Volume 2008, Article ID 324575, 12 pagesdoi:10.1155/2008/324575 Research Article Implicit Iteration Process for Common Fixed Points of Strictly Asymptotically Pseudocontractive Mappings
Trang 1Volume 2008, Article ID 324575, 12 pages
doi:10.1155/2008/324575
Research Article
Implicit Iteration Process for Common Fixed
Points of Strictly Asymptotically Pseudocontractive Mappings in Banach Spaces
You Xian Tian, 1 Shih-sen Chang, 2 Jialin Huang, 2
Xiongrui Wang, 2 and J K Kim 3
1 College of Mathematics and Physics, Chongqing University of Post Telecommunications,
Chongqing 400065, China
2 Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
3 Department of Mathematics, Kyungnam University, Masan 631-701, South Korea
Correspondence should be addressed to You Xian Tian,tianyx@cqupt.edu.cn
Received 25 May 2008; Accepted 3 September 2008
Recommended by Nanjing Huang
In this paper, a new implicit iteration process with errors for finite families of strictly asymptot-ically pseudocontractive mappings and nonexpansive mappings is introduced By using the iterative process, some strong convergence theorems to approximating a common fixed point of strictly asymptotically pseudocontractive mappings and nonexpansive mappings are proved The results presented in the paper are new which extend and improve some recent results of Osilike
et al.2007, Liu 1996, Osilike 2004, Su and Li 2006, Gu 2007, Xu and Ori 2001
Copyrightq 2008 You Xian Tian et al 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 and preliminaries
Throughout this paper, we assume that E is a real Banach space, C is a nonempty closed convex subset of E, E∗is the dual space of E, and J : E → E∗is the normalized duality mapping defined by
Jx
f ∈ E∗:x, f ||x||2 ||f||2
Recall that a set C ⊂ E is said to be closed, convex, and pointed cone if it is a closed set
and satisfies the following conditions:1 C C ⊂ C; 2 λC ⊂ C for each λ ≥ 0; 3 if x ∈ C with x / 0, then −x /∈ C.
Trang 22 Fixed Point Theory and Applications
Definition 1.1 Let T : C → C be a mapping:
1 T is said to be λ, {k n }-strictly asymptotically pseudocontractive if there exist a constant λ ∈ 0, 1 and a sequence {k n } ⊂ 1, ∞ with k n→ 1 such that for all
x, y ∈ C, and for all jx − y ∈ Jx − y,
T n x − T n y, jx − y
≤ k n ||x − y||2− λx − T n x −
y − T n y2 ∀n ≥ 1, 1.2
2 T is said to be λ-strictly pseudocontractive in the terminology of Browder-Petryshyn 1
if there exist a constant λ ∈ 0, 1 such that for all x, y ∈ C,
Tx − Ty, jx − y
≤ ||x − y||2− λx − Tx − y − Ty2 ∀jx − y ∈ Jx − y, 1.3
3 T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that
T n x − T n y ≤ L||x − y|| ∀n ≥ 1. 1.4
The class of λ, {k n}-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Liu2 In the case of Hilbert spaces, it is shown by 2 that
1.2 is equivalent to the inequality
T n x − T n y2≤ k n ||x − y||2 λI − T n
x −
I − T n
y2
Concerning the convergence problem of iterative sequences for strictly pseudocontrac-tive mappings has been studied by several authorssee, e.g., 1,3 7 Concerning the class
of strictly asymptotically pseudocontractive mappings, Liu2 and Osilike et al 8 proved the following results
Theorem 1.2 Liu 2 Let H be a real Hilbert space, let C be a nonempty closed convex and bounded subset of H, and let T : C → C be a completely continuous uniformly L-Lipschitzian λ, {k n }-strictly asymptotically pseudocontractive mapping such that ∞
n1 k2
n − 1 < ∞ Let {α n } ⊂ 0, 1 be a sequence satisfying the following condition:
0 < ≤ α n ≤ 1 − λ − ∀n ≥ 1 and some > 0. 1.6
Then, the sequence {x n } generated from an arbitrary x1 ∈ C by
x n11− α n
converges strongly to a fixed point of T.
In 2007, Oslike et al.8 proved the following theorem
Trang 3Theorem 1.3 Oslike et al 8 Let E be a real q-uniformly smooth Banach space which is also uniformly convex, let C be a nonempty closed convex subset of E, let T : C → C be a λ, {k n }-strictly asymptotically pseudocontractive mapping such that ∞
n1 k n − 1 < ∞, and let FT / ∅ Let {α n } ⊂ 0, 1 be a real sequence satisfying the following condition:
0 < a ≤ α q−1 n ≤ b < q1 − k
2c q 1 L −q−2 ∀n ≥ 1. 1.8
Let {x n } be the sequence defined by 1.7 Then,
1 limn → ∞ ||x n − p|| exists ∀ p ∈ FT,
2 limn → ∞ ||x n − Tx n || 0,
3 {x n } converges weakly to a fixed point of T.
It is our purpose in this paper to introduce the following new implicit iterative process with errors for a finite family of strictly asymptotically pseudocontractive mappings{T i} and
a finite family of nonexpansive mappings{S i}:
x1∈ C,
x n α n S n x n−11− α n
T n x n u n ∀n ≥ 1, 1.9
where C is a closed convex cone of E, S n S nmod N , T n T n
nmod N, and{u n} is a bounded
sequence in C Also, we aim to prove some strong convergence theorems to approximating
a common fixed point of{S i } and {T i} The results presented in the paper are new which extend and improve some recent results of2 8
In order to prove our main results, we need the following lemmas
Lemma 1.4 see 9 Let E be a real Banach space, let C be a nonempty subset of E, and let T : C → C
be a λ, {k n }-strictly asymptotically pseudocontractive mapping, then T is uniformly L-Lipschitzian.
Lemma 1.5 Let E be a real Banach space, let C be a nonempty closed convex subset of E, and let
T i : C → C be a λ i , {k n i }-strictly asymptotically pseudocontractive mapping, i 1, 2, , N, then there exist a constant λ ∈ 0, 1, a constant L > 0, and a sequence {k n } ⊂ 1, ∞ with lim n → ∞ k n 1
such that for any x, y ∈ C and for each i 1, 2, , N and each n ≥ 1, the following hold:
T i n x − T i n y, jx − y
≤ k n ||x − y||2− λx − T n
i x −
y − T i n y2 1.10
for each jx − y ∈ Jx − y and
T n
Proof Since for each i 1, 2, , N, T iisλ i , {k i n }-strictly asymptotically pseudocontractive,
where λ i ∈ 0, 1 and {k i n } ⊂ 1, ∞ with lim n → ∞ k n i 1 ByLemma 1.4, T i is L i-Lipschitzian
Trang 44 Fixed Point Theory and Applications
Taking k n max{k i n , i 1, 2, , N} and λ min{λ i , i 1, 2, , N}, hence, for each i
1, 2, , N, we have
T n
i x − T n
i y, jx − y
≤ k n i ||x − y||2− λ ix − T n
i x −
y − T n
i y2
≤ k n ||x − y||2− λx − T n
i x −
y − T n
i y2
The conclusion1.10 is proved Again, taking L max{L i : i 1, 2, N} for any x, y ∈ C,
we have
T n
i x − T i n y ≤ L i ||x − y|| ≤ L||x − y|| ∀n ≥ 1. 1.13
This completes the proof ofLemma 1.5
Lemma 1.6 see 9 Let {a n }, {b n }, and {c n } be three nonnegative real sequences satisfying the following condition:
a n1≤1 b n
where n0 is some nonnegative integer such that ∞
n1 b n < ∞ and ∞
n1 c n < ∞, then lim n → ∞ a n
exists.
In addition, if there exists a subsequence {a n i } ⊂ {a n } such that a n i → 0, then a n→
0n → ∞.
2 Main results
We are now in a position to prove our main results in this paper
Theorem 2.1 Let E be a real Banach space, let C be a nonempty closed pointed convex cone of E, let
T i : C → C, i 1, 2, , N, be a finite family of λ i , {k i n }-strictly asymptotically pseudocontractive mappings, and let S i : C → C, i 1, 2, , N, be a finite family of nonexpansive mappings with
F
N i1
F
S i N i1
F
T i
/
(the set of common fixed points of {S i } and {T i }) Let {α n } be a sequence in 0, 1, let {u n } be a bounded sequence in C, let λ min{λ i : i 1, 2, , N}, k n max{k n i , i 1, 2, , N}, and let
L max{L i : i 1, 2, , N} > 0 be positive numbers defined by 1.10 and 1.11, respectively If the following conditions are satisfied:
i 0 < max{λ, 1 − 1/L} < lim inf n → ∞ α n ≤ α n < 1,
ii ∞n1 1 − α n ∞,
iii ∞
n1 k n − 1 < ∞ and 1 ≤ k n < 1 − λ/1 − lim inf n → ∞ α n ,
iv ∞
n1 ||u n || < ∞,
Trang 5then the iterative sequence {x n } with errors defined by 1.9 has the following properties:
1 limn → ∞ ||x n − p|| exists for each p ∈ F,
2 limn → ∞ dx n , F exists,
3 lim infn → ∞ ||x n − T n x n || 0,
4 the sequence {x n } converges strongly to a common fixed point p ∈ F if and only if
lim inf
n → ∞ d
x n , F
Proof We divide the proof ofTheorem 2.1into four steps
I First, we prove that the mapping G n : C → C, n 1, 2, , defined by
G n x α n S n x n−11− α n
is a Banach contractive mapping
Indeed, it follows from conditioni that 1 − 1/L < α n, that is,1 − α n L < 1 Hence,
G n x − G n y α n S n x n−11− α n
T n n x u n−α n S n x n−11− α n
T n n y u n
1 − α nT n
n x − T n n y
≤ 1 − α n L||x − y||, n 1, 2, ,
2.4
that is, for each n 1, 2, , G n : C → C is a Banach contraction mapping Therefore, there exists a unique fixed point x n ∈ C such that x n Gx n This shows that the sequence {x n} defined by1.9 is well defined
II The proof of conclusions 1 and 2
For any given p ∈ F and for any jx n − p ∈ Jx n − y fromLemma 1.5, we have
x n − p2α n
S n x n−1 − p1− α n
T n n x n − p u n2
α n
S n x n−1 − p, jx n − p1− α n
T n n x n − p, jx n − pu n , j
x n − p
≤ α nx n−1 − px n − p 1 − α n
k nx n − p2− λx n − T n
n x n2
u nx n − p.
2.5 Simplifying it, we have
x n − p ≤ α n
1−1− α n
k n
x n−1 − p u n
1−1− α n
k n
−
1− α n
λ
1−1− α n
k n
·x n − T n x n2
x n − p
2.6
By virtue of conditionsi and iii, we have
1− lim infn → ∞ α n ≤ 1− λ
Trang 66 Fixed Point Theory and Applications and so
0 < λ ≤ 1 −
1− α n
It follows from2.6 and 2.8 that
x n − p ≤ α n
1−1− α n
k n
x n−1 − p u n
λ −1− α n
λ ·x n − T n x n2
x n − p
1
1− α n
k n− 1
1−1− α n
k n
x n−1 − p u n
λ −1− α n
λ ·x n − T n x n2
x n − p
2.9
Letting b n 1 − α n k n − 1/1 − 1 − α n k n and c n u n /λ, then we have
x n − p ≤ 1 b nx n−1 − p c n ∀n ≥ 1. 2.10
By using2.8,
b n≤
1− α n
k n− 1
k n− 1
By conditions iii and iv, ∞
n1 b n < ∞ and ∞
n1 c n < ∞ By virtue of Lemma 1.6, limn → ∞ ||x n − p|| exists; and so {x n } is a bounded sequence in C Denote
M sup
n≥1
From2.10, we have
d
x n , F
≤1 b n
d
x n−1 , F
By usingLemma 1.6again, we know that limn → ∞ dx n , F exists.
The conclusions1 and 2 are proved
III The proof of conclusion 3
It follows from2.9 that
x n − p ≤ 1 b n
x n−1 − p c n−1− α n
λ ·x n − T n x n2
M
≤x n−1 − p b n M c n−1− α n
λ ·x n − T n x n2
2.14
that is,
1− α n
λ ·x n − T n
n x n2
M ≤x n−1 − p − x n − p b n M c n 2.15
Trang 7For any positive number n1, we have
λ
M
n1
n1
1− α nx n − T n x n2≤x0− p − x n1− p n1
n1
b n M c n
≤x0− p n1
n1
b n M c n
.
2.16
Letting n1→ ∞, we have
λ M
∞
n1
1− α nx n − T n
n x n2≤x0− p ∞
n1
b n M c n
< ∞. 2.17
By conditionii, we have
lim inf
n → ∞ x n − T n
IV Next, we prove the conclusion 4
Necessity
If{x n } converges strongly to some point p ∈ F, then from 0 ≤ dx n , F ≤ x n − p → 0, we
have
lim inf
n → ∞ d
x n , F
Sufficiency
If lim infn → ∞ dx n , F 0, it follows from the conclusion 2 that lim n → ∞ dx n , F 0.
Next, we prove that{x n } is a Cauchy sequence in C In fact, since for any t > 0, 1 t ≤
expt, therefore, for any m, n ≥ 1 and for given p ∈ F, from 2.10, we have
x nm − p ≤ 1 b nmx nm−1 − p c nm
≤ expb nmx nm−1 − p c nm
≤ expb nm
exp
b nm−1x nm−2 − p c nm−1
c nm
expb nm b nm−1}x nm−2 − p expb nm
c nm−1 c nm
≤ · · ·
≤ exp
nm
in1
b i
x n − p nm
in1
exp
nm
ji1
b j
c i
≤ K
x n − p nm
in1
c i
< ∞,
2.20
Trang 88 Fixed Point Theory and Applications
where K exp{ ∞
j1 b j } < ∞ Since
lim
n → ∞ d
x n , F
n1
for any given > 0, there exists a positive integer n1such that
d
x n , F
< 4K 1 ,
∞
in1
c i <
Hence, there exists p1 ∈ F such that
x n − p1<
Consequently, for any n ≥ n1and m ≥ 1, from 2.20, we have
x nm − x n ≤ x nm − p1 x n − p1
≤ K
x n − p1 nm
in1
c i
x n − p1
≤ K 1x n − p1 K nm
in1
c i
≤ K 12K 1 K
2K .
2.24
This implies that{x n } is a Cauchy sequence in C Let x n → x∗∈ C Since lim n → ∞ dx n , F 0, and so dx∗, F 0 Again, since {S n } is a finite family of nonexpansive mappings and {T n}
is a finite family of strictly asymptotically pseudocontractive mappings, byLemma 1.5, it is
a finite family of uniformly Lipschitzian mappings Hence, the set F of common fixed points
of{S n } and {T n } is closed and so x∗∈ F.
This completes the proof ofTheorem 2.1
Remark 2.2. Theorem 2.1is a generalization and improvement of the corresponding results in Osilike et al.8 and Liu 2 which is also an improvement of the corresponding results in
3,5 7
The following theorem can be obtained fromTheorem 2.1immediately
Theorem 2.3 Let E be a real Banach space, let C be a nonempty closed pointed convex cone of E, let
T : C → C be a λ, {k n }-strictly asymptotically pseudocontractive mappings, and let {S i : C → C, i
1, 2, , N} be a finite family of nonexpansive mappings with
F
N i1
F
Trang 9(the set of common fixed points of {S i } and T) Let {α n } be a sequence in 0, 1, let {u n } be a bounded sequence in C If the following conditions are satisfied:
i 0 < max{λ, 1 − 1/L} < lim inf n → ∞ α n ≤ α n < 1, where L > 0 is a constant appeared in
ii ∞n1 1 − α n ∞,
iii ∞n1 k n − 1 < ∞ and 1 ≤ k n < 1 − λ/1 − lim inf n → ∞ α n ,
iv ∞n1 ||u n || < ∞,
then the conclusions in Theorem 2.1 still hold.
Theorem 2.4 Let E be a real Banach space, let C be a nonempty closed convex subset of E, and
{T i : C → C, i 1, 2, , N} be a finite family of λ i , {k i n }-strictly asymptotically pseudocontractive mappings, and let {S i : C → C, i 1, 2, , N} be a finite family of nonexpansive mappings with
F
N i1
F
S i N i1
(the set of common fixed points of {S i } and {T i }) Let {x n } be the sequence defined by the following: for any given x1∈ C,
x n α n S n x n−1 β n T n n x n γ n u n ∀n ≥ 1, 2.27
where S n S nmod N , T n T n
nmod N , {α n }, {β n }, and {γ n } are sequences in 0, 1 with α n β n
γ n 1, {u n } is a bounded sequence in C, λ min{λ i : i 1, 2, , N}, k n max{k i n , i
1, 2, , N}, and L max{L i : i 1, 2, , N} > 0 are positive numbers defined by 1.10 and
1.11, respectively If the following conditions are satisfied:
i 0 < λ < lim inf n → ∞ α n ≤ α n < 1,
ii ∞n1 1 − α n ∞,
iii 0 < β n≤ lim supn → ∞ β n ≤ min{1 − λ, 1/L} < 1,
iv ∞n1 k n − 1 < ∞ and 1 ≤ k n < 1 − λ/1 − lim inf n → ∞ α n ,
v ∞n1 γ n < ∞,
then the conclusions of Theorem 2.1 for sequence {x n } defined by 2.27 still hold.
Proof By the same method as given in the proof of Theorem 2.1, we can prove that the
mapping W n : C → C defined by
W n x α n S n x n−1 β n T n n x γ n u n , x ∈ C, n ≥ 1, 2.28
is a Banach contractive mapping Hence, there exists a unique x n ∈ C such that x n Wx n This implies that the sequence{x n} defined by 2.27 is well defined
Trang 1010 Fixed Point Theory and Applications
For each p ∈ F, we have
x n − p2 α n
S n x n−1 − p, jx n − p β n
T n n x n − p, jx n − p γ n
u n − p, jx n − p
≤ α nx n−1 − px n − p β n
k nx n − p2− λx n − T n
n x n2
γ nu n − px n − p.
2.29 Simplifying it, we have
x n − p2≤ α nx n−1 − px n − p
1− β n k n − β n λ
1− β n k n
x n − T n
n x n2 γ n
1− β n k n
u n − px n − p.
2.30 Since
lim sup
n → ∞
β n lim sup
n → ∞
1− α n − γ n
≤ lim sup
n → ∞
1− α n
1 − lim inf
n → ∞ α n , 2.31
by conditionsi, iii, and iv, we have
k n≤ 1− λ
1− lim infn → ∞ α n ≤ 1− λ
lim supn → ∞ β n ≤ 1− λ
that is, 1− β n k n ≥ λ > 0 Hence, we have
x n − p ≤ α nx n−1 − p
1− β n k n γ n
λu n − p
1β n k n − β n − γ n
1− β n k n
x
n−1 − p γ n
λu n − p
≤
1β n k n − β n
1− β n k n
x n−1 − p γ n
λu n − p.
2.33
By conditioniv,
∞
n1
β n k n − β n
1− β n k n ≤ 1
λ
∞
n1
Again, since{||u n − p||} is bounded, by condition v, we have
∞
n1
γ nu n − p
It follows from2.33 andLemma 1.6that limn → ∞ ||x n − p|| exists, and so {x n} is bounded Since{T i } is uniformly Lipschitzian, {T n
n x n} is bounded
... is a finite family of nonexpansive mappings and {T n}is a finite family of strictly asymptotically pseudocontractive mappings, byLemma 1.5, it is
a finite... class="page_container" data-page ="9 ">
(the set of common fixed points of {S i } and T) Let {α n } be a sequence in 0, 1, let {u n } be a bounded sequence in. .. n} defined by 2.27 is well defined
Trang 1010 Fixed Point Theory and Applications
For each