Iterative procedure for uniform continuous mapping.

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.8, 2013
53
Iterative Procedure for Uniform Continuous Mapping.
Chika Moore1
Nnubia Agatha C1*
and. Mogbademu Adesanmi2
1. Department of Mathematics, Nnamdi Azikiwe University P.M.B 5025 Awka, Nigeria.
2. Department of Mathematics, University of Lagos, Lagos Nigeria.
* E-mail of the corresponding author: obijiakuagatha@ymail.com
Abstract
Let K be a closed convex nonempty subset of a normed linear space E and let
{ }N
iiT 1= be a finite family of self
maps on K such that T1 is a uniformly continuous uniformly hemicontractive map and T1(K) is a bounded set
with φ≠= = ))(( 1 i
N
i TFF I , sufficient conditions for the strong convergence of an N-step iteration process to a
fixed common point of the family are proved
Keywords: key words, uniformly continuous, uniformly hemicontractive, finite family, common fixed point,
Noor iteration, strong convergence.
1. Introduction
{ } ExfxfxEfxJ ∈∀==∈= ;,:)(
22*
Where E*
denotes the dual space of E and .,. denotes the
generalized duality pairing between E and E*
. The single-valued normalized duality mapping is denoted by j. A
mapping
*
: EET → is called strongly pseudocontractive if for all Eyx ∈, , there exist
)()( yxJyxj −∈− and a constant )1,0(∈K such that
2
)1()(, yxkyxjTyTx −−≤−− .
T is called strongly eractractivpseudocont−φ if for all Eyx ∈, , there exist )()( yxJyxj −∈− and
a strictly increasing function [ ) [ )1,01,0: →φ with 0)0( =φ such that
( ) .)(,
2
yxyxyxyxjTyTx −−−−≤−− φ
It is called generalized strongly eractractivpseudocont−ψ or uniformly pseudocontractive if for all
Eyx ∈, , there exist )()( yxJyxj −∈− and a strictly increasing function [ ) [ )1,01,0: →ψ with
0)0( =ψ such that
( )yxyxyxjTyTx −−−≤−− ψ
2
)(, .
Every strongly eractractivpseudocont−φ operator is a uniformly eractractivpseudocont−ψ
operator with [ ) [ )1,01,0: →ψ defined by sss )()( φψ = , but not conversely (see [13]).
These classes of operators have been studied by several authors (see, for example [3], [4], [7], [13], [19], [23],
[24] and references therein).
If I denotes the identity operator, then T is strongly pseudocontractive, strongly eractractivpseudocont−φ ,
generalized strongly eractractivpseudocont−ψ if and only if )( TI − is strongly accretive, strongly
accretive−φ , generalized strongly accretive−φ operators respectively. The interest in pseudocontractive
mappings is mainly due to their connection with the important class of nonlinear accretive operators. In recent
years, many authors have given much attention to approximate the fixed points of non-linear operators in Banach
space using the Ishikawa and Mann iterative schemes (see, for example [8], [10], [11] and references therein).
Noor [14] introduced the three-step iteration process for solving nonlinear operator equations in real Banach as
follows;
Let E be a real Banach space, K a nonempty convex subset of E and KKT →: , a mapping. For an arbitrary
Kx ∈0 , the sequence { } Kx nn ⊂
∞
=0
defined by

Vol.3, No.8, 2013
54
= 1 −∝ +∝ ,
= 1 − + ,
= 1 − + ,
(1)
Where {∝ },{ }and { }are three sequences in [0,1] is called the three-step iteration (or the Noor iteration).
When = 0, then the three-step iteration reduces to the Ishikawa iterative sequence if = = 0, then the
three-step iteration reduces to the Mann iteration.
Rafiq [21], recently introduced a new type of iteration-the modified three-step iteration process which is defined
as follows;
Let ∶ → be three mappings for any given ∈ , the modified three-step iteration is defined by
= 1 −∝ +∝ ,
= 1 − +
= 1 − +
(2)
Where {∝ },{ }and { } are three real sequences staisfying some conditions. It is clear that the iteration
scheme (2) includes Noor as special case.
Glowinski and Le Tallec [5] used the three-step iteration schemes to solve elastoviscoplasiticty, liquid crystal
and eigen-value problems. They have shown that the three-step approximation scheme performs better than the
two-step and one-step iteration methods. Haubruge et al. [6] have studied the convergence analysis of three-step
iteration schemes and applied these three-step iteration to obtain new splitting type algorithms for solving
variational inequalities, separable convex programming and minimization of a sum of convex functions. They
also proved that three-step iterations also lead to highly parallelized algorithms under certain conditions. Thus, it
is clear that three-step schemes play an important part in solving various problems, which arise in pure and
applied sciences.
Recently, Xue and Fan [23] used the iteration procedure define by (2) in their theorem as stated below.
Theorem 1.1 Let X be a real Banach space and K be a nonempty closed convex subset of X. Let T1, T2 and T3
be strongly pseudocontractive self maps of K with T1(K) bounded and T1, T2 and T3 uniformly continuous. Let
{ } be defined by (2),where { }, { }and { } are three real sequences in [0,1] such that (i) , →
0 → ∞. (ii) ∑∞
# = ∞, and $ ∩ $ ∩ $ ≠ 0 then the sequence{ } converges strongly to the
common fixed point of T1, T2 and T3
Olaleru and Mogbademu [17] established the strong convergence of a modified Noor iterative process when
applied to three generalized strongly eractractivpseudocont−φ , operators or generalized strongly
accretive−φ operators in Banach space. Thus, generalizing the recent results of Fan and Xue (2009). In fact
the stated and proved the following result.
Theorem 1.2 let E be real Banach space, K a nonempty closed convex subset of E,
eractractivpseudocont−φ mappings such that ' bounded. Let { } be a sequence defined by (2)
where {( }, { } and { } are three sequences in [0,1] satisfiying the following conditions:
lim →∞ ( = lim →∞ = lim →∞ = 0
∞=∑
∞
≥0n
nα if F(T1) F(T2) F(T3) ≠ ∅, then the sequence { } converges to the unique common fixed
points T1, T2 and T3 .
Remark 1.1 In theorem 1.2, it is required that;
All the 3 maps be generalized strongly eractractivpseudocont−φ with the same function ϕ (which is
rather strong function).
All the 3 maps are required to be uniformly continuous (and thus bounded).
Our purpose in this paper is to extend is to extend and generalized the result of Olaleru and Mogbademu (17) in
the following ways:
We introduce m-step iteration scheme
We extend the result to any finite family of m-maps.
The conditions of our theorems are less restrictive and more general than the one used in (17), (23). For instance,
the demand that the three maps must be uniformly continuous is weakened by allowing some of the maps to be
free.

Vol.3, No.8, 2013
55
1.1. The M-Step Iteration Process
Let K be a non-empty convex subset of a normed linear space E and let T: K K be a map. For any
given ∈ . The m-step iteration process is defined by
, =
,- = .1 − ( ,-/ + ( ,- ,-0 ; ' = 1, 2, … , 4
,5 = = , 6ℎ898 + 1 = ' 4:; 4; ≥ 0 (3)
For a finite family
m
iiT 1}{ = of m-maps, the m-step iterative process becomes
, =
,- = .1 − ( ,-/ + ( ,- 5 0- ,-0 ; ' = 1, 2, … , 4
,5 = = , (4)
6ℎ898 + 1 = ' 4:; 4 =:9 ' = >8 ?
+ 1
4
@ = 4 A
+ 1
4
B − A
+ 1
4
BC , ≥ 0
In the case where at least one of the maps in the finite family has some asymptotic behaviour (satisfies an
asymptotic condition) then the iterative process becomes:
, =
,- = .1 − ( ,-/ + ( ,-
r
imT −+1 ,-0 ; ' = 1, 2, … , 4
,5 = = , (5)
With n and m as in equation (4) and 9 = 1 − D
5
E
We need the following lemma in this work:
Lemma 1.1 [13] Let {F }, { }, { } be sequence of nonnegative numbers satisfying the conditions:
,
0
∞=∑
∞
≥n
nβ → 0 as n→ ∞ and = 0{ }. Suppose that
;)( 1
22
1 nnnnn γµψβµµ +−≤ ++ = 1, 2, … …
Where H:[0,1) [0,1) is a strictly increasing function with H(0) = 0. Then F → 0 as → ∞.
2. Main Result
2.1 Theorem 2.1 Let E be a normed linear space and K a nonempty closed convex subset of E, let {Ti}
m
i 1= be
a finite family of self maps on the K such that :
• T1(K), T2(K) are bounded
• T1 is a uniformly continuous uniformly hemicontractive map on K
•
m
iF 1== I F(Ti) ∅ where F(Ti) is the set of fixed points of Ti in K
Starting with an arbitrary ∈ , let { } be the iterative sequence defined by
, =
,- = .1 − ( ,-/ + ( ,- imT −+1 ,-0 ; ' = 1, 2, … , 4 − 1
,5 = (6)
Where {( ,-} ⊂ 0,1 is a finite family of real sequence such that },...2,1{;0lim , miin
n
∈∀=
∞→
α and
.
0
, ∞=∑
∞
≥n
mnα Then, { nx } converges strongly to a common fixed point of the finite family.
Proof Let ∗
J $. It suffices to prove that:
• { } is bounded.
• Let K = L ,50 − L Then K → 0 as n→ ∞
• converges to ∗
.
Now, since T1(K) is bounded, let D1 = M − ∗M + NO # L ,50 − ∗
L < ∞. We establish by
induction that M − ∗M ≤ R ∀ ≥ 0. The case n = 0 is trivial, so assume it is true for n = v +1
M T − ∗M ≤ .1 − (T,5/M T − ∗M + (T,5L T,50 − ∗
L ≤ D1
Thus, M − ∗M ≤ R ∀ ≥ 0 which gives { } are bounded.
More so, since T1(K), T2(K) and { } are bounded sets, let

Vol.3, No.8, 2013
56
D2 = NO # {M − ∗M + L ,50 − ∗
L + L ,50 − ∗
L < ∞
Then,
L ,50 − L ≤ ( ,50 L − ,50 L + ( ,5L − ,50 L
≤ ( ,50 M − ∗M + L ,50 − ∗
L + ( ,5M − ∗M + L ,50 − ∗
L)
≤ ( ,50 + ( ,5 M − ∗M + max {L ,50 − ∗
L, L ,50 − ∗
L})
≤ ( ,50 + ( ,5 D2
Thus, 011, →− +− nmn xy as ∞→n .
Then by uniform continuity of T1, 0→nδ as ∞→n . Thus proving (ii).
Also,
))(,( *
1
*
1
2*
1 xxjxxxx nnn −−=− +++
))(,())(,)(1( *
1
*
1,1,
*
1
*
1, xxjxyTxxjxx nmnmnnnmn −−+−−−= +−++ αα
))(,()1( *
1
*
11,
*
1111,1,
*
1
*
, xxjxxTxxxTyTxxxx nnmnnnmnmnnnmn −−+−−+−−−≤ ++++−+ ααα
)()1( *
11
2*
1,
*
1111,1,
*
1
*
, xxxxxxxTyTxxxx nnmnnnmnmnnnmn −−−+−−+−−−≤ ++++−+ ψααα
)()1(
2
1
])1[(
2
1 *
11
2*
1,
2*
1,
2*
1
2*2
, xxxxxxxxxx nnmnnnmnnnmn −−−+−++−+−−≤ ++++ ψαδαα
So that
2*
1,
2*
1,,
2*
1
2*2
,
2*
1 2
2
1
)1(2 xxxxxxxxxx nmnnnmnnmnnnmnn −+−++−+−−≤− ++++ αδαδαα
)( *
11, xxnmn −− +ψα
Hence,
2*
11,,
2*
1
2
,
2*
,, 2)1()21( xxxxxx nmnnmnnmnnnmnmn −−+−−≤−−− ++ ψαδααδαα
(7)
Since
0, →inalpha as ,in ∞∀→ there exist Nn ∈0 such that 1)2(1
2
1 ,0 <+−<≥∀ nmnn n δα
)(
)2(1
2
)2(1)2(1
)1( 2*
11
,
,
,
,2*
,
2
,2*
1 xxxxxx n
nmn
mn
nmn
nmn
n
nmn
mn
n −
+−
−
+−
+−
+−
−
≤− ++ ψ
δα
α
δα
δα
δα
α
)(22)(2
2*
11,,,,
2*
1 xxDxx nmnnmnnmnmnn −−+++−≤ ++ ψαδαδαα
Let
*
xxnn −=µ ; mnn ,2αβ = ,
2
, )( Dnmnn δαδυ ++= . Then, we have that
)( 11
22
1 ++ −+≤ nnnnnn µψβυβµµ (8)
By Lemma, 0→nµ as ∞→n . Hence, the theorem.
2.2 Theorem 2.2
Let E be a normal linear space, K a non-empty closed convex subset of E, and let
m
iiT 1}{ = be a finite family
of self maps on K such that:
• T1(K) is bounded
• T1is a uniformly continuous uniformly hemicontractive map on K
• T2, …,Tm are bounded maps
• φ≠= = )(1 i
m
i TFF I where F(Ti) is the set of fixed points of Ti in K
Starting with an arbitrary Kx ∈0 , let { nx } be the iterative sequence defined by Equation 6. Then, { nx }

Vol.3, No.8, 2013
57
converges strongly to a common fixed point of the finite family.
Proof Let Fx ∈*
. Then
))(,( *
1
*
1
2*
1 xxjxxxx nnn −−=− +++
))(,())(,)(1( *
1
*
1,1,
*
1
*
1, xxjxyTxxjxx nmnmnnnmn −−+−−−= +−++ αα
))(,()1( *
1
*
11,
*
1111,1,
*
1
*
, xxjxxTxxxTyTxxxx nnmnnnmnmnnnmn −−+−−+−−−≤ ++++−+ ααα
)()1( *
11
2*
1,
*
1111,1,
*
1
*
, xxxxxxxTyTxxxx nnmnnnmnmnnnmn −−−+−−+−−−≤ ++++−+ ψααα
)()1(
2
1
])1[(
2
1 *
11
2*
1,
2*
1,
2*
1
2*2
, xxxxxxxxxx nnmnnnmnnnmn −−−+−++−+−−≤ ++++ ψαδαα
Where 111,1 +− −= nmnn xTyTδ ; So that
2*
1,
2*
1,,
2*
1
2*2
,
2*
1 2
2
1
)1(2 xxxxxxxxxx nmnnnmnnmnnnmnn −+−++−+−−≤− ++++ αδαδαα
)( *
11, xxnmn −− +ψα
So that
2*
11,,
2*
1
2
,
2*
,, 2)1()21( xxxxxx nmnnmnnmnnnmnmn −−+−−≤−−− ++ ψαδααδαα (9)
Now
L ,50 − L ≤ ( ,50 L − ,50 L + ( ,5L − ,50 L
≤ ( ,50 M − ∗M + L ,50 − ∗
L + ( ,5M − ∗M + L ,50 − ∗
L)
≤ ( ,50 + ( ,5 M − ∗M + max {L ,50 − ∗
L, L ,50 − ∗
L})
Since T1(K) is bounded by the same argument in the prove of part (i) of theorem 2.1 we establish that { nx }
is bounded. Since T2,…,Tn are bounded maps, we have that { nT nx } is bounded. Let D2 =
max {R , NO # M 5 − ∗M < ∞
L , − ∗
L ≤ .1 − ( , /M − ∗M + ( , M 5 − ∗M ≤ D ∀ ≥ 0 (10)
Thus, { 1−mT 1,ny }is bounded. Let D3 = = max {R , NO # L 50 , − ∗
L < ∞
L , − ∗
L ≤ .1 − ( , /M − ∗M + ( , L 50 , − ∗
L ≤ D ∀ ≥ 0 (11)
So, { 2−mT 2,ny }is bounded. Let D4 = max {R , NO # L 50 , − ∗
L < ∞. Proceeding thus, we obtain
that if { iny , }is bounded then { imT − iny , }is bounded, and Di+2 = max {R , NO # L 50- ,- − ∗
L < ∞
so that
L ,- − ∗
L ≤ .1 − ( ,- /M − ∗M + ( ,- L 50- ,- − ∗
L ≤ DX ∀ ≥ 0 (12)
Hence, {yn,i+1} and {Tm-1-iyn,i+1} are bounded. We have thus established that there exists a constant Do>0
such that
}.,...,2,1{},,max{ *
,
*
,
*
0 mxyTxyxxD iiniminn ∈∀−−−≥ − Thus, let D=2D0
0)( ,1,1, →+≤− −+− Dxy mnmnnimn αα as ∞→n (13)
So that 0→nδ as 0→n . There exists Nn ∈0 such that 0nn ≥∀
)(
)2(1
2
)2(1)2(1
)1( 2*
1
,
,
,
,2*
,
2
,2*
1 xxxxxx n
nmn
mn
nmn
nmn
n
nmn
mn
n −
+−
−
+−
+−
+−
−
≤− ++ ψ
δα
α
δα
δα
δα
α
)(22)(2
2*
11,,
2
,,
2*
1 xxDxx nmnnmnnmnmnn −−+++−≤ ++ ψαδαδαα
Let ;*
xxnn −=µ mnn ,2αβ = ,
2
, )( Dnmnn δαδυ ++= . Then, we have that

Vol.3, No.8, 2013
58
)( 11
22
1 ++ −+≤ nnnnnn µψβυβµµ (14)
By Lemma 1.1, 0→nµ as ∞→n . Hence, the theorem.
A map EEA →: is said to be accretive if Eyx ∈∀ , and 0>∀α
yxAyAxyx −≥−+− )(α . (15)
If the above holds, Eyx ∈∀ , and }0|{:)( =∈=∈∀ AwEwAZy (the zero set of A), then A is said to
be quasi-accretive. It is easy to see that T is hemicontractive if and if A=I-T is quasi-accretive. We have the
following theorem as an easy corollary to Theorem 2.2 and Theorem 2.1
2.3 Theorem 2.3
Let E be a normed liner space and let },...,2,1{;: miEEAi ∈→ be a finite family of maps such that:
• The simultaneous nonlinear equations },...,2,1{;0 mixAi ∈= have a common solution Ex ∈*
• R(I-Ai) is bounded.
• (I-A2), …, (I-Am) are bounded maps.
Starting with an arbitrary Ex ∈0 ,define the iterative sequence { nx }by
nn xy =0,
miyAxy iniminninin ,...,1;)1()1( 1,1,,, =−+−= −−+αα
miyTx iniminnin ,...,1;)1( 1,1,, =+−= −++αα
1, += nmn xy where in ≡+1 mod m; (16)
Where { in,α } )1,0(⊂ is a family of real sequences such that },...,2,1{;0lim , miin
n
∈∀=
∞→
α and
∑
∞
≥
∞=
0
,
n
mnα . Then, {xn} converges strongly to a solution of the simultaneous nonlinear equations.
Proof: Let Ti = I-Ai. Then Ti is a uniform continuous uniformly hemicontraction. Further,
nn xy =0,
miyAxy iniminninin ,...,1;)1()1( 1,1,,, =−+−= −−+αα
miyTx iniminnin ,...,1;)1( 1,1,, =+−= −++αα
1, += nmn xy where in ≡+1 mod m;
Thus, Theorem 2.2 applies and we have the stated results. Similarly
Theorem 2.4 Let E be a normed linear space and let },...,2,1{;: miEEAi ∈→ be a finite family of
maps such that:
• The simultaneous nonlinear equations Aix= 0 {1,2,…,m} have a common solution Ex ∈*
that is
φ≠= )(1 i
m
i AZI .
• R(I-A1), R(I-A2) are bounded
• Ai is a uniformly continuous quasi-accretive map
Starting with an arbitrary ,0 Ex ∈ define the iterative sequence {xn} by equation (16). Then {xn} converges
strongly to a solution of the simultaneous nonlinear equations.
3. ConclusionRemark
A Theorem 2.1 extends theorem 1.2 in the following ways; in Equation (6) of Theorem 2.1, let m = 3, ( , =
( , ( , ≡ , ( , ≡ , , ≡ , 0 ≡ , then we have Equation (2) of Theorem 1.2. Theorem 2.1 is
proved for any finite family of maps, so if they are just three in the family, we have Theorem 1.2. More so, the
conditions of Theorem 2.1. For instance, whereas the three maps in Theorem 1.2 are required to be uniformly
continuous and uniformly pseudocontractive with the same function H, Theorem 2.1 requires that only one map

Vol.3, No.8, 2013
59
in the family satisfies such conditions. Also, Theorem 2.2 extends Theorem 1.2 in a similar manner. Hence,
Theorem 2.1, Theorem 2.2 and their corollaries are rather interesting.
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Iterative procedure for uniform continuous mapping.

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Iterative procedure for uniform continuous mapping.