Convergence theorems of multi-step iterative algorithm with errors for generalized asymptotically quasi-nonexpansive mappings in Banach spaces

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ISSN: 2008-6822 (electronic) http://www.ijnaa.com

CONVERGENCE THEOREMS OF MULTI-STEP ITERATIVE ALGORITHM WITH ERRORS

FOR GENERALIZED ASYMPTOTICALLY

QUASI-NONEXPANSIVE MAPPINGS IN BANACH SPACES

GURUCHARAN SINGH SALUJA

Abstract. The purpose of this paper is to study and give the necessary and

sufficient condition of strong convergence of the multi-step iterative algorithm with errors for a finite family of generalized asymptotically quasi-nonexpansive mappings to converge to common fixed points in Banach spaces. Our results extend and improve some recent results in the literature (see, e.g. [2,3,5,6,7,8, 11,14,19]).

1. Introduction

It is well known that the concept of asymptotically nonexpansive mappings was in-troduced by Goebel and Kirk [4] who proved that every asymptotically nonexpansive self-mapping of nonempty closed bounded and convex subset of a uniformly convex Banach space has fixed point. In 1973, Petryshyn and Williamson [11] gave necessary and sufficient conditions for Mann iterative sequence [9] to converge to fixed points of quasi-nonexpansive mappings. In 1997, Ghosh and Debnath [3] extended the re-sults of Petryshyn and Williamson [11] and gave necessary and sufficient conditions for Ishikawa iterative sequence to converge to fixed points for quasi-nonexpansive mappings.

Liu [8] extended results of [3, 11] and gave necessary and sufficient conditions for Ishikawa iterative sequence with errors to converge to fixed point of asymptotically quasi-nonexpansive mappings.

In 2003, Zhou et al. [21] introduced a new class of generalized asymptotically nonexpansive mapping and gave a necessary and sufficient condition for the modi-fied Ishikawa and Mann iterative sequences to converge to fixed points for the class of mappings. Atsushiba [1] studied the necessary and sufficient condition for the convergence of iterative sequences to a common fixed point of the finite family of asymptotically nonexpansive mappings in Banach spaces. Suzuki [15], Zeng and

Date:Received:June 2010; Revised:December 2010.

2000Mathematics Subject Classification. Primary 47H09; Secondary 47H10.

Key words and phrases. Generalized asymptotically quasi–nonexpansive mapping; multi–step iterative algorithm with bounded errors; common fixed point; Banach space; strong convergence.

∗ _{Corresponding author}_.

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Yao [20] discussed a necessary and sufficient condition for common fixed points of two nonexpansive mappings and a finite family of nonexpansive mappings, and proved some convergence theorems for approximating a common fixed point, respec-tively.

In 2006, Lan [6] introduced a new class of generalized asymptotically quasi-nonexpansive mappings and gave necessary and sufficient condition for the 2-step modified Ishikawa iterative sequences to converge to fixed points for the class of mappings.

Recently, Tang and Peng [18] give the necessary and sufficient condition of strong convergence of common fixed points for a finite family of uniformly quasi-Lipschitzian mappings in banach spaces for the following iteration scheme:

Let {Ti : i = 1,2, . . . , k}: K → K, where K is a nonempty subset of a Banach

space E, be a finite family of uniformly quasi-Lipschitzian mappings. Let x1 ∈K,

then the sequence {xn} is defined by

xn+1 = aknxn+bknTkny(k−1)n+cknukn,

y(k−1)n = a(k−1)nxn+b(k−1)nTkn−1y(k−2)n+c(k−1)nu(k−1)n, y(k−2)n = a(k−2)nxn+b(k−2)nTkn−2y(k−3)n+c(k−2)nu(k−2)n,

.. .

y2n = a2nxn+b2nT₂ny1n+c2nu2n

y1n = a1nxn+b1nT1nxn+c1nu1n, n ≥1, (1.1)

where {ain}, {bin}, {cin} are sequences in [0,1] with ain +bin +cin = 1 for all i= 1,2, . . . , k and n≥1,{uin, i= 1,2, . . . , k, n≥1}are bounded sequences inK.

Remark 1.1. The iterative algorithm (1.1) is called multi-step iterative algorithm with errors. It contains well known iterations as special case. Such as, the modified Mann iteration (see, [13]), the modified Ishikawa iteration (see, [17]), the three-step iteration (see, [19]), the multi-step iteration (see, [5]).

The purpose of this paper is study the multi-step iterative algorithm with bounded errors (1.1) for a finite family of generalized asymptotically quasi-nonexpansive map-pings to converge to common fixed points in Banach spaces. The results obtained in this paper extend and improve the corresponding results of [2,3,5,6,7,8,11,14,19] and many others.

2. Preliminaries

In the sequel, we need the following definitions and lemmas for our main results in this paper.

Definition 2.1.(see [10]) LetE be a real Banach space,Kbe a nonempty subset of

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(1) nonexpansive if

kT x−T yk ≤ kx−yk, (2.1)

for all x, y ∈K,

(2) quasi-nonexpansive if

kT x−T yk ≤ kx−yk, (2.2)

for all x∈K and p∈F(T),

(3) asymptotically nonexpansive if there exists a sequence {rn} ⊂ [0,∞) with rn→0 as n → ∞such that

kTnx−Tnyk ≤ (1 +rn)kx−yk, (2.3)

for all x, y ∈K,

(4) asymptotically quasi-nonexpansive if (3) holds for all x∈K and y∈F(T);

(5) generalized quasi-nonexpansive with respect to{sn}, if there exists a sequence

{sn} ⊂[0,1) withsn→0 as n → ∞such that

kTnx−pk ≤ kx−pk+snkx−Tnxk, (2.4)

for all x∈K, p∈F(T) and n≥1,

(6) generalized asymptotically quasi-nonexpansive with respect to{rn}and{sn} ⊂

[0,1) with rn→0 and sn →0 as n→ ∞ such that

kTnx−pk ≤ (1 +rn)kx−pk+snkx−Tnxk, (2.5)

for all x∈K, p∈F(T) and n≥1.

Remark 2.1. From the above definition, it is clear that:

(i) a nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping;

(ii) a quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping;

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(iv) a generalized quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping.

However, the converse of the above statements are not true.

Lemma 2.1. (see [16]) Let {an}, {bn} and {δn} be sequences of nonnegative real

numbers satisfying the inequality

an+1 ≤(1 +δn)an+bn, n≥1.

If P∞

n=1δn < ∞ and

P∞

n=1bn < ∞, then limn→∞an exists. In particular, if {an}

has a subsequence converging to zero, then limn→∞an = 0.

Lemma 2.2. (see [6]) Let K be nonempty closed subset of a Banach spaceE and

T: K → K be a generalized asymptotically quasi-nonexpansive mapping with the fixed point setF(T)6=∅. Then F(T) is closed subset in K.

3. _{Main Results}

In this section, we prove strong convergence theorems of multi-step iterative al-gorithm with bounded errors for a finite family of generalized asymptotically quasi-nonexpansive mappings in a real Banach space.

Theorem 3.1. Let E be a real arbitrary Banach space, K be a nonempty closed convex subset of E. Let {Ti :i = 1,2, . . . , k}: K → K be a finite family of

gener-alized asymptotically quasi-nonexpansive mappings with respect to{rin} and {sin}

such that P∞

n=1

rin+2sin

1−sin <∞ for all i∈ {1,2, . . . , k}. Let{xn} be the sequence

de-fined by (1.1) with P∞

n=1cin <∞, i= 1,2, . . . , k. If F =∩ k

i=1F(Ti)6=∅. Then the

sequence{xn} converges strongly to a common fixed point of{Ti :i= 1,2, . . . , k}if

and only if lim inf

n→∞ d(xn,F) = 0, where d(x,F) denotes the distance between x and

the setF.

Proof. The necessity is obvious and it is omitted. Now we prove the sufficiency. Letp∈ F, then it follows from (2.5) and for all i∈ {2,3, . . . , k}, we have

y_(i−1)n−Tiny(i−1)n

≤

y_(i−1)n−p

+

T_iny_(i−1)n−p

≤

y_(i−1)n−p

+ (1 +r_i_n)

y_(i−1)n−p

+sin

y(i−1)n−Tiny(i−1)n

≤ (2 +rin)

y(i−1)n−p

+si_n

y(i−1)n−Tiny(i−1)n

which implies that

y_(i−1)n−Tiny(i−1)n

≤

2 +r_in 1−sin

y_(i−1)n−p

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and

kxn−T1nxnk ≤ kxn−pk+kT1nxn−pk

≤ kxn−pk+ (1 +r1n)kxn−pk+s1nkxn−T1nxnk

≤ (2 +r1n)kxn−pk+s1nkxn−T1nxnk

which implies that

kxn−T1nxnk ≤

2 +r_1n 1−s1n

kxn−pk. (3.2)

Since {uin, i = 1,2, . . . , k, n ≥ 1} are bounded sequences in K, therefore there

exists a M >0, such that

M = max

sup

n≥1

kuin−pk, i= 1,2, . . . , k

.

Using (1.1), (3.1) and (3.2), we note that

ky1n−pk = ka1nxn+b1nT₁nxn+c1nu1n−pk

= ka1n(xn−p) +b1n(T1nxn−p) +c1n(u1n−p)k

≤ a1nkxn−pk+b1nkT1nxn−pk+c1nku1n−pk

≤ a1nkxn−pk+b1n

h

(1 +r1n)kxn−pk

+s1nkxn−T1nxnk

i

+c1nku1n−pk

≤ a1nkxn−pk+b1n.

1 +r1n+s1n 1−s1n

kxn−pk

+c1nku1n−pk

≤ (a1n+b1n)

1 +r_1n+s_1n 1−s1n

kxn−pk

+c1nku1n−pk

= (1−c1n)

1 +r_1n+s_1n 1−s1n

kxn−pk

+c1nku1n−pk

≤ 1 +r1n+s1n

1−s1n

kxn−pk+c1nM

= h1 + r1n+ 2s1n 1−s1n

i

kxn−pk+c1nM

= (1 +t1n)kxn−pk+c1nM (3.3)

where t1n = r1₁n−+2ss1n1n. Since

P∞

n=1

rin+2sin

1−sin < ∞ for all i ∈ {1,2, . . . , k}, it follows

that P∞

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ky2n−pk = ka2nxn+b2nT2ny1n+c2nu2n−pk

= ka2n(xn−p) +b2n(T2ny1n−p) +c2n(u2n−p)k

≤ a2nkxn−pk+b2nkT₂ny1n−pk+c2nku2n−pk

≤ a2nkxn−pk+b2n

h

(1 +r2n)ky1n−pk

+s2nky1n−T2ny1nk

i

+c2nku2n−pk

1 +r_2n+s_2n 1−s2n

ky1n−pk

+c2nku2n−pk

1 + r2n+ 2s2n 1−s2n

h

(1 +t1n)kxn−pk

+c1nM

i

+c2nku2n−pk

≤ a2nkxn−pk+b2n.(1 +t2n)

h

(1 +t1n)kxn−pk

+c1nM

i

+c2nku2n−pk

≤ a2nkxn−pk+b2n.(1 +t2n)

h

(1 +t1n)kxn−pk+c1nM

i

+c2nku2n−pk

≤ (a2n+b2n)(1 +t1n)(1 +t2n)kxn−pk+b2nc1n(1 +t2n)M

+c2nM

= (1−c2n)(1 +t1n)(1 +t2n)kxn−pk+b2nc1n(1 +t2n)M

+c2nM

≤ (1 +t1n)(1 +t2n)kxn−pk+c1n(1 +t2n)M+c2nM

≤ (1 +t1n+t2n)kxn−pk+ (c1n+c2n)M

= h1 +

2

X

k=1 tkn

i

kxn−pk+M 2

X

k=1

ckn (3.4)

where t2n = r2₁n₋+2s_s 2n

2n . Since

P∞

n=1

rin+2sin

1−sin < ∞ for all i ∈ {1,2, . . . , k}, it follows

that P∞

n=1t2n <∞.

Repeating the above process, we get

kyjn−pk ≤

h 1 +

j

X

k=1 tkn

i

kxn−pk+M j

X

k=1

ckn, (3.5)

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ky(j+1)n−pk=ka(j+1)n(xn−p) +b(j+1)n(T(j+1)n yjn−p)

+c(j+1)n(u(j+1)n−p)k

≤a(j+1)nkxn−pk+b(j+1)nkT(j+1)n yjn−pk

+c(j+1)nku(j+1)n−pk

≤a(j+1)nkxn−pk+b(j+1)n

h

(1 +r(j+1)n)kyjn−pk

+s(j+1)nkyjn−T(j+1)n yjnk

i

1 +r_(j+1)n+s_(j+1)n 1−s(j+1)n

kyjn−pk

1 + r(j+1)n+ 2s(j+1)n 1−s(j+1)n

( 1 + j X k=1 tkn

kxn−pk+M j

X

k=1 ckn

)

≤a(j+1)nkxn−pk+b(j+1)n(1 +t(j+1)n)

( 1 + j X k=1 tkn

kxn−pk+M j

X

k=1 ckn

)

1 +t(j+1)n

( 1 + j X k=1 tkn

kxn−pk+M j

X

k=1 ckn

)

≤a(j+1)n+b(j+1)n

1 + j X k=1 tkn

1 +t(j+1)n

kxn−pk

+M b(j+1)n

1 +t(j+1)n

_Xj

k=1

ckn+c(j+1)nM

≤1 +

j

X

k=1 tkn

1 +t(j+1)n

kxn−pk

+M1 +t(j+1)n

_Xj

k=1

ckn+c(j+1)nM

≤1 +

j

X

k=1

tkn+t(j+1)n

kxn−pk

+M(

j

X

k=1

ckn+c(j+1)n)

= 1 + j+1 X k=1 tkn

kxn−pk+M

_Xj+1

k=1 ckn

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Hence (3.5) holds. It follows from (1.1) and (3.5) that

kxn+1−pk =

a_kn(x_n−p) +b_kn(T_kny_(k−1)n−p) +ckn(ukn−p)

≤ aknkxn−pk+bkn

T_kny_(k−1)n−p

+c_knku_kn−pk

≤ aknkxn−pk+bkn.

1 +r_kn+s_kn 1−skn

y_(k−1)n−p

+cknkukn−pk

1 + rkn+ 2skn 1−skn

y_(k−1)n−p

+cknkukn−pk

1 +tkn

h 1 +

k−1

X

l=1 tln

kxn−pk

+M k−1

X

l=1 cln

i

+cknM

≤ akn+bkn

1 +tkn

1 +

k−1

X

l=1 tln

kxn−pk

+M1 +tkn

_Xk−1

l=1 cln

+cknM

=

1−ckn

1 +tkn

1 +

k−1

X

l=1 tln

kxn−pk

+M1 +tkn

Xk−1

l=1 cln

+cknM

≤ 1 +

k−1

X

l=1

tln+tkn

kxn−pk

+M k−1

X

l=1

cln+ckn

= h1 +

k

X

l=1 tln

i

kxn−pk+M

_Xk

l=1 cln

= (1 +θn)kxn−pk+M λn (3.7)

where θn=Pk

l=1tln and λn=

Pk

l=1cln. Since

P∞

n=1tln<∞ and

P∞

n=1cln<∞ for

alll = 1,2, . . . , k, it follows that P∞

n=1θn <∞ and

P∞

n=1λn <∞. Therefore from

Lemma 2.1, we know that lim

n→∞d(xn,F) = 0.

Next, we will prove that {xn} is a Cauchy sequence. Notice that when x > 0,

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kxn+m−pk ≤ (1 +θn+m−1)kxn+m−1−pk+M λn+m−1

≤ eθn+m−1_k_x

n+m−1−pk+M λn+m−1

≤ eθn+m−1 h

eθn+m−2_k_x

n+m−2−pk+M λn+m−2

+M λn+m−1

≤ e

h

θn+m−1+θn+m−2 i

kxn+m−2−pk

+M eθn+m−1

λn+m−1 +λn+m−2

≤ . . .

≤ ePnk=+nm−1θk

kxn−pk

+M

ePnk=+nm+1−1θk

n+m_X−1

k=n λk

≤ ePnk=+nm−1θk

kxn−pk

+MePnk=+nm−1θk

n+m_X−1

k=n λk

≤ M0kxn−pk+M M0 n+m−1

X

k=n

λk (3.8)

where M0 = ePkn=+mn−1θk _{and for all} _p ∈ F_, _{m, n} ∈

N. Since lim

n→∞d(xn,F) = 0, for

given ε >0, there exists a natural number n1 such that forn ≥n1,

d(xn,F)< ε

4(M0_{+ 1)} and

n+m

X

k=n1+1

λk < ε

2M M0. (3.9)

Hence, there exists a point q∈ F such that

kxn1 −qk<

ε

2(M0_{+ 1)}. (3.10)

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kxn+m−xnk ≤ kxn+m−qk+kxn−qk

≤ M0kxn1 −qk+M M

0

n+m−1

X

k=n1

λk+kxn1 −qk

≤ (M0+ 1)kxn1 −qk+M M

0

n+m−1

X

k=n1

λk

< (M0+ 1). ε

2(M0_{+ 1)} +M M 0

. ε

2M M0 =ε. (3.11)

This implies that {xn} is a Cauchy sequence. Thus the completeness of E implies

that {xn} must be convergent. Let limn→∞xn = p, that is, {xn} converges to p.

Then p ∈K, because K is a closed subset of E. By Lemma 2.2 we know that the setF is closed. From the continuity ofd(xn,F) with

d(xn,F)→0 and xn →p as n→ ∞,

we get

d(p,F) = 0

and sop∈ F =∩k

i=1F(Ti), that is,pis a common fixed point of{Ti :i= 1,2, . . . , k}.

This completes the proof.

such that P∞

n=1

rin+2sin

1−sin < ∞ for all i ∈ {1,2, . . . , k}. Let {xn} be the sequence

defined by (1.1) with P∞

n=1cin < ∞, i = 1,2, . . . , k. If F = ∩ki=1F(Ti) 6= ∅. Then

the sequence {xn} converges strongly to a common fixed point p of the mappings

{Ti : i = 1,2, . . . , k} if and only if there exists a subsequence {xnj} of {xn} which

converges to p.

Proof. The proof of Theorem 3.2 follows from Lemma 2.1 and Theorem 3.1.

such that P∞

n=1

rin+2sin

1−sin < ∞ for all i ∈ {1,2, . . . , k}. Let {xn} be the sequence

defined by (1.1) withP∞

n=1cin<∞,i= 1,2, . . . , k. IfF =∩ k

i=1F(Ti)6=∅. Suppose

that the mappings{Ti :i= 1,2, . . . , k} satisfy the following conditions:

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(C2) there exists a constant A > 0 such that {kxn−Tixnk} ≥ Ad(xn,F) for all i∈ {1,2, . . . , k} and for all n ≥1.

Then{xn}converges strongly to a common fixed point of the mappings {Ti :i=

1,2, . . . , k}.

Proof. From condition (C1) and (C2), we have limn→∞d(xn,F) = 0, it follows as

in the proof of Theorem 3.1, that {xn} must converges strongly to a common fixed point of the mappings {Ti :i= 1,2, . . . , k}.

Remark 3.1. Theorem 3.1 extends the corresponding result of Khan et al. [5] and Tang and Peng [18] to the case of more general class of asymptotically quasi-nonexpansive or uniformly quasi-Lipschitzian mappings considered in this paper.

Remark 3.2. Theorem 3.1 also extend and improve the corresponding results of [7, 8, 12, 14, 19]. Especially Theorem 3.1 extend and improve Theorem 1 and 2 in [8], Theorem 1 in [7] and Theorem 3.2 in [14] in the following ways:

(1) The asymptotically quasi-nonexpansive mapping in [7], [8] and [14] is replaced by finite family of generalized asymptotically quasi-nonexpansive mappings.

(2) The usual Ishikawa iteration scheme in [7], the usual modified Ishikawa itera-tion scheme with errors in [8] and the usual modified Ishikawa iteration scheme with errors for two mappings are extended to the multi-step iteration scheme with errors for a finite family of mappings.

Remark 3.3. Theorem 3.2 extend and improve Theorem 3 in [8] and Theorem 3.3 extend and improve Theorem 3 in [7] in the following aspects:

(1) The asymptotically quasi-nonexpansive mapping in [7] and [8] is replaced by finite family of generalized asymptotically quasi-nonexpansive mappings.

(2) The usual Ishikawa iteration scheme in [7] and the usual modified Ishikawa iteration scheme with errors in [8] are extended to the multi-step iteration scheme with errors for a finite family of mappings.

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Department of Mathematics & Information Technology, Govt. Nagarjun P.G.

College of Science, Raipur (C.G.), India.