Volume 2008, Article ID 428241, 11 pagesdoi:10.1155/2008/428241 Research Article Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings Chao
Trang 1Volume 2008, Article ID 428241, 11 pages
doi:10.1155/2008/428241
Research Article
Convergence Theorems for Common
Fixed Points of Nonself Asymptotically
Quasi-Non-Expansive Mappings
Chao Wang and Jinghao Zhu
Department of Applied Mathematics, Tongji University, Shanghai 200092, China
Correspondence should be addressed to Chao Wang,wangchaoxj20002000@yahoo.com.cn
Received 1 April 2008; Revised 12 June 2008; Accepted 19 July 2008
Recommended by Simeon Reich
We introduce a new three-step iterative scheme with errors Several convergence theorems of this scheme are established for common fixed points of nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces Our theorems improve and generalize recent known results in the literature
Copyrightq 2008 C Wang and J Zhu 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 K be a nonempty closed convex subset of real normed linear space E Recall that a mapping T : K → K is called asymptotically nonexpansive if there exists a sequence{r n} ⊂
0, ∞, with lim n→∞ r n 0 such that T n x − T n y ≤ 1 r n x − y, for all x, y ∈ K and n ≥ 1 Moreover, it is uniformly L-Lipschitzian if there exists a constant L > 0 such
that T n x − T n y ≤ Lx − y, for all x, y ∈ K and each n ≥ 1 Denote and define by FT {x ∈ K : Tx x} the set of fixed points of T Suppose FT / ∅ A mapping T
is called asymptotically quasi-non-expansive if there exists a sequence{r n } ⊂ 0, ∞, with
limn→∞ r n 0 such that T n x − p ≤ 1 r n x − p, for all x, y ∈ K, p ∈ FT, and n ≥ 1.
It is clear from the above definitions that an asymptotically nonexpansive mapping
must be uniformly L-Lipschitzian as well as asymptotically quasi-non-expansive, but the
converse does not hold Iterative technique for asymptotically nonexpansive self-mapping
in Hilbert spaces and Banach spaces including Mann-type and Ishikawa-type iteration processes has been studied extensively by many authors; see, for example,1 6
Recently, Chidume et al.7 have introduced the concept of nonself asymptotically nonexpansive mappings, which is the generalization of asymptotically nonexpansive mappings Similarly, the concept of nonself asymptotically quasi-non-expansive mappings
Trang 22 Fixed Point Theory and Applications can also be defined as the generalization of asymptotically quasi-non-expansive mappings and nonself asymptotically nonexpansive mappings These mappings are defined as follows
Definition 1.1 Let K be a nonempty closed convex subset of real normed linear space E, let
P : E → K be the nonexpansive retraction of E onto K, and let T : K → E be a nonself
mapping
i T is said to be a nonself asymptotically nonexpansive mapping if there exists a
sequence{r n } ⊂ 0, ∞, with lim n→∞ r n 0 such that
TP T n−1 x − TP T n−1 y ≤ 1 r n
for all x, y ∈ K and n ≥ 1.
ii T is said to be a nonself uniformly L-Lipschitzian mapping if there exists a constant
L > 0 such that
TP T n−1
x − TP T n−1 y ≤ Lx − y, 1.2
for all x, y ∈ K and n ≥ 1.
iii T is said to be a nonself asymptotically quasi-non-expansive mapping if FT / ∅
and there exists a sequence{r n } ⊂ 0, ∞, with lim n→∞ r n 0 such that
TP T n−1
x − p ≤ 1 r n
for all x, y ∈ K, p ∈ FT, and n ≥ 1.
By studying the following iteration processMann-type iteration:
x1 ∈ K, x n1 P1− α n
x n α n TP T n−1 x n
where{α n } ⊂ 0, 1, Chidume et al 7 obtained many convergence theorems for the fixed
points of nonself asymptotically nonexpansive mapping T Later on, Wang 8 generalized the iteration process1.4 as follows Ishikawa-type iteration:
x1 ∈ K,
x n1 P1− α n
x n α n T1
P T1n−1
y n
,
y n P1− β n
x n β n T2
P T2n−1
x n
, ∀n ≥ 1
1.5
where T1, T2 : K → E are nonself asymptotically nonexpansive mappings and {α n }, {β n} ⊂
0, 1 Also, he got several convergence theorems of the iterative scheme 1.5 under proper conditions
In 2000, Noor 9 first introduced a three-step iterative sequence and studied the approximate solutions of variational inclusion in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle Glowinski and Tallec10 showed that the three-step iterative schemes perform better than the Mann-type and Ishikawa-type iterative schemes On the other hand, Xu and Noor11 introduced and studied a three-step scheme
to approximate fixed points of asymptotically nonexpansive mappings in Banach spaces Cho et al 12 and Plubtieng et al 13 extended the work of Xu and Noor to the three-step iterative scheme with errors, and gave weak and strong convergence theorems for asymptotically nonexpansive mappings in Banach spaces
Trang 3Inspired and motivated by these facts, a new class of three-step iterative schemes with errors, for three nonself asymptotically quasi-non-expansive mappings, is introduced and studied in this paper This scheme can be viewed as an extension for1.4, 1.5, and others This scheme is defined as follows
Let K be a nonempty convex subset of real normed linear space X, let P : E → K be the nonexpansive retraction of E onto K, and let T1, T2, T3: K → E be three nonself asymptotically
quasi-non-expansive mappings Compute the sequences{xn }, {y n }, and {z n} by
x1 ∈ K,
x n1 Pα n T1
P T1n−1
y n β n x n γ n w n
,
y n Pα n T2
P T2n−1
z n β n x n γ n v n
,
z n Pα n T3 PT3
n−1
x n β n x n γ n u n
, ∀n ≥ 1
1.6
where{α n }, {α n }, {α n }, {β n }, {β n }, {β n }, {γ n }, {γ n }, and {γ n } are real sequences in 0, 1 with
α n β n γ n α n β n γ n α n β n γ n 1, and {u n }, {v n }, and {w n} are bounded sequences
in K.
Remark 1.2 i If T1 T2 T3 : T, γn γ n γ n 0, and α n α n 0, then scheme 1.6 reduces to the Mann-type iteration1.4
ii If T2 T3, γ n γ n γ n 0, and α n 0, then scheme 1.6 reduces to the Ishikawa-type iteration1.5
iii If T1, T2, and T3 are three self-asymptotically nonexpansive mappings, then scheme1.6 reduces to the three-step iteration with errors defined by 12,13, and others The purpose of this paper is to study the iterative sequences 1.6 to converge to a common fixed point of three nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces Our results extend and improve the corresponding results
in5,7,8,11–13, and many others
2 Preliminaries and lemmas
In this section, we first recall some well-known definitions
A real Banach space E is said to be uniformly convex if the modulus of convexity of E:
δ E ε inf
1−x y
2 :x y 1, x − y ε
> 0, 2.1
for all 0 < ε ≤ 2 i.e., δ E ε is a function 0, 2 → 0, 1.
A subset K of E is said to be a retract if there exists continuous mapping P : E → K such that P x x, for all x ∈ K, and every closed convex subset of a uniformly convex Banach space is a retract A mapping P : E → E is said to be a retraction if P2 P.
A mapping T : K → E with FT / ∅ is said to satisfy condition A see 14 if there
exists a nondecreasing function f : 0, ∞ → 0, ∞ with f0 0, for all r ∈ 0, ∞, such that
x − Tx ≥ fd
x, FT
for all x ∈ K, where dx, FT inf{x − x∗ : x∗∈ FT}.
We modify this condition for three mappings T1, T2, T3 : K → E as follows Three mappings T1, T2, T3: K → E, where K is a subset of E, are said to satisfy condition B if there
Trang 44 Fixed Point Theory and Applications
exist a real number α > 0 and a nondecreasing function f : 0, ∞ → 0, ∞ with f0 0, for all r ∈ 0, ∞, such that
x − T1x ≥ αfdx,F or x − T2x ≥ αfdx,F or x − T3x ≥ αfdx,F,
2.3
for all x ∈ K, where F FT1∩FT2∩FT3 / ∅ Note that condition B reduces to condition
A when T1 T2 T3and α 1.
A mapping T : K → E is said to be semicompact if, for any sequence {x n } in K such
thatx n −Tx n → 0 n → ∞, there exists subsequence {x n j } of {x n } such that {x n j} converges
strongly to x∗∈ K.
Next we state the following useful lemmas
Lemma 2.1 see 5 Let {a n }, {b n }, and {c n } be sequences of nonnegative real numbers satisfying
the inequality
a n1≤1 c n
If∞
n1 c n < ∞ and∞
n1 b n < ∞, then lim n→∞ a n exists.
Lemma 2.2 see 15 Let E be a real uniformly convex Banach space and 0 ≤ k ≤ t n ≤ q < 1,
for all positive integer n ≥ 1 Suppose that {x n } and {y n } are two sequences of E such that
lim supn→∞ x n ≤ r, lim sup n→∞ y n ≤ r, and lim n→∞ t n x n 1 − t n y n r hold, for some
r ≥ 0; then lim n→∞ x n − y n 0.
3 Main results
In this section, we will prove the strong convergence of the iteration scheme 1.6 to a
common fixed point of nonself asymptotically quasi-non-expansive mappings T1, T2, and T3
We first prove the following lemmas
Lemma 3.1 Let K be a nonempty closed convex subset of a real normed linear space E Let T1, T2, T3:
K → E be nonself asymptotically quasi-non-expansive mappings with sequences {r n i } such that
∞
n1 r i n < ∞, for all i 1, 2, 3 Suppose that {x n } is defined by 1.6 with∞n1 γ n < ∞,∞
n1 γ n <
∞, and∞n1 γ n < ∞ If F FT1 ∩ FT2 ∩ FT3 / ∅, then lim n→∞ x n − p exists, for all p ∈ F.
Proof Let p ∈ F Since {u n }, {v n }, and {w n } are bounded sequences in K, therefore there exists M > 0 such that
M max
sup
n≥1
u n − p, sup
n≥1
v n − p, sup
n≥1
Let r n max{r n1, r n2, r n3} and k n max{γ n , γ n , γ n } Then∞n1 r n < ∞ and∞
n1 k n < ∞ By
1.6, we have
x n1 − p Pα n T1
P T1n−1
y n β n x n γ n w n
− Pp
≤α n T1
P T n−1
1
y n β n x n γ n w n−α n β n γ n
p
≤α n
T1
P T1n−1
y n − p β n
x n − p γ n
w n − p
≤ α n
1 r ny n − p β nx n − p k nw n − p,
3.2
y n − p Pα n T2
P T2n−1
z n β n x n γ n v n
− Pp
≤α n T2
P T2n−1
z n β n x n γ n v n−α n β n γ np
≤ α n1 r nz n − p β nx n − p k nv n − p, 3.3
Trang 5and similarly, we also have
z n − p ≤ α n
1 r nx n − p β nx n − p k nu n − p. 3.4 Substituting3.4 into 3.3, we obtain
y n − p ≤ α n
1 r n
α n
1 r nx n − p β nx n − p k nu n − p
β nx n − p k nv n − p
≤ α n α n
1 r n
2x n − p α n β n
1 r nx n − p β nx n − p
α n k n
1 r nu n − p k nv n − p
≤1− β n − γ nα n
1 r n
2x n − p 1 − β n − γ nβ n
1 r nx n − p
β nx n − p k n
1 r nu n − p k nv n − p
≤1− β n − γ nα n β n1 r n
2x n − p β nx n − p m n
≤1− β n1 r n
2x n − p β n
1 r n
2x n − p m n
≤1 r n
2x n − p m n ,
3.5
where m n k n 2r n M Since∞
n1 r n < ∞ and∞
n1 k n < ∞, then∞
n1 m n < ∞ Substituting
3.5 into 3.2, we have
x n1 − p ≤ α n
1 r n
1 r2
nx n − p m n
β nx n − p γ nw n − p
≤α n
1 r n
3
β n x n − p α n
1 r n
m n γ nw n − p
≤α n β n
1 r n
3x n − p 1 r n
m n k nw n − p
≤1 r n
3x n − p 1 r n
m n k n M
≤1 c nx n − p b n ,
3.6
where c n 1 r n3 − 1 and b n 1 r n m n k n M Since ∞
n1 r n < ∞, ∞
n1 k n < ∞,
and ∞
n1 m n < ∞, then∞
n1 c n < ∞ and∞
n1 b n < ∞ It follows from Lemma 2.1 that limn→∞ x n − p exists This completes the proof.
Lemma 3.2 Let K be a nonempty closed convex subset of a real uniformly convex Banach space E Let
T1, T2, T3 : K → E be uniformly L-Lipschitzian nonself asymptotically quasi-non-expansive mappings
with sequences {r n i } such that∞n1 r n i < ∞, for all i 1, 2, 3 Suppose that {x n } is defined by 1.6
with∞
n1 γ n < ∞,∞
n1 γ n < ∞, and∞
n1 γ n < ∞, where α n , α n , and α n are three sequences in
ε, 1 − ε, for some ε > 0 If F FT1 ∩ FT2 ∩ FT3 / ∅, then
lim
n→∞x n − T1xn limn→∞x n − T2xn limn→∞x n − T3xn 0. 3.7
Proof For any p ∈ F, byLemma 3.1, we see that limn→∞ x n − p exists Assume lim n→∞ x n−
p a, for some a ≥ 0 For all n ≥ 1, let r n max{r1n , r n2, r n3} and k n max{γ n , γ n , γ n }.
Trang 66 Fixed Point Theory and Applications Then,∞
n1 r n < ∞ and∞
n1 k n < ∞ From 3.5, we have
y n − p ≤ 1 r n
Taking lim supn→∞on both sides in3.8, since∞n1 r n < ∞ and∞
n1 m n < ∞, we obtain
lim sup
n→∞
y n − p ≤ limsup
n→∞
x n − p lim
so that
lim sup
n→∞
T1 PT1n−1
y n − p ≤ limsup
n→∞
1 r ny n − p limsup
n→∞
y n − p ≤ a. 3.10
Next consider
T1
P T1n−1
y n − p γ n
w n − x n ≤ T1
P T1n−1
y n − p k nw n − x n. 3.11 Since limn→∞ k n 0, we have
lim sup
n→∞
T1
P T1n−1
y n − p γ n
In addition,
x n − p γ n
w n − x n ≤ x n − p k nw n − x n. 3.13 This implies that
lim sup
n→∞
x n − p γ n
Further, observe that
a lim
n→∞x n − p
lim
n→∞α n T1
P T1n−1
y n β n x n γ n w n − p
lim
n→∞α n T1
P T1n−1
y n1− α n
x n − γ n x n γ n w n−1− α n
p − α n p
lim
n→∞α n T1
P T1n−1
y n − α n p α n γ n w n − α n γ n x n1− α n
x n
−1− α n
p − γ n x n γ n w n − α n γ n w n α n γ n x n
lim
n→∞α n
T1
P T1n−1
y n − p γ n
w n − x n
1− α n
x n − p γ n
w n − x n .
3.15
lim
n→∞T1
P T1n−1
Trang 7Next we will prove that limn→∞ T2PT2n−1
z n − x n 0 Since
x n − p ≤ T1
P T1n−1
y n − x n T1
P T1n−1
y n − p
≤T1
P T1n−1
y n − x n 1 r ny n − p 3.17 and limn→∞ T1PT1n−1
y n − x n 0 limn→∞ r n, we obtain
a lim
n→∞x n − p ≤ liminf n→∞ y n − p. 3.18 Thus, it follows from3.10 and 3.18 that
lim
On the other hand, from3.4, we have
z n − p ≤ α n
1 r n
β n x n − p k nu n − p
≤1 r nx n − p k nu n − p. 3.20
By boundedness of the sequence{u n} and by limn→∞ r n limn→∞ k n 0, we have
lim sup
n→∞
z n − p ≤ limsup
n→∞
so that
lim sup
n→∞
T2
P T2n−1
z n − p ≤ limsup
n→∞
1 r nz n − p ≤ a. 3.22
Next consider
T2
P T2n−1
z n − p γ nv n − x n ≤ T2
P T2n−1
z n − p k nv n − x n. 3.23 Thus, we have
lim sup
n→∞
T2
P T2n−1
z n − p γ nv n − x n ≤ a,
x n − p γ n
v n − x n ≤ x n − p k nv n − x n. 3.24 This implies that
lim sup
n→∞
x n − p γ n
Note that
a lim
n→∞y n − p
lim
n→∞α n T2
P T2n−1
z n β n x n γ n v n − p
lim
n→∞α n
T2
P T2n−1
z n − p γ nv n − x n
1− α nx n − p γ nv n − x n .
3.26
Trang 88 Fixed Point Theory and Applications
It follows fromLemma 2.2,3.24, and 3.25 that
lim
n→∞T2
P T2n−1
Similarly, by using the same argument as in the proof above, we obtain
lim
n→∞T3
P T3n−1
Hence,
lim
n→∞T1
P T1n−1
y n − x n lim
n→∞T2
P T2n−1
z n − x n lim
n→∞T3
P T3n−1
x n − x n 0,
3.29 and this implies that
x n1 − x n ≤ α nT1
P T1n−1
y n − x n k nw n − x n −→ 0 as n −→ ∞. 3.30
Since T1is uniformly L-Lipschitzian mapping, then we have
T1
P T1n−1
x n − x n
≤T1
P T1n−1
x n − T1
P T1n−1
y n T1
P T1n−1
y n − x n
≤ Lx n − y n T1
P T1n−1
y n − x n
≤ Lx n − α n T2
P T2n−1
z n − β n x n − γ n v n T1
P T1n−1
y n − x n
≤ Lα nT2
P T2n−1
z n − x n Lk nv n − x n T1
P T1n−1
y n − x n −→ 0 as n −→ ∞,
3.31
x n − T1xn
≤x n1 −x n x n1 −T1
P T1n
x n1 T1
P T1n
x n1 −T1
P T1n
x n T1
P T1n
x n −T1xn
≤x n1 − x n x n1 − T1
P T1n
x n1 Lx n1 − x n LT1
P T1n−1
x n − x n.
3.32
It follows from3.30, 3.31, and 3.32 that
lim
Next consider
T2
P T2n−1
x n − x n
≤T2
P T2n−1
x n − T2
P T2n−1
z n T2
P T2n−1
z n − x n
≤ Lx n − z n T2
P T2n−1
z n − x n
≤ Lα nT3
P T3n−1
x n − x n Lk nu n − x n T2
P T2n−1
z n − x n −→ 0 as n −→ ∞,
3.34
x n − T2xn
≤x n1 −x n x n1 −T2
P T2n
x n1 T2
P T2n
x n1 −T2
P T2n
x n T2
P T2n
x n −T2xn
≤x n1 − x n x n1 − T2
P T2n
x n1 Lx n1 − x n LT2
P T2n−1
x n − x n.
3.35
Trang 9It follows from3.30, 3.34, and 3.35 that
lim
Finally, we consider
x n − T3xn
≤x n1 −x n x n1 −T3
P T3n
x n1 T3
P T3n
x n1 −T3
P T3n
x n T3
P T3n
x n −T3xn
≤x n1 − x n x n1 − T3
P T3n
x n1 Lx n1 − x n LT3
P T3n−1
x n − x n.
3.37
It follows from3.29, 3.30, and 3.37 that
lim
Therefore,
lim
n→∞x n − T1xn limn→∞x n − T2xn limn→∞x n − T3xn 0. 3.39 This completes the proof
Now, we give our main theorems of this paper
Theorem 3.3 Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.
Let T1, T2, T3 : K → E be uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive
mappings with sequences {r n i } such that∞
n1 r n i < ∞, for all i 1, 2, 3, satisfying condition (B) Suppose that {x n } is defined by 1.6 with∞n1 γ n < ∞,∞
n1 γ n < ∞, and∞
n1 γ n < ∞, where
α n , α n , and α n are three sequences in ε, 1 − ε, for some ε > 0 If F FT1 ∩ FT2 ∩ FT3 / ∅,
then {x n } converges strongly to a common fixed point of T1, T2, and T3.
Proof It follows fromLemma 3.2that limn→∞ x n − T1xn limn→∞ x n − T2xn limn→∞ x n−
T3x n 0 Since T1, T2, and T3satisfy conditionB, we have limn→∞ dx n , F 0.
sequence in K Assume that lim n→∞ x n p ∈ K Since lim n→∞ x n − T1xn limn→∞ x n −
T2x n limn→∞ x n − T3xn 0, by the continuity of T1, T2, and T3, we have p ∈ F, that is, p is
a common fixed point of T1, T2, and T3 This completes the proof
Corollary 3.4 Let K be a nonempty closed convex subset of a real uniformly convex Banach space
E Let T1, T2, T3 : K → E be nonself asymptotically nonexpansive mappings with sequences {r n i}
such that∞
n1 r n i < ∞, for all i 1, 2, 3, satisfying condition (B) Suppose that {x n } is defined by
1.6 with∞n1 γ n < ∞,∞
n1 γ n < ∞, and∞
n1 γ n < ∞, where α n , α n , and α n are three sequences
in ε, 1 − ε, for some ε > 0 If F FT1 ∩ FT2 ∩ FT3 / ∅, then {x n } converges strongly to a
common fixed point of T1, T2, and T3
Proof Since every nonself asymptotically nonexpansive mapping is uniformly L-Lipschitzian
and nonself asymptotically quasi-non-expansive, the result can be deduced immediately
Trang 1010 Fixed Point Theory and Applications
Theorem 3.5 Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.
Let T1, T2, T3 : K → E be uniformly L-Lipschitzian and nonself asymptotically quasi-non-expansive
mappings with sequences {r n i } such that∞
n1 r n i < ∞, for all i 1, 2, 3 Suppose that {x n } is
defined by1.6 with∞
n1 γ n < ∞,∞
n1 γ n < ∞, and∞
n1 γ n < ∞, where α n , α n , and α n are three sequences in ε, 1 − ε, for some ε > 0 If F FT1 ∩ FT2 ∩ FT3 / ∅ and one of T1, T2, and T3is demicompact, then {x n } converges strongly to a common fixed point of T1, T2, and T3.
Proof Without loss of generality, we may assume that T1is demicompact Since limn→∞ x n−
T1x n 0, there exists a subsequence {x n j } ⊂ {x n } such that x n j → x∗∈ K Hence, from 3.39,
we have
x∗− T i x∗ lim
n→∞x n
j − T i x n j 0, i 1,2,3. 3.40
This implies that x∗ ∈ F By the arbitrariness of p ∈ F, fromLemma 3.1, and taking p x∗, similarly we can prove that
lim
where d ≥ 0 is some nonnegative number From x n j → x∗, we know that d 0, that is,
x n → x∗ This completes the proof
Corollary 3.6 Let K be a nonempty closed convex subset of a real uniformly convex Banach space
E Let T1, T2, T3 : K → E be nonself asymptotically nonexpansive mappings with sequences {r n i}
such that∞
n1 r n i < ∞, for all i 1, 2, 3 Suppose that {x n } is defined by 1.6 with∞n1 γ n < ∞,
∞
n1 γ n < ∞, and∞
n1 γ n < ∞, where α n , α n , and α n are three sequences in ε, 1 − ε, for some
ε > 0 If F FT1 ∩ FT2 ∩ FT3 / ∅ and one of T1, T2, and T3 is demicompact, then {x n}
converges strongly to a common fixed point of T1, T2, and T3.
Acknowledgments
The authors would like to thank the referee and the editor for their careful reading of the manuscript and their many valuable comments and suggestions This paper was supported
by the National Natural Science Foundation of ChinaGrant no 10671145
References
1 W R Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.
4, no 3, pp 506–510, 1953
2 S Ishikawa, “Fixed points and iteration of a nonexpansive mapping in a Banach space,” Proceedings
of the American Mathematical Society, vol 59, no 1, pp 65–71, 1967.
3 K K Tan and H K Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa
iteration process,” Journal of Mathematical Analysis and Applications, vol 178, no 2, pp 301–308, 1993.
4 J Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,”
Bulletin of the Australian Mathematical Society, vol 43, no 1, pp 153–159, 1991.
5 Q H Liu, “Iterative sequences for asymptotically quasi-nonexpansive mappings with error member,”
Journal of Mathematical Analysis and Applications, vol 259, no 1, pp 18–24, 2001.
6 N Shahzad and A Udomene, “Approximating common fixed points of two asymptotically
quasi-nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol 2006, Article ID
18909, 10 pages, 2006
7 C E Chidume, E U Ofoedu, and H Zegeye, “Strong and weak convergence theorems for
asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol 280,
no 2, pp 364–374, 2003