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Strong convergence theorem for total quasiϕasymptotically nonexpansive mappings in a Banach space
Fixed Point Theory and Applications volume 2013, Article number: 297 (2013)
Abstract
In this paper, we prove strong convergence theorems to a point which is a fixed point of multivalued mappings, a zero of an αinversestrongly 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 multivalued 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
αinversestrongly monotone if there exists a constant $\alpha >0$ such that
LLipschitz continuous if there exists a constant $L>0$ such that
If A is αinverse strongly monotone, then it is $\frac{1}{\alpha}$Lipschitz continuous, i.e.,
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
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 $\stackrel{\u02c6}{z}\in C$ such that
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 $\u3008x,{x}^{\ast}\u3009$ or ${x}^{\ast}(x)$. For each $p>1$, the generalized duality mapping ${J}_{p}:E\to {2}^{{E}^{\ast}}$ is defined by
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
where J is the normalized duality mapping. It is obvious from the definition of ϕ that
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
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 inversestrongly monotone operator A in a 2uniformly convex and uniformly smooth Banach space E: ${x}_{1}=x\in C$ and
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 2uniformly convex and uniformly smooth Banach space E:
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 ϕnonexpansive [7, 8] if
T is said to be quasiϕnonexpansive [7, 8] if $F(T)\ne \mathrm{\varnothing}$ and
T is said to be total quasiϕasymptotically 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
Let ${2}^{C}$ be the family of all nonempty subsets of C, and let $S:C\to {2}^{C}$ be a multivalued mapping. For a point $q\in C$, $n\ge 1$ define an iterative sequence as follows:
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
The asymptotic fixed point set of S is denoted by $\stackrel{\u02c6}{F}(S)$.
A multivalued mapping S is said to be total quasiϕasymptotically 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 $x\in C$, $p\in F(S)$,
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 multivalued mapping S is said to be uniformly asymptotically regular on C if
Every quasiϕasymptotically nonexpansive multivalued mapping implies a quasiϕasymptotically nonexpansive mapping but the converse is not true.
In 2012, Chang et al. [9] introduced the concept of total quasiϕasymptotically nonexpansive multivalued 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
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, minmax 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 $f(x,y)=\u3008Ax,yx\u3009$ for all $x,y\in C$. Then $x\in \mathit{EP}(f)$ if and only if $\u3008Tx,yx\u3009\ge 0$ 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 quasiϕasymptotically nonexpansive multivalued 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 2uniformly 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
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 puniformly 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 puniform convex is uniformly convex. Every uniformly convex Banach space E has the KadecKlee 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 normtonorm continuous on each bounded subset of E.

If E is reflexive smooth and strictly convex, then the normalized duality mapping J is singlevalued, onetoone 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 singlevalued, 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 $\u3008x,Jy\u3009={\parallel x\parallel}^{2}={\parallel Jy\parallel}^{2}$. 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 2uniformly convex Banach space, then, for all $x,y\in E$, we have
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 puniformly convex constant of E.
Lemma 2.4 (Beauzamy [24] and Zalinescu [26])
If E is a puniformly 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 ${J}_{y}\in {J}_{p}(y)$,
where ${J}_{p}$ is the generalized duality mapping of E and $\frac{1}{c}$ is the puniformly 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
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
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 KadecKlee property. Let $S:C\to {2}^{C}$ be a closed and total quasiϕasymptotically nonexpansive multivalued 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 $x,y,z\in C$,
$$\underset{t\downarrow 0}{lim}f(tz+(1t)x,y)\le f(x,y);$$ 
(A4)
for each $x\in C$, $y\mapsto f(x,y)$ is convex and lower semicontinuous.
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
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:
Then the following hold:

(1)
${T}_{r}$ is singlevalued;

(2)
${T}_{r}$ is a firmly nonexpansivetype mapping [29], that is, for all $x,y\in E$,
$$\u3008{T}_{r}x{T}_{r}y,J{T}_{r}xJ{T}_{r}y\u3009\le \u3008{T}_{r}x{T}_{r}y,JxJy\u3009;$$ 
(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 $q\in F({T}_{r})$,
Let A be an inversestrongly 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 $h(t)=A(tx+(1t)y)$, 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,
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:
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
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]:
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
Lemma 2.16 (Beauzamy [24] and Xu [25])
If E is a 2uniformly convex Banach space, then, for all $x,y\in E$, we have
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
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 2uniformly convex Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)(A4), and let A be an αinversestrongly monotone mapping of E into ${E}^{\ast}$. Let $S:C\to {2}^{C}$ be a closed and total quasiϕasymptotically nonexpansive multivalued 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
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 2uniformly 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
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
Since A is an αinversestrongly monotone mapping, we get
It follows from Lemma 2.17 that
Replacing (3.2) by (3.3) and (3.4), we get
From Lemma 2.11, we know that
Since S is a total quasiϕasymptotically nonexpansive multivalued mapping and ${w}_{n}\in {S}^{n}{x}_{n}$, it follows that
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
that is, $\{\varphi ({x}_{n},{x}_{1})\}$ is nondecreasing. By Lemma 2.7, we get
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
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
we also get that
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
Since ${lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})$ exists, we get
It follows from Lemma 2.5 that
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
From ${lim}_{n\to \mathrm{\infty}}{x}_{n}=p$, we also have
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 normtonorm 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
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
From Lemma 2.17, we have
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
Since ${J}^{1}$ is uniformly normtonorm continuous, we obtain
From (3.10) it follows that
For ${w}_{n}\in {S}^{n}{x}_{n}$, generate a sequence $\{{s}_{n}\}$ by
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
we have
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
Since J is uniformly normtonorm continuous, we also have
Hence, from the definition of the sequence $\{{z}_{n}\}$, it follows that
From (3.23) and the definition of the sequence $\{{\lambda}_{n}\}$, we have
that is,
Since A is Lipschitz continuous, it follows from (3.10) that
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
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
and
Since J is uniformly normtonorm continuous on bounded subsets of E, we obtain
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
From (A2), the fact that
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 ${y}_{t}=ty+(1t)p$. Then ${y}_{t}\in C$, which implies that $f({y}_{t},p)\le 0$. Thus it follows from (A1) that
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 $\u3008J{x}_{1}J{x}_{n},{x}_{n}z\u3009\ge 0$ for all $z\in {C}_{n}$. Since $F\subset {C}_{n}$, we also have
Taking limit $n\to \mathrm{\infty}$, we obtain
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 AyAu\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 2uniformly 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 quasiϕnonexpansive 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 2uniformly convex Banach space E. Let f be a bifunction from $C\times C$ to ℝ satisfying conditions (A1)(A4), and let A be an αinversestrongly monotone mapping of C into ${E}^{\ast}$ satisfying $\parallel Ay\parallel \le \parallel AyAu\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 quasiϕasymptotically nonexpansive multivalued 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
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 2uniformly 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 quasiϕnonexpansive mapping, we have
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 $wAz\in {N}_{C}(z)$.
From ${z}_{n}\in C$, we have
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
and thus
It follows from (3.31) and (3.32) that
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 $\u3008zp,w\u3009\ge 0$. 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,
which implies that
It follows that
hence A is αinversestrongly 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 2uniformly 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 quasiϕasymptotically nonexpansive multivalued 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
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 2uniformly 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 2uniformly 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 AyAu\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 quasiϕasymptotically nonexpansive multivalued 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
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 2uniformly 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 αinverse 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 2uniformly 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 quasiϕasymptotically nonexpansive multivalued 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
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 2uniformly convex constant $\frac{1}{c}$ of E,

3.
$\{{r}_{n}\}\subset [d,\mathrm{\infty})$ for some $d>0$,
then $\{{x}_{n}\}$ converges strongly to ${\mathrm{\Pi}}_{F}{x}_{1}$.
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Acknowledgements
This work was supported by Thaksin University Research Fund. Moreover, the author also would like to thank Faculty of Science, Thaksin University.
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Saewan, S. Strong convergence theorem for total quasiϕasymptotically nonexpansive mappings in a Banach space. Fixed Point Theory Appl 2013, 297 (2013). https://doi.org/10.1186/168718122013297
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Keywords
 total quasiϕasymptotically nonexpansive multivalued mappings
 hybrid scheme
 equilibrium problem
 variational inequality problems
 inversestrongly monotone operator