Theoretical Mathematics & Applications, vol.2, no.1, 2012, 161-177
ISSN: 1792- 9687 (print), 1792-9709 (online)
International Scientific Press, 2012
Approximation of Common Fixed Points of a Finite
Family of Asymptotically   Demicontractive Maps
Using a Composite Implicit Iteration Process
Donatus I. Igbokwe1 and Uko S. Jim2
Abstract
We prove that the modified form of the composite implicit iteration process
introduced by Su and Li [1] can be used to approximate the common fixed
points of a finite family of asymptotically   demicontractive maps in real
Hilbert spaces. Our results compliment the results of Su and Li [1], Osilike
and Isiogugu [2], Igbokwe and Udofia [3], Igbokwe and Udo-Utun [4, 5],
Igbokwe and Ini [6] and extend several others from asymptotically
demicontractive maps to the more general class of asymptotically
  demicontractive maps (see for example [7-10]).
1
2
Department of Mathematics, University of Uyo, Uyo, Nigeria,
e-mail: igbokwedi@yahoo.com
Department of Mathematics, University of Uyo, Uyo, Nigeria,
e-mail: ukojim@yahoo.com
Article Info: Received : January 5, 2012. Revised : February 17, 2012
Published online : March 5, 2012
162
Approximation of Common Fixed Points of a Finite Family ...
Mathematics Subject Classification: 47H10, 37C25, 55M20
  demicontractive maps, composite implicit
Keywords: Asymptotically
iteration, fixed point, uniformly Lipschitzian, Hilbert space
1 Introduction
Let K be a nonempty subset of a real Hilbert space H . A mapping
T :K K
is
 k n   1,  ,
called
  demicontractive with sequence
asymptotically
lim k n  1 (see for example [3]), if F T    x  K : Tx  x  
n 
and there exists an increasing continuous function  :  0,     0,   with
 0  0 such that
T nx  p
2

2
2

 kn x  p  x  T n x   x  T n x ,
(1)
x  K , p  F T  and n  1.
A mapping T : K  K is called asymptotically demicontractive with
sequence
 a n   1,  ,
lim a n  1 (see for example [10]), if F T    and
n 
x  K , p  F T  ,  a k   0, 1 
T nx  p
T
is
k  strictly
 a n   1,  ,

2

2
 a n2 x  p  k I  T n x ,
asymptotically
pseudocontractive
(2)
with
a
sequence
lim a n  1 if x  K, n  N ,  a k   0, 1 
n 
T nx T n y
where I
2
2
2

 

2
 a n2 x  p  k I  T n x  I  T n y ,
(3)
is the identity operator. The class of k  strictly asymptotically
pseudocontractive and asymptotically demicontractive maps were first introduced
in Hilbert spaces by Qihou [10]. Observe that a k  strictly asymptotically
pseudocontractive map with a nonempty fixed point set F T  is asymptotically
Donatus I. Igbokwe and Uko S. Jim
163
demicontractive. An example of a k  strictly asymptotically pseudocontractive
map is given in [11].
Furthermore, T is uniformly L  Lipschitzian if there exists a constant
L  0  T nx T n y  L x  y .
Let K be a subset of an arbitrary real Banach space E , a mapping T : DT   E
is
said
to
 a n   1,  ,
be
  demicontractive
asymptotically
with
sequence
lim a n  1 ( See for example [2] ), if F T    and there exists a
n 
strictly increasing continuous function  :  0,     0,   with  0  0 such
that

 

x  T n x, j x  p    x  T n x  12 a n2  1 x  p
2
(4)
x  K , p  F T  and n  N . j is the single – valued duality mapping from E
to E  given by

j  x   f  E  : x, f  x
2
; x
2
 f
2
,
which holds in strictly convex dual spaces. E  denotes the dual space of E and
,
denotes
the generalized duality paring. The class of asymptotically
  demicontractive maps was first introduced in arbitrary real Banach spaces by
Osilike and Isiogugu [2]. It is shown in [2] that the class of asymptotically
demicontractive map is a proper subclass of the class of asymptotically
  demicontractive map while in [3], it is shown that every asymptotically
demicontractive
map
is
asymptotically
  demicontractive
with
 :  0,     0,   given by
 t    1  k t 2  12 a n2  1 x  p .
2
These classes of operators have been studied by several authors (See for example
[2,3, 7-10, 12] ). In [2] Osilike and Isiogugu proved the convergence of the
modified averaging iteration process of Mann [13] to the fixed points of
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Approximation of Common Fixed Points of a Finite Family ...
asymptotically   demicontractive maps. Specifically they proved the following:
Theorem 1.1 ([2], p.65) 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 a sequence
 a n   1,  

a
2
n

 1   . Let  an  be a real sequence satisfying
(i) 0   n  1 (ii)

Then the sequence
 x n n 1
n
  (iii)

2
n
 .
generated from arbitrary x1  K
by the modified
averaging Mann iteration process
x n 1  1   n x n   n T n x n , n  1
(5)
converges strongly to a common fixed point of T .
Similarly, in [3], using the modified averaging implicit iteration scheme
 xn  of
Sun [14], generated from an x1  K , by
x n   n x n 1  1   n Ti k x n , n  1
where n   k  1N  i, i  I  1, 2, 3 . . . , N , Igbokwe and Udofia [3] proved
that under certain conditions on the iteration sequence  n  , the above iteration
process  x n
 converges strongly to the common fixed point of the family Ti iN1
of N uniformly Li  Lipschitzian asymptotically   demicontractive self maps
of nonempty closed convex subset of a Hilbert space H .
Recently, Su and Li [1] introduced the following iteration scheme and called it
Composite Implicit Iteration Process. From x1  K , the sequence
 x n n 1
is
generated by
xn   n xn 1  1   n Tn yn
yn   n xn 1  1   n Tn xn , n  1
where  n ,   n
   0,1 , Tn
(6)
 Tn mod N .
Motivated by the results of Su and Li [1], very recently, Igbokwe and Ini [6]
Donatus I. Igbokwe and Uko S. Jim
165
modified the iteration process (6) and applied the modified iteration process to
approximate the common fixed points of a finite family of k-strictly
asymptotically pseudocontractive maps. In compact form, the modified composite
implicit iteration process of Igbokwe and Ini is expressed as follows:
x n   n x n 1  1   n Ti k y n
y n   n x n 1  1   n Ti k x n , n  1
(7)
where n   k  1N  i, i  I  1, 2, 3 . . . , N .  n ,   n
Observe that, if T : K  K
   0,1 .
is uniformly L  Lipschitzian asymptotically
  Demicontractive map with sequence  a n   1,   such that lim a n  1 then
n 
for every fixed u  K and t   1LL ,1  , the operator S t , s ,n : K  K defined for all
x  K by
S t , s , n x  tu   1  t T n [ su   1  s T n x ]
satisfies
S t , s , n x  S t , s ,n y   1  t  1  s L2 x  y , x , y  K .
Thus, the composite implicit iteration process (7) is defined in K for the family
Ti iN1
of N
uniformly L  Lipschitzian asymptotically   Demicontractive
mappings of nonempty closed convex subset K of a Hilbert space provided that
 n ,   n     ,1  for all
n  1 , where  
L
and L  max Li  .
1 L
1i  N
It is our purpose in this paper to prove that the iteration process (7) converges to
common fixed points of finite family of
N
uniformly L  Lipschitzian
asymptotically   Demicontractive mappings in Hilbert space. We show that the
recent results of Osililke [2], Osililke, Aniagbosor and Akucku [9] concerning the
iterative approximation of fixed points of asymptotically   demicontractive and
asymptotically demicontractive maps which are themselves generalizations of a
theorem of Qihou [10], a result of Osililke [7] and a result of Osililke and
166
Approximation of Common Fixed Points of a Finite Family ...
Aniagbosor [8] will be special cases of our results. Moreover, in Hilbert spaces,
our present results extend the recent results of Igbokwe and Ini [6] from k-strictly
asymptotically pseudocontraction to the much more general asymptotically
  demicontractive maps.
2 Preliminary Notes
In the sequel, we need the following:
 a n n 1 ,  bn n 1
Lemma 2.1 ([8], p.80) Let
and
 n n 1
be sequences of
nonnegative real numbers satisfying the inequality
an1  1   n an  bn , n  1.

If
  n   and
n 1

b
n 1
n
 , Then lim an exists. If in addition {a n }n 1 has a
n 
subsequence which converges strongly to zero, then lim an  0.
n 
Definition 2.1 Let K be a closed subset of a real Banach space E and
T : K  K be a mapping. T is said to be semicompact (see for example [1]) if for
any bounded sequence
 xn 
in K such that x n  Tn x n  0 as n   , there
exists a subsequence  x nk    x n
 such that
x nk  x   K .
3 Main Results
Let K be a subset of a real Hilbert space H . We call the mapping T : K  K
asymptotically   demicontractive with sequence
F T   
and
there
exists
a
strictly
 :  0,     0,   with  0  0 such that
ain   0,  ,
increasing
lim a n  1 if
n
continuous
function
Donatus I. Igbokwe and Uko S. Jim
T nx T n p

2
167
 x  p

 1  1 a n2  1
2
2
2
 x T nx

 x T nx

(8)
for all x  K , p  F T  and n  1.
Since in a Hilbert space, j is the identity map, for all x  K , p  F T  and
n  1. Using (4), we have:
T nx T n p
2

 

2

x  p  [ I  T n x  I  T n p]

x  p  2 I  T n x  I  T n p,  x  p   I  T n x  I  T n p

2
 


 x  p  2 x  T n x, x  p   x  T n x
2

 
2

2
2
 x  p   x  T n x  1 a n2  1 x  p  x  T n x
2




2

2
2
 1  1 a n2  1 x  p  x  T n x   x  T n x , proving (8).
2


Observe that if we set 1  1 a n2  1  k n in (8), then k n  [1,  ) and lim k n  1 .
2
n 
So that equivalently, we have,
T nx T n p
2
2
2

 kn x  p  x  T n x   x  T n x

(9)
Lemma 3.1 Let E be a normed space and K a nonempty convex subset of E .
Let Ti iN1
be N
L  maxLi  ,
Li
uniformly Li  Lipschitzian self mappings of K such that
Ti , i  1, 2, . . . , N .
the Lipschitzian constant of
 n ,   n  be sequences in such that
(i)

 1     
n 1
n
(ii)

 1   
n 1
n
2
  (iii)

 1    
n 1
For arbitrary x1  K , generate the  x n  by
x n   n x n 1  1   n Ti k y n
y n   n x n 1  1   n Ti k x n , n  1
Then,
n
.
Let
2
168
Approximation of Common Fixed Points of a Finite Family ...


2 2 L3  2  L   1  L2

Ti x n  x n
 T
k
i
xn  xn
 2 L Ti k 1 x n 1  x n 1  41   n L2 1  2 L 1  L  x n  x n 1
Proof. Let in  x n  Ti k x n . Then, applying (7) and the fact that L  max( Li ) we
obtain :
x n  Ti x n

x n  Ti k x n  L Ti k 1 x n 1  x n

in + L2 x n  x n 1  L Ti k 1 x n  x n
1  1   L  + L  21   L x
 1   L T x  x  21   L y  x

2
n
n
 x n 1
n
.
2
in 1
in
n
2
3
n
i
n
n
n
n
That is
1  1   L  x
2
n
n
 Ti x n
1  L 

+ Lin 1  21   n L2 x n  x n 1
2
in
 21   n L3 y n  x n .
(10)
Observe that

yn  xn
 2  L in  21  L 
x n  x n 1 .
(11)
Substituting (11) into (10), we obtain
Ti x n  x n

1
1  1   n L2


  2 L3  2  L   1  L2   Ti k xn  xn
 L Ti k 1 x n 1  x n 1  21   n L2 1  2 L 1  L  x n  x n 1
.
Since from condition (ii) lim 1   n   0 , then there exists an N1  0 such that
n
n  N1 ,
1  1   n L2 
1
.
2
Therefore,
Ti x n  x n



2 2 L3  2  L   1  L2
 T
i
k
xn  xn
 2 L Ti k 1 x n 1  x n 1  41   n L2 1  2 L 1  L  x n  x n 1 ,
Donatus I. Igbokwe and Uko S. Jim
169
completing the proof.

Theorem 3.1 Let H be a real Hilbert space and Let K be a nonempty closed
convex subset of H . Let Ti iN1 be N uniformly L  Lipschitzian asymptotically
  demicontractive self maps of K with sequence ain   [1,  ) such that

N
n 1
i 1
 ain  1   for all i  I , and F   F Ti   
where F Ti   x  K : Ti x  x. Let one member of the family
Ti iN1
be
semicompact. Let {  n } n 1 , {  n } n 1  [  ,1] be two real sequences satisfying the
conditions;

 1     
(i)
n
n 1

 1   
(ii)
n 1
(iii)
2
n

  1  
n 1
where  
n
L
1 L
 
 , 1   n  1   n L2  1 , 0     n    1,
and
L  maxLi  ,
Li
uniform Lipschitz constants of
Ti , i  1, 2, . . . , N . Let x1  K be arbitrary, the modified implicit iteration
sequence { xn } n 1 generated by
x n   n x n 1  1   n Ti k y n
y n   n x n 1  1   n Ti k x n , n  1
(12)
exists in K and converges strongly to a common fixed point p of the family
Ti iN1 .
Proof. We use the well known result of Reinermann [15] (See also Osilike and
Igbokwe [16]).
tx  1  t  y
2
 t x
2
 1  t  y
2
 t 1  t  x  y
2
(13)
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Approximation of Common Fixed Points of a Finite Family ...
which holds for all x, y  H , t  [0,1].
Using (12) and (13), we obtain
xn  p
2
  n xn1  p   1   n Ti k y n  p 
2
  n xn 1  p  1   n  Ti k y n  p   n 1   n  xn 1  Ti k y n
2
  n xn 1  p  L2 1   n  y n  p   n 1   n  x n1  Ti k y n
2
  n xn 1  p  L2 1   n  y n  p   n 1   n  xn 1  Ti k y n
2
2
2
2
2
2
2
  n x n 1  p  L2 1   n   n x n1  p   1   n Ti k x n  p 
2
  n 1   n  xn 1  Ti k yn
2
  n x n 1  p  L2 1   n   n x n 1  p
2
2
 1   n  Ti k xn  p   n 1   n  xn 1  Ti k xn
2
  n 1   n  xn1  Ti k y n
2

2
  n xn 1  p  L2  n 1   n  xn 1  p  L2 1   n 1   n  Ti k x n  p
2
2
 L2  n 1   n n 1   n  xn 1  Ti k xn
x n 1  Ti k y n

2
L  1  L 1   L  1 x
 L 1   L  1  L 1     x
2
2
n
n
2
n
2
n 1
n
n
p
2
2
2
4
n
 1   n  x n1  Ti k y n . (14)
n
p .
(15)
Substitute (15) into (14) to obtain
xn  p
2



[ n  L2  n 1   n   1   n L n  1  L2 1   n L n  1 ] x n 1  p
2
 L2 1   n 1   n  Ti k xn  p  L2  n 1   n 1   n  x n1  Ti k x n
2


 1   n  L2 1   n L n  1  L4 1   n  xn  p .
2
2
2
2
(16)
2
Donatus I. Igbokwe and Uko S. Jim
171
From (8), we have
Ti k xn  p
2

L  1
2




2
 1  1 a in2  1 x n  p   x n  Ti k x n .
2
(17)
Substitute (17) into (16) to obtain
xn  p
2



[ n  L2  n 1   n   1   n L n  1  L2 1   n L n  1 ] x n 1  p
2

2

 L2 1   n 1   n  x n  Ti k x n  L2  n 1   n 1   n  x n1  Ti k x n
 1   n  L2 1   n L n  1  L4 1   n 

  x

2
 L2 1   n L  1  1  1 ain2  1
2
2
2
2
n
p .
That is
 1  1   n L2 L n  1  L4 1   n   L2 L  12  1  1 2 ain2  1




2
 1   n  L2 L n  1  L4 1   n   L2 L  1  1  1 a in2  1  n   n
2
 1   n [ n  L2  n  L n  12  L2 1   n L n  1] x n 1  p




 xn  p 2
2
2
 L2 1   n 1   n   x n  Ti k x n  L2  n 1   n 1   n  xn 1  Ti k xn

2
2
(18)


2
Observed that L2 L n  1  L4 1   n   L2 L  1  1  1 ain2  1  1 . Using the
2
fact that 1   n   1, then,
1   n   1   n L2 L n  1  L4 1   n   L2 L  12  1  1 2 ain2  1  n   n 




2
 1   n  L2 L n  1  L4 1   n   L2 L  1  1  1 ain2  1  n   n .
2
So we obtain,
1  1  
n
 in  1   n 2  n in  1   n  n  x n  p 2

1   n [ n  L2  n  L n  12  L2 1   n L n  1] x n 1  p 2


2
 L2 1   n 1   n   x n  Ti k x n  L2  n 1   n 1   n  x n1  Ti k x n
2
2
,
172
Approximation of Common Fixed Points of a Finite Family ...



2
where  in  L2 L n  1  L4 1   n   L2 L  1  1  1 ain2  1
2
 0.
So that

xn  p
2

 1   n  L2  n  L n  12  L2 1   n L n  1 



  1     1   2  


n
in
n
n in
 1 
 x n 1  p
1  1   n  in 1  (1   n )  n    n 






L2 1   n 1   n 
 x n  Ti k x n
1  1   n  in 1  (1   n )  n    n 

2

2

2
L2  n 1   n 1   n 

x n 1  Ti k x n .
1  1   n  in 1  (1   n )  n    n 
2
(19)
Since lim 1   n   0, then there exists a natural number N 2 , such that  n  N 2
n
1  1   n  in 1  (1   n )  n    n  
1
.
2
Therefore, it follows from (19) that
xn  p
2


 [1  2[1   n  L2  n  L n  12  L2 1   n L n  1
 1   n  in  1   n   n  in ] x n 1  p
2

 L2 1   n 1   n   x n  Ti k x n
2
2

 L2  n 1   n 1   n  x n1  Ti k x n .
2
 [1   in ] x n 1  p
2
2

 L2 1   n 1   n   x n  Ti k x n
2
 L2  n 1   n 1   n  x n 1  Tn x n ,
2
2

(20)
where


 in  2[1   n  L2  n   L n  1  L2 1   n  L n  1
2
 1   n  in  1   n   nin ].
2
(21)
Donatus I. Igbokwe and Uko S. Jim
Since from condition (iii)


n 1
in
173
  , it follows from (21) and Lemma 2.1 that
lim xn1  p exists. Hence lim x n  p exists. So there exists M  0 such that
n 
n 
x n  p  M ,  n  1 , we obtain from (20) that

L2 1   n 1   n   x n  Ti k x n
2

1   in  xn1  p 2 

2
2
 1     x

j
j  N 1
j
 T jk x j

n 1
Since
2
xN  p  M 2

ij

2
n 1
  1      , it follows that lim inf   x
n 1

j  N 1
x N  p  M 2   in   .


n

2

  1   n   xn  Ti k xn

2
x n 1  p  x n  p  M 2 in

L2 1   
xn  p
n
n

 Ti k x n  0 .
Since  is an increasing and continuous function, then lim inf x n  Ti k x n  0 .
n
Furthermore, since x n  is bounded and lim 1   n   0 , it follows from
n
Lemma 2.1 that lim inf x n  Ti k x n  0 for all i  I .
n
Since one member of the family Ti iN1 is semicompact, there exists a subsequence
x 

nj
j 1
of
xn n 1
which converges strongly to
u  Ti u  Lim x n  Ti x n  0 for all i  I . Thus
n
u
and furthermore,
 
u  F . Since x n j

j 1
converges to u and Lim u  Ti u exists, It follows from Lemma 2.1, that xn n 1
n
converges strongly to u and hence the proof.

Since every asymptotically demicontractive maps T is asymptotically
  demicontractive map (see for example [2]), we have the following:
174
Approximation of Common Fixed Points of a Finite Family ...
Corollary 3.1 Let K be a nonempty closed convex subset of a real Hilbert space
H . Let Ti iN1 be N uniformly L  Lipschitzian asymptotically demicontractive
self maps of K with sequence ain   [1,  ) such that
i  I . Let F 
N
 F T   

 a
n 1
in
 1   for all
where F Ti   x  K , Ti x  x . Let one member of
i
i 1
the family Ti iN1 be semicompact. Let {  n } n 1 , {  n } n 1  [  ,1] be two real
sequences satisfying the conditions:
(i)

 1   n   
n 1
(ii)

2
 1   n 
  (iii)
n 1

 1    
n 1
n
.
(iv)  1   n
 1   n L2  1 ,
where  
L
and L  maxLi  , Li the Lipschitzian constants of Ti iN1 . Let
1 i  N
1 L
0     n    1,
x n  be the implicit iteration sequence generated by (12). Then { xn } n 1 exists in
K and converges strongly to a common fixed point p of the family Ti iN1 .
Remark
If we set  n  1 , the iteration scheme takes the non-implicit form:
x n   n x n 1  1   n T k x n 1 .
(22)
In the case of N=1, (22) becomes the modified Mann iteration process [13] given
by
x n   n x n 1  1   n T k x n 1 .
(23)
The conclusion of Theorem 3.1 and Corollary 3.1 are still valid. Hence, we state
the following theorem without proof:
Theorem 3.2 Let K be a nonempty closed convex subset of real Hilbert space H .
Let T be an L  Lipschitzian asymptotically   demicontractive self map of K
Donatus I. Igbokwe and Uko S. Jim
175
with sequence ain   [1,  ) such that

 a
n 1
in
 1   and F T   x  K , Tx  x   .
Let {  n } n 1  0,1 be a real sequence satisfying the conditions:
(i)

 1     
n 1
(ii)
n

 1 
n 1
n
2
 ,
0     n    1.
For arbitrary x1  K , let
x n 
be the averaging
Mann iteration process
generated by (23). If T is semicompact, Then { xn } n 1 exists in K and converges
strongly to a common fixed point p of the family T .
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