Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 169062, pptx
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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 169062, 12 pages
doi:10.1155/2010/169062
Research Article
Comparison of the Rate of Convergence among
Picard, Mann, Ishikawa, and Noor Iterations
Applied to Quasicontractive Maps
B. E. Rhoades
1
and Zhiqun Xue
2
1
Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
2
Department of Mathematics and Physics, Shijiazhuang Railway University, Shijiazhuang 050043, China
Correspondence should be addressed to Zhiqun Xue,
Received 12 October 2010; Accepted 14 December 2010
Academic Editor: Juan J. Nieto
Copyright q 2010 B. E. Rhoades and Z. Xue. 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.
We provide sufficient conditions for Picard iteration to converge faster than Krasnoselskij,
Mann, Ishikawa, or Noor iteration for quasicontractive operators. We also compare the rates of
convergence between Krasnoselskij and Mann iterations for Zamfirescu operators.
1. Introduction
Let X, d be a complete metric space, and let T be a self-map of X.IfT has a unique fixed
point, which can be obtained as the limit of the sequence {p
n
},wherep
n
T
n
p
0
,p
0
any point
of X,thenT is called a Picard operator see, e.g., 1, and the iteration defined by {p
n
} is
called Picard iteration.
One of the most general contractive conditions for which a map T is a Picard operator
is that of
´
Ciri
´
c 2see also 3. A self-map T is called quasicontractive if it satisfies
d
Tx,Ty
≤ δ max
d
x, y
,d
x, Tx
,d
y, Ty
,d
x, Ty
,d
y, Tx
, 1.1
for each x, y ∈ X,whereδ is a real number satisfying 0 ≤ δ<1.
Not every map which has a unique fixed point enjoys the Picard property. For example,
let X 0, 1 with the absolute value metric, T : X → X defined by Tx 1 − x. Then, T has a
unique fixed point at x 1/2, but if one chooses as a starting point x
0
a for any a
/
1/2, then
successive function iterations generate the bounded divergent sequence {a, 1 − a, a, 1 − a, }.
2 Fixed Point Theory and Applications
To obtain fixed points for some maps for which Picard iteration fails, a number of fixed
point iteration procedures have been developed. Let X be a Banach space, the corresponding
quasicontractive mapping T : X → X is defined by
Tx − Ty
≤ δ max
x − y
,
x − Tx
,
y − Ty
,
x − Ty
,
y − Tx
. 1.2
In this paper, we will consider the following four iterations.
Krasnoselskij:
∀v
0
∈ X, v
n1
1 − λ
v
n
λTv
n
,n≥ 0, 1.3
where 0 <λ<1.
Mann:
∀u
0
∈ X, u
n1
1 − a
n
u
n
a
n
Tu
n
,n≥ 0, 1.4
where 0 <a
n
≤ 1forn ≥ 0, and
∞
n0
a
n
∞.
Ishikawa:
∀x
0
∈ X,
x
n1
1 − a
n
x
n
a
n
Ty
n
,n≥ 0,
y
n
1 − b
n
x
n
b
n
Tx
n
,n≥ 0,
1.5
where {a
n
}⊂0, 1, {b
n
}⊂0, 1.
Noor:
∀w
0
∈ X,
z
n
1 − c
n
w
n
c
n
Tw
n
,n≥ 0,
y
n
1 − b
n
w
n
b
n
Tz
n
,n≥ 0,
w
n1
1 − a
n
w
n
a
n
Ty
n
,n≥ 0,
1.6
where {a
n
}⊂0, 1, {b
n
}, {c
n
}⊂0, 1.
Three of these iteration schemes have also been used to obtain fixed points for some
Picard maps. Consequently, it is reasonable to try to determine which process converges the
fastest.
In this paper, we will discuss this question for the above quasicontractions and for
Zamfirescu operators. For this, we will need the following result, which is a special case of
the Theorem in 4.
Theorem 1.1. Let C be any nonempty closed convex subset of a Banach space X,andletT be a
quasicontractive self-map of C.Let{x
n
} be the Ishikawa iteration process defined by 1.5,whereeach
a
n
> 0 and
∞
n0
a
n
∞.then{x
n
} converges strongly to the fixed point of T.
Fixed Point Theory and Applications 3
2. Results for Quasicontractive Operators
To avoid trivialities, we shall always assume that p
0
/
q,whereq denotes the fixed point of
the map T.
Let {f
n
}, {g
n
} be two convergent sequences with the same limit q,then{f
n
} is said to
converge faster than {g
n
} see, e.g., 5 if
lim
n →∞
f
n
− q
g
n
− q
0. 2.1
Theorem 2.1. Let E be a Banach space, D a closed convex subset of E,andT a quasicontractive self-
map of D,then,for0 <λ<1 − δ
2
, Picard iteration converges faster than Krasnoselskij iteration.
Proof. From Theorem 1 of 2 and 1.2,
p
n1
− q
T
n1
p
0
− q
≤
δ
n1
1 − δ
Tp
0
− p
0
≤
δ
n1
1 − δ
Tp
0
− Tq
p
0
− q
≤
δ
n1
1 − δ
δ max
p
0
− q
,
p
0
− q
Tp
0
− Tq
p
0
− q
≤
δ
n1
1 − δ
δ
p
0
− q
δ
1 − δ
p
0
− q
p
0
− q
≤
δ
n1
1 − δ
2
p
0
− q
,
2.2
where q is the fixed point of T.
From 1.3,withv
0
/
q,
v
n1
− q
≥
1 − λ
v
n
− q
− λ
Tv
n
− Tq
≥
1 −
λ
1 − δ
v
n
− q
≥···
≥
1 −
λ
1 − δ
n1
v
0
− q
.
2.3
By setting each β
n
0andeachα
n
λ, it follows from Theorem 1.1 that {v
n
} converges
to q.
4 Fixed Point Theory and Applications
Therefore,
p
n1
− q
v
n1
− q
≤
δ
1 − δ − λ
n1
1 − δ
n−1
p
0
− q
v
0
− q
−→ 0
, 2.4
as n →∞,sinceλ<1 − δ
2
.
Theorem 2.2. Let E, D,andT be as in Theorem 2.1.Andlet0 <a
n
<θ1 − δ,b
n
,c
n
∈ 0, 1 for all
n>0.
A If the constant 0 <θ<1 − δ, then Picard iteration converges faster than Mann iteration.
B If the constant 0 <θ<1 − δ
2
/1 − δ δ
2
, then Picard iteration converges faster than
Ishikawa iteration.
C If the constant 0 <θ<1 − δ
3
/1 − 2δ 2δ
2
, then Picard iteration converges faster than
Noor iteration.
Proof. We have the following cases
Case A Mann Iteration.UsingTheorem 1.1 with each β
n
0, {u
n
} converges to q.Using
1.4,
u
n1
− q
≥
1 − a
n
u
n
− q
− a
n
Tu
n
− Tq
≥
1 −
a
n
1 − δ
u
n
− q
≥···≥
n
i0
1 −
a
i
1 − δ
u
0
− q
.
2.5
Therefore,
p
n1
− q
u
n1
− q
≤
δ
n1
p
0
− q
1 − δ
2
n
i0
1 − a
i
/
1 − δ
u
0
− q
−→ 0
, 2.6
as n →∞,sincea
n
<θ1 − δ for each n>0.
Case B Ishikawa Iteration.FromTheorem 1.1, {x
n
} converges to q.Using1.5,
x
n1
− q
≥
1 − a
n
x
n
− q
− a
n
Ty
n
− Tq
≥
1 − a
n
x
n
− q
−
a
n
δ
1 − δ
y
n
− q
≥
1 − a
n
x
n
− q
−
a
n
δ
1 − δ
x
n
− q
b
n
Tx
n
− Tq
Fixed Point Theory and Applications 5
≥
1 − a
n
−
a
n
δ
1 − δ
x
n
− q
−
a
n
b
n
δ
2
1 − δ
2
x
n
− q
≥
1 − a
n
−
a
n
δ
1 − δ
−
a
n
δ
2
1 − δ
2
x
n
− q
≥···
≥
n
i0
1 − a
i
−
a
i
δ
1 − δ
−
a
i
δ
2
1 − δ
2
x
0
− q
.
2.7
Hence,
p
n1
− q
x
n1
− q
≤
δ
n1
p
0
− q
1 − δ
2
n
i0
1 − a
i
− a
i
δ/
1 − δ
− a
i
δ
2
/
1 − δ
2
x
0
− q
−→ 0,
2.8
as n →∞,sincea
n
<θ1 − δ for each n>0.
Case C Noor Iteration. First we must show that {w
n
} converges to q. The proof will follow
along the lines of that of Theorem 1.1.
Lemma 2.3. Define
A
n
{
z
i
}
n
i0
∪
y
i
n
i0
∪
{
w
i
}
n
i0
∪
{
Tz
i
}
n
i0
∪
Ty
i
n
i0
∪
{
Tw
i
}
n
i0
,
α
n
diam
A
n
,
β
n
max
max
{
w
0
− Tw
i
:0≤ i ≤ n
}
, max
w
0
− Ty
i
:0≤ i ≤ n
,
max
{
w
0
− Tz
i
:0≤ i ≤ n
}}
,
2.9
then {A
n
} is bounded.
Proof.
Case 1. Suppose that α
n
Tz
i
− Tz
j
for some 0 ≤ i, j ≤ n, then, from 1.2 and the definition
of α
n
,
α
n
Tz
i
− Tz
j
≤ δ max
z
i
− z
j
,
z
i
− Tz
i
,
z
j
− Tz
j
,
z
i
− Tz
j
,
z
j
− Tz
i
≤ δα
n
,
2.10
a contradiction, since δ<1.
Similarly, α
n
/
Ty
i
− Ty
j
, α
n
/
Tw
i
− Tw
j
, α
n
/
Tz
i
− Ty
j
, α
n
/
Tz
i
− Tw
j
,and
α
n
/
Ty
i
− Tw
j
for any 0 ≤ i, j ≤ n.
6 Fixed Point Theory and Applications
Case 2. Suppose that α
n
w
i
− w
j
, without loss of generality we let 0 ≤ i<j≤ n. Then,
from 1.6,
α
n
w
i
− w
j
≤
1 − a
j−1
w
i
− w
j−1
a
j−1
w
i
− Ty
j−1
≤
1 − a
j−1
w
i
− w
j−1
a
j−1
α
n
.
2.11
Hence, α
n
≤w
i
− w
j−1
≤α
n
,thatis,α
n
w
i
− w
j−1
. By induction on j,weobtainα
n
w
i
− w
i
0, a contradiction.
Case 3. Suppose that α
n
w
i
− Tw
j
for some 0 ≤ i, j ≤ n.Ifi>0, then we have, using 1.6,
α
n
w
i
− Tw
j
≤
1 − a
i−1
w
i−1
− Tw
j
a
i−1
Ty
i−1
− Tw
j
≤
1 − a
i−1
w
i−1
− Tw
j
a
i−1
α
n
,
2.12
which implies that α
n
≤w
i−1
− Tw
j
, and by induction on i,wegetα
n
w
0
− Tw
j
.
Case 4. Suppose that α
n
w
i
− z
j
or α
n
z
i
− z
j
, y
i
− z
j
, z
i
− Ty
j
, y
i
− y
j
for some
0 ≤ i, j ≤ n,then
α
n
w
i
− z
j
≤
1 − c
j
w
i
− w
j
c
j
w
i
− Tw
j
≤ max
w
i
− w
j
,
w
i
− Tw
j
.
2.13
From Cases 2 and 3, w
i
− w
j
<α
n
,andw
i
− Tw
j
≤w
0
− Tw
m
for some m ≤ j,thatis,
α
n
w
0
− Tw
m
.Ifα
n
z
i
− z
j
,weobtainthatα
n
≤w
i
− z
j
. Therefore; α
n
w
0
− Tw
m
,
other cases, omitting.
Case 5. Suppose that α
n
w
i
−Tz
j
or α
n
z
i
−Tz
j
, w
i
−y
j
, y
i
−Tz
j
for some 0 ≤ i, j ≤ n,
then if i>0,
α
n
w
i
− Tz
j
≤
1 − a
i−1
w
i−1
− Tz
j
a
i−1
Ty
i−1
− Tz
j
≤
1 − a
i−1
w
i−1
− Tz
j
a
i−1
α
n
,
2.14
it leads to α
n
≤w
i−1
− Tz
j
. Again by induction on i,wehaveα
n
w
0
− Tz
j
. Similarly, if
α
n
z
i
− Tz
j
or, α
n
w
i
− y
j
,wealsogetα
n
w
0
− Tz
j
; other cases, omitting.
Fixed Point Theory and Applications 7
Case 6. Suppose that α
n
z
i
− Tw
j
or α
n
y
i
− Tw
j
for some 0 ≤ i, j ≤ n, then, using
Case 1,
α
n
z
i
− Tw
j
≤
1 − c
i
w
i
− Tw
j
c
i
Tw
i
− Tw
j
≤
1 − c
i
w
i
− Tw
j
c
i
α
n
,
2.15
or
α
n
y
i
− Tw
j
≤
1 − b
i
w
i
− Tw
j
b
i
Tz
i
− Tw
j
≤
1 − b
i
w
i
− Tw
j
b
i
α
n
,
2.16
these imply that α
n
≤w
i
− Tw
j
.ByCase3,weobtainthatα
n
w
0
− Tw
j
.
Case 7. Suppose that α
n
w
i
− Ty
j
or α
n
y
i
− Ty
j
for some 0 ≤ i, j ≤ n,thenifi>0,
using Case 2,
α
n
w
i
− Ty
j
≤
1 − a
i−1
w
i−1
− Ty
j
a
i−1
Ty
i−1
− Ty
j
≤
1 − a
i−1
w
i−1
− Ty
j
a
i−1
α
n
,
2.17
which implies that α
n
≤w
i−1
− Ty
j
. Using induction on i,wehaveα
n
w
0
− Ty
j
.
In view of the a bove cases, so we have shown that α
n
β
n
. It remains to show that
{α
n
} is bounded.
Indeed, suppose that α
n
w
0
− Tw
j
for some 0 ≤ j ≤ n, then, using Case 1,
α
n
w
0
− Tw
j
≤
w
0
− Tw
0
Tw
0
− Tw
j
≤ B δα
n
,
2.18
where B : w
0
− Tw
0
,thenα
n
≤ B/1 − δ.
Similarly, if α
n
w
0
− Ty
j
,orα
n
w
0
− Tz
j
we again get α
n
≤ B/1 − δ.Hence,
{α
n
} is bounded, that is, {A
n
} is bounded.
Lemma 2.4. Let E, D,andT be as in Theorem 2.1,andthat
a
n
∞,then{w
n
},asdefinedby
1.6, converges strongly to the unique fixed point q of T.
Proof. From
´
Ciri
´
c 2, T has a unique fixed point q.Foreachn ∈
,define
B
n
{
w
i
}
i≥n
∪
y
i
i≥n
∪
{
z
i
}
i≥n
∪
{
Tw
i
}
i≥n
∪
Ty
i
i≥n
∪
{
Tz
i
}
i≥n
.
2.19
8 Fixed Point Theory and Applications
Then, using the same proof as that of Lemma 2.3,itcanbeshownthat
r
n
: diam
B
n
max
sup
w
n
− Tw
j
: j ≥ n
, sup
w
n
− Ty
j
: j ≥ n
, sup
w
n
− Tz
j
: j ≥ n
.
2.20
Using 1.2 and 1.6,
r
n
w
n
− Tw
j
≤
1 − a
n−1
w
n−1
− Tw
j
a
n−1
Tw
n−1
− Tw
j
≤
1 − a
n−1
r
n−1
a
n−1
δr
n−1
1 − a
n−1
1 − δ
r
n−1
≤···
≤ r
0
n−1
i0
1 −
1 − δ
a
i
,
2.21
lim r
n
0, since
a
n
∞.
For any m, n > 0withj ≥ 0,
w
n
− w
m
≤
w
n
− Tw
j
Tw
j
− w
m
r
n
r
m
,
2.22
and {w
n
} is Cauchy sequence. Since D is closed, there exists w
∞
∈ D such that lim w
n
w
∞
.
Also, lim w
n
− Tw
n
0.
Using 1.2,
Tw
∞
− w
∞
Tw
∞
− Tw
n
Tw
n
− w
n
w
n
− w
∞
lim
Tw
∞
− Tw
n
≤ lim sup δ max
{
w
∞
− w
n
,
w
∞
− Tw
∞
,
w
n
− Tw
n
,
w
∞
− Tw
n
,
w
n
− Tw
∞
}
δ
w
∞
− Tw
∞
.
2.23
Since δ<1, it follows that w
∞
Tw
∞
,andw
∞
is a fixed point of T.Butthefixedpoint
is unique. Therefore, w
∞
q.
Fixed Point Theory and Applications 9
Returning to the proof of Case C, from 1.6,
w
n1
− q
≥
1 − a
n
w
n
− q
− a
n
Ty
n
− Tq
≥
1 − a
n
w
n
− q
−
a
n
δ
1 − δ
y
n
− q
≥
1 − a
n
w
n
− q
−
a
n
δ
1 − δ
w
n
− q
b
n
Tz
n
− Tq
≥
1 − a
n
−
a
n
δ
1 − δ
w
n
− q
−
a
n
δ
2
1 − δ
2
z
n
− q
≥
1 − a
n
−
a
n
δ
1 − δ
−
a
n
δ
2
1 − δ
2
−
a
n
δ
3
1 − δ
3
w
n
− q
≥···
≥
n
i0
1 − a
i
−
a
i
δ
1 − δ
−
a
i
δ
2
1 − δ
2
−
a
i
δ
3
1 − δ
3
w
0
− q
.
2.24
So,
p
n1
− q
w
n1
− q
≤
δ
n1
p
0
− q
1 − δ
2
n
i0
1 − a
i
− a
i
δ/1 − δ − a
i
δ
2
/
1 − δ
2
− a
i
δ
3
/
1 − δ
3
w
0
− q
−→ 0,
2.25
as n →∞,sincea
n
<θ1 − δ for n>0.
It is not possible to compare the rates of convergence between the Krasnoselskij, Mann,
and Noor iterations for quasicontractive maps. However, if one considers Zamfirescu maps,
then some comparisons can be made.
3. Zamfirescu Maps
AselfmapT is called a Zamfirescu operator if there exist real numbers a, b, c sa tisfying 0 <
a<1, 0 <b,c < 1/2suchthat,foreachx, y ∈ X at least one of the following cond itions is
true:
1 dTx,Ty ≤ adx, y,
2 dTx,Ty ≤ bdx, Txdy, Ty,
3 dTx,Ty ≤ cdx, Tydy, Tx.
10 Fixed Point Theory and Applications
In 6 it was shown that the above set of conditions is equivalent to
d
Tx,Ty
≤ δ max
d
x, y
,
d
x, Tx
d
y, Ty
2
,
d
x, Ty
d
y, Tx
2
, 3.1
for some 0 <δ<1.
In the following results, we shall use the representation 3.1.
Theorem 3.1. Let E,andD be as in Theorem 2.1, T a Zamfirescu selfmap of D,thenifa
n
<λ1 −
δθ/1 δ with the constant 0 <θ<1 δ for each n>0, Krasnoselskij iteration converges faster
than Mann, Ishikawa, or Noor iteration.
Proof. Since Zamfirescu maps are special cases of quasicontractive maps, from Theorem 1.1
{v
n
}, {x
n
},and{w
n
} converge to the unique fixed point of T, which we will call q.
Using 1.2,
v
n1
− q
≤
1 − λ
v
n
− q
λ
Tv
n
− q
. 3.2
Using 3.1,
Tv
n
− q
≤ δ max
v
n
− q
,
v
n
− Tv
n
0
2
,
v
n
− q
q − Tv
n
2
δ
v
n
− q
.
3.3
Therefore,
v
n1
− q
≤
1 − λ
1 − δ
v
n
− q
≤···
≤
1 − λ
1 − δ
n1
v
0
− q
,
3.4
and
u
n1
− q
≥
1 − a
n
1 δ
u
n
− q
≥···
≥
n
i0
1 − a
i
1 δ
u
0
− q
.
3.5
Thus,
v
n1
− q
u
n1
− q
≤
1 − λ
1 − δ
n1
v
0
− q
n
i0
1 − a
i
1 δ
u
0
− q
−→ 0
, 3.6
as n →∞,sincea
n
<λ1 − δ.
The proofs for Ishikawa and Noor iterations are similar.
Fixed Point Theory and Applications 11
Theorem 3.2. Let E, D,andT be as in Theorem 3.1,thenifλ1 δθ/1 − δ <a
n
< 1 with the
constant 0 <θ<1 − δ for any n, Mann iteration converges faster than Krasnoselskij iteration.
Proof. Using 1.4 and 3.1,
u
n1
− q
≤
1 − a
n
u
n
− q
a
n
Tu
n
− q
≤
1 − a
n
1 − δ
u
n
− q
≤···
≤
n
i0
1 − a
i
1 − δ
u
0
− q
.
3.7
And again using 1.3, 3.1,wehave
v
n1
− q
≥
1 − λ
v
n
− q
− λ
Tv
n
− Tq
≥
1 − λ
1 δ
v
n
− q
≥···
≥
1 − λ
1 δ
n1
u
0
− q
.
3.8
Thus,
u
n1
− q
v
n1
− q
≤
n
i0
1 − a
i
1 − δ
u
0
− q
1 − λ
1 δ
n1
v
0
− q
−→ 0
, 3.9
as n →∞,sinceλ1 δθ/1 − δ <a
n
< 1.
It is not possible to compare the rates of convergence for Mann, Ishikawa, and Noor
iterations, even for Zamfirescu maps.
Remark 3.3. It has been noted in 7 that the principal result in 8 is incorrect.
Remark 3.4. Krasnoselskij and Mann iterations were developed to obtain fixed point iteration
methods which conver ge for some operators, such as nonexpansive ones, for which Picard
iteration fails. Ishikawa iteration was invented to obtain a convergent fi xed p oint iteration
procedure for continuous pseudocontractive maps, for which Mann iteration failed. To date,
there is no example of any operator that requires Noor iteration; that is, no example of
an operator for which Noor ite ration c onverges, but for which neither Mann n or Ishikawa
converges.
Acknowledgments
The authors would like to thank the reviewers for valuable suggestions, and the National
Natural Science Foundation of China Grant 10872136 for the financial support.
12 Fixed Point Theory and Applications
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c, “A generalization of Banach’s contraction principle,” Proceedings of the American
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