∎

11institutetext: ∗Corresponding author
Mostafa Ghadampour 22institutetext: Department of Mathematics, Lorestan
University, Lorestan, Khoramabad, Iran.
22email<EMAIL_ADDRESS>33institutetext: Donal O′Regan
44institutetext: School of Mathematics, Statistics, National University of
Ireland, Galway, Ireland.
44email<EMAIL_ADDRESS>55institutetext: Ebrahim Soori
66institutetext: Department of Mathematics, Lorestan University, Lorestan,
Khoramabad, Iran.
66email<EMAIL_ADDRESS>77institutetext: Ravi P. Agarwal88institutetext:
Department of Mathematics Texas A$\&$M University-Kingsville 700 University
Blvd., MSC 172 Kingsville, Texas, USA.
88email<EMAIL_ADDRESS>
#  Two generalized strong convergence algorithms for the variational
inequality problems in Banach spaces

Mostafa Ghadampour1 Ebrahim Soori∗,2

(Received: date / Accepted: date)

###### Abstract

In this paper, two generalized algorithms for solving the variational
inequality problem in Banach spaces are proposed. Then the strong convergence
of the sequences generated by these algorithms will be proved under the
suitable conditions. Finally, using MATLAB software, we provide some numerical
examples to illustrate our results.

###### Keywords:

Variational inequality Relatively nonexpansive mapping Monotone mapping
Asymptotical fixed point

##  1 Introduction

Let $C$ be a nonempty closed convex subset of a Banach space $E$ with norm
$\|.\|$ and let $E^{*}$ denotes the dual of $E$. The variational inequality
problem (VIP) is to find a point $x\in C$ such that

$\langle Ax,y-x\rangle\geq 0\;\;\;\forall\;\;y\in C,$ (1)

where $A$ is a mapping of $C$ into $E^{*}$ and $\langle.,.\rangle$ denotes the
pairing between $E$ and $E^{*}$. The solutions set of (1) is denoted by
$VI(A,C)$.
It is well known that variational inequalities cover a variety of fields in
optimal control, optimization, mathematical programming, operational research,
partial differential equations, engineering, and equilibrium models and hence,
it have been studied by many authors in the recent yearsjlotaa ; tdvh ; vpts ;
cgga .

The operator $A$ of $C$ to $E^{*}$ is said to be
(i) monotone if

$\langle x-y,Ax-Ay\rangle\geq 0,\;\;\forall x,y\in C;$

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

$\langle x-y,Ax-Ay\rangle\geq\alpha\|Ax-Ay\|^{2}\;\;\forall x,y\in C;$

(iii) $L$-Lipchitz continuous if there exists $L>0$ such that

$\displaystyle\|Ax-Ay\|\leq L\|x-y\|,\;\;\forall x,y\in C.$

Let $f:C\times C\rightarrow\mathbb{R}$ be a bifunction. The equilibrium
problem (GEP) is as follows: Find $x\in C$ such that

$f(x,y)+\langle Ax,y-x\rangle\geq 0,\;\;\forall\;y\in C.$ (2)

The set of solutions of (2) is denoted by $GEP(f,A)$. Clearly, the problem (2)
is equivalent to (VIP) if $f\equiv 0$.

Korpelevichko  proposed the following algorithm for solving the problem (VIP)
that is known as extragradient method as (3). Let $x_{1}$ be an arbitrarily
element in $H$:

$\begin{cases}y_{n}=P_{C}(x_{n}-\lambda Ax_{n}),\\\
x_{n+1}=P_{C}(x_{n}-\lambda Ay_{n}),\end{cases}$ (3)

Tseng tp  proposed the following algorithm which was introduced using the
modified front-to-back (F-B) method.

$\begin{cases}y_{n}=P_{C}(x_{n}-\lambda Ax_{n}),\\\
x_{n+1}=P_{X}(y_{n}-\lambda(Ay_{n}-Ax_{n})),\end{cases}$ (4)

where $X=C$ and $X=H$ if $A$ is Lipschitz continuous. Thong et al th  proposed
the following convergent algorithm based on the Tseng algorithm.

$\begin{cases}y_{n}=P_{C}(x_{n}-\lambda_{n}Ax_{n}),\\\
z_{n}=y_{n}-\lambda_{n}(Ay_{n}-Ax_{n}),\\\
x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})z_{n},\end{cases}$ (5)

where the operator $A$ is monotone and Lipschitz continuous,
$\gamma>0,\;l\in(0,1),\;\mu\in(0,1)$ and $\lambda_{n}$ is chosen to be the
largest $\lambda\in\\{\gamma,\gamma l,\gamma l^{2},...\\}$ satisfying

$\displaystyle\lambda\|Ax_{n}-Ay_{n}\|\leq\mu\|x_{n}-y_{n}\|.$ (6)

In this paper, we present our algorithms in Banach spaces motivated by the
Thong algorithm and prove the strong convergence of the sequences generated by
these algorithms. Finally, using MATLAB software, we provide some numerical
examples to illustrate our claims.

##  2 Preliminaries

Let $E$ be a real Banach space with norm $\|.\|$ and let $E^{*}$ be the dual
space of $E$. The strong convergence and the weak convergence of the sequence
$\\{x_{n}\\}$ to $x$ in $E$ are denoted by $x_{n}\rightarrow x$ and
$x_{n}\rightharpoonup x$ through in the paper, respectively. The modulus
$\delta$ of convexity of $E$ is defined by

$\delta(\epsilon)=\inf\\{1-\frac{\|x+y\|}{2}:\|x\|\leq 1,\|y\|\leq
1,\|x-y\|\geq\epsilon\\}$

for every $\epsilon\in[0,2]$. $A$ Banach space $E$ is said to be uniformly
convex if $\delta(0)=0$ and $\delta(\epsilon)>0$ for every $\epsilon>0$. It is
known that a Banach space $E$ is uniformly convex if and only if for any two
sequences $\\{x_{n}\\}$ and $\\{y_{n}\\}$ in $E$ such that

$\lim_{n\rightarrow\infty}\|x_{n}\|=\lim_{n\rightarrow\infty}\|y_{n}\|=1\;and\;\lim_{n\rightarrow\infty}\|x_{n}+y_{n}\|=2,$

$\lim_{n\rightarrow\infty}\|x_{n}-y_{n}\|=0$ holds. Suppose that $p$ is a
fixed real number with $p\geq 2$. A Banach space $E$ is said to be
$p$-uniformly convextyhk , if there exists a constant $c>0$ such that
$\delta\geq c\epsilon^{p}$ for all $\epsilon\in[0,2]$. It is also known that a
uniformly convex Banach space has the Kadec-Klee property, that is,
$x_{n}\rightharpoonup u$ and $\|x_{n}\|\rightarrow\|u\|$ imply that
$x_{n}\rightarrow u$(see cii ; res ).
The normalized duality mapping $J:E\rightarrow E^{*}$ is defined by

$\displaystyle J(x)=\\{f\in E^{*}:\langle x,f\rangle=\|x\|^{2}=\|f\|^{2}\\},$

for each $x\in E$. Suppose that $S(E)=\\{x\in E:\|x\|=1\\}$. A Banach space
$E$ is called smooth if for all $x\in S(E)$, there exists a unique functional
$j_{x}\in E^{*}$ such that $\langle x,j_{x}\rangle=\|x\|$ and $\|j_{x}\|=1$(
see Ag ).

The norm of $E$ is said to be $G\hat{a}teaux$ differentiable if for each
$x,y\in S(E)$, the limit

$\lim_{t\rightarrow 0}\frac{\|x+ty\|-\|x\|}{t}$ (7)

exists. In this case, $E$ is called smooth and $E$ is said to be uniformly
smooth if the limit (7) is attained uniformly for all $x,y\in S(E)$tn . If a
Banach space $E$ is uniformly convex, then $E$ is reflexive and strictly
convex, and $E^{*}$ is uniformly smoothAg . It is well known that if $E$ is a
reflexive, strictly convex and smooth Banach space and $J^{*}:E^{*}\rightarrow
E$ be the duality mapping on $E^{*}$, then $J^{-1}=J^{*}$, also, if $E$ is a
uniformly smooth Banach space, then $J$ is uniformly norm to norm continuous
on bounded sets of $E$ and $J^{-1}=J^{*}$ is also uniformly norm to norm
continuous on bounded sets of $E^{*}$. Let $E$ be a smooth Banach space and
let $J$ be the duality mapping on $E$. The function $\phi:E\times
E\rightarrow\mathbb{R}$ is define by

$\phi(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y\|^{2},\;\;\;\forall x,y\in E.$
(8)

Clearly, from (8), it is concluded that

$(\|x\|-\|y\|)^{2}\leq\phi(x,y)\leq(\|x\|+\|y\|)^{2}.$ (9)

If $E$ is a reflexive, strictly convex and smooth Banach space, then for all
$x,y\in E$

$\phi(x,y)=0\Leftrightarrow x=y.$ (10)

Also, It is obvious from the definition of the function $\phi$ that the
following conditions hold for all $x,y,z,w\in E$,

$\phi(x,y)=\phi(x,z)+\phi(z,y)+2\langle x-z,Jz-Jy\rangle,$ (11) $2\langle
x-y,Jz-Jw\rangle=\phi(x,w)+\phi(y,z)-\phi(x,z)-\phi(y,w).$ (12)
$\phi(x,y)=\langle x,Jx-Jy\rangle+\langle y-x,Jy\rangle\leq\|x\|\|Jx-
Jy\|+\|y-x\|\|y\|.$ (13)

Now, the function $V:E\times E^{*}\rightarrow\mathbb{R}$ is defined as follows

$V(x,x^{*})=\|x\|^{2}-2\langle x,x^{*}\rangle+\|x^{*}\|^{2},$

for all $x\in E$ and $x^{*}\in E$. Moreover, $V(x,x^{*})=\phi(x,J^{-1}x^{*})$
for all $x\in E$ and $\in E$. If $E$ is a reflexive strictly convex and smooth
Banach space with $E^{*}$ as its dual, it is concluded that

$V(x,x^{*})+2\langle J^{-1}x^{*}-x,y^{*}\rangle\leq V(x,x^{*}+y^{*}),$ (14)

for all $x\in E$ and all $x^{*},y^{*}\in E^{*}$kt .

An operator $A:C\rightarrow E^{*}$ is hemicontinuous at $x_{0}\in C$, if for
any sequence $\\{x_{n}\\}$ converging to $x_{0}$ along a line implies that
$Tx_{n}\rightharpoonup Tx_{0}$, i.e., $Tx_{n}=T(x_{0}+t_{n}x)\rightharpoonup
Tx_{0}$ as $t_{n}\rightarrow 0$ for all $x\in C$.
The generalized projection $\Pi_{C}:E\rightarrow C$ is a mapping that assigns
to an arbitrary point $x\in E$, the minimum point of the functional
$\phi(y,x)$; that is, $\Pi_{C}x=x_{0}$, where $x_{0}$ is the solution of the
minimization problem

$\phi(x_{0},x)=\min_{y\in C}\phi(y,x).$ (15)

The existence and uniqueness of the operator $\Pi_{C}$ follows from the
properties of the functional $\phi(x,y)$ and strict monotonicity of the
mapping $J$ayl . Suppose that $C$ is a nonempty closed convex subset of $E$,
and $T$ is a mapping from $C$ into itself. A point $p\in C$ is called an
asymptotically fixed point of $T$ if $C$ contains a sequence $\\{x_{n}\\}$
which converges weakly to $p$ such that $Tx_{n}-x_{n}\rightarrow 0$Ag . The
set of asymptotical fixed points of $T$ will be denoted by $\hat{F}(T)$. A
mapping $T$ from $C$ into itself is said to be relatively nonexpansive if
$\hat{F}(T)=F(T)$ and $\phi(p,Tx)\leq\phi(p,x)$ for all $x\in C$ and $p\in
F(T)$. The asymptotic behavior of a relatively nonexpansive mapping was
studied in bdrs ; bdrs1 ; cyrs .
We need the following lemmas for the proof of our main results.

###### Lemma 1

(ktw ) Let $E$ be a smooth and uniformly convex Banach space and let
$\\{x_{n}\\}$ and $\\{y_{n}\\}$ be two sequences of $E$. If
$\phi(x_{n},y_{n})\rightarrow 0$ and either $\\{x_{n}\\}$ or $\\{y_{n}\\}$ is
bounded, then $x_{n}-y_{n}\rightarrow 0$.

###### Lemma 2

(ayl ) Let $C$ be a nonempty closed convex subset of a smooth, strictly convex
and reflexive Banach space $E$, let $x\in E$ and let $z\in C$. Then
$z=\Pi_{C}x\Leftrightarrow\langle y-z,Jx-Jz\rangle\leq 0$, for all $y\in C$.

###### Lemma 3

(ayl ) Let $C$ be a nonempty closed convex subset of a smooth, strictly convex
and reflexive Banach space $E$ and let $y\in E$. Then
$\phi(x,\Pi_{C}y)+\phi(\Pi_{C}y,y)\leq\phi(x,y),\;\;\forall x\in C$.

###### Lemma 4

(bb ; xuh ) Let $E$ be a 2-uniformly convex and smooth Banach space. Then, for
all $x$, $y\in E$, it is concluded that
$\|x-y\|\leq\frac{2}{c^{2}}\|Jx-Jy\|$,
where $\frac{1}{c}(0\leq c\leq 1)$is the 2-uniformly convex constant of $E$.

###### Lemma 5

(Xu xuh ). Let $E$ be a uniformly convex Banach space and $r>0$. Then there
exists a continuous strictly increasing convex function
$g:[0,2r]\rightarrow[0,\infty)$ such that $g(0)=0$ and

$\displaystyle\|tx+(1-t)y\|^{2}\leq
t\|x\|^{2}+(1-t)\|y\|^{2}-t(1-t)g(\|x-y\|),$

for all $x,y\in B_{r}(0)=\\{z\in E:\|z\|\leq r\\}$ and $t\in[0,1]$.

###### Lemma 6

(ktw ). Let $E$ be a uniformly convex Banach space and $r>0$. Then there
exists a continuous strictly increasing convex function
$g:[0,2r]\rightarrow[0,\infty)$ such that $g(0)=0$ and

$\displaystyle g(\|x-y\|)\leq\phi(x,y),$

for all $x,y\in B_{r}(0)=\\{z\in E:\|z\|\leq r\\}$.

Throughout this paper, we assume that $f:C\times C\rightarrow\mathbb{R}$ be a
bifunction satisfying the following conditions

  1. (A1)

$f(x,x)=0$ for all $x\in C$,

  2. (A2)

f is monotone, i.e. $f(x,y)+f(y,x)\leq 0$, for all $x,y\in C$,

  3. (A3)

$\displaystyle\lim_{t\downarrow 0}f(tz+(1-t)x,y)\leq f(x,y)$, for all
$x,y,z\in C$,

  4. (A4)

for each $x\in C,y\mapsto f(x,y)$ is convex and lower semicontinuous.

###### Lemma 7

(lyc ) Let $C$ be a nonempty closed convex subset of a smooth, strictly convex
and reflexive Banach space $E$. Let $A:C\longrightarrow E^{*}$ be an
$\alpha-$inverse-strongly monotone operator and $f$ be a bifunction from
$C\times C$ to $\mathbb{R}$ satisfying $(A_{1})-(A_{4})$. Then for all $r>0$
hold the following

  1. (i)

for $x\in E$, there exists $u\in C$ such that

$f(u,x)+\langle Au,y-u\rangle+\frac{1}{r}\langle y-u,Ju-Jx\rangle\geq
0,\;\;\;\forall y\in C,$

  2. (ii)

if $E$ is additionally uniformly smooth and $K_{r}:E\longrightarrow C$ is
defined as

$K_{r}(x)=\\{u\in C\;\;:\;\;f(u,y)+\langle Au,y-u\rangle+\frac{1}{r}\langle
y-u,Ju-Jx\rangle\geq 0,\;\;\;\forall y\in C\\}$

Then, the following conditions hold:

  1. (1)

$K_{r}$ is single-valued,

  2. (2)

$K_{r}$ is firmly nonexpansive, i.e., for all $x,y\in E$,

$\langle K_{r}x-K_{r}y,JK_{r}x-JK_{r}y\rangle\leq\langle K_{r}x-K_{r}y,Jx-
Jy\rangle,$

  3. (3)

$F(K_{r})=\hat{F(K_{r})}=GEP(f,A)$,

  4. (4)

$GEP$ is a closed convex subset of $C$,

  5. (5)

$\phi(p,K_{r}x)+\phi(K_{r}x,x)\leq\phi(p,x),\;\;\forall\;\;p\in F(K_{r})$.

The normal cone for $C$ at a point $\upsilon\in C$ is denoted by
$N_{C}(\upsilon)$, that is $N_{C}(\upsilon):=\\{x^{*}\in
E^{*}:\langle\upsilon-y,x^{*}\rangle\geq 0,\forall y\in C\\}$.

###### Lemma 8

(rrt ) Let $C$ be a nonempty closed convex subset of a Banach space $E$ and
let $T$ be monotone and hemicontinuous operator of $C$ into $E^{*}$ with
$C=D(T)$. Let $B\subset E\times E^{*}$ be an operator define as follows:

$Bv=\left\\{\begin{array}[]{lr}Tv+N_{C}v,\qquad v\in C,\\\
\emptyset,\qquad\qquad\qquad v\notin C.\end{array}\right.$

Then $B$ is maximal monotone and $B^{-1}(0)=$ SOL$(T,C)$.

##  3 Main results

In this section, we introduce a new iterative algorithms for solving monotone
variational inequality problems which are based on Tseng’s intergradient
method.We prove strong convergence theorems for generated sequences by
presented intergradient algorithms, under suitable conditions.

Throughout this section, we assume that $C$ is a nonempty closed convex subset
of a real 2-uniformly convex and uniformly smooth Banach space $E$ and $E^{*}$
is the dual space of $E$, $A:C\rightarrow E^{*}$ is an $\alpha$-inverse
strongly monotone operator. Assume that $\\{\lambda_{n}\\}$ is a sequence of
real numbers such that $0<\lambda_{n}<\frac{c^{2}\alpha}{2}$ for all
$n\in\mathbb{N}$, where $\frac{1}{c}$ is the 2-uniformly convexity constant of
$E$.

###### Theorem 3.1

Let $x_{0}\in C$, $\Gamma:=VI(C,A)\cap F(f)\neq\emptyset$ and

$\begin{cases}y_{n}=\Pi_{C}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}),\\\
z_{n}=J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}),\\\
x_{n+1}=\Pi_{C}J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}),\end{cases}$
(16)

where $\\{\lambda_{n}\\}\subseteq[0,1]$ such that
$\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$. Let
$\\{\alpha_{n,i}\\}\subset(0,1)$ for $i=1,2,3$,
$\alpha_{n,1}+\alpha_{n,2}+\alpha_{n,3}=1$ and
$\displaystyle\liminf_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$. Let f
be a relatively nonexpansive self-mapping on $C$ and $\|Ax\|\leq\|Ax-Au\|$ for
all $x\in C$ and $u\in\Gamma$. Consider the sequence $\\{x_{n}\\}$ generated
by the algorithm (16). Then the sequence $\\{x_{n}\\}$ converges strongly to
$q=\Pi_{VI(C,A)}\circ f(q)$, where $P_{VI(C,A)}\circ f:H\rightarrow VI(C,A)$
is the mapping defined by $P_{VI(C,A)}\circ f(x)=P_{VI(C,A)}(f(x))$ for each
$x\in H$.

###### Proof

Let $\hat{u}\in\Gamma$. From the definition of function $V$ and the inequality
(14), it is concluded that

$\displaystyle\phi(\hat{u},z_{n})=$
$\displaystyle\phi(\hat{u},J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}))$ $\displaystyle=$
$\displaystyle V(\hat{u},Jy_{n}-\lambda_{n}Ay_{n})$ $\displaystyle\leq$
$\displaystyle V(\hat{u},Jy_{n})-2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-\hat{u},\lambda_{n}Ay_{n}\rangle$
$\displaystyle=$ $\displaystyle\phi(\hat{u},y_{n})+2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n}),-\lambda_{n}Ay_{n}\rangle$
$\displaystyle-2\langle y_{n}-\hat{u},\lambda_{n}Ay_{n}\rangle,$ (17)

then from Lemma 4 and the condition $\|Ax\|\leq\|Ax-A\hat{u}\|$ for all $x\in
C$, it is followed that

$\displaystyle 2\langle J^{-1}(Jy_{n}-$
$\displaystyle\lambda_{n}Ay_{n})-J^{-1}(Jy_{n}),-\lambda_{n}Ay_{n}\rangle$
$\displaystyle\leq$ $\displaystyle
2\|J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n})\|\|-\lambda_{n}Ay_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ay_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ay_{n}-A\hat{u}\|^{2}.$ (18)

Since $A$ is $\alpha$-inverse strongly monotone and the fact that $\hat{u}\in
VI(C,A)$, we have

$\displaystyle-2\langle y_{n}-\hat{u},$
$\displaystyle\lambda_{n}Ay_{n}\rangle$ $\displaystyle=$
$\displaystyle-2\lambda_{n}\langle
y_{n}-\hat{u},Ay_{n}-A\hat{u}\rangle-2\lambda_{n}\langle
y_{n}-\hat{u},A\hat{u}\rangle$ $\displaystyle\leq$
$\displaystyle-2\lambda_{n}\langle y_{n}-\hat{u},Ay_{n}-A\hat{u}\rangle$
$\displaystyle\leq$ $\displaystyle-2\lambda_{n}\alpha\|Ay_{n}-A\hat{u}\|^{2},$
(19)

substituting (3) and (3) in (3) and using our assumptions, we obtain

$\displaystyle\phi(\hat{u},z_{n})\leq$
$\displaystyle\phi(\hat{u},y_{n})+(\frac{4\lambda_{n}^{2}}{c^{2}}-2\lambda_{n}\alpha)\|Ay_{n}-Au\|^{2}$
$\displaystyle=$
$\displaystyle\phi(\hat{u},y_{n})+2\lambda_{n}(\frac{2\lambda_{n}}{c^{2}}-\alpha)\|Ay_{n}-A\hat{u}\|^{2}$
$\displaystyle\leq$ $\displaystyle\phi(\hat{u},y_{n}),$

hence,

$\phi(\hat{u},z_{n})\leq\phi(\hat{u},y_{n}).$ (20)

From Lemma 3 and the inequality (14), we have

$\displaystyle\phi(\hat{u},y_{n})=$
$\displaystyle\phi(\hat{u},\Pi_{C}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))$
$\displaystyle\leq$
$\displaystyle\phi(\hat{u},J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))=V(\hat{u},Jx_{n}-\lambda_{n}Ax_{n})$
$\displaystyle\leq$ $\displaystyle V(\hat{u},Jx_{n})-2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-\hat{u},\lambda_{n}Ax_{n}\rangle$
$\displaystyle=$ $\displaystyle\phi(\hat{u},x_{n})-2\lambda_{n}\langle
x_{n}-\hat{u},Ax_{n}\rangle$ $\displaystyle+2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n}),-\lambda_{n}Ax_{n}\rangle,$
(21)

since $A$ is $\alpha-$inverse strongly monotone and $\hat{u}\in VI(C,A)$, it
follows that

$\displaystyle-2\lambda_{n}\langle x_{n}-\hat{u},$ $\displaystyle
Ax_{n}\rangle$ $\displaystyle=$ $\displaystyle-2\lambda_{n}\langle
x_{n}-\hat{u},Ax_{n}-A\hat{u}\rangle-2\lambda_{n}\langle
x_{n}-\hat{u},A\hat{u}\rangle$ $\displaystyle\leq$
$\displaystyle-2\lambda_{n}\langle x_{n}-\hat{u},Ax_{n}-A\hat{u}\rangle$
$\displaystyle\leq$ $\displaystyle-2\lambda_{n}\alpha\|Ax_{n}-A\hat{u}\|^{2}.$
(22)

From Lemma 4 and our assumptions, we can conclude

$\displaystyle 2\langle J^{-1}(Jx_{n}-\lambda_{n}$ $\displaystyle
Ax_{n})-J^{-1}(Jx_{n}),-\lambda_{n}Ax_{n}\rangle$ $\displaystyle\leq$
$\displaystyle
2\|J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n})\|\|-\lambda_{n}Ax_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ax_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ax_{n}-A\hat{u}\|^{2}.$ (23)

By applying (3) and (3) in (3) and our assumptions, it is implied that

$\phi(\hat{u},y_{n})\leq\phi(\hat{u},x_{n})+2\lambda_{n}(\frac{2\lambda_{n}}{c^{2}}-\alpha)\|Ay_{n}-Ax_{n}\|^{2}\leq\phi(\hat{u},x_{n}).$
(24)

Hence, from (20) and (24), we have

$\phi(\hat{u},z_{n})\leq\phi(\hat{u},x_{n}).$ (25)

Next, it will be shown that the sequence $\\{\phi(\hat{u},x_{n})\\}$ is
decreasing. From the relatively nonexpansiveness condition of $f$, convexity
of $\|.\|^{2}$, Lemma 3 and the inequality (25), it is implied that

$\displaystyle\phi(\hat{u},x_{n+1})\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n})$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},x_{n})$
$\displaystyle=$ $\displaystyle\phi(\hat{u},x_{n}),$ (26)

so $\\{\phi(\hat{u},x_{n})\\}$ is decreasing. Then it is implied that
$\\{\phi(\hat{u},x_{n})\\}$ is bounded, hence
$\displaystyle\lim_{n\rightarrow\infty}\phi(\hat{u},x_{n})$ exists. Then from
(9), $\\{x_{n}\\}$ is bounded. It follows from the relatively nonexpansiveness
condition of $f$, (24) and (25) that $\\{f(x_{n})\\}$, $\\{y_{n}\\}$ and
$\\{z_{n}\\}$ are bounded. From Lemmas 3, 4, the inequality (14) and the
condition $\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$, we have

$\displaystyle\phi(x_{n},y_{n})$
$\displaystyle\leq\phi(x_{n},J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))$
$\displaystyle=$ $\displaystyle V(x_{n},Jx_{n}-\lambda_{n}Ax_{n})$
$\displaystyle\leq$ $\displaystyle V(x_{n},Jx_{n})-2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-x_{n},\lambda_{n}Ax_{n})$ $\displaystyle=$
$\displaystyle\phi(x_{n},x_{n})-2\langle
J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n}),\lambda_{n}Ax_{n})\rangle$
$\displaystyle\leq$ $\displaystyle
2\|J^{-1}(Jx_{n}-\lambda_{n}Ax_{n})-J^{-1}(Jx_{n})\|\|\lambda_{n}Ax_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ax_{n}\|^{2}\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (27)

By Lemma 1, it is implied that

$\displaystyle\lim_{n\rightarrow\infty}\|x_{n}-y_{n}\|=0.$ (28)

Next, from (13), (28), the boundedness of the sequences $\\{x_{n}\\}$ and
$\\{y_{n}\\}$, and using uniformly norm-to-norm continuity of $J$ on bounded
sets, it is obvious that

$\phi(y_{n},x_{n})\leq\|y_{n}\|\|Jy_{n}-Jx_{n}\|+\|x_{n}-y_{n}\|\|x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (29)

By Lemmas 3, 4, the inequality (14) and the condition
$\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$, we have

$\displaystyle\phi(y_{n},z_{n})$
$\displaystyle=\phi(y_{n},J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}))$ $\displaystyle=$
$\displaystyle V(y_{n},Jy_{n}-\lambda_{n}Ay_{n})$ $\displaystyle\leq$
$\displaystyle V(y_{n},Jy_{n})-2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-y_{n},\lambda_{n}Ay_{n})$ $\displaystyle=$
$\displaystyle\phi(y_{n},y_{n})-2\langle
J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n}),\lambda_{n}Ay_{n})\rangle$
$\displaystyle\leq$ $\displaystyle
2\|J^{-1}(Jy_{n}-\lambda_{n}Ay_{n})-J^{-1}(Jy_{n})\|\|\lambda_{n}Ay_{n}\|$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Ay_{n}\|^{2}\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (30)

By Lemma 1, it is implied that

$\displaystyle\lim_{n\rightarrow\infty}\|y_{n}-z_{n}\|=0.$ (31)

Since $\\{f(x_{n})\\}$ and $\\{z_{n}\\}$ are bounded. Now, setting
$r_{1}=sup\\{\|f(x_{n})\|,\|z_{n}\|\\}$, by Lemma 5 there exists a continuous
strictly increasing and convex function
$g_{1}:[0,2r_{1}]\longrightarrow[0,\infty]$ with $g_{1}(0)=0$. From (25),
Lemmas 3, 5 and the condition relatively nonexpansiveness of $f$, it is
concluded for each $\hat{u}\in\Gamma$ that

$\displaystyle\phi(\hat{u},x_{n+1})\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n})$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},x_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle\phi(\hat{u},x_{n})-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|),$

therefore

$\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)\leq\phi(\hat{u},x_{n})-\phi(\hat{u},x_{n+1}).$

Since $\liminf_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$, we have

$\lim_{n\rightarrow\infty}g_{1}(\|Jf(x_{n})-Jz_{n}\|)=0,$ (32)

because $\\{\phi(\hat{u},x_{n})\\}$ is Cauchy and
$\displaystyle\lim_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$. Since
$g_{1}$ is a continuous function, so

$g_{1}(\lim_{n\rightarrow\infty}\|Jf(x_{n})-Jz_{n}\|)=\lim_{n\rightarrow\infty}g_{1}(\|Jf(x_{n})-Jz_{n}\|)=0=g_{1}(0),$
(33)

and also $g_{1}$ is strictly increasing, hence

$\lim_{n\rightarrow\infty}\|Jf(x_{n})-Jz_{n}\|)=0.$ (34)

On the other hand, since $J^{-1}$ is uniformly norm-to-norm continuous on
bounded sets, we obtain that

$\lim_{n\rightarrow\infty}\|f(x_{n})-z_{n}\|=\lim_{n\rightarrow\infty}\|J^{-1}(Jf(x_{n}))-J^{-1}(Jz_{n})\|=0.$
(35)

Next, from (13) and (35), we have

$\displaystyle\lim_{n\rightarrow\infty}\phi(z_{n},f(x_{n}))=0.$ (36)

Similarly, from (13), (28) and (31), we obtain

$\displaystyle\lim_{n\rightarrow\infty}\phi(z_{n},x_{n})=0.$ (37)

Moreover, from Lemma 3, the inequalities (36), (37) and the convexity of
$\|.\|^{2}$, it is concluded that

$\displaystyle\phi(z_{n},x_{n+1})\leq$
$\displaystyle\phi(z_{n},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}))$
$\displaystyle=$ $\displaystyle\|z_{n}\|^{2}-2\langle
z_{n},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}\|^{2}$
$\displaystyle\leq$ $\displaystyle\|z_{n}\|^{2}-2\alpha_{n,1}\langle
z_{n},Jx_{n}\rangle-2\alpha_{n,2}\langle
z_{n},Jf(x_{n})\rangle-2\alpha_{n,3}\langle z_{n},Jz_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n}))+\alpha_{n,3}\phi(z_{n},z_{n})$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n}))\rightarrow
0\;\;\;as\;n\rightarrow\infty,$

then using Lemma 1, we get

$\displaystyle\lim_{n\rightarrow\infty}\|z_{n}-x_{n+1}\|=0.$ (38)

It follows from (28), (31) and (38) that

$\|x_{n+1}-x_{n}\|\leq\|x_{n+1}-z_{n}\|+\|z_{n}-y_{n}\|+\|y_{n}-x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty.$ (39)

So $\\{x_{n}\\}$ is a Cauchy sequence, thus $\\{x_{n}\\}$ converges strongly
to a point $q\in C$. It follows from (28) and (31) that the sequences
$\\{y_{n}\\}$ and $\\{z_{n}\\}$ are convergent to $q$. Next, it will be shown
that $q\in VI(C,A)$. Let $B\subset E\times E^{*}$ be an operator defined as
follows:

$Bv=\left\\{\begin{array}[]{lr}\lambda_{n}Av+N_{C}v,\qquad v\in C,\\\
\emptyset,\qquad\qquad\qquad v\notin C.\end{array}\right.$ (40)

Since $\lambda_{n}A$ is $\lambda_{n}\alpha$-inverse strongly monotone, it is
followed that $\lambda_{n}A$ is $\frac{1}{\lambda_{n}\alpha}$-Lipschitz
continuous, hence $\lambda_{n}A$ is hemicontinuous. Therefore, by Lemma 8 $B$
is maximal monotone and $B^{-1}(0)=VI(C,\lambda_{n}A)=VI(C,A)$. Let
$(\upsilon,w)\in G(B)$ with $w\in
B\upsilon=\lambda_{n}A\upsilon+N_{C}(\upsilon)$. Then
$w-\lambda_{n}A\upsilon\in N_{C}(\upsilon)$, hence

$\langle\upsilon-y_{n},w-\lambda_{n}A\upsilon\rangle\geq 0,$ (41)

because $y_{n}\in C$. On the other hand by Lemma 2, it is concluded that

$\langle\upsilon-y_{n},J(J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}))-Jy_{n}\rangle\leq
0,$

so

$\langle\upsilon-y_{n},\lambda_{n}Ax_{n}+Jy_{n}-Jx_{n}\rangle\geq 0.$ (42)

From (41), (42) and using the definition $A$, we get

$\displaystyle\langle\upsilon-y_{n},w\rangle$ $\displaystyle\geq$
$\displaystyle\lambda_{n}\langle\upsilon-
y_{n},A\upsilon\rangle-\langle\upsilon-
y_{n},\lambda_{n}Ax_{n}+Jy_{n}-Jx_{n}\rangle$ $\displaystyle=$
$\displaystyle\lambda_{n}\langle\upsilon-y_{n},A\upsilon-
Ay_{n}\rangle+\lambda_{n}\langle\upsilon-y_{n},Ay_{n}\rangle$
$\displaystyle-\langle\upsilon-y_{n},\lambda_{n}Ax_{n}+Jy_{n}-Jx_{n}\rangle$
$\displaystyle\geq$ $\displaystyle\lambda_{n}\langle\upsilon-
y_{n},Ay_{n}-Ax_{n}\rangle-\langle\upsilon-y_{n},Jy_{n}-Jx_{n}\rangle$
$\displaystyle\geq$ $\displaystyle-\lambda_{n}\|\upsilon-
y_{n}\|\|Ax_{n}-Ay_{n}\|-\|\upsilon-y_{n}\|\|Jx_{n}-Jy_{n}\|.$ (43)

Hence, using uniformly norm-to-norm continuity of $J$ on bounded sets and
(28), $\langle\upsilon-y_{n},w\rangle\geq 0$ as $n\rightarrow\infty$, i.e.
$\langle\upsilon-q,w\rangle\geq 0$. Therefore $\langle
q-\upsilon,0-w\rangle\geq 0$, it is concluded from Lemma 8 that $q\in
B^{-1}(0)=VI(C,A)$, because $B$ is a maximal monotone operator.

Next, we show that $q\in F(f)$. From (28), (31) and (35), we have

$\|f(x_{n})-x_{n}\|\leq\|f(x_{n})-z_{n}\|+\|z_{n}-y_{n}\|+\|y_{n}-x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty,$ (44)

and since $x_{n}\rightharpoonup q$, then $q$ is an asymptotic fixed point of
$f$. Moreover, $\hat{F}(f)=F(f)$, because $f$ is a relatively nonexpansive
mapping, hence $q\in F(f)$. Therefore,
$\Pi_{VI(C,A)}of(q)=\Pi_{VI(C,A)}(q)=q$.

###### Theorem 3.2

Suppose that $\tilde{F}$ is a bifunction from $C\times C$ to $\mathbb{R}$
which satisfies the conditions $(A_{1})-(A_{4})$. Let f be a relatively
nonexpansive self-mapping on $C$ and $\|Ax\|\leq\|Ax-Au\|$ for all $x\in C$
and $u\in\Omega:=VI(C,A)\cap GEP(\tilde{F},A)\cap F(f)$. Let $x_{0}$ be an
arbitrary point in $C$ and $\\{x_{n}\\}$ be a sequence generated by

$\left\\{\begin{array}[]{lr}u_{n}\in C\;\;s.t\;\;\tilde{F}(u_{n},y)+\langle
Au_{n},y-u_{n}\rangle+\frac{1}{r_{n}}\langle y-u_{n},Ju_{n}-Jx_{n}\rangle\geq
0,\\\ w_{n}=\Pi_{C}J^{-1}(Ju_{n}-\lambda_{n}Au_{n}),\\\
y_{n}=\Pi_{C}J^{-1}(Jx_{n}-\lambda_{n}Ax_{n}),\\\ C_{n}=\\{v\in
C:\phi(v,w_{n})\leq\phi(v,x_{n})\\},\\\
z_{n}=\Pi_{C_{n}}J^{-1}(Jy_{n}-\lambda_{n}Ay_{n}),\\\
x_{n+1}=\Pi_{C}J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}).\end{array}\right.$
(45)

where $\\{\lambda_{n}\\}\subseteq[0,1]$ such that
$\displaystyle\lim_{n\rightarrow\infty}\lambda_{n}=0$, and
$\\{r_{n}\\}\subset[a,\infty)$ for some $a>0$. If
$\\{\alpha_{n,i}\\}\subset[0,1]$ for $i=1,2,3,4$ such that
$\sum_{i=1}^{4}\alpha_{n,i}=1$ and
$\displaystyle\liminf_{n\to\infty}\alpha_{n,2}\alpha_{n,3}>0$ and
$\displaystyle\liminf_{n\to\infty}\alpha_{n,2}\alpha_{n,4}>0$. Then the
sequence $\\{x_{n}\\}$ generated by (45) converges strongly to
$q=\Pi_{VI(C,A)\cap GEP(\tilde{F},A)}\circ f(q)$.

###### Proof

Clearly, by part $(i)$ of lemma 7, the sequence $\\{u_{n}\\}$ exists. Now, it
will be checked that $C_{n}$ is closed and convex for each $n\geq 1$.
Obviously, by the definition of $C_{n}$, it is clear that $C_{n}$ is closed.
Applying the definition of $\phi$, the inequality
$\phi(v,w_{n})\leq\phi(v,x_{n})$ is equivalent to

$2\langle v,Jx_{n}-Jw_{n}\rangle\leq\|x_{n}\|^{2}-\|w_{n}\|^{2}.$ (46)

It is obvious from (46) that $C_{n}$ is convex for all $n\geq 1$.

Now, it will be verified that $\\{x_{n}\\}$ is well defined. Suppose that
$p\in\Omega$. By Lemma 7, it may be put $u_{n}=K_{r_{n}}x_{n}$. So, by the
condition (5) of Lemma 7, it is concluded that

$\phi(p,u_{n})=\phi(p,K_{r_{n}}x_{n})\leq\phi(p,x_{n}).$ (47)

Moreover, from Lemma 3 and the inequality (14), it follows that

$\displaystyle\phi(p,w_{n})=$
$\displaystyle\phi(p,\Pi_{C}J^{-1}(Ju_{n}-\lambda_{n}Au_{n}))$
$\displaystyle\leq$ $\displaystyle\phi(p,J^{-1}(Ju_{n}-\lambda_{n}Au_{n}))$
$\displaystyle\leq$ $\displaystyle V(p,Ju_{n}-\lambda_{n}Au_{n})$
$\displaystyle\leq$ $\displaystyle V(p,Ju_{n})-2\langle
J^{-1}(Ju_{n}-\lambda_{n}Au_{n})-p,\lambda_{n}Au_{n}\rangle$ $\displaystyle=$
$\displaystyle\phi(p,u_{n})-2\lambda_{n}\langle u_{n}-p,Au_{n}\rangle$
$\displaystyle+2\langle
J^{-1}(Ju_{n}-\lambda_{n}Au_{n})-J^{-1}(Ju_{n}),-\lambda_{n}Au_{n}\rangle,$
(48)

since $A$ is an $\alpha-$inverse strongly monotone operator, it is proved that

$\displaystyle-2\lambda_{n}\langle u_{n}-p,$ $\displaystyle Au_{n}\rangle$
$\displaystyle=$ $\displaystyle-2\lambda_{n}\langle
u_{n}-p,Au_{n}-Ap\rangle-2\lambda_{n}\langle u_{n}-p,Ap\rangle$
$\displaystyle\leq$ $\displaystyle-2\lambda_{n}\alpha\|Au_{n}-Ap\|^{2}.$ (49)

From Lemma 4 and the condition $\|Ax\|\leq\|Ax-Ap\|$ for all $x\in C$, it is
demonstrated that

$\displaystyle 2\langle J^{-1}(Ju_{n}-\lambda_{n}$ $\displaystyle
Au_{n})-J^{-1}(Ju_{n}),-\lambda_{n}Au_{n}\rangle$ $\displaystyle\leq$
$\displaystyle
2\|J^{-1}(Ju_{n}-\lambda_{n}Au_{n})-J^{-1}(Ju_{n})\|\|\lambda_{n}Au_{n}\|$
$\displaystyle=$ $\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Au_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{4\lambda_{n}^{2}}{c^{2}}\|Au_{n}-Ap\|^{2}.$ (50)

By substituting (3) and (3) in (3) and the assumption
$0<\lambda_{n}<\frac{c^{2}\alpha}{2}$, it is implied that

$\phi(p,w_{n})\leq\phi(p,u_{n})+2\lambda_{n}(\frac{2}{c^{2}}\lambda_{n}-\alpha)\|Au_{n}-Ap\|^{2}\leq\phi(p,u_{n}).$
(51)

From (47) and (51), it is evident that

$\phi(p,w_{n})\leq\phi(p,x_{n}).$ (52)

Then $p\in C_{n}$ and hence $\\{x_{n}\\}$ is well defined.

Let $\Omega\neq\emptyset$ and $\hat{u}\in\Omega$. From Lemma 3, the convexity
of $\|.\|^{2}$ and the relatively nonexpansiveness of $f$, it follows

$\displaystyle\phi(\hat{u},x_{n+1}$ $\displaystyle)$ $\displaystyle\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}))$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle-2\alpha_{n,4}\langle\hat{u},Jw_{n}\rangle$
$\displaystyle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}+\alpha_{n,4}\|w_{n}\|^{2}$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle=$
$\displaystyle(\alpha_{n,1}+\alpha_{n,2})\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n}).$

Similarly, using Lemma 3, the inequality (25) holds for the algorithm (45),
too. Hence, from (25) and (52), it is implied that

$\phi(\hat{u},x_{n+1})\leq\phi(\hat{u},x_{n}).$ (53)

It is concluded that $\\{\phi(\hat{u},x_{n})\\}$ is decreasing, so from the
boundedness of the sequence $\\{\phi(\hat{u},x_{n})\\}$,
$\displaystyle\lim_{n\rightarrow\infty}\phi(\hat{u},x_{n})$ exists. Also from
(9), $\\{x_{n}\\}$ is bounded and hence from (47) and the relatively
nonexpansiveness of $f$, $\\{u_{n}\\}$ and $\\{f(x_{n})\\}$ are bounded.
Similarly, using Lemma 3, the inequalities (28) and (31) hold for the
algorithm (45). Hence, it is concluded from (28) and (31) that the sequences
$\\{y_{n}\\}$ and $\\{z_{n}\\}$ are bounded. Now, let
$r_{1}=\sup\\{\|z_{n}\|,\|f(x_{n})\|\\}$, by Lemma 5, there exists a
continuous strictly increasing and convex function
$g_{1}:[0,2r_{1}]\longrightarrow[0,\infty)$ with $g_{1}(0)=0$. We get

$\displaystyle\phi(\hat{u},x_{n+1}$ $\displaystyle)$ $\displaystyle\leq$
$\displaystyle\phi(\hat{u},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}))$
$\displaystyle=$
$\displaystyle\|\hat{u}\|^{2}-2\langle\hat{u},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\|^{2}$
$\displaystyle\leq$
$\displaystyle\|\hat{u}\|^{2}-2\alpha_{n,1}\langle\hat{u},Jx_{n}\rangle-2\alpha_{n,2}\langle\hat{u},Jf(x_{n})\rangle-2\alpha_{n,3}\langle\hat{u},Jz_{n}\rangle$
$\displaystyle-2\alpha_{n,4}\langle\hat{u},Jw_{n}\rangle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle+\alpha_{n,4}\|w_{n}\|^{2}-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},f(x_{n}))+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle\leq$
$\displaystyle\alpha_{n,1}\phi(\hat{u},x_{n})+\alpha_{n,2}\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)$
$\displaystyle=$
$\displaystyle(\alpha_{n,1}+\alpha_{n,2})\phi(\hat{u},x_{n})+\alpha_{n,3}\phi(\hat{u},z_{n})+\alpha_{n,4}\phi(\hat{u},w_{n})$
$\displaystyle-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|).$

Now from (25) and (52), we have

$\phi(\hat{u},x_{n+1})\leq\phi(\hat{u},x_{n})-\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|).$
(54)

So

$\alpha_{n,2}\alpha_{n,3}g_{1}(\|Jf(x_{n})-Jz_{n}\|)\leq\phi(\hat{u},x_{n})-\phi(\hat{u},x_{n+1}).$

Since $\liminf_{n\rightarrow\infty}\alpha_{n,2}\alpha_{n,3}>0$, using the
method as in the proof of Theorem 3.1, we conclude that the inequality (36)
and (37) hold.

By Lemma 3 and convexity of $\|.\|^{2}$, it is obtained that

$\displaystyle\phi(z_{n},x_{n+1}$ $\displaystyle)$ $\displaystyle\leq$
$\displaystyle\phi(z_{n},J^{-1}(\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}))$
$\displaystyle=$ $\displaystyle\|z_{n}\|^{2}-2\langle
z_{n},\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\rangle$
$\displaystyle+\|\alpha_{n,1}Jx_{n}+\alpha_{n,2}Jf(x_{n})+\alpha_{n,3}Jz_{n}+\alpha_{n,4}Jw_{n}\|^{2}$
$\displaystyle\leq$ $\displaystyle\|z_{n}\|^{2}-2\alpha_{n,1}\langle
z_{n},Jx_{n}\rangle-2\alpha_{n,2}\langle
z_{n},Jf(x_{n})\rangle-2\alpha_{n,3}\langle z_{n},Jz_{n}\rangle$
$\displaystyle-2\alpha_{n,4}\langle
z_{n},Jw_{n}\rangle+\alpha_{n,1}\|x_{n}\|^{2}+\alpha_{n,2}\|f(x_{n})\|^{2}+\alpha_{n,3}\|z_{n}\|^{2}$
$\displaystyle+\alpha_{n,4}\|w_{n}\|^{2}$ $\displaystyle=$
$\displaystyle\alpha_{n,1}\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n}))+\alpha_{n,3}\phi(z_{n},z_{n})+\alpha_{n,4}\phi(z_{n},w_{n})$
$\displaystyle\leq$
$\displaystyle(\alpha_{n,1}+\alpha_{n,4})\phi(z_{n},x_{n})+\alpha_{n,2}\phi(z_{n},f(x_{n})),$

because $z_{n}\in C_{n}$. Using (36), (37) and taking the limit in the above
as $n\rightarrow\infty$, it is deduced that

$\phi(z_{n},x_{n+1})\rightarrow 0.$

Then, from Lemma 1, we have

$\displaystyle\lim_{n\rightarrow\infty}\|x_{n+1}-z_{n}\|=0,$

therefore, it follows from (28), (31) that

$\displaystyle\|x_{n+1}-x_{n}\|\leq\|x_{n+1}-z_{n}\|+\|z_{n}-y_{n}\|+\|y_{n}-x_{n}\|\rightarrow
0\;\;as\;\;n\rightarrow\infty,$

hence, $\\{x_{n}\\}$ is a cauchy sequence. Thus, $\\{x_{n}\\}$ converges
strongly to a point $q\in C$. Obviously, the relations (40), (41), (42) and
(3) are valid for the algorithm (45). Hence, as in the proof of Theorem 3.1,
it is understood that $q\in VI(C,A)$.

Now, it will be proved that $q\in GEP(\tilde{F},A)$. From (37) and the fact
that $z_{n}\in C_{n}$, it is induced that $\phi(z_{n},w_{n})\rightarrow 0$ as
$n\rightarrow\infty$. Therefore, by Lemma 1, we have

$\displaystyle\lim_{n\rightarrow\infty}\|z_{n}-w_{n}\|=0.$ (55)

From (28), (31) and (55), it is evident that

$\displaystyle\lim_{n\rightarrow\infty}\|x_{n}-w_{n}\|=0.$ (56)

Assume that $r_{2}=\sup\\{\|u_{n}\|,\|x_{n}\|\\}$. From Lemma 6, there exists
a continuous, convex and strictly increasing function
$g_{2}:[0,2r_{2}]\longrightarrow[0,\infty)$ such that $g_{2}(0)=0$ and

$\displaystyle g_{2}(\|u_{n}-x_{n}\|)\leq\phi(u_{n},x_{n}).$ (57)

Since $u_{n}=K_{r_{n}}(x_{n})$ and by using (51), (57) and condition $(5)$ of
Lemma 7, it is implied that

$\displaystyle g_{2}(\|u_{n}-x_{n}\|)\leq$ $\displaystyle\phi(u_{n},x_{n})$
$\displaystyle\leq$ $\displaystyle\phi(u,x_{n})-\phi(u,u_{n})$
$\displaystyle\leq$ $\displaystyle\phi(u,x_{n})-\phi(u,w_{n})$
$\displaystyle=$ $\displaystyle\|u\|^{2}-2\langle
u,Jx_{n}\rangle+\|x_{n}\|^{2}-\|u\|^{2}+2\langle
u,Jw_{n}\rangle-\|w_{n}\|^{2}$ $\displaystyle=$
$\displaystyle\|x_{n}\|^{2}-\|w_{n}\|^{2}+2\langle u,Jw_{n}-Jx_{n}\rangle$
$\displaystyle\leq$
$\displaystyle\|x_{n}\|^{2}-\|w_{n}\|^{2}+2\|u\|\|Jw_{n}-Jx_{n}\|$
$\displaystyle\leq$
$\displaystyle(\|x_{n}-w_{n}\|+\|w_{n}\|)^{2}-\|w_{n}\|^{2}+2\|u\|\|Jw_{n}-Jx_{n}\|$
$\displaystyle\leq$
$\displaystyle\|x_{n}-w_{n}\|^{2}+2\|w_{n}\|\|x_{n}-w_{n}\|+2\|u\|\|Jw_{n}-Jx_{n}\|,$

from (56) and the condition uniformly norm-to-norm continuity of $J$ on
bounded sets, we have
$\displaystyle\lim_{n\rightarrow\infty}g_{2}(\|u_{n}-x_{n}\|)=0$. Then it is
followed from the conditions that $g_{2}$ is a strictly increasing and
continuous function that $\|u_{n}-x_{n}\|\rightarrow 0$ as
$n\rightarrow\infty$. Then

$\lim_{n\rightarrow\infty}\|Ju_{n}-Jx_{n}\|\rightarrow 0.$ (58)

Since $u_{n}=K_{r_{n}}x_{n}$, we concluded that

$\tilde{F}(u_{n},y)+\langle Au_{n},y-u_{n}\rangle+\frac{1}{r_{n}}\langle
y-u_{n},Ju_{n}-Jx_{n}\rangle\geq 0,$ (59)

for all $y\in C$. From the condition $(A_{2})$, we have

$\tilde{F}(y,u_{n})\leq-\tilde{F}(u_{n},y)\;\;\text{for}\;\text{all}\;y\in C.$
(60)

From (59) and (60), it is implied that

$\displaystyle\tilde{F}(y,u_{n})\leq-\tilde{F}(u_{n},y)\leq\langle
Au_{n},y-u_{n}\rangle+\frac{1}{r_{n}}\langle y-u_{n},Ju_{n}-Jx_{n}\rangle,$

for all $y\in C$. Letting $n\rightarrow\infty$, using condition $(A_{4})$ and
by (58), it can be concluded that

$\tilde{F}(y,q)\leq\langle Aq,y-q\rangle\;\;\;\text{for}\;\text{all}\;y\in C.$
(61)

Put $y_{\lambda}=\lambda y+(1-\lambda)q$ for all $y\in C$ and
$\lambda\in(0,1)$. Now from the conditions $(A_{1})$, $(A_{4})$, the
inequality (61), the monotonicity of $A$ and the convexity of $\tilde{F}$, we
have

$\displaystyle 0=$ $\displaystyle\tilde{F}(y_{\lambda},y_{\lambda})+\langle
Ay_{\lambda},y_{\lambda}-y_{\lambda}\rangle$ $\displaystyle\leq$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+(1-\lambda)\tilde{F}(y_{\lambda},q)+\langle
Ay_{\lambda},\lambda y+(1-\lambda)q-y_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+(1-\lambda)\tilde{F}(y_{\lambda},q)+\lambda\langle
Ay_{\lambda},y-y_{\lambda}\rangle+(1-\lambda)\langle
Ay_{\lambda},q-y_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+(1-\lambda)\tilde{F}(y_{\lambda},q)+\lambda\langle
Ay_{\lambda},y-y_{\lambda}\rangle+(1-\lambda)\langle
Ay_{\lambda}-Aq,q-y_{\lambda}\rangle$ $\displaystyle+(1-\lambda)\langle
Aq,q-y_{\lambda}\rangle$ $\displaystyle\leq$
$\displaystyle\lambda\tilde{F}(y_{\lambda},y)+\lambda\langle
Ay_{\lambda},y-y_{\lambda}\rangle,$

for all $y\in C$. So $0\leq\tilde{F}(y_{\lambda},y)+\langle
Ay_{\lambda},y-y_{\lambda}\rangle$. Now by taking limit as $\lambda\rightarrow
0$ and by using the condition $(A_{3})$, it is followed that
$0\leq\tilde{F}(q,y)+\langle Aq,y-q\rangle$ for all $y\in C$. Therefore $q\in
GEP(\tilde{F},A)$.

Now, we show that $q\in F(f)$. Let $r_{3}=\sup\\{\|w_{n}\|,\|f(x_{n})\|\\}$,
hence, in a similar way with (54), there exists a continuous, convex and
strictly increasing function $g_{3}:[0,2r_{3}]\longrightarrow[0,\infty)$ whit
$g_{3}(0)=0$, such that

$\phi(\hat{u},x_{n+1})\leq\phi(\hat{u},x_{n})-\alpha_{n,2}\alpha_{n,4}g_{3}(\|Jf(x_{n})-Jw_{n}\|),$

hence

$\alpha_{n,2}\alpha_{n,4}g_{3}(\|Jf(x_{n})-Jw_{n}\|)\leq\phi(\hat{u},x_{n+1})-\phi(\hat{u},x_{n}).$

Taking the limit as $n\longrightarrow\infty$ and using our assumptions, we
obtain

$\displaystyle\lim_{n\rightarrow\infty}g_{3}(\|Jf(x_{n})-Jw_{n}\|)=0,$

since $g_{1}$ is a continuous function, it is easy to see that

$\displaystyle\lim_{n\rightarrow\infty}\|Jf(x_{n})-Jw_{n}\|=0.$ (62)

Therefore

$\lim_{n\rightarrow\infty}\|f(x_{n})-w_{n}\|=\lim_{n\rightarrow\infty}\|J^{-1}(Jf(x_{n}))-J^{-1}(Jw_{n})\|=0,$
(63)

because $J^{-1}$ is uniformly norm-to-norm continuous on bounded sets. From
(56) and (63), it is concluded that

$\|f(x_{n})-x_{n}\|\leq\|f(x_{n})-w_{n}\|+\|w_{n}-x_{n}\|\rightarrow
0\;\;\;as\;n\rightarrow\infty,$

and since $x_{n}\rightharpoonup q$, then $q\in\hat{F(f)}=F(f)$. Hence
$\\{x_{n}\\}$ is strongly convergent to a point $q\in\Omega$, and also we have
$q=\Pi_{VI(C,A)\cap GEP(\tilde{F},A)}\circ f(q)$.

##  4 Numerical example

Now, some examples are given to illustrate Theorem 3.2. Then the behaviors of
the sequences $\\{x_{n}\\},\\{y_{n}\\},\\{z_{n}\\}$ and $\\{w_{n}\\}$ are
investigated which were generated by the algorithm (45).

###### Example 1

Let $E=\mathbb{R}$, $C=[-5,5]$, $A=I$, $\lambda_{n}=\frac{1}{n}$, $c=1$,
$\alpha=1$ and $f$ be a self-mapping on $C$ defined by $f(x)=\frac{x}{3}$ for
all $x\in C$. Consider the function $\tilde{F}:C\times C\rightarrow\mathbb{R}$
defined by

$\tilde{F}(u,y):=16y^{2}+9uy-25u^{2},$

for all $u$, $y\in C$. We see that $f$ satisfies in the conditions (A1) - (A4)
as follows:
(A1) $\tilde{F}(u,u)=16u^{2}+9u^{2}-25u^{2}=0$ for all $u\in[-5,5]$,
(A2) $\tilde{F}$ is monotone, because
$\tilde{F}(u,y)+\tilde{F}(y,u)=-9(u-y)^{2}\leq 0$ for all $y,u\in[-5,5]$,
(A3) for each $u,y,z\in[-5,5],$

$\displaystyle\lim_{\lambda\to 0}\tilde{F}(\lambda z+(1$
$\displaystyle-\lambda)u,y)$ $\displaystyle=$ $\displaystyle\lim_{\lambda\to
0}(16y^{2}+9(\lambda z+(1-\lambda)u)y-25(\lambda z+(1-\lambda)u)^{2})$
$\displaystyle=$ $\displaystyle 16y^{2}+9uy-25u^{2}$ $\displaystyle=$
$\displaystyle\tilde{F}(u,y).$

(A4) Obviously, for each $u\in[-5,5]$, $y\rightarrow(16y^{2}+9uy-25u^{2})$ is
convex and lower semicontinuous.
Let $u\in K_{r}x$, hence, it is concluded from Lemma 7 that

$\displaystyle\tilde{F}(u,y)+\langle Au,y-u\rangle+\frac{1}{r}\langle y-u,Ju-
Jx\rangle\geq 0,$

for all $y\in[-5,5]$ and $r>0$, i.e.,

$\displaystyle 0\leq 16ry^{2}+9ruy-25ru^{2}+$ $\displaystyle ruy-
ru^{2}+uy-u^{2}+ux-xy$ $\displaystyle=$ $\displaystyle
16ry^{2}+(10ru+u-x)y-26ru^{2}-u^{2}+ux.$

Let $a=16r$, $b=10ru+u-x$ and $c=-26ru^{2}-u^{2}+ux$. Then, it is implied that
$\triangle=b^{2}-4ac\leq 0$, i.e.,

$\displaystyle 0\geq(10ru+u-x)^{2}-64r(-26$ $\displaystyle ru^{2}-u^{2}+ux)$
$\displaystyle=$ $\displaystyle 1764r^{2}u^{2}+84ru^{2}+u^{2}-84rux-2ux+x^{2}$
$\displaystyle=$ $\displaystyle((42r+1)u-x)^{2}.$

It follows that $u=\frac{x}{42r+1}$. It is concluded from Lemma 7 that $K_{r}$
is single valued. Hence, $K_{r}x=\frac{x}{42r+1}$. Now by applying in theorem
3.2, it is implied that $u_{n}=\frac{x_{n}}{42r_{n}+1}$ where $\\{x_{n}\\}$ is
a sequence generated by the algorithm (45). Since $F(K_{r_{n}})=\\{0\\}$, from
condition $(3)$ of Lemma 7, we have $GEP(\tilde{F},I)=\\{0\\}$.
Obviously, $F(f)=\\{0\\}$ and $\phi(0,f(x))\leq\phi(0,x)$, for all $x\in C$.
Now, let $x_{n}\rightharpoonup q$ and also
$\lim_{n\to\infty}(f(x_{n})-x_{n})=0$, hence $q=0$ and
$\hat{F(f)}=\\{0\\}=F(f)$. Therefore, $f$ is a relatively nonexpansive
mapping. Moreover, it is obvious that $0\in VI(C,I)$. Therefore,
$0=\Pi_{\\{0\\}}of(0)=\Pi_{VI(C,I)\cap GEP(\tilde{F},I)}of(0)$.

Next, assume that
$\alpha_{n,1}=\frac{1}{4}+\frac{1}{4n},\alpha_{n,2}=\frac{1}{4}-\frac{1}{6n},\alpha_{n,3}=\frac{1}{4}+\frac{1}{12n},\alpha_{n,4}=\frac{1}{4}-\frac{1}{6n}$,
for all $n\in\mathbb{N}$ and $u_{0}=0$, so clearly $\alpha_{n}$, $\beta_{n}$
and $\gamma_{n}$ satisfy in the conditions of Theorem 3.2. Since $x_{n}\in C$,
we have

$\left\\{\begin{array}[]{lr}w_{n}=\Pi_{C}J^{-1}(u_{n}-\frac{1}{n}u_{n})=\frac{n-1}{n}u_{n}=\frac{n-1}{2n}x_{n},\\\
y_{n}=\Pi_{C}J^{-1}(x_{n}-\frac{1}{n}x_{n})=\Pi_{C}\frac{n-1}{n}x_{n}=\frac{n-1}{n}x_{n},\\\
C_{n}=\\{v\in C:|v-w_{n}|\leq|v-x_{n}|\\},\\\
z_{n}=\Pi_{C_{n}}J^{-1}(y_{n}-\frac{1}{n}y_{n})=\frac{n-1}{n}y_{n}=(\frac{n-1}{n})^{2}x_{n},\\\
x_{n+1}=\Pi_{C}J^{-1}((\frac{1}{4}+\frac{1}{4n})x_{n}+(\frac{1}{4}-\frac{1}{6n})\frac{1}{3}x_{n}+(\frac{1}{4}+\frac{1}{12n})(\frac{n-1}{n})^{2}x_{n}\\\
\hskip
31.2982pt+(\frac{1}{4}-\frac{1}{6n})\frac{n-1}{2n}x_{n}).\end{array}\right.$

See the table LABEL:tableexample1 and Figure LABEL:pp1 with the initial point
$x_{1}=5$ of the sequence $\\{x_{n}\\}$.

## Declarations

Not applicable.

## Funding

No funding is applicable to this article.

## Conflict of interest

The authors declare that they have no conflict of interest.

## Authors’ contributions

The two authors equally contributed, read, and approved the final manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Acknowledgements

The authors would like to thank the referees for their esteemed comments and
suggestions.

## References

  * (1) Agarwal, R. P., Oregan, D. and Sahu, D. R.: Fixed point theory for Lipschitzian-type mappings with applications, vol. 6, Springer, New York, (2009)
  * (2) Alber, Ya.I.: Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications. In: Kartsatos AG (ed.) Theory Appl Nonlinear Operat Accretive Monotone Type. 178, 15-50. Marcel Dekker, New York (1996)
  * (3) Beauzamy, B.: Introduction to Banach Spaces and Their Geometry. In North-Holland Mathematics Studies, vol. 68,North-Holland Publishing Co., Amsterdam (1985)
  * (4) Butnariu, D., Reich, S., Zaslavski, A.J.: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J Appl Anal. 7, 151-174 (2001)
  * (5) Butanriu, D., Reich, S., ZasIavaski A. J.: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer Funct. Anal. Optim. 24, 489-508 (2003)
  * (6) Cai, G., Gibali, A., Iyiola, O.S.: A New Double-Projection Method for Solving Variational Inequalities in Banach Spaces. J Optim Theory Appl 178, 219-239 (2018).
  * (7) Cens, Y., Reich, S.: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization. 37, 323–339 (1996). doi:10.1080/02331939608844225
  * (8) Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990)
  * (9) Censor, Y., Gibali, A., Reich, S.: Two extensions of Korpelevich’s extragradient method for solving the variational inequality problem in Euclidean space. Technical Report. (2010)
  * (10) Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148 318-335 (2011)
  * (11) Kamimura, S., Takahashi, W.: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945 (2002)
  * (12) Kohsaka, F., Takahashi, W.: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr Appl Anal. 3, 239-249 (2004)
  * (13) Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747-756 (1976)
  * (14) Jolaoso, L.O., Taiwo, A., Alakoya, T.O.: A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods. J Optim Theory Appl 185, 744-766 (2020). https://doi.org/10.1007/s10957-020-01672-3
  * (15) Liou, Y. C.: Shrinking projection method of proximal-type for a generalized equilibrium problem, a maximal monotone operator and a pair of relatively nonexpansive mappings. Taiwanese J. Math. 14, 517-540 (2010)
  * (16) Reich, S.: Book Review: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Bull. Amer. Math. Soc. 26, 367-370 (1992)
  * (17) Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J Control Optim. 14, 877-898 (1976). doi:10.1137/0314056
  * (18) Takahashi, W.:Nonlinear Functional Analysis: Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, (2000)
  * (19) Takahashi, Y., Hashimoto, K., Kato M.: On sharp uniform convexity, smoothness, and strong type, cotype inegualities. J. Nonlinear Convex Anal. 3, 267-281 (2002)
  * (20) Thong, D. V., Hieu, D. V.: Some extragradient-viscosity algorithms for solving variational inequality problems and fixed point problems. Numer Algor 82, 761-789(2019)
  * (21) Thong, D. V., Hieu, D. V.: Weak and strong convergence theorems for variational inequality problems, Numerical Algorithms. 78, 1045-1060(2018)
  * (22) Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431-446 (2000)
  * (23) Vuong, P.T., Shehu, Y.:Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer Algor 81, 269-291 (2019)
  * (24) Xu, HK.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127-1138 (2009)