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A new iteration process for equilibrium, variational inequality, fixed point problems, and zeros of maximal monotone operators in a Banach space
Journal of Inequalities and Applications volume 2013, Article number: 23 (2013)
Abstract
In this article, a new iterative process is introduced to approximate a common element of a fixed point set, the solutions of equilibrium problems, the solution set of variational inequality problems, and the set of zeros of maximal monotone operators in a uniformly smooth and strictly convex Banach space by using a hybrid projection method. Also, we prove new strong convergence theorems for this proposed iterative precess in a Banach space.
MSC:47H05, 47H09, 47H10.
1 Introduction
Let E be a real Banach space, {E}^{\ast} be the dual space of E. A setvalued mapping A:D(A)\subset E\to {E}^{\ast} with graph G(A)=\{(x,{x}^{\ast}):{x}^{\ast}\in Ax\}, domain D(A)=\{x\in E:Ax\ne \mathrm{\varnothing}\}, and range R(A)=\cup \{Ax:x\in D(A)\}. A is said to be monotone if \u3008xy,{x}^{\ast}{y}^{\ast}\u3009\ge 0 whenever {x}^{\ast}\in Ax, {y}^{\ast}\in Ay. A monotone operator A is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. Let A\subset E\times {E}^{\ast} be a maximal monotone operator. We consider the problem for finding x\in E
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. We know that if A is maximal monotone, then the solution set {A}^{1}0=\{x\in D(A):0\in Tx\} is closed and convex. One popular algorithm for approximating a solution of this problem is called the proximal point algorithm which was first proposed by Martinet [1] and studied further by Rockafellar [2] in Hilbert spaces. Since the proximal point algorithm weakly converges in general which is the proximal point algorithm is defined by {x}_{0}\in E and
where \{{r}_{n}\}\subset (0,\mathrm{\infty}) and {J}_{{r}_{n}} are the resolvent of A. Solovov and Svaitor [3] proposed a modified proximal point algorithm which converges strongly to a solution of the equation {A}^{1}0 by using the projection method. Many problems in nonlinear analysis and optimization can be formulated by the proximal point algorithm (see [4–9]).
Let E be a real Banach space with dual {E}^{\ast} and let C be a nonempty closed and convex subset of E. Let f:C\times C\to \mathbb{R} be a bifunction. The equilibrium problem is to find x\in C such that
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.
A mapping A:D(A)\subset E\to {E}^{\ast} is said to be αinversestrongly monotone if there exists a constant \alpha >0 such that
If A is αinverse strongly monotone, then it is \frac{1}{\alpha}Lipschitz continuous, i.e.,
Let C be a nonempty closed and convex subset of a real Banach space E. Let A be a monotone operator from C into E . The variational inequality problem for an operator A is to find \stackrel{\u02c6}{z}\in C such that
The set of solutions of (1.4) is denoted by \mathit{VI}(A,C).
Let C be a nonempty closed and convex subset of E. A mapping T from C into itself is said to be nonexpansive if
T is said to be total asymptotically nonexpansive if 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
A point x\in C is a fixed point of T provided Tx=x. Denote by F(T) the fixed point set of T; that is, F(T)=\{x\in C:Tx=x\}. A point p in C is called an asymptotic fixed point of T [12] if C contains a sequence \{{x}_{n}\} which converges weakly to p such that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0. The asymptotic fixed point set of T is denoted by \stackrel{\u02c6}{F}(T).
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
If E is a Hilbert space, then \varphi (y,x)={\parallel yx\parallel}^{2}. It is obvious from the definition of ϕ that
T is said to be ϕnonexpansive [13, 14] if
T is said to be quasiϕnonexpansive [13, 14] if F(T)\ne \mathrm{\varnothing} and
T is said to be asymptotically ϕnonexpansive [14] if there exists a sequence \{{k}_{n}\}\subset [0,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that
T is said to be quasiϕasymptotically nonexpansive [14] if F(T)\ne \mathrm{\varnothing} and there exists a sequence \{{k}_{n}\}\subset [0,\mathrm{\infty}) with {k}_{n}\to 1 as n\to \mathrm{\infty} such that
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
A mapping T is said to be uniformly LLipschitz continuous, if there exists a constant L>0 such that
T is said to be closed if for any sequence \{{x}_{n}\}\subset C such that {lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0} and {lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}, T{x}_{0}={y}_{0}.
Remark 1.1 Every quasiϕnonexpansive mapping implies a quasiϕasymptotically nonexpansive mapping and a quasiϕasymptotically nonexpansive mapping implies a total quasiϕasymptotically nonexpansive mapping, but the converse is not true.
On the other hand, Alber [15] 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
The existence and uniqueness of the operator {\mathrm{\Pi}}_{C} follow from the properties of the functional \varphi (x,y) and strict monotonicity of the mapping J. 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 relatively quasinonexpansive mapping from E onto C with F({\mathrm{\Pi}}_{C})=C.
Matsushita and Takahashi [16] proposed the following hybrid iteration method with a generalized projection for a relatively nonexpansive mapping T in a Banach space E:
They proved that \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)}{x}_{0}. Many authors studied the methods for approximating fixed points of a countable family of (relatively quasi) nonexpansive mappings (see [17–19]).
Recently, Qin et al. [20] considered a pair of asymptotically quasiϕnonexpansive mappings. To be more precise, they proved the following results.
Theorem QCK Let E be a uniformly smooth and uniformly convex Banach space and C be a nonempty closed and convex subset of E. Let T:C\to C be a closed and asymptotically quasiϕnonexpansive mapping with the sequence \{{k}_{n}^{(t)}\}\subset [1,\mathrm{\infty}) such that {k}_{n}^{(t)}\to 1 as n\to \mathrm{\infty} and S:C\to C be a closed and asymptotically quasiϕnonexpansive mapping with the sequence \{{k}_{s}^{(t)}\}\subset [1,\mathrm{\infty}) such that {k}_{n}^{(s)}\to 1 as n\to \mathrm{\infty}. Let \{{\alpha}_{n}\}, \{{\beta}_{n}\}, \{{\gamma}_{n}\}, and \{{\delta}_{n}\} be real number sequences in [0,1]. Assume that T and S are uniformly asymptotically regular on C and \mathrm{\Omega}=F(T)\cap F(S) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {k}_{n}=max\{{k}_{n}^{(t)},{k}_{n}^{(s)}\} for each n\ge 1, J is the duality mapping on E, {M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathrm{\Omega}\} for each n\ge 1. Assume that the control sequences \{{\alpha}_{n}\}, \{{\beta}_{n}\}, \{{\gamma}_{n}\}, and \{{\delta}_{n}\} satisfy the following restrictions:

(a)
{\beta}_{n}+{\gamma}_{n}+{\delta}_{n}=1, \mathrm{\forall}n\ge 1;

(b)
{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\gamma}_{n}{\delta}_{n}, {lim}_{n\to \mathrm{\infty}}{\beta}_{n}=0;

(c)
0\le {\alpha}_{n}<1 and {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\alpha}_{n}<1.
In 2008, Alber et al. [21] proved the strong convergence theorems to approximate a fixed point of a total asymptotically nonexpansive mapping in a Hilbert space. In 2011, Chang et al. [22, 23] proved the strong convergence theorems for finding the set of fixed points of a total quasiϕasymptotically nonexpansive mapping in the framework of Banach spaces.
Motivated and inspired by the work mentioned above, in this paper, we introduce a new hybrid projection algorithm for a pair of total quasiϕasymptotically nonexpansive mappings for finding a set of solutions of the equilibrium problem, a zero point of maximal monotone operators, and a set of solutions of the variation inequality in a uniformly smooth and strictly convex Banach space.
2 Preliminaries
In this article, we denote the strong convergence and weak convergence of a sequence \{{x}_{n}\} by {x}_{n}\to x and {x}_{n}\rightharpoonup x, respectively.
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 puniformly convex is uniformly convex. One should note that no Banach space is puniformly convex for 1<p<2.
Remark 2.1 The basic properties of E, J, and {J}^{1} are as follows (see [24]).

If E is an arbitrary Banach space, then J is monotone and bounded;

If E is strictly convex, then J is strictly monotone;

If E is smooth, then J is singlevalued and semicontinuous;

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};

If E is uniformly smooth, then E is smooth and reflexive;

If E is a reflexive and strictly convex Banach space, then {J}^{1} is normweak^{∗}continuous.
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 (1.5), 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 [24–26] for more details).
Recall that a Banach space E has the KadecKlee property [24, 25, 27] if for any sequence \{{x}_{n}\}\subset E and x\in E with {x}_{n}\rightharpoonup x and \parallel {x}_{n}\parallel \to \parallel x\parallel, then \parallel {x}_{n}x\parallel \to 0 as n\to \mathrm{\infty}. It is well known that if E is a uniformly convex Banach space, then E has the KadecKlee property.
The generalized projection [15] from E into C is defined by {\mathrm{\Pi}}_{C}(x)={argmin}_{y\in C}\varphi (y,x). The existence and uniqueness of the operator {\mathrm{\Pi}}_{C} follow from the properties of the functional \varphi (y,x) and the strict monotonicity of the mapping J (see, for example, [4, 15, 24, 25, 28]). If E is a Hilbert space, then \varphi (x,y)={\parallel xy\parallel}^{2} and {\mathrm{\Pi}}_{C} becomes the metric projection {P}_{C}:H\to C. If C is a nonempty closed and convex subset of a Hilbert space H, then {P}_{C} is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. We also need the following lemmas for the proof of our main results.
Lemma 2.3 (Alber [15])
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 2.4 (Alber [15])
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 2.5 (Change et al. [22])
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 C be a closed and total quasiϕasymptotically nonexpansive mapping with nonnegative real sequences {\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 \zeta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with \zeta (0)=0. If {\mu}_{n}=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,
(A4) for each x\in C, y\mapsto f(x,y) is convex and lower semicontinuous.
The following result is in Blum and Oettli [8].
Lemma 2.6 (Blum and Oettli [8])
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 2.7 (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 {K}_{r}:E\to C as follows:
Then the following hold:

(1)
{K}_{r} is singlevalued;

(2)
{K}_{r} is a firmly nonexpansivetype mapping [29], that is, for all x,y\in E,
\u3008{K}_{r}x{K}_{r}y,J{K}_{r}xJ{K}_{r}y\u3009\le \u3008{K}_{r}x{K}_{r}y,JxJy\u3009; 
(3)
F({K}_{r})=EP(f);

(4)
EP(f) is closed and convex.
Lemma 2.8 (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({K}_{r}),
Lemma 2.9 [30]
Let E be a uniformly convex Banach space and {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.
Let E be a smooth strictly convex and reflexive Banach space, C be a nonempty closed convex subset of E and A\subset E\times {E}^{\ast} be a monotone operator satisfying D(A)\subset C\subset {J}^{1}({\bigcap}_{\lambda >0}R(J+\lambda A)). Then the resolvent {J}_{\lambda}:C\to D(A) of A is defined by
{J}_{\lambda} is a singlevalued mapping from E to D(A). For any \lambda >0, the Yosida approximation {A}_{\lambda}:C\to {E}^{\ast} of A is defined by {A}_{\lambda}x=\frac{JxJ{J}_{\lambda}x}{\lambda} for all x\in C. We know that {A}_{\lambda}x\in A({J}_{\lambda}x) for all \lambda >0 and x\in E.
Lemma 2.10 (Kohsaka and Takahashi [29])
Let E be a smooth strictly convex and reflexive Banach space, C be a nonempty closed convex subset of E and A\subset E\times {E}^{\ast} be a monotone operator satisfying D(A)\subset C\subset {J}^{1}({\bigcap}_{\lambda >0}R(J+\lambda A)). For any \lambda >0, let {J}_{\lambda} and {A}_{\lambda} be the resolvent and the Yosida approximation of A, respectively. Then the following hold:

(a)
\varphi (p,{J}_{\lambda}x)+\varphi ({J}_{\lambda}x,x)\le \varphi (p,x) for all x\in C and p\in {A}^{1}0;

(b)
({J}_{\lambda}x,{A}_{\lambda}x)\in A for all x\in C;

(c)
F({J}_{\lambda})={A}^{1}0.
Lemma 2.11 (Rockafellar [31])
Let E be a reflexive strictly convex and smooth Banach space. Then an operator A\subset E\times {E}^{\ast} is maximal monotone if and only if R(J+\lambda A)={E}^{\ast} for all \lambda >0.
3 Main result
Theorem 3.1 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly uniformly convex Banach space E with the KadecKlee property. Let f be a bifunction from C\times C to ℝ satisfying the conditions (A1)(A4) and let A\subset E\times {E}^{\ast} be a maximal monotone operator satisfying D(A)\subset C and {J}_{{r}_{n}}={(J+{r}_{n}A)}^{1}J for all {r}_{n}>0. Let S:C\to C be a closed and total quasiϕasymptotically nonexpansive mapping with nonnegative real sequences {\nu}_{n}^{S}, {\mu}_{n}^{S} with {\nu}_{n}^{S}\to 0, {\mu}_{n}^{S}\to 0 as n\to \mathrm{\infty} and a strictly increasing continuous function {\psi}^{S}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with {\psi}^{S}(0)=0. Let T:C\to C be a closed and total quasiϕasymptotically nonexpansive mapping with nonnegative real sequences {\nu}_{n}^{T}, {\mu}_{n}^{T} with {\nu}_{n}^{T}\to 0, {\mu}_{n}^{T}\to 0 as n\to \mathrm{\infty} and a strictly increasing continuous function {\psi}^{T}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with {\psi}^{T}(0)=0. Assume that S and T are uniformly LLipschitz continuous and F=F(S)\cap F(T)\cap 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 \{{\alpha}_{n}\}, \{{\beta}_{n}\}, and \{{\gamma}_{n}\} are sequences in (0,1) such that {\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1, \{{r}_{n}\}\subset [d,\mathrm{\infty}) for some d>0 {\mu}_{n}=sup\{{\mu}_{n}^{S},{\mu}_{n}^{T}\}, {\nu}_{n}=sup\{{\nu}_{n}^{S},{\nu}_{n}^{T}\}, \psi =sup\{{\psi}^{S},{\psi}^{T}\} for all n\ge 1, \zeta ={\nu}_{n}{sup}_{q\in \mathcal{F}}\psi (\varphi (q,{x}_{n}))+{\mu}_{n}. If {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}{\beta}_{n}=0 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n}<1, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F}{x}_{1}.
Proof First, we show that {C}_{n} is closed and convex for all n\in \mathbb{N} since {C}_{1}=C is convex. Suppose that {C}_{n} is 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})+{\zeta}_{n} is equivalent to
That is, {C}_{n+1} is convex for all n\in \mathbb{N}. By the definition of {C}_{n}, it is obvious that {C}_{n} is closed for all n\in \mathbb{N}.
We show that \{{x}_{n}\} is well defined. It is obvious that F\subset {C}_{1}=C. Suppose F\subset {C}_{n} for n\in \mathbb{N}, from Lemma 2.8 and Lemma 2.10, S, T are total quasiϕasymptotically nonexpansive mappings. For each q\in F\subset {C}_{n}, it follows that
where {\zeta}_{n}={\nu}_{n}{sup}_{q\in F}\psi (\varphi (q,{x}_{n}))+{\mu}_{n}. This shows that q\in {C}_{n+1}, thus F\subset {C}_{n+1}. Hence, F⊂ {C}_{n} for all n\ge 1. This implies that the sequence \{{x}_{n}\} is well defined.
We show that {lim}_{n\to \mathrm{\infty}}{x}_{n}=p. 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
By Lemma 2.4, we get
From (3.3) and (3.4), we have that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1}) exists. In particular, it follows from (1.6) that the sequence \{{x}_{n}\} is bounded and so are \{{z}_{n}\}, \{{u}_{n}\}, and \{{y}_{n}\}. Since {x}_{n}\in {C}_{n}\subset E and E is reflexive, the sequence \{{x}_{n}\} converges weakly to an element of E, we assume that {x}_{n}\rightharpoonup p. Note that {C}_{n} is closed and convex and {x}_{n}\in {C}_{n}. We have that p\in {C}_{n}, that is,
For p\in {C}_{n}, we have
On the other hand, {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, we have
It follows that
This implies that {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1})=\varphi (p,{x}_{1}). Hence, we get
From (3.5), (3.6), and the KadecKlee property of E, we have
Therefore,
From (3.7), it follows that
and hence
We show that p\in F(S)\cap F(T)\cap {A}^{1}0\cap EP(f).
Now, we show that p\in EP(f). For {x}_{n+1}\in {C}_{n+1}\subset {C}_{n} and {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{1}, it follows that
Since {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{x}_{1}) exists, we have
Since {x}_{n+1}\subset {C}_{n} and the definition of {C}_{n+1}, we have \varphi ({x}_{n+1},{y}_{n})\le \varphi ({x}_{n+1},{x}_{n})+{\zeta}_{n}. From (3.11), we also have
From (1.6) and (3.7), it follows that
and hence
This implies that \{\parallel J{y}_{n}\parallel \} is bounded. Note that E is reflexive and {E}^{\ast} is also reflexive, we can assume that J{y}_{n}\rightharpoonup {x}^{\ast}\in {E}^{\ast}. Since E is reflexive, we see that J(E)={E}^{\ast}. Hence, there exists x\in E such that Jx={x}^{\ast} and we have
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on the both sides of the equality above, in view of the weak lower semicontinuity of the norm \parallel \cdot \parallel, it follows that
From Remark 2.2, we have p=x, which implies thatJ{y}_{n}\rightharpoonup Jp as n\to \mathrm{\infty}. From the KadecKlee property of {E}^{\ast}, we obtain that
Note that {J}^{1}:{E}^{\ast}\to E is demicontinuous, that is, {y}_{n}\rightharpoonup p as n\to \mathrm{\infty}. From the KadecKlee property of E, it follows that
From (3.2), (3.7), and (3.16), it follows that {lim}_{n\to \mathrm{\infty}}\varphi (q,{u}_{n})=\varphi (q,p). Since {u}_{n}={K}_{{r}_{n}}{x}_{n} and from Lemma 2.8, we have
From (1.6), it follows that
Since \{{u}_{n}\} is bounded and E is also reflexive, we can assume that {u}_{n}\rightharpoonup u\in E and we have
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on the both sides of the equality above, in view of the weak lower semicontinuity of the norm \parallel \cdot \parallel, it follows that
From Remark 2.2, we have u=p, that is, {u}_{n}\rightharpoonup p as n\to \mathrm{\infty}. From the KadecKlee property of E, we obtain that
Since {lim}_{n\to \mathrm{\infty}}{u}_{n}=p and {lim}_{n\to \mathrm{\infty}}{x}_{n}=p, we have that
Since J is uniformly normtonorm continuous, we obtain
From {r}_{n}>0, we have \frac{\parallel J{u}_{n}J{x}_{n}\parallel}{{r}_{n}}\to 0 as n\to \mathrm{\infty} and
By (A2),
and {u}_{n}\to p, we get f(y,p)\le 0 for all y\in C. For 0<t<1, define {y}_{t}=ty+(1t)p. Then {y}_{t}\in C, which implies that f({y}_{t},p)\le 0. From (A1), we obtain that
Thus f({y}_{t},y)\ge 0. From (A3), we have f(p,y)\ge 0 for all y\in C. Hence, p\in EP(f).
Next, we show that p\in {A}^{1}0. From (3.2), (3.7), (3.16), and (3.18), it follows that {lim}_{n\to \mathrm{\infty}}\varphi (q,{z}_{n})=\varphi (q,p). Since {z}_{n}={J}_{{r}_{n}}{x}_{n} and from Lemma 2.10, we have
From (1.6), it follows that
Since \{{u}_{n}\} is bounded and E is also reflexive, we can assume that {z}_{n}\rightharpoonup z\in E and we have
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on the both sides of the equality above, in view of the weak lower semicontinuity of the norm \parallel \cdot \parallel, it follows that
From Remark 2.2, we have z=p, that is, {u}_{n}\rightharpoonup p as n\to \mathrm{\infty}. From the KadecKlee property of E, we obtain that
Since {lim}_{n\to \mathrm{\infty}}{z}_{n}=p and {lim}_{n\to \mathrm{\infty}}{x}_{n}=p, we have that
and hence
From the condition \{{r}_{n}\}\subset [d,\mathrm{\infty}) for some d>0, we have
Thus, since {z}_{n}={J}_{{r}_{n}}{x}_{n}, we have
For any (w,{w}^{\ast})\in G(A), it follows from the monotonicity of A that \u3008w{z}_{n},{w}^{\ast}{A}_{{r}_{n}}{x}_{n}\u3009\ge 0 for all n\ge 0. Letting n\to \mathrm{\infty}, we get \u3008wp,{w}^{\ast}\u3009\ge 0. Therefore, since A is maximal monotone, we obtain p\in {A}^{1}0.
On the other hand, we have
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 2.9, we have
It follows from {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\beta}_{n}>0, (3.23), (3.8), and the property of g that
Since {x}_{n}\to p as n\to \mathrm{\infty} and J is uniformly continuous, it yields that J{x}_{n}\to Jp, we have
Since {J}^{1} is demicontinuous, we also have
On the other hand, we observe that
we obtain that \parallel {S}^{n}{z}_{n}\parallel \to \parallel p\parallel. Since E has the KadeeKlee property, we get
By the assumption that S is uniformly LLipschitz continuous, we have
Since {lim}_{n\to \mathrm{\infty}}{z}_{n}=p and {lim}_{n\to \mathrm{\infty}}{S}^{n}{z}_{n}=p, it yields that \parallel {S}^{n+1}{z}_{n}{S}^{n}{z}_{n}\parallel \to 0, n\to \mathrm{\infty}. From {S}^{n}{z}_{n}\to p, we get {S}^{n+1}{z}_{n}\to p, that is, S{S}^{n}{z}_{n}\to p. In view of the closeness of S, we have Sp=p. This implies that p\in F(S). By the same way, we have that p\in F(T).
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}v\u3009\ge 0, \mathrm{\forall}v\in {C}_{n}. Since F\subset {C}_{n}, we also have
By taking limit n\to \mathrm{\infty}, we obtain that
By Lemma 2.3, we can conclude that p={\mathrm{\Pi}}_{F}{x}_{1} and {x}_{n}\to p as n\to \mathrm{\infty}. The proof is completed. □
Let A be a continuous and monotone operator of C into {E}^{\ast}. Then we can find a solution of \mathit{VI}(A,C) in a uniformly smooth and strictly convex Banach space E with the KadecKlee property by using the following lemma.
Lemma 3.2 (Zegeye and Shahzad [32])
Let C be a nonempty closed convex subset of a uniformly smooth strictly convex real Banach space E. Let A:C\to {E}^{\ast} be a continuous monotone mapping. For any r>0, define a mapping {W}_{r}:E\to C as follows:
for all x\in C. Then the following hold:

(1)
{W}_{r} is singlevalued;

(2)
F({W}_{r})=\mathit{VI}(A,C);

(3)
\mathit{VI}(A,C) is a closed and convex subset of C;

(4)
\varphi (q,{W}_{r}x)+\varphi ({W}_{r}x,x)\le \varphi (q,x) for all q\in F({W}_{r}).
Corrollary 3.3 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly uniformly convex Banach space E with the KadecKlee property. Let f be a bifunction from C\times C to ℝ satisfying the conditions (A1)(A4) and let A be a continuous and monotone operator of C into {E}^{\ast}. Let S:C\to C be a closed and total quasiϕasymptotically nonexpansive mapping with nonnegative real sequences {\nu}_{n}^{S}, {\mu}_{n}^{S} with {\nu}_{n}^{S}\to 0, {\mu}_{n}^{S}\to 0 as n\to \mathrm{\infty} and a strictly increasing continuous function {\psi}^{S}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with {\psi}^{S}(0)=0. Let T:C\to C be a closed and total quasiϕasymptotically nonexpansive mapping with nonnegative real sequences {\nu}_{n}^{T}, {\mu}_{n}^{T} with {\nu}_{n}^{T}\to 0, {\mu}_{n}^{T}\to 0 as n\to \mathrm{\infty} and a strictly increasing continuous function {\psi}^{T}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with {\psi}^{T}(0)=0. Assume that S and T are uniformly LLipschitz continuous and F=F(S)\cap F(T)\cap EP(f)\cap \mathit{VI}(A,C)\ne \mathrm{\varnothing}. For an initial point {x}_{1}\in E, {C}_{1}=C, define the sequence \{{x}_{n}\} by
where \{{\alpha}_{n}\}, \{{\beta}_{n}\} and \{{\gamma}_{n}\} are sequences in (0,1) such that {\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1, \{{r}_{n}\}\subset [d,\mathrm{\infty}) for some d>0 {\mu}_{n}=sup\{{\mu}_{n}^{S},{\mu}_{n}^{T}\}, {\nu}_{n}=sup\{{\nu}_{n}^{S},{\nu}_{n}^{T}\}, \psi =sup\{{\psi}^{S},{\psi}^{T}\} for all n\ge 1, \zeta ={\nu}_{n}{sup}_{q\in \mathcal{F}}\psi (\varphi (q,{x}_{n}))+{\mu}_{n}. If {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}{\beta}_{n}=0 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n}<1, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F}{x}_{1}.
Proof From the proof of Theorem 3.1, we known that {lim}_{n\to \mathrm{\infty}}{z}_{n}=p and {lim}_{n\to \mathrm{\infty}}{x}_{n}=p. We obtain that
and hence
Since \{{r}_{n}\}\subset [d,\mathrm{\infty}) for some d>0, we have
From the definition of W, it follows that
For 0<t<1, define {y}_{t}=ty+(1t)p, then {y}_{t}\in C. We have
It follows that
Since {A}_{n} are continuous and monotone mappings, we have \u3008{y}_{t}{z}_{n},{A}_{n}{y}_{t}{A}_{n}{z}_{n}\u3009\ge 0. From (3.31) and (3.32), it follows that \u3008{y}_{t}{z}_{n},{A}_{n}{y}_{t}\u3009\ge 0. Take the limit as n\to \mathrm{\infty} and {z}_{n}\to p. We get \u3008{y}_{t}p,{A}_{n}{y}_{t}\u3009\ge 0 for all {y}_{t}\in C. Therefore, \u3008yp,{A}_{n}{y}_{t}\u3009\ge 0 for all y\in C. If t\to 0, we have that \u3008yp,{A}_{n}p\u3009\ge 0 for all y\in C. Hence, p\in \mathit{VI}(A,C). The proof is completed. □
Corrollary 3.4 Let C be a nonempty closed and convex subset of a uniformly smooth and strictly uniformly convex Banach space E with the KadecKlee property. Let A be a continuous and monotone operator of C into {E}^{\ast} and let B\subset E\times {E}^{\ast} be a maximal monotone operator satisfying D(B)\subset C and {J}_{{r}_{n}}={(J+{r}_{n}B)}^{1}J for all {r}_{n}>0. Let S:C\to C be a closed and total quasiϕasymptotically nonexpansive mapping with nonnegative real sequences {\nu}_{n}^{S}, {\mu}_{n}^{S} with {\nu}_{n}^{S}\to 0, {\mu}_{n}^{S}\to 0 as n\to \mathrm{\infty} and a strictly increasing continuous function {\psi}^{S}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with {\psi}^{S}(0)=0. Let T:C\to C be a closed and total quasiϕasymptotically nonexpansive mapping with nonnegative real sequences {\nu}_{n}^{T}, {\mu}_{n}^{T} with {\nu}_{n}^{T}\to 0, {\mu}_{n}^{T}\to 0 as n\to \mathrm{\infty} and a strictly increasing continuous function {\psi}^{T}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} with {\psi}^{T}(0)=0. Assume that S and T are uniformly LLipschitz continuous and F=F(S)\cap F(T)\cap \mathit{VI}(A,C)\cap {B}^{1}0\ne \mathrm{\varnothing}. For an initial point {x}_{1}\in E, {C}_{1}=C, define the sequence \{{x}_{n}\} by
where \{{\alpha}_{n}\}, \{{\beta}_{n}\}, and \{{\gamma}_{n}\} are sequences in (0,1) such that {\alpha}_{n}+{\beta}_{n}+{\gamma}_{n}=1, \{{r}_{n}\}\subset [d,\mathrm{\infty}) for some d>0 {\mu}_{n}=sup\{{\mu}_{n}^{S},{\mu}_{n}^{T}\}, {\nu}_{n}=sup\{{\nu}_{n}^{S},{\nu}_{n}^{T}\}, \psi =sup\{{\psi}^{S},{\psi}^{T}\} for all n\ge 1, \zeta ={\nu}_{n}{sup}_{q\in \mathcal{F}}\psi (\varphi (q,{x}_{n}))+{\mu}_{n}. If {lim}_{n\to \mathrm{\infty}}{\alpha}_{n}{\beta}_{n}=0 and {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}{\gamma}_{n}<1, then \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F}{x}_{1}.
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The authors would like to express their thanks to the reviewer for helpful suggestions and comments for the improvement of this paper. This work was supported by Thaksin University.
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Saewan, S., Kumam, P. A new iteration process for equilibrium, variational inequality, fixed point problems, and zeros of maximal monotone operators in a Banach space. J Inequal Appl 2013, 23 (2013). https://doi.org/10.1186/1029242X201323
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DOI: https://doi.org/10.1186/1029242X201323