Boletı́n de la Asociación Matemática Venezolana, Vol. XVI, No. 1 (2009)
31
On the stability of some fixed point iteration
procedures with errors
J. O. Olaleru and A. A. Mogbademu
Abstract. Several results have been obtained on the stability of
Mann, Ishikawa and Kirk iteration procedures (without errors) when
dealing with different classes of quasi-contractive operators. In this
paper, we consider the stability of Mann, Ishikawa and Kirk iteration procedures with errors and show that they are almost stable
with respect to some classes of quasi-contractive maps. The results
obviously generalise the results of several authors.
Resumen. Varios resultados se han obtenido sobre la estabilidad
del proceso de iteración de Mann, Ishikawa y Kirk (sin errores)
cuando se trata con clases diferentes de operadores cuasi-contractivos.
En este trabajo, consideramos la estabilidad del proceso de la iteración de Mann, Kirk Ishikawa con errores y se prueba que son semi estable con respecto a algunas clases de mapas cuasi-contractivo. Los
resultados, obviamente, generalizar los resultados de varios autores.
1. Introduction
Suppose X is a normed linear space and T is a self map of X. Suppose xo ∈
X and xn+1 = f (T, xn ) defines an iteration procedure which yields a sequence
of points {xn } in X. Suppose F (T ) = {x ∈ X|T x = x} =
6 φ and that {xn }∞
n=0
∞
converges strongly to p ∈ F (T ). Suppose {yn }n=0 is a sequence in X and {n } is
a sequence in [0, ∞) given by n = kyn+1 − f (T, yn )k. If limn→∞ n = 0 implies
that limn→∞ yn = p, then the iteration procedure defined
by xn+1 = f (T, xn )
P∞
is said to be T − stable or stable with respect to T . If n=o n < ∞ implies
yn → p, then the iteration procedure is said to be almost T − stable. Clearly
any T-stable iteration procedure is almost T stable, but an almost T-stable
iteration procedure may fail to be T-stable.
The stability of several iteration procedures for certain contractive operators
has been studied by several authors. Harder and Ricks [2] proved the stability
of Picard and Mann iteration procedures for the quasi-contractive operators,
called Zamfirescu operators, satisfying the following condition:
2000 AMS Mathematics Classification: 47H10
32
J. O. Olaleru and A. A. Mogbademu
Let (X, d) be a complete metric space and T : X → X a map for which there
exists the real numbers a, b and c satisfying 0 < a < 1, 0 < b, c < 1/2 such
that for each pair x, y ∈ X, at least one of the following is true:
(i) d(T x, T y) ≤ ad(x, y);
(ii) d(T x, T y) ≤ b[d(x, T x) + d(y, T y)];
(iii) d(T x, T y) ≤ c[d(x, T y) + d(y, T x)].
In [13], Rhoades considered the stability of some iteration procedures for the
contractive operators satisfying the following definition:
there exists a constant c ∈ [0, 1) such that for each x, y ∈ X,
1
d(T x, T y) ≤ c max{d(x, y), [d(x, T x) + d(y, T y)], d(x, T y), d(y, T x)}. (0.1)
2
The Zamfirescu operators are independent of the operators satisfying (0.1).
However, it was shown by Rhoades, B. E. 1976 [Comments on fixed point iteration methods. J. Math. Anal. Appl., 56:741-750], that if T satisfies (0.1) or if
T is a Zamfirescu operator, then for δ ∈ [0, 1)
d(T x, T y) ≤
δ
d(x, T x) + δd(x, y).
1−δ
(0.2)
Osilike [11] considered a more general contractive definition: there exists L ≥ 0,
a ∈ [0, 1) such that, for each x, y ∈ X
d(T x, T y) ≤ Ld(x, T x) + ad(x, y).
(0.3)
Recently, Imoru and Olatinwo [3] proved the stability of the Picard and the
Mann iteration process for the following operator which is more general than the
one introduced by Osilike [11]. The operator satisfies the following contractive
definition: there exist a constant q ∈ [0, 1) and a monotone increasing function
Φ : R+ → R+ with Φ(0) = 0 such that for each x, y ∈ X,
kT x − T yk ≤ Φ(kx − T xk) + qkx − yk.
(0.4)
Olaleru [9] also considered the stability of Ishikawa and Kirk iteration procedures when the operator satisfies (0.4).
Osilike [10] proved that the Ishikawa iteration is almost T − stable when the
operator is a Lipschitz strongly pseudocontractive operator. It is the purpose of
this paper to generalise previous results on the stability of iteration procedures
in literature by studying the stability of those iteration procedures with errors
when the operators satisfy (0.4).
Stability of some fixed point iteration procedures
33
2. Preliminaries
The idea of considering fixed point iteration procedures with errors comes
from practical numerical computations.
Definition 1 [1]. Let K be a nonempty convex subset of a Banach space X
and T : K → X a mapping. The sequence {xn }∞
n=0 defined iteratively by
xo ∈ K,
(0.5)
xn+1 = an xn + bn T yn + cn un ,
yn =
a0n xn
+
b0n T xn
+
c0n vn ,
n≥0
(0.6)
(0.7)
where {un }, {vn } are bounded sequences in K and {an }, {bn }, {cn }, {a0n }, {b0n }
and {c0n } are sequences in [0, 1] such that
an + bn + cn = a0n + b0n + c0n = 1, n ≥ 0
(0.8)
is called the Ishikawa iteration sequence with errors.
Remark 1. If un and vn are null sequences, then we have the usual Ishikawa
iteration [4]. If b0n = c0n = 0, n ≥ 0, then the sequence {xn } will be called
M ann iteration with errors. For recent results on Ishikawa iteration with
errors, see [6] and [14]. If in addition, un is a null sequence, then we have the
usual Mann iteration procedure [7-8].
Definition 2. The Kirk iteration with errors is defined as the sequence
{xn }∞
n=1 given iteratively by
xn+1 = ao xn + a1 T xn + a2 T 2 xn + ... + ak T k xn + wn un , n ≥ 0
(0.9)
where {un } is a bounded sequence in X, {an } and {wn } are sequences in [0,1],
Pk
k ≥ 1 an integer, i = 0, 1, ..., k such that a1 > 0 and i=0 ai + wn = 1 for each
n.
Remark 2. If un is a null sequence, then we have the usual Kirk iteration ([1],
[11]).
In the sequel we shall require the following Lemma.
Lemma [1]. Let {rn }, {sn }, and {kn } be sequences of nonnegative numbers
satisfying
rn+1 ≤ (1 − tn )rn + sn + kn
P∞
∞
for
n=0 tn = ∞, sn = o(tn ) and
P∞all n ≥ 0, where {tn }n=0 ⊂ [0, 1]. If
k
<
∞
hold,
then
lim
r
=
0.
n
n→∞
n
n=o
3. Main Results
Theorem 1. Let X be a normed linear space, T : X → X a self map of
X satisfying (0.4) and {yn }∞
n=0 ⊂ X. For arbitrary xo ∈ X, define sequence
{xn }∞
iteratively
by
n=0
xn+1 = f (T, xn ) = an xn + bn T xn + cn un , n ≥ 0
34
J. O. Olaleru and A. A. Mogbademu
where {an }, {bn } and {cn } are sequences in [0,1] such that an + bn + cn = 1 and
{un } is a bounded sequence in X. Suppose n = kyn+1 −(an yn +bn T yn +cn un )k
and suppose T has a fixed point p. Assume that M = supn≥0 kun − pk. If
limP
n→∞ cn = 0, then
∞
(i) n=o n < ∞ implies limn→∞ yn = p, i.e., the Mann iteration with errors
is almost T-stable;
(ii) limn→∞ yn = P
p implies limn→∞ n = 0.
∞
Proof. (i) Let n=o n < ∞. We shall establish that limn→∞ yn = p.
In view of (0.4) and the fact that kT yn − pk = kT yn − T pk,
kyn+1 − pk
≤
+
≤
≤
+
=
=
kyn+1 − (an yn + bn T yn + cn un )k
k(an yn + bn T yn + cn un ) − (an + bn + cn )pk
n + an kyn − pk + bn kT yn − pk + cn kun − pk
n + an kyn − pk + bn [Φkp − T pk + qkp − yn k]
cn kun − pk
n + an kyn − pk + bn qkyn − pk] + cn kun − pk
n + (an + bn q)kyn − pk] + M cn .
Thus
kyn+1 − pk ≤ (an + bn q)kyn − pk + M cn + n .
(0.10)
In view of the Lemma and considering tha fact that 0 ≤ an + bn q < 1, (0.10)
yields limn→∞ kyn − pk = 0. Consequently, limn→∞ yn = p.
(ii) Suppose limn→∞ yn = p. Following the same method as above, it is easy to
see that,
n
=
=
−
≤
≤
+
=
=
kyn+1 − (an yn + bn T yn + cn un )k
kyn+1 − (an + bn + cn )p + (an + bn + cn )p
(an yn + bn T yn + cn un )k
kyn+1 − pk + an kyn − pk + bn kT yn − pk + cn kun − pk
kyn+1 − pk + an kyn − pk + bn [Φkp − T pk + qkp − yn k]
cn kun − pk
kyn+1 − pk + an kyn − pk + bn qkyn − pk] + cn kun − pk
kyn+1 − pk + (an + bn q)kyn − pk + M cn .
If n → ∞, it is clear that n → 0 and this completes the proof.
Remark 3. If cn ≡ 0 in Theorem 1(ii), we obtain a corresponding stability
result for the Mann iteration procedure in [3] and the corresponding stability
result in [11] if specifically, Φ(d(x, T x)) is Ld(x, T x), L ≥ 0.
Example 1. Let X and T be defined as in Theorem 1. Suppose
1
n
2
1
, an = n+3
, bn = n+3
, cn = n+3
, n ≥ 0. Then, Theorem 1 is
kun k = n+7
satisfied.
Theorem 2. Let X be a normed linear space, T : X → X a self map of X
satisfying (0.4). Suppose T has a fixed point p. For arbitrary xo ∈ X, define
Stability of some fixed point iteration procedures
35
sequence {xn }∞
n=0 iteratively by
zn = a0n xn + b0n T xn + c0n vn
xn+1 = f (T, xn ) = an xn + bn T zn + cn un , n ≥ 0
where {an }, {bn }, {cn }, {a0n }, {b0n } and {c0n } are sequences in [0,1] such that
an + bn + cn = 1, a0n + b0n + c0n = 1 and {un }, {vn } are bounded sequences in X.
Let {yn } be any sequence in X and define
sn = a0n yn + b0n T yn + c0n vn
and
n = kyn+1 − an yn + bn T sn + cn un k, n ≥ 0
Assume that M = supn≥0 kun − pk and N = supn≥0 kvn − pk. If ∀ n, cn c0n and
0, then
c0n →
P∞
(i) n=o n < ∞ implies limn→∞ yn = p, i.e. Ishikawa iteration with errors is
almost T-stable;
(ii) limn→∞ yn = p implies
limn→∞ n = 0.
P∞
Proof. (i) Suppose n=o n < ∞, we shall establish that limn→∞ yn = p. In
view of (0.4), it follows that
kyn+1 − pk
= kyn+1 − (an yn + bn T sn + cn un )
+ (an yn + bn T sn + cn un ) − (an + bn + cn )pk
≤ n + an kyn − pk + bn kT sn − pk + cn kun − pk
≤ n + an kyn − pk + bn [Φkp − T pk + qkp − sn k]
+ cn kun − pk
= n + an kyn − pk + bn qkp − sn k + cn kun − pk
= n + an kp − yn k
+ bn qka0n yn + b0n T yn + c0n vn − (a0n + b0n + c0n )pk + cn kun − pk
i.e.
kyn+1 − pk
≤ n + an kp − yn k + bn qk(a0n (yn − p) + b0n (T yn − p)
+ c0n (vn − p)k + cn kun − pk
≤ n + an kp − yn k + bn qa0n kyn − pk + bn qb0n kT yn − pk
+ bn qc0n kvn − pk + cn kun − pk
≤ n + an kp − yn k + bn qa0n kyn − pk
+ bn qb0n [Φ(p − T p) + qkp − yn k] + bn qc0n kvn − pk + cn kun − pk
= n + an kp − yn k + bn qa0n kyn − pk
+ bn q 2 b0n kp − yn k + bn qc0n kvn − pk + cn kun − pk
= n + (an + bn qa0n + bn q 2 b0n )kp − yn k + bn qc0n kvn − pk
+ cn kun − pk
≤ (an + bn q)kyn − pk + (M cn + bn qc0n N ) + n .
36
J. O. Olaleru and A. A. Mogbademu
Hence
kyn+1 − pk ≤ (an + bn q)kyn − pk + (M cn + bn qc0n N ) + n .
(0.11)
In view of the Lemma, (0.11) yields limn→∞ yn = p.
(ii) Suppose limn→∞ yn = p.
n
=
=
−
≤
≤
+
=
=
+
=
+
≤
+
≤
+
=
+
=
+
≤
kyn+1 − (an yn + bn T sn + cn un )k
kyn+1 − (an + bn + cn )p + (an + bn + cn )p
(an yn + bn T sn + cn un )k
kyn+1 − pk + an kyn − pk + bn kT sn − pk + cn kun − pk
kyn+1 − pk + an kyn − pk + bn [Φkp − T pk + qkp − sn k]
cn kun − pk
kyn+1 − pk + an kyn − pk + bn qkp − sn k + cn kun − pk
kyn+1 − pk + an kp − yn k
bn qk(a0n yn + b0n T yn + c0n vn − (a0n + b0n + c0n )pk + cn kun − pk
kyn+1 − pk + an kp − yn k
bn qk(a0n (yn − p) + b0n (T yn − p) + c0n (vn − p)k + cn kun − pk
kyn+1 − pk + an kp − yn k + bn qa0n kyn − pk
bn qb0n kT yn − pk + bn qc0n kvn − pk + cn kun − pk
kyn+1 − pk + an kp − yn k + bn qa0n kyn − pk
bn qb0n [Φ(p − T p) + qkp − yn k] + bn qc0n kvn − pk + cn kun − pk
kyn+1 − pk + an kp − yn k + bn qa0n kyn − pk
bn q 2 b0n kp − yn k + bn qc0n kvn − pk + cn kun − pk
kyn+1 − pk + (an + bn qa0n + bn q 2 b0n )kp − yn k + bn qc0n kvn − pk
cn kun − pk
kyn+1 − pk + (an + bn q)kyn − pk + M cn + bn qc0n N.
Thus n → 0 as n → ∞.
Remark 4. If Φ(d(x, T x)) is Ld(x, T x), L ≥ 0 and ∀ n, cn = c0n = 0, then
Theorem 2(ii) gives the corresponding result of [11].
1
Example 2. Let X and T be defined as in Theorem 2. Suppose kun k = n+7
,
1
n+2
n
1
1
n
2
0
0
0
kvn k = n+1 , an = n+3 , an = 2n+3 , bn = n+3 , bn = 2n+3 , cn = n+3 , cn = 2n+3 ,
n ≥ 0. Then, Theorem 2 is satisfied.
Theorem 3. Let X be a normed linear space and let T : X → X be a selfmap
of X satisfying (0.4). Suppose T has a fixed point p. Let xo ∈ X and suppose
that
xn+1 = f (T, xn ) = ao xn + a1 T xn + a2 T 2 xn + ... + ak T k xn + wn un , n ≥ 0,
k ≥ 1 an integer, where {un } is a bounded sequence
P in X, {an } and {wn }
are sequences in [0,1] such that limn→∞ un = 0,
wn = ∞, a1 > 0 and
ao + a1 + .... + ak + wn = 1 for any n ≥ 0. Let {yn } be any sequences in X and
Pk
suppose n = kyn+1 − i=0 ai T i yn + wn un k. If wn → 0, then
Stability of some fixed point iteration procedures
37
P∞
(i) n=o n < ∞ implies limn→∞ yn = p i.e the Kirk iteration with errors is
almost T-stable;
(ii) limn→∞ yn = pPimplies limn→∞ n = 0.
∞
Proof. (i) Let n=o n < ∞. We shall establish that limn→∞ yn = p. It is
clear that kT yn − pk = kT p − T yn k ≤ qkyn − pk and hence for i = 1, 2, ..., k,
kT i yn − pk ≤ q i kyn − pk. Thus
Pk
kyn+1 − pk ≤ kyn+1 − ( i=0 ai T i yn + wn un )k
Pk
Pk
+ k i=0 ai T i yn + wn un − p( i=0 ai + wn )k
Pk
≤ n + ( i=0 ai )kT i yn − T i pk + wn kun − pk
Pk
≤ ( i=0 ai q i )kyn − pk + wn kun − pk + n .
Pk
Since i=0 ai q i < 1, in view of the Lemma , it follows that limn→∞ yn = p.
(ii) If limn→∞ yn = p, it follows easily that
k
k
X
X
ai q i )kyn − pk + wn kun k
ai T i yn + wn un )k ≤ kyn+1 − pk + (
n = kyn+1 − (
i=0
i=0
and n → 0 as n → ∞. Thus the proof of the Theorem is complete.
Remark 5. If Φ(d(x, T x)) is Ld(x, T x), L ≥ 0 and wn ≡ 0, then Theorem 2(ii)
is the same as Theorem 7 of [11].
It is still an open problem to show that our Theorems can be proved for Tstability instead of almost T-stability for the iteration processes with errors.
References
[1] V. Berinde, Iterative approximation of fixed points, Editura Efemeride,
Baia Mare, 2002.
[2] A. M. Harder and T. L. Ricks, Stability results for fixed point iteration
procedures, Math. Japonica 33(1998), No. 5, 693-706.
[3] C. O. Imoru and M. O. Olatinwo, On the stability of Picard and Mann
iteration processes, Carpathian J. Math. 19(2003), No. 2, 155-160.
[4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math.
Soc.44(1974), 147-150.
[5] G. E. Kim, H. Kiuchi, W. Takahashi, Weak and strong convergence of
Ishikawa iterations for asympotically nonexpansive mappings in the intermediate sense, Sci. Math. Japon. Online 10 (2004), 81-92.
[6] 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.
38
J. O. Olaleru and A. A. Mogbademu
[7] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc.
125(5)(1997), 1363-1366.
[8] J. O. Olaleru, On the convergence of Mann iteration scheme in locally
convex spaces, Carpathian Journal of Mathematics, 22(2006) No.1-2, 115120.
[9] J. O. Olaleru, On the stability of Ishikawa and Kirk iteration processes, to
appear in Nigerian Journal of Mathematics and Applications.
[10] M. O. Osilike, Stability of the Mann and the Ishikawa iteration procedures
for strongly pseudocontractions and nonlinear equations of the strongly accretive type, J. Math. Anal. Appl. 227 (1998), No.2, 319-334.
[11] M. O. Osilike, Short proofs of stability results for fixed point iteration procedures for a class of quasi-contractive mappings, Indian J. Pure Appl. Math.
30 (1999), No. 12, 1229-1234.
[12] B. E. Rhoades, Some fixed point iteration procedures, Internat. J. Math.
Math. Sci. 14(1991), No. 1, 1-16.
[13] B. E. Rhoades, Fixed point theorems and stability results for fixed point
iteration procedures, Indian J. Pure Appl. Math. 24(11), (1993), 691-703.
[14] Y. G. Xu, Ishikawa and Mann iterative processes with errors for nonlinear
strongly accretive operators equations, J. Math. Anal. Appl. 224 (1998),
91-101.
J. O. Olaleru and A. A. Mogbademu
Mathematics Department, University of Lagos,
Lagos, Nigeria
email: olaleru1@yahoo.co.uk, prinsmo@yahoo.com