NOTE ON SOME RESULTS FOR ASYMPTOTICALLY
PSEUDOCONTRACTIVE MAPPINGS AND
ASYMPTOTICALLY NONEXPANSIVE MAPPINGS
YUCHAO TANG AND LIWEI LIU
Received 26 May 2006; Revised 5 August 2006; Accepted 13 August 2006
We discuss convergence theorems of modified Ishikawa and Mann iterative sequences
with errors for asymptotically pseudocontractive and asymptotically nonexpansive mappings in Banach spaces, and the boundedness of the domain and range can be dropped,
generalizing theorems of Chang.
Copyright © 2006 Y. Tang and L. Liu. 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, E∗ is the topological dual
space of E, ·, · is the dual between E and E∗ , D(T) and F(T) denote the domain of T
∗
and the set of all fixed points of T, respectively, and J : E → 2E is the normalized duality
mapping defined by
J(x) = f ∈ E∗ : x, f = x · f , f = x ,
x ∈ E.
(1.1)
Definition 1.1. Let T : D(T) ⊂ E → E be a mapping.
(1) T is said to be asymptotically nonexpansive if there exists a sequence {kn } in (0, ∞)
with limn→∞ kn = 1 such that
n
T x − T n y ≤ kn x − y (1.2)
for all x, y ∈ D(T) and n = 1,2,....
(2) T is said to be asymptotically pseudocontractive if there exists a sequence {kn }
in (0, ∞) with limn→∞ kn = 1, and for any x, y ∈ D(T) there exists j(x − y) ∈ J(x − y)
such that
T n x − T n y, j(x − y) ≤ kn x − y 2
for all n = 1,2,....
Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2006, Article ID 24978, Pages 1–7
DOI 10.1155/FPTA/2006/24978
(1.3)
2
Some results for asymptotically pseudocontractive mappings
(3) T is said to be uniformly L-Lipschitzian if there exists L > 0 such that
n
T x − T n y ≤ L x − y (1.4)
for all x, y ∈ D(T) and n = 1,2,....
The following proposition follows from Definition 1.1 immediately.
Proposition 1.2. (1) If T : D(T) ⊂ E → E is nonexpansive, then T is an asymptocially
nonexpansive mapping with a constant sequence {1}.
(2) If T : D(T) ⊂ E → E is asymptotically nonexpansive, then T is a uniformly L-Lipschitzian, where L = supn≥1 {kn } and asymptotically pseudocontractive mapping.
Definition 1.3. (1) Let T : D(T) ⊂ E → E be a mapping, let D(T) be a nonempty convex
subset of E, let x0 ∈ D(T) be a given point, and let {αn }, {βn }, {γn }, and {δn } be four
sequences in [0,1]. Then the sequence {xn } defined by
xn+1 = 1 − αn − γn xn + αn T n yn + γn un ,
yn = 1 − βn − δn xn + βn T n xn + δn vn ,
∀n ≥ 0,
(1.5)
is called the modified Ishikawa iterative sequence with errors of T, where {un } and {vn }
are two bounded sequences in D(T).
(2) In (1.5) if βn = 0 and δn = 0, n = 0,1,2,..., then yn = xn . The sequence {xn } defined
by
xn+1 = 1 − αn − γn xn + αn T n xn + γn un ,
∀n ≥ 0,
(1.6)
is called the modified Mann iterative sequence with errors of T.
In this paper, we discuss convergence theorems of modified Ishikawa and Mann iterative sequences with errors for asymptotically pseudocontractive and asymptotically nonexpansive mappings in Banach spaces, and the boundedness of the domain and range can
be dropped, generalizing theorems of Chang [1].
Lemma 1.4 [4]. Let {An }, {Bn }, and {Cn } be sequences of nonnegative real numbers satisfying the inequality
An+1 ≤ 1 + Bn An + Cn ,
If
∞
n =0 B n
< +∞ and
∞
n=0 Cn
∀n ≥ 0.
(1.7)
< +∞, then limn→∞ An exists.
2. Main results
Lemma 2.1. Let E be an arbitrary real Banach space, let D be a nonempty closed convex
L-Lipschitzian asymptotically pseudocontractive
subset of E, let T : D → D be a uniformly mapping with a sequence {kn } ⊂ [1, ∞), ∞
n=0 (kn − 1) < +∞. Let {αn }, {βn }, {γn }, and
Y. Tang and L. Liu 3
{δn } be four sequences in [0,1] satisfying the following conditions:
(i) αn + γn ≤ 1, βn + δn ≤ 1;
≤ α , δ ≤ γn , for all n ≥ 0;
(ii) β
n∞ n n
(iii) n=0 γn < +∞;
2
(iv) ∞
n=0 αn < +∞.
Let x0 ∈ D be any given point and let {xn } and { yn } be the modified Ishikawa iterative sequence with errors defined by (1.5). If F(T) = ∅, then for any given q ∈ F(T),
limn→∞ xn − q exists.
Proof. Set M = max{supn≥0 un − q, supn≥0 vn − q}.
Since T is asymptotically pseudocontractive, for all x, y ∈ D, there exists j(x − y) ∈
J(x − y) such that
T n x − T n y, j(x − y) ≤ kn x − y 2 .
(2.1)
Then from inequality (2.1), we obtain
kn I − T n x − kn I − T n y, j(x − y) = kn x − y 2 − T n x − T n y, j(x − y) ≥ 0,
(2.2)
and it follows from Kato [2] that
x − y ≤ x − y + λ kn I − T n x − kn I − T n y ,
∀x, y ∈ D, λ > 0.
(2.3)
Set an := αn + γn . Then from the recursive formula (1.5), we have xn+1 = (1 − an )xn +
an T n yn − γn (T n yn − un ). It follows that
xn = 1 + an xn+1 + an kn I − T n xn+1 − an kn xn + a2n 1 + kn xn − T n yn
+ an T n xn+1 − T n yn + γn 1 + an 1 + kn
(2.4)
T n y n − un .
Observe that
q = 1 + an q + an kn I − T n q − an kn q.
(2.5)
So that
xn − q = 1 + an xn+1 − q + an kn I − T n xn+1 − kn I − T n q − an kn xn − q
+ a2n 1 + kn xn − T n yn + an T n xn+1 − T n yn + γn 1 + an 1 + kn
T n y n − un .
(2.6)
4
Some results for asymptotically pseudocontractive mappings
Hence
n
n
xn − q ≥ 1 + an xn+1 − q + an
kn I − T xn+1 − kn I − T q − an kn xn − q
1 + an
− a2n 1 + kn xn − T n yn − an T n xn+1 − T n yn − γn 1 + an 1 + kn T n yn − un ≥ 1 + an xn+1 − q − an kn xn − q − a2n 1 + kn xn − T n yn − an T n xn+1 − T n yn − γn 1 + an 1 + kn T n yn − un .
(2.7)
So
xn+1 − q ≤ 1 +
an kn − 1 xn − q + a2n 1 + kn xn − T n yn 1 + an
+ an T n xn+1 − T n yn + γn 1 + an 1 + kn T n yn − un .
(2.8)
Furthermore, set bn := βn + δn . Then from recursive formula (1.5), we have yn = (1 −
bn )xn + bn T n xn − δn (T n xn − vn ). By condition (ii), we have bn ≤ an , for all n ≥ 0. We make
the following estimates:
yn − q = 1 − bn xn − q + bn T n xn − q − δn T n xn − vn ≤ 1 + bn L − 1 xn − q + δn T n xn − vn ≤ Lxn − q + δn Lxn − q + δn M
= L 1 + δn xn − q + δn M,
x n − T n y n ≤ x n − q + L y n − q ≤ xn − q + L2 1 + δn xn − q + Lδn M
≤ 1 + L2 1 + δn xn − q + Lδn M,
n
T y n − u n ≤ L y n − q + u n − q ≤ L2 1 + δn xn − q + 1 + Lδn M,
n
T xn+1 − T n yn ≤ Lxn+1 − yn = Lxn − yn + an T n yn − xn − γn T n yn − un ≤ Lxn − yn + Lan T n yn − xn + Lγn T n yn − un = Lbn xn − T n xn + δn T n xn − vn + Lan T n yn − xn + Lγn T n yn − un (2.9)
Y. Tang and L. Liu 5
≤ Lbn xn − T n xn + Lδn T n xn − vn + Lan T n yn − xn + Lγn T n yn − un ≤ Lan (1 + L)xn − q + L2 δn xn − q
+ Lδn M + Lan 1 + L2 1 + δn xn − q
+ L2 an δn M + L3 γn 1 + δn xn − q + Lγn 1 + Lδn M.
(2.10)
Using (2.9) and (2.10) in (2.8), we obtain the following estimation:
xn+1 − q ≤ 1 +
an kn − 1 xn − q + a2n 1 + kn 1 + L2 1 + δn xn − q
1 + an
+ a2n 1 + kn Lδn M + La2n 1 + L xn − q + L2 an δn xn − q + Lan δn M
+ La2n 1 + L2 1 + δn xn − q + L2 a2n δn M + L3 an γn 1 + δn xn − q
+ Lan γn 1 + Lδn M + γn 1 + an 1 + kn L2 1 + δn xn − q
+ γn 1 + an 1 + kn
= 1+
1 + Lδn M
an kn − 1 + a2n 1 + kn 1 + L2 1 + δn + La2n 1 + L + L2 an δn
1 + an
+ La2n 1 + L2 1 + δn
+ L 3 an γ n 1 + δ n
+ γn 1 + an 1 + kn L2 1 + δn
x n − q + a2n 1 + kn Lδn M + Lan δn M + L2 a2n δn M
+ Lan γn 1 + Lδn M + γn 1 + an 1 + kn
1 + Lδn M .
(2.11)
Set
An := xn − q,
Bn :=
an kn − 1 + a2n 1 + kn 1 + L2 1 + δn + La2n 1 + L + L2 an δn
1 + an
+ La2n 1 + L2 1 + δn
Cn := a2n 1 + kn Lδn M + Lan δn M + L2 a2n δn M + Lan γn 1 + Lδn M
+ γn 1 + an 1 + kn
+ L3 an γn 1 + δn + γn 1 + an 1 + kn L2 1 + δn ,
1 + Lδn M,
(2.12)
6
Some results for asymptotically pseudocontractive mappings
then inequality (2.11) is equal to
An+1 ≤ 1 + Bn An + Cn .
By conditions (ii), (iii), and (iv), we know
we know limn→∞ xn − q exists.
∞
n =0 B n
< +∞,
(2.13)
∞
n=0 Cn
< +∞. By Lemma 1.4,
Theorem 2.2. Let E be a real uniformly smooth Banach space, let D be a nonempty closed
L-Lipschitzian asymptotically pseuconvex subset of E, and let T : D → D be a uniformly docontractive mapping with a sequence {kn } ⊂ [1, ∞), ∞
n=0 (kn − 1) < +∞. Let {αn }, {βn },
{γn }, and {δn } be four sequences in [0,1] satisfying the conditions (i)–(iv) in Lemma 2.1.
Let x0 ∈ D be any given point and let {xn }, { yn } be the modified Ishikawa iterative
sequence with errors defined by (1.5).
(1) If {xn } converges strongly to a fixed point q of T in D, then there exists a nondecreasing function φ : [0, ∞) → [0, ∞), φ(0) = 0 such that
T n yn − q, J yn − q
2
≤ kn yn − q − φ yn − q ,
(∗ )
for all n ≥ 0.
(2) Conversely, if there exists a strictly increasing function φ : [0, ∞) → [0, ∞), φ(0) = 0
satisfying condition (∗ ), then xn → q ∈ F(T).
Proof. Since E is uniformly smooth, the normalized duality mapping J : E → E∗ is singlevalued and uniformly continous on any bounded subset of E.
(1) Let xn → q ∈ F(T). From conditions (ii)–(iv) in Lemma 2.1, we have βn → 0, δn → 0.
Besides noticing yn − q ≤ L(1 + δn )xn − q + δn M in (2.9) of Lemma 2.1, we have
yn −→ q
(n −→ ∞).
(2.14)
The rest of the proof is the same as Chang’s [1, Theorem 2.1].
(2) By Lemma 2.1, we know limn→∞ xn − q exists. So {xn } is bounded. And by the
proof of Lemma 2.1, we can also get {T n yn − yn }, {xn − T n xn }, {xn − vn }, and {un − yn };
all are bounded. So the rest of the proof is the same as Chang’s [1, Theorem 2.1].
Remark 2.3. (1) Theorem 2.2 removes the restriction on D which is bounded in Chang
[1, Theorem 2.1].
(2) Respectively, we can get Chang [1, Theorems 2.2, 2.3, and 2.4], but without the
restriction on D which is bounded.
(3) In Osilike and Akuchu [3], they discussed the common fixed points of a family of
asymptotically pseudocontractive maps. Our paper is different from it.
Acknowledgments
This project is supported by National Natural Science Foundations of China (10561007)
and the Natural Science Foundations of JiangXi Province (0411036).
Y. Tang and L. Liu 7
References
[1] S. S. Chang, Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings, Proceedings of the American Mathematical Society 129 (2001), no. 3, 845–
853.
[2] T. Kato, Nonlinear semigroups and evolution equations, Journal of the Mathematical Society of
Japan 19 (1967), 508–520.
[3] M. O. Osilike and B. G. Akuchu, Common fixed points of a finite family of asymptotically pseudocontractive maps, Fixed Point Theory and Applications 2004 (2004), no. 2, 81–88.
[4] K.-K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa
iteration process, Journal of Mathematical Analysis and Applications 178 (1993), no. 2, 301–308.
Yuchao Tang: Department of Mathematics, NanChang University, NanChang, JiangXi 330047, China
E-mail address: hhaaoo1331@yahoo.com.cn
Liwei Liu: Department of Mathematics, NanChang University, NanChang, JiangXi 330047, China
E-mail address: liulws@yahoo.com.cn