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Strong convergence theorem for total quasiϕasymptotically nonexpansive mappings in a Banach space
 Siwaporn Saewan^{1}Email author
https://doi.org/10.1186/168718122013297
© Saewan; licensee Springer. 2013
 Received: 11 July 2013
 Accepted: 9 October 2013
 Published: 11 November 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.
Keywords
 total quasiϕasymptotically nonexpansive multivalued mappings
 hybrid scheme
 equilibrium problem
 variational inequality problems
 inversestrongly monotone operator
1 Introduction
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.
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.
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 $Ax=0$.
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.
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.
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].
The asymptotic fixed point set of S is denoted by $\stackrel{\u02c6}{F}(S)$.
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 $w\in Sx$.
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.
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 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])
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])
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])
Lemma 2.7 (Alber [1])
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.
 (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])
Lemma 2.10 (Takahashi and Zembayashi [11])
 (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])
Theorem 2.12 (Rockafellar [30])
Then B is maximal monotone and ${B}^{1}0=\mathit{VI}(A,C)$.
Theorem 2.13 (Takahashi [31])
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.
that is, $V(x,{x}^{\ast})=\varphi (x,{J}^{1}({x}^{\ast}))$.
Lemma 2.15 (Alber [1])
Lemma 2.16 (Beauzamy [24] and Xu [25])
where J is the normalized duality mapping of E and $0<c\le 1$.
Lemma 2.17 (Cho et al. [32])
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
 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 ${\mathrm{\Pi}}_{F}{x}_{1}$.
That is, ${C}_{n+1}$ is closed and convex, hence ${C}_{n}$ is closed and convex for all $n\in \mathbb{N}$.
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.
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.
Next, we show that $p\in F:=F(S)\cap {A}^{1}0\cap \mathit{EP}(f)$.
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 $p\in {A}^{1}0$.
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 $p\in \mathit{EP}(f)$.
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, $p\in F(S)\cap {A}^{1}0\cap \mathit{EP}(f)$.
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 $F({\mathrm{\Pi}}_{C})=C$.
 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 ${\mathrm{\Pi}}_{F}{x}_{1}$.
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 $wAz\in {N}_{C}(z)$.
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. □
hence A is αinversestrongly monotone with $\alpha =\frac{k}{L}$. Therefore, we have the following corollaries.
 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 ${\mathrm{\Pi}}_{F}{x}_{1}$.
 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.
where ${\mu}_{n}=sup\{{\mu}_{n}^{S},{\mu}_{n}^{T}\}$, ${\nu}_{n}=sup\{{\nu}_{n}^{S},{\nu}_{n}^{T}\}$, $\psi =sup\{{\psi}^{S},{\psi}^{T}\}$, ${k}_{n}={\nu}_{n}{sup}_{q\in \mathcal{F}}\psi (\varphi (q,{x}_{n}))+{\mu}_{n}$.
 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 ${\mathrm{\Pi}}_{F}{x}_{1}$.
Declarations
Acknowledgements
This work was supported by Thaksin University Research Fund. Moreover, the author also would like to thank Faculty of Science, Thaksin University.
Authors’ Affiliations
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