Journal of Inequalities in Pure and
Applied Mathematics
APPROXIMATION OF FIXED POINTS OF ASYMPTOTICALLY
DEMICONTRACTIVE MAPPINGS IN ARBITRARY
BANACH SPACES
volume 3, issue 1, article 3,
2002.
D.I. IGBOKWE
Department of Mathematics
University of Uyo
Uyo, NIGERIA.
EMail: epseelon@aol.com
Received 14 May, 2001;
accepted 17 July, 2001.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
043-01
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Abstract
Let E be a real Banach Space and K a nonempty closed convex (not necessarily bounded) subset of E. Iterative methods for the approximation of fixed points
of asymptotically demicontractive mappings T : K → K are constructed using
the more general modified Mann and Ishikawa iteration methods with errors.
Our results show that a recent result of Osilike [3] (which is itself a generalization of a theorem of Qihou [4]) can be extended from real q-uniformly
smooth Banach spaces, 1 < q < ∞, to arbitrary real Banach spaces, and to the
more general Modified Mann and Ishikawa iteration methods with errors. Furthermore, the boundedness assumption imposed on the subset K in ([3, 4]) are
removed in our present more general result. Moreover, our iteration parameters
are independent of any geometric properties of the underlying Banach space.
Approximation of Fixed Points
of Asymptotically
Demicontractive Mappings in
Arbitrary Banach Spaces
D.I. Igbokwe
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2000 Mathematics Subject Classification: 47H06, 47H10, 47H15, 47H17.
Key words: Asymptotically Demicontractive Maps, Fixed Points, Modified Mann and
Ishikawa Iteration Methods with Errors.
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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1.
Introduction
Let E be an arbitrary real Banach space and let J denote the normalized duality
∗
mapping from E into 2E given by J(x) = {f ∈ E ∗ : hx, f i = kxk2 = kf k2 },
where E ∗ denotes the dual space of E and h , i denotes the generalized duality
pairing. If E ∗ is strictly convex, then J is single-valued. In the sequel, we shall
denote the single-valued duality mapping by j.
Let K be a nonempty subset of E. A mapping T : K → K is called
k-strictly asymptotically pseudocontractive mapping, with sequence {kn } ⊆
[1, ∞), lim kn = 1 (see for example [3, 4]), if for all x, y ∈ K there exists
n
j(x − y) ∈ J(x − y) and a constant k ∈ [0, 1) such that
(1.1)
h(I − T n )x − (I − T n )y, j(x − y)i
1
1
≥ (1 − k) k(I − T n )x − (I − T n )yk2 − (kn2 − 1)kx − yk2 ,
2
2
for all n ∈ N. T is called an asymptotically demicontractive mapping with
sequence kn ⊆ [0, ∞), lim kn = 1 (see for example [3, 4]) if F (T ) = {x ∈ K :
n
∗
T x = x} 6= ∅ and for all x ∈ K and x ∈ F (T ), there exists k ∈ [0, 1) and
j(x − x∗ ) ∈ J(x − x∗ ) such that
1
1
(1.2) hx − T x, j(x − x )i ≥ (1 − k)kx − T n xk2 − (kn2 − 1)kx − x∗ k2
2
2
n
D.I. Igbokwe
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∗
for all n ∈ N. Furthermore, T is uniformly L-Lipschitzian, if there exists a
constant L > 0, such that
(1.3)
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kT n x − T n yk ≤ Lkx − yk,
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for all x , y ∈ K and n ∈ N.
It is clear that a k-strictly asymptotically pseudocontrative mapping with a
nonempty fixed point set F (T ) is asymptotically demicontrative. The classes
of k-strictly asymptotically pseudocontractive and asymptotically demicontractive maps were first introduced in Hilbert spaces by Qihou [4]. In a Hilbert
space, j is the identity and it is shown in [3] that (1.1) and (1.2) are respectively
equivalent to the inequalities:
(1.4)
kT n x − T n yk ≤ kn2 kx − yk2 + kk(I − T n )x − (I − T n )yk2
and
(1.5)
kT n x − T n yk2 ≤ kn2 kx − yk2 + kx − T n xk2
which are the inequalities considered by Qihou [4].
In [4], Qihou using the modified Mann iteration method introduced by Schu
[5], proved convergence theorem for the iterative approximation of fixed points
of k-strictly asymptotically pseudocontractive mappings and asymptotically demicontractive mappings in Hilbert spaces. Recently, Osilike [3], extended the theorems of Qihou [4] concerning the iterative approximation of fixed points of kstrictly asymptotically demicontractive mappings from Hilbert spaces to much
more general real q-uniformly smooth Banach spaces, 1 < q < ∞, and to
the much more general modified Ishikawa iteration method. More precisely, he
proved the following:
Theorem 1.1. (Osilike [3, p. 1296]): Let q > 1 and let E be a real q-uniformly
smooth Banach space. Let K be a closed convex and bounded subset of E and
Approximation of Fixed Points
of Asymptotically
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Arbitrary Banach Spaces
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T : K → K a completely continuous uniformly L-Lipschitzian asymptotically
P
2
demicontractive mapping with a sequence kn ⊆ [1, ∞) satisfying ∞
n=1 (kn −
1) < ∞. Let {αn } and {βn } be real sequences satisfying the conditions.
(i) 0 ≤ αn , βn ≤ 1, n ≥ 1;
(ii) 0 <∈≤ cq αnq−1 (1 + Lβn )q ≤ 12 {q(1 − k)(1 + L)−(q−2) }− ∈, for all n ≥ 1
and for some ∈> 0; and
P∞
(iii) n=0 βn < ∞.
Then the sequence {xn } generated from an arbitrary x1 ∈ K by
yn = (1 − βn )xn + βn T n xn , n ≥ 1,
xn+1 = (1 − αn )xn + αn T n yn , n ≥ 1
converges strongly to a fixed point of T .
In Theorem 1.1, cq is a constant appearing in an inequality which characterizes q-uniformly smooth Banach spaces. In Hilbert spaces, q = 2, cq = 1 and
with βn = 0 ∀n, Theorems 1 and 2 of Qihou [4] follow from Theorem 1.1 (see
Remark 2 of [3]).
It is our purpose in this paper to extend Theorem 1.1 from real q-uniformly
smooth Banach spaces to arbitrary real Banach spaces using the more general
modified Ishikawa iteration method with errors in the sense of Xu [7] given by
(1.6)
n
yn = an xn + bn T xn + cn un , n ≥ 1,
xn+1 = a0n xn + b0n T n yn + c0n vn , n ≥ 1,
Approximation of Fixed Points
of Asymptotically
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Arbitrary Banach Spaces
D.I. Igbokwe
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where {an }, {bn }, {cn }, {a0n }, {b0n }, {c0n } are real sequences in [0, 1]. an +
bn + cn = 1 = a0n + b0n + c0n , {un } and {vn } are bounded sequences in K. If
we set bn = cn = 0 in (1.6) we obtain the modified Mann iteration method with
errors in the sense of Xu [7] given by
(1.7)
xn+1 = a0n xn + b0n T n xn + c0n vn , n ≥ 1.
Approximation of Fixed Points
of Asymptotically
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Arbitrary Banach Spaces
D.I. Igbokwe
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2.
Main Results
In the sequel, we shall need the following:
Lemma 2.1. Let E be a normed space, and K a nonempty convex subset of E.
Let T : K → K be uniformly L-Lipschitzian mapping and let {an }, {bn }, {cn },
{a0n }, {b0n } and {c0n } be sequences in [0, 1] with an +bn +cn = a0n +b0n +c0n = 1.
Let {un }, {vn } be bounded sequences in K. For arbitrary x1 ∈ K, generate
the sequence {xn } by
y n = an x n + b n T n x n + c n u n , n ≥ 1
xn+1 = a0n xn + b0n T n yn + c0n vn , n ≥ 0.
Then
Approximation of Fixed Points
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Arbitrary Banach Spaces
D.I. Igbokwe
n
2
n−1
kxn − T xn k ≤ kxn − T xn k + L(1 + L) kxn+1 − T
xn−1 k
2
0
+ L(1 + L)cn−1 kvn−1 − xn−1 k + L (1 + L)cn−1 kun−1 − xn k
(2.1)
+ Lc0n−1 kxn−1 − T n−1 xn−1 k.
Proof. Set λn = kxn − T n xn k. Then
kxn − T xn k ≤ kxn − T n xn k + LkT n−1 xn − xn k
≤ λn + L2 kxn − xn−1 k + LkT n−1 xn−1 − xn k
= λn + L2 ka0n−1 xn + b0n−1 T n−1 yn−1 + c0n−1 vn−1 − xn−1 k
+ Lka0n−1 xn−1 + b0n−1 T n−1 yn−1 + c0n−1 vn−1 − T n−1 xn−1 k
= λn + L2 kb0n−1 (T n−1 yn−1 − xn−1 ) + c0n−1 (vn−1 − xn−1 )k
+ Lka0n−1 (xn−1 − T n−1 xn−1 ) + b0n−1 (T n−1 yn−1 − T n−1 xn−1 )
+ c0n−1 (vn−1 − T n−1 xn−1 )k
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≤ λn + L2 kT n−1 yn−1 − xn−1 k + L2 c0n−1 kvn−1 − xn−1 k
+ Lkxn−1 − T n−1 xn−1 k + L2 kyn−1 − xn−1 k + Lc0n−1 kvn−1 − xn−1 k
+ Lc0n−1 kxn−1 − T n−1 xn−1 k
= λn + Lλn−1 + L(1 + L)c0n−1 kvn−1 − xn−1 k + L2 kT n−1 yn−1 − xn−1 k
+ L2 kyn−1 − xn−1 k + Lc0n−1 kxn−1 − T n−1 xn−1 )k
≤ λn + 2Lλn−1 + L(1 + L)c0n−1 kvn−1 − xn−1 k + L2 (1 + L)kyn−1 − xn−1 k
+ L2 kT n−1 xn−1 − xn−1 k + Lc0n−1 kxn−1 − T n−1 xn−1 )k
= λn + L(1 + L)λn−1 + L(1 + L)c0n−1 kvn−1 − xn−1 k
+ L2 (1 + L)kbn−1 (T n−1 xn−1 − xn−1 ) + cn−1 (un−1 − xn−1 )k
+ Lc0n−1 kxn−1 − T n−1 xn−1 k
≤ λn + L(L2 + 2L + 1)λn−1 + L(1 + L)c0n−1 kvn−1 − xn−1 k
+ L2 (1 + L)cn−1 kun−1 − xn−1 k + Lc0n−1 kxn−1 − T n−1 xn−1 k,
completing the proof of Lemma 1.
Approximation of Fixed Points
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Lemma 2.2. Let {an }, {bn } and {δn } be sequences of nonnegative real numbers satisfying
(2.2)
an+1 ≤ (1 + δn )an + bn , n ≥ 1.
P
P∞
If ∞
lim an exists. If in addition {an } has
n=1 δn < ∞ and
n=1 bn < ∞ then n→∞
a subsequence which converges strongly to zero then lim an = 0.
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Proof. Observe that
an+1 ≤ (1 + δn )an + bn
≤ (1 + δn )[(1 + δn−1 )an−1 + bn−1 ] + bn
n
n
n
Y
Y
X
≤ ... ≤
(1 + δj )a1 + (1 + δj )
bj
≤ ... ≤
j=1
∞
Y
j=1
∞
Y
j=1
∞
X
j=1
j=1
j=1
(1 + δj )a1 +
(1 + δj )
bj < ∞.
Hence {an } is bounded. Let M > 0 be such that an ≤ M, n ≥ 1. Then
an+1 ≤ (1 + δn )an + bn ≤ an + M δn + bn = an + σn
where σn = M δn + bn . It now follows from Lemma 2.1 of ([6, p. 303])
that lim an exists. Consequently, if {an } has a subsequence which converges
n
strongly to zero then lim an = 0 completing the proof of Lemma 2.2.
n
Lemma 2.3. Let E be a real Banach space and K a nonempty convex subset of
E. Let T : K → K be uniformly L-Lipschitzian asymptotically demicontractive
P
2
mapping with a sequence {kn } ⊆ [1, ∞), such that lim kn = 1, and ∞
n=1 (kn −
Approximation of Fixed Points
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n
1) < ∞. Let {an }, {bn }, {cn },
{a0n } , {b0n } , {c0n } be real sequences in [0, 1] satisfying:
(i) an + bn + cn = 1 = a0n + b0n + c0n ,
P
0
(ii) ∞
n=1 bn = ∞,
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P∞
0 2
n=1 (bn )
(iii)
< ∞,
P∞
0
n=1 cn
< ∞,
P∞
n=1 bn
< ∞, and
P∞
n=1 cn
< ∞.
Let {un } and {vn } be bounded sequences in K and let {xn } be the sequence
generated from an arbitrary x1 ∈ K by
yn = an xn + bn T n xn + cn un , n ≥ 1,
xn+1 = a0n + b0n T n yn + c0n vn , n ≥ 1,
then lim inf kxn − T xn k = 0.
n
Proof. It is now well-known (see e.g. [1]) that for all x, y ∈ E, there exists
j(x + y) ∈ J(x + y) such that
Approximation of Fixed Points
of Asymptotically
Demicontractive Mappings in
Arbitrary Banach Spaces
D.I. Igbokwe
(2.3)
kx + yk2 ≤ kxk2 + 2hy, j(x + y)i.
Let x∗ ∈ F (T ) and let M > 0 be such that kun − x∗ k ≤ M, kvn − x∗ k ≤
M, n ≥ 1. Using (1.6) and (2.3) we obtain
kxn+1 − x∗ k2
= k(1 − b0n − c0n )xn + b0n T n yn + c0n vn − x∗ k2
= k(xn − x∗ ) + b0n (T n yn − xn ) + c0n (vn − xn )k2
≤ k(xn − x∗ )k2 + 2hb0n (T n yn − xn ) + c0n (vn − xn ), j(xn+1 − x∗ )i
= k(xn − x∗ )k2 − 2b0n hxn+1 − T n xn+1 , j(xn+1 − x∗ )i
+ 2b0n hxn+1 − T n xn+1 , j(xn+1 − x∗ )i
+ 2b0n hT n yn − xn , j(xn+1 − x∗ )i + 2c0n hvn − xn , j(xn+1 − x∗ )i
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(2.4)
= k(xn − x∗ )k2 − 2b0n hxn+1 − T n xn+1 , j(xn+1 − x∗ )i
+ 2b0n hxn+1 − xn , j(xn+1 − x∗ )i
+ 2b0n hT n yn − T n xn+1 , j(xn+1 − x∗ )i
+ 2c0n hvn − xn , j(xn+1 − x∗ )i.
Observe that
xn+1 − xn = b0n (T n yn − xn ) + c0n (vn − xn ).
Using this and (1.2) in (2.4) we have
∗ 2
(2.5)
kxn+1 − x k
≤ k(xn − x∗ )k2 − b0n (1 − k)kxn+1 − T n xn+1 k2
+ b0n (kn2 − 1)kxn+1 − x∗ k2
+ 2(b0n )2 hT n yn − xn , j(xn+1 − x∗ )i
+ 2b0n hT n yn − T n xn+1 , j(xn+1 − x∗ )i
+ 3c0n hvn − xn , j(xn+1 − x∗ )i
≤ k(xn − x∗ )k2 − b0n (1 − k)kxn+1 − T n xn+1 k2
+ (kn2 − 1)kxn+1 − x∗ k2
+ 2(b0n )2 kT n yn − xn kkxn+1 − x∗ k
+ 2b0n Lkxn+1 − yn kkxn+1 − x∗ k
+ 3c0n kvn − xn kkxn+1 − x∗ k
= k(xn − x∗ )k2 − b0n (1 − k)kxn+1 − T n xn+1 k2
+ (kn2 − 1)kxn+1 − x∗ k2 + [2(b0n )2 kT n yn − xn k
+ 2b0n Lkxn+1 − yn k + 3c0n kvn − xn k]kxn+1 − x∗ k.
Approximation of Fixed Points
of Asymptotically
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Arbitrary Banach Spaces
D.I. Igbokwe
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Observe that
kyn − x∗ k = kan (xn − x∗ ) + bn (T n xn − x∗ ) + cn (un − x∗ )k
≤ kxn − x∗ k + Lkxn − x∗ k + M
(2.6)
= (1 + L)kxn − x∗ k + M,
so that
(2.7)
kT n yn − xn k ≤ Lkyn − x∗ k + kxn − x∗ )k
≤ L[(1 + L)kxn − x∗ k + M ] + kxn − x∗ k
≤ [1 + L(1 + L)]kxn − x∗ k + M L,
kxn+1 − x∗ k =
≤
≤
(2.8)
=
ka0n (xn − x∗ ) + b0n (T n yn − x∗ ) + c0n (vn − x∗ )k
kxn − x∗ k + Lkyn − x∗ k + M
kxn − x∗ k + L[(1 + L)kxn − x∗ k + M ] + M
[1 + L(1 + L)]kxn − x∗ k + (1 + L)M,
and
kxn+1 − yn k = ka0n (xn − yn ) + b0n (T n yn − yn ) + c0n (vn − yn )k
≤ kxn − yn k + b0n [kT n yn − x∗ k + kyn − x∗ k]
+ c0n [kvn − x∗ k + kyn − x∗ k]
= kbn (T n xn − xn ) + cn (un − xn )k + b0n [Lkyn − xn k + kyn − x∗ k]
+ c0n M + c0n kyn − x∗ k
≤ bn (1 + L)kxn − x∗ k + cn M + cn kxn − x∗ k
+ [b0n (1 + L) + c0n ]kyn − x∗ k + c0n M
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(2.9)
≤ [bn (1 + L) + cn ]kxn − x∗ k + cn M
+ [b0n (1 + L) + c0n ][(1 + L)kxn − x∗ k + M ] + c0n M
≤ {[bn (1 + L) + cn ] + [b0n (1 + L) + c0n ](1 + L)}kxn − x∗ k
+ M [b0n (1 + L) + 2c0n + cn ].
Substituting (2.7)-(2.9) in (2.5) we obtain,
kxn+1 − x∗ k2
≤ k(xn − x∗ )k2 − b0n (1 − k)kxn+1 − T n xn+1 k2
+ (kn2 − 1){[1 + L(1 + L)]kxn+1 − x∗ k + M (1 + L)}2
+ {(b0n [[1 + L(1 + L)]kxn − x∗ k + M L] + 3c0n [M + kxn − x∗ k]
+ 2b0n L[[bn (1 + L) + cn ] + [b0n (1 + L + c0n ](1 + L)]kxn − x∗ k
+ M [b0n (1 + L) + 2c0n + cn ]}{[1 + L(1 + L)]kxn − x∗ k + M (1 + L)}
≤ k(xn − x∗ )k2 − b0n (1 − k)kxn+1 − T n xn+1 k2
+ (kn2 − 1){[1 + L(1 + L)]2 kxn+1 − x∗ k2
+ 2M (1 + L)[1 + L(1 + L)]kxn − x∗ k + M 2 (1 + L)2 }
+ 2(b0n )2 [[1 + L(1 + L)]kxn − x∗ k + M L][[1 + L(1 + L)]kxn − x∗ k
+ M (1 + L)] + 3c0n [M + kxn − x∗ k][[1 + L(1 + L)]kxn − x∗ k
+ M (1 + L)] + 2b0n L{[[bn (1 + L) + cn ]
+ [b0n (1 + L) + c0n ](1 + L)]kxn − x∗ k
+ M [b0n (1 + L) + 2c0n + cn ]}{[1 + L(1 + L)]kxn − x∗ k + M (1 + L)}.
Since kxn − x∗ k ≤ 1 + kxn − x∗ k2 , we have
(2.10) kxn+1 − x∗ k2 ≤ [1 + δn ]kxn − x∗ k2 + σn − b0n (1 − k)kxn+1 − T n xn+1 k2 ,
Approximation of Fixed Points
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where
δn = (kn2 − 1){[1 + L(1 + L)]2 + 2M (1 + L)[1 + L(1 + L)]}
+ 2(b0n )2 {[1 + L(1 + L)]2 + M (1 + L)[1 + L(1 + L)]
+ M L[1 + L(1 + L)]}
+ 3c0n {[1 + L(1 + L)] + M [1 + L(1 + L)] + M (1 + L)}
+ 2b0n L{{[bn (1 + L) + cn ] + [b0n (1 + L) + c0n ](1 + L)}
× {[1 + L(1 + L)] + M (1 + L)}
+ M [b0n (1 + L) + 2c0n + cn ][1 + L(1 + L)]}
and
σn = (kn2 − 1){2M (1 + L)[1 + L(1 + L)] + M 2 (1 + L)2 }
+ 2(b0n )2 {[1 + L(1 + L)]M (1 + L) + M L[1 + L(1 + L)] + M 2 L(1 + L)}
+ 3c0n {M [1 + L(1 + L)] + M 2 (1 + L) + M (1 + L)
+ 2b0n L{[[bn (1 + L) + cn ] + [b0n (1 + L) + c0n ](1 + L)][M (1 + L)]
+ M [b0n (1 + L) + 2c0n + cn ][[1 + L(1 + L)] + M (1 + L)]}.
P∞
P∞
2
Since
n=1 (kn − 1) < ∞, condition (iii) implies that
n=1 δn < ∞ and
P∞
n=1 σn < ∞. From (2.10) we obtain
kxn+1 − x∗ k2 ≤ [1 + δn ]kxn − x∗ k + σn
n
n
n
Y
Y
X
∗ 2
≤ ... ≤
[1 + δj ]kx1 − x k + [1 + δj ]
σj
j=1
≤
j=1
j=1
∞
∞
∞
Y
Y
X
[1 + δj ]kx1 − x∗ k2 + [1 + δj ]
σj < ∞,
j=1
j=1
j=1
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P∞
P
∗
∞
since ∞
n=1 σn < ∞. Hence {kxn − x k}n=1 is bounded.
n=1 δn < ∞ and
Let kxn − x∗ k ≤ M, n ≥ 1. Then it follows from (2.10) that
(2.11) kxn+1 − x∗ k2 ≤ kxn − x∗ k2 + M 2 δn + σn
− b0n (1 − k)kxn+1 − T n xn+1 k2 , n ≥ 1
Hence,
b0n (1 − k)kxn+1 − T n xn+1 k2 ≤ kxn − x∗ k2 − kxn+1 − x∗ k2 + µn ,
2
where µn = M δn + σn so that,
(1 − k)
n
X
b0j kxj+1
j
2
∗ 2
− T xj+1 k ≤ kx1 − x k +
j=1
Hence,
n
X
µj < ∞,
D.I. Igbokwe
j=1
∞
X
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b0n kxn+1 − T n xn+1 k2 < ∞,
n=1
and condition (ii) implies that lim inf kxn+1 − T n xn+1 k = 0. Observe that
n→∞
kxn+1 − T n xn+1 k2 = k(1 − b0n − c0n )xn + b0n T n yn + c0n vn − T n xn+1 k2
= kxn − T n xn + b0n (T n yn − xn ) + T n xn − T n xn+1
(2.12)
+c0n (vn − xn )k2 .
For arbitrary u, v ∈ E, set x = u + v and y = −v in (2.3) to obtain
(2.13)
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kv + uk2 ≥ kuk2 + 2hv, j(u)i.
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From (2.12) and (2.13), we have
kxn+1 − T n xn+1 k2
= kxn − T n xn + b0n (T n yn − xn ) + T n xn − T n xn+1 + c0n (vn − xn )k2
≥ kxn − T n xn k2 + 2hb0n (T n yn − xn ) + T n xn − T n xn+1
+ c0n (vn − xn ), j(xn − T n xn )i.
Hence
kxn − T n xn k2
≤ kxn+1 − T n xn+1 k2 + 2kb0n (T n yn − xn )
+ T n xn − T n xn+1 + c0n (vn − xn )kkxn − T n xn k
≤ kxn+1 − T n xn+1 k2 + 2{b0n kT n yn − xn k + Lkxn+1 − xn k
+ c0n kvn − xn k}kxn − T n xn k
≤ kxn+1 − T n xn+1 k2 + 2{b0n kT n yn − xn k + Lb0n kT n yn − xn k
+ Lc0n kvn − xn k + c0n kvn − xn k}kxn − T n xn k
≤ kxn+1 − T n xn+1 k2 + 2(1 + L)kxn − x∗ k
× {(1 + L)b0n kT n yn − xn k + (1 + L)c0n kvn − xn k}
≤ kxn+1 − T n xn+1 k2 + 2(1 + L)kxn − x∗ k
× {(1 + L)b0n [[1 + L(1 + L)]kxn − x∗ k + M L]
+ (1 + L)c0n [M + kxn − x∗ k], (using (2.6))
≤ kxn+1 − T n xn+1 k2
+ 2(1 + L)M {(1 + L)b0n [[1 + L(1 + L)]M + M L]
+ (1 + L)c0n [M + M ]}, (since kxn − x∗ k ≤ M )
Approximation of Fixed Points
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= kxn+1 − T n xn+1 k2 + 2b0n (1 + L)4 M 2 + 4c0n (1 + L)2 M.
(2.14)
Since lim b0n = 0, lim c0n = 0 and lim inf kxn+1 − T n xn+1 k = 0, it follows
n→∞
n→∞
n→∞
from (2.14) that lim inf kxn − T n xn k = 0. It then follows from Lemma 1 that
n→∞
lim inf kxn − T xn k = 0, completing the proof of Lemma 2.3.
n→∞
Corollary 2.4. Let E be a real Banach space and K a nonempty convex subset
of E. Let T : K → K be a k-strictly asymptotically pseudocontractive
P∞ map
with F (T ) 6= ∅ and sequence {kn } ⊂ [1, ∞) such that lim kn = 1, n=1 (kn2 −
n
1) < ∞. Let {an }, {bn }, {cn }, {a0n }, {b0n }, {c0n }, {un }, and {vn } be as in
Lemma 2.3 and let {xn } be the sequence generated from an arbitrary x1 ∈ K
by
yn = an xn + bn T n xn + cn un , n ≥ 1,
xn+1 = a0n + b0n T n yn + c0n vn , n ≥ 1,
Then lim inf kxn − T xn k = 0.
n→∞
Proof. From (1.1) we obtain
k(I − T n )x − (I − T n )ykkx − yk
1
≥ {(1 − k)k(I − T n )x − (I − T n )yk2 − (kn2 − 1)kx − yk2 }
2
1√
= [ 1 − kk(I − T n )x − (I − T n )yk
2p
√
√
+ kn2 − 1kx − yk][ 1 − kk(I − T n )x − (I − T n )yk − k 2 − 1kx − yk]
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1√
≥ [ 1 − kk(I − T n )x − (I − T n )yk]
2
√
√
[ 1 − kk(I − T n )x − (I − T n )yk − k 2 − 1kx − yk]
so that
√
√
1√
1 − k[ 1 − kk(I − T n )x − (I − T n )yk] − k 2 − 1kx − yk ≤ kx − yk.
2
Hence
k(I − T n )x − (I − T n )yk ≤ [
2+
p
k)(kn2
{(1 −
1−k
− 1)}
]kx − yk.
D.I. Igbokwe
Furthermore,
kT n x − T n yk − kx − yk ≤ k(I − T n )x − (I − T n )yk
p
2 + {(1 − k)(kn2 − 1)}
≤ [
]kx − yk,
1−k
from which it follows that
kT n x − T n yk ≤ [1 +
Approximation of Fixed Points
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2+
p
{(1 − k)(kn2 − 1)}
]kx − yk.
1−k
Since {kn } is bounded, let kn ≤ D, ∀ n ≥ 1. Then
p
2 + {(1 − k)(D2 − 1)}
n
n
kT x − T yk ≤ [1 +
]kx − yk
1−k
≤ Lkx − yk,
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where
p
{(1 − k)(D2 − 1)}
L=1+
.
1−k
Hence T is uniformly L-Lipschitzian. Since F (T ) 6= ∅, then T is uniformly
L-Lipschitzian and asymptotically demicontractive and hence the result follows
from Lemma 2.3.
2+
Remark 2.1. It is shown in [3] that if E is a Hilbert space and T : K → K is
k-asymptotically pseudocontractive with sequence {kn } then
√
D+ k
n
n
√ kx − yk ∀ x, y ∈ K, where kn ≤ D, ∀ n ≥ 1.
kT x − T yk ≤
1− k
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Theorem 2.5. Let E be a real Banach space and K a nonempty closed convex subset of E. Let T : K → K be a completely continuous uniformly
L-Lipschitzian asymptotically demicontractive
mapping with sequence {kn } ⊂
P∞
2
[1, ∞) such that lim kn = 1 and n=1 (kn −1) < ∞. Let {an }, {bn }, {cn }, {a0n },
{b0n },
{c0n },
n
{un }, and {vn } be as in Lemma 2.3. Then the sequence {xn } generated from an arbitrary x1 ∈ K by
yn = an xn + bn T n xn + cn un , n ≥ 1,
xn+1 = a0n xn + b0n T n yn + c0n vn , n ≥ 1,
converges strongly to a fixed point of T .
Proof. From Lemma 2.3, lim inf kxn − T n xn k = 0, hence there exists a subsen
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quence {xnj } of {xn } such that lim kxnj − T xnj k = 0.
n
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Since {xnj } is bounded and T is completely continuous, then {T xnj } has a
subsequence {T xjk } which converges strongly. Hence {xnjk } converges strongly.
Suppose lim xnjk = p. Then lim T xnjk = T p. lim kxnjk − T xnjk k
k→∞
k→∞
k→∞
= kp − T pk = 0 so that p ∈ F (T ). It follows from (2.11) that
kxn+1 − pk2 ≤ kxn − pk2 + µn
Lemma 2.2 now implies lim kxn − pk = 0 completing the proof of Theorem
n→∞
2.5.
Corollary 2.6. Let E be an arbitrary real Banach space and K a nonempty
closed convex subset of E. let T : K → K be a k-strictly asymptotically pseudocontractive mapping
F (T ) 6= ∅ and sequence {kn } ⊂ [1, ∞) such that
P∞ with
2
lim kn = 1, and n=1 (kn − 1) < ∞. Let {an }, {bn }, {cn }, {a0n }, {b0n }, {c0n },
Approximation of Fixed Points
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n
{un }, and {vn } be as in Lemma 2.3. Then the sequence {xn } generated from
an arbitrary x1 ∈ K by
yn = an xn + bn T n xn + cn un , n ≥ 1,
xn+1 = a0n xn + b0n T n yn + c0n vn , n ≥ 1,
converges strongly to a fixed point of T .
Proof. As shown in Corollary 2.4, T is uniformly L-Lipschitzian and since
F (T ) 6= ∅ then T is asymptotically demicontractive and the result follows from
Theorem 2.5.
Remark 2.2. If we set bn = cn = 0, ∀ n ≥ 1 in Lemma 2.3, Theorem 2.5 and
Corollaries 2.4 and 2.6, we obtain the corresponding results for the modified
Mann iteration method with errors in the sense of Xu [7].
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Remark 2.3. Theorem 2.5 extends the results of Osilike [3] (which is itself a
generalization of a theorem of Qihou [4]) from real q-uniformly smooth Banach
space to arbitrary real Banach space.
Furthermore, our Theorem 2.5 is proved without the boundedness condition imposed on the subset K in ([3, 4]) and using the more general modified
Ishikawa Iteration method with errors in the sense of Xu [7]. Also our iteration
parameters {an }, {bn }, {cn }, {a0n }, {b0n }, {c0n }, {un }, and {vn } are completely
independent of any geometric properties of underlying Banach space.
Remark 2.4. Prototypes for our iteration parameters are:
1
1
, c0n =
, a0 = 1 − (b0n + c0n ),
3(n + 1)
3(n + 1)2 n
1
1
= cn =
, an = 1 −
, n ≥ 1.
2
3(n + 1)
3(n + 1)2
b0n =
bn
The proofs of the following theorems and corollaries for the Ishikawa iteration method with errors in the sense of Liu [2] are omitted because the proofs
follow by a straightforward modifications of the proofs of the corresponding
results for the Ishikawa iteration method with errors in the sense of Xu [7].
Theorem 2.7. Let E be a real Banach space and let T : E → E be a uniformly
L-Lipschitzian asymptotically demicontractive
mapping with sequence {kn } ⊂
P
2
[1, ∞) such that lim kn = 1, and ∞
(k
−
1)
< ∞. Let {un } and {vn } be
n=1 n
n
P∞
P∞
sequences in E such that n=1 kun k < ∞ and n=1 kvn k < ∞, and let {αn }
and {βn } be sequences in [0, 1] satisfying the conditions:
(i) 0 ≤ αn , βn ≤ 1, n ≥ 1;
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(ii)
P∞
αn = ∞
(iii)
P∞
αn2 < ∞ and
n=1
n=1
P∞
n=1
βn < ∞.
Let {xn } be the sequence generated from an arbitrary x1 ∈ E by
yn = (1 − βn )xn + βn T n xn + un , n ≥ 1,
xn+1 = (1 − αn )xn + αn T n yn + vn , n ≥ 1,
Then lim inf kxn − T xn k = 0.
n→∞
Corollary 2.8. Let E be a real Banach space and let T : E → E be a k-strictly
asymptotically pseudocontractive P
map with F (T ) 6= ∅ and sequence {kn } ⊂
2
[1, ∞) such that lim kn = 1, and ∞
n=1 (kn − 1) < ∞. Let {un }, {vn }, {αn }
n
and {βn } be as in Theorem 2.7 and let {xn } be the sequence generated from an
arbitrary x1 ∈ E by
yn = (1 − βn )xn + βn T n xn + un , n ≥ 1,
xn+1 = (1 − αn )xn + αn T n yn + vn , n ≥ 1,
Then lim inf kxn − T xn k = 0.
n→∞
Theorem 2.9. Let E, T , {un }, {vn }, {αn } and {βn } be as in Theorem 2.7.
If in addition T : E → E is completely continuous then the sequence {xn }
generated from an arbitrary x1 ∈ E by
yn = (1 − βn )xn + βn T n xn + un , n ≥ 1,
xn+1 = (1 − αn )xn + αn T n yn + vn , n ≥ 1,
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converges strongly to a fixed point of T .
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Corollary 2.10. Let E, T , {un }, {vn }, {αn } and {βn } be as in Corollary 2.8. If
in addition T is completely continuous, then the sequence {xn } generated from
an arbitrary x, y ∈ E by
yn = (1 − βn )xn + βn T n xn + un , n ≥ 1,
xn+1 = (1 − αn )xn + αn T n yn + vn , n ≥ 1,
converges strongly to a fixed point of T .
Remark 2.5. (a) If K is a nonempty closed convex subset of E and T : K →
K, then Theorems 2.7 and 2.9 and Corollaries 2.8 and 2.10 also hold
provided that in each case the sequence {xn } lives in K.
(b) If we set βn = 0, ∀ n ≥ 1 in Theorems 2.7 and 2.9 and Corollaries 2.8
and 2.10, we obtain the corresponding results for modified Mann iteration
method with errors in the sense of Liu [2].
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References
[1] S.S. CHANG, Some problems and results in the study of nonlinear analysis,
Nonlinear Analysis, 30 (1997), 4197–4208.
[2] L. LIU, Ishikawa and Mann iteration processes with errors for nonlinear
strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194
(1995), 114–125.
[3] M.O. OSILIKE, Iterative approximations of fixed points of asympotically
demicontractive mappings, Indian J. Pure Appl. Math., 29(12) (1998),
1291–1300.
[4] L. QIHOU, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Analysis,
26(11) (1996), 1835–1842.
[5] J. SCHU, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158 (1991), 407–413.
[6] K.K. TAN AND H.K. XU, Approximating fixed points of nonexpansive
mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178
(1993), 301–308.
[7] Y. XU, Ishikawa and Mann iterative processes with errors for nonlinear
strongly accretive operator equations, J. Math. Anal. Appl., 224 (1998), 91–
101.
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