Internat. J. Math. & Math. Sci.
Vol. 24, No. 1 (2000) 49–53
S0161171200003380
© Hindawi Publishing Corp.
ISHIKAWA ITERATION PROCESS WITH ERRORS FOR
NONEXPANSIVE MAPPINGS IN UNIFORMLY
CONVEX BANACH SPACES
DENG LEI and LI SHENGHONG
(Received 6 May 1999)
Abstract. We shall consider the behaviour of Ishikawa iteration with errors in a uniformly
convex Banach space. Then we generalize the two theorems of Tan and Xu without the
restrictions that C is bounded and lim supn sn < 1.
Keywords and phrases. Uniformly convex Banach space, nonexpansive mapping, Ishikawa
iteration process with errors.
2000 Mathematics Subject Classification. Primary 47H10; Secondary 40A05.
1. Introduction. Let C be a closed convex subset of a Banach space X and T : C → C
be nonexpansive (that is, T x −T y ≤ x −y for all x, y in C). In 1974, Ishikawa [1]
introduced a new iteration process as
(1.1)
xn+1 = tn T sn T xn + 1 − sn xn + 1 − tn xn , n = 0, 1, 2, . . . ,
where {tn } and {sn } are sequences in [0, 1] satisfying certain restrictions. The Mann
iteration process is a special case of Ishikawa where sn = 0 for all n ≥ 0 [4].
In 1993, Tan and Xu [7] obtained following result: let C be a bounded closed convex subset of a uniformly convex Banach space X, T : C → C a nonexpansive mapping. If for any initial guess x0 in C, {xn } defined by (1.1), with the restrictions that
∞
∞
n=0 tn (1−tn ) = ∞,
n=0 sn (1−tn ) < ∞, and lim supn sn < 1, then limn→∞ xn−T xn = 0.
Let C be a closed convex subset of a Banach space X and T : C → C be nonexpansive.
For any given x0 ∈ C the sequence {xn } defined by
xn+1 = αn xn + βn T yn + γn un ,
yn = α̂n xn + β̂n T xn + γ̂n vn ,
n ≥ 0.
(1.2)
is called the Ishikawa iteration sequence with errors. Here {un } and {vn } are two
bounded sequences in C, and {αn }, {βn }, {γn }, {α̂n }, {β̂n }, and {γ̂n } are six sequences
in [0, 1] satisfying the conditions
αn + βn + γn = α̂n + β̂n + γ̂n = 1
for all n ≥ 0.
(1.3)
In particular, if β̂n = γ̂n = 0 for all n ≥ 0, the {xn } defined by
x0 ∈ C,
xn+1 = αn xn + βn T xn + γn un ,
is called the Mann iteration sequence with errors.
n ≥ 0,
(1.4)
50
D. LEI AND L. SHENGHONG
Remark 1.1. Note the Ishikawa and Mann iterative processes are all special cases
of the Ishikawa and Mann iterative processes with errors.
It has been shown that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space X, then every nonexpansive mapping T : C → C has a
fixed point (see [2]). In this paper, we first extend [7, Lemma 2.3] to the Ishikawa iteration sequence with errors (1.2), without the restrictions that C is bounded and
lim supn sn < 1. Then we generalize [7, Theorems 3.1, 3.2, and 3.4].
2. Lemmas
Lemma 2.1. Suppose that {an }, {bn }, and {cn } are three sequences of nonnegative
numbers such that
an+1 ≤ 1 + bn an + cn
If
∞
n=1 bn
and
∞
n=1 cn
for all n ≥ 1.
(2.1)
converges, then limn→∞ an exists.
Proof. For n, m ≥ 1, we have
an+m+1 ≤ 1 + bn+m an+m + cn+m
≤
n+m
n+m
n+m
1 + bi an +
1 + bj ci
i=n
≤ ··· ≤
(2.2)
i=n j=i+1
n+m
i=n
1 + bi an +
n+m
j=n
1 + bj
n+m
ci .
i=n
It follows that
lim sup am ≤
m →∞
∞
∞
∞
ci .
1 + bi an +
1 + bj
i=n
j=n
(2.3)
i=n
Hence, lim supm→∞ am ≤ lim inf n→∞ an . This completes the proof.
Lemma 2.2. Let C be a closed convex subset of a Banach space X, and let T : C → X
a nonexpansive mapping. Then for any initial guess x0 in C, {xn } defined by (1.2),
xn+1 − p ≤ xn − p + γn un − p + βn γ̂n vn − p ,
(2.4)
for all n ≥ 1 and for all p ∈ F (T ), where F (T ), denotes the set of fixed points of T .
Proof. For all p ∈ F (T ), we have
xn+1 − p ≤ αn xn − p + βn T yn − p + γn un − p ≤ αn xn − p + βn α̂n xn − p + β̂n T xn − p + γ̂n vn − p + γn un − p ≤ xn − p + γn un − p + βn γ̂n vn − p .
(2.5)
This completes the proof.
Lemma 2.3 [3]. Let C be a closed convex subset of a uniformly convex Banach space X,
and let T : C → X a nonexpansive mapping. Then the mapping I −T is demiclosed on C.
ISHIKAWA ITERATION PROCESS WITH ERRORS FOR . . .
51
3. Main Results
Theorem 3.1. Let C be a closed convex subset of a uniformly convex Banach space
X, T : C → C a nonexpansive mapping with a fixed point. If for any initial guess x0
∞
∞
in C, {xn } defined by (1.2), with the restrictions that n=0 αn βn = ∞, n=0 αn β̂n <
∞
∞
∞, n=0 γn < ∞ and n=0 γ̂n < ∞, then limn→∞ xn − T xn = 0.
Proof. By Lemma 2.2 and T with a fixed point, we set
M = sup T xn − un , xn − un , T yn − vn , yn − un , xn − vn n≥0
(3.1)
It follows from (1.2) that
xn+1 − T xn+1 ≤ αn xn − T xn+1 + βn T yn − T xn+1 + γn T xn+1 − un ≤ αn xn − T xn + T xn − T xn+1 + βn αn yn − xn + βn yn − T yn + γn yn − un + γn M
≤ αn xn − T xn + βn xn − T yn + αn βn β̂n xn − T xn + β2n α̂n xn − T yn + β̂n T xn − T yn + γn M + βn γn yn − un + αn γn xn − un + αn βn γ̂n xn − vn + β2n γ̂n T yn − vn ≤ αn xn − T xn + αn βn xn − T yn + αn βn β̂n xn − T xn + β2n α̂n xn − T yn + β2n β̂n xn − yn + 2γn M + βn γ̂n M
≤ αn + αn βn β̂n + β2n β̂2n xn − T xn + αn βn + β2n α̂n xn − T xn + T xn − T yn + 2γn M + βn γ̂n M + β2n β̂n γ̂n M
≤ αn + αn βn β̂n + β2n β̂2n + αn βn + β2n α̂n + αn βn β̂n + β2n α̂n β̂n
× xn − T xn + 2γn M + βn γ̂n M + αn βn + β2n α̂n + β2n β̂n γ̂n M
≤ 1 + 2αn βn β̂n xn − T xn + 2 γn + βn γ̂n M.
(3.2)
Setting an = T xn − xn , bn = 2αn βn β̂n , and cn = 2(γn + βn γ̂n )M, it follows from
Lemma 2.1 that limn→∞ an exists.
Let r (x0 ) = limn→∞ xn − T xn . To reach the desired conclusion, it suffices to show
∗
that r (x0 ) is independent of the initial value x0 . We let {xn
} denote iteration (1.2) com∗
∗
∗
mencing at x0 . Since xn+1 − xn+1 ≤ xn − xn , we may assume that limn→∞ xn −
∗
xn
= d > 0. Then, we obtain
xn+1 − x ∗ = αn xn − x ∗ + βn T yn − T y ∗ n
n
n+1
xn − x ∗ − T yn − T y ∗ (3.3)
n
n
xn − x ∗ ,
≤ 1 − 2αn βn δ
n
∗
xn − xn
∗
since T yn − T yn∗ ≤ xn − xn
. Thus,
n
xi − x ∗ − T yi − T y ∗ i
i
xi − x ∗ ≤ x0 − x ∗ − xn+1 − x ∗ . (3.4)
2αi βi δ
0
n+1
i
xi − x ∗ i
i=0
52
D. LEI AND L. SHENGHONG
It follows that
∞
n=0
By condition
xn − x ∗ − T yn − T y ∗ n
n
αn βn δ
< ∞.
∗
xn − xn
ˆ < ∞, we have ∞ αn βn β̂n < ∞. Thus,
n=0
∞
xn − x ∗ − T yn − T y ∗ n
n
+
β̂
αn βn δ
n < ∞.
∗
xn − xn
n=0
(3.5)
∞
n=0 αn βn
(3.6)
It follows that
∗
− T yn − T yn∗ + β̂n = 0
lim inf xn − xn
n →0
(3.7)
∞
since n=0 αn βn = ∞ and δ is the modulus of convexity of uniformly convex Banach
space X. Hence, there is a sequence {nk } ⊂ {n} such that
∗
− T ynk − T yn∗k = 0,
(3.8)
lim β̂nk = 0.
lim xnk − xn
k
k→∞
k→∞
On the other hand, we have
xn −T xn − x ∗ − T x ∗ nk
nk
k
k
∗
∗ ≤ xnk − T xnk − xn
− T xn
k
k
∗
∗
≤xnk − xn
− T ynk − T yn∗k + T xnk − T ynk + T xn
− T yn∗k k
k
∗
∗
∗ ≤xnk − xn
− T ynk − T yn∗k + β̂nk xnk − T xnk + β̂nk xn
− T xn
+ 2γ̂n M.
k
k
k
(3.9)
Setting k → ∞ in (3.9), it follows from (3.8) that
∗
∗ − T xn
= 0.
lim xnk − T xnk − xn
k
k
k→∞
(3.10)
Thus,
∗
∗ − T xn
= 0,
lim xn − T xn − xn
n→∞
(3.11)
that is, r (x0 ) = r (x0∗ ). This completes the proof.
Recall that a Banach space X is said to satisfy Opial’s condition [5] if the condition
xn → x0 weakly implies
(3.12)
lim sup xn − x0 < lim sup xn − y for all y ≠ x0 .
n →∞
n →∞
Theorem 3.2. Let C be a closed convex subset of a uniformly convex Banach space
X which satisfies Opial’s condition, T : C → C a nonexpansive mapping with a fixed
point, and {xn } as in Theorem 3.1. Then {xn } converges weakly to a fixed point of T .
Proof. Let ωw (xn ) be the weak limit ω-set of {xn }. By Lemma 2.3 and
Theorem 3.1, ωw (xn ) is contained in F (T ), the fixed point set of T .
The remainder of the proof is similar to that of [7, Theorem 3.1], so the details are
omitted.
ISHIKAWA ITERATION PROCESS WITH ERRORS FOR . . .
53
Remark 3.3. Theorem 3.2 generalizes [7, Theorem 3.1].
Recall that a mapping T : C → C with a nonempty fixed points set F (T ) in C will be
said to satisfy condition A [6] if there is a nondecreasing function f : [0, ∞) → [0, ∞)
with f (0) = 0, f (r ) > 0 for r ∈ (0, ∞), such that x − T x ≥ f (d(x, F (T ))) for all
x ∈ C, where d(x, F (T )) = inf {x − z : z ∈ F (T )}.
The following two theorems generalize Theorem 3.2 and [7, Theorem 3.4] respectively. Since a similar proof is in [7], we omit their proof here.
Theorem 3.4. Let X, C, T , and {xn } be as in Theorem 3.1. If the range of C under
T is contained in a compact subset of X, then {xn } converges strongly to a fixed point
of T .
Theorem 3.5. Let X, C, T and {xn } be as in Theorem 3.1. If T with a nonempty
fixed points set F (T ) satisfies condition A, then {xn } converges strongly to a fixed point
of T .
Acknowledgement. Research partially supported by NNSF(79790130) and
ZJPNSF(198013).
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Deng Lei: Department of Mathematics, Southwest China Normal University, Beibei,
Chongqing 400715, China
Li Shenghong: Department of Mathematics, Zhejiang University Hangzhou, 310027,
China
E-mail address: lsh@math.zju.edu.cn