Research Inventy: International Journal of Engineering And Science Vol.4, Issue 8 (August 2014), PP 49-55 Issn (e): 2278-4721, Issn (p):2319-6483, www.researchinventy.com
Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite Family Of Generalized Asymptotically Quasi-Nonexpansive Mappings In Convex Metric Spaces Renu Chugh1 and Preety2 and Sanjay Kumar 3 1,2,3
Department of Mathematics, Maharshi Dayanand University, Rohtak-124001(INDIA)
ABSTRACT: In this paper, we prove the strong convergence of an implicit iterative scheme with errors to a common fixed point for a finite family of generalized asymptotically quasi-nonexpansive mappings in convex metric spaces. Our results refine and generalize several recent and comparable results in uniformly convex Banach spaces. With the help of an example we compare implicit iteration used in our result and some other implicit iteration.
KEYWORDS: Implicit iteration process with errors, Generalized asymptotically quasi-nonexpansive mappings, Common fixed point, Strong convergence, Convex metric spaces.
I.
INTRODUCTION AND PRELIMINARIES
In 1970, Takahashi [1] introduced the notion of convexity in metric space and studied some fixed point theorems for nonexpansive mappings in such spaces.
W : X 2 [0,1] X is a convex structure in X if d (u,W ( x, y, )) d (u, x) (1 )d (u, y) for all x, y, u X and [0,1]. A metric space ( X , d ) together with a convex structure W is known as convex metric space and is denoted by ( X , d , W ). A nonempty subset C of a convex metric space is convex if W ( x, y, ) C for all x, y C and [0,1] . All normed spaces and their subsets are the examples of Definition 1.1 [1] A map
convex metric spaces. Remark
1.2
Every
normed
space
is
a
convex
metric
space,
where
a
convex
structure
W ( x, y, z; , , ) x y z, for all x, y, z X and , , [0,1] with 1. In
fact,
d (u,W ( x, y, z; , , )) u ( x y z) u x u y u z d (u, x) d (u, y) d (u, z), for all
u X . But there exists some convex metric spaces which cannot be embedded into normed spaces. Throughout this paper, we assume that X is a metric space and let F (Ti ) (i N ) be the set of all fixed
points of mappings
Ti respectively, that is, F (Ti ) {x X : Ti x x}. The set of common fixed points of N
Ti (1, 2,......N ) denoted by F i 1 F (Ti ). Definition 1.3 [2, 3] Let is said to be (1)
T : X X be a mapping and F(T) denotes the fixed point of T. Then the mapping T
d (Tx, Ty) d ( x, y), x, y X .
(2)
nonexpansive if quasi-nonexpansive if F (T ) and
d (Tx, p) d ( x, p), x X , p F (T ). (3) asymptotically nonexpansive [4] if there exists a sequence
49
{un } in [0, ) with lim un 0 such that n
Convergence Theorems for Implicit...
d (T n x, T n y ) (1 un )d ( x, y ), x , y X and n 1. (4) asymptotically quasi-nonexpansive if F (T ) and there exists a sequence
{un } in [0, ) with
lim un 0 such that n
d (T n x, p ) (1 un )d ( x, p ), x X , p F (T ) and n 1. (5)
generalized
asymptotically
quasi-nonexpansive
[5]
if
F (T ) and there exist two sequences of real numbers {un } and {cn } with lim un 0 lim cn n
n
such that
d (T n x, p ) (1 un )d ( x, p ) cn , x X , p F (T ) and n 1. (6)
Uniformly
L-
Lipschitzian if there exists a positive constant L such that
d (T n x, T n y ) Ld ( x, y ), x, y X and n 1. Remark 1.4 From definition 1.2, it follows that if F(T) is nonempty, then a nonexpansive mapping is quasinonexpansive and an asymptotically nonexpansive mapping is asymptotically quasi-nonexpansive. But the converse does not hold. From (5), if
cn 0 for all n 1, then T becomes asymptotically quasi-nonexpansive.
Hence the class of generalized asymptotically quasi-nonexpansive maps includes the class of asymptotically quasi-nonexpansive maps. The Mann and Ishikawa iteration processes have been used by a number of authors to approximate the fixed point of nonexpansive, asymptotically nonexpansive mappings and quasi-nonexpansive mappings on Banach spaces (see e.g., [9-16]. In 1998, Xu [9] gave the following definitions: For a nonempty subset C of a normed space E and
T : C C , the Ishikawa iteration process with errors is the iterative sequence { xn } defined by x1 x C ,
xn 1 n xn nT n yn nun ,
(1.1)
yn n' xn n' T n xn n' vn , n 1, where
{un }, {vn } are bounded sequence in C and { n }, { n }, { n }, { n' }, { n' }, { n' } are sequences in
[0,1] such that If
n n n 1 n' n' n' for all n 1.
n' 0 n' for all n 1, then (1.3) reduces to Mann iteration process with errors. The normal Ishikawa
and Mann iteration processes are special cases of the Ishikawa iteration process with errors. In 2001, Xu and Ori [6] have introduced an implicit iteration process for a finite family of nonexpansive mappings in a Hilbert space H. Let C be a nonempty subset of H. Let selfmappings of C and suppose that
T1 , T2 ,......., TN be
F i 1 F (Ti ) , the set of common fixed points of N
Ti , i 1, 2,.....N . An implicit iteration process for a finite family of nonexpansive mappings is defined as
{t n } a real sequence in (0, 1), x0 C : x1 t1 x0 (1 t1 )T1 x1 ,
follows: Let
x2 t2 x1 (1 t2 )T2 x2 , ... .... xN t N xn 1 (1 t N )TN xN , xN 1 t N 1 xN (1 t N 1 )T1 xN 1 , .... .... which can be written in the following compact form:
50
Convergence Theorems for Implicit...
xn tn xn1 (1 tn )Tn xn ,
n 1,
(1.2)
where Tk Tk (mod N ) . (Here the mod N function takes values in the set {1,2,….N}). They proved the weak convergence of the iterative scheme (1.2). In 2003, Sun [7] extend the process (1.2) to a process for a finite family of asymptotically quasinonexpansive mappings, with follows:
{ n } a real sequence in (0,1) and an initial point x0 C , which is defined as
x1 1 x0 (1 1 )T1 x1 , x2 2 x1 (1 2 )T2 x2 , ... .... xN N xN 1 (1 N )TN xN , xN 1 N 1 xN (1 N 1 )T12 xN 1 , .... .... x2 N 2 N x2 N 1 (1 2 N )TN2 x2 N , x2 N 1 2 N 1 x2 N (1 2 N 1 )T13 x2 N 1 , .... ....
which can be written in the following compact form:
xn n xn 1 (1 n )Ti k xn , n 1, Where n (k 1) N i, i {1, 2,....N}.
(1.3)
Sun [7] proved the strong convergence of the process (1.3) to a common fixed point in real uniformly
{Ti , i 1, 2,.....N } to be semi compact. Definition 1.5 Let C be a convex subset of a convex metric space ( X , d ,W ) with a convex structure W and let I is the indexing set i.e. I {1, 2,....N}. Let {Ti : i I } be N generalized asymptotically quasiconvex Banach spaces, requiring only one member T in the family
nonexpansive mappings on C. For any given
x0 C , the iteration process {xn } defined by
x1 W ( x0 , T1 x1 , u1;1 , 1 , 1 ), x2 W ( x1 , T2 x2 , u2 ; 2 , 2 , 2 ), ... ... xN W ( xN 1 , TN xN , u N ; N , N , N ), xN 1 W ( xN , T12 xN 1 , u N 1 ; N 1 , N 1 , N 1 ), ... .... x2 N W ( x2 N 1 , TN2 x2 N , u2 N ; 2 N , 2 N , 2 N ), x2 N 1 W ( x2 N , T13 x2 N 1 , u2 N 1; 2 N , 2 N , 2 N ), ... ... where {un } is a
bounded sequence in C and
{ n }, { n }, { n } are sequences in [0,1] such that
n n n 1. The above sequence can be written in compact form as
xn W ( xn 1 , Ti k xn , un ; n , n , n ), n 1,
(1.4)
with n ( k 1) N i, i I and Tn Ti (mod N ) Ti . Let
{Ti : i I } be N uniformly L-Lipschitzian generalized asymptotically quasi-nonexpansive selfmappings of
C. Then clearly (1.5) exists. If
un 0 in 1.5 then, 51
Convergence Theorems for Implicit...
xn W ( xn 1 , Ti k xn ; n , n ), n 1,
(1.5)
with n ( k 1) N i, i I and Tn Ti (mod N ) Ti
and
{ n }, { n } are sequences in [0,1] such that
n n 1. The purpose of this paper is to prove the strong convergence of implicit iteration with errors defined by (1.5) for generalized asymptotically quasi-nonexpansive in the setting of convex metric spaces. The following Lemma will be used to prove the main results: Lemma 1.6 [8] Let
{an } and {bn } be sequences of nonnegative real numbers satisfying the inequality
an 1 an bn , n 1. If
b , then lim an exists. In particular, if {an } has a subsequence converging to zero, then
n 1 n
n
lim an 0. n
Now we prove our main results: 2. Main Results Theorem 2.1 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
Ti : C C (i I {1, 2.....N }) be N uniformly L-Lipschitzian generalized quasi-nonexpansive mappings with
{uin }, {cin } [0, ) such that
uin and n 1
c n 1
in
. Suppose that F i 1 F (Ti ) . Let {xn } N
be the implicit iteration process with errors defined by (1.4) with the restrictions
n 1
n
and
1 { n } ( s,1 s) for some s (0, ). Then lim d ( xn , F) exists. n 2 Proof: Let p F
N i 1
F (Ti ), using (1.5) and (5), we have
d ( xn , p ) d (W ( xn 1 , Ti k xn , un ; n , n , n ), p ) n d ( xn 1 , p ) n d (Ti k xn , p ) n d (un , p )
n d ( xn1 , p) n [(1 uik )d ( xn , p ) cik ] n d (un , p ) n d ( xn 1 , p) (1 n n )[(1 uik )d ( xn , p ) cik ] n d (un , p ) n d ( xn1 , p) (1 n )[(1 uik )d ( xn , p ) cik ] n d (un , p ) n d ( xn1 , p) (1 n )(1 uik )d ( xn , p ) (1 n )cik n d (un , p ) n d ( xn 1 , p) (1 n uik )d ( xn , p) (1 n )cik n d (un , p) s Since lim n 0, there exists a natural number n1 such that for n n1 , n . n 2
(2.1.1)
Hence,
s 2
n 1 n n 1 (1 s) for
s 2
(2.1.2)
n n1. Thus, We have from (2.1.1) that n d ( xn , p) n d ( xn1 , p) uik d ( xn , p) (1 n )cik n d (un , p)
(2.1.3)
and
1 d ( xn , p) 1 cik n d (un , p) n n n 2u 2 2 d ( xn 1 , p) ik d ( xn , p) 1 cik n d (un , p) s s s
d ( xn , p) d ( xn1 , p)
uik
52
(2.1.4)
Convergence Theorems for Implicit...
s s 2M 2 s d ( xn , p) n d ( xn1 , p) 1 cik s s 2ik s 2ik s 2ik s 2ik 2ik 2ik 2M 2 1 n d ( xn1 , p) 1 1 cik 1 s 2 s s 2 s 2 s ik ik ik (2.1.5) where
M sup n1{d (un , p)}, since {un } is a bounded sequence in C. Since
This gives that there exists a natural number
u n 1
ik
for all i I .
n0 (as k (n0 / N ) 1) such that s 2ik 0 and
cik . Since k 1 uik and cik for all i I , it follows that vik and tik . Therefore from (2.1.5) we k 1 k 1 k 1
ik s / 4
for all
n n0 . Let vik
2ik 2ik 2 and tik 1 1 s 2ik s s 2ik
get,
d ( xn , p) (1 vik )d ( xn1 , p) tik
2M (1 vik ) n s
(2.1.6)
This further implies that
2M (1 vik ) n (2.1.7) s Since by assumptions, vik , tik and ik . Therefore, applying Lemma (1.6) k 1 k 1 k 1 d ( xn , F ) (1 vik )d ( xn1 , F ) tik
to the inequalities (2.1.6) and (2.1.7), we conclude that both
lim d ( xn , p) and lim d ( xn , F) exists. n
n
Theorem 2.2 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
Ti : C C (i I {1, 2.....N }) be N uniformly L-Lipschitzian generalized asymptotically nonexpansive mappings with
{uin }, {cin } [0, ) such that
uin and n 1
c n 1
in
quasi-
. Suppose that
F i 1 F (Ti ) . Let { xn } be the implicit iteration process with errors defined by (1.4) with the N
1 and { n } ( s,1 s) for some s (0, ). Then the sequence {xn } converges 2 n 1 strongly to a common fixed point p of the mapping {Ti : i I } if and only if restrictions
n
liminf d ( xn , F) 0. n
Proof: From Theorem 2.1 Since by hypothesis
lim d ( xn , F) exists. The necessity is obvious. Now we only prove the sufficiency. n
liminf d ( xn , F) 0, so by Lemma 1.6, we have n
lim d ( xn , F) 0.
(2.2.1)
n
{xn } is a Cauchy sequence in C. Note that when x 0, 1 x e x . It follows from (2.1.6) that for any m 1, for all n n0 and for any p F, we have Next we prove that
N N d ( xnm , p) exp vik d ( xn , p ) tik i 1 k 1 i 1 k 1 N 2M exp vik n s i 1 k 1 n1
53
Convergence Theorems for Implicit... N
Qd ( xn , p ) tik i 1 k 1
2QM n, s n 1
(2.2.2)
N Q exp vik 1 . i 1 k 1 Let 0. Also lim d ( xn , p) exists, therefore for 0, there exists a natural number n1 such that where,
n
d ( xn , p ) / 6(1 Q),
N
tik / 3 and i 1 k 1
n 1
such that d ( xn1 , p ) / 3(1 Q ). Hence, for all *
n
s / 6QM for all n n1. So we can find p* F
n n1 and m 1, we have that
d ( xn m , xn ) d ( xn m , p ) d ( xn , p ) *
*
N
Qd ( xn1 , p* ) tik i 1 k n1
2QM s
n n1
n
d ( xn1 , p* )
2QM n s n n1 i 1 k n1 2QM s (2.2.3) (1 Q). . . 3(1 Q) 3 s 6QM This proves that { xn } is a Cauchy sequence in C. And, the completeness of X implies that { xn } must be N
(1 Q)d ( xn1 , p* ) tik
convergent. Let us assume that mappings with
lim xn p. Now, we show that p is common fixed point of the mappings the n
{Ti : i I }. And, we know that F i 1 F (Ti ) is closed. From the continuity of d ( x, F) 0 N
lim d ( xn , F) 0 and lim xn p, we get n
n
d ( p, F) 0,
(2.2.4)
and so p F , that is p is a common fixed point of the mappings {T } . Hence the proof. N i i 1
If
un 0 , in above theorem, we obtain the following result:
Theorem 2.3 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
Ti : C C (i I {1, 2.....N }) be N uniformly L-Lipschitzian generalized asymptotically nonexpansive mappings with
{cin } [0, ) such that
c n 1
in
quasi-
. Suppose that F i 1 F (Ti ) . Let N
{xn } be the implicit iteration process with errors defined by (1.5) with { n } ( s,1 s) for some 1 s (0, ). Then the sequence {xn } converges strongly to a common fixed point p of the mapping 2 {Ti : i I } if and only if liminf d ( xn , F) 0. n
From Lemma 1.6 and Theorem 2.2, we can easily obtain the result. Corollary 2.4 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
Ti : C C (i I {1, 2.....N }) be N uniformly L-Lipschitzian generalized asymptotically nonexpansive mappings with
{uin }, {cin } [0, ) such that 54
u n 1
in
and
c n 1
in
quasi-
. Suppose that
Convergence Theorems for Implicit...
F i 1 F (Ti ) . Let { xn } be the implicit iteration process with errors defined by (1.4) with the N
1 and { n } ( s,1 s) for some s (0, ). Then the sequence {xn } converges 2 n 1 strongly to a common fixed point p of the mapping {Ti : i I } if and only if there exists a subsequence {xnj } restrictions
n
{xn } which converges to p.
of
Numerical Example: Let defined by,
Tn x
{Ti }in1
be the family of generalized asymptotically quasi- non expansive mappings
x , n 1 , we know that for n 1, given family is non expansive and since zero is n2
the only common fixed point of given family, so it is quasi-nonexpansive and so generalized asymptotically quasi-nonexpansive. Now the initial values used in C++ program for formation of our result are x0=.5 and
n 0 , we have following observations about the common fixed point of family of generalized
asymptotically quasi non expansive mappings and with help of these observations we prove that the implicit iteration (1.2) and (1.3) has same rate of convergence and converges fast as
n 0 .From the table given
below we can prove the novelty of our result. Tabular observations of Implicit Iteration (1.2) and (1.3), where initial approximation is x0==0.5. N
xn
0 1e-011
1 1.5e-022
2 2e-033
3 2.5e-044
-----
-----
27 1.45e-307
28 1.5e-318
5.02488e012
5.04988e023
5.075e034
5.10025e045
---
---
5.74538e309
5.77415e320
29 0
30 0
0
0
for implicit iteration (1.2)
xn for implicit k-step iteration (1.3)
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