On the Convergence of an Implicit Iterative Process for Generalized Asymptotically Quasi-Nonexpansive Mappings

Fixed Point Theory and Applications Volume 2010, Article ID 714860,19pages doi:10.1155/2010/714860

Research Article

On the Convergence of an Implicit

Iterative Process for Generalized Asymptotically

Quasi-Nonexpansive Mappings

Xiaolong Qin,

_{Shin Min Kang,}

_{and Ravi P. Agarwal}

1_{School of Mathematics and Information Sciences, North China University of Water Resources and}

Electric Power, Zhengzhou 450011, China

2_{Department of Mathematics and the RINS, Gyeongsang National University,}

Jinju 660-701, Republic of Korea

3_{Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA} 4_{Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals,}

Dhahran 31261, Saudi Arabia

Correspondence should be addressed to Ravi P. Agarwal,[email protected]

Received 27 June 2010; Revised 16 October 2010; Accepted 10 November 2010

Academic Editor: Naseer Shahzad

Copyrightq2010 Xiaolong Qin 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.

The purpose of this paper is to introduce and consider a general implicit iterative process which includes Schu’s explicit iterative processes and Sun’s implicit iterative processes as special cases for a finite family of generalized asymptotically quasi-nonexpansive mappings. Strong convergence of the purposed iterative process is obtained in the framework of real Banach spaces.

1. Introduction and Preliminaries

LetEbe a real Banach space andUE{x∈E:x1}.Eis said to beuniformly convexif for

any∈0,2there existsδ >0 such that for anyx, y∈UE,

x−y≥ implies xy 2

≤1−δ. 1.1

It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall thatTis said to be nonexpansiveif

_{Tx−Ty≤x−y, ∀x, y∈C. 1.2}

Tis said to bequasi-nonexpansiveifFT/∅and

Tx−y≤x−y, ∀x∈C, y∈FT. 1.3

A nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive; however, the inverse may be not true. See the following example1.

Example 1.1. LetER1_{and define a mapping byT :E → Eby}

Tx

⎧ ⎨ ⎩

x

2sin 1

x ifx /0,

0 ifx0.

1.4

ThenTis quasi-nonexpansive but not nonexpansive.

T is said to be asymptotically nonexpansiveif there exists a positive sequence {kn} ⊂

1,∞withkn → 1 asn → ∞such that

Tnx−Tny≤knx−y, ∀x, y∈C, n≥1. 1.5

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence {1}. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk2in 1972. It is known that if Cis a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive mapping onChas a fixed point. Further, the setFTof fixed points ofT is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.

T is said to beasymptotically quasi-nonexpansiveifFT/∅, and there exists a positive sequence{kn} ⊂1,∞withkn → 1 asn → ∞such that

_Tn_x−y≤k

nx−y, ∀x∈C, y∈FT, n≥1. 1.6

Tis said to be asymptotically nonexpansive in the intermediate senseif it is continuous and the following inequality holds:

lim sup

n→ ∞ x,ysup∈C

_Tn_x−Tn_{y−x−y≤0.}

1.7

Puttingξn max{0,sup_x,y∈CTnx−Tny − x−y}, we see thatξn → 0 asn → ∞. Then

1.7is reduced to the following:

_Tn_x−Tn_y≤x−yξ

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Kirk 3 see also Bruck et al. 4 as a generalization of the class of asymptotically nonexpansive mappings. It is known that if Cis a nonempty closed convex and bounded subset of a real Hilbert space, then every asymptotically nonexpansive self-mapping in the intermediate sense has a fixed point; see5more details.

T is said to be asymptotically quasi-nonexpansive in the intermediate sense if it is continuous,FT/∅, and the following inequality holds:

lim sup

n→ ∞ x∈C,ysup∈FT

_Tn_{x−y−x−y≤0.}

1.9

Puttingξnmax{0,supx∈C,y∈FTTnx−y − x−y}, we see thatξn → 0 asn → ∞. Then

1.9is reduced to the following:

Tn_{x−y≤x−yξ}

n, ∀x∈C, y∈FT, n≥1. 1.10

T is said to begeneralized asymptotically nonexpansive if there exist two positive sequences {kn} ⊂ 1,∞ with kn → 1 and {ξn} ⊂ 0,∞ with ξn → 0 as n → ∞ such

that

_Tn_x−Tn_y≤k

nx−yξn, ∀x, y∈C, n≥1. 1.11

It is easy to see that the class of generalized asymptotically nonexpansive includes the class of asymptotically nonexpansive as a special case.

T is said to begeneralized asymptotically quasi-nonexpansiveifFT/∅, and there exist two positive sequences{kn} ⊂1,∞withkn → 1 and{ξn} ⊂0,∞withξn → 0 asn → ∞

such that

Tnx−y≤knx−yξn, ∀x∈C, y∈FT, n≥1. 1.12

The class of generalized asymptotically quasi-nonexpansive was considered by Shahzad and Zegeye6; see6,7for more details.

Recall that the modified Mann iteration which was introduced by Schu8generates a sequence{xn}in the following manner:

x1 ∈C, xn1 1−αnxnαnTnxn, ∀n≥1, 1.13

where {αn} is a sequence in the interval 0,1 and T:C → C is an asymptotically

nonexpansive mapping.

In 1991, Schu8obtained the following results.

Theorem Schu 1. Let E be a uniformly convex Banach space, ∅/C ⊂ E closed bounded and convex, and T : C → C asymptotically nonexpansive with sequence {kn} ⊂ 1,∞ for which

_∞

n1kn−1 <∞and{αn} ∈ 0,1is bounded away. Let{xn}be a sequence generated in1.13.

Theorem Schu 2. LetEbe a uniformly convex Banach space,∅/C⊂Eclosed bounded and convex, andT :C → Casymptotically nonexpansive with sequence{kn} ⊂1,∞for which∞n1kn−1<

∞and{αn} ∈0,1is bounded away. Let{xn}be a sequence generated in1.13. Suppose thatTm

is compact for some positive integerm≥1. Then the sequence{xn}converges strongly to some fixed

point ofT.

Theorem Schu 3. LetEbe a uniformly convex Banach space,∅/C⊂Eclosed bounded and convex, andT :C → Casymptotically nonexpansive with sequence{kn} ⊂1,∞for which∞n1kn−1<

∞and{αn} ∈0,1is bounded away. Let{xn}be a sequence generated in1.13. Suppose that there

exists a nonempty compact and convex subsetKofEandλ∈0,1such that

dTx, K≤λdx, K, ∀x∈C. 1.14

Then the sequence{xn}converges strongly to some fixed point ofT.

In 2007, Shahzad and Zegeye6considered the following implicit iterative process for a finite family of generalized asymptotically quasi-nonexpansive mappings{T1, T2, . . . , TN}:

x1α1x0 1−α1T1x1,

x2α2x1 1−α2T2x2,

.. .

xNαNxN−1 1−αNTNxN,

xN1αN1xN 1−αN1T12xN1,

.. .

x2Nα2Nx2N−1 1−α2NT_N2x2N,

x2N1α2N1x2N 1−α2N1T13x2N1,

.. .,

1.15

where x0 is the initial value and {αn}is a sequence 0,1. Since for each n ≥ 1, it can be

written asn h−1Ni, wherei in∈ {1,2, . . . , N},hhn≥ 1 is a positive integer, andhn → ∞asn → ∞. Hence the above table can be rewritten in the following compact form:

xnαnxn−1αnT_ih_nnxn, ∀n≥1. 1.16

Shahzad and Zegeye6obtained the following results.

Theorem SZ 1. LetEbe a real uniformly convex Banach space andCbe a nonempty closed convex subset of E. Let {Ti : i ∈ J}, where J {1,2, . . . , N}, be N uniformly Lipschitz, generalized

asymptotically quasi-nonexpansive self-mappings ofCwith{kin} ⊂1,∞,{ξn} ⊂0,∞such that

_∞

n1kin−1<∞and

_∞

n1ξin<∞for alli∈J. Suppose thatF∩N_i1FTi/∅and there exists one

memberT in{Ti:i∈J}which is either semicompact or satisfies conditionC. Let{αn} ⊂δ,1−δ

for someδ∈0,1. From arbitraryx1 ∈C, define the sequence{xn}by1.16. Then{xn}converges

strongly to a common fixed point of the mappings{Ti:i∈J}.

Theorem SZ 2. LetE be a real uniformly convex Banach space and C a nonemptyclosed convex subset ofE. Let {Ti : i ∈ J}, where J {1,2, . . . , N}, be N generalized asymptotically

quasi-nonexpansive self-mappings ofCwith{kin} ⊂1,∞,{ξin} ⊂0,∞such that

_∞

n1kin−1< ∞

and∞_n1ξin < ∞for alli∈ J. Suppose thatF ∩N_i1FTi/∅is closed. Let{αn} ⊂δ,1−δfor

someδ ∈ 0,1. From arbitrary x1 ∈ C, define the sequence{xn}by 1.16. Then{xn}converges

strongly to a common fixed point of the mappings{Ti:i∈J}if and only iflim infn→ ∞dxn, F 0.

In this paper, motivated by the above results, we consider the following implicit iterative process for two finite families of generalized asymptotically quasi-nonexpansive mappings{S1, S2, . . . , SN}and{T1, T2, . . . , TN}:

x1 α1x0β1S1x0γ1T1x1δ1u1,

x2 α2x1β2S2x1γ2T2x2δ2u2,

.. .

xN αNxN−1βNSNxN−1γNTNxNδNuN,

xN1 αN1xNβN1S21xNγN1T12xN1δN1uN1,

.. .

x2N α2Nx2N−1β2NS2_Nx2N−1γ2NT_N2x2Nδ2Nu2N,

x2N1 α2N1x2Nβ2N1S31x2Nγ2N1T13x2N1δ2N1u2N1,

.. .,

1.17

wherex0is the initial value,{un}is a bounded sequence inC, and{αn},{βn},{γn}, and{δn}

are sequences0,1such thatαnβnγnδn 1 for eachn≥1. Since for eachn≥1, it can

be written asn h−1Ni, whereiin∈ {1,2, . . . , N},hhn≥1 is a positive integer andhn → ∞asn → ∞. Hence the above table can be rewritten in the following compact form:

We remark that our implicit iterative process1.18which includes the explicit iterative process1.13and the implicit iterative process1.16as special cases is general.

IfSi I, whereI denotes the identity mapping, for eachi ∈ {1,2, . . . , N}, then the

implicit iterative process1.18is reduced to the following implicit iterative process:

xn

αnβn

xn−1γnT_ih_nnxnδnun, ∀n≥1. 1.19

IfTi I, whereI denotes the identity mapping, for eachi ∈ {1,2, . . . , N}, then the

implicit iterative process1.18is reduced to the following explicit iterative process:

xn

αn

1−γnxn−1

βn

1−γnS hn

inxn−1

δn

1−γnun, ∀n≥1. 1.20

The purpose of this paper is to study the convergence of the implicit iteration process

1.18 for two finite families of generalized asymptotically quasi-nonexpansive mappings. Strong convergence theorems are obtained in the framework of real Banach spaces. The results presented in this paper improve and extend the corresponding results in Shahzad and Zegeye6, Sun9, Chang et al.10, Chidume and Shahzad11, Guo and Cho12, Kim et al.13, Qin et al.14, Thianwan and Suantai15, Xu and Ori16, and Zhou and Chang17.

In order to prove our main results, we also need the following lemmas.

Lemma 1.2 see 18. Let {rn}, {sn}, and {tn} be three nonnegative sequences satisfying the

following condition:

rn1≤1snrntn, ∀n≥n0, 1.21

wheren0is some positive integer. If

_∞

n1sn<∞and

_∞

n1tn<∞, thenlimn→ ∞rnexists.

Lemma 1.3see19. LetEbe a real uniformly convex Banach space,s > 0 a positive number, andBs0a closed ball ofE. Then there exists a continuous, strictly increasing, and convex function

g:0,∞ → 0,∞withg0 0such that

axbyczdw2≤ax2by2cz2dw2−abgx−y 1.22

for allx, y, z, w∈Bs0 {x∈E:x ≤s}anda, b, c, d∈0,1such thatabcd1.

2. Main Results

Lemma 2.1. LetEbe a real uniformly convex Banach space andCa nonempty closed convex subset ofE. LetTi :C → Cbe a uniformlyLt,i-Lipschitz and generalized asymptotically quasi-nonexpansive

mapping with sequences{kn,t,i} ⊂ 1,∞and{ξn,t,i} ⊂0,∞such that

_∞

n1kn,t,i−1 < ∞and

_∞

n1ξn,t,i < ∞for each1 ≤ i ≤ N and Si : C → Ca uniformly Ls,i-Lipschitz and generalized

asymptotically quasi-nonexpansive mapping with sequences{kn,s,i} ⊂1,∞and {ξn,s,i} ⊂ 0,∞

∩N

i1FTi ∩N_i1FSiis nonempty. Let {un} be a bounded sequence inC, kn max{kn,t, kn,s},

wherekn,t max{kn,t,i : 1 ≤i≤N}andkn,s max{kn,s,i: 1 ≤i≤N}andξn max{ξn,t, ξn,s},

whereξn,t max{ξn,t,i : 1≤ i≤N}andξn,s max{ξn,s,i : 1≤i≤N}. Let{αn},{βn},{γn}, and

{δn}be sequences in0,1such thatαnβnγnδn 1for eachn≥ 1. Let{xn}be a sequence

generated in1.18. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c, d∈0,1such thata≤αn,b≤βn, andc≤ γn ≤d < 1/Lt,

whereLtmax{Lt,i: 1≤i≤N}, for alln≥1;

b∞_n1δn<∞.

Then

lim

n→ ∞xn−Trxnnlim→ ∞xn−Srxn0, ∀r∈ {1,2, . . . , N}. 2.1

Proof . First, we show that the sequence{xn}generated in 1.18 is well defined. For each

n≥1, define a mappingCn:C → Cas follows:

Cnxαnxn−1βnSh_innxn−1γnT_ih_nnxδnun, ∀x∈C. 2.2

Notice that

_{Cnx−Cny≤γnT}hn

inx−Tihnny

≤dLtx−y, ∀x, y∈C.

2.3

From the restriction a, we see that Cn is a contraction for each n ≥ 1. From Banach

contraction mapping principle, we can prove that the sequence {xn} generated in1.18is

well defined.

Fixingp∈F, we see that

_{xn−p≤αnxn−1−pβnS}hn

inxn−1−p

γnT_ih_nnxn−pδnun−p

≤αnxn−1−pβn

khnxn−1−pξhnγn

khnxn−pξhn

δnun−p

≤αnβnkhnxn−1−p

1−αn−βn

khnxn−p2ξhn

δnun−p.

2.4

Notice that ∞_n1kn−1 < ∞. We see from the restrictionsaand b that there exists a

positive integern0such that

1−αn−βn

whereR 1−ab1 ab/2−2ab. It follows from2.4that

xn−p≤

αnβnkhn

1−1−αn−βn

khn

xn−1−p

δn

1−1−αn−βn

khn

un−p

2ξhn

1−1−αn−βn

khn

≤

1khn−1 1−R

xn−1−p δn

1−Run−p

2ξhn 1−R

≤

1khn−1 1−R

xn−1−pM1

δnξhn, ∀n≥n0,

2.6

whereM1is an appropriate constant such thatM1max{supn≥1{un−p/1−R},2/1−R}.

In view of the restrictionsaandb, we obtain fromLemma 1.2that limn→ ∞xn−pexists.

It follows that the sequence{xn}is bounded. In view ofLemma 1.3, we see that

_xn−p2_≤

αnxn−1−p2βnSh_innxn−1−p 2

γnT_ih_nnxn−p

2

δnun−p2−αnβngS_ih_nnxn−1−xn−1

≤αnxn−1−p2βn

khnxn−1−pξhn2γn

khnxn−pξhn2

δnun−p2−αnβngS_ih_nnxn−1−xn−1

≤αnxn−1−p2βn

k2

hnxn−1−p

2_ξ2

hn2khnξhnxn−1−p

γn

k2

hnxn−p

2_ξ2

hn2khnξhnxn−p

δnun−p2−αnβngS_ih_nnxn−1−xn−1

≤αnβnk_h2_n

xn−1−p2γnk2_hnxn−p22ξ_h2_n

2khnξhnM2δnM3−αnβngSh_innxn−1−xn−1

,

2.7

whereM2 andM3are appropriate constants such thatM2 sup_n≥1{xn−pxn−1−p} andM3sup_n≥1{un−p2}. This implies that

αnβngS_ih_nnxn−1−xn−1

≤αnβnk_h2_n

xn−1−p2−xn−p2

k2_hn−1xn−p2

2ξ2_hn2khnξhnM2δnM3.

In view of the restrictionsaandb, we obtain that

lim

n→ ∞g

_Shn

inxn−1−xn−1

0. 2.9

Since g : 0,∞ → 0,∞ is a continuous, strictly increasing, and convex function with

g0 0, we obtain that

lim

n→ ∞

Sh_innxn−1−xn−10. 2.10

Next, we show that

lim

n→ ∞

T_ih_nnxn−xn−10. 2.11

FromLemma 1.3, we also see that

_xn−p2_≤

αnxn−1−p2βnSh_innxn−1−p 2

γnT_ih_nnxn−p

2

δnun−p2−αnγngT_ih_nnxn−xn−1

≤αnxn−1−p2βn

khnxn−1−pξhn2γn

khnxn−pξhn2

δnun−p2−αnγngT_ih_nnxn−xn−1

≤αnxn−1−p2βn

k2_hnxn−1−p2ξ2hn2khnξhnxn−1−p

γn

k2

hnxn−p2ξ2hn2khnξhnxn−p

δnun−p2−αnγngT_ih_nnxn−xn−1

≤αnβnk2_hn

xn−1−p2γnk_h2_nxn−p22ξ_h2_n

2khnξhnM2δnM3−αnγngT_ih_nnxn−xn−1

.

2.12

This implies that

αnγngT_ih_nnxn−xn−1

≤αnβnk_h2_n

xn−1−p2−xn−p2

k2_hn−1xn−p2

2ξ2_hn2khnξhnM2δnM3.

In view of the restrictionsaandb, we obtain that

lim

n→ ∞g

_Thn

inxn−xn−1

0. 2.14

Since g : 0,∞ → 0,∞ is a continuous, strictly increasing, and convex function with

g0 0, we obtain that2.11holds. Notice that

xn−xn−1 ≤S_ih_nnxn−1−xn−1T_ih_nnxn−xn−1δnun−xn−1. 2.15

In view of2.10and2.11, we see from the restrictionbthat

lim

n→ ∞xn−xn−10, 2.16

which implies that

lim

n→ ∞xn−xnj0, ∀j ∈ {1,2, . . . , N}. 2.17

Since for any positive integern > N, it can be written asn hn−1Nin, where

in∈ {1,2, . . . , N}, observe that

xn−1−Tnxn ≤xn−1−T_ih_nnxnT_ih_nnxn−Tnxn

≤xn−1−T_ih_nnxnLtT_ih_nn−1xn−xn

≤xn−1−T_ih_nnxn

LtT_ih_nn−1xn−T_ih_nn_−N−1xn−NT_ih_nn_−N−1xn−N−xn−N−1

xn−N−1−xn.

2.18

Since for eachn > N,n n−NmodN, on the other hand, we obtain fromn hn−

1Ninthatn−N hn−1−1Nin hn−N−1Nin−N. That is,

hn−N hn−1, in−N in. 2.19

Notice that

T_ih_nn−1xn−T_ih_nn_−N−1xn−NT_ih_nn−1xn−T_ih_nn−1xn−N

≤Ltxn−xn−N,

T_ih_nn_−N−1xn−N−xn−N−1T_ih_nn₋−_NNxn−N−xn−N−1.

Substituting2.20into2.18, we arrive at

xn−1−Tnxn ≤xn−1−T_ih_nnxn

Lt

Ltxn−xn−NT_ih_nn₋−_NNxn−N−xn−N−1xn−N−1−xn.

2.21

In view of2.11and2.17, we obtain that

lim

n→ ∞xn−1−Tnxn0. 2.22

Notice that

xn−Tnxn ≤ xn−xn−1xn−1−Tnxn. 2.23

It follows from2.16and2.22that

lim

n→ ∞xn−Tnxn0. 2.24

Notice that

_{xn−Tnjxn≤xn−xnjxnj−TnjxnjTnjxnj−Tnjxn}

≤1Ltxn−xnjxnj−Tnjxnj, ∀j ∈ {1,2, . . . , N}.

2.25

From2.17and2.24, we arrive at

lim

n→ ∞xn−Tnjxn0, ∀j∈ {1,2, . . . , N}. 2.26

Note that any subsequence of a convergent number sequence converges to the same limit. It follows that

lim

n→ ∞xn−Trxn0, ∀r ∈ {1,2, . . . , N}. 2.27

LettingLsmax{Ls,i: 1≤i≤N}, we have

Sh_innxn−xn−1≤S_ih_nnxn−Sh_innxn−1Sh_innxn−1−xn−1

≤Lsxn−xn−1Sh_innxn−1−xn−1.

2.28

In view of2.10and2.16, we see that

lim

n→ ∞

Observe that

xn−1−Snxn−1 ≤xn−1−Sh_innxn−1S_ih_nnxn−1−Snxn−1

≤xn−1−Sh_innxn−1LsS_ih_nn−1xn−1−xn−1

≤xn−1−Sh_innxn−1

LsS_ih_nn−1xn−1−Sh_inn_−N−1xn−NS_ih_nn_−N−1xn−N−xn−N−1

xn−N−1−xn−1.

2.30

In view of

Sh_inn−1xn−1−S_ih_nn₋−_N1xn−NSh_inn−1xn−1−S_ih_nn−1xn−N

≤Lsxn−1−xn−N,

Sh_inn₋−_N1xn−N−xn−N−1Sh_inn₋−_NNxn−N−xn−N−1,

2.31

we arrive at

xn−1−Snxn−1 ≤xn−1−Sh_innxn−1

Ls

Lsxn−1−xn−NS_ih_nn₋−_NNxn−N−xn−N−1xn−N−1−xn−1.

2.32

In view of2.10,2.17, and2.29, we obtain that

lim

n→ ∞xn−1−Snxn−10. 2.33

Notice that

xn−Snxn ≤ xn−xn−1xn−1−Snxn−1Snxn−1−Snxn

≤1Lsxn−xn−1xn−1−Snxn−1.

2.34

From2.16and2.33, we see that

lim

n→ ∞xn−Snxn0. 2.35

On the other hand, we have

xn−Snjxn≤xn−xnjxnj−SnjxnjSnjxnj−Snjxn

≤1Lsxn−xnjxnj−Snjxnj, ∀j∈ {1,2, . . . , N}.

It follows from2.17and2.35that

lim

n→ ∞xn−Snjxn0, ∀j∈ {1,2, . . . , N}. 2.37

Note that any subsequence of a convergent number sequence converges to the same limit. It follows that

lim

n→ ∞xn−Srxn0, ∀r∈ {1,2, . . . , N}. 2.38

This completes the proof.

Recall that a mappingT :C → Cis said to be semicompactif for any bounded sequence

{xn}inCsuch thatxn−Txn → 0 asn → ∞, then there exists a subsequence{xni} ⊂ {xn} such thatxni → x∈C.

Next, we give strong convergence theorems with the help of the semicompactness.

Theorem 2.2. LetEbe a real uniformly convex Banach space andCa nonempty closed convex subset ofE. LetTi :C → Cbe a uniformlyLt,i-Lipschitz and generalized asymptotically quasi-nonexpansive

mapping with sequences {kn,t,i} ⊂ 1,∞ and {ξn,t,i} ⊂ 0,∞such that ∞n1kn,t,i −1 < ∞

and ∞_n1ξn,t,i < ∞ for each 1 ≤ i ≤ N, and let Si : C → C be a uniformly Ls,i-Lipschitz

and generalized asymptotically quasi-nonexpansive mapping with sequences{kn,s,i} ⊂ 1,∞and

{ξn,s,i} ⊂0,∞such that∞n1kn,s,i−1<∞and∞n1ξn,s,i<∞for each1≤i≤N. Assume that

F∩N

i1FTi ∩Ni1FSiis nonempty. Let{un}be a bounded sequence inC,knmax{kn,t, kn,s},

wherekn,t max{kn,t,i : 1 ≤i≤N}andkn,s max{kn,s,i: 1 ≤i≤N}andξn max{ξn,t, ξn,s},

whereξn,t max{ξn,t,i : 1≤ i≤N}andξn,s max{ξn,s,i : 1≤i≤N}. Let{αn},{βn},{γn}, and

{δn}be sequences in0,1such thatαnβnγnδn 1for eachn≥ 1. Let{xn}be a sequence

generated in1.18. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c, d∈0,1such thata≤αn,b≤βn, andc≤ γn ≤d < 1/Lt,

whereLtmax{Lt,i: 1≤i≤N}, for alln≥1;

b∞_n1δn<∞.

If one of{S1, S2, . . . , SN}or one of{T1, T2, . . . , TN}is semicompact, then the sequence{xn}converges

strongly to some point inF.

Proof. Without loss of generality, we may assume that S1 is semicompact. From 2.38, we see that there exits a subsequence{xni}of{xn}converging strongly tox ∈ C. For eachr ∈

{1,2, . . . , N}, we get that

x−Srx ≤ x−xnixni−SrxniSrxni−Srx. 2.39

SinceSris Lipshcitz continuous, we obtain from2.38thatx∈ ∩N_r1FSr. Notice that

x−Trx ≤ x−xnixni−TrxniTrxni−Trx. 2.40

SinceTr is Lipshcitz continuous, we obtain from2.27thatx∈ ∩N_r1FTr. This means that

x∈F. In view ofLemma 2.1, we obtain that limn→ ∞xn−xexists. Therefore, we can obtain

If Si I, where I denotes the identity mapping, for each i ∈ {1,2, . . . , N}, then

Theorem 2.2is reduced to the following.

Corollary 2.3. LetEbe a real uniformly convex Banach space andCa nonempty closed convex subset ofE. LetTi :C → Cbe a uniformlyLt,i-Lipschitz and generalized asymptotically quasi-nonexpansive

mapping with sequences{kn,t,i} ⊂ 1,∞and{ξn,t,i} ⊂0,∞such that

_∞

n1kn,t,i−1 < ∞and

_∞

n1ξn,t,i <∞for each1≤i≤N. Assume thatF∩N_i1FTiis nonempty. Let{un}be a bounded

sequence inC,kn,t max{kn,t,i : 1 ≤i≤ N}, andξn,t max{ξn,t,i : 1≤ i≤N}. Let{αn},{βn},

{γn}, and{δn}be sequences in0,1such thatαnβnγnδn 1for eachn≥ 1. Let{xn}be a

sequence generated in1.19. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c∈0,1such thata≤αnβn andb≤γn ≤c <1/Lt, where

Ltmax{Lt,i: 1≤i≤N}, for alln≥1;

b∞_n1δn<∞.

If one of{T1, T2, . . . , TN}is semicompact, then the sequence converges{xn}strongly to some point in

F.

If Ti I, where I denotes the identity mapping, for each i ∈ {1,2, . . . , N}, then

Theorem 2.2is reduced to the following.

Corollary 2.4. LetEbe a real uniformly convex Banach space andCa nonempty closed convex subset ofE. LetSi:C → Cbe a uniformlyLs,i-Lipschitz and generalized asymptotically quasi-nonexpansive

mapping with sequences{kn,s,i} ⊂1,∞and{ξn,s,i} ⊂0,∞such that

_∞

n1kn,s,i−1< ∞and

_∞

n1ξn,s,i <∞for each1≤i≤N. Assume thatF ∩N_i1FSiis nonempty. Let{un}be a bounded

sequence inC,kn,s max{kn,s,i : 1≤ i≤ N}andξn,s max{ξn,s,i: 1 ≤i≤N}. Let{αn},{βn},

{γn}, and{δn}be sequences in0,1such thatαnβnγnδn 1for eachn≥ 1. Let{xn}be a

sequence generated in1.20. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c, d∈0,1such thata≤αn,b≤βn, andc≤γn, for alln≥1;

b∞_n1δn<∞.

If one of{S1, S2, . . . , SN}is semicompact, then the sequence{xn}converges strongly to some point in

F.

In 2005, Chidume and Shahzad11introduced the following conception. Recall that a family{Ti}Ni1:C → CwithF∩Ni1FTi/∅is said to satisfy ConditionBonCif there is

a nondecreasing functionf :0,∞ → 0,∞withf0 0 andfm>0 for allm∈0,∞

such that for allx∈C

max

1≤i≤N{x−Tix} ≥fdx, F. 2.41

Based on ConditionB, we introduced the following conception for two finite families of mappings. Recall that two families {Si}Ni1 : C → C and {Ti}Ni1 : C → C with

F∩N

i1FSi ∩Ni1FTi/∅are said to satisfy ConditionBonCif there is a nondecreasing

functionf :0,∞ → 0,∞withf0 0 andfm> 0 for allm∈0,∞such that for all

x∈C

max

Next, we give strong convergence theorems with the help of ConditionB.

Theorem 2.5. LetEbe a real uniformly convex Banach space andCa nonempty closed convex subset ofE. LetTi :C → Cbe a uniformlyLt,i-Lipschitz and generalized asymptotically quasi-nonexpansive

mapping with sequences {kn,t,i} ⊂ 1,∞ and {ξn,t,i} ⊂ 0,∞such that ∞n1kn,t,i −1 < ∞

and ∞_n1ξn,t,i < ∞ for each 1 ≤ i ≤ N, and let Si : C → C be a uniformly Ls,i-Lipschitz

and generalized asymptotically quasi-nonexpansive mapping with sequences{kn,s,i} ⊂ 1,∞and

{ξn,s,i} ⊂0,∞such that∞n1kn,s,i−1<∞and∞n1ξn,s,i<∞for each1≤i≤N. Assume that

F∩N_i1FTi ∩N_i1FSiis nonempty. Let{un}be a bounded sequence inC,knmax{kn,t, kn,s},

wherekn,t max{kn,t,i : 1 ≤i≤N}andkn,s max{kn,s,i: 1 ≤i≤N}andξn max{ξn,t, ξn,s},

whereξn,t max{ξn,t,i : 1≤ i≤N}andξn,s max{ξn,s,i : 1≤i≤N}. Let{αn},{βn},{γn}, and

{δn}be sequences in0,1such thatαnβnγnδn 1for eachn≥ 1. Let{xn}be a sequence

generated in1.18. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c, d∈0,1such thata≤αn,b≤βn, andc≤ γn ≤d < 1/Lt,

whereLtmax{Lt,i: 1≤i≤N}, for alln≥1;

b∞_n1δn<∞.

If{S1, S2, . . . , SN}and{T1, T2, . . . , TN}satisfy ConditionB, then the sequence converges strongly

to some point inF.

Proof. In view of ConditionB, we obtain from2.27and2.38thatfdxn, F → 0, which

impliesdxn, F → 0. Next, we show that the sequence{xn}is Cauchy. In view of2.6, for

any positive integersm, n, wherem > n > n0, we see that

_{xm−p≤Bxn−pB} ∞

in1

M1

δiξhiM1

δmξhm, 2.43

whereBexp{∞_n1khn−1/1−R}. It follows that

xn−xm ≤xn−pxm−p

≤1Bxn−pB

∞

in1

M1

δiξhi

M1

δmξhm

. 2.44

It follows that{xn}is a Cauchy sequence inCand so{xn}converges strongly to someq∈C.

SinceTr andSr are Lipschitz for eachr ∈ {1,2, . . . , N}, we see thatF is closed. This in turn

implies thatq∈F. This completes the proof.

If Si I, where I denotes the identity mapping, for each i ∈ {1,2, . . . , N}, then

Theorem 2.2is reduced to the following.

Corollary 2.6. LetEbe a real uniformly convex uniformly convex Banach space andCa nonempty closed convex subset ofE. LetTi:C → Cbe a uniformlyLt,i-Lipschitz and generalized asymptotically

quasi-nonexpansive mapping with sequences {kn,t,i} ⊂ 1,∞ and {ξn,t,i} ⊂ 0,∞ such that

_∞

n1kn,t,i −1 < ∞ and

_∞

n1ξn,t,i < ∞ for each 1 ≤ i ≤ N. Assume thatF ∩N_i1FTi is

nonempty. Let {un} be a bounded sequence in C, kn,t max{kn,t,i : 1 ≤ i ≤ N} and where

αnβnγnδn 1for eachn ≥ 1. Let{xn}be a sequence generated in1.19. Assume that the

following restrictions are satisfied:

athere exist constantsa, b, c∈0,1such thata≤αnβn andb≤γn ≤c <1/Lt, where

Ltmax{Lt,i: 1≤i≤N}, for alln≥1;

b∞_n1δn<∞.

If{T1, T2, . . . , TN}satisfies ConditionB, then the sequence{xn}converges strongly to some point

inF.

If Ti I, where I denotes the identity mapping, for each i ∈ {1,2, . . . , N}, then

Theorem 2.2is reduced to the following.

Corollary 2.7. LetEbe a real uniformly convex Banach space andCa nonempty closed convex subset ofE. LetSi:C → Cbe a uniformlyLs,i-Lipschitz and generalized asymptotically quasi-nonexpansive

mapping with sequences{kn,s,i} ⊂1,∞and{ξn,s,i} ⊂0,∞such that

_∞

n1kn,s,i−1< ∞and

_∞

n1ξn,s,i <∞for each1≤i≤N. Assume thatF ∩N_i1FSiis nonempty. Let{un}be a bounded

sequence inC,kn,s max{kn,s,i : 1≤i≤N}, andξn,smax{ξn,s,i : 1≤i≤N}. Let{αn},{βn},

{γn}, and{δn}be sequences in0,1such thatαnβnγnδn 1for eachn≥ 1. Let{xn}be a

sequence generated in1.20. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c, d∈0,1such thata≤αn,b≤βnandc≤γn, for alln≥1;

b∞_n1δn<∞.

If{S1, S2, . . . , SN}satisfies ConditionB, then the sequence{xn}converges strongly to some point

inF.

Finally, we give a strong convergence theorem criterion.

Theorem 2.8. Let E be a real Banach space and C a nonempty closed convex subset of E. Let

Ti : C → C be a uniformly Lt,i-Lipschitz and generalized asymptotically quasi-nonexpansive

mapping with sequences {kn,t,i} ⊂ 1,∞ and {ξn,t,i} ⊂ 0,∞such that ∞n1kn,t,i −1 < ∞

and ∞_n1ξn,t,i < ∞ for each 1 ≤ i ≤ N, and let Si : C → C be a uniformly Ls,i-Lipschitz

and generalized asymptotically quasi-nonexpansive mapping with sequences{kn,s,i} ⊂ 1,∞and

{ξn,s,i} ⊂0,∞such that∞n1kn,s,i−1<∞and∞n1ξn,s,i<∞for each1≤i≤N. Assume that

F∩N_i1FTi ∩N_i1FSiis nonempty. Let{un}be a bounded sequence inC,knmax{kn,t, kn,s},

wherekn,t max{kn,t,i : 1 ≤i≤N}andkn,s max{kn,s,i: 1 ≤i≤N}andξn max{ξn,t, ξn,s},

whereξn,t max{ξn,t,i : 1≤ i≤N}andξn,s max{ξn,s,i : 1≤i≤N}. Let{αn},{βn},{γn}, and

{δn}be sequences in0,1such thatαnβnγnδn 1for eachn≥ 1. Let{xn}be a sequence

generated in1.18. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c, d∈0,1such thata≤αn,b≤βn, andc≤ γn ≤d < 1/Lt,

whereLtmax{Lt,i: 1≤i≤N}, for alln≥1;

b∞_n1δn<∞.

Then{xn}converges strongly to some point inFif and only iflim infn→ ∞dxn, F 0.

Proof. The necessity is obvious. We only show the sufficiency. Assume that

lim inf

For eachp∈F, we see that

xn−p≤αnxn−1−pβnSh_innxn−1−pγnT_ih_nnxn−pδnun−p

≤αnxn−1−pβn

khnxn−1−pξhnγn

khnxn−pξhn

δnun−p

≤αnβnkhnxn−1−p

1−αn−βn

khnxn−p2ξhn

δnun−xn.

2.46

Notice that ∞_n1kn−1 < ∞. We see from the restrictionsaand b that there exists a

positive integern0such that

1−αn−βn

khn≤R <1, ∀n≥n0, 2.47

whereR 1−ab1 ab/2−2ab. Notice that the sequence{xn}is bounded.

It follows from2.46that

xn−p≤

αnβnkhn 1−1−αn−βn

khn

xn−1−p

δn

1−1−αn−βn

khnun−xn

2ξhn

1−1−αn−βn

khn

≤

1khn−1 1−R

xn−1−p δn

1−Run−xn

2ξhn 1−R

≤

1khn−1 1−R

xn−1−pM4

δnξhn

, ∀n≥n0,

2.48

whereM4is an appropriate constant such thatM4 max{sup_n≥1{un−xn/1−R},2/1−

R}. In view of the restrictionsaandb, we obtain fromLemma 1.2that limn→ ∞dxn,F

exists. This implies that

lim

n→ ∞dxn,F 0. 2.49

In view ofTheorem 2.5, we can conclude the desired conclusion easily.

If Si I, where I denotes the identity mapping, for each i ∈ {1,2, . . . , N}, then

Theorem 2.2is reduced to the following.

Corollary 2.9. LetEbe a real Banach space andCa nonempty closed convex subset ofE. LetTi :

C → Cbe a uniformlyLt,i-Lipschitz and generalized asymptotically quasi-nonexpansive mapping

with sequences{kn,t,i} ⊂1,∞and{ξn,t,i} ⊂0,∞such that

_∞

n1kn,t,i−1<∞and

_∞

n1ξn,t,i<

∞for each1≤ i≤ N. Assume thatF ∩N

i1FTiis nonempty. Let{un}be a bounded sequence in

and{δn}be sequences in0,1such thatαnβnγnδn1for eachn≥1. Let{xn}be a sequence

generated in1.19. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c∈0,1such thata≤αnβn andb≤γn ≤c <1/Lt, where

Ltmax{Lt,i: 1≤i≤N}, for alln≥1;

b∞_n1δn<∞.

Then{xn}converges strongly to some point inFif and only iflim infn→ ∞dxn, F 0.

If Ti I, where I denotes the identity mapping, for each i ∈ {1,2, . . . , N}, then

Theorem 2.2is reduced to the following.

Corollary 2.10. LetEbe a real Banach space andCa nonempty closed convex subset ofE. LetSi :

C → Cbe a uniformlyLs,i-Lipschitz and generalized asymptotically quasi-nonexpansive mapping

with sequences{kn,s,i} ⊂1,∞and{ξn,s,i} ⊂0,∞such that

_∞

n1kn,s,i−1<∞and

_∞

n1ξn,s,i<

∞for each1≤i≤N. Assume thatF∩N_i1FSiis nonempty. Let{un}be a bounded sequence inC,

kn,smax{kn,s,i: 1≤i≤N}, andξn,smax{ξn,s,i: 1≤i≤N}. Let{αn},{βn},{γn}, and{δn}be

sequences in0,1such thatαnβnγnδn1for eachn≥1. Let{xn}be a sequence generated in

1.20. Assume that the following restrictions are satisfied:

athere exist constantsa, b, c, d∈0,1such thata≤αn,b≤βn, andc≤γn, for alln≥1;

b∞_n1δn<∞.

Then{xn}converges strongly to some point inFif and only iflim infn→ ∞dxn, F 0.

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