Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 3 Issue 1(2011), Pages 89-101.
THREE-STEP ITERATION PROCESS FOR A FINITE FAMILY
OF ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS IN
THE INTERMEDIATE SENSE IN CONVEX METRIC SPACES
(COMMUNICATED BY DANIEL PELLEGRINO)
G. S. SALUJA
Abstract. The purpose of this paper is to establish some strong convergence
theorems of three-step iteration process with errors for approximating common
fixed points for a finite family of asymptotically quasi-nonexpansive mappings
in the intermediate sense in the setting of convex metric spaces. The results
obtained in this paper generalize, improve and unify some main results of
[1]-[7], [9]-[11], [13]-[17], [19] and [21].
1. Introduction
Throughout this paper, we assume that E is a metric space, F (Ti ) = {x ∈ E :
Ti x = x} is the set of all fixed points of the mappings Ti (i = 1, 2, . . . , N ), D(T ) is
the domain of T and N is the set of all positive integers. The set of common fixed
points of Ti (i = 1, 2, . . . , N ) denoted by F , that is, F = ∩N
i=1 F (Ti ).
Definition 1. ([1]) Let T : D(T ) ⊂ E → E be a mapping.
(i) The mapping T is said to be L-Lipschitzian if there exists a constant L > 0
such that
d(T x, T y) ≤ Ld(x, y),
∀x, y ∈ D(T ).
(1.1)
(ii) The mapping T is said to be nonexpansive if
d(T x, T y) ≤ d(x, y),
∀x, y ∈ D(T ).
(1.2)
2000 Mathematics Subject Classification. 47H09, 47H10.
Key words and phrases. Asymptotically quasi-nonexpansive mapping in the intermediate
sense, three-step iteration process with errors, common fixed point, strong convergence, convex
metric space.
c
2011
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted July 29, 2010. Published January 25, 2011.
89
90
G. S. SALUJA
(iii) The mapping T is said to be quasi-nonexpansive if F (T ) 6= ∅ and
d(T x, p) ≤ d(x, p),
∀x ∈ D(T ), ∀p ∈ F (T ).
(1.3)
(iv) The mapping T is said to be asymptotically nonexpansive if there exists a
sequence {kn } ⊂ [1, ∞) with limn→∞ kn = 1 such that
d(T n x, T n y) ≤ kn d(x, y),
∀x, y ∈ D(T ), ∀n ∈ N.
(1.4)
(v) The mapping T is said to be asymptotically quasi-nonexpansive if F (T ) 6= ∅
and there exists a sequence {kn } ⊂ [1, ∞) with limn→∞ kn = 1 such that
d(T n x, p) ≤ kn d(x, p),
∀x ∈ D(T ), ∀p ∈ F (T ), ∀n ∈ N.
(1.5)
(vi) T is said to be asymptotically nonexpansive type, if
(
lim sup
sup
n→∞
x,y∈D(T )
n
n
d(T x, T y) − d(x, y)
)
≤ 0.
(1.6)
(vii) T is said to be asymptotically quasi-nonexpansive type, if F (T ) 6= ∅ and
(
lim sup
sup
n→∞
x∈D(T ), p∈F (T )
d(T x, p) − d(x, p)
n
)
≤
0.
(1.7)
Remark 1. It is easy to see that if F (T ) is nonempty, then nonexpansive mapping,
quasi-nonexpansive mapping, asymptotically nonexpansive mapping, asymptotically
quasi-nonexpansive mapping and asymptotically nonexpansive type mapping all are
the special cases of asymptotically quasi-nonexpansive type mappings.
Now, we define asymptotically quasi-nonexpansive mapping in the intermediate
sense in convex metric space.
T is said be asymptotically quasi-nonexpansive mapping in the intermediate
sense provided that T is uniformly continuous and
(
)
n
lim sup
sup
d(T x, p) − d(x, p)
≤ 0.
(1.8)
n→∞
x∈D(T ), p∈F (T )
In recent years, the problem concerning convergence of iterative sequences (and
sequences with errors) for asymptotically nonexpansive mappings or asymptotically
quasi-nonexpansive mappings converging to some fixed points in Hilbert spaces or
Banach spaces have been considered by many authors.
In 1973, Petryshyn and Williamson [13] obtained a necessary and sufficient condition for Picard iterative sequences and Mann iterative sequences to converge to a
fixed point for quasi-nonexpansive mappings. In 1994, Tan and Xu [16] also proved
THREE-STEP ITERATION PROCESS FOR A. . . . . .
91
some convergence theorems of Ishikawa iterative sequences satisfying Opial’s condition [12] or having Frečhet differential norm. In 1997, Ghosh and Debnath [4]
extended the result of Petryshyn and Williamson [13] and gave a necessary and
sufficient condition for Ishikawa iterative sequences to converge to a fixed point
of quasi-nonexpansive mappings. Also in 2001 and 2002, Liu [9, 10, 11] obtained
some necessary and sufficient conditions for Ishikawa iterative sequences or Ishikawa
iterative sequences with errors to converge to a fixed point for asymptotically quasinonexpansive mappings.
In 2004, Chang et al. [1] extended and improved the result of Liu [11] in convex
metric space. Further in the same year, Kim et al. [7] gave a necessary and sufficient
conditions for asymptotically quasi-nonexpansive mappings in convex metric spaces
which generalized and improved some previous known results.
Very recently, Tian and Yang [18] gave some necessary and sufficient conditions
for a new Noor-type iterative sequences with errors to approximate a common fixed
point for a finite family of uniformly quasi-Lipschitzian mappings in convex metric
spaces.
The purpose of this paper is to establish some strong convergence theorems of
three-step iteration process with errors to approximate a common fixed point for
a finite family of asymptotically quasi-nonexpansive mappings in the intermediate
sense in the setting of convex metric spaces. The results obtained in this paper
generalize, improve and unify some main results of [1]-[7], [9]-[11], [13]-[17], [19]
and [21].
Let T be a given self mapping of a nonempty convex subset C of an arbitrary
normed space. The sequence {xn }∞
n=0 defined by:
x0 ∈ C,
xn+1
= αn xn + βn T yn + γn un , n ≥ 0,
yn
= an xn + bn T zn + cn vn ,
zn
= dn xn + en T xn + fn wn ,
(1.9)
is called the Noor-type iterative procedure with errors [2], where αn , βn , γn , an , bn , cn ,
dn , en and fn are appropriate sequences in [0, 1] with αn + βn + γn = an + bn + cn =
dn + en + fn = 1, n ≥ 0 and {un }, {vn } and {wn } are bounded sequences in C. If
dn = 1(en = fn = 0), n ≥ 0, then (1.9) reduces to the Ishikawa iterative procedure
with errors [20] defined as follows:
x0 ∈ C,
xn+1
yn
= αn xn + βn T yn + γn un , n ≥ 0,
(1.10)
= an xn + bn T xn + cn vn .
If an = 1(bn = cn = 0), then (1.10) reduces to the following Mann type iterative
procedure with errors [20]:
x0 ∈ C,
xn+1
=
αn xn + βn T yn + γn un , n ≥ 0.
(1.11)
92
G. S. SALUJA
For the sake of convenience, we first recall some definitions and notation.
Definition 2. (see [1]): Let (E, d) be a metric space and I = [0, 1]. A mapping
W : E 3 × I 3 → E is said to be a convex structure on E if it satisfies the following
condition:
d(u, W (x, y, z; α, β, γ)) ≤ αd(u, x) + βd(u, y) + γd(u, z),
for any u, x, y, z ∈ E and for any α, β, γ ∈ I with α + β + γ = 1.
If (E, d) is a metric space with a convex structure W , then (E, d) is called a
convex metric space and is denoted by (E, d, W ). Let (E, d) be a convex metric
space, a nonempty subset C of E is said to be convex if
W (x, y, z, λ1 , λ2 , λ3 ) ∈ C,
∀(x, y, z, λ1 , λ2 , λ3 ) ∈ C 3 × I 3 .
Remark 2. It is easy to prove that every linear normed space is a convex metric
space with a convex structure W (x, y, z; α, β, γ) = αx + βy + γz, for all x, y, z ∈ E
and α, β, γ ∈ I with α + β + γ = 1. But there exist some convex metric spaces which
can not be embedded into any linear normed spaces (see, Takahashi [15]).
Definition 3. Let (E, d, W ) be a convex metric space and Ti : E → E be a finite
family of asymptotically quasi-nonexpansive mappings in the intermediate sense
with i = 1, 2, . . . , N . Let {αn }, {βn }, {γn }, {an }, {bn }, {cn }, {dn }, {en } and {fn }
be nine sequences in [0, 1] with
αn + βn + γn = an + bn + cn = dn + en + fn = 1, n = 0, 1, 2, . . . .
(1.12)
For a given x0 ∈ E, define a sequence {xn } as follows:
xn+1
= W (xn , Tnn yn , un ; αn , βn , γn ), n ≥ 0,
yn
= W (g(xn ), Tnn zn , vn ; an , bn , cn ),
zn
= W (g(xn ), Tnn xn , wn ; dn , en , fn ),
(1.13)
n
where Tnn = Tn(mod
N ) , g : E → E is a Lipschitz continuous mapping with a Lipschitz constant ξ > 0 and {un }, {vn }, {wn } are any given three sequences in E.
Then {xn } is called the Noor-type iterative sequence with errors for a finite family
of asymptotically quasi-nonexpansive type mappings {Ti }N
i=1 . If g = I (the identity
mapping on E) in (1.13), then the sequence {xn } defined by (1.13) can be written
as follows:
xn+1
= W (xn , Tnn yn , un ; αn , βn , γn ), n ≥ 0,
yn
= W (xn , Tnn zn , vn ; an , bn , cn ),
zn
= W (xn , Tnn xn , wn ; dn , en , fn ),
(1.14)
THREE-STEP ITERATION PROCESS FOR A. . . . . .
93
If dn = 1(en = fn = 0) for all n ≥ 0 in (1.13), then zn = g(xn ) for all n ≥ 0 and
the sequence {xn } defined by (1.13) can be written as follows:
xn+1
yn
= W (xn , Tnn yn , un ; αn , βn , γn ), n ≥ 0,
= W (g(xn ), Tnn g(xn ), vn ; an , bn , cn ).
(1.15)
If g = I and dn = 1(en = fn = 0) for all n ≥ 0, then the sequence {xn } defined by
(1.13) can be written as follows:
xn+1
yn
= W (xn , Tnn yn , un ; αn , βn , γn ), n ≥ 0,
= W (xn , Tnn xn , vn ; an , bn , cn ),
(1.16)
which is the Ishikawa type iterative sequence with errors considered in [17]. Further,
if g = I and dn = an = 1(en = fn = bn = cn = 0) for all n ≥ 0, then zn = yn = xn
for all n ≥ 0 and (1.13) reduces to the following Mann type iterative sequence with
errors [17]:
xn+1
=
W (xn , Tnn xn , un ; αn , βn , γn ), n ≥ 0.
(1.17)
Lemma 1.1. (see [10]): Let {pn }, {qn }, {rn } be three nonnegative sequences of
real numbers satisfying the following conditions:
pn+1 ≤ (1 + qn )pn + rn , n ≥ 0,
∞
X
qn < ∞,
∞
X
rn < ∞.
(1.18)
n=0
n=0
Then
(1) limn→∞ pn exists.
(2) In addition, if lim inf n→∞ pn = 0, then limn→∞ pn = 0.
2. Main Results
Now we state and prove our main results of this paper.
Lemma 2.1. Let (E, d, W ) be a complete convex metric space and C be a nonempty
closed convex subset of E. Let Ti : C → C be a finite family of asymptotically
quasi-nonexpansive mappings in the intermediate sense for i = 1, 2, . . . , N such
that F = ∩N
i=1 F (Ti ) 6= ∅ and g : C → C a contractive mapping with a contractive
constant ξ ∈ (0, 1). Put
n
Gn = max
sup
d(Tnn xn , p) − d(xn , p) ∨ sup
d(Tnn yn , p) − d(yn , p)
p∈F, n≥0
p∈F, n≥0
∨
sup
p∈F, n≥0
d(Tnn zn , p)
o
− d(zn , p) ∨ 0 ,
(2.1)
P∞
such that n=0 Gn < ∞. Let {xn } be the iterative sequence with errors defined
by (1.13) and {un }, {vn }, {wn } be three bounded sequences in C. Let {αn }, {βn },
{γn }, {an }, {bn }, {cn }, {dn }, {en }, {fn } be sequences in [0, 1] satisfying the following conditions:
(i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0;
94
G. S. SALUJA
(ii)
P∞
n=0 (βn
+ γn ) < ∞.
Then the following conclusions hold:
(1) for all p ∈ F and n ≥ 0,
d(xn+1 , p) ≤ (1 + 3βn )d(xn , p) + 3Gn + M θn ,
(2.2)
where θn = βn + γn for all n ≥ 0 and
M=
n
o
d(un , p) + d(vn , p) + d(wn , p) + 2d(g(p), p) .
sup
p∈F, n≥0
(2) there exists a constant M1 > 0 such that
d(xn+m , p) ≤ M1 d(xn , p) + 3M1
n+m−1
X
Gk
k=n
+M M1
n+m−1
X
θk , ∀p ∈ F,
(2.3)
k=n
for all n, m ≥ 0.
Proof. For any p ∈ F , using (1.13) and (2.1), we have
d(xn+1 , p)
= d(W (xn , Tnn yn , un ; αn , βn , γn ), p)
≤ αn d(xn , p) + βn d(Tnn yn , p) + γn d(un , p)
≤ αn d(xn , p) + βn [d(yn , p) + Gn ] + γn d(un , p)
≤ αn d(xn , p) + βn d(yn , p) + βn Gn + γn d(un , p)
≤ αn d(xn , p) + βn d(yn , p) + Gn + γn d(un , p),
(2.4)
and
d(yn , p)
= d(W (g(xn ), Tnn zn , vn ; an , bn , cn ), p)
≤ an d(g(xn ), p) + bn d(Tnn zn , p) + cn d(vn , p)
≤ an d(g(xn ), g(p)) + an d(g(p), p)
+bn [d(zn , p) + Gn ] + cn d(vn , p)
≤ an ξd(xn , p) + an d(g(p), p) + bn d(zn , p)
+bn Gn + cn d(vn , p)
≤ an ξd(xn , p) + an d(g(p), p) + bn d(zn , p)
+Gn + cn d(vn , p),
(2.5)
THREE-STEP ITERATION PROCESS FOR A. . . . . .
95
and
d(zn , p)
=
d(W (g(xn ), Tnn xn , wn ; dn , en , fn ), p)
≤
dn d(g(xn ), p) + en d(Tnn xn , p) + fn d(wn , p)
≤
dn d(g(xn ), g(p)) + dn d(g(p), p)
+en [d(xn , p) + Gn ] + fn d(wn , p)
≤ dn ξd(xn , p) + dn d(g(p), p) + en d(xn , p)
+en Gn + fn d(wn , p)
≤ (dn ξ + en )d(xn , p) + dn d(g(p), p)
+en Gn + fn d(wn , p)
≤ (dn ξ + en )d(xn , p) + dn d(g(p), p)
+Gn + fn d(wn , p).
(2.6)
Substituting (2.5) into (2.4) and simplifying it, we have
h
d(xn+1 , p) ≤ αn d(xn , p) + βn an ξd(xn , p) + an d(g(p), p)
i
+bn d(zn , p) + Gn + cn d(vn , p) + Gn + γn d(un , p)
≤ (αn + an βn ξ)d(xn , p) + an βn d(g(p), p) + βn Gn
+bn βn d(zn , p) + cn βn d(vn , p) + Gn + γn d(un , p)
=
(αn + an βn ξ)d(xn , p) + an βn d(g(p), p) + (1 + βn )Gn
+bn βn d(zn , p) + cn βn d(vn , p) + γn d(un , p)
≤ (αn + an βn ξ)d(xn , p) + an βn d(g(p), p) + 2Gn
+bn βn d(zn , p) + cn βn d(vn , p) + γn d(un , p).
(2.7)
96
G. S. SALUJA
Substituting (2.6) into (2.7) and simplifying it, we have
d(xn+1 , p) ≤ (αn + an βn ξ)d(xn , p) + an βn d(g(p), p) + 2Gn
h
+bn βn (dn ξ + en )d(xn , p) + dn d(g(p), p) + Gn
i
+fn d(wn , p) + cn βn d(vn , p) + γn d(un , p)
h
i
≤ αn + an βn ξ + bn βn (dn ξ + en ) d(xn , p)
+βn (an + bn dn )d(g(p), p) + Gn (2 + bn βn )
+bn βn fn d(wn , p) + cn βn d(vn , p) + γn d(un , p)
h
i
≤ αn + βn (an ξ + bn dn ξ + bn en ) d(xn , p)
+2βn d(g(p), p) + 3Gn + βn d(wn , p)
+βn d(vn , p) + γn d(un , p)
≤ (1 + 3βn )d(xn , p) + 2βn d(g(p), p)
+2γn d(g(p), p) + 3Gn + βn d(wn , p)
+γn d(wn , p) + βn d(vn , p) + γn d(vn , p)
+βn d(un , p) + γn d(un , p)
=
(1 + 3βn )d(xn , p) + 3Gn + 2(βn + γn )d(g(p), p)
h
i
+(βn + γn ) d(un , p) + d(vn , p) + d(wn , p)
h
= (1 + 3βn )d(xn , p) + 3Gn + (βn + γn ) d(un , p)
i
+d(vn , p) + d(wn , p) + 2d(g(p), p)
≤ (1 + 3βn )d(xn , p) + 3Gn + M θn , ∀n ≥ 0, p ∈ F,
(2.8)
where
M=
sup
n
o
d(un , p) + d(vn , p) + d(wn , p) + 2d(g(p), p) ,
p∈F n≥0
This completes the proof of part (1).
θn = βn + γn .
THREE-STEP ITERATION PROCESS FOR A. . . . . .
97
(2) Since 1 + x ≤ ex for all x ≥ 0, it follows from (2.8) that, for n, m ≥ 0 and
p ∈ F , we have
d(xn+m , p) ≤ (1 + 3βn+m−1 )d(xn+m−1 , p) + 3Gn+m−1 + M θn+m−1
≤ e3βn+m−1 d(xn+m−1 , p) + 3Gn+m−1 + M θn+m−1
h
i
≤ e3βn+m−1 e3βn+m−2 d(xn+m−2 , p) + 3Gn+m−2 + M θn+m−2
+3Gn+m−1 + M θn+m−1
h
≤ e3(βn+m−1 +βn+m−2 ) d(xn+m−2 , p) + 3 e3βn+m−1 Gn+m−2
i
h
i
+Gn+m−1 + M e3βn+m−1 θn+m−2 + θn+m−1
≤ e3(βn+m−1 +βn+m−2 ) d(xn+m−2 , p) + 3e3βn+m−1 Gn+m−2
+Gn+m−1 + M e3βn+m−1 θn+m−2 + θn+m−1
≤ ...
≤ ...
≤ M1 d(xn , p) + 3M1
n+m−1
X
Gk + M M1
k=n
n+m−1
X
θk ,
(2.9)
k=n
where
M1 = e3
P∞
k=0
βk
.
This completes the proof of part (2).
Theorem 2.2. Let (E, d, W ) be a complete convex metric space and C be a nonempty
closed convex subset of E. Let Ti : C → C be a finite family of uniformly LLipschitzian asymptotically quasi-nonexpansive mappings in the intermediate sense
for i = 1, 2, . . . , N such that F = ∩N
i=1 F (Ti ) 6= ∅ and g : C → C a contractive
mapping with a contractive constant ξ ∈ (0, 1). Put
n
Gn = max
sup
d(Tnn xn , p) − d(xn , p) ∨ sup
d(Tnn yn , p) − d(yn , p)
p∈F, n≥0
p∈F, n≥0
∨
sup
p∈F, n≥0
o
d(Tnn zn , p) − d(zn , p) ∨ 0 ,
P∞
such that n=0 Gn < ∞. Let {xn } be the iterative sequence with errors defined
by (1.13) and {un }, {vn }, {wn } be three bounded sequences in C, and {αn }, {βn },
{γn }, {an }, {bn }, {cn }, {dn }, {en } and {fn } be nine sequences in [0, 1] satisfying
the following conditions:
(i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0;
P∞
(ii) n=0 (βn + γn ) < ∞.
Then the sequence {xn } converges strongly to a common fixed point p of the
mappings {Ti }N
i=1 if and only if
lim inf d(xn , F ) = 0,
n→∞
98
G. S. SALUJA
where d(x, F ) = inf p∈F d(x, p).
Proof. The necessity is obvious. Now, we prove the sufficiency. In fact, from Lemma
2.1, we have
d(xn+1 , F ) ≤ (1 + 3βn )d(xn , F ) + 3Gn + M θn , ∀n ≥ 0,
(2.10)
where θn = βn + γn . By assumption and conditions (i) and (ii), we know that
∞
X
∞
X
θn < ∞,
n=0
∞
X
βn < ∞,
n=0
Gn < ∞.
(2.11)
n=0
It follows from Lemma 1.1 that limn→∞ d(xn , F ) exists. Since lim inf n→∞ d(xn , F )
= 0, we have
lim d(xn , F ) = 0.
(2.12)
n→∞
Next, we prove that {xn } is a Cauchy sequence in C. In fact, for any given ε > 0,
there exists a positive integer n1 such that for any n ≥ n1 , we have
d(xn , F ) <
ε
,
12M1
∞
X
ε
, ∀n ≥ 0.
18M1
Gn <
n=n1
(2.13)
and
∞
X
θn <
n=n1
ε
, ∀n ≥ 0.
6M M1
(2.14)
From (2.13), there exists p1 ∈ F and positive integer n2 ≥ n1 such that
ε
.
d(xn2 , p1 ) <
6M1
(2.15)
Thus Lemma 2.1(2) implies that, for any positive integer n, m with n ≥ n2 , we
have
d(xn+m , xn ) ≤ d(xn+m , p1 ) + d(xn , p1 )
≤ M1 d(xn2 , p1 ) + 3M1
n+m−1
X
Gk + M M1
k=n2
+M1 d(xn2 , p1 ) + 3M1
n+m−1
X
n+m−1
X
Gk + M M1
n+m−1
X
k=n2
≤ 2M1 d(xn2 , p1 ) + 6M1
n+m−1
X
<
ε.
θk
k=n2
Gk + 2M M1
k=n2
< 2M1 .
θk
k=n2
n+m−1
X
θk
k=n2
ε
ε
ε
+ 6M1 .
+ 2M M1 .
6M1
18M1
6M M1
(2.16)
This shows that {xn } is a Cauchy sequence in a nonempty closed convex subset C
of a complete convex metric space E. Without loss of generality, we can assume
that limn→∞ xn = q ∈ E. Now we will prove that q ∈ F . Since xn → q and
THREE-STEP ITERATION PROCESS FOR A. . . . . .
99
d(xn , F ) → 0 as n → ∞, for any given ε1 > 0, there exists a positive integer
n3 ≥ n2 such that for n ≥ n3 , we have
d(xn , q) < ε1 ,
d(xn , F ) < ε1 .
(2.17)
Again from the second inequality of (2.17), there exists q1 ∈ F such that
d(xn3 , q1 ) < 2ε1 .
(2.18)
Moreover, it follows from (2.1) that for any n ≥ n3 , we have
d(Tnn q, q1 ) − d(q, q1 ) < Gn .
(2.19)
Thus for any i = 1, 2, . . . , N , from (2.17) - (2.19) and for any n ≥ n3 , we have
d(Tin q, q) ≤
d(Tin q, q1 ) + d(q1 , q)
≤
d(q, q1 ) + Gn + d(q1 , q)
=
Gn + 2d(q, q1 )
≤
Gn + 2[d(q, xn3 ) + d(xn3 , q1 )]
<
Gn + 2(ε1 + 2ε1 ) = Gn + 6ε1 = ε0 ,
(2.20)
0
where ε = Gn + 6ε1 , since Gn → 0 as n → ∞ and ε1 > 0, it follows that ε0 > 0.
By the arbitrariness of ε0 > 0, we know that Tin q = q for all i = 1, 2, . . . , N .
Again since for any n ≥ n3 , we have
d(Tin q, Ti q) ≤ d(Tin q, q1 ) + d(Ti q, q1 )
≤ d(q, q1 ) + Gn + d(Ti q, q1 )
≤ d(q, q1 ) + Gn + Ld(q, q1 )
=
(1 + L)d(q, q1 ) + Gn
≤ (1 + L)[d(q, xn3 ) + d(xn3 , q1 )] + Gn
< (1 + L)[ε1 + 2ε1 ] + Gn
=
3(1 + L)ε1 + Gn = ε00 ,
(2.21)
00
where ε = 3(1 + L)ε1 + Gn , since Gn → 0 as n → ∞ and ε1 > 0, it follows
that ε00 > 0. By the arbitrariness of ε00 > 0, we know that Tin q = Ti q for all
i = 1, 2, . . . , N . From the uniqueness of limit, we have q = Ti q for all i = 1, 2, . . . , N ,
that is, q ∈ F . This shows that q is a common fixed point of the mappings {Ti }N
i=1 .
This completes the proof.
Taking g = I in Theorem 2.2, then we have the following theorem.
Theorem 2.3. Let (E, d, W ) be a complete convex metric space and C be a nonempty
closed convex subset of E. Let Ti : C → C be a finite family of uniformly LLipschitzian asymptotically quasi-nonexpansive mappings in the intermediate sense
for i = 1, 2, . . . , N such that F = ∩N
i=1 F (Ti ) 6= ∅. Put
n
n
Gn = max
sup
d(Tn xn , p) − d(xn , p) ∨ sup
d(Tnn yn , p) − d(yn , p)
p∈F, n≥0
p∈F, n≥0
∨
sup
p∈F, n≥0
P∞
o
d(Tnn zn , p) − d(zn , p) ∨ 0 ,
such that n=0 Gn < ∞. Let {xn } be the iterative sequence with errors defined
by (1.14) and {un }, {vn }, {wn } be three bounded sequences in C and {αn }, {βn },
100
G. S. SALUJA
{γn }, {an }, {bn }, {cn }, {dn }, {en } and {fn } be nine sequences in [0, 1] satisfying
the following conditions:
(i) αn + βn + γn = an + bn + cn = dn + en + fn = 1, ∀n ≥ 0;
P∞
(ii) n=0 (βn + γn ) < ∞.
Then the sequence {xn } converges strongly to a common fixed point p of the
mappings {Ti }N
i=1 if and only if
lim inf d(xn , F ) = 0,
n→∞
where d(x, F ) = inf p∈F d(x, p).
Taking dn = 1(en = fn = 0) for all n ≥ 0 in Theorem 2.2, then we have the
following theorem.
Theorem 2.4. Let (E, d, W ) be a complete convex metric space and C be a nonempty
closed convex subset of E. Let Ti : C → C be a finite family of uniformly LLipschitzian asymptotically quasi-nonexpansive mappings in the intermediate sense
for i = 1, 2, . . . , N such that F = ∩N
i=1 F (Ti ) 6= ∅ and g : C → C a contractive
mapping with a contractive constant ξ ∈ (0, 1). Put
n
Gn = max
sup
d(Tnn xn , p) − d(xn , p) ∨
p∈F, n≥0
sup
p∈F, n≥0
o
d(Tnn yn , p) − d(yn , p) ∨ 0 ,
P∞
such that n=0 Gn < ∞. Let {xn } be the iterative sequence with errors defined by
(1.15) and {un }, {vn } be two bounded sequences in C and {αn }, {βn }, {γn }, {an },
{bn } and {cn } be six sequences in [0, 1] satisfying the following conditions:
(i) αn + βn + γn = an + bn + cn = 1, for all n ≥ 0;
P∞
(ii) n=0 (βn + γn ) < ∞.
Then the sequence {xn } converges to a common fixed point p in F if and only if
lim inf d(xn , F ) = 0,
n→∞
where d(x, F ) = inf p∈F d(x, p).
Remark 3. Theorems 2.2 - 2.4 generalize, improve and unify some corresponding
result in [1]-[7], [9]-[11], [13]-[17], [19] and [21].
Remark 4. Our results also extend the corresponding results of [18] to the case
of more general class of uniformly quasi-Lipschitzian mappings considered in this
paper.
Acknowledgement: The author thanks the referee for his valuable suggestions
and comments on the manuscript.
THREE-STEP ITERATION PROCESS FOR A. . . . . .
101
References
[1] S.S. Chang, J.K. Kim and D.S. Jin, Iterative sequences with errors for asymptotically quasinonexpansive type mappings in convex metric spaces, Archives of Inequality and Applications
2 (2004), 365–374.
[2] Y.J. Cho, H. Zhou and G. Guo, Weak and strong convergence theorems for three step iterations with errors for asymptotically nonexpansive mappings, Computers and Math. with
Appl. (47) (2004), 707–717.
[3] H. Fukhar-ud-din and S.H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl. (328 2)
(2007), 821–829.
[4] M.K. Ghosh and L. Debnath, Convergence of Ishikawa iterates of quasinonexpansive mappings, J. Math. Anal. Appl. (207) (1997), 96–103.
[5] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive
mappings, Proc. Amer. Math. Soc. (35) (1972), 171–174.
[6] J.U. Jeong and S.H. Kim, Weak and strong convergence of the Ishikawa iteration process with errors for two asymptotically nonexpansive mappings, Appl. Math.
Comput. (181 2) (2006), 1394–1401.
[7] J.K. Kim, K.H. Kim and K.S. Kim, Three-step iterative sequences with errors
for asymptotically quasi-nonexpansive mappings in convex metric spaces, Nonlinear
Anal. Convex Anal. RIMS Vol. (1365) (2004), 156–165.
[8] W.A. Kirk, Fixed point theorems for non-lipschitzian mappings of asymptotically
nonexpansive type, Israel J. Math. (17) (1974), 339–346.
[9] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J.
Math. Anal. Appl. (259) (2001), 1–7.
[10] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with
error member, J. Math. Anal. Appl. (259) (2001), 18–24.
[11] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with
error member of uniformly convex Banach spaces, J. Math. Anal. Appl. (266)
(2002), 468–471.
[12] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.
[13] W.V. Petryshyn and T.E. Williamson, Strong and weak convergence of the sequence
of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl.
(43) (1973), 459–497.
[14] G.S. Saluja, Convergence of fixed point of asymptotically quasi-nonexpansive type
mappings in convex metric spaces, J. Nonlinear Sci. Appl. (1 3) (2008), 132–144.
[15] W. Takahashi, A convexity in metric space and nonexpansive mappings I, Kodai
Math. Sem. Rep. (22) (1970), 142–149.
[16] K.K. Tan and H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. (122) (1994), 733–739.
[17] Y.-X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically
quasi-nonexpansive mappings, Compt. Math. Appl. (49 11-12) (2005), 1905–1912.
[18] Y.-X. Tian and Chun-de Yang, Convergence theorems of three-step iterative scheme
for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces,
Fixed Point Theory and Applications, Vol. 2009, Article ID 891965, 12 pages,
doi:10.1155/2009/891965.
[19] B.L. Xu and M.A. Noor, Fixed point iterations for asymptotically nonexpansive
mappings in Banach spaces, J. Math. Anal. Appl. (267 2) (2002), 444–453.
[20] Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly
accretive operator equations, J. Math. Anal. Appl. (224 1) (1998), 91–101.
[21] H. Zhou; J.I. Kang; S.M. Kang and Y.J. Cho, Convergence theorems for uniformly
quasi-lipschitzian mappings, Int. J. Math. Math. Sci. (15) (2004), 763–775.
G. S. Saluja
Department of Mathematics & Information Technology, Govt. Nagarjuna P.G. College
of Science, Raipur - 492010 (C.G.), India.
E-mail address: saluja 1963@rediffmail.com, saluja1963@gmail.com