Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 2 Issue 1(2010), Pages 1-11.
CONVERGENCE THEOREMS OF FIXED POINT FOR
GENERALIZED ASYMPTOTICALLY QUASI-NONEXPANSIVE
MAPPINGS IN BANACH SPACES
GURUCHARAN SINGH SALUJA, HEMANT KUMAR NASHINE
Abstract. In this paper, we study the convergence of common fixed point for
generalized asymptotically quasi-nonexpansive mappings in real Banach spaces
and give the necessary and sufficient condition for convergence of three-step
iterative sequence with errors for such maps. The results obtained in this
paper extend and improve some recent known results.
1. INTRODUCTION
It is well known that the concept of asymptotically nonexpansive mappings was
introduced by Goebel and Kirk [3] 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 [8] gave
necessary and sufficient conditions for Mann iterative sequence (cf. [6]) to converge
to fixed points of quasi-nonexpansive mappings. In 1997, Ghosh and Debnath [2]
extended the results of Petryshyn and Williamson [8] and gave necessary and sufficient conditions for Ishikawa iterative sequence to converge to fixed points for
quasi-nonexpansive mappings.
Qihou [10] extended results of [2, 8] 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. [16] introduced a new class of generalized asymptotically
nonexpansive mapping and gave a necessary and sufficient condition for the modified 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 [12], Zeng and
Yao [15] discussed a necessary and sufficient condition for common fixed points
of two nonexpansive mappings and a finite family of nonexpansive mappings, and
2000 Mathematics Subject Classification. 47H09, 47H10.
Key words and phrases. Generalized asymptotically quasi-nonexpansive mapping; three-step
iteration scheme with bounded errors; common fixed point; Banach space.
c
⃝2010
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted May, 2009. Published January, 2010.
1
2
G. S. SALUJA, H. K. NASHINE
proved some convergence theorems for approximating a common fixed point, respectively.
Recently, Lan [5] introduced a new class of generalized asymptotically quasinonexpansive 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.
More recently, Nantadilok [7] extend and improve the result of Lan [5] and gave
a necessary and sufficient condition for convergence of common fixed point for
three-step iterative sequence with errors {𝑥𝑛 } for generalized asymptotically quasinonexpansive mappings, which was defined as follows:
𝑥1 ∈ 𝐶;
𝑧𝑛
=
(1 − 𝑐𝑛 )𝑥𝑛 + 𝑐𝑛 𝑇3𝑛 𝑥𝑛 + 𝑣𝑛 ,
𝑦𝑛
=
(1 − 𝑏𝑛 )𝑥𝑛 + 𝑏𝑛 𝑇2𝑛 𝑧𝑛 + 𝑢𝑛 ,
𝑥𝑛+1
=
(1 − 𝑎𝑛 )𝑥𝑛 + 𝑎𝑛 𝑇1𝑛 𝑦𝑛 + 𝑤𝑛 ,
𝑛 ≥ 1,
(1)
where {𝑢𝑛 }, {𝑣𝑛 } and {𝑤𝑛 } are sequences in 𝐶 and {𝑎𝑛 }, {𝑏𝑛 }, {𝑐𝑛 } are sequences
in [0, 1] satisfying some conditions.
The aim of this paper is to obtain a convergence result of three-step iteration scheme with bounded errors for generalized asymptotically quasi-nonexpansive
mappings in real Banach spaces which extend and improve some recent known results.
Let 𝑋 be a normed space, 𝐶 be a nonempty closed convex subset of 𝑋, and
𝑇𝑖 : 𝐶 → 𝐶, (𝑖 = 1, 2, 3) be three generalized ∑
asymptotically quasi-nonexpansive
∑∞𝑠
∞𝑟
mappings with respect to {𝑟𝑖𝑛 } and {𝑠𝑖𝑛 } with 𝑛=1𝑛 < ∞ and 𝑛=1𝑛 < ∞ where
𝑟𝑛 = max{𝑟1𝑛 , 𝑟2𝑛 , 𝑟3𝑛 }, 𝑠𝑛 = max{𝑠1𝑛 , 𝑠2𝑛 , 𝑠3𝑛 }. Define a sequence {𝑥𝑛 } in 𝐶 as
follows:
𝑥1 ∈ 𝐶;
𝑧𝑛
=
(1 − 𝛾𝑛 − 𝜈𝑛 )𝑥𝑛 + 𝛾𝑛 𝑇3𝑛 𝑥𝑛 + 𝜈𝑛 𝑢𝑛 ,
𝑦𝑛
=
(1 − 𝛽𝑛 − 𝜇𝑛 )𝑥𝑛 + 𝛽𝑛 𝑇2𝑛 𝑧𝑛 + 𝜇𝑛 𝑣𝑛 ,
𝑥𝑛+1
=
(1 − 𝛼𝑛 − 𝜆𝑛 )𝑥𝑛 + 𝛼𝑛 𝑇1𝑛 𝑦𝑛 + 𝜆𝑛 𝑤𝑛 ,
𝑛 ≥ 1,
(2)
∑∞
∑∞
where
𝑛=1 𝜆𝑛 < +∞,
𝑛=1 𝜇𝑛 < +∞ and
∑∞ 0 < 𝑎 ≤ 𝛼𝑛 , 𝛽𝑛 , 𝛾𝑛 ≤ 𝑏 < 1,
𝑛=1 𝜈𝑛 < +∞, {𝑢𝑛 }, {𝑣𝑛 } and {𝑤𝑛 } are bounded sequences in 𝐶.
2. PRELIMINARIES
In the sequel, we need the following definitions and lemmas for our main results
in this paper.
CONVERGENCE THEOREMS OF FIXED POINT
3
Definition 2.1 (See [5]). Let 𝑋 be a real Banach space, 𝐶 be a nonempty subset of 𝑋 and 𝐹 (𝑇 ) denotes the set of fixed points of 𝑇 . A mapping 𝑇 : 𝐶 → 𝐶 is
said to be
(1) asymptotically nonexpansive if there exists a sequence {𝑟𝑛 } ⊂ [0, ∞) with
𝑟𝑛 → 0 as 𝑛 → ∞ such that
∥𝑇 𝑛 𝑥 − 𝑇 𝑛 𝑦∥ ≤ (1 + 𝑟𝑛 )∥𝑥 − 𝑦∥,
(3)
(2) asymptotically quasi-nonexpansive if (4) holds for all 𝑥 ∈ 𝐶 and 𝑦 ∈ 𝐹 (𝑇 );
(3) generalized quasi-nonexpansive with respect to {𝑠𝑛 }, if there exists a sequence {𝑠𝑛 } ⊂ [0, 1) with 𝑠𝑛 → 0 as 𝑛 → ∞ such that
∥𝑇 𝑛 𝑥 − 𝑝∥ ≤ ∥𝑥 − 𝑝∥ + 𝑠𝑛 ∥𝑥 − 𝑇 𝑛 𝑥∥,
(4)
for all 𝑥 ∈ 𝐶, 𝑝 ∈ 𝐹 (𝑇 ) and 𝑛 ≥ 1,
(4) generalized asymptotically quasi-nonexpansive with respect to {𝑟𝑛 } and
{𝑠𝑛 } ⊂ [0, 1) with 𝑟𝑛 → 0 and 𝑠𝑛 → 0 as 𝑛 → ∞ such that
∥𝑇 𝑛 𝑥 − 𝑝∥ ≤ (1 + 𝑟𝑛 )∥𝑥 − 𝑝∥ + 𝑠𝑛 ∥𝑥 − 𝑇 𝑛 𝑥∥,
(5)
for all 𝑥 ∈ 𝐶, 𝑝 ∈ 𝐹 (𝑇 ) and 𝑛 ≥ 1.
Remark 2.2. It is easy to see that,
(i) if 𝑠𝑛 = 0 for all 𝑛 ≥ 1, then the generalized asymptotically quasi-nonexpansive
mapping reduces to the usual asymptotically quasi-nonexpansive mapping.
(ii) if 𝑟𝑛 = 𝑠𝑛 = 0 for all 𝑛 ≥ 1, then the generalized asymptotically quasinonexpansive mapping reduces to the usual quasi-nonexpansive mapping.
(iii) if 𝑟𝑛 = 0 for all 𝑛 ≥ 1, then the generalized asymptotically quasi-nonexpansive
mapping reduces to the generalized quasi-nonexpansive mapping.
Lan [5] has shown that the generalized asymptotically quasi-nonexpansive mapping is not a generalized quasi-nonexpansive mapping.
We have the following example shows that a generalized asymptotically quasinonexpansive mapping is not a generalized quasi-nonexpansive mapping.
4
G. S. SALUJA, H. K. NASHINE
Example [11]. Let 𝑋 = ℓ∞ with the norm ∥.∥ defined by
∥𝑥∥ = sup ∣𝑥𝑖 ∣,
∀𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , . . . ) ∈ 𝑋,
𝑖∈𝑁
and 𝐶 = {𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , . . . ) ∈ 𝑋 : 𝑥𝑖 ≥ 0, 𝑥1 ≥ 𝑥𝑖 , ∀ 𝑖 ∈ 𝑁 𝑎𝑛𝑑 𝑥2 = 𝑥1 }.
Then 𝐶 is a nonempty subset of 𝑋.
Now, for any 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , . . . ) ∈ 𝐶, define a mapping 𝑇 : 𝐶 → 𝐶 as
follows
𝑇 (𝑥) = (0, 2𝑥1 , 0, . . . , 0, . . . ).
It is easy to see that 𝑇 is a generalized asymptotically quasi-nonexpansive mapping.
In fact, for any 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , . . . ) ∈ 𝐶, taking 𝑇 (𝑥) = 𝑥, that is,
(0, 2𝑥1 , 0, . . . , 0, . . . ) = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , . . . ).
Then we have 𝐹 (𝑇 ) = {0} and 𝑇 𝑛 (𝑥) = (0, 0, . . . , 0, . . . ), ∀𝑛 = 2, 3, . . . . for all
𝑟1 , 𝑠1 ∈ [0, 1) with 𝑟1 + 𝑠1 ≥ 1, we have
∥𝑇 (𝑥) − 𝑝∥ − (1 + 𝑟1 )∥𝑥 − 𝑝∥ − 𝑠1 ∥𝑥 − 𝑇 (𝑥)∥
= ∥(0, 2𝑥1 , 0, . . . , 0, . . . )∥−(1+𝑟1 )∥(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , . . . )∥−𝑠1 ∥(𝑥1 , 𝑥2 , . . . , 𝑥𝑛 , . . . )∥
= 2𝑥1 − (1 + 𝑟1 )𝑥1 − 𝑠1 𝑥1 ≤ 0.
And
∥𝑇 𝑛 (𝑥) − 𝑝∥ − (1 + 𝑟𝑛 )∥𝑥 − 𝑝∥ − 𝑠𝑛 ∥𝑥 − 𝑇 𝑛 (𝑥)∥
=
0 − (1 + 𝑟𝑛 )𝑥1 − 𝑠𝑛 𝑥1
≤
0.
For all 𝑛 = 2, 3, . . . , {𝑟𝑛 } and {𝑠𝑛 } ⊂ [0, 1) with 𝑟𝑛 → 0 and 𝑠𝑛 → 0 as 𝑛 → ∞,
and so 𝑇 is a generalized asymptotically quasi-nonexpansive mapping. However, 𝑇
is not a generalized quasi-nonexpansive mapping. Since
∥𝑇 (𝑥) − 𝑝∥ − ∥𝑥 − 𝑝∥ − 𝑠1 ∥𝑥 − 𝑇 (𝑥)∥
=
2𝑥1 − 𝑥1 − 𝑠1 𝑥1
> 0, ∀𝑠1 ∈ [0, 1).
Lemma 2.3 (See [13]). Let {𝑎𝑛 }, {𝑏𝑛 } and {𝛿𝑛 } be sequences of nonnegative
real numbers satisfying the inequality
𝑎𝑛+1 ≤ (1 + 𝛿𝑛 )𝑎𝑛 + 𝑏𝑛 ,
𝑛 ≥ 1.
∑∞
∑∞
If 𝑛=1 𝛿𝑛 < ∞ and 𝑛=1 𝑏𝑛 < ∞, then lim𝑛→∞ 𝑎𝑛 exists. In particular, if {𝑎𝑛 }
has a subsequence converging to zero, then lim𝑛→∞ 𝑎𝑛 = 0.
Lemma 2.4 (See [5]). Let 𝐶 be nonempty closed subset of a Banach space 𝑋
and 𝑇 : 𝐶 → 𝐶 be a generalized asymptotically quasi-nonexpansive mapping with
the fixed point set 𝐹 (𝑇 ) ∕= ∅. Then 𝐹 (𝑇 ) is closed subset in 𝐶.
CONVERGENCE THEOREMS OF FIXED POINT
5
3. MAIN RESULTS
In this section, we establish some strong convergence theorems of three-step iteration scheme with bounded errors for generalized asymptotically quasi-nonexpansive
mappings in a real Banach space.
Lemma 3.1. Let 𝑋 be a real arbitrary Banach space, 𝐶 be a nonempty
closed convex subset of 𝑋. Let 𝑇𝑖 : 𝐶 → 𝐶, (𝑖 = 1, 2, 3) be generalized asymptotically
quasi-nonexpansive mappings with respect to {𝑟𝑖𝑛 } and {𝑠𝑖𝑛 } such that
∑∞ 𝑟𝑛 +2𝑠
𝑛
= max{𝑠1𝑛 , 𝑠2𝑛 , 𝑠3𝑛 }. Let
𝑛=1 1−𝑠𝑛 < ∞ where 𝑟𝑛 = max{𝑟1𝑛 , 𝑟2𝑛 , 𝑟3𝑛 }, 𝑠𝑛 ∑
∑∞
∞
{𝑥𝑛 } be the
sequence defined by (2) with the restrictions 𝑛=1 𝜆𝑛 < +∞, 𝑛=1 𝜇𝑛 <
∑∞
+∞ and 𝑛=1 𝜈𝑛 < +∞. If ℱ = ∩3𝑖=1 𝐹 (𝑇𝑖 ) ∕= ∅. Then
(i) ∥𝑥𝑛+1 − 𝑝∥ ≤ (1 + ℎ𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝜃𝑛 ,
for all 𝑝 ∈ ℱ and 𝑛 ≥ 1, where ℎ𝑛 = 𝑘𝑛3 − 1, 𝑘𝑛 =
∑∞
∑∞
𝑀2 (𝜆𝑛 + 𝜇𝑛 + 𝜈𝑛 ) with 𝑛=1 ℎ𝑛 < ∞ and 𝑛=1 𝜃𝑛 < ∞.
1+𝑟𝑛 +𝑠𝑛
1−𝑠𝑛
and 𝜃𝑛 =
(ii) there exists a constant 𝑀 ′ > 0 such that
∥𝑥𝑛+𝑚 − 𝑝∥ ≤ 𝑀 ′ ∥𝑥𝑛 − 𝑝∥ + 𝑀 ′
𝑛+𝑚−1
∑
𝜃𝑘 ,
𝑘=𝑛
for all 𝑝 ∈ ℱ and 𝑛, 𝑚 ≥ 1.
(iii) lim𝑛→∞ ∥𝑥𝑛 − 𝑝∥ exists.
Proof. Let 𝑝 ∈ ℱ, then it follows from (5), we have
∥𝑥𝑛 − 𝑇3𝑛 𝑥𝑛 ∥
≤
∥𝑥𝑛 − 𝑝∥ + ∥𝑇3𝑛 𝑥𝑛 − 𝑝∥
≤
∥𝑥𝑛 − 𝑝∥ + (1 + 𝑟3𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝑠3𝑛 ∥𝑥𝑛 − 𝑇3𝑛 𝑥𝑛 ∥
≤ (2 + 𝑟3𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝑠3𝑛 ∥𝑥𝑛 − 𝑇3𝑛 𝑥𝑛 ∥
≤ (2 + 𝑟𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝑠𝑛 ∥𝑥𝑛 − 𝑇3𝑛 𝑥𝑛 ∥
which implies that
∥𝑥𝑛 − 𝑇3𝑛 𝑥𝑛 ∥ ≤
2 + 𝑟𝑛
∥𝑥𝑛 − 𝑝∥.
1 − 𝑠𝑛
(6)
Similarly, we have
∥𝑦𝑛 − 𝑇1𝑛 𝑦𝑛 ∥
≤
∥𝑦𝑛 − 𝑝∥ + ∥𝑇1𝑛 𝑦𝑛 − 𝑝∥
≤
∥𝑦𝑛 − 𝑝∥ + (1 + 𝑟1𝑛 )∥𝑦𝑛 − 𝑝∥ + 𝑠1𝑛 ∥𝑦𝑛 − 𝑇1𝑛 𝑦𝑛 ∥
≤ (2 + 𝑟1𝑛 )∥𝑦𝑛 − 𝑝∥ + 𝑠1𝑛 ∥𝑦𝑛 − 𝑇1𝑛 𝑦𝑛 ∥
≤
(2 + 𝑟𝑛 )∥𝑦𝑛 − 𝑝 + 𝑠𝑛 ∥𝑦𝑛 − 𝑇1𝑛 𝑦𝑛 ∥
which implies that
∥𝑦𝑛 − 𝑇1𝑛 𝑦𝑛 ∥ ≤
2 + 𝑟𝑛
∥𝑦𝑛 − 𝑝∥,
1 − 𝑠𝑛
(7)
∥𝑧𝑛 − 𝑇2𝑛 𝑧𝑛 ∥ ≤
2 + 𝑟𝑛
∥𝑧𝑛 − 𝑝∥.
1 − 𝑠𝑛
(8)
and also
6
G. S. SALUJA, H. K. NASHINE
Since {𝑢𝑛 }, {𝑣𝑛 } and {𝑤𝑛 } are bounded sequences in 𝐶, for any given 𝑝 ∈ ℱ, we
can set
𝑀 = max{sup ∥𝑢𝑛 − 𝑝∥, sup ∥𝑣𝑛 − 𝑝∥, sup ∥𝑤𝑛 − 𝑝∥}.
𝑛≥1
𝑛≥1
𝑛≥1
It follows from (2), (5) and (6) that
∥𝑧𝑛 − 𝑝∥
= ∥(1 − 𝛾𝑛 − 𝜈𝑛 )𝑥𝑛 + 𝛾𝑛 𝑇3𝑛 𝑥𝑛 + 𝜈𝑛 𝑢𝑛 − 𝑝∥
= ∥(1 − 𝛾𝑛 − 𝜈𝑛 )(𝑥𝑛 − 𝑝) + 𝛾𝑛 (𝑇3𝑛 𝑥𝑛 − 𝑝) + 𝜈𝑛 (𝑢𝑛 − 𝑝)∥
≤ (1 − 𝛾𝑛 − 𝜈𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛾𝑛 ∥𝑇3𝑛 𝑥𝑛 − 𝑝∥ + 𝜈𝑛 ∥𝑢𝑛 − 𝑝∥
≤ (1 − 𝛾𝑛 − 𝜈𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛾𝑛 [(1 + 𝑟3𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝑠3𝑛 ∥𝑥𝑛 − 𝑇3𝑛 𝑥𝑛 ∥]
+𝜈𝑛 𝑀
≤ (1 − 𝛾𝑛 − 𝜈𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛾𝑛 [(1 + 𝑟𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝑠𝑛 ∥𝑥𝑛 − 𝑇3𝑛 𝑥𝑛 ∥]
+𝜈𝑛 𝑀
≤ (1 − 𝛾𝑛 − 𝜈𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛾𝑛 [(1 + 𝑟𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝑠𝑛
2 + 𝑟𝑛
∥𝑥𝑛 − 𝑝∥]
1 − 𝑠𝑛
+𝜈𝑛 𝑀
≤ (1 − 𝛾𝑛 − 𝜈𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛾𝑛 (
≤
≤
≤
1 + 𝑟𝑛 + 𝑠𝑛
)∥𝑥𝑛 − 𝑝∥ + 𝜈𝑛 𝑀
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
− 1) − 𝜈𝑛 }∥𝑥𝑛 − 𝑝∥ + 𝜈𝑛 𝑀
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
{1 + 𝛾𝑛 (
− 1)}∥𝑥𝑛 − 𝑝∥ + 𝜈𝑛 𝑀
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
∥𝑥𝑛 − 𝑝∥ + 𝜈𝑛 𝑀.
(9)
1 − 𝑠𝑛
{1 + 𝛾𝑛 (
Again from (2), (7) and (9), we have
∥𝑦𝑛 − 𝑝∥
≤
(1 − 𝛽𝑛 − 𝜇𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 ∥𝑇2𝑛 𝑧𝑛 − 𝑝∥ + 𝜇𝑛 ∥𝑣𝑛 − 𝑝∥
≤
(1 − 𝛽𝑛 − 𝜇𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 [(1 + 𝑟2𝑛 )∥𝑧𝑛 − 𝑝∥ + 𝑠2𝑛 ∥𝑧𝑛 − 𝑇2𝑛 𝑧𝑛 ∥] + 𝜇𝑛 𝑀
≤
(1 − 𝛽𝑛 − 𝜇𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 [(1 + 𝑟𝑛 )∥𝑧𝑛 − 𝑝∥ + 𝑠𝑛 ∥𝑧𝑛 − 𝑇2𝑛 𝑧𝑛 ∥] + 𝜇𝑛 𝑀
2 + 𝑟𝑛
(1 − 𝛽𝑛 − 𝜇𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 [(1 + 𝑟𝑛 )∥𝑧𝑛 − 𝑝∥ + 𝑠𝑛
∥𝑧𝑛 − 𝑝∥] + 𝜇𝑛 𝑀
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
(1 − 𝛽𝑛 − 𝜇𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 (
)∥𝑧𝑛 − 𝑝∥ + 𝜇𝑛 𝑀
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛 1 + 𝑟𝑛 + 𝑠𝑛
(1 − 𝛽𝑛 − 𝜇𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 (
){
∥𝑥𝑛 − 𝑝∥ + 𝜈𝑛 𝑀 }
1 − 𝑠𝑛
1 − 𝑠𝑛
+𝜇𝑛 𝑀
≤
≤
≤
CONVERGENCE THEOREMS OF FIXED POINT
≤ (1 − 𝛽𝑛 − 𝜇𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 (
7
1 + 𝑟𝑛 + 𝑠𝑛 2
) ∥𝑥𝑛 − 𝑝∥
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
)𝜈𝑛 𝑀 + 𝜇𝑛 𝑀
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛 2
1 + 𝑟𝑛 + 𝑠𝑛
[1 + 𝛽𝑛 {(
) − 1} − 𝜇𝑛 ]∥𝑥𝑛 − 𝑝∥ + 𝛽𝑛 (
)𝜈𝑛 𝑀
1 − 𝑠𝑛
1 − 𝑠𝑛
+𝜇𝑛 𝑀
1 + 𝑟𝑛 + 𝑠𝑛 2
1 + 𝑟𝑛 + 𝑠𝑛
[1 + 𝛽𝑛 {(
) − 1}]∥𝑥𝑛 − 𝑝∥ + (
)𝜈𝑛 𝑀
1 − 𝑠𝑛
1 − 𝑠𝑛
+𝜇𝑛 𝑀
1 + 𝑟𝑛 + 𝑠𝑛 2
1 + 𝑟𝑛 + 𝑠𝑛
(
) ∥𝑥𝑛 − 𝑝∥ + (𝜇𝑛 + 𝜈𝑛 )𝑀 (
)
1 − 𝑠𝑛
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛 2
) ∥𝑥𝑛 − 𝑝∥ + (𝜇𝑛 + 𝜈𝑛 )𝑀1 ,
(10)
(
1 − 𝑠𝑛
+𝛽𝑛 (
≤
≤
≤
≤
𝑛 +𝑠𝑛
where 𝑀1 = sup𝑛≥1 { 1+𝑟
1−𝑠𝑛 }𝑀 and from (2), (8) and (10), we have
∥𝑥𝑛+1 − 𝑝∥
≤ (1 − 𝛼𝑛 − 𝜆𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛼𝑛 ∥𝑇1𝑛 𝑦𝑛 − 𝑝∥ + 𝜆𝑛 ∥𝑤𝑛 − 𝑝∥
≤ (1 − 𝛼𝑛 − 𝜆𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛼𝑛 [(1 + 𝑟1𝑛 )∥𝑦𝑛 − 𝑝∥ + 𝑠1𝑛 ∥𝑦𝑛 − 𝑇1𝑛 𝑦𝑛 ∥ + 𝜆𝑛 𝑀
≤ (1 − 𝛼𝑛 − 𝜆𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛼𝑛 [(1 + 𝑟𝑛 )∥𝑦𝑛 − 𝑝∥ + 𝑠𝑛 ∥𝑦𝑛 − 𝑇1𝑛 𝑦𝑛 ∥ + 𝜆𝑛 𝑀
2 + 𝑟𝑛
∥𝑦𝑛 − 𝑝∥] + 𝜆𝑛 𝑀
≤ (1 − 𝛼𝑛 − 𝜆𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛼𝑛 [(1 + 𝑟𝑛 )∥𝑦𝑛 − 𝑝∥ + 𝑠𝑛
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
≤ (1 − 𝛼𝑛 − 𝜆𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛼𝑛 (
)∥𝑦𝑛 − 𝑝∥ + 𝜆𝑛 𝑀
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛 2
≤ (1 − 𝛼𝑛 − 𝜆𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝛼𝑛 (
){(
) ∥𝑥𝑛 − 𝑝∥
1 − 𝑠𝑛
1 − 𝑠𝑛
+(𝜇𝑛 + 𝜈𝑛 )𝑀1 } + 𝜆𝑛 𝑀
1 + 𝑟𝑛 + 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛 3
) − 1} − 𝜆𝑛 ]∥𝑥𝑛 − 𝑝∥ + 𝛼𝑛 (
)(𝜇𝑛 + 𝜈𝑛 )𝑀1
≤ [1 + 𝛼𝑛 {(
1 − 𝑠𝑛
1 − 𝑠𝑛
+𝜆𝑛 𝑀
1 + 𝑟𝑛 + 𝑠𝑛 3
1 + 𝑟𝑛 + 𝑠𝑛
≤ [1 + 𝛼𝑛 {(
) − 1}]∥𝑥𝑛 − 𝑝∥ + (
)(𝜇𝑛 + 𝜈𝑛 )𝑀1
1 − 𝑠𝑛
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛
+(
)𝜆𝑛 𝑀1
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛 3
1 + 𝑟𝑛 + 𝑠𝑛
≤ (
) ∥𝑥𝑛 − 𝑝∥ + (𝜆𝑛 + 𝜇𝑛 + 𝜈𝑛 )(
)𝑀1
1 − 𝑠𝑛
1 − 𝑠𝑛
1 + 𝑟𝑛 + 𝑠𝑛 3
≤ (
) ∥𝑥𝑛 − 𝑝∥ + (𝜆𝑛 + 𝜇𝑛 + 𝜈𝑛 )𝑀2
1 − 𝑠𝑛
= (1 + ℎ𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝜃𝑛 ,
(11)
3
𝑛 +𝑠𝑛
where 𝑀2 = sup𝑛≥1 { 1+𝑟
1−𝑠𝑛 }𝑀1 , ℎ𝑛 = 𝑘𝑛 − 1, 𝑘𝑛 =
𝜈𝑛 )𝑀2 . This completes the prove of part (i).
1+𝑟𝑛 +𝑠𝑛
1−𝑠𝑛
and 𝜃𝑛 = (𝜆𝑛 + 𝜇𝑛 +
(ii) If 𝑥 ≥ 0 then 1+𝑥 ≤ 𝑒𝑥 . Thus, from part (i) for any positive integer 𝑚, 𝑛 ≥ 1,
we have
8
G. S. SALUJA, H. K. NASHINE
∥𝑥𝑛+𝑚 − 𝑝∥
≤ (1 + ℎ𝑛+𝑚−1 )∥𝑥𝑛+𝑚−1 − 𝑝∥ + 𝜃𝑛+𝑚−1
≤ 𝑒ℎ𝑛+𝑚−1 ∥𝑥𝑛+𝑚−1 − 𝑝∥ + 𝜃𝑛+𝑚−1
≤ 𝑒ℎ𝑛+𝑚−1 {𝑒ℎ𝑛+𝑚−2 ∥𝑥𝑛+𝑚−2 − 𝑝∥ + 𝜃𝑛+𝑚−2 } + 𝜃𝑛+𝑚−1
≤ 𝑒(ℎ𝑛+𝑚−1 +ℎ𝑛+𝑚−2 ) ∥𝑥𝑛+𝑚−2 − 𝑝∥ + 𝑒ℎ𝑛+𝑚−1 𝜃𝑛+𝑚−2 + 𝜃𝑛+𝑚−1
≤ 𝑒(ℎ𝑛+𝑚−1 +ℎ𝑛+𝑚−2 ) ∥𝑥𝑛+𝑚−2 − 𝑝∥ + 𝑒ℎ𝑛+𝑚−1 (𝜃𝑛+𝑚−1 + 𝜃𝑛+𝑚−2 )
≤ ...
≤ ...
≤ 𝑒(ℎ𝑛+𝑚−1 +ℎ𝑛+𝑚−2 +⋅⋅⋅+ℎ𝑛 ) ∥𝑥𝑛 − 𝑝∥
+𝑒(ℎ𝑛+𝑚−1 +ℎ𝑛+𝑚−2 +⋅⋅⋅+ℎ𝑛 ) (𝜃𝑛+𝑚−1 + 𝜃𝑛+𝑚−2 + ⋅ ⋅ ⋅ + 𝜃𝑛 )
∑𝑛+𝑚−1
≤ 𝑒(
𝑘=𝑛
ℎ𝑘 )
∥𝑥𝑛 − 𝑝∥ + 𝑒(
∑𝑛+𝑚−1
𝑘=𝑛
ℎ𝑘 )
𝑛+𝑚−1
∑
𝜃𝑘 .
𝑘=𝑛
∑𝑛+𝑚−1
Setting 𝑀 ′ = 𝑒(
𝑘=𝑛
ℎ𝑘 )
. Hence the above inequality reduces to
∥𝑥𝑛+𝑚 − 𝑝∥ ≤ 𝑀 ′ ∥𝑥𝑛 − 𝑝∥ + 𝑀 ′
𝑛+𝑚−1
∑
𝜃𝑘 .
𝑘=𝑛
This completes the proof of part (ii).
(iii) From (i), we have
∥𝑥𝑛+1 − 𝑝∥ ≤ (1 + ℎ𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝜃𝑛 .
3
𝑛 +𝑠𝑛
𝑛 +𝑠𝑛
= 1+𝑟
and 𝜃𝑛 = (𝜆𝑛 + 𝜇𝑛 +
where 𝑀2 = sup𝑛≥1 { 1+𝑟
1−𝑠𝑛 }𝑀1 , ℎ𝑛 = 𝑘𝑛 − 1, 𝑘𝑛∑
1−𝑠𝑛
∞ 𝑟𝑛 +2𝑠𝑛
𝑟𝑛 +2𝑠𝑛
𝜈𝑛 )𝑀2 . Since 𝑘𝑛 − 1 = 1−𝑠𝑛 , the assumption 𝑛=1 1−𝑠𝑛 < ∞ implies that
∑∞
∑∞
lim𝑛→∞ 𝑘𝑛 = 1. By assumptions, it follows that 𝑛=1 ℎ𝑛 < ∞ and 𝑛=1 𝜃𝑛 < ∞.
It follows from Lemma 2.3 that the limit lim𝑛→∞ ∥𝑥𝑛 − 𝑝∥ exists. This completes
the proof of part (iii).
Theorem 3.2. Let 𝑋 be a real arbitrary Banach space, 𝐶 be a nonempty
closed convex subset of 𝑋. Let 𝑇𝑖 : 𝐶 → 𝐶, (𝑖 = 1, 2, 3) be generalized asymptotically
quasi-nonexpansive mappings with respect to {𝑟𝑖𝑛 } and {𝑠𝑖𝑛 } such that
∑∞ 𝑟𝑛 +2𝑠
𝑛
𝑛=1 1−𝑠𝑛 < ∞ where 𝑟𝑛 = max{𝑟1𝑛 , 𝑟2𝑛 , 𝑟3𝑛 }, 𝑠𝑛 = max{𝑠1𝑛 , 𝑠2𝑛 , 𝑠3𝑛 }. Let
{𝑥𝑛 } be the sequence defined by (2) and some 𝑎, 𝑏 ∈ (0, 1) with the following restrictions:
(i) 0 < 𝑎 ≤ 𝛼𝑛 , 𝛽𝑛 , 𝛾𝑛 ≤ 𝑏 < 1;
(ii)
∑∞
𝑛=1
𝜆𝑛 < +∞,
∑∞
𝑛=1
𝜇𝑛 < +∞,
∑∞
𝑛=1
𝜈𝑛 < +∞.
If ℱ = ∩3𝑖=1 𝐹 (𝑇𝑖 ) ∕= ∅. Then the iterative sequence {𝑥𝑛 } converges strongly to
a common fixed point 𝑝 of 𝑇1 , 𝑇2 and 𝑇3 if and only if
lim inf 𝑑(𝑥𝑛 , ℱ) = 0,
𝑛→∞
CONVERGENCE THEOREMS OF FIXED POINT
9
where 𝑑(𝑥, ℱ) denotes the distance between 𝑥 and the set ℱ.
Proof. The necessity is obvious and it is omitted. Now we prove the sufficiency.
From Lemma 3.1 (i) we have
∥𝑥𝑛+1 − 𝑝∥ ≤ (1 + ℎ𝑛 )∥𝑥𝑛 − 𝑝∥ + 𝜃𝑛 ,
𝑛 ≥ 1.
Therefore
𝑑(𝑥𝑛+1 , ℱ) ≤ (1 + ℎ𝑛 )𝑑(𝑥𝑛 , ℱ) + 𝜃𝑛 ,
+2𝑠𝑛
𝑛 +𝑠𝑛
where ℎ𝑛 = 𝑘𝑛3 −1, 𝑘𝑛 = 1+𝑟
and 𝜃𝑛 = (𝜆𝑛 +𝜇𝑛 +𝜈𝑛 )𝑀2 . Since 𝑘𝑛 −1 = 𝑟𝑛1−𝑠
,
1−𝑠𝑛
𝑛
∑∞ 𝑟𝑛 +2𝑠
𝑛
the assumption 𝑛=1 1−𝑠𝑛 < ∞ implies that lim𝑛→∞ 𝑘𝑛 = 1. By assumptions,
∑∞
∑∞
it follows that 𝑛=1 ℎ𝑛 < ∞ and 𝑛=1 𝜃𝑛 < ∞. By Lemma 2.3 and lim inf 𝑛→∞
𝑑(𝑥𝑛 , ℱ) = 0, we get that lim𝑛→∞ 𝑑(𝑥𝑛 , ℱ) = 0.
Next, we prove that {𝑥𝑛 } is a Cauchy sequence. From Lemma 3.1 (ii), we have
∥𝑥𝑛+𝑚 − 𝑝∥ ≤ 𝑀 ′ ∥𝑥𝑛 − 𝑝∥ + 𝑀 ′
𝑛+𝑚−1
∑
𝜃𝑘 ,
(12)
𝑘=𝑛
for all 𝑝 ∈ ℱ and 𝑛, 𝑚 ≥ 1. Since lim𝑛→∞ 𝑑(𝑥𝑛 , ℱ) = 0 for each 𝜀 > 0, there exists
a natural number 𝑛1 such that for all 𝑛 ≥ 𝑛1 ,
𝑑(𝑥𝑛 , ℱ) <
𝜀
,
8𝑀 ′
∞
∑
𝜃𝑛 <
𝑛=𝑛1
𝜀
.
2𝑀 ′
(13)
By the first inequality of (13), we know that there exists 𝑝1 ∈ ℱ such that
∥𝑥𝑛1 − 𝑝1 ∥ <
𝜀
.
4𝑀 ′
(14)
From (12), (13) and (14), for all 𝑛 ≥ 𝑛1 , we have
∥𝑥𝑛+𝑚 − 𝑥𝑛 ∥
≤
∥𝑥𝑛+𝑚 − 𝑝1 ∥ + ∥𝑥𝑛 − 𝑝1 ∥
≤
[𝑀 ′ ∥𝑥𝑛1 − 𝑝1 ∥ + 𝑀 ′
𝑛1 +𝑚−1
∑
𝜃𝑘 ] + 𝑀 ′ ∥𝑥𝑛1 − 𝑝1 ∥
𝑘=𝑛1
=
2𝑀 ′ ∥𝑥𝑛1 − 𝑝1 ∥ + 𝑀 ′
𝑛1 +𝑚−1
∑
𝜃𝑘
𝑘=𝑛1
𝜀
𝜀
+ 𝑀 ′.
= 𝜀,
4𝑀 ′
2𝑀 ′
which shows that {𝑥𝑛 } is a Cauchy sequence in 𝑋. Thus the completeness of 𝑋
implies that {𝑥𝑛 } must be convergent. Let lim𝑛→∞ 𝑥𝑛 = 𝑝, that is, {𝑥𝑛 } converges
to 𝑝. Then 𝑝 ∈ 𝐶, because 𝐶 is a closed subset of 𝑋. By Lemma 2.4 we know that
<
2𝑀 ′ .
10
G. S. SALUJA, H. K. NASHINE
the set ℱ is closed. From the continuity of 𝑑(𝑥𝑛 , ℱ) with
𝑑(𝑥𝑛 , ℱ) → 0
𝑎𝑛𝑑
𝑥𝑛 → 𝑝
𝑎𝑠
𝑛 → ∞,
we get
𝑑(𝑝, ℱ) = 0
and so 𝑝 ∈ ℱ = ∩3𝑖=1 𝐹 (𝑇𝑖 ), that is, 𝑝 is a common fixed point of 𝑇1 , 𝑇2 and 𝑇3 .
This completes the proof.
In Theorem 3.2, if 𝑇1 = 𝑇2 = 𝑇3 = 𝑇 , we obtain the following result:
Theorem 3.3. Let 𝑋 be a real arbitrary Banach space, 𝐶 be a nonempty
closed convex subset of 𝑋. Let 𝑇 : 𝐶 → 𝐶 be generalized asymptotically
quasi∑∞
+2𝑠𝑛
nonexpansive mapping with respect to {𝑟𝑛 } and {𝑠𝑛 } such that 𝑛=1 𝑟𝑛1−𝑠
< ∞.
𝑛
Let {𝑥𝑛 } be the sequence defined as:
𝑥1 ∈ 𝐶;
𝑧𝑛
=
(1 − 𝛾𝑛 − 𝜈𝑛 )𝑥𝑛 + 𝛾𝑛 𝑇 𝑛 𝑥𝑛 + 𝜈𝑛 𝑢𝑛 ,
𝑦𝑛
=
(1 − 𝛽𝑛 − 𝜇𝑛 )𝑥𝑛 + 𝛽𝑛 𝑇 𝑛 𝑧𝑛 + 𝜇𝑛 𝑣𝑛 ,
𝑥𝑛+1
=
(1 − 𝛼𝑛 − 𝜆𝑛 )𝑥𝑛 + 𝛼𝑛 𝑇 𝑛 𝑦𝑛 + 𝜆𝑛 𝑤𝑛 ,
𝑛 ≥ 1,
where {𝑢𝑛 }, {𝑣𝑛 } and {𝑤𝑛 } are bounded sequences in 𝐶 and some 𝑎, 𝑏 ∈ (0, 1) with
the following restrictions:
(i) 0 < 𝑎 ≤ 𝛼𝑛 , 𝛽𝑛 , 𝛾𝑛 ≤ 𝑏 < 1;
(ii)
∑∞
𝑛=1
𝜆𝑛 < +∞,
∑∞
𝑛=1
𝜇𝑛 < +∞,
∑∞
𝑛=1
𝜈𝑛 < +∞.
If 𝐹 (𝑇 ) ∕= ∅. Then the iterative sequence {𝑥𝑛 } converges strongly to a fixed
point 𝑝 of 𝑇 if and only if
lim inf 𝑑(𝑥𝑛 , 𝐹 (𝑇 )) = 0.
𝑛→∞
Remark 3.4. Our results extend and improve some recent corresponding results
announced by Lan [5] and Nantadilok [7].
Acknowledgments. The authors would like to thank the referee for his comments
that helped us improve this article.
References
[1] S. Atsushiba, Strong convergence of iterative sequences for asymptotically nonexpansive mappings in Banach spaces, Sci. Math. Japan. 57 (2003), 2, 377-388.
[2] M. K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings,
J. Math. Anal. Appl. 207 (1997), 96-103.
[3] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings,
Proc. Amer. Math. Soc. 35 (1972), 171-174.
CONVERGENCE THEOREMS OF FIXED POINT
11
[4] S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44 (1974),
147-150.
[5] H. Y. Lan, Common fixed point iterative processes with errors for generalized asymptotically
quasi-noexpansive mappings, Comput. Math. Appl. 52 (2006), 1403-1412.
[6] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
[7] J. Nantadilok, Three-step iteration scheme with errors for generalized asymptotically quasinonexpansive mappings, Thai J. Math. 6 (2008), 2, 297-308.
[8] W. V. Petryshyn, 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.
[9] L. Qihou, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal.
Appl. 259 (2001), 1-7.
[10] L. Qihou, Iterative sequences for asymptotically quasi-nonexpansive mappings with error
member, J. Math. Anal. Appl. 259 (2001), 18-24.
[11] S. Suantai, W. Cholamjiak, Approximating common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings, Thai J. Math. 6 (2008), 2, 317-331.
[12] T. Suzuki, Common fixed points of two nonexpansive mappings in Banach spaces, Bull.
Austral. Math. Soc. 69 (2004), 1-18.
[13] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa
iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.
[14] K.K. Tan, H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994), 733-739.
[15] L. C. Zeng, J. C. Yao, Implicit iteration scheme with perturbed mapping for common fixed
points of a finite family of nonexpansive mappings, Nonlinear Anal. Series A: Theory and
Applications 64 (2006), 1, 2507-2515.
[16] H. Y. Zhou, Y. J. Cho and M. Grabiec, Iterative processes for generalized asymptotically
nonexpansive mappings in Banach spaces, Pan Amer. Math. J. 13 (2003), 99-107.
Gurucharan Singh Saluja
Department of Mathematics & Information Technology, Govt. Nagarjun P.G. College
of Science, Raipur (Chhattisgarh), India
E-mail address: saluja 1963@rediffmail.com
Hemant Kumar Nashine
Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha - Chandrakhuri Marg (Baloda Bazar Road), Mandir
Hasaud, Raipur - 492101(Chhattisgarh), India
E-mail address: hemantnashine@gmail.com, hnashine@rediffmail.com