ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 4 Issue 1(2012), Pages 13-23.
AN IMPLICIT ITERATION SCHEME WITH ERRORS FOR COMMON FIXED POINTS OF GENERALIZED ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS
(COMMUNICATED BY VIJAY GUPTA)
GURUCHARAN SINGH SALUJA, HEMANT KUMAR NASHINE
Abstract. In this paper, we study an implicit iteration scheme with errors for a finite family of generalized asymptotically quasi-nonexpansive mappings and give the necessary and sufficient condition to converge to common fixed points for the said scheme in Banach spaces. The results presented in this paper generalize, improve and unify the corresponding results in [3, 4, 7, 9, 11, 12, 13, 15, 16, 17, 18].
1. Introduction
It is well known that the concept of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [8] who proved that every asymptotically nonex-pansive self-mapping of nonempty closed bounded and convex subset of a uniformly convex Banach space has fixed point. In 1973, Petryshyn and Williamson [16] gave necessary and sufficient conditions for Mann iterative sequence [14] to converge to fixed points of quasi-nonexpansive mappings. In 1997, Ghosh and Debnath [7] extended the results of Petryshyn and Williamson [16] and gave necessary and suf-ficient conditions for Ishikawa iterative sequence to converge to fixed points for quasi-nonexpansive mappings.
Liu [13] extended results of [7, 16] and gave necessary and sufficient conditions for Ishikawa iterative sequence with errors to converge to fixed point of asymptotically quasi-nonexpansive mappings.
In 2003, Zhou et al. [26] introduced a new class of generalized asymptotically nonexpansive mapping and gave a necessary and sufficient condition for the modi-fied Ishikawa and Mann iterative sequences to converge to fixed points for the class of mappings. Atsushiba [1] studied the necessary and sufficient condition for the convergence of iterative sequences to a common fixed point of the finite family of
2000Mathematics Subject Classification. 47H09, 47H10, 47J25.
Key words and phrases. Generalized asymptotically quasi-nonexpansive mapping, implicit it-eration scheme with bounded errors, common fixed point, Banach space.
c
⃝2012 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted Jan 28, 2011. Published December 3, 2011.
asymptotically nonexpansive mappings in Banach spaces. Suzuki [21], Zeng and Yao [25] discussed a necessary and sufficient condition for common fixed points of two nonexpansive mappings and a finite family of nonexpansive mappings, and proved some convergence theorems for approximating a common fixed point, re-spectively (see, also [2, 5, 6, 19, 27]).
Recently, Lan [11] introduced a new class of generalized asymptotically quasi-nonexpansive mappings and gave necessary and sufficient condition for the 2-step modified Ishikawa iterative sequences to converge to fixed points for the class of mappings.
More recently, Nantadilok [15] (Thai J. Math. 6(2) (2008), 297-308) extended and improved the result of Lan [11] and gave the necessary and sufficient condition for convergence of common fixed point for three-step iteration scheme with errors for generalized asymptotically quasi-nonexpansive mappings.
In 2001, Xu and Ori [24] have introduced an implicit iteration process for a finite family of nonexpansive mappings in a Hilbert spaceH. LetCbe a nonempty subset ofH. LetT1, T2, . . . , TN be self-mappings ofCand suppose thatF=∩Ni=1F(Ti)̸=
∅, the set of common fixed points ofTi, i= 1,2, . . . , N. An implicit iteration process
for a finite family of nonexpansive mappings is defined as follows, with{tn}a real
sequence in (0,1), x0∈C:
x1 = t1x0+ (1−t1)T1x1,
x2 = t2x1+ (1−t2)T2x2,
.. .
xN = tNxN−1+ (1−tN)TNxN,
xN+1 = tN+1xN + (1−tN+1)T1xN+1,
.. .
which can be written in the following compact form:
xn = tnxn−1+ (1−tn)Tnxn, n≥1 (1.1)
where Tk = Tkmod N. (Here the mod N function takes values in N). And they
proved the weak convergence of the process (1.1).
In 2003, Sun [20] extended the process (1.1) to a process for a finite family of asymptotically quasi-nonexpansive mappings, with {αn} a real sequence in (0,1)
x1 = α1x0+ (1−α1)T1x1,
.. .
xN = αNxN−1+ (1−αN)TNxN,
xN+1 = αN+1xN+ (1−αN+1)T12xN+1,
.. .
x2N = α2Nx2N−1+ (1−α2N)TN2x2N,
x2N+1 = α2N+1x2N + (1−α2N+1)T13x2N+1,
.. .
which can be written in the following compact form:
xn = αnxn−1+ (1−αn)Tikxn, n≥1 (1.2)
wheren= (k−1)N+i,i∈ N.
Sun [20] proved the strong convergence of the process (1.2) to a common fixed point, requiring only one memberT in the family{Ti:i∈ N }to be semi-compact.
The result of Sun [20] generalized and extended the corresponding main results of Wittmann [23] and Xu and Ori [24].
The purpose of this paper is to study an implicit iteration process with errors which converges strongly to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. Also prove some strong convergence theorems for said mappings.
Let X be a normed space, C be a nonempty closed convex subset of X, and
Ti: C→C,i={1,2, . . . , N}=N beNgeneralized asymptotically quasi-nonexpansive
mappings with respect to{rni}and{sni}with
∑∞
n=1rni <∞and
∑∞
n=1sni <∞. Define a sequence{xn}in Cas follows:
x1 = α1x0+β1T1x1+γ1u1,
x2 = α2x1+β2T2x2+γ2u2,
.. .
xN = αNxN−1+βNTNxN +γNuN,
xN+1 = αN+1xN +βN+1T12xN+1+γN+1uN+1,
.. .
x2N = α2Nx2N−1+β2NTN2x2N +γ2Nu2N,
x2N+1 = α2N+1x2N +β2N+1T13x2N+1+γ2N+1u2N+1,
.. .
which can be written in the following compact form:
wheren= (k−1)N+i,i∈ N and{αn},{βn}and{γn}are real sequences in [0,1]
such thatαn+βn+γn= 1 and{un} is a bounded sequence inC.
2. Preliminaries
In the sequel, we need the following definitions and lemmas for our main results in this paper.
Definition 2.1. (see [15]) Let X be a real Banach space, C be a nonempty subset ofX andF(T) denotes the set of fixed points ofT. A mappingT:C→C
is said to be
(1) nonexpansive if
∥T x−T y∥ ≤ ∥x−y∥ (2.1) for allx, y∈C,
(2) quasi-nonexpansive ifF(T)̸=∅ and
∥T x−p∥ ≤ ∥x−p∥ (2.2) for allx∈C andp∈F(T),
(3) asymptotically nonexpansive if there exists a sequence {rn} ⊂ [0,∞) with
rn→0 asn→ ∞such that
∥Tnx−Tny∥ ≤ (1 +rn)∥x−y∥ (2.3)
for allx, y∈C,
(4) asymptotically quasi-nonexpansive if (3) holds for allx∈C andy∈F(T);
(5) generalized quasi-nonexpansive with respect to{sn}, if there exists a sequence
{sn} ⊂[0,1) with sn→0 asn→ ∞such that
∥Tnx−p∥ ≤ ∥x−p∥+sn∥x−Tnx∥ (2.4)
for allx∈C,p∈F(T) andn≥1,
(6) generalized asymptotically quasi-nonexpansive with respect to{rn}and{sn} ⊂
[0,1) with rn →0 andsn →0 asn→ ∞such that
∥Tnx−p∥ ≤ (1 +rn)∥x−p∥+sn∥x−Tnx∥ (2.5)
for allx∈C,p∈F(T) andn≥1.
Remark 2.1.From the above definitions, it is clear that:
(i) a nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping;
(ii) a quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping;
(iv) a generalized quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive mapping.
However, the converse of the above statements are not true.
Lemma 2.1. (see [22]) Let {an},{bn} and{δn} be sequences of nonnegative real numbers satisfying the inequality
an+1≤(1 +δn)an+bn, n≥1.
If ∑∞n=1δn<∞ and
∑∞
n=1bn<∞, thenlimn→∞an exists. In particular, if{an} has a subsequence converging to zero, thenlimn→∞an = 0.
Lemma 2.2. (see[11]) LetC be nonempty closed subset of a Banach spaceX and
T: C → C be a generalized asymptotically quasi-nonexpansive mapping with the fixed point set F(T)̸=∅. ThenF(T)is closed subset inC.
3. Main Results
In this section, we prove strong convergence theorems of an implicit iteration scheme with bounded errors for a finite family of generalized asymptotically quasi-nonexpansive mappings in a real Banach space.
Theorem 3.1. Let X be a real arbitrary Banach space, C be a nonempty closed convex subset of X. Let Ti:C → C, i = {1,2, . . . , N} = N be N generalized asymptotically quasi-nonexpansive mappings with respect to{rni}and{sni}for all i∈ N such that∑∞n=1rni+2sni
1−sni <∞. Let{xn}be the sequence defined by (1.3) with
∑∞
n=1γn <∞. If F =∩Ni=1F(Ti)̸=∅ and {βn} ⊂(s,1−s) for some s∈(0,12). Then the sequence{xn} converges strongly to a common fixed point of{Ti:i∈ N } if and only if lim inf
n→∞ d(xn,F) = 0, where d(x,F) denotes the distance between x and the setF.
Proof. The necessity is obvious and it is omitted. Now we prove the sufficiency. Letp∈ F, then it follows from (2.5), we have
∥xn−Tikxn∥ ≤ ∥xn−p∥+∥Tikxn−p∥
≤ ∥xn−p∥+ (1 +rki)∥xn−p∥+ski∥xn−T
k ixn∥
≤ (2 +rki)∥xn−p∥+ski∥xn−T
k i xn∥
≤ (2 +rki)∥xn−p∥+ski∥xn−T
k i xn∥
which implies that
∥xn−Tikxn∥ ≤
2 +rki 1−ski
∥xn−p∥. (3.1)
Again for any p∈ F, wheren = (k−1)N +i, Tn =Tn(mod N) =Ti, i ∈ N, it
∥xn−p∥ = ∥αnxn−1+βnTikxn+γnun−p∥
= ∥αn(xn−1−p) +βn(Tikxn−p) +γn(un−p)∥
≤ αn∥xn−1−p∥+βn∥Tikxn−p∥+γn∥un−p∥
≤ αn∥xn−1−p∥+βn
[
(1 +rki)∥xn−p∥+ski∥xn−T
k ixn∥
]
+γn∥un−p∥
≤ αn∥xn−1−p∥+βn
[
(1 +rki)∥xn−p∥+ski.
(2 +rk
i 1−ski
)
∥xn−p∥
]
+γn∥un−p∥
≤ αn∥xn−1−p∥+βn
(1 +r
ki+ski 1−ski
)
∥xn−p∥+γn∥un−p∥
≤ αn∥xn−1−p∥+ (1−αn)
(
1 +rki+ 2ski 1−ski
)
∥xn−p∥+γn∥un−p∥
≤ αn∥xn−1−p∥+ (1−αn)(1 +wki)∥xn−p∥+γn∥un−p∥
≤ αn∥xn−1−p∥+ (1−αn+wki)∥xn−p∥+γn∥un−p∥ (3.2)
wherewki =
rki+2ski
1−ski . Since limn→∞γn = 0, there exists a natural numbern1 such
that forn > n1,γn≤ s2. Hence
αn = 1−βn−γn≥1−(1−s)−
s
2 =
s
2
forn > n1. Thus, we have from (3.2) that
αn∥xn−p∥ ≤αn∥xn−1−p∥+wki∥xn−p∥+γn∥un−p∥
and
∥xn−p∥ ≤ ∥xn−1−p∥+ wki
αn∥
xn−p∥+
γn
αn∥
un−p∥
≤ ∥xn−1−p∥+
2
swki∥xn−p∥+ 2
sγn∥un−p∥. (3.3)
Since ∑∞k=1rki+2ski
1−ski < ∞, it follows that
∑∞
k=1wki < ∞ for all i ∈ N and lim
n→∞wni= 0 for eachi∈ N. Hence there exists a natural numbern2, asn > n2
N +1,
that is,n > n2 such that
wni ≤ s
4, ∀ i∈ N. (3.4)
Then (3.3) becomes
∥xn−p∥ ≤
s s−2wki
∥xn−1−p∥+
2γn
s−2wki
∥un−p∥. (3.5)
Let
1 +tki = s s−2wki
= 1 + 2wki s−2wki
.
Then
tki = 2wki s−2wki
Therefore
∞
∑
k=1 tki <
4
s ∞
∑
k=1
wki <∞, ∀ i∈ N (3.6)
and (3.5) becomes
∥xn−p∥ ≤ (1 +tki)∥xn−1−p∥+ 2γn
s−2wki
∥un−p∥
≤ (1 +tki)∥xn−1−p∥+ 4
sγnK, (3.7)
whereK= supn≥1{∥un−p∥}, since{un}is a bounded sequence inC. This implies
that
d(xn,F) ≤ (1 +tki)d(xn−1,F) + 4
sγnK. (3.8)
Since ∑∞
k=1
tki < ∞ and ∞
∑
n=1
γn < ∞, it follows from Lemma 2.1, we know that
lim
n→∞d(xn,F) = 0.
Next, we will prove that{xn} is a Cauchy sequence. Notice that when x >0,
1 +x≤ex, from (3.7) we have
∥xn+m−p∥ ≤ (1 +tki)∥xn+m−1−p∥+ 4K
s γn+m
≤ (1 +tki)
[
(1 +tki)∥xn+m−2−p∥+ 4K
s γn+m−1 ]
+4K
s γn+m
≤ (1 +tki)
2∥x
n+m−2−p∥+
4K
s (1 +tki)
[
γn+m−1+γn+m
]
≤ (1 +tki)
2[(1 +t
ki)∥xn+m−3−p∥+ 4K
s γn+m−2 ]
+4K
s (1 +tki)
[
γn+m−1+γn+m
]
≤ (1 +tki)
3∥
xn+m−3−p∥
+4K
s (1 +tki)
[
γn+m−2+γn+m−1+γn+m
] ≤ .. . ≤ ≤ exp {N ∑ i=1 ∞ ∑ k=1 tki
}
∥xn−p∥+
4K s exp {N ∑ i=1 ∞ ∑ k=1 tki
} n+m ∑
j=n+1 γj
≤ K′∥xn−p∥+
4KK′ s
n∑+m
j=n+1
γj, (3.9)
for allp∈ Fandm, n∈N, whereK′= exp
{∑N
i=1
∞
∑
k=1 tki
}
<∞. Since lim
n→∞d(xn,F) =
0 and ∑∞
k=1
d(xn,F)<
ε
4K′ and
n∑+m
j=n1+1 γj ≤
s.ε
16KK′. (3.10)
Thus there exists a point q ∈ F such that d(xn1, q)<
ε
4K′. It follows from (3.9)
that for alln≥n1 andm≥1, we have
∥xn+m−xn∥ ≤ ∥xn+m−q∥+∥xn−q∥
≤ K′∥xn1−q∥+
4KK′ s
n∑+m
j=n1+1 γj
+K′∥xn1−q∥+
4KK′ s
n∑+m
j=n1+1 γj
< K′. ε
4K′ +
4KK′ s .
s.ε
16KK′
+K′. ε
4K′ +
4KK′ s .
s.ε
16KK′
= ε. (3.11)
This implies that{xn}is a Cauchy sequenceX. Thus the completeness ofXimplies
that {xn} must be convergent. Let limn→∞xn =p, that is, {xn} converges top.
Thenp∈C, becauseC is a closed subset ofX. By Lemma 2.2 we know that the setF is closed. From the continuity ofd(xn,F) with
d(xn,F)→0 and xn→p as n→ ∞,
we get
d(p,F) = 0
and sop∈ F =∩N
i=1F(Ti), that is,pis a common fixed point of{Ti :i∈ N }. This
completes the proof.
Theorem 3.2. Let X be a real arbitrary Banach space, C be a nonempty closed convex subset of X. Let Ti:C → C, i = {1,2, . . . , N} = N be N generalized asymptotically quasi-nonexpansive mappings with respect to {rni} and {sni} for
all i ∈ N such that ∑∞n=1rni+2sni
1−sni < ∞. Let {xn} be the sequence defined by (1.3) with ∑∞n=1γn <∞. IfF =∩Ni=1F(Ti)̸=∅ and {βn} ⊂ (s,1−s) for some
s∈(0,12). Then{xn}converges strongly to a common fixed pointpof the mappings
{Ti:i∈ N }if and only if there exists a subsequence{xnj}of{xn}which converges
top.
Proof. The proof of Theorem 3.2 follows from Lemma 2.1 and Theorem 3.1.
Theorem 3.3. Let X be a real arbitrary Banach space, C be a nonempty closed convex subset of X. Let Ti:C → C, i = {1,2, . . . , N} = N be N generalized asymptotically quasi-nonexpansive mappings with respect to{rni}and{sni}for all i∈ N such that∑∞n=1rni+2sni
∑∞
n=1γn <∞. If F =∩Ni=1F(Ti)̸=∅ and {βn} ⊂(s,1−s) for some s∈(0,12). Suppose that the mappings {Ti:i∈ N }satisfy the following conditions:
(C1) limn→∞∥xn−Tixn∥= 0 for alli∈ {1,2, . . . , N}=N;
(C2)there exists a constant A >0 such that{∥xn−Tixn∥} ≥Ad(xn,F)for all
i∈ {1,2, . . . , N}=N and for alln≥1.
Then{xn}converges strongly to a common fixed point of the mappings{Ti:i∈
N }.
Proof. From condition (C1) and (C2), we have limn→∞d(xn,F) = 0, it follows as
in the proof of Theorem 3.1, that{xn}must converges strongly to a common fixed
point of the mappings{Ti:i∈ N }.
Remark 3.1.Theorem 3.1 extends, improves and unifies the corresponding re-sults of [3, 7, 11, 12, 13, 15, 16, 18]. Especially Theorem 3.1 extends, improves and unifies Theorem 1 and 2 in [13], Theorem 1 in [12] and Theorem 3.2 in [18] in the following ways:
(1) The asymptotically quasi-nonexpansive mapping in [12], [13] and [18] is re-placed by finite family of generalized asymptotically quasi-nonexpansive mappings.
(2) The usual Ishikawa [10] iteration scheme in [12], the usual modified Ishikawa iteration scheme with errors in [13] and the usual modified Ishikawa iteration scheme with errors for two mappings are extended to the implicit iteration scheme with errors for a finite family of mappings.
Remark 3.2. Theorem 3.2 extends, improves and unifies Theorem 3 in [13] and Theorem 3.3 extends, improves and unifies Theorem 3 in [12] in the following aspects:
(1) The asymptotically quasi-nonexpansive mapping in [12] and [13] is replaced by finite family of generalized asymptotically quasi-nonexpansive mappings.
(2) The usual Ishikawa [10] iteration scheme in [12] and the usual modified Ishikawa iteration scheme with errors in [13] are extended to the implicit iteration scheme with errors for a finite family of mappings.
Remark 3.3.Theorem 3.1 also extends the corresponding results of [11] and [15] to the case of implicit iteration scheme with errors for a finite family of mappings considered in this paper.
Remark 3.4.Our results also extend the corresponding results of [9, 17] and [4] to the case of more general class of asymptotically nonexpansive mappings and implicit iteration scheme with errors respectively, considered in this paper.
Acknowledgments. The authors would like to thank the referee for his valuable suggestions on the manuscript.
References
[2] S.S. Chang, Y.J. Cho and Y.X. Tian,Strong convergence theorems of Reich type iterative sequence for non-self asymptotically nonexpansive mappings, Taiwan. J. Math., 11(2007) 729-743.
[3] C.E. Chidume and Bashir Ali, Convergence theorems for finite families of asymptotically quasi-nonexpansive mappings, J. Inequalities and Applications, 2007(2010), Article ID 68616, 10 pages.
[4] Y.J. Cho, J.K. Kim and H.Y. Lan,Three step procedure with errors for generalized asymptoti-cally quasi-nonexpansive mappings in Banach spaces, Taiwan. J. Math.,12(2008) 2155-2178. [5] Y.J. Cho, S.M. Kang and X. Qin,Strong convergence of an implicit iterative process for an
infinite family of strict pseudocontraction, Bull. Korean Math. Soc.,47(2010) 1259-1268. [6] Y.J. Cho, N. Hussain and H.K. Pathak, Approximation of nearest common fixed points of
asymptotically I-nonexpansive mappings in Banach spaces, Commun. Korean Math. Soc.,
26(2011) 483-498.
[7] M.K. Ghosh and L. Debnath,Convergence of Ishikawa iterates of quasi-nonexpansive map-pings, J. Math. Anal. Appl.,207(1997) 96-103.
[8] K. Goebel and W.A. Kirk,A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc.,35(1972) 171-174.
[9] W. Guo and Y.J. Cho,On the strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings, Appl. Math. Lett., 21(2008) 1046-1052.
[10] S. Ishikawa,Fixed point by a new iteration method, Proc. Amer. Math. Soc.,44(1974) 147-150.
[11] H.Y. Lan,Common fixed point iterative processes with errors for generalized asymptotically quasi-noexpansive mappings, Comput. Math. Appl.,52(2006) 1403-1412.
[12] Q.H. Liu,Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl.,259(2001) 1-7.
[13] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl.,259(2001) 18-24.
[14] W.R. Mann,Mean value methods in iteration, Proc. Amer. Math. Soc.,4(1953) 506-510. [15] J. Nantadilok,Three-step iteration scheme with errors for generalized asymptotically
quasi-nonexpansive mappings, Thai J. Math.,6(2) (2008) 297-308.
[16] W.V. Petryshyn and T.E. Williamson,Strong and weak convergence of the sequence of suc-cessive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl., 43(1973) 459-497.
[17] X. Qin, Y.J. Cho and M. Shang, Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings, Appl. Math. Comput.,219(2009) 542-550.
[18] N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications,
2006(2006), Article ID 18909, Pages 1-10.
[19] Y.S. Song and Y.J. Cho,Averaged iterates for non-expansive nonself mappings in Banach spaces, J. Comput. Anal. Appl.,11(2009) 451-460.
[20] Z.H. Sun,Strong convergence of an implicit iteration process for a finite family of asymptot-ically quasi-nonexpansive mappings, J. Math. Anal. Appl.,286(2003) 351-358.
[21] T. Suzuki, Common fixed points of two nonexpansive mappings in Banach spaces, Bull. Austral. Math. Soc.,69(2004) 1-18.
[22] K.K. Tan and H.K. Xu,Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl.,178(1993) 301-308.
[23] R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math.,
58(1992) 486-491.
[24] H.K. Xu and R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim.,22(2001) 767-773.
[25] L.C. Zeng and J.C. Yao,Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal. Series A: Theory and Applications,64(1)(2006) 2507-2515.
[27] H.Y. Zhou, Y.J. Cho and S.M. Kang, A new iterative algorithm for approximating com-mon fixed points for asymptotically nonexpansive mappings, Fixed Point Theory Appl.,
2007(2007), Article ID 64974, 10 pages, Gurucharan Singh Saluja
Department of Mathematics and Information Technology, Govt. Nagarjuna P.G. Col-lege of Science, Raipur (C.G.), India.
E-mail address:saluja [email protected] Hemant Kumar Nashine
Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha - Chandrakhuri Marg, Mandir Hasaud, Raipur - 492101(Chhattis-garh), India
