Volume 2009, Article ID 483497,25pages doi:10.1155/2009/483497

*Research Article*

**Strong Convergence Theorems of Modified**

**Ishikawa Iterations for Countable Hemi-Relatively** **Nonexpansive Mappings in a Banach Space**

**Narin Petrot,**

^{1, 2}**Kriengsak Wattanawitoon,**

^{3, 4}**and Poom Kumam**

^{2, 3}*1**Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand*

*2**Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand*

*3**Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi*
*(KMUTT), Bangmod, Bangkok 10140, Thailand*

*4**Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology,*
*Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand*

Correspondence should be addressed to Poom Kumam,poom.kum@kmutt.ac.th Received 17 March 2009; Accepted 12 September 2009

Recommended by Lech G ´orniewicz

We prove some strong convergence theorems for fixed points of modified Ishikawa and Halpern iterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space by using the hybrid projection methods. Moreover, we also apply our results to a class of relatively nonexpansive mappings, and hence, we immediately obtain the results announced by Qin and Su’s result2007, Nilsrakoo and Saejung’s result2008, Su et al.’s result2008, and some known corresponding results in the literatures.

Copyrightq2009 Narin Petrot et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**1. Introduction**

Let*C*be a nonempty closed convex subset of a real Banach space*E. A mappingT*:*C* → *C*is
*said to be nonexpansive if*Tx−Ty ≤ x−yfor all*x, y*∈*C.*We denote by*FT*the set of fixed
points of*T*, that is*FT* {x ∈ *C* : *x* *Tx}. A mappingT* *is said to be quasi-nonexpansive*
if *FT/*∅ and Tx−*y ≤ x*−*y*for all*x* ∈ *C*and *y* ∈ *FT*. It is easy to see that if*T*
is nonexpansive with*FT/*∅, then it is quasi-nonexpansive. Some iterative processes are
often used to approximate a fixed point of a nonexpansive mapping. The Mann’s iterative
algorithm was introduced by Mann 1 in 1953. This iterative process is now known as
Mann’s iterative process, which is defined as

*x*_{n1}*α**n**x**n* 1−*α**n*Tx*n**, n*≥0, 1.1

where the initial guess*x*0is taken in*C*arbitrarily and the sequence{α*n*}^{∞}* _{n0}*is in the interval
0,1.

In 1976, Halpern2first introduced the following iterative scheme:

*x*_{0}*u*∈*C,* chosen arbitrarily,

*x*_{n1}*α**n**u* 1−*α**n*Tx*n**,* 1.2

see also Browder3. He pointed out that the conditions lim*n*→ ∞*α**n*0 and_{∞}

*n1**α**n* ∞are
necessary in the sence that, if the iteration1.2converges to a fixed point of*T, then these*
conditions must be satisfied.

In 1974, Ishikawa4introduced a new iterative scheme, which is defined recursively by

*y*_{n}*β*_{n}*x** _{n}*
1−

*β*

_{n}*Tx*_{n}*,*

*x*_{n1}*α**n**x**n* 1−*α**n*Ty*n**,* 1.3

where the initial guess*x*_{0}is taken in*C*arbitrarily and the sequences{α*n*}and{β*n*}are in the
interval0,1.

Concerning a family of nonexpansive mappings it has been considered by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings; see, for example, 5. The problem of finding an optimal point that minimizes a given cost function over common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importancesee6.

Zhang and Su7introduced the following implicit hybrid method for a finite family
of nonexpansive mappings{T*i*}^{N}* _{i1}*in a real Hilbert space:

*x*_{0}∈*C* is arbitrary,
*y*_{n}*α*_{n}*x** _{n}* 1−

*α*

*T*

_{n}*n*

*z*

_{n}*,*

*z*

*n*

*β*

*n*

*y*

*n*

1−*β**n*
*T**n**y**n**,*
*C*_{n}

*z*∈*C*:*y** _{n}*−

*z*≤

*x*

*−*

_{n}*z*

*,*

*Q*

*n*{z∈

*C*: x

*n*−

*z, x*

_{0}−

*x*

*n*≥0},

*x*

_{n1}*P*

*C*

*n*∩Q

*n*x0, n0,1,2, . . . ,

1.4

where*T** _{n}*≡

*T*

_{n}_{mod}

*,{α*

_{N}*n*}and{β

*n*}are sequences in0,1and{α

*n*} ⊂0, afor some

*a*∈0,1 and{β

*n*} ⊂b,1for some

*b*∈0,1.

In 2008, Nakprasit et al.8 established weak and strong convergence theorems for finding common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. In the same year, Cho et al.9introduced the normal Mann’s iterative process and proved some strong convergence theorems for a finite family nonexpansive mapping in the framework Banach spaces.

To find a common fixed point of a family of nonexpansive mappings, Aoyama et al.

10introduced the following iterative sequence. Let*x*_{1}*x*∈*C*and

*x*_{n1}*α**n**x* 1−*α**n*T*n**x**n**,* 1.5

for all *n* ∈ N, where*C* is a nonempty closed convex subset of a Banach space, {α*n*} is a
sequence of0,1,and{T*n*}is a sequence of nonexpansive mappings. Then they proved that,
under some suitable conditions, the sequence{x*n*}defined by1.5converges strongly to a
common fixed point of{T*n*}.

In 2008, by using anewhybrid method, Takahashi et al.11proved the following theorem.

**Theorem 1.1**Takahashi et al. 11. Let*H* *be a Hilbert space and letCbe a nonempty closed*
*convex subset ofH. Let*{T*n*}*andT* *be families of nonexpansive mappings ofCinto itself such that*

∩^{∞}_{n1}*FT**n*: *FT/*∅*and letx*0 ∈ *H. Suppose that*{T*n*}*satisfies the NST-condition*I*with*T.

*ForC*_{1}*Candx*_{1}*P*_{C}_{1}*x*_{0}*, define a sequence*{x*n*}*ofCas follows:*

*y**n* *α**n**x**n* 1−*α**n*T*n**x**n**,*
*C*_{n1}

*z*∈*C** _{n}*:

*y*

*−*

_{n}*z*≤

*x*

*−*

_{n}*z*

*,*

*x*

_{n1}*P*

*C*

_{n1}*x*

_{0}

*, n*∈N,

1.6

*where 0* ≤*α**n* *<* *1 for alln*∈N*and*{T*n*}*is said to satisfy the NST-condition*I*with*T*if for each*
*bounded sequence*{z*n*} ⊂*C, lim**n*→ ∞z*n*−*T**n**z**n* *0 implies that lim**n*→ ∞z*n*−*Tz**n**0 for all*
*T* ∈ T. Then,{x*n*}*converges strongly toP*_{FT}*x*_{0}*.*

Note that, recently, many authors try to extend the above result from Hilbert spaces to a Banach space setting.

Let*E*be a real Banach space with dual*E*^{∗}. Denote by ·,·the duality product. The
*normalized duality mappingJ*from*E*to 2^{E}^{∗}is defined by*Jx*{f ∈*E*^{∗}: x, fx^{2} f^{2}},
for all*x*∈*E. The functionφ*:*E*×*E* → Ris defined by

*φ*
*x, y*

x^{2}−2

*x, Jy* *y*^{2}*,* ∀x, y∈*E.* 1.7

A mapping*Tis said to be hemi-relatively nonexpansive*see12if*FT/*∅and
*φ*

*p, Tx*

≤*φ*
*p, x*

*,* ∀x∈*C, p*∈*FT.* 1.8

A point*p*in*Cis said to be an asymptotic fixed point ofT*13if*C*contains a sequence
{x*n*} which converges weakly to *p* such that the strong lim_{n}_{→ ∞}x*n* − *Tx** _{n}* 0. The set
of asymptotic fixed points of

*T*will be denoted by

*FT*. A hemi-relatively nonexpansive mapping

*T*from

*Cinto itself is called relatively nonexpansive ifFT*

*FT*; see14–16for more details.

On the other hand, Matsushita and Takahashi17introduced the following iteration.

A sequence{x*n*},defined by

*x** _{n1}* Π

*C*

*J*

^{−1}α

*n*

*Jx*

*1−*

_{n}*α*

*JTx*

_{n}*n*, n0,1,2, . . . , 1.9

where the initial guess element*x*_{0} ∈ *C*is arbitrary,{α*n*} is a real sequence in0,1,*T* is a
relatively nonexpansive mapping, andΠ*C* denotes the generalized projection from*E*onto a
closed convex subset*C*of*E. Under some suitable conditions, they proved that the sequence*
{x*n*}converges weakly to a fixed point of*T*.

Recently, Kohsaka and Takahashi 18 extended iteration 1.9 to obtain a weak
convergence theorem for common fixed points of a finite family of relatively nonexpansive
mappings{T*i*}^{m}* _{i1}*by the following iteration:

*x** _{n1}* Π

*C*

*J*

^{−1}

_{m}

*i1*

*w**n,i*α*n,i**Jx**n* 1−*α**n,i*JT*i**x**n*

*, n*0,1,2, . . . , 1.10

where*α**n,i*⊂0,1and*w**n,i*⊂0,1with_{m}

*i1**w**n,i*1, for all*n*∈N. Moreover, Matsushita and
Takahashi14proposed the following modification of iteration1.9in a Banach space*E:*

*x*_{0}*x*∈*C,* chosen arbitrarily,
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JTx*n*,
*C*_{n}

*z*∈*C*:*φ*
*z, y*_{n}

≤*φz, x**n*
*,*
*Q**n*{z∈*C*: x*n*−*z, Jx*−*Jx**n* ≥0},
*x** _{n1}* Π

*C*

*n*∩Q

*n*

*x, n*0,1,2, . . . ,

1.11

and proved that the sequence{x*n*}converges strongly toΠ*FT**x.*

Qin and Su 15 showed that the sequence {x*n*}, which is generated by relatively
nonexpansive mappings*T*in a Banach space*E, as follows:*

*x*_{0}∈*C,* chosen arbitrarily,
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JTz*n*,
*z*_{n}*J*^{−1}

*β*_{n}*Jx** _{n}*
1−

*β*

_{n}*JTx*_{n}*,*
*C**n*

*v*∈*C*:*φ*
*v, y**n*

≤*α**n**φv, x**n* 1−*α**n*φv, z*n*
*,*
*Q** _{n}*{v∈

*C*: Jx0−

*Jx*

_{n}*, x*

*−*

_{n}*v ≥*0},

*x** _{n1}* Π

*C*

*n*∩Q

*n*

*x*

_{0}

1.12

converges strongly toΠ_{FT}*x*_{0}*.*

Moreover, they also showed that the sequence{x*n*}, which is generated by

*x*_{0}∈*C,* chosen arbitrarily,
*y**n**J*^{−1}α*n**Jx*0 1−*α**n*JTx*n*,
*C*_{n}

*v*∈*C*:*φ*
*v, y*_{n}

≤*α*_{n}*φv, x*0 1−*α** _{n}*φv, x

*n*

*,*

*Q*

*n*{v∈

*C*: Jx0−

*Jx*

*n*

*, x*

*n*−

*v ≥*0},

*x** _{n1}* Π

*C*

*n*∩Q

*n*

*x*

_{0}

*,*

1.13

converges strongly toΠ_{FT}*x*_{0}*.*

In 2008, Nilsrakoo and Saejung19used the following Mann’s iterative process:

*x*_{0}∈*C* is arbitrary,
*C*_{−1}*Q*_{−1}*C,*

*y*_{n}*J*^{−1}α*n**Jx** _{n}* 1−

*α*

*JT*

_{n}*n*

*x*

*,*

_{n}*C*

*n*

*v*∈*C**n*:*φ*
*v, y**n*

≤*φv, x**n*
*,*
*Q** _{n}*{v∈

*C*: Jx0−

*Jx*

_{n}*, x*

*−*

_{n}*v ≥*0},

*x*

*Π*

_{n1}*C*

*n*∩Q

*n*

*x*0

*, n*0,1,2, . . .

1.14

and showed that the sequence {x*n*} converges strongly to a common fixed point of a
countable family of relatively nonexpansive mappings.

Recently, Su et al. 12 extended the results of Qin and Su 15, Matsushita and Takahashi 14 to a class of closed hemi-relatively nonexpansive mapping. Note that, since the hybrid iterative methods presented by Qin and Su 15 and Matsushita and Takahashi 14 cannot be used for hemi-relatively nonexpansive mappings. Thus, as we know, Su et al.12showed their results by using the method as a monotoneCQhybrid method.

In this paper, motivated by Qin and Su15, Nilsrakoo and Saejung19, we consider the modified Ishikawa iterative 1.12 and Halpern iterative processes 1.13, which is diﬀerent from those of1.12–1.14, for countable hemi-relatively nonexpansive mappings.

By using the shrinking projection method, some strong convergence theorems in a uniformly convex and uniformly smooth Banach space are provided. Our results extend and improve the recent results by Nilsrakoo and Saejung’s result 19, Qin and Su 15, Su et al. 12, Takahashi et al.’s theorem11, and many others.

**2. Preliminaries**

In this section, we will recall some basic concepts and useful well-known results.

A Banach space*Eis said to be strictly convex if*
*xy*

2

*<*1, 2.1

for all*x, y* ∈ *E*withx y 1 and*x /y. It is said to be uniformly convex if for any two*
sequences{x*n*} and{y*n*}in*E*such thatx*n*y*n*1 and

*n*lim→ ∞*x*_{n}*y** _{n}*2, 2.2

lim_{n}_{→ ∞}x*n*−*y** _{n}*0 holds.

Let*U*{x∈*E*:x 1}be the unit sphere of*E. Then the Banach spaceE*is said to
*be smooth if*

lim*t→*0

*xty*− *x*

*t* 2.3

exists for each*x, y*∈*U.It is said to be uniformly smooth if the limit is attained uniformly for*
*x, y* ∈*E. In this case, the norm ofEis said to be Gˆateaux diﬀerentiable. The spaceE*is said to
*have uniformly Gˆateaux diﬀerentiable if for eachy*∈*U, the limit*2.3is attained uniformly for
*y*∈*U. The norm ofEis said to be uniformly Fr´echet diﬀerentiable*and*E*is said to be uniformly
smoothif the limit2.3is attained uniformly for*x, y*∈*U.*

In our work, the concept duality mapping is very important. Here, we list some known
facts, related to the duality mapping*J*, as follows.

a*E*E^{∗}, resp.is uniformly convex if and only if*E*^{∗}E, resp.is uniformly smooth.

b*Jx/*∅for each*x*∈*E.*

cIf*E*is reflexive, then*J*is a mapping of*E*onto*E*^{∗}.
dIf*E*is strictly convex, then*Jx*∩*Jy/*∅for all*x /y.*

eIf*E*is smooth, then*J*is single valued.

fIf*E*has a Fr´echet diﬀerentiable norm, then*J*is norm to norm continuous.

gIf *E*is uniformly smooth, then*J* is uniformly norm to norm continuous on each
bounded subset of*E.*

hIf*E*is a Hilbert space, then*J*is the identity operator.

For more information, the readers may consult20,21.

If*C*is a nonempty closed convex subset of a real Hilbert space*H*and*P** _{C}* :

*H*→

*C*is

*the metric projection, thenP*

*C*is nonexpansive. Alber22

*has recently introduced a generalized*

*projection operator*Π

*C*in a Banach space

*E*which is an analogue representation of the metric projection in Hilbert spaces.

The generalized projectionΠ*C* : *E* → *C*is a map that assigns to an arbitrary point
*x*∈*E*the minimum point of the functional*φy, x, that is,*Π*C**xx*^{∗}, where*x*^{∗}is the solution
to the minimization problem

*φx*^{∗}*, x *min

*y∈C* *φ*
*y, x*

*.* 2.4

Notice that the existence and uniqueness of the operatorΠ*C*is followed from the properties of
the functional*φy, x*and strict monotonicity of the mapping*J, and moreover, in the Hilbert*
spaces setting we haveΠ*C* *P** _{C}*. It is obvious from the definition of the function

*φ*that

*y*− *x*_{2}

≤*φ*
*y, x*

≤*y*x_{2}

*,* ∀x, y∈*E.* 2.5

*Remark 2.1. IfE*is a strictly convex and a smooth Banach space, then for all*x, y*∈*E,φy, x *
0 if and only if*xy, see Matsushita and Takahashi*14.

To obtain our results, following lemmas are important.

**Lemma 2.2** Kamimura and Takahashi23. Let*Ebe a uniformly convex and smooth Banach*
*space and let* *r >* *0. Then there exists a continuous strictly increasing and convex functiong* :
0,2r → 0,∞*such thatg0 0 and*

*gx*−*y*≤*φ*
*x, y*

*,* 2.6

*for allx, y*∈*B** _{r}* {z∈

*E*:z ≤

*r}.*

**Lemma 2.3**Kamimura and Takahashi23. Let*Ebe a uniformly convex and smooth real Banach*
*space and let*{x*n*},{y*n*}*be two sequences ofE. Ifφx**n**, y**n* → *0 and either*{x*n*}*or*{y*n*}*is bounded,*
*then*x*n*−*y** _{n}* →

*0.*

**Lemma 2.4**Alber22. Let*Cbe a nonempty closed convex subset of a smooth real Banach space E*
*andx*∈*E. Then,x*0 Π*C**xif and only if*

*x*_{0}−*y, Jx*−*Jx*_{0} ≥0, ∀y∈*C.* 2.7

**Lemma 2.5**Alber22. Let*Ebe a reflexive strict convex and smooth real Banach space, letCbe a*
*nonempty closed convex subset of E and letx*∈*E. Then*

*φ*

*y,*Π*C**x*

*φΠ**C**x, x*≤*φ*
*y, x*

*,* ∀y∈*C.* 2.8

**Lemma 2.6**Matsushita and Takahashi14. Let*Ebe a strictly convex and smooth real Banach*
*space, letCbe a closed convex subset of E, and let T be a hemi-relatively nonexpansive mapping from*
*C into itself. Then F(T) is closed and convex.*

*LetCbe a subset of a Banach spaceEand let*{T*n*}*be a family of mappings fromCintoE. For*
*a subsetBofC, one says that*

a {T*n*}, B*satisfies condition AKTT if*

∞
*n1*

sup{T_{n1}*z*−*T*_{n}*z*:*z*∈*B}<*∞, 2.9

b {T*n*}, B*satisfies condition*^{∗}*AKTT if*

∞
*n1*

sup{JT*n1**z*−*JT**n**z*:*z*∈*B}<*∞. 2.10

*For more information, see Aoyama et al. [10].*

**Lemma 2.7**Aoyama et al.10. Let*Cbe a nonempty subset of a Banach spaceEand let*{T*n*}*be a*
*sequence of mappings fromCintoE. LetBbe a subset ofCwith*{T*n*}, B*satisfying condition AKTT,*
*then there exists a mappingT*:*B* → *Esuch that*

*Tx* lim

*n*→ ∞*T**n**x,* ∀x∈*B* 2.11

*and lim sup*_{n}_{→ ∞}{*Tz* −*T**n**z*:*z*∈*B}*0.

Inspired byLemma 2.7, Nilsrakoo and Saejung19prove the following results.

**Lemma 2.8**Nilsrakoo and Saejung19. Let*Ebe a reflexive and strictly convex Banach space*
*whose norm is Fr´echet diﬀerentiable, letCbe a nonempty subset of a Banach spaceE, and let*{T*n*}*be a*
*sequence of mappings fromCintoE. LetBbe a subset ofCwith*{T*n*}, B*satisfies condition*^{∗}*AKTT,*
*then there exists a mappingT*:*B* → *Esuch that*

*Tx* lim

*n*→ ∞*T**n**x,* ∀x∈*B* 2.12

*and lim sup*_{n}_{→ ∞}{J*Tz* −*JT**n**z*:*z*∈*B}*0.

**Lemma 2.9**Nilsrakoo and Saejung19. Let*Ebe a reflexive and strictly convex Banach space*
*whose norm is Fr´echet diﬀerentiable, letCbe a nonempty subset of a Banach spaceE,and let*{T*n*}*be*
*a sequence of mappings fromCintoE. Suppose that for each bounded subsetBofC, the ordered pair*
{T*n*}, B*satisfies either condition AKTT or condition*^{∗}*AKTT. Then there exists a mappingT* :*B* →
*Esuch that*

*Tx* lim

*n*→ ∞*T**n**x,* ∀*x*∈*C.* 2.13

**3. Modified Ishikawa Iterative Scheme**

In this section, we establish the strong convergence theorems for finding common fixed
points of a countable family of hemi-relatively nonexpansive mappings in a uniformly
convex and uniformly smooth Banach space. It is worth mentioning that our main theorem
generalizes recent theorems by Su et al.12 from relatively nonexpansive mappings to a
more general concept. Moreover, our results also improve and extend the corresponding
results of Nilsrakoo and Saejung19. In order to prove the main result, we recall a concept
as follows. An operator *T* in a Banach space is closed if *x**n* → *x* and *Tx**n* → *y, then*
*Txy.*

**Theorem 3.1. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *C* *be a*
*nonempty bounded closed convex subset ofE. Let*{T*n*}*be a sequence of hemi-relatively nonexpansive*
*mappings fromCinto itself such that*_{∞}

*n0**FT**n**is nonempty. Assume that*{a*n*}^{∞}_{n0}*and*{β*n*}^{∞}_{n0}*are*
*sequences in*0,1*such that lim sup*_{n}_{→ ∞}*α*_{n}*<1 and lim*_{n→ ∞}*β*_{n}*1 and let a sequence*{x*n*}*inCby*
*the following algorithm be:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JT*n**z**n*,
*z*_{n}*J*^{−1}

*β*_{n}*Jx** _{n}*
1−

*β*

_{n}*JT*_{n}*x*_{n}*,*
*C*_{n1}

*v*∈*C**n*:*φ*
*v, y**n*

≤*φv, x**n*
*,*
*x** _{n1}* Π

*C*

*n1*

*x*

_{0}

*,*

3.1

*for* *n* ∈ N ∪ {0}, where *J* *is the single-valued duality mapping on* *E. Suppose that for each*
*bounded subset* *B* *of* *C, the ordered pair* {T*n*}, B *satisfies either condition AKTT or condition*

∗*AKTT. LetT* *be the mapping fromC* *into itself defined byTv* lim_{n}_{→ ∞}*T*_{n}*vfor allv* ∈ *Cand*
*suppose that* *T* *is closed and* *FT * _{∞}

*n0**FT**n*. If *T**n* *is uniformly continuous for all* *n* ∈ N,
*then* {x*n*} *converges strongly to* Π_{FT}*x*_{0}*, where* Π_{FT}*is the generalized projection from* *C* *onto*
*FT*.

*Proof. We first show that* *C** _{n1}* is closed and convex for each

*n*≥ 0. Obviously, from the definition of

*C*

*, we see that*

_{n1}*C*

*is closed for each*

_{n1}*n*≥0. Now we show that

*C*

*is convex for any*

_{n1}*n*≥0. Since

*φ*
*v, y**n*

≤*φv, x**n*⇐⇒2

*v, Jx**n*−*Jy**n* *y**n*^{2}− *x**n*^{2}≤0, 3.2
this implies that*C** _{n1}*is a convex set. Next, we show that

_{∞}

*n0**FT**n*⊂*C** _{n}*for all

*n*≥0. Indeed,

let*p*∈_{∞}

*n0**FT**n*, we have
*φ*

*p, y*_{n}*φ*

*p, J*^{−1}α*n**Jx** _{n}* 1−

*α*

*JT*

_{n}*n*

*z*

_{n}*p*

^{2}−2

*p, α**n**Jx**n* 1−*α**n*JT*n**z**n* α*n**Jx**n* 1−*α**n*JT*n**z**n*^{2}

≤*p*^{2}−2α_{n}

*p, Jx** _{n}* −21−

*α*

_{n}*p, JT*_{n}*z*_{n}*α*_{n}*x*_{n}^{2} 1−*α*_{n}*T*_{n}*z*_{n}^{2}
*α**n**p*^{2}−2

*p, Jx**n* x*n*^{2}

1−*α**n**p*^{2}−2

*p, JT**n**z**n* T*n**z**n*^{2}

≤*α**n**φ*
*p, x**n*

1−*α**n*φ

*p, T**n**z**n*

≤*α**n**φ*
*p, x**n*

1−*α**n*φ
*p, z**n*

*,*

3.3

*φ*
*p, z*_{n}

*φ*
*p, J*^{−1}

*β*_{n}*Jx** _{n}*
1−

*β*

_{n}*JT*_{n}*x*_{n}*p*^{2}−2

*p, β**n**Jx**n*
1−*β**n*

*JT**n**x**n* *β**n**Jx**n*
1−*β**n*

*JT**n**x**n*^{2}
*p*^{2}−2β*n*

*p, Jx**n* −2

1−*β**n*

*p, JT**n**x**n* *β**n*x*n*^{2}
1−*β**n*

T*n**x**n*^{2}
*β**n**p*^{2}−2

*p, Jx**n* x*n*^{2}

1−*β**n**p*^{2}−2

*p, JT**n**x**n* T*n**x**n*^{2}

≤*β**n**φ*
*p, x**n*

1−*β**n*

*φ*

*p, T**n**x**n*

≤*β**n**φ*
*p, x**n*

1−*β**n*

*φ*
*p, x**n*

≤*φ*
*p, x*_{n}

*.*

3.4

Substituting3.4into3.3, we have
*φ*

*p, y**n*

≤ *φ*
*p, x**n*

*.* 3.5

This means that,*p* ∈ *C** _{n1}* for all

*n*≥ 0. Consequently, the sequence{x

*n*} is well defined.

Moreover, since*x**n* Π*C**n**x*_{0}and*x** _{n1}*∈

*C*

*⊂*

_{n1}*C*

*n*, we get

*φx**n**, x*0≤*φx**n1**, x*0, 3.6

for all*n*≥0. Therefore,{φx*n**, x*_{0}}is nondecreasing.

By the definition of*x**n*andLemma 2.5, we have
*φx**n**, x*0 *φΠ**C**n**x*0*, x*0≤*φ*

*p, x*0

−*φ*

*p,*Π*C**n**x*0

≤*φ*
*p, x*0

*,* 3.7

for all*p*∈_{∞}

*n0**FT**n*⊂*C**n*. Thus,{φx*n**, x*0}is a bounded sequence. Moreover, by2.5, we
know that{x*n*}is bounded. So, lim_{n→ ∞}*φx**n**, x*_{0}exists. Again, byLemma 2.5, we have

*φx**n1**, x**n* *φx**n1**,*Π*C**n**x*0

≤*φx*_{n1}*, x*_{0}−*φΠ**C**n**x*_{0}*, x*_{0}
*φx**n1**, x*_{0}−*φx**n**, x*_{0},

3.8

for all*n*≥0. Thus,*φx**n1**, x**n* → 0 as*n* → ∞.

Next, we show that{x*n*}is a Cauchy sequence. UsingLemma 2.2, for*m, n*such that
*m > n, we have*

*gx**m*−*x** _{n}*≤

*φx*

*m*

*, x*

*≤*

_{n}*φx*

*m*

*, x*

_{0}−

*φx*

*n*

*, x*

_{0}, 3.9 where

*g*:0,∞ → 0,∞is a continuous stricly increasing and convex function with

*g0*0. Then the properties of the function

*g*yield that{x

*n*}is a Cauchy sequence. Thus, we can say that{x

*n*}converges strongly to

*p*for some point

*p*in

*C. However, since lim*

_{n}_{→ ∞}

*β*

*1 and{x*

_{n}*n*}is bounded, we obtain

*φx**n1**, z**n* *φ*

*x*_{n1}*, J*^{−1}

*β**n**Jx**n*
1−*β**n*

*JT**n**x**n*
x*n1*^{2}−2

*x*_{n1}*, β**n**Jx**n*
1−*β**n*

*JT**n**x**n* *β**n**Jx**n*
1−*β**n*

*JT**n**x**n*^{2}

≤ *x*_{n1}^{2}−2β*n* x*n1**, Jx**n* −2
1−*β**n*

x*n1**, JT**n**x**n**β**n*x*n*^{2}
1−*β**n*

T*n**x**n*^{2}
*β*_{n}*φx*_{n1}*, x*_{n}

1−*β*_{n}

*φx*_{n1}*, T*_{n}*x** _{n}*.

3.10
Therefore*φx*_{n1}*, z** _{n}* → 0 as

*n*→ ∞.

Since*x** _{n1}* Π

*C*

*n1*

*x*

_{0}∈C

*, from the definition of*

_{n1}*C*

*, we have*

_{n}*φ*

*x*_{n1}*, y*_{n}

≤*φx*_{n1}*, x** _{n}*, 3.11

for all*n*≥0. Thus

*φ*

*x*_{n1}*, y**n*

−→0, as*n*−→ ∞. 3.12

By usingLemma 2.3, we also have

*n*lim→ ∞*x** _{n1}*−

*y*

*n*lim

*n*→ ∞x*n1*−*x**n* lim

*n*→ ∞x*n1*−*z**n*0. 3.13

Since*J*is uniformly norm-to-norm continuous on bounded sets, we have

*n*lim→ ∞*Jx** _{n1}*−

*Jy*

*n*lim

*n*→ ∞Jx* _{n1}*−

*Jx*

*lim*

_{n}*n*→ ∞Jx* _{n1}*−

*Jz*

*0. 3.14*

_{n}For each*n*∈N∪ {0},we observe that

*Jx** _{n1}*−

*Jy*

_{n}*Jx*

*−*

_{n1}*α*

_{n}*Jx*

*1−*

_{n}*α*

_{n}*JT*

_{n}*z*

_{n}α*n*Jx* _{n1}*−

*Jx*

*1−*

_{n}*α*

*Jx*

_{n}*−*

_{n1}*JT*

_{n}*z*

*1−*

_{n}*α*

*n*Jx

*n1*−

*JT*

*n*

*z*

*n*−

*α*

*n*Jx

*n*−

*Jx*

_{n1}≥1−*α** _{n}*Jx

*−*

_{n1}*JT*

_{n}*z*

*−*

_{n}*α*

*Jx*

_{n}*n*−

*Jx*

*.*

_{n1}3.15

It follows that

Jx* _{n1}*−

*JT*

_{n}*z*

*≤ 1*

_{n}1−*α*_{n}*Jx** _{n1}*−

*Jy*

_{n}*α*

*Jx*

_{n}*n*−

*Jx*

_{n1}*.* 3.16

By3.14and lim sup_{n}_{→ ∞}*α**n* *<*1, we obtain

*n*lim→ ∞Jx* _{n1}*−

*JT*

_{n}*z*

*0. 3.17*

_{n}Since*J*^{−1}is uniformly norm-to-norm continuous on bounded sets, we have

*n*lim→ ∞x* _{n1}*−

*T*

_{n}*z*

*0. 3.18*

_{n}By3.13, we have

z*n*−*x**n* ≤ z*n*−*x** _{n1}*x

*n1*−

*x*

*n*−→0, as

*n*−→ ∞. 3.19

Since*T**n*is uniformly continuous, by3.13and3.18, we obtain

x*n*−*T*_{n}*x** _{n}* ≤ x

*n*−

*x*

*x*

_{n1}*−*

_{n1}*T*

_{n}*z*

*T*

_{n}*n*

*z*

*−*

_{n}*T*

_{n}*x*

*−→0, 3.20*

_{n}as*n* → ∞, and so

*n→ ∞*limJx*n*−*JT*_{n}*x** _{n}*0. 3.21

Based on the hypothesis, we now consider the following two cases.

*Case 1.* {T*n*},{x*n*}satisfies condition^{∗}AKTT. ApplyingLemma 2.8to get
Jx*n*−*JTx** _{n}* ≤ Jx

*n*−

*JT*

_{n}*x*

*JT*

_{n}*n*

*x*

*−*

_{n}*JTx*

_{n}≤ *Jx**n*−*JT**n**x**n*sup{JT*n**z*−*JTz*:*z*∈ {x*n*}} −→0. 3.22

*Case 2.* {T*n*},{x*n*}satisfies condition AKTT. ApplyLemma 2.7to get
x*n*−*Tx** _{n}* ≤ x

*n*−

*T*

_{n}*x*

*T*

_{n}*n*

*x*

*−*

_{n}*Tx*

_{n}≤ *x**n*−*T**n**x**n*sup{T*n**z*−*Tz*:*z*∈ {*x**n*}} −→0. 3.23

Hence

*n*lim→ ∞x*n*−*Tx**n* lim

*n*→ ∞

J^{−1}Jx*n*−*J*^{−1}JTx*n*0. 3.24

Therefore, from the both two cases, we have

*n*lim→ ∞x*n*−*Tx**n*0. 3.25

Since*T* is closed and*x**n* → *p, we havep*∈*FT.*Moreover, by3.7, we obtain
*φ*

*p, x*_{0}
lim

*n*→ ∞*φx**n**, x*_{0}≤*φ*
*p, x*_{0}

*,* 3.26

for all*p*∈*FT*. Therefore,*p* Π_{FT}*x*_{0}*.*This completes the proof.

Since every relatively nonexpansive mapping is a hemi-relatively nonexpansive mapping, we obtain the following result for a countable family of relatively nonexpansive mappings of modified Ishikawa iterative process.

**Corollary 3.2. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *Cbe a*
*nonempty bounded closed convex subset of* *E. Let* {T*n*} *be a sequence of relatively nonexpansive*
*mappings from* *Cinto itself such that*_{∞}

*n0**FT**n**is nonempty. Assume that*{α*n*}^{∞}_{n0}*and* {β*n*}^{∞}_{n0}*are sequences in*0,1*such that lim sup*_{n}_{→ ∞}*α*_{n}*<1 and lim*_{n}_{→ ∞}*β*_{n}*1 and let a sequence*{x*n*}*in*
*Cbe defined by the following algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y*_{n}*J*^{−1}α*n**Jx** _{n}* 1−

*α*

*JT*

_{n}*n*

*z*

*,*

_{n}*z*

_{n}*J*

^{−1}

*β*_{n}*Jx** _{n}*
1−

*β*

_{n}*JT*_{n}*x*_{n}*,*
*C*_{n1}

*v*∈*C** _{n}*:

*φ*

*v, y*

_{n}≤*φv, x**n*
*,*
*x** _{n1}* Π

*C*

*n1*

*x*

_{0}

*,*

3.27

*forn*∈ N∪ {0}, where*J* *is the single-valued duality mapping onE. Suppose that for each bounded*
*subsetBofC, the ordered pair*{T*n*}, B*satisfies either condition AKTT or condition*^{∗}*AKTT. LetTbe*
*the mapping fromCinto itself defined byTv*lim*n*→ ∞*T**n**vfor allv*∈*Cand suppose thatTis closed*

*andFT* _{∞}

*n0**FT**n*. If*T**n**is uniformly continuous for alln*∈N, then{x*n*}*converges strongly to*
Π_{FT}*x*_{0}*, where*Π_{FT}*is the generalized projection fromContoFT*.

**Theorem 3.3. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *C* *be a*
*nonempty bounded closed convex subset ofE. Let*{T*n*}*be a sequence of hemi-relatively nonexpansive*
*mappings fromCinto itself such that*_{∞}

*n0**FT**n**is nonempty. Assume that*{α*n*}^{∞}_{n0}*is a sequence in*
0,1*such that lim sup*_{n}_{→ ∞}*α**n**<1 and let a sequence*{x*n*}*inCbe defined by the following algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y*_{n}*J*^{−1}α*n**Jx** _{n}* 1−

*α*

*JT*

_{n}*n*

*x*

*,*

_{n}*C*

_{n1}*v*∈*C** _{n}*:

*φ*

*v, y*

_{n}≤*φv,x*_{n}*,*
*x** _{n1}* Π

*C*

*n1*

*x*

_{0}

*,*

3.28

*forn*∈ N∪ {0}, where*J* *is the single-valued duality mapping onE. Suppose that for each bounded*
*subsetBofC, the ordered pair*{T*n*}, B*satisfies either condition AKTT or condition*^{∗}*AKTT. LetT*
*be the mapping fromCinto itself defined byTv* lim_{n}_{→ ∞}*T*_{n}*vfor allv*∈ *Cand suppose thatT* *is*
*closed andFT* _{∞}

*n0**FT**n*. Then{x*n*}*converges strongly to*Π*FT**x*_{0}*.*

*Proof. In*Theorem 3.1, if*β**n*1 for all*n*∈N∪ {0}then3.1reduced to3.28.

**Corollary 3.4. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *Cbe a*
*nonempty bounded closed convex subset of* *E. Let* {T*n*} *be a sequence of relatively nonexpansive*
*mappings fromCinto itself such that*_{∞}

*n0**FT**n**is nonempty. Assume that*{α*n*}^{∞}_{n0}*is a sequence*
*in* 0,1 *such that lim sup*_{n}_{→ ∞}*α*_{n}*<* *1 and let a sequence* {x*n*} *in* *Cbe defined by the following*
*algorithm:*

*x*0∈*C,* *chosen arbitrarity,* *C*0*C,*
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JT*n**x**n*,
*C*_{n1}

*v*∈*C**n*:*φ*
*v, y**n*

≤*φv, x**n*
*,*
*x** _{n1}* Π

*C*

_{n1}*x*

_{0}

*,*

3.29

*forn*∈ N∪ {0}, where*J* *is the single-valued duality mapping onE. Suppose that for each bounded*
*subsetBofC, the ordered pair*{T*n*}, B*satisfies either condition AKTT or condition*^{∗}*AKTT. LetT*
*be the mapping fromCinto itself defined byTv* lim*n*→ ∞*T**n**vfor allv*∈ *Cand suppose thatT* *is*
*closed andFT* _{∞}

*n0**FT**n*. Then{x*n*}*converges strongly to*Π_{FT}*x*_{0}*.*

Notice that every uniformly continuous mapping must be a continuous and closed
mapping. Then setting*T**n* ≡*T* for all*n*∈N, in Theorems3.1and3.3, we immediately obtain
the following results.

**Corollary 3.5. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *Cbe a*
*nonempty bounded closed convex subset ofE. LetT* :*C* → *Cbe a closed hemi-relatively nonexpansive*
*mapping such that* *FT/*∅. Assume that {α*n*}^{∞}_{n0}*and* {β*n*}^{∞}_{n0}*are sequences in* 0,1 *such that*

lim sup_{n}_{→ ∞}*α**n* *<* *1 and lim**n*→ ∞*β**n* *1 and let a sequence*{x*n*}*in* *Cbe defined by the following*
*algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JTz*n*,
*z**n**J*^{−1}

*β**n**Jx**n*
1−*β**n*

*JTx**n*
*,*
*C*_{n1}

*v*∈*C**n*:*φ*
*v, y**n*

≤*φv, x**n*
*,*
*x** _{n1}* Π

*C*

*n1*

*x*

_{0}

*,*

3.30

*forn*∈N∪{0}, where*Jis the single-valued duality mapping onE. IfTis uniformly continuous, then*
{x*n*}*converges strongly to*Π_{FT}*x*_{0}*.*

**Corollary 3.6. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *Cbe a*
*nonempty bounded closed convex subset ofE. LetT* : *C* → *Cbe a closed relatively nonexpansive*
*mapping such that* *FT/*∅. Assume that {α*n*}^{∞}_{n0}*and* {β*n*}^{∞}_{n0}*are sequences in* 0,1 *such that*
lim sup_{n}_{→ ∞}*α**n* *<* *1 and lim**n*→ ∞*β**n* *1 and let a sequence*{x*n*}*in* *Cbe defined by the following*
*algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity, C*_{0}*C,*
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JTz*n*,
*z*_{n}*J*^{−1}

*β*_{n}*Jx** _{n}*
1−

*β*

_{n}*JTx*_{n}*,*
*C*_{n1}

*v*∈*C** _{n}*:

*φ*

*v, y*

_{n}≤*φv, x**n*
*,*
*x** _{n1}* Π

*C*

*n1*

*x*

_{0}

*,*

3.31

*forn*∈N∪{0}, where*Jis the single-valued duality mapping onE. IfTis uniformly continuous, then*
{x*n*}*converges strongly to*Π_{FT}*x*_{0}*.*

*Proof. Since a closed relatively nonexpansive mapping is a closed hemi-relatively one,*
Corollary 3.6is implied byCorollary 3.5.

**Corollary 3.7. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *Cbe a*
*nonempty bounded closed convex subset ofE. LetT* :*C* → *Cbe a closed hemi-relatively nonexpansive*
*mapping fromCinto itself such thatFT/*∅. Assume that{α*n*}^{∞}_{n0}*is a sequence in*0,1*such that*
lim sup_{n}_{→ ∞}*α*_{n}*<1 and let a sequence*{x*n*}*inCbe defined by the following algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y*_{n}*J*^{−1}α*n**Jx** _{n}* 1−

*α*

*JTx*

_{n}*n*,

*C*

_{n1}*v*∈*C** _{n}*:

*φ*

*v, y*

_{n}≤*φv, x**n*
*,*
*x** _{n1}* Π

*C*

*n1*

*x*

_{0}

*,*

3.32

*forn*∈N∪ {0}, where*Jis the single-valued duality mapping onE. Then*{x*n*}*converges strongly to*
Π_{FT}*x*_{0}*.*

**Corollary 3.8. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *Cbe a*
*nonempty bounded closed convex subset ofE. LetT* : *C* → *Cbe a closed relatively nonexpansive*
*mapping fromCinto itself such thatFT/*∅. Assume that{α*n*}^{∞}_{n0}*is a sequence in*0,1*such that*
lim sup_{n}_{→ ∞}*α**n* *<1 and let a sequence*{x*n*}*inCbe defined by the following algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JTx*n*,
*C*_{n1}

*v*∈*C** _{n}*:

*φ*

*v, y*

_{n}≤*φv, x**n*
*,*
*x** _{n1}* Π

*C*

_{n1}*x*

_{0}

*,*

3.33

*forn*∈N∪ {0}, where*Jis the single-valued duality mapping onE. Then*{x*n*}*converges strongly to*
Π_{FT}*x*_{0}*.*

Similarly, as in the proof ofTheorem 3.1, we obtain the following results.

**Theorem 3.9. Let***E* *be a uniformly convex and uniformly smooth Banach space and let* *C* *be a*
*nonempty bounded closed convex subset ofE. Let*{T*n*}*be a sequence of hemi-relatively nonexpansive*
*mappings fromCinto itself such that*_{∞}

*n0**FT**n**is nonempty. Assume that*{α*n*}^{∞}_{n0}*and*{β*n*}^{∞}_{n0}*are*
*sequences in*0,1*such that lim sup*_{n}_{→ ∞}*α*_{n}*<1 and lim*_{n}_{→ ∞}*β*_{n}*<1 and let a sequence*{x*n*}*inCbe*
*defined by the following algorithm:*

*x*0∈*C,* *chosen arbitrarity,* *C*0*C,*
*y*_{n}*J*^{−1}α*n**Jx** _{n}* 1−

*α*

*JT*

_{n}*n*

*z*

*,*

_{n}*z*

*n*

*J*

^{−1}

*β**n**Jx**n*
1−*β**n*

*JT**n**x**n*
*,*
*C*_{n}

*v*∈*C*:*φ*
*v, y*_{n}

≤*φv, x**n*
*,*
*Q**n*{v∈*C*: v−*x**n**, Jx**n*−*Jx*_{0} ≥0},
*x** _{n1}* Π

*C*

*n*∩Q

*n*

*x*0

*,*

3.34

*forn*∈ N∪ {0}, where*J* *is the single-valued duality mapping onE. Suppose that for each bounded*
*subsetBofC, the ordered pair*{T*n*}, B*satisfies either condition AKTT or condition*^{∗}*AKTT. LetTbe*
*the mapping fromCinto itself defined byTv*lim*n*→ ∞*T*_{n}*v* *for allv*∈*Cand suppose thatTis closed*

*andFT* _{∞}

*n0**FT**n*. If*T*_{n}*is uniformly continuous for alln*∈N, then{x*n*}*converges strongly to*
Π*FT**x*0*, where*Π*FT**is the generalized projection fromContoFT*.

**Corollary 3.10. Let**E*be a uniformly convex and uniformly smooth Banach space and letC* *be a*
*nonempty bounded closed convex subset ofE. LetT* :*C* → *Cbe closed hemi-relatively nonexpansive*
*mappings fromCinto itself such thatFT/*∅. Assume that{α*n*}^{∞}_{n0}*and*{β*n*}^{∞}_{n0}*are sequences in*
0,1*such that lim sup*_{n}_{→ ∞}*α*_{n}*<1 and lim sup*_{n}_{→ ∞}*β*_{n}*1 and let a sequence*{x*n*}*inCbe defined*

*by the following algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JTz*n*,
*z*_{n}*J*^{−1}

*β*_{n}*Jx** _{n}*
1−

*β*

_{n}*JTx*_{n}*,*
*C**n*

*v*∈*C*:*φ*
*v, y**n*

≤*φv, x**n*
*,*
*Q** _{n}*{

*v*∈

*C*: v−

*x*

_{n}*, Jx*

*−*

_{n}*Jx*

_{0}≥0}

*,*

*x*

*Π*

_{n1}*C*

*n*∩Q

*n*x0,

3.35

*forn*∈N∪{0}, where*Jis the single-valued duality mapping onE. IfTis uniformly continuous, then*
{x*n*}*converges strongly to*Π_{FT}*x*_{0}*.*

**Theorem 3.11. Let**E*be a uniformly convex and uniformly smooth Banach space and let* *C* *be a*
*nonempty bounded closed convex subset of* *E. Let* {T*n*} *be a sequence of relatively nonexpansive*
*mappings fromCinto itself such that*_{∞}

*n0**FT**n**is a nonempty. Assume that*{α*n*}^{∞}_{n0}*is a sequence in*
0,1*such that lim sup*_{n}_{→ ∞}*α**n**<1 and let a sequence*{x*n*}*inCbe defined by the following algorithm:*

*x*_{0}∈*C,* *chosen arbitrarity,* *C*_{0}*C,*
*y**n**J*^{−1}α*n**Jx**n* 1−*α**n*JT*n**x**n*,
*C*_{n}

*v*∈*C*:*φ*
*v, y*_{n}

≤*φv, x**n*
*,*
*Q**n*{v∈*C*: v−*x**n**, Jx**n*−*Jx*_{0} ≥0},
*x** _{n1}* Π

*C*

*n*∩Q

*n*

*x*0

*,*

3.36

*forn*∈ N∪ {0}, where*J* *is the single-valued duality mapping onE. Suppose that for each bounded*
*subsetBofC, the ordered pair*{T*n*}, B*satisfies either condition AKTT or condition*^{∗}*AKTT. LetT*
*be the mapping fromCinto itself defined byTv* lim*n*→ ∞*T*_{n}*vfor allv*∈ *Cand suppose thatT* *is*
*closed andFT* _{∞}

*n0**FT**n*. Then{x*n*}*converges strongly to*Π_{FT}*x*_{0}*.*

*Proof. Puttingβ**n*1, for all*n*∈N∪{0}, inTheorem 3.9we immediately obtainTheorem 3.11.

**Corollary 3.12. Let**E*be a uniformly convex and uniformly smooth Banach space and letC* *be a*
*nonempty bounded closed convex subset ofE. LetT* :*C* → *Cbe closed hemi-relatively nonexpansive*
*mappings fromCinto itself such thatF*T*/*∅. Assume that{α*n*}^{∞}_{n0}*is a sequence in*0,1*such that*