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CONVERGENCE THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE
MAPPINGS IN BANACH SPACES
YONGFU SU, XIAOLONG QIN and MEIJUAN SHANG
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Abstract. Let E be a uniformly convex Banach space, and let K be a nonempty convex closed subset which is also a nonexpansive retract of E. Let T : K → Ebe an asymptotically nonexpansive mapping
with
{kn }
⊂
[1, ∞)
such
that
P∞
n=1 (kn − 1) < ∞ and let F (T ) be nonempty, where F (T ) denotes the fixed points set of T . Let
0 }, {γ 0 }, {α00 }, {β 00 } and {γ 00 } be real sequences in [0, 1] such that
{αn }, {βn }, {γn }, {α0n }, {βn
n
n
n
n
0
0
0
00
00
0
00
αn + βn + γn = αn + βn + γn = α00
n + βn + γn = 1 and ε ≤ αn , αn , αn ≤ 1 − ε for all n ∈ N and some
ε > 0, starting with arbitrary x1 ∈ K, define the sequence {xn } by setting
8
00
n−1 x + β 00 x + γ 00 w ),
>
n
n n
n n
<zn = P (αn T (P T )
0
0 x + γ 0 v ),
yn = P (αn T (P T )n−1 zn + βn
n
n
n
>
:
xn+1 = P (αn T (P T )n−1 yn + βn xn + γn un ),
P∞
P∞
P
0
00
with the restrictions ∞
n=1 γn < ∞ and
n=1 γn < ∞,
n=1 γn < ∞, where {wn }, {vn } and {un } are
bounded sequences in K. (i) If E is real uniformly convex Banach space satisfying Opial0 s condition,
then weak convergence of {xn } to some p ∈ F (T ) is obtained; (ii) If T satisfies condition (A), then {xn }
convergence strongly to some p ∈ F (T ).
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Received September 11, 2006.
2000 Mathematics Subject Classification. Primary 47H09; 47H10; 47J25.
Key words and phrases. asymptotically nonexpansive; non-self map; composite iterative with errors; Kadec-Klee
property; Uniformly convex Banach space.
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1. Introduction and Preliminaries
Let E be a real Banach space, K be a nonempty subset of X and F (T ) denote the set of fixed
points of T . A mapping T : K → K is said to be asymptotically nonexpansive if there exists a
sequence {kn } of positive real numbers with kn → 1 as n → ∞ such that
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kT n x − T n yk ≤ kn kx − yk
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for all x, y ∈ K.
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This class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972.
They proved that, if K is a nonempty bounded closed convex subset of a uniformly convex Banach
space E, then every asymptotically nonexpansive self-mapping T of K has a fixed point. Moreover,
the fixed point set F (T ) of T is closed and convex.
Recently, Chidume et al. have introduced another new concept about asymptotically nonexpansive mappings
Definition 1.1 ([1]). Let E be a real normed linear space, K a nonempty subset of E. Let
P : E → K be the nonexpansive retraction of E onto K. A map T : K → E is said to be an
asymptotically nonexpansive if there exists a sequence {kn } ⊂ [1, ∞) and kn → 1 as n → ∞ such
that the following inequality holds:
kT (P T )n−1 x − T (P T )n−1 yk ≤ kn kx − yk,
∀x, y ∈ K, n ≥ 1.
T is called uniformly L-lipschitzian if there exists L > 0 such that
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kT (P T )n−1 x − T (P T )n−1 yk ≤ Lkx − yk,
∀x, y ∈ K, n ≥ 1.
Many authors have contributed by their efforts to investigate the problem of finding a fixed point
of asymptotically nonexpansive mappings and non-self asymptotically nonexpansive mappings. In
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[5], [6], Schu introduced a modified Mann iteration process to approximate fixed points of asymptotically nonexpansive self-maps defined on nonempty closed convex and bounded subsets of a Hilbert
space H. More precisely, he proved the following theorems.
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Theorem JS1 ([5]). Let H be a Hilbert space, K a nonempty closed convex and bounded subset of
H, and T : K → K be a completely
continuous asymptotically nonexpansive mapping with sequence
P∞
{kn } ⊂ [1, ∞), kn → 1 and n=1 (kn2 − 1) < ∞. Let {αn }∞
n=1 be a real sequence in [0,1] satisfying
the condition ε ≤ αn ≤ 1 − ε for all n ≥ 1 and for some ε > 0. Then the sequence {xn } generated
from arbitrary x1 ∈ K by
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xn+1 = (1 − αn )xn + αn T n xn , n ≥ 1,
converges strongly to a fixed point of T.
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Theorem JS2 ([6]). Let E be a uniformly convex Banach space satisfying Opial’s condition, K
a nonempty closed convex and bounded subset of E, and
PT∞: K → K an asymptotically nonexpansive
mapping with sequence {kn } ⊂ [1, ∞), kn → 1 and n=1 (kn2 − 1) < ∞. Let {αn }∞
n=1 be a real
sequence in [0, 1] satisfying the condition 0 < a ≤ αn ≤ b < 1, for all n ≥ 1 and some a, b ∈ (0, 1).
Then the sequence {xn } generated from arbitrary x1 ∈ K by
xn+1 = (1 − αn )xn + αn T n xn , n ≥ 1,
converges weakly to a fixed point of T .
In [4], Rhoades extended Theorem JS1 to a uniformly convex Banach space using a modified
Ishikawa iteration method. In [3], Osilike and Aniagbosor proved that the theorems of Schu and
Rhoades remain true without the boundedness condition imposed on K, provided that F (T ) =
{x ∈ K : T x = x} =
6 ∅.
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In [9], Tan and Xu introduced a modified Ishikawa processes to approximate fixed points of
nonexpansive mappings defined on nonempty closed convex bounded subsets of a uniformly convex
Banach space E. More precisely, they proved the following theorem.
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Theorem TX ([9]). Let E be a uniformly convex Banach space which satisfies Opial’s condition or
has a Frechet differentiable norm. Let C be a nonempty closed convex bounded subset
P∞ of E, T : C →
C aP
nonexpansive mapping and {αn }, {βn } be real sequences in [0, 1] such that n=1 αn (1 − αn ) =
∞
∞, n=1 βn (1 − αn ) = ∞. Then the sequence {xn } generated from arbitrary x1 ∈ C by
(1.1)
xn+1 = (1 − αn )xn + αn T [(1 − βn )xn + βn T xn ], n ≥ 1
converges weakly to a fixed point of T .
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In the above results, T remains a self-mapping of a nonempty closed convex subset K of a
uniformly convex Banach space, however if, the domain K of T is a proper subset of E (and this
is the case in several applications), and T maps K into E, then iteration processes of Mann and
Ishikawa may fail to be well defined.
In 2003, Chidume et al. [1] studied the iteration scheme defined by
x1 ∈ K,
xn+1 = P ((1 − αn )xn + αn T (P T )n−1 xn ),
n ≥ 1.
In the framework of a uniformly convex Banach space, where K is a nonempty closed convex
nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction.
T : K → E is an asymptotically nonexpansive non-self map with sequence {kn } ⊂ [1, ∞), kn → 1.
{αn }∞
n=1 is a real sequence in [0, 1] satisfying the condition ε ≤ αn ≤ 1 − ε for all n ≥ 1 and for
some ε > 0. They proved strong and weak convergence theorems for asymptotically nonexpansive
nonself-maps.
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Recently, Naseer Shahzad [7] studied the sequence {xn } defined by
x1 ∈ K,
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xn+1 = P ((1 − αn )xn + αn T P [(1 − βn )xn + βn T xn ]),
where K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach
space E with P as a nonexpansive retraction. He proved weak and strong convergence theorems
for non-self nonexpansive mappings in Banach spaces.
Motivated by the Chidume et al. [1], Nasser Shahzad [7] and some others, the purpose of
this paper is to construct an iterative scheme for approximating a fixed point of asymptotically
nonexpansive non-self maps (provided that such a fixed point exists ) and to prove some strong and
weak convergence theorems for such maps.
Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. In this
paper, the following iteration scheme is studied

x1 ∈ K




zn = P (αn00 T (P T )n−1 xn + βn00 xn + γn00 wn )
(1.2)

yn = P (αn0 T (P T )n−1 zn + βn0 xn + γn0 vn )



xn+1 = P (αn T (P T )n−1 yn + βn xn + γn un )
where {αn }, {βn }, {γn }, {αn0 }, {βn0 }, {γn0 }, {αn00 }, {βn00 } and {γn00 } are real sequences in (0, 1) such
that αn + βn + γn = αn0 + βn0 + γn0 = αn00 + βn00 + γn00 = 1.
Our theorems improve and generalize some previous results to some extent.
Let E be a real Banach space. A subset K of E is said to be a retract of E if there exists a
continuous map P : E → E such that P x = x for all x ∈ K. A map P : E → E is said to be a
retraction if P 2 = P . It follows that if a map P is a retraction, then P y = y for all y in the range
of P .
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A mapping T with domain D(T ) and range R(T ) in E is said to be demiclosed at p if whenever
{xn } is a sequence in D(T ) such that {xn } converges weakly to x∗ ∈ D(T ) and {T xn } converges
strongly to p, then T x∗ = p.
Recall that the mapping T : K → E with F (T ) 6= ∅ where K is a subset of E, is said to satisfy
condition A [8] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0 and f (r) > 0
for all r ∈ (0, ∞) such that for all x ∈ K
kx − T xk ≥ f (d(x, F (T )),
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where d(x, F (T )) = inf{kx − pk : p ∈ F (T )}.
In order to prove our main results, we shall make use of the following Lemmas.
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Lemma 1.1 (Schu [6].). Suppose that E is a uniformly convex Banach space and 0 < p ≤ tn ≤
q < 1 for all n ∈ N . Suppose further that {xn } and {yn } are sequences of E such that
lim sup kxn k ≤ r, lim sup kyn k ≤ r
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n→∞
n→∞
and
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lim ktn xn + (1 − tn )yn k = r
n→∞
hold for some r ≥ 0. Then limn→∞ kxn − yn k = 0.
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Lemma 1.2 ([1] Demiclosed principle for nonself-map). Let E be a uniformly convex Banach
space, K a nonempty closed convex subset of E. Let T : K → E be an asymptotically nonexpansive
mapping with {kn } ⊂ [1, ∞) and kn → 1 as n → ∞. Then I − T is demiclosed with respect to zero.
Lemma 1.3 (Tan and Xu [9]). Let {rn }, {sn } and {tn } be three nonnegative sequences satisfying
the following condition
rn+1 ≤ (1 + sn )rn + tn ,
∀n ≥ 1.
P∞
P∞
If n=1 sn < ∞ and n=1 tn < ∞, then limn→∞ rn exists.
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2. Main results
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Lemma 2.1. Let E be a uniformly convex Banach space and K a nonempty closed convex subset
which is also a nonexpansive retract of E. Let T : K → E be an asymptotically nonexpansive mapping
with
{kn }
⊂
[1, ∞)
such
that
P∞
taking
arbitrary
x
∈
K,
(k
−1)
<
∞.
Let
{x
}
be
the
sequence
defined
by
the
recursion
(1.2)
1
n
n
n=1
P∞ 00
P∞ 0
P∞
with the restrictions
<
∞,
<
∞
and
γ
γ
γ
<
∞.
Then
lim
kx
−
pk
n
n→∞
n
n=1 n
n=1 n
n=1
exists, for any p ∈ F (T ), where F (T ) denotes the nonempty fixed point set of T .
Proof. Since {wn }, {vn } and {un } are bounded sequences in C, for any given p ∈ F (T ), we can
set
M1 = sup{kun − pk : n ≥ 1},
M2 = sup{kvn − pk : n ≥ 1},
M3 = sup{kwn − pk : n ≥ 1},
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M = max{Mi : i = 1, 2, 3}.
It follows from (1.2) that
kzn − pk = kP (αn00 T (P T )n−1 xn + βn00 xn + γn00 wn ) − pk
≤ kαn00 T (P T )n−1 xn + βn00 xn + γn00 wn − pk
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≤ αn00 kT (P T )n−1 xn − pk + βn00 kxn − pk + γn00 kwn − pk
= αn00 kT (P T )n−1 xn − T (P T )n−1 pk + βn00 kxn − pk + γn00 kwn − pk
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≤ αn00 kn kxn − pk + βn00 kxn − pk + γn00 kwn − pk
≤ αn00 kn kxn − pk + (1 − αn00 )kxn − pk + γn00 kwn − pk
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≤ kn kxn − pk + γn00 M,
which implies that
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(2.1)
kzn − pk ≤ kn kxn − pk + γn00 M.
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From (1.2) and (2.1) we get
kyn − pk = kP (αn0 T (P T )n−1 zn + βn0 xn + γn0 vn ) − pk
≤ kαn0 T (P T )n−1 zn + βn0 xn + γn0 vn − pk
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≤ αn0 kT (P T )n−1 zn − pk + βn0 kxn − pk + γn0 kvn − pk
= αn0 kT (P T )n−1 zn − T (P T )n−1 pk + βn0 kxn − pk + γn0 kvn − pk
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≤ αn0 kn kzn − pk + βn0 kxn − pk + γn0 kvn − pk
≤ αn0 kn kzn − pk + (1 − αn0 )kxn − pk + γn0 kvn − pk
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≤ αn0 kn (kn kxn − pk + γn00 M ) + (1 − αn0 )kxn − pk + γn0 kvn − pk
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≤ kn2 kxn − pk + kn γn00 M + γn0 M,
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which inplies that
(2.2)
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kyn − pk ≤ kn2 kxn − pk + kn γn00 M + γn0 M.
Again, from (1.2) and (2.2) we have
kxn+1 − pk = kP (αn T (P T )n−1 yn + βn xn + γn un ) − pk
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= kαn T (P T )n−1 yn + βn xn + γn un − pk
≤ αn kT (P T )n−1 yn − pk + βn kxn − pk + γn kun − pk
≤ αn kT (P T )n−1 yn − T (P T )n−1 pk + βn kxn − pk + γn kun − pk
Close
≤ αn kn kyn − pk + βn kxn − pk + γn kun − pk
≤ αn kn kyn − pk + (1 − αn )kxn − pk + γn kun − pk
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≤ αn kn (kn2 kxn − pk + kn γn00 M + γn0 M ) + (1 − αn0 )kxn − pk + γn M
≤ kn3 kxn − pk + kn2 γn00 M + kn γn0 M + γn M.
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Therefore
kxn+1 − pk ≤ (1 + (kn3 − 1))kxn − pk + (kn2 γn00 + kn γn0 + γn0 )M.
P∞
P∞
3
Note that
n=1 (kn − 1) < ∞ is equivalent to
n=1 (kn − 1) < ∞, therefore by Lemma 1.3,
limn→∞ kxn − pk exists for all p ∈ F (T ). This completes the proof.
(2.3)
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Lemma 2.2. Let E be a normed linear space, K a nonempty closed convex subset which is also
a nonexpansive retract of E, T : K → E a uniformly L-Lipschitzian mapping. Let
P∞{xn }00 be the
taking
arbitrary
x
∈
K,
with
the
restrictions
sequence
defined
by
the
recursion
(1.2)
1
n=1 γn < ∞,
P∞
P∞ 0
n−1
xn k, ∀n ≥ 1. If limn→∞ Cn = 0,
n=1 γn < ∞ and set Cn = kxn −T (P T )
n=1 γn < ∞ and
then limn→∞ kxn − T xn k = 0.
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Proof. Since {un }, {vn } and {wn } are bounded, it follows from Lemma 2.1 that {un − xn },
{vn − xn }, {wn − xn } are all bounded, now, we set
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r1 = sup{kun − xn k : n ≥ 1},
r2 = sup{kvn − xn k : n ≥ 1},
r3 = sup{kwn − xn k : n ≥ 1},
r4 = sup{kvn−1 − xn k : n ≥ 1},
n−2
r5 = sup{kun−1 − T (P T )
xn k : n ≥ 1},
r = max{ri : i = 1, 2, 3, 4, 5}.
It follows from (1.2) that
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kxn+1 − xn k ≤ kαn T (P T )n−1 yn + βn xn + γn un − xn k
≤ kT (P T )n−1 yn − xn k + γn r
Close
≤ kT (P T )n−1 xn − xn k + kT (P T )n−1 yn − T (P T )n−1 xn k + γn r
≤ Cn + Lkyn − xn k + γn r
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≤ Cn + Lkαn0 T (P T )n−1 zn + βn0 xn + γn0 vn − xn k + γn r
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≤ Cn + LkT (P T )n−1 zn − xn k + γn0 Lr + γn r
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≤ Cn + LkT (P T )n−1 xn − xn k + LkT (P T )n−1 zn − T (P T )n−1 xn k
+ γn0 Lr + γn r
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≤ Cn + LCn + L2 kzn − xn k + γn0 Lr + γn r
≤ Cn + LCn + L2 kαn00 T (P T )n−1 xn + βn00 xn + γn00 wn − xn k
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+ γn0 Lr + γn r
(2.4)
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= Cn (1 + L + L2 ) + γn00 L2 r + γn0 Lr + γn r
and
0
0
0
kyn−1 − xn k ≤ kαn−1
T (P T )n−2 zn−1 + βn−1
xn−1 + γn−1
vn−1 − xn k
0
≤ kT (P T )n−2 zn−1 − xn k + kxn−1 − xn k + γn−1
r
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≤ kT (P T )n−2 xn−1 − xn−1 k + kT (P T )n−2 zn−1 − T (P T )n−2 xn−1 k
0
+ 2kxn−1 − xn k + γn−1
r
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(2.5)
00
0
≤ Cn−1 + LCn−1 + Lγn−1
r + 2kxn−1 − xn k + γn−1
r.
Substituting (2.4) into (2.5) we obtain
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(2.6)
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00
0
kyn−1 − xn k ≤ Cn−1 (3 + 3L + 2L2 ) + (1 + 2L)r(Lγn−1
+ γn−1
)
0
+ 2γn−1
r.
On the other hand, from (2.4) and (2.6) we have
kxn − (P T )n−1 xn k
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≤ kαn−1 T (P T )n−2 yn−1 + βn−1 xn−1 + γn−1 un−1 − T (P T )n−2 xn k
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≤ kT (P T )n−2 yn−1 − T (P T )n−2 xn k + kxn−1 − T (P T )n−2 xn k
+ γn−1 r
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≤ Lkyn−1 − xn k + kxn−1 − T (P T )n−2 xn−1 k
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+ kT (P T )n−2 xn−1 − T (P T )n−2 xn k + γn−1 r
≤ Lkyn−1 − xn k + Cn−1 + Lkxn−1 − xn k + γn−1 r
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00
≤ LCn−1 (4 + 4L + 3L2 ) + Cn−1 + L2 rγn−1
(1 + 3L)
(2.7)
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0
+ 3Lrγn−1
(1 + L) + (1 + L)rγn−1 .
It follows from (2.7) that
kxn − T xn k ≤ kxn − T (P T )n−1 xn k + kT (P T )n−1 xn − T xn k
≤ Cn + Lk(P T )n−1 xn − xn k
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00
≤ Cn + L2 Cn−1 (4 + 4L + 3L2 ) + LCn−1 + L3 rγn−1
(1 + 3L)
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0
+ 3L2 rγn−1
(1 + L) + L(1 + L)rγn−1 .
P∞ 00
P∞ 0
P∞
It follows from limn→∞ Cn = 0,
n=1 γn < ∞ that
n=1 γn < ∞,
n=1 γn < ∞ and
lim kxn − T xn k = 0.
n→∞
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This completes the proof.
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Theorem 2.1. Let E be a uniformly convex Banach space and K a nonempty closed convex
subset which is also a nonexpansive retract of E. P
Let T : K → E be an asymptotically non∞
expansive mapping with {kn } ⊂ [1, ∞) such that
Let
n=1 (kn − 1) < ∞ and F (T ) 6= ∅.
0
0
0
00
00
{αn }, {βn }, {γn }, {αn }, {βn }, {γn }, {αn }, {βn } and {γn00 } be real sequences in [0, 1] such that
αn + βn + γn = αn0 + βn0 + γn0 = αn00 + βn00 + γn00 = 1 and ε ≤ αn , αn0 , αn00 ≤ 1 − ε for all n ∈ N and
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some ε > 0. Let {xn } be the sequence defined by the recursion (1.2) taking arbitrary x1 ∈ K. Then
limn→∞ kxn − T xn k = 0.
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Proof. Take p ∈ F (T ), by Lemma 2.1 we know, limn→∞ kxn −pk exists. Let limn→∞ kxn −pk = c.
If c = 0, then by the continuity of T the conclusion follows. Now suppose c > 0. We claim
limn→∞ kT xn − xn k = 0. Taking limsup on both the sides in the inequality (2.1), we have
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lim sup kzn − pk ≤ c.
(2.8)
n→∞
Similarly, taking limsup on both sides of the inequality (2.2), we have
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lim sup kyn − pk ≤ c.
(2.9)
n→∞
Next, we consider
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kT (P T )n−1 yn − p + γn (un − xn )k ≤ kT (P T )n−1 yn − pk + γn kun − xn k
≤ kn kyn − pk + γn r.
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Taking limsup on both the sides in the above inequality and using (2.9) we get
lim sup kT (P T )n−1 yn − p + γn (un − xn )k ≤ c.
n→∞
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and
kxn − p + γn (un − xn )k ≤ kxn − pk + γn kun − xn k
Close
≤ kxn − pk + γn r,
which imply that
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lim sup kxn − p + γn (un − xn )k ≤ c.
n→∞
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Again, lim kxn+1 − pk = c means that
n→∞
(2.10)
lim inf kαn (T (P T )n−1 yn − p + γn (un − xn ))
n→∞
+ (1 − αn )(xn − p + γn (un − xn ))k ≥ c.
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On the other hand, using (2.1) yields
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kαn (T (P T )n−1 yn − p + γn (un − xn )) + (1 − αn )(xn − p + γn (un − xn ))k
≤ αn kT (P T )n−1 yn − pk + (1 − αn )kxn − pk + γn kun − xn k
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≤ αn kn kyn − pk + (1 − αn )kxn − pk + γn kun − xn k
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≤ αn kn (kn2 kxn − pk + kn γn00 r + γn0 r) + (1 − αn )kxn − pk + γn kun − xn k
≤ kn3 kxn − pk + kn2 γn00 r + kn γn0 r + γn r.
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Therefore,
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lim sup kαn (T (P T )n−1 yn − p + γn (un − xn ))
(2.11)
n→∞
+ (1 − αn )(xn − p + γn (un − xn ))k ≤ c.
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Combining (2.10) with (2.11) we obtain
lim kαn (T (P T )n−1 yn − p + γn (un − xn ))
n→∞
+ (1 − αn )(xn − p + γn (un − xn ))k = c.
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By applying Lemma 1.1, we have
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(2.12)
lim kT (P T )n−1 yn − xn k = 0.
n→∞
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Notice that
kxn − pk ≤ kT (P T )n−1 yn − xn k + kT (P T )n−1 yn − pk
≤ kT (P T )n−1 yn − xn k + kn kyn − pk,
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which yields
c ≤ lim inf kyn − pk ≤ lim sup kyn − pk ≤ c.
n→∞
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n→∞
This implies that
lim kyn − pk = c.
n→∞
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Again, lim kyn − pk = c gives
n→∞
lim inf kαn0 (T zn − p + γn0 (vn − xn ))
n→∞
(2.13)
+ (1 − αn0 )(xn − p + γn0 (vn − xn ))k ≥ c.
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Similarly, we have
kαn0 (T (P T )n−1 zn − p + γn0 (vn − xn )) + (1 − αn0 )(xn − p + γn0 (vn − xn ))k
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≤ αn0 kT (P T )n−1 zn − pk + (1 − αn0 )kxn − pk + γn0 kvn − xn k
≤ αn0 kn kzn − pk + (1 − αn0 )kxn − pk + γn0 kvn − xn k
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≤ αn0 kn (kn kxn − pk + γn00 r) + (1 − αn0 )kxn − pk + γn0 kvn − xn k
≤ kn2 kxn − pk + kn γn00 r + γn0 r.
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Therefore,
(2.14)
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lim sup kαn0 (T (P T )n−1 zn − p + γn0 (vn − xn ))
n→∞
+ (1 − αn0 )(xn − p + γn0 (vn − xn ))k ≤ c.
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Combining (2.13) with (2.14) yields that
(2.15)
lim kαn0 (T (P T )n−1 zn − p + γn0 (vn − xn ))
n→∞
+ (1 − αn0 )(xn − p + γn0 (vn − xn ))k = c.
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On the other hand, we have
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kT (P T )n−1 zn − p + γn0 (vn − xn )k ≤ kT (P T )n−1 zn − pk + γn0 kvn − xn k
≤ kn kzn − pk + γn0 r
Taking limsup on both sides of the above inequality and using (2.1), we have
(2.16)
lim sup kT (P T )n−1 zn − p + γn0 (vn − xn )k ≤ c
n→∞
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and
kxn − p + γn0 (vn − xn )k ≤ kxn − pk + γn0 kvn − xn k
≤ kxn − pk + γn0 r,
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which yields
(2.17)
lim sup kxn − p + γn0 (vn − xn )k ≤ c.
n→∞
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Applying Lemma 1.1, it follows from (2.15), (2.16) and (2.17) that
(2.18)
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lim kT (P T )n−1 zn − xn k = 0.
n→∞
Notice that
kxn − pk ≤ kT (P T )n−1 zn − xn k + kT (P T )n−1 zn − pk
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≤ kT (P T )n−1 zn − xn k + kn kzn − pk.
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We have
c ≤ lim inf kzn − pk ≤ lim sup kzn − pk ≤ c.
n→∞
n→∞
That implies that
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lim kzn − pk = c.
(2.19)
n→∞
By the same method, we have
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lim kαn00 (T (P T )n−1 xn − p + γn00 (wn − xn ))
n→∞
(2.20)
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+ (1 − αn00 )(xn − p + γn00 (wn − xn )) − pk = c.
Moreover,
kT (P T )n−1 xn − p + γn00 (wn − xn )k ≤ kT (P T )n−1 xn − pk + γn00 kwn − xn k
≤ kn kxn − pk + γn00 r
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which implies that
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(2.21)
lim sup kT (P T )n−1 xn − p + γn00 (wn − xn )k ≤ c.
n→∞
It follows from
kxn − p + γn00 (wn − xn )k ≤ kxn − pk + γn00 kwn − xn k
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≤ kxn − pk + γn00 r.
we obtain
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(2.22)
lim sup kxn − p + γn00 (wn − xn )k ≤ c.
n→∞
Combining (2.20), (2.21) with (2.22) yields
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(2.23)
lim kT (P T )n−1 xn − xn k = 0.
n→∞
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Since T is uniformly L-Lipschitzian for some L > 0, it follows form Lemma 2.2 that
lim kxn − T xn k = 0.
n→∞
This completes the proof.
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Theorem 2.2. Let K be a nonempty closed convex subset of a uniformly convex Banach space E
satisfying Opial’s condition. Suppose that
P∞T : K → E is an asymptotically nonexpansive mapping
with sequence {kn } ⊂ [1, ∞) such that n=1 (kn − 1) < ∞, kn → 1 as n → ∞. Let {xn } be defined
by (1.2), where {αn }, {βn }, {γn }, {αn0 }, {βn0 }, {γn0 }, {αn00 }, {βn00 } and {γn00 } are real sequences in
[0, 1] such that αn + βn + γn = αn0 + βn0 + γn0 = αn00 + βn00 + γn00 = 1 and ε ≤ αn , αn0 , αn00 ≤ 1 − ε for all
n ∈ N and some ε > 0. Then {xn } converges weakly to a fixed point of F (T ).
Proof. For any p ∈ F (T ), it follows from Lemma 2.1 that limn→∞ kxn − pk exists. We now prove
that {xn } has a unique weak subsequential limit in F (T ). Firstly, let p1 and p2 be weak limits
of subsequences {xnk } and {xnj } of {xn }, respectively. By Lemmas 2.1 and 2.2, we know that
p ∈ F (T ). Secondly, let us assume p1 6= p2 , then by Opial’s condition, we obtain
lim kxn − p1 k = lim kxnk − p1 k < lim kxnk − p2 k = lim kxnj − p2 k
n→∞
k→∞
k→∞
j→∞
< lim kxnk − p1 k = lim kxn − p1 k
k→∞
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n→∞
which is a contradiction. Hence p1 = p2 . Then {xn } converges weakly to a fixed point of T . The
proof is complete.
Next, we shall prove a strong convergence theorem.
Theorem 2.3. Let E be a uniformly convex Banach space and K a nonempty closed convex
subset which is also a nonexpansive retract of E. Let T : K → E be a nonexpansive mapping with
p ∈ F (T ) := {x ∈ K : T x = x}. Let {αn }, {βn }, {γn }, {αn0 }, {βn0 }, {γn0 }, {αn00 }, {βn00 } and {γn00 }
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be real sequences in [0, 1] such that αn + βn + γn = αn0 + βn0 + γn0 = αn00 + βn00 + γn00 = 1 and and
ε ≤ αn , αn0 , αn00 ≤ 1 − ε for all n ∈ N and some ε > 0. Let {xn } be the sequence defined by the
recursion (1.2) taking arbitrary x1 ∈ K. Suppose T satisfies condition (A). Then {xn } converges
strongly to a fixed point of T .
Proof. By Lemma 2.1, limn→∞ kxn − pk exists for all p ∈ F = F (T ). Let limn→∞ kxn − pk = c
for some c ≥ 0. If c = 0, there is nothing to prove. Suppose c > 0. By Theorem 2.1, limn→∞ kT xn −
xn k = 0, and (2.5) give
inf kxn+1 − pk ≤ inf (1 + (kn3 − 1))kxn − pk + (kn2 γn00 + kn γn0 + γn )M.
p∈F
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p∈F
This means that
d(xn+1 , F ) ≤ (1 + (kn3 − 1))d(xn , F ) + (kn2 γn00 + kn γn0 + γn )M.
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Thus limn→∞ d(xn , F ) exists by virtue of Lemma 1.3. Now by condition (A), limn→∞ f (d(xn , F )) =
0. Since f is a nondecreasing function and f (0) = 0, therefore limn→∞ d(xn , F ) = 0. Now we can
take a subsequence {xnj } of {xn } and sequence {yj } ⊂ F such that kxnj − yj k < 2−j . Then
following the method in the proof of Tan and Xu [9], we get that {yj } is a Cauchy sequence in
F and so it converges. Let yj → y. Since F is closed, therefore y ∈ F and then xnj → y. As
limn→∞ kxn − pk exists, xn → y ∈ F = F (T ) thereby completing the proof.
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1. Chidume C. E., Ofoedu E. U. and Zegeye H., Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math, Anal. Appl. 280 (2003), 364–374.
2. Goebel K. and Kirk W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math.
Soc. 35 (1972), 171–174.
3. Osilike M. O. and Aniagbosor S. C., Weak and strong convergence theorems for fixed points of asymptotically
nonexpansive mappings, Math. Comput. Modelling 32 (2000), 1181–1191.
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4. Rhoades B. E., Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl. 183 (1994), 118–120.
5. Schu J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl.
158 (1991), 407–413.
6. Schu J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral.
Math. Soc. 43 (1991), 153–159.
7. Shahzad N., Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal.
61 (2005), 1031–1039.
8. Senter H. F. and Doston W. G., Approximating fixed points of nonexpansive mapping, Proc. Amer. Math. Soc.
44(2) (1974), 375–380.
9. Tan K. K. and Xu H. K., Approximating fixed points of nonexpansive mappings by the Ishikawaiteration process,
J. Math. Anal. Appl. 178 (1993), 301–308.
Yongfu Su, Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China,
e-mail: qxlxajh@163.com
Xiaolong Qin, Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
Meijuan Shang, Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China and Department
of Mathematics, Shijiazhuang University, Shijiazhuang 300160, China
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