{rcl}\varphi (q,y_n)& =& \varphi (q,J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n))\\ =& {\parallel q\parallel }^2-2〈q,{\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n〉+{\parallel {\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n\parallel }^2\\ \le & {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,w_n)+{\gamma }_n\varphi (q,u_n)\\ \le & {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,x_n)+{\beta }_n{\nu }_n\psi (\varphi (q,x_n))+{\beta }_n{\mu }_n+{\gamma }_n\varphi (q,u_n)\\ \le & {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,x_n)+{\nu }_n\underset{q\in F}{sup}\psi (\varphi (q,x_n))+{\mu }_n+{\gamma }_n\varphi (q,u_n)\\ =& {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,x_n)+{\gamma }_n\varphi (q,u_n)+K_n\\ \le & {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,x_n)+{\gamma }_n\varphi (q,T_{r_n}{z}_n)+K_n\\ \le & {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,x_n)+{\gamma }_n\varphi (q,z_n)+K_n\\ \le & {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,x_n)+{\gamma }_n\varphi (q,x_n)+K_n\\ \le & \varphi (q,x_n)+K_n,\end{array}$$

(3.7)

where $K_n={\nu }_n{sup}_{q\in F}\psi (\varphi (q,x_n))+{\mu }_n$.

This shows that $q\in C_{n+1}$, which implies that $F\subset C_{n+1}$. Hence $F\subset C_n$ for all $n\in \mathbb{N}$ and the sequence $\{x_n\}$ is well defined.

From the definition of $C_{n+1}$ with $x_n={\mathrm{\Pi }}_{C_n}{x}_1$ and $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1\in C_{n+1}\subset C_n$, it follows that

$$\varphi (x_n,x_1)\le \varphi (x_{n+1},x_1),\mathrm{\forall }n\ge 1,$$

(3.8)

that is, $\{\varphi (x_n,x_1)\}$ is nondecreasing. By Lemma 2.7, we get

$$\begin{array}{rcl}\varphi (x_n,x_1)& =& \varphi ({\mathrm{\Pi }}_{C_n}{x}_1,x_1)\\ \le & \varphi (q,x_1)-\varphi (q,x_n)\\ \le & \varphi (q,x_1),\mathrm{\forall }q\in F.\end{array}$$

(3.9)

This implies that $\{\varphi (x_n,x_1)\}$ is bounded and so ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_1)$ exists. In particular, by (1.4), the sequence $\{{(\parallel x_n\parallel -\parallel x_1\parallel )}^2\}$ is bounded. This implies $\{x_n\}$ is also bounded. So, we have $\{u_n\}$, $\{z_n\}$ and $\{y_n\}$ are also bounded.

Since $x_m={\mathrm{\Pi }}_{C_m}{x}_1\in C_m\subset C_n$ for all $m,n\ge 1$ with $m>n$, by Lemma 2.7, we have

$$\begin{array}{rcl}\varphi (x_m,x_n)& =& \varphi (x_m,{\mathrm{\Pi }}_{C_n}{x}_1)\\ \le & \varphi (x_m,x_1)-\varphi ({\mathrm{\Pi }}_{C_n}{x}_1,x_1)\\ =& \varphi (x_m,x_1)-\varphi (x_n,x_1),\end{array}$$

taking $m,n\to \mathrm{\infty }$, we have $\varphi (x_m,x_n)\to 0$. This implies that $\{x_n\}$ is a Cauchy sequence. From Lemma 2.5, it follows that $\parallel x_n-x_m\parallel \to 0$ and $\{x_n\}$ is a Cauchy sequence. By the completeness of E and the closedness of C, we can assume that there exists $p\in C$ such that

$$\underset{n\to \mathrm{\infty }}{lim}{x}_n=p,$$

(3.10)

we also get that

$$\underset{n\to \mathrm{\infty }}{lim}{K}_n=\underset{n\to \mathrm{\infty }}{lim}{\nu }_n\underset{q\in F}{sup}\psi (\varphi (q,x_n))+{\mu }_n=0.$$

(3.11)

Next, we show that $p\in F:=F(S)\cap A^{-1}0\cap \mathit{EP}(f)$.

(a) We show that $p\in F(S)$. By the definition of ${\mathrm{\Pi }}_{C_n}{x}_1$, we have

$$\begin{array}{rcl}\varphi (x_{n+1},x_n)& =& \varphi (x_{n+1},{\mathrm{\Pi }}_{C_n}{x}_1)\\ \le & \varphi (x_{n+1},x_1)-\varphi ({\mathrm{\Pi }}_{C_n}{x}_1,x_1)\\ =& \varphi (x_{n+1},x_1)-\varphi (x_n,x_1).\end{array}$$

Since ${lim}_{n\to \mathrm{\infty }}\varphi (x_n,x_1)$ exists, we get

$$\underset{n\to \mathrm{\infty }}{lim}\varphi (x_{n+1},x_n)=0.$$

(3.12)

It follows from Lemma 2.5 that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0.$$

(3.13)

From the definition of $C_{n+1}$ and $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1\in C_{n+1}\subset C_n$, we have $\varphi (x_{n+1},y_n)\le \varphi (x_{n+1},x_n)+K_n\to 0$ as $n\to \mathrm{\infty }$. By Lemma 2.5, it follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-y_n\parallel =0.$$

(3.14)

From ${lim}_{n\to \mathrm{\infty }}{x}_n=p$, we also have

$$\underset{n\to \mathrm{\infty }}{lim}{y}_n=p.$$

(3.15)

By using the triangle inequality, we get $\parallel x_n-y_n\parallel \le \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-y_n\parallel \to 0$ as $n\to \mathrm{\infty }$. Since J is uniformly norm-to-norm continuous, we obtain $\parallel J{x}_n-J{y}_n\parallel \to 0$ as $n\to \mathrm{\infty }$. On the other hand, we note that

$$\begin{array}{rcl}\varphi (q,x_n)-\varphi (q,y_n)& =& {\parallel x_n\parallel }^2-{\parallel y_n\parallel }^2-2〈q,J{x}_n-J{y}_n〉\\ \le & \parallel x_n-y_n\parallel (\parallel x_n+y_n\parallel )+2\parallel q\parallel \parallel J{x}_n-J{y}_n\parallel .\end{array}$$

In view of $\parallel x_n-y_n\parallel \to 0$ and $\parallel J{x}_n-J{y}_n\parallel \to 0$ as $n\to \mathrm{\infty }$, we obtain that

$$\varphi (q,x_n)-\varphi (q,y_n)\to 0\text{as }n\to \mathrm{\infty }.$$

(3.16)

From Lemma 2.17, we have

$$\begin{array}{rcl}\varphi (q,y_n)& =& \varphi (q,J^{-1}[{\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n])\\ \le & {\parallel q\parallel }^2-2〈q,{\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n〉+{\parallel {\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n\parallel }^2\\ -{\alpha }_n{\beta }_{n}g(\parallel J{x}_n-J{w}_n\parallel )\\ =& {\alpha }_n\varphi (q,x_n)+{\beta }_n\varphi (q,w_n)+{\gamma }_n\varphi (q,u_n)-{\alpha }_n{\beta }_{n}g(\parallel J{x}_n-J{w}_n\parallel )\\ \le & \varphi (q,x_n)+K_n-{\alpha }_n{\beta }_{n}g(\parallel J{x}_n-J{w}_n\parallel ).\end{array}$$

(3.17)

It follows from ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$, (3.16), (3.11) and the property of g that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{x}_n-J{w}_n\parallel =0.$$

Since $J^{-1}$ is uniformly norm-to-norm continuous, we obtain

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

(3.18)

From (3.10) it follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel w_n-p\parallel =0.$$

(3.19)

For $w_n\in S^n{x}_n$, generate a sequence $\{s_n\}$ by

$$\begin{array}{c}{s}_2\in S{w}_1\subset S^2{x}_1,\hfill \\ s_3\in S{w}_2\subset S^3{x}_2,\hfill \\ s_4\in S{w}_3\subset S^4{x}_3,\hfill \\ ⋮\hfill \\ s_{n+1}\in S{w}_n\subset S^{n+1}{x}_n.\hfill \end{array}$$

On the other hand, we have $\parallel s_{n+1}-p\parallel \le \parallel s_{n+1}-w_n\parallel +\parallel w_n-p\parallel$. Since S is uniformly asymptotically regular, it follows that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel s_{n+1}-p\parallel =0,$$

(3.20)

we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel S^{n+1}{x}_n-p\parallel =0,$$

(3.21)

that is, $S{S}^n{x}_n\to p$ as $n\to \mathrm{\infty }$. From the closedness of S, we have $p\in F(S)$.

(b) We show that $p\in A^{-1}0$.

From the definition of $C_{n+1}$ and $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1\in C_{n+1}\subset C_n$, we have $\varphi (x_{n+1},z_n)\le \varphi (x_{n+1},x_n)+K_n\to 0$ as $n\to \mathrm{\infty }$. By Lemma 2.5, it follows that ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-z_n\parallel =0$. By the triangle inequality, we get $\parallel x_n-z_n\parallel \le \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-z_n\parallel \to 0$ as $n\to \mathrm{\infty }$. From ${lim}_{n\to \mathrm{\infty }}\parallel z_n-x_n\parallel =0$ and from (3.10), it follows that

$$\underset{n\to \mathrm{\infty }}{lim}{z}_n=p.$$

(3.22)

Since J is uniformly norm-to-norm continuous, we also have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{z}_n-J{x}_n\parallel =0.$$

(3.23)

Hence, from the definition of the sequence $\{z_n\}$, it follows that

$$\parallel A{x}_n\parallel =\frac{\parallel J{z}_n-J{x}_n\parallel }{{\lambda }_n}.$$

(3.24)

From (3.23) and the definition of the sequence $\{{\lambda }_n\}$, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel A{x}_n\parallel =0,$$

(3.25)

that is,

$$\underset{n\to \mathrm{\infty }}{lim}A{x}_n=0.$$

(3.26)

Since A is Lipschitz continuous, it follows from (3.10) that

$$Ap=0.$$

(3.27)

Again, since A is Lipschitz continuous and monotone so it is maximal monotone. It follows from Lemma 2.18 that $p\in A^{-1}0$.

(c) We show that $p\in \mathit{EP}(f)$.

From $x_n,y_n\to 0$ and $K_n\to 0$ as $n\to \mathrm{\infty }$ and applying (3.7) for any $q\in F$, we get ${lim}_{n\to \mathrm{\infty }}\varphi (q,u_n)\to \varphi (q,p)$, it follows that

$$\begin{array}{rcl}\varphi (u_n,x_n)& =& \varphi (T_{r_n},x_n)\\ \le & \varphi (q,x_n)-\varphi (q,T_{r_n}{x}_n)\\ =& \varphi (q,x_n)-\varphi (q,u_n).\end{array}$$

Taking limit as $n\to \mathrm{\infty }$ on the both sides of the inequality, we have ${lim}_{n\to \mathrm{\infty }}\varphi (u_n,x_n)=0$. From Lemma 2.5, it follows that

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

(3.28)

and

$$\underset{n\to \mathrm{\infty }}{lim}{u}_n=p.$$

(3.29)

Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel J{u}_n-J{z}_n\parallel =0.$$

Since $r_n>0$ for all $n\ge 1$, we have $\frac{\parallel J{u}_n-J{z}_n\parallel }{r_n}\to 0$ as $n\to \mathrm{\infty }$ and

$$f(u_n,y)+\frac{1}{r_n}〈y-u_n,J{u}_n-J{z}_n〉\ge 0,\mathrm{\forall }y\in C.$$

From (A2), the fact that

$$\begin{array}{rcl}\parallel y-u_n\parallel \frac{\parallel J{u}_n-J{z}_n\parallel }{r_n}& \ge & \frac{1}{r_n}〈y-u_n,J{u}_n-J{z}_n〉\\ \ge & -f(u_n,y)\\ \ge & f(y,u_n),\mathrm{\forall }y\in C,\end{array}$$

taking the limit as $n\to \mathrm{\infty }$ in the above inequality and from the fact that $u_n\to p$ as $n\to \mathrm{\infty }$, it follows that $f(y,p)\le 0$ for all $y\in C$. For any $0<t<1$, define $y_t=ty+(1-t)p$. Then $y_t\in C$, which implies that $f(y_t,p)\le 0$. Thus it follows from (A1) that

$$0=f(y_t,y_t)\le tf(y_t,y)+(1-t)\theta (y_t,p)\le tf(y_t,y),$$

and so $f(y_t,y)\ge 0$. From (A3) we have $f(p,y)\ge 0$ for all $y\in C$ and so $p\in \mathit{EP}(f)$. Hence, by (a), (b) and (c), that is, $p\in F(S)\cap A^{-1}0\cap \mathit{EP}(f)$.

Finally, we show that $p={\mathrm{\Pi }}_F{x}_1$. From $x_n={\mathrm{\Pi }}_{C_n}{x}_1$, we have $〈J{x}_1-J{x}_n,x_n-z〉\ge 0$ for all $z\in C_n$. Since $F\subset C_n$, we also have

$$〈J{x}_1-J{x}_n,x_n-\hat{p}〉\ge 0,\mathrm{\forall }\hat{p}\in F.$$

Taking limit $n\to \mathrm{\infty }$, we obtain

$$〈J{x}_1-Jp,p-\hat{p}〉\ge 0,\mathrm{\forall }\hat{p}\in F.$$

By Lemma 2.6, we can conclude that $p={\mathrm{\Pi }}_F{x}_1$ and $x_n\to p$ as $n\to \mathrm{\infty }$. The proof is completed. □

Next, we define $z_n={\mathrm{\Pi }}_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)$ and assume that $\parallel Ay\parallel \le \parallel Ay-Au\parallel$ for all $y\in C$ and $u\in \mathit{VI}(A,C)\ne \mathrm{\varnothing }$. We can prove the strong convergence theorem for finding the set of solutions of the variational inequality problem in a real uniformly smooth and 2-uniformly convex Banach space.

Remark 3.2 (Qin et al. [7])

Let ${\mathrm{\Pi }}_C$ be the generalized projection from a smooth strictly convex and reflexive Banach space E onto a nonempty closed convex subset C of E. Then ${\mathrm{\Pi }}_C$ is a closed quasi-ϕ-nonexpansive mapping from E onto C with $F({\mathrm{\Pi }}_C)=C$.

Corollary 3.3 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of C into $E^{\ast }$ satisfying $\parallel Ay\parallel \le \parallel Ay-Au\parallel$ for all $y\in C$ and $u\in \mathit{VI}(A,C)\ne \mathrm{\varnothing }$. Let $S:C\to 2^C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ${\nu }_n$, ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\psi (0)=0$. Assume that S is uniformly asymptotically regular on C with ${\mu }_1=0$ and $F:=F(S)\cap \mathit{EP}(f)\cap \mathit{VI}(A,C)\ne \mathrm{\varnothing }$. For arbitrary $x_1\in C$, $C_1=C$, generate a sequence $\{x_n\}$ by

$$\{\begin{array}{c}{z}_n={\mathrm{\Pi }}_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ u_n=T_{r_n}{x}_n,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n),w_n\in S^n{x}_n,\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,z_n)\le \varphi (v,x_n)+K_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,n\in \mathbb{N},\hfill \end{array}$$

(3.30)

where $K_n={\nu }_n{sup}_{q\in F}\psi (\varphi (q,x_n))+{\mu }_n$. Assume that the control sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{{\lambda }_n\}$ and $\{r_n\}$ satisfy the following conditions:

• 1. $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\gamma }_n\}$ are sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$,

• 2. $\{{\lambda }_n\}\subset [a,b]$ for some a, b with $0<a<b<\frac{c^2\alpha }{2}$ and $\frac{1}{c}$ is the 2-uniformly convex constant of E,

• 3. $\{r_n\}\subset [d,\mathrm{\infty })$ for some $d>0$,

then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_F{x}_1$.

Proof For $q\in F$ and ${\mathrm{\Pi }}_C$ is quasi-ϕ-nonexpansive mapping, we have

$$\varphi (q,z_n)=\varphi (q,{\mathrm{\Pi }}_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n))\le \varphi (q,J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)).$$

So, we can show that $p\in \mathit{VI}(A,C)$.

Define $B\subset E\times E^{\ast }$ by Theorem 2.14, B is maximal monotone and $B^{-1}0=\mathit{VI}(A,C)$. Let $(z,w)\in G(B)$. Since $w\in Bz=Az+N_C(z)$, we get $w-Az\in N_C(z)$.

From $z_n\in C$, we have

$$〈z-z_n,w-Az〉\ge 0.$$

(3.31)

On the other hand, since $z_n={\mathrm{\Pi }}_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n)$. Then, by Lemma 2.6, we have

$$〈z-z_n,J{z}_n-(J{x}_n-{\lambda }_{n}A{x}_n)〉\ge 0,$$

and thus

$$〈z-z_n,\frac{J{x}_n-J{z}_n}{{\lambda }_n}-A{x}_n〉\le 0.$$

(3.32)

It follows from (3.31) and (3.32) that

$$\begin{array}{rcl}〈z-z_n,w〉& \ge & 〈z-z_n,Az〉\\ \ge & 〈z-z_n,Az〉+〈z-z_n,\frac{J{x}_n-J{z}_n}{{\lambda }_n}-A{x}_n〉\\ =& 〈z-z_n,Az-A{x}_n〉+〈z-z_n,\frac{J{x}_n-J{z}_n}{{\lambda }_n}〉\\ =& 〈z-z_n,Az-A{z}_n〉+〈z-z_n,A{z}_n-A{x}_n〉+〈z-z_n,\frac{J{x}_n-J{z}_n}{{\lambda }_n}〉\\ \ge & -\parallel z-z_n\parallel \frac{\parallel z_n-x_n\parallel }{\alpha }-\parallel z-z_n\parallel \frac{\parallel J{x}_n-J{z}_n\parallel }{a}\\ \ge & -M(\frac{\parallel z_n-x_n\parallel }{\alpha }+\frac{\parallel J{x}_n-J{z}_n\parallel }{a}),\end{array}$$

where $M={sup}_{n\ge 1}\parallel z-z_n\parallel$. From $\parallel x_n-z_n\parallel \to 0$ as $n\to \mathrm{\infty }$ and (3.23), taking ${lim}_{n\to \mathrm{\infty }}$ on the both sides of the equality above, we have $〈z-p,w〉\ge 0$. By the maximality of B, we have $p\in B^{-1}0$, that is, $p\in \mathit{VI}(A,C)$. From Theorem 3.1, we have $p\in F(S)\cap \mathit{EP}(f)\cap \mathit{VI}(A,C)$. The proof is completed. □

Let A be a strongly monotone mapping with constant k, Lipschitz with constant $L>0$, that is,

$$\parallel Ax-Ay\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in D(A),$$

which implies that

$$\frac{1}{L}\parallel Ax-Ay\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in D(A).$$

It follows that

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

hence A is α-inverse-strongly monotone with $\alpha =\frac{k}{L}$. Therefore, we have the following corollaries.

Corollary 3.4 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)-(A4), and let $A:E\to E^{\ast }$ be a strongly monotone mapping with constant k, Lipschitz with constant $L>0$. Let $S:C\to 2^C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ${\nu }_n$, ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\psi (0)=0$. Assume that S is uniformly asymptotically regular on C with ${\mu }_1=0$ and $F:=F(S)\cap \mathit{EP}(f)\cap A^{-1}0\ne \mathrm{\varnothing }$. For arbitrary $x_1\in C$, $C_1=C$, a sequence $\{x_n\}$ is generated by

$$\{\begin{array}{c}{z}_n=J^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ u_n=T_{r_n}{z}_n,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n),w_n\in S^n{x}_n,\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,z_n)\le \varphi (v,x_n)+K_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,n\in \mathbb{N},\hfill \end{array}$$

(3.33)

where $K_n={\nu }_n{sup}_{q\in F}\psi (\varphi (q,x_n))+{\mu }_n$. Assume that the control sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{{\lambda }_n\}$ and $\{r_n\}$ satisfy the following conditions:

• 1. $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\gamma }_n\}$ are sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$,

• 2. $\{{\lambda }_n\}\subset [a,b]$ for some a, b with $0<a<b<\frac{c^{2}k}{2L}$ and $\frac{1}{c}$ is the 2-uniformly convex constant of E,

• 3. $\{r_n\}\subset [d,\mathrm{\infty })$ for some $d>0$,

then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_F{x}_1$.

Corollary 3.5 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)-(A4), and let $A:C\to E^{\ast }$ be a strongly monotone mapping with constant k,Lipschitz with constant $L>0$ satisfying $\parallel Ay\parallel \le \parallel Ay-Au\parallel$ for all $y\in C$ and $u\in \mathit{VI}(A,C)\ne \mathrm{\varnothing }$. Let $S:C\to 2^C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ${\nu }_n$, ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\psi (0)=0$. Assume that S is uniformly asymptotically regular on C with ${\mu }_1=0$ and $F:=F(S)\cap \mathit{EP}(f)\cap \mathit{VI}(A,C)\ne \mathrm{\varnothing }$. For arbitrary $x_1\in C$, $C_1=C$, generate a sequence $\{x_n\}$ by

$$\{\begin{array}{c}{z}_n={\mathrm{\Pi }}_C{J}^{-1}(J{x}_n-{\lambda }_{n}A{x}_n),\hfill \\ u_n=T_{r_n}{x}_n,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n),w_n\in S^n{x}_n,\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,z_n)\le \varphi (v,x_n)+K_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,n\in \mathbb{N},\hfill \end{array}$$

(3.34)

where $K_n={\nu }_n{sup}_{q\in F}\psi (\varphi (q,x_n))+{\mu }_n$. Assume that the control sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{{\lambda }_n\}$ and $\{r_n\}$ satisfy the following conditions:

• 1. $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\gamma }_n\}$ are sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$,

• 2. $\{{\lambda }_n\}\subset [a,b]$ for some a, b with $0<a<b<\frac{c^{2}k}{2L}$ and $\frac{1}{c}$ is the 2-uniformly convex constant of E,

• 3. $\{r_n\}\subset [d,\mathrm{\infty })$ for some $d>0$,

then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_F{x}_1$.

Let F be a Fréchet differentiable functional in a Banach space E and ∇F be the gradient of F, denote ${(\mathrm{\nabla }F)}^{-1}0=\{x\in E:F(x)={min}_{y\in E}F(y)\}$. Baillon and Haddad [34] proved the following lemma.

Lemma 3.6 (Baillon and Haddad [34])

Let E be a Banach space. Let F be a continuously Fréchet differentiable convex functional on E and ∇F be the gradient of F. If ∇F is $\frac{1}{\alpha }$-Lipschitz continuous, then ∇F is an α-inverse strongly monotone mapping.

We replace A in Theorem 3.1 by ∇F, then we can obtain the following corollary.

Corollary 3.7 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)-(A4). Let F be a continuously Fréchet differentiable convex functional on E and ∇F be $\frac{1}{\alpha }$-Lipschitz continuous. Let $S:C\to 2^C$ be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ${\nu }_n$, ${\mu }_n$ with ${\nu }_n\to 0$, ${\mu }_n\to 0$ as $n\to \mathrm{\infty }$ and a strictly increasing continuous function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with $\psi (0)=0$. Assume that S is uniformly asymptotically regular on C with ${\mu }_1=0$ and $F:=F(S)\cap F(T)\cap \mathit{EP}(f)\cap A^{-1}0\ne \mathrm{\varnothing }$. For an initial point $x_1\in E$, $C_1=C$, define the sequence $\{x_n\}$ by

$$\{\begin{array}{c}{z}_n=J^{-1}(J{x}_n-{\lambda }_n\mathrm{\nabla }F{x}_n),\hfill \\ u_n=T_{r_n}{x}_n,\hfill \\ y_n=J^{-1}({\alpha }_{n}J{x}_n+{\beta }_{n}J{w}_n+{\gamma }_{n}J{u}_n),w_n\in S^n{x}_n,\hfill \\ C_{n+1}=\{v\in C_n:\varphi (v,y_n)\le \varphi (v,z_n)\le \varphi (v,x_n)+K_n\},\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_1,n\in \mathbb{N},\hfill \end{array}$$

(3.35)

where ${\mu }_n=sup\{{\mu }_n^S,{\mu }_n^T\}$, ${\nu }_n=sup\{{\nu }_n^S,{\nu }_n^T\}$, $\psi =sup\{{\psi }^S,{\psi }^T\}$, $k_n={\nu }_n{sup}_{q\in \mathcal{F}}\psi (\varphi (q,x_n))+{\mu }_n$.

Assume that the control sequences $\{{\alpha }_n\}$, $\{{\beta }_n\}$, $\{{\gamma }_n\}$, $\{{\lambda }_n\}$ and $\{r_n\}$ satisfy the following conditions:

• 1. $\{{\alpha }_n\}$, $\{{\beta }_n\}$ and $\{{\gamma }_n\}$ are sequences in $(0,1)$ such that ${\alpha }_n+{\beta }_n+{\gamma }_n=1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\beta }_n>0$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_n{\gamma }_n>0$,

• 2. $\{{\lambda }_n\}\subset [a,b]$ for some a, b with $0<a<b<\frac{c^2\alpha }{2}$ and the 2-uniformly convex constant $\frac{1}{c}$ of E,

• 3. $\{r_n\}\subset [d,\mathrm{\infty })$ for some $d>0$,

then $\{x_n\}$ converges strongly to ${\mathrm{\Pi }}_F{x}_1$.