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 : DT 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
164
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 1N 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 iN1
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 1N 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 1LL ,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 iN1
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
1i 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
an1 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 iN1
be N
L maxLi ,
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 41 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 21 L x
1 L T x x 21 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
+ Lin 1 21 n L2 x n x n 1
2
in
21 n L3 y n x n .
(10)
Observe that
yn xn
2 L in 21 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 21 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 41 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 iN1 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 iN1
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 maxLi ,
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 iN1 .
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)
170
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 xn1 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 n1 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 n1 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 xn1 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 n1 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 n1 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 n1 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 12 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 12 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 12 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 12 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 n1 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 12 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 12 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 n1 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 nin ].
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 xn1 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 xn1 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 iN1 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 iN1 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 iN1 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 maxLi , Li the Lipschitzian constants of Ti iN1 . 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 iN1 .
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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