CONVERGENCE OF PARALLEL ITERATIVE ALGORITHMS FOR A SYSTEM OF NONLINEAR VARIATIONAL INEQUALITIES IN BANACH SPACES†

Journal of Applied Mathematics & Informatics.
2016.
Jan,
34(1_2):
61-73

- Received : June 30, 2015
- Accepted : October 10, 2015
- Published : January 30, 2016

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In this paper, we consider the problems of convergence of parallel iterative algorithms for a system of nonlinear variational inequalities and nonexpansive mappings. Strong convergence theorems are established in the frame work of real Banach spaces.
AMS Mathematics Subject Classification: 47J20; 65K10; 65K15; 90C33.
E
, ║ · ║) be a Banach space and
C
be a nonempty closed convex subset of
E
. This paper deals with the problems of convergence of iterative algorithms for a system of nonlinear variational inequalities: Find (
x
^{∗}
,
y
^{∗}
) ∈
C
×
C
such that
where
T
_{1}
,
T
_{2}
:
C
×
C
→
E
,
g
_{1}
,
g
_{2}
:
C
→
C
are nonlinear mappings,
J
is the normalized duality mapping,
j
∈
J
and
ρ
_{1}
,
ρ
_{2}
are two positive real numbers.
If
T
_{1}
,
T
_{2}
:
C
→
E
are nonlinear mappings and
g
_{1}
=
g
_{2}
=
I
(
I
denotes the identity mapping), then (1.1) reduces to finding (
x
^{∗}
,
y
^{∗}
) ∈
C
×
C
such that
which was considered by Yao et al.
[13]
.
If
E
=
H
is a real Hilbert space and
T
_{1}
,
T
_{2}
:
C
→
H
are nonlinear mappings and
g
_{1}
=
g
_{2}
=
g
, then (1.1) reduces to finding (
x
^{∗}
,
y
^{∗}
) ∈
C
×
C
such that
which was studied by Yang et al.
[12]
.
If
g
=
I
, then (1.3) reduces to finding (
x
^{∗}
,
y
^{∗}
) ∈
C
×
C
such that
which was introduced by Ceng et al.
[2]
.
In particular, if
T
_{1}
=
T
_{2}
=
T
, then (1.4) reduces to finding (
x
^{∗}
,
y
^{∗}
) ∈
C
×
C
such that
which is defined by Verma
[9]
.
Further, if
x
^{∗}
=
y
^{∗}
, then (1.5) reduces to the following classical variational inequality (VI(
T,C
)) of finding
x
^{∗}
∈
C
such that
We can see easily that the variational inequality (1.6) is equivalent to a fixed point problem. An element
x
^{∗}
∈
C
is a solution of the variational inequality (1.6) if and only if
x
^{∗}
∈
C
is a fixed point of the mapping
P_{C}
(
I − λT
), where
P_{C}
is the metric projection and
λ
is a positive real number. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.
Recent development of the variational inequality is to design efficient iterative algorithms to compute approximate solutions for variational inequalities and their generalization. Up to now, many authors have presented implementable and significant numerical methods such as projection method and it’s variant forms, linear approximation, descent method, Newton’s method and the method based on auxiliary principle technique.
However, these sequential iterative methods are only suitable for implementing on the traditional single-processor computer. To satisfy practical requirements of modern multiprocessor systems, efficient iterative methods having parallel characteristics need to be further developed for the system of variational inequalities (see
[1
,
4
,
5
,
6
,
12
,
14]
).
Motivated and inspired by the research work going on this field, in this paper, we construct an parallel iterative algorithm for approximating the solution of a new system of variational inequalities involving four different nonlinear mappings. Finally, we prove the strong convergence of the purposed iterative scheme in 2-uniformly smooth Banach spaces.
C
be a nonempty closed convex subset of a Banach space
E
with the dual space
E
^{∗}
. Let ⟨·, ·⟩ denote the dual pair between
E
and
E
^{∗}
. Let 2
^{E}
denote the family of all the nonempty subsets of
E
. For
q
> 1, the generalized duality mapping
J_{q}
:
E
→ 2
^{E∗}
is defined by
In particular,
J
=
J
_{2}
is the normalized duality mapping. It is known that
J_{q}
(
x
) = ║
x
║
^{q−2}
J
(
x
) for all
x
∈
E
and
J_{q}
is single-valued if
E
^{∗}
is strictly convex or
E
is uniformly smooth. If
E
=
H
is a Hilbert space,
J
=
I
, the identity mappings.
Let
B
= {
x
∈
E
: ║
x
║ = 1}. A Banach space
E
is said to be smooth if the limit
exists for all
x, y
∈
B
. The modulus of smoothness of
E
is the function
ρ_{E}
: [0,∞) → [0,∞) defined by
A Banach space
E
is called uniformly smooth if
E
is called
q
-uniformly smooth if there exists a constant
c
> 0 such that
If
E
is
q
-uniformly smooth, then
q
≤ 2 and
E
is uniformly smooth.
Definition 2.1.
Let
T
:
C
×
C
→
E
be a mapping.
T
is said to be
(i) (
δ, ξ
)-relaxed cocoercive with respect to the first argument if there exist
j
(
x − y
) ∈
J
(
x − y
) and constants
δ, ξ
> 0 such that
for all
x, y
∈
C
;
(ii)
μ
-Lipschitz continuous with respect to the first argument if there exists a constant
μ
> 0 such that
for all
x, y
∈
C
;
(iii)
γ
-Lipschitz continuous with respect to the second argument if there exists a constant
γ
> 0 such that
for all
x, y
∈
C
.
Definition 2.2.
Let
g
:
C
→
C
be a mapping.
g
is said to be
(i)
ζ
-strongly accretive if there exists a constant
ζ
> 0 such that
for all
x, y
∈
C
.
(ii)
η
-Lipschitz continuous if there exists a constant
η
> 0 such that
for all
x, y
∈
C
.
Let
D
be a subset of
C
and
Q
be a mapping of
C
into
D
. Then
Q
is said to be sunny if
whenever
Q
(
x
) +
t
(
x
−
Q
(
x
)) ∈
C
for
x
∈
C
and
t
≥ 0. A mapping
Q
of
C
into itself is called a retraction if
Q
^{2}
=
Q
. If a mapping
Q
of
C
into itself is a retraction, then
Q
(
z
) =
z
for all
z
∈
R
(
Q
), where
R
(
Q
) is the range of
Q
. A subset
D
of
C
is called a sunny nonexpansive retract of
C
if there exists a sunny nonexpansive retraction from
C
onto
D
.
In order to prove our main results, we also need the following lemmas.
Lemma 2.3
(
[11]
).
Let E be a real 2-uniformly smooth Banach space. Then
where K is the 2-uniformly smooth constant of E
.
Lemma 2.4
(
[7]
).
Let C be a nonempty closed convex subset of a smooth Banach space E and let Q_{C} be a retraction from E onto C. Then the following are equivalent:
(i) Q_{C} is both sunny and nonexpansive;
(ii)
⟨
x − Q_{C}
(
x
),
j
(
y − Q_{C}
(
x
))⟩ ≤ 0
for all x
∈
E and y
∈
C
.
Lemma 2.5
(
[10]
).
Suppose
{
δ_{n}
}
is a nonnegative sequence satisfying the fol- lowing inequality:
with λ_{n}
∈ [0, 1],
and σ_{n}
= 0(
λ_{n}
).
Then
lim
_{n→∞}
δ_{n}
= 0.
Lemma 2.6
(
[3]
).
Let
{
c_{n}
} and {
k_{n}
}
be two real sequences of nonnegative num-bers that satisfy the following conditions:
(i)
0 ≤
k_{n}
≤ 1
for n
= 1, 2, · · ·
and
lim sup
_{n}
k_{n}
< 1
;
(ii)
c
_{n+1}
≤
k_{n}c_{n} for n
= 1, 2, · · · .
Then c_{n} converges to
0
as n
→ ∞.
Lemma 3.1.
Let C be a nonempty closed convex subset of a smooth Banach space E. Let Q_{C}
:
E
→
C be a sunny nonexpansive retraction
,
T_{i}
:
C
×
C
→
E and g_{i}
:
C
→
C be mappings for i
= 1, 2.
Then
(
x
^{∗}
,
y
^{∗}
)
with x
^{∗}
,
y
^{∗}
∈
C is a solution of problem (1.1) if and only if
Proof
. Applying Lemma 2.4, we have that
That is,
This completes the proof. □
This fixed point formulation allow us to suggest the following parallel iterative algorithms.
Algorithm 3.1.
For any given
x
_{0}
,
y
_{0}
∈
C
, computer the sequences {
x_{n}
} and {
y_{n}
} defined by
where
ρ
_{1}
,
ρ
_{2}
are positive real numbers.
Also, we propose a relaxed parallel algorithm which can be applied to the approximation of solution of the problem (1.1) and common fixed point of two mappings.
Algorithm 3.2.
For any given
x
_{0}
,
y
_{0}
∈
C
, compute the sequences {
x_{n}
} and {
y_{n}
} defined by
where
S
_{1}
,
S
_{2}
:
C
→
C
are nonexpansive mappings, {
α_{n}
}, {
β_{n}
} are sequences in [0,1],
κ
∈ (0, 1) and
ρ
_{1}
,
ρ
_{2}
are positive real numbers.
If
T
_{1}
,
T
_{2}
:
C
→
E
are nonlinear mappings and
g
_{1}
=
g
_{2}
=
I
, then the algorithm 3.1 reduces to the following parallel iterative method for solving problem (1.2).
Algorithm 3.3.
For any given
x
_{0}
,
y
_{0}
∈
C
, compute the sequences {
x_{n}
} and {
y_{n}
} defined by
where
ρ
_{1}
,
ρ
_{2}
are positive real numbers.
If
E
=
H
is a Hilbert space,
T
_{1}
,
T
_{2}
:
C
→
H
are nonlinear mappings and
g
_{1}
=
g
_{2}
=
g
, Algorithm 3.1 reduces to the following parallel iterative method for solving problem (1.3).
Algorithm 3.4.
For any given
x
_{0}
,
y
_{0}
∈
C
, compute the sequences {
x_{n}
} and {
y_{n}
} defined by
where
ρ
_{1}
,
ρ
_{2}
are positive real numbers.
Theorem 4.1.
Let E be a
2
-uniformly smooth Banach space with the
2-
uniformly smooth constant K, C be a nonempty closed convex subset of E and Q_{C} be a sunny nonexpansive retraction from E onto C. Let T_{i}
:
C
×
C
→
E be a non-linear mapping such that
(
δ_{i}, ξ_{i}
)
-relaxed cocoercive, μ_{i}-Lipschitz continuous with respect to the first argument and γ_{i}-Lipschitz continuous with respect to the sec-ond argument for i
= 1, 2.
Let g_{i}
:
C
→
C be a η_{i}-Lipschitz continuous and ζ_{i}-strongly accretive mapping for i
= 1, 2.
Assume that the following assump-tions hold:
where τ
_{1}
=
m
_{1}
+
m
_{2}
+
ρ
_{2}
γ
_{2}
,
τ
_{2}
=
m
_{1}
+
m
_{2}
+
ρ
_{1}
γ
_{1}
,
Then there exist x
^{∗}
, y
^{∗}
∈
E, which solves the problem (1.1). Moreover, the parallel iterative sequences
{
x_{n}
}
and
{
y_{n}
}
generated by the Algorithm 3.1 con-verge to x
^{∗}
and y
^{∗}
, respectively
.
Proof
. To proof the result, we first need to evaluate ║
x
_{n+1}
−
x_{n}
║ for all
n
≥ 0. From Algorithm 3.1 and the nonexpansive property of the sunny nonexpansive retraction
Q_{C}
, we can get
Using the strongly accretivity and Lipschitz continuity of
g
_{1}
and Lemma 2.3, we find that
and
which imply that
and
where
. Since
T
_{1}
is (
δ
_{1}
,
ξ
_{1}
)-relaxed cocoercive and
μ
_{1}
-Lipschitz continuous with respect to the first argument, we have
Also,using the Lipschitz continuity of
T
_{1}
with respect to second argument,
Combining (4.3)-(4.7), we have
where
Similarly, since
g
_{2}
is
η
_{2}
-Lipschitz continuous and
ζ
_{2}
-strongly accretive,
T
_{2}
is (
δ
_{2}
,
ξ
_{2}
)-relaxed cocoercive,
μ
_{2}
-Lipschitz continuous with respect to the first argument and
γ
_{2}
-Lipschitz continuous with respect to the second argument, we obtain
where
. It follows from (4.8) and (4.9) that
where
k
= max{
m
_{1}
+
m
_{2}
+
θ
_{2}
+
ρ
_{1}
γ
_{1}
,
m
_{1}
+
m
_{2}
+
θ
_{1}
+
ρ
_{2}
γ
_{2}
}. From (4.1) and (4.2), we know that 0 ≤
k
< 1. Let
c_{n}
= ║
x_{n}
−
x
_{n−1}
║ + ║
y_{n}
−
y
_{n−1}
║. Then (4.10) can be rewritten as
It follows from Lemma 2.6 that {
x_{n}
} and {
y_{n}
} are both Cauchy sequences in
E
. There exist
x
^{∗}
,
y
^{∗}
∈
E
such that
x_{n}
→
x
^{∗}
and
y_{n}
→
y
^{∗}
as
n
→ ∞. By continuity, we know that
x
^{∗}
,
y
^{∗}
satisfy
It follows from Lemma 3.1 that (
x
^{∗}
,
y
^{∗}
) is a solution of problem (1.1). This completes the proof. □
If
T
_{1}
,
T
_{2}
:
C
→
E
are nonlinear mappings and
g
_{1}
=
g
_{2}
=
I
, the the following corollary follows immediately from Theorem 4.1.
Corollary 4.2.
Let E be a
2
-uniformly smooth Banach space with the
2-
uniformly smooth constant K, C be a nonempty closed convex subset of E and Q_{C} be a sunny nonexpansive retraction from E onto C. Let T_{i}
:
C
→
E be a
(
δ_{i}, ξ_{i}
)
-relaxed cocoercive and μ_{i}-Lipschitz continuous mapping for i =
1, 2.
Assume that the following assumptions hold:
Then there exist x
^{∗}
,
y
^{∗}
∈
E, which solves the problem (1.2). Moreover, the par-allel iterative sequences
{
x_{n}
}
and
{
y_{n}
}
generated by the Algorithm 3.3 converge to x
^{∗}
and y
^{∗}
, respectively
.
Remark 4.1.
(i) We note that Hilbert spaces and
L^{p}
(
p
≥ 2) spaces are 2-uniformly smooth.
(ii) If
E
=
H
is a Hilbert space, then a sunny nonexpansive retraction
Q_{C}
is coincident with the metric projection
P_{C}
from
H
onto
C
.
(iii) It is well known that the 2-uniformly smooth constant
in Hilbert spaces.
We can obtain the following result immediately.
Corollary 4.3.
Let C be a nonempty closed convex subset of a real Hilbert space H. Let T_{i}
:
C
→
H be a
(
δ_{i}, ξ_{i}
)
-relaxed cocoercive and μ_{i}-Lipschitz continuous mapping for i =
1, 2.
Let g
:
C
→
C be a η-Lipschitz continuous and ζ-strongly monotone mapping. Assume that the following assumptions hold:
where
Then there exist x
^{∗}
,
y
^{∗}
∈
H, which solve the problem (1.3). Moreover, the par-allel iterative sequences
{
x_{n}
}
and
{
y_{n}
}
generated by the Algorithm 3.4 converge to x
^{∗}
and y
^{∗}
, respectively
.
Let
Fix
(
S_{i}
) denote the set of fixed points of the mapping
S_{i}
, i.e.,
Fix
(
S_{i}
) = {
x
∈
C
:
S_{i}x
=
x
} and Ω the set of solutions of the problem (1.1).
Theorem 4.4.
Let E be a
2-
uniformly smooth Banach space with the
2-
uniformly smooth constant K, C be a nonempty closed convex subset of E and Q_{C} be a sunny nonexpansive retraction from E onto C. Let T_{i}
:
C
×
C
→
E be a non-linear mapping such that
(
δ_{i}, ξ_{i}
)
-relaxed cocoercive, μ_{i}-Lipschitz continuous with respect to the first argument and γ_{i}-Lipschitz continuous with respect to the sec-ond argument for i
= 1, 2.
Let g_{i}
:
C
→
C be a η_{i}-Lipschitz continuous and ζ_{i}-strongly accretive mapping for i
= 1, 2.
Let S_{i}
:
C
→
C be a nonexpansive mapping with a fixed point for i
= 1, 2.
Let
{
α_{n}
}, {
β_{n}
}
be sequences in
[0, 1].
Assume that the following assumptions hold:
(C1)
0 < Θ
_{1,n}
=
α_{n}
(1 −
κ
− (1 −
κ
)(
m
_{1}
+
ρ
_{1}
γ
_{1}
)) −
β_{n}
(1 −
κ
)(
m
_{2}
+
θ
_{2}
) < 1,
(C2)
0 < Θ
_{2,n}
=
β_{n}
(1 −
κ
− (1 −
κ
)(
m
_{2}
+
ρ
_{2}
γ
_{2}
)) −
α_{n}
(1 −
κ
)(
m
_{1}
+
θ
_{1}
) < 1,
(C3)
,
where
and
If
Ω ∩
Fix
(
S
_{1}
) ∩
Fix
(
S
_{2}
) ≠
ϕ, then the sequences
{
x_{n}
}
and
{
y_{n}
}
generated by the Algorithm 3.2 converge to x
^{∗}
and y
^{∗}
, respectively, where
(
x
^{∗}
,
y
^{∗}
) ∈ Ω
and x
^{∗}
,
y
^{∗}
∈
Fix
(
S
_{1}
) ∩
Fix
(
S
_{2}
).
Proof
. Letting (
x
^{∗}
,
y
^{∗}
) ∈ Ω, we obtain from Lemma 3.1 that
Since
x
^{∗}
,
y
^{∗}
∈
Fix
(
S
_{1}
) ∩
Fix
(
S
_{2}
), we have
Putting
e
_{1,n}
=
κ
S
_{1}
(
x_{n}
) + (1 −
κ
)(
x_{n}
−
g
_{1}
(
x_{n}
) +
Q_{C}
[
g
_{1}
(
y_{n}
) −
ρ
_{1}
T
_{1}
(
y_{n}
,
x_{n}
)]) for each
n
= 0, 1, 2, · · · , we arrive at
Using the arguments as in the proof of Theorem 4.1, we obtain
and
where
From (4.11), we have
It follows that
Similarly, we obtain
where
Now (4.12) and (4.13) imply
where
Define the norm ║·║
_{∗}
on
E
×
E
by
Then (
E
×
E
, ║·║
_{∗}
) is a Banach space. Hence, (4,14) implies that
From the conditions (C1)-(C3) and Lemma 2.5 to (4.15), we obtain that
Therefore, the sequences {
x_{n}
} and {
y_{n}
} converge to
x
^{∗}
and
y
^{∗}
, respectively. This completes the proof. □
Remark 4.2.
Theorem 4.1 and 4.4 extend the solvability of the systems of variational inequalities (1.2)-(1.6) to the more general system of variational inequalities (1.1). The underlying mapping
T_{i}
:
C
×
C
→
E
(
i
= 1, 2) in our paper needs to be relaxed (
δ_{i}, ξ_{i}
)-relaxed cocoercive while the underlying operators
A,B
in
[13]
needs to inverse strongly accretive. Hence, Theorem 4.1 and 4.4 extend and improve the main results of
[9
,
12
,
13]
.
Jae Ug Jeong received M.Sc. from Busan National University and Ph.D at Gyeongsang National University. Since 1982 he he has been at Dongeui University. His research interests include fixed point theory and variational inequality problems.
Department of Mathematics, Dongeui University, Busan 614-714, South Korea.
e-mail: jujeong@deu.ac.kr

1. Introduction

Let (
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2. Preliminaries

Let
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3. Iterative algorithms

In this section, we suggest a parallel iterative algorithm for solving the system of nonlinear variational inequality (1.1). First of all, we establish the equivalence between the system of variational inequalities and fixed point problems.
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4. Main results

We now state and prove the main results of this paper.
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Citing 'CONVERGENCE OF PARALLEL ITERATIVE ALGORITHMS FOR A SYSTEM OF NONLINEAR VARIATIONAL INEQUALITIES IN BANACH SPACES†
'

@article{ E1MCA9_2016_v34n1_2_61}
,title={CONVERGENCE OF PARALLEL ITERATIVE ALGORITHMS FOR A SYSTEM OF NONLINEAR VARIATIONAL INEQUALITIES IN BANACH SPACES†}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2016.061}, DOI={10.14317/jami.2016.061}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={JEONG, JAE UG}
, year={2016}
, month={Jan}