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 operator1 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
References
 Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos A. Dekker, New York; 1996:15–50.Google Scholar
 Iiduka H, Takahashi W: Weak convergence of a projection algorithm for variational inequalities in a Banach space. J. Math. Anal. Appl. 2008, 339: 668–679. 10.1016/j.jmaa.2007.07.019MathSciNetView ArticleGoogle Scholar
 Iiduka H, Takahashi W: Strong convergence studied by a hybrid type method for monotone operators in a Banach space. Nonlinear Anal. 2008, 68: 3679–3688. 10.1016/j.na.2007.04.010MathSciNetView ArticleGoogle Scholar
 Ceng LC, Latif A, Yao JC: On solutions of system of variational inequalities and fixed point problems in Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 176 10.1186/168718122013176Google Scholar
 Kangtunyakarn A: A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudocontractive mappings and two sets of variational inequalities in uniformly convex and 2smooth Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 157 10.1186/168718122013157Google Scholar
 Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002MathSciNetView ArticleGoogle Scholar
 Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleGoogle Scholar
 Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi ϕ asymptotically nonexpansive mappings. J. Appl. Math. Comput. 2010, 32(2):453–464. 10.1007/s1219000902634MathSciNetView ArticleGoogle Scholar
 Chang SS, Wang L, Tang YK, Zhao YH, Ma ZL: Strong convergence theorems of nonlinear operator equations for countable family of multivalued total quasi ϕ asymptotically nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 69 10.1186/16871812201269Google Scholar
 Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory Appl. 2008., 2008: Article ID 528476Google Scholar
 Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleGoogle Scholar
 Cholamjiak P: Strong convergence of projection algorithms for a family of relatively quasinonexpansive mappings and an equilibrium problem in Banach spaces. Optimization 2011, 60(4):495–507. 10.1080/02331930903477192MathSciNetView ArticleGoogle Scholar
 Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
 Iusema AN, Sosa W: Iterative algorithms for equilibrium problems. Optimization 2003, 52(3):301–316. 10.1080/0233193031000120039MathSciNetView ArticleGoogle Scholar
 Saewan S, Kumam P, Wattanawitoon K: Convergence theorem based on a new hybrid projection method for finding a common solution of generalized equilibrium and variational inequality problems in Banach spaces. Abstr. Appl. Anal. 2010., 2010: Article ID 734126Google Scholar
 Ceng LC, AlHomidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed points problems of strict pseudocontraction mappings. J. Comput. Appl. Math. 2009, 223(2):967–974. 10.1016/j.cam.2008.03.032MathSciNetView ArticleGoogle Scholar
 Zeng LC, Ansari QH, Shyu DS, Yao JC: Strong and weak convergence theorems for common solutions of generalized equilibrium problems and zeros of maximal monotone operators. Fixed Point Theory Appl. 2010., 2010: Article ID 590278Google Scholar
 Ceng LC, Ansari QH, Yao JC: Strong and weak convergence theorems for asymptotically strict pseudocontractive mappings in intermediate sense. J. Nonlinear Convex Anal. 2010, 11(2):283–308.MathSciNetGoogle Scholar
 Ceng LC, Ansari QH, Yao JC: Hybrid proximaltype and hybrid shrinking projection algorithms for equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31(7):763–797. 10.1080/01630563.2010.496697MathSciNetView ArticleGoogle Scholar
 Zeng LC, AlHomidan S, Ansari QH: Hybrid proximaltype algorithms for generalized equilibrium problems, maximal monotone operators and relatively nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 973028Google Scholar
 Zeng LC, Ansari QH, Schaible S, Yao JC: Hybrid viscosity approximate method for zeros of m accretive operators in Banach spaces. Numer. Funct. Anal. Optim. 2012, 33(2):142–165. 10.1080/01630563.2011.594197MathSciNetView ArticleGoogle Scholar
 Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht; 1990.View ArticleMATHGoogle Scholar
 Reich S: Geometry of Banach spaces, duality mappings and nonlinear problems. Bull. Am. Math. Soc. 1992, 26: 367–370. 10.1090/S027309791992002872View ArticleGoogle Scholar
 Beauzamy B: Introduction to Banach Spaces and Their Geometry. 2nd edition. NorthHolland, Amsterdam; 1985.MATHGoogle Scholar
 Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362546X(91)90200KMathSciNetView ArticleGoogle Scholar
 Zalinescu C: On uniformly convex functions. J. Math. Anal. Appl. 1983, 95: 344–374. 10.1016/0022247X(83)901129MathSciNetView ArticleGoogle Scholar
 Kamimura S, Takahashi W: Strong convergence of a proximaltype algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611XMathSciNetView ArticleGoogle Scholar
 Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
 Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansivetype mappings in Banach spaced. SIAM J. Optim. 2008, 19(2):824–835. 10.1137/070688717MathSciNetView ArticleGoogle Scholar
 Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 1970, 149: 75–88. 10.1090/S00029947197002822725MathSciNetView ArticleGoogle Scholar
 Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.MATHGoogle Scholar
 Cho YJ, Zhou HY, Guo G: Weak and strong convergence theorems for threestep iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S08981221(04)900582MathSciNetView ArticleGoogle Scholar
 Pascali D, Sburlan S: Nonlinear Mappings of Monotone Type. Editura Academiae Bucaresti, Romania; 1978.MATHView ArticleGoogle Scholar
 Baillon JB, Haddad G: Quelques propriétés des opérateurs andlebornés et n cycliquement monotones. Isr. J. Math. 1977, 26: 137–150. 10.1007/BF03007664MathSciNetView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.