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GENERAL ITERATIVE ALGORITHMS FOR NONEXPANSIVE
MAPPINGS IN BANACH SPACESJONG SOO JUNG

DEPARTMENT OF MATHEMATICS, DONG‐A UNIVERSITY

ABSTRACT. In this paper, we introduce two general iterative algorithms (one implicit
algorithm and other explicit algorithm) for nonexpansive mappings in a reflexive Banach
space with a uniformly Gâteaux differentiable norm. Strong convergence theorems for thesequences generated by the proposed algorithms are established.

1. INTRODUCTION

Let E be a real Banach space with the norm \Vert . \Vert , and let E^{*} be the dual space of E.
Let J denote the normalized duality mapping from E into 2^{E^{*}} defined by

J(x)=\{f\in E^{*} : \langle x, f\}=\Vert x\Vert\Vert f\Vert, \Vert f\Vert=\Vert x\Vert\}, \forall x\in E,
where \langle\cdot, } denotes the generalized duality pair between E and E^{*} . Let C be a nonempty
closed convex subset of E . For the mapping T:Carrow C , we denote the fixed point set of T
by Fix (T) , that is, Fix (T)=\{x\in C : Tx=x\} . Recaıl that the mapping T:Carrow C is
\cdot

said to be nonexpansive if\Vert Tx-Ty\Vert\leq\Vert x-y\Vert, \forall x, y\in C.
E having a single‐valued normalized duality mapping J , we say that

In a Banach spacean operator A is strongly positive on E if there exists a \overline{\gamma}>0 with the property{ Ax , J(x)\rangle\geq\overline{\gamma}\Vert x\Vert^{2}

(1.1)

and

\Vert aI-bA\Vert=\sup|\{(aI-bA)x, J(x)\rangle|, a\in[0,1], b\in[-1.1],\Vert x\Vert\leq 1

for all x\in E , where I is the identity mapping. If E :=H is a real Hilbert space, then theinequality (1.1) reduce to\langle Ax, x\rangle\geq\overline{\gamma}\Vert x\Vert^{2}, \forall x\in H.
One classical way to study nonexpansive mappings it to use contractions to approximate
a nonexpansive mapping. More precisely, take t\in(0,1) and define a contraction T_{t}:Earrow EbyT_{t}x=tu+(1-t)Tx, \forall x\in E,
where
Banach’s contraction mapping principle guar‐antees that T_{t} has unique a fixed point x_{t} in E , which uniquely solves the following fixed
u\in E is an arbitrarily chosen point.

point equation:x_{t}=tu+(1-t)Tx_{t},(Such a path \{x_{t}\} is said to be an approximating fixed point of T since it posesesses theproperty that if \{x_{t}\} is bounded, then \lim_{tarrow 0}\Vert Tx_{t}-x_{t}\Vert=0 ). It is unclear, in general, what2010 Mathematics Subject Classification. Primary 47H10 Secondary 47H09.
Key words and phrases. Nonexpansive mapping, general iterative algorithms, strong positive linear op‐
erator, strongly pseudocontractive mapping, fixed points, uniformly Gâteaux differentiable norm.
The results presented in this lecture are collected mainly from the work [8] by the author of this report.

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JONG SOO JUNG

is the behavior of x_{t} as tarrow 0 , even if T has a fixed point. However, in the case of T having

a fixed point, Browder [3] proved that if E is a Hilbert space, then x_{t} converges strongly
to a fixed point of T . Reich [11] extended Browder’s result to the setting of Banach spaces
and proved that if E is a uniformly smooth Banach space, then \{x_{t}\} converges strongly toa fixed point of T and the limit defines the (umique) sunny nonexpansive retraction from E
onto Fix (T) . Xu [17] proved Reich’s results hold in reflexive Banach space having a weakly
continuous duality mapping.

In a real Hilbert space H , in 2000, Moudafi [10] introduced the following viscosity ap‐
proximation methods for nonexpansive mapping T on C in an implicit way and an explicit
way, respectively:x_{n}=\alpha_{n}f(x_{n})+(1-\alpha_{n})Tx_{n}, n\geq 0,and

x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})Tx_{n}, n\geq 0 ,(1.2)

where \{\alpha_{n}\} is a sequence in (0,1) ; and f : Carrow C is a contractive mapping (i.e., there
exists a constant k\in(0,1) such that \Vert f(x)-f(y)\Vert\leq k\Vert x-y\Vert, \forall x, y\in H ).
In 2006, Marino and Xu [9] considered the following general iterative algorithm for non‐expansive mapping T on H in an implicit way:x_{t}=t \gamma f(x_{t})+(I-tA)Tx_{t}, \forall t\in(0, \min\{1, \Vert A\Vert^{- 1}\}) ,(1.3)

where A:Harrow H is a strongly positive linear bounded operator with a coefficient \overline{\gamma}>0 ;f : Harrow H is a contractive mapping; and \gamma>0 . In 2011, Wangkeeree et al. [14]extended the result of Marino and Xu [9] to a reflexive Banach space having a weakly
continuous duality mapping. The results of Marino and Xu [9] and Wangkeeree et al. [14]improved upon the corresponding results of Browder [3], Moudafi [10], Reich [11] and Xu[17] to a general approximating fixed point \{x_{t}\} defined by (1.3). Combining the Moudafi’s
method (1.2) with Xu’s method [16], Marino and Xu [9] also considered the following general
iterative algorithm for a nonexpansive mapping T in an explicit way:x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}A)Tx_{n}, \forall n\geq 0 ,where f is a contractive mapping on

H;

(1.4)

and \gamma>0 . They proved that if the sequence\{\alpha_{n}\} in (0,1) satisfies appropriate conditions, then the sequence \{x_{n}\} generated by (1.4)

converges strongly to the unique solution of a certain variational inequality related to A.
In this paper, as a continuation of study in this direction, we present new general iterativealgorithms for the nonexpansive mapping in a reflexive Banach space with a uniformly
Gâteaux differentiable norm. First, we introduce a general implicit iterative algorithm.
Consequently, by discretizing the continuous implicit method, we provide a general explicit
iterative aıgorithm for finding a fixed point of the nonexpansive mapping. Under some
control conditions, we establish the strong convergence of the proposed explicit algorithm
to a fixed point of the mapping, which solves a ceratin variational inequality.
1. PRELIMINARIES AND LEMMASLet E be a real Banach space with norm \Vert\cdot\Vert and let E^{*} be its dual.
A Banach space E is called strictly convex if its unit sphere U=\{x\in E : \Vert x\Vert=1\} does
not contain any linear segment. For every \varepsilon with 0\leq\varepsilon\leq 2 , the modulus \delta(\varepsilon) of convexityof E is defined by\delta(\varepsilon)=\inf\{1-\Vert\frac{x+y}{2}\Vert : \Vert x\Vert\leq 1, \Vert y\Vert\leq 1, \Vert x-y\Vert\geq\varepsilon\}.
E is said to be uniformly convex if \delta(\varepsilon)>0 for every \varepsilon>0 .
E is reflexive and strictly convex.

If E is uniformly convex, then

23
GENERAL ITERATIVE ALGORITHMS FOR NPNEXPANSIVE MAPPINGSThe norm of E is said to be Gâteaux differentiable (and E is said to be smooth) if\lim_{tar ow 0}\frac{\Vert x+ty\Vert-\Vert x\Vert}{t}exists for eachx,y in its unit

(2.1)

sphere U=\{x\in E : \Vert x\Vert=1\} . It is said to be uniformly

Gâteaux differentiable if for each y\in U , this limit is attained uniformly for x\in U . Finally,
the norm is said to be uniformly Fréchet differentiable (and E is said to be uniformly
smooth) if the limit in (2.1) is attained uniformly for (x, y)\in U\cross U . Since the dual E^{*}of E is uniformly convex if and only if the norm of E is uniformly Fréchet differentiable,
every Banach space with a umiformly convex dual is reflexive and has a uniformly Gâteauxdifferentiable norm. The converse implication is false. A discussion of these and related

concepts may be found in [5].

Let J be the normalized duaıity mapping from E into 2^{E^{*}} It is welı‐known that J issingle valued if and only if E is smooth, and that if E has a uniformly Gâteaux differentiable
norm, J is uniformly continuous on bounded subsets of E from the strong topology of E

to the weak topology of E^{*} . For these facts, see [5, 13].
*

Let LIM be a linear continuous functional on \ell\infty . According to time and circumstances,
we use LIM_{n}(a_{n}) instead of LIM(a) for every a=\{a_{n}\}\in\ell\infty . LIM is called a Banach
limit if \Vert LIM\Vert=LIM(1)=1 and LIM_{n}(a_{n+1})=LIM_{n}(a_{n}) for every a=\{a_{n}\}\in\ell\infty.
Recall that a closed convex subset C of E is said to have the fixed point property fornonexpansive self‐mappings (FPP for short) if every nonexpansive mapping T :Carrow C

has a fixed point, that is, there is a point p\in C such that Tp=p . It is well‐known thatevery bounded closed convex subset of a uniformly smooth Banach space has the FPP ([7,p. 45]).
The mapping T:Carrow C is said to be pseudocontractive if there exists j(x-y)\in J(x-y)
such that\{Tx-Ty, j(x-y)\rangle\leq\Vert x-y\Vert^{2}, \forall x, y\in C,andT

is said to be strongly pseudocontractive it there exists a constant k\in(0,1) andj(x-y)\in J(x-y) such that\{Tx-Ty, j(x-y)\rangle\leq k\Vert x-y\Vert^{2}, \forall x, y\in C.
We need the following lemmas for the proof of our main results.

Lemma 2.1. ([5]) Let E be a Banach space, let C be a nonempty closed convex subset ofand let T:Carrow C be a continuous strongly pseudocontractive mapping. Then T has afixed point in C.
E,

Lemma 2.2 ([4]) Assume that A is a strongly positive linear bounded operator on a smooth
Banach space E with coefficient \overline{\gamma}>0 and 0<\rho<\Vert A\Vert^{-1} . Then \Vert I-pA\Vert\leq 1-\rho\overline{\gamma}.
Lemma 2.3 ([15]) Let \{s_{n}\} be a sequence of nonnegative real numbers satisfyings_{n+1}\leq(1-\lambda_{n})s_{n}+\lambda_{n}\delta_{n}+\omega_{n}, \forall n\geq 1,
where \{\lambda_{n}\}, \{\delta_{n}\} and \omega_{m} satisfy the following conditions:(i) \{\lambda_{n}\}\subset[0,1] and \sum_{n=1}^{\infty}\lambda_{n}=\infty or, equivalently, \prod_{n={\imath}}^{\infty}(1-\lambda_{n})=0 ;(ii) ıim \sup_{narrow\infty}\delta_{n}\leq 0 or \sum_{n=1}^{\infty}\lambda_{n}|\delta_{n}|<\infty ;(iii) \omega_{n}\geq 0 and \sum_{n=1}^{\infty}\omega_{n}<\infty.
Then\lim_{narrow\infty}s_{n}=0.

Lemma 2.4. Let \{x_{n}\} and \{y_{n}\} be bounded sequences in a Banach space E such thatx_{n+1}=\lambda_{n}x_{n}+(1-\lambda_{n})y_{n}, \forall n\geq 0,
where \{\lambda_{n}\} is a sequence in [0,1] such that0<1 i_{M}\inf_{narrow\infty}\lambda_{n}\leq\lim_{narrow}\sup_{\infty} \lambda_{n}<1.

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JONG SOO JUNG

Assume that

1 \dot{ \imath} m\sup_{narrow\infty}(\Vert y_{n+1}-y_{n}\Vert-\Vert x_{n+1}- x_{n}\Vert)\leq 0.
\lim_{narrow\infty}\Vert y_{n}-x_{n}\Vert=0.

Then

Lemma 2.5. ([1, 2]) Let C be a closed convex of a reflexive and strictly convex Banach
space E. Then C^{o}= \{x\in C:\Vert x\Vert=\inf\{\Vert y\Vert : y\in C\}\} is a singleton.
Lemma 2.6. Let E be a smooth Banach space. Then there holds\Vert x+y\Vert^{2}\leq\Vert x\Vert^{2}+2\{y, J(x+y)\rangle , \forall x, y\in E.

2. MAIN RESULTS

Throughout the rest of this paper, we always assume the following:\bulletE is a real smooth Banach space;\bulletC is a nonempty closed subspace of E ;\bullet

A:Carrow C is a strongly positive linear bounded operator with a constant \overline{\gamma}>0 ;\bulleth:Carrow C is a continuous bounded strongly pseudocontractive mapping with aeP^{seudocontractonstant\dot{ \imath} vecoef\gamma>0
fic\dot{ \imsatisfiesath} entk\in(0,1_{\f0rac{)}< \gamma<\frac{}{k}{\gamma}};}Thec;\bullet

T:Carrow C is a nonexpansive mapping with Fix (T)\neq\emptyset.

In this section, first, we introduce the folıowing general iterative algorithm that generates

a net \{x_{t}\}, t\in ( 0, min{ı, \Vert A\Vert^{-1}\} ) in an implicit way:x_{t}=t\gamma h(x_{t})+(I-tA)Tx_{t} ,(3.1)

Now, for t \in(0, \min\{1, \Vert A\Vert^{-1}\}) , consider the mapping G_{t} : Carrow C defined by

G_{t}(x):=t\gamma h(x)+(I-tA)Tx, x\in C.
Then G_{t} is a continuous strongly pseudocontractive mapping with a pseudocontractive
coefficient 1-t(\overline{\gamma}-\gamma k)\in(0,1) . Indeed, from Lemma 2.2 we have for each x, y\in C,\{G_{t}x-G_{t}y, J(x-y)\rangle=t\gamma\{h(x)-h(y), J(x-y)\rangle+\langle(I-tA) (Tx—Ty), J(x-y)\rangle\leq t\gamma k\Vert x-y\Vert^{2}+\Vert I-tA\Vert\Vert Tx-Ty \Vert\Vert x-y\Vert\leq t\gamma k\Vert x-y\Vert^{2}+(1-t\overline{\gamma})\Vert x-y\Vert^{2}=(1-t(\overline{\gamma}-\gamma k))\Vert x-y\Vert^{2}.
Thus, by Lemma 2.1, G_{t} has a unique fixed point, denotcd by x_{t} , which uniquely solves the
fixed point equation (3.1).
We summarize the basic properties of \{x_{t}\}.

Proposition 3.1. Let \{x_{t}\} be defined via (3.1). Then the following hold:(a) x_{t} is a unique path t\mapsto x_{t}\in C, t \in(0, \min\{1, \Vert A\Vert^{-1}\}) .
(b) If v is a fixed point of T , then for each t \in(0, \min\{1, \Vert A\Vert^{-1}\})\{(A-\gamma h)x_{t}, J(x_{t}-v)\rangle\leq\{A(I-T)x_{t}, J(x_{t}-v)\rangle.
(c) If T has a fixed point in C , then the path \{x_{t}\} is bounded and \Vert x_{t}-Tx_{t}\Vertarrow 0 as
tarrow 0.

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GENERAL ITERATIVE ALGORITHMS FOR NPNEXPANSIVE MAPPINGS

Using Proposition 3.1, we establish strong convergence of \{x_{t}\}.
Theorem 3.2. Let E be a a reflexive Banach space with a uniformly Gâteaux differentiable
norm. Assume that every weakly compact convex subset of E has the FPP for nonexpansive

mappings. Let \{x_{t}\} be defined via (3.1). Then, as tarrow 0, \{x_{t}\} converges strongly to a fixed
point p of T , which is the unique solution in Fix (T) of the variational inequality\{(A-\gamma h)p, J(p-q)\rangle\leq 0, \forall q\in Fix(T) .

(3.2)

Next, we substitute the fixed point property assumption, mentioned in Theorem 3.2, byassuming that the space E is strict convex.

Theorem 3.3. Let E be a a reflexive and strictly convex Banach space with a uniformly

Gâteaux differentiable norm. Let \{x_{t}\} be defined via (3.1). Then, as tarrow 0, \{x_{t}\} converges
strongly to a fixed point p of T , which is the unique solution in Fix (T) of the variational
inequality (3.2).
Now, we propose the following general iterative algorithm which generates a sequence inan explicit way:\{\begin{ar ay}{l} x_{1}=x\in C x_{n+1}=\alpha_{n}\gamma h(x_{n})+(I-\alpha_{n}A)Tx_{n}, n\geq 1, \end{ar ay}(3.3)

where \{\alpha_{n}\} is a sequence in (0,1) .

Using Theorem 3.2 and Theorem 3.3, we obtain strong convergence of the sequence \{x_{n}\}generated by (3.3).

Theorem 3.4. Let \{x_{n}\} be a sequence generated by the explicit algorithm (3.3). Let \{\alpha_{n}\}satisfy the following conditions:(C1) \lim_{narrow\infty}\alpha_{n}=0 and \sum_{n=1}^{\infty}\alpha_{n}=\infty ;(C2) |\alpha_{n+1}-\alpha_{n}|\leq o(\alpha_{n+1})+\sigma_{n}, \sum_{n=1}^{\infty}\sigma_{n}<\infty.
If one of the following assumptions holds:

(H1)

(H2)

E

is a reflexive Banach space with a uniformly Gâteaux differentiable norm, andevery weakly compact convex subset of E has the FPP for nonexpansive mappings;
E is a reflexive and strictly convex Banach space with a uniformly

Gâteaux differ‐entiable norm,

then \{x_{n}\} converges strongly to a fixed point p of T , which is the unique solution in Fix (T)of the variational inequality (3.2).
be a uniformly smooth Banach space. Let \{x_{n}\} be a sequence
generated by the explicit algorithm (3.3). Let \{\alpha_{n}\} satisfy the conditions (C1) and (C2) in
Theorem 3.4. Then \{x_{n}\} converges strongly to a fixed point p of T , which is the uniquesolution in Fix (T) of the variational inequality (3.2).
Corollary 3.5.

Let

E

Removing the condition |\alpha_{n+1}-\alpha_{n}|\leq o(\alpha_{n+1})+\sigma_{n}, \sum_{n=1}^{\infty}\sigma_{n}<\infty on the sequence\{\alpha_{n}\} in Theorem 3.4, we have the following result.
Theorem 3.6. Let \{x_{n}\} be a sequence generated by the following explicit algorithm :\{\begin{ar ay}{l } x_{1}=x\in C x_{n+1}=\alpha_{n}\gamma h(x_{n})+\beta_{n}x_{n}+( 1-\beta_{n})I-\alpha_{n}A)Tx_ {n}, n\geq 1, \end{ar ay}

where \{\alpha_{n}\} and \{\beta_{n}\} are sequences in (0,1) , which satisfy the following conditions:(C1) {\imath} im_{narrow\infty}\alpha_{n}=0 and \sum_{n=1}^{\infty}\alpha_{n}=\infty ;(C2) 0< \lim\inf_{narrow\infty}\beta_{n}\leq\lim\sup_{narrow\infty}\beta_{n}<1.
If one of the following assumptions holds:

(3.4)

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JONG SOO JUNG

(H1)

E

is a reflexive Banach space with a uniformly Gâteaux differentiable norm, andevery weakly compact convex subset of E has the FPP for nonexpansive mappings;(H2)

E is a reflexive and strictly convex Banach space with a uniformly

Gâteaux differ‐entiable norm,

then \{x_{n}\} converges strongly to a fixed point p of T , which is the unique solution in Fix (T)of the variational inequality (3.2).

Proof. By conditions (C1) and (C2), we may assume, without loss of generality, that \frac{\alpha_{n} {1-\beta_{n} <\Vert A\Vert^{-1} for all n\geq 1 . By Lemma 2.2, we have \Vert(1-\beta_{n})I-\alpha_{n}A\Vert\leq(1-\beta_{n}-\alpha_{n} \overline{\gamma}) .
Step 1. We show that \{x_{n}\}, \{h(x_{n})\}, \{Tx_{n}\} and \{ATx_{n}\} are bounded. Indeed, pick any
p\in Fix(T) to obtain\Vert x_{n+}{\imath}-p\Vert\leq\alpha_{n}\gamma k\Vert x_{n}-p\Vert+\alpha_{n} \Vert\gamma h(p)-Ap\Vert+\beta_{n}\Vert x_{n}-p\Vert+(1-\beta_{n}-\alpha_{n} \overline{\gamma})\Vert x_{n}-p\Vert
It follows from induction that

\Vert x_{n}-p\Vert\leq\max\{\Vert x_{1}-p\Vert, \frac{\Vert\gamma h(p)- Ap\Ve\{h(x_{n})\}rt}{\overline{\gamma}-\gamma k}\},\forall n\geq 1 . Hence \{x_{n}\}is bounded. Moreover, since h is a bounded mapping,is bounded. Also, \{Tx_{n}\}\{ATx_{n}\} are bounded.
Step 2. We show that \lim_{narrow\infty}\Vert x_{n+1}-x_{n}\Vert=0 . To this end, define a sequence \{z_{n}\} byz_{n}=(x_{n+1}-\beta_{n}x_{n})/(1-\beta_{n}) so thatand

x_{n+1}=\beta_{n}x_{n}+(1-\beta_{n})z_{n} .

(3.5)

We now observe thatz_{n+1}-z_{n}= \frac{\alpha_{n+1}}{1-\beta_{n+1}}(\gamma h(x_{n+1})-ATx_{n+1})+Tx_{n+1}- Tx_{n}+\frac{\alpha_{n} {1-\beta_{n} (ATx_{n})-\gamma h(x_{n}) .

(3.6)

It follows from (3.6) that
\Vert z_{n+1}-z_{n}\Vert-\Vert x_{n+1}-x_{n}\Vert\leq\frac{\alpha_{n+1}}{1-\beta_{n+1}}(\Vert\gamma h(x_{n+1})\Vert+\Vert ATx_{n+1}\Vert)+\frac{\alpha_{n} {1-\beta_{n} (\Vert\gamma h(x_{n})\Vert+\Vert ATx_{n}\Vert) .

(3.7)

By conditions (C1), (C2) and (3.7), we obtain that

\lim_{narrow}\sup_{\infty}(\Vert z_{n+1}-z_{n}\Vert-\Vert x_{n+1}-x_{n}\Vert) \leq 0.
Hence by Lemma 2.4, we have\lim_{narrow\infty}\Vert z_{n}-x_{n}\Vert=0 .

(3.8)

It then folıows from condition (C2), (3.5) and (3.8) thatn arrow\infty 1\dot{ \imath} m\Vert x_{n+1}-x_{n}\Vert=\lim_{narrow\infty}(1- \beta_{n})\Vert z_{n}-x_{n}\Vert=0.
Step 3. We show that \lim_{narrow\infty}\Vert x_{n}-Tx_{n}\Vert=0 . In fact, from (3.4) it follows that

\Vert Tx_{n}-x_{n}\Vert\leq\leq\Vert\alpha_{n}\gamma h(x_{n})-\alpha_{n}ATx_{n} \Vert+\beta_{n}\Vert x_{n}-Tx_{n}\Vert+\Vert x_{n+1}-x_{n}\Vert
This implies that(1-\beta_{n})\Vert Tx_{n}-x_{n}\Vert\leq\alpha_{n}(\gamma\Vert h(x_{n})\Vert+ \Vert ATx_{n}\Vert)+\Vert x_{n+1}-x_{n}\Vert.
Thus, by conditions (C1) and (C2) and Step 2, we have \lim_{narrow\infty}\Vert Tx_{n}-x_{n}\Vert=0.
Step 4. We show that \lim\sup_{narrow\infty}\{\gamma h(p)-Ap, J(x_{n}-p)\rangle\leq 0 , where p= \lim_{tarrow 0}x_{t} and x_{t}is defined by (3.1). In fact, let x_{t}=t\gamma h(x_{t})+(I-tA)Tx_{t} . Then, it follows from Theorem
3.2 or Theorem 3.3 that \{x_{t}\} converges strongly to p\in Fix(T) which is the unique solutionof the variational inequality (3.2). Noting thatx_{t}-x_{n}=t(\gamma h(x_{t})-Ax_{t})+(Tx_{t}-Tx_{n})+(Tx_{n}-x_{n})+t^{2} A(\gamma h(x_{t})-ATx_{t}),27
GENERAL ITERATIVE ALGORITHMS FOR NPNEXPANSIVE MAPPINGS

we have\Vert x_{t}-x_{n}\Vert^{2}\leq t\{\gamma h(x_{t})-Ax_{t}, J(x_{t}-x_{n})\}+ \Vert x_{t}-x_{n}\Vert^{2}+\Vert Tx_{n}-x_{n}\Vert\Vert x_{t}-x_{n}\Vert+t^{2}\Vert A(\gamma h(x_{t})- ATx_{t})\Vert\Vert x_{t}-x_{n}\Vert,
which implies that\{\gamma h(x_{t})-Ax_{t}, J(x_{n}-x_{t}) \leq\frac{\Vert Tx_{n}-x_{n}\Vert}{t} M+tL ,(3.9)

where M= \sup { \Vert x_{t}-x_{n}\Vert : n\geq 1 and t \in(0, \min\{1, \Vert A\Vert^{-1}\}) } and L= \sup\{\Vert A(\gamma h(x_{t})ATx_{t})\Vert\Vert x_{t}-x_{n}\Vert : n\geq 1 and t \in(0, \min\{1, \Vert A\Vert^{-1}\}) }. Since x_{n}-Tx_{n}arrow 0 by Step 3 ,taking the upper limit as narrow\infty in (3.9), we derive

1 \dot{ \imath} m\sup_{narrow\infty}\langle\gamma h(x_{t})-Ax_{t}, J(x_{n}- x_{t})\}\leq tL ,(3.10)Taking the \lim\sup as tarrow 0 in (3.10) and noticing that the fact that the two limits areinterchangeable due to the fact that J is uniformly continuous on bounded subsets of E
from the strong topology of E to the weak topology of E^{*} , we obtain*

1 \dot{ \imath} m\sup_{narrow\infty}\langle\gamma h(p)-Ap, J(x_{n}-p)\}\leq 0.
Step 5. We show that \lim_{narrow\infty}x_{n}=p, where p= \lim_{tarrow 0}x_{t}\in Fix(T), x_{t} being defined by(3.1), which is the unique solution of the variational inequality (3.2). Indeed, from (3.4),

observe thatx_{n+1}-p=\alpha_{n}(\gamma h(x_{n})-Ap)+\beta_{n}(x_{n}-p)+((1-\beta_{n})I- \alpha_{n}A)(Tx_{n}-p).

By Lemma 2.2 and Lemma 2.6, we derive\Vert x_{n+1}-p\Vert^{2}\leq(1-\alpha_{n}\overline{\gamma})^{2}\Vert x_{n}- p\Vert^{2}+\alpha_{n}\gamma k(\Vert x_{n}-p\Vert^{2}+\Vert x_{n+1}-p\Vert^{2})+2\alpha_{n}\{\gamma h(p) —Ap, J(x_{n+1}-p)\rangle.
This implies that

\Vertx_{n+1}-p\Vert^{2}\leq(1-\frac{2\alpha_{n}(\overline{\gam a}-\gam ak)} {1-\alpha_{n}\gam ak})\Vertx_{n}-p\Vert^{2}+\frac{2\alpha_{n} (\overline{\gam a}-\gam ak)}{1-\alpha_{n}\gam ak}\cdot\frac{\alpha_{n} \overline{\gam a}^{2} {2(\overline{\gam a}-\gam ak)}K+ \frac{2\alpha_{n}(\overline{\gamma}-\gamma k)}{1-\alpha_{n}\gamma k} \cdot\frac{1}{\overline{},\gamma-\gamma k}\langle\gamma h(p)-Ap, J(x_{n+1}-p) \rangle,

where(3.11)

K= \sup\{||x_{n}-p\Vert : n\geq 1\} . Put \lambda_{n}=\frac{2\alpha_{n}(\overline{\gam a}-\gam ak)}{1-\alpha_{n}\gam a k} and\delta_{n}=\frac{\alpha_{n}\overline{\gamma}^{2} {2(\overline{\gamma}-\gamma k)}L+\frac{1}{\overline{\gamma}-\gamma k}\{\gamma h(p)-Ap, J(x_{n+1}-p)\}.
Then it follows from the condition (C1) and Step 4 that \lim_{narrow\infty}\lambda_{n}=0, \sum_{n=1}^{\infty}\lambda_{n}=\infty,and \lim\sup_{narrow\infty}\delta_{n}\leq 0. (3.11) reduces to\Vert x_{n+1}-p\Vert^{2}\leq(1-\lambda_{n})\Vert x_{n}-p\Vert^{2}+\lambda_{n} \delta_{n} .

(3.11)

Thus, applying Lemma 2.3 together with \omega_{n}=0 to (3.11), we conclude that \lim_{narrow\infty}x_{n}=p.
This completes the proof.

\squareRemark Our results in this paper extend, improve and develop the corresponding resultsin [9, 10, 11, 14] and the references therein.

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JONG SOO JUNG

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DEPARTMENT OF MATHEMATICS, DONG‐A UNIVERSITY, BUSAN 49315, KOREA
E‐mail address: jungjs@dau.ac.kr