Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 8 (2015), 710–718
Research Article
On the Ulam stability of the Cauchy-Jensen
equation and the additive-quadratic equation
Jae-Hyeong Baea , Won-Gil Parkb,∗
a
Humanitas College, Kyung Hee University, Yongin 446-701, Republic of Korea.
b
Department of Mathematics Education, College of Education, Mokwon University, Daejeon 302-729, Republic of Korea.
Abstract
In this paper, we investigate the Ulam stability of the functional equations
z+w
= f (x, z) + f (x, w) + f (y, z) + f (y, w)
2f x + y,
2
and
f (x + y, z + w) + f (x + y, z − w) = 2f (x, z) + 2f (x, w) + 2f (y, z) + 2f (y, w)
c
in paranormed spaces. 2015
All rights reserved.
Keywords: Cauchy-Jensen mapping, additive-quadratic mapping, paranormed space.
2010 MSC: 39B52, 39B82.
1. Introduction
In 1940, S. M. Ulam proposed the stability problem (see [10]):
Let G1 be a group and let G2 be a metric group with the metric d(·, ·). Given ε > 0, does there exist a
δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y ∈ G1
then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x ∈ G1 ?
In 1941, this problem was solved by D. H. Hyers [3] in the case of Banach space. Thereafter, we call that
type the Hyers-Ulam stability. In 1978, Th. M. Rassias [9] extended the Hyers-Ulam stability by considering
variables. It also has been generalized to the function case by P. Găvruta [2]. For more details on this topic,
we also refer to [1, 4, 6] and references therein.
We recall some basic facts concerning Fréchet spaces (see [11]).
∗
Corresponding author
Email addresses: jhbae@khu.ac.kr (Jae-Hyeong Bae), wgpark@mokwon.ac.kr (Won-Gil Park)
Received 2015-2-17
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
711
Definition 1.1. Let X be a vector space. A paranorm on X is a function P : X → R such that for all
x, y ∈ X
(i) P (0) = 0;
(ii) P (−x) = P (x);
(iii) P (x + y) ≤ P (x) + P (y) (triangle inequality);
(iv) If {tn } is a sequence of scalars with tn → t and {xn } ⊂ X with P (xn − x) → 0, then P (tn xn − tx) → 0
(continuity of scalar multiplication).
The pair (X, P ) is called a paranormed space if P is a paranorm on X. Note that
P (nx) ≤ nP (x)
for all n ∈ N and all x ∈ (X, P ). The paranorm P on X is called total if, in addition, P satisfies (v) P (x) = 0
implies x = 0. A Fréchet space is a total and complete paranormed space. Note that each seminorm P on
X is a paranorm, but the converse need not be true. In recent, C. Park [5] obtained some stability results
in paranormed spaces.
Let X and Y be vector spaces. A mapping f : X × X → Y is called a Cauchy-Jensen mapping
(respectively, additive-quadratic mapping) if it satisfies the system of equations
y + z
f (x + y, z) = f (x, z) + f (y, z), 2f x,
= f (x, y) + f (x, z)
2
(respectively, f (x + y, z) = f (x, z) + f (y, z), f (x, y + z) + f (x, y − z) = 2f (x, y) + 2f (x, z)).
The authors [7, 8] considered the following functional equations:
z+w
2f x + y,
= f (x, z) + f (x, w) + f (y, z) + f (y, w)
2
(1.1)
and
f (x + y, z + w) + f (x + y, z − w) = 2f (x, z) + 2f (x, w) + 2f (y, z) + 2f (y, w).
(1.2)
It is easy to show that the functions f (x, y) = ax2 + bx and f (x, y) = axy 2 satisfy the functional equations
(1.1) and (1.2), respectively. Also, they solved the solutions of (1.1) and (1.2).
From now on, assume that (X, P ) is a Fréchet space and (Y, k · k) is a Banach space.
In this paper, we investigate the Ulam stability of the functional equations (1.1) and (1.2) in paranormed
spaces.
2. Ulam stability of the Cauchy-Jensen functional equation (1.1)
Theorem 2.1. Let r, θ be positive real numbers with r > log2 6, and let f : Y × Y → X be a mapping
satisfying f (x, 0) = 0 for all x ∈ Y such that
z + w
P 2f x + y,
− f (x, z) − f (x, w) − f (y, z) − f (y, w) ≤ θ(kxkr + kykr + kzkr + kwkr )
(2.1)
2
for all x, y, z, w ∈ Y . Then there exists a unique mapping F : Y × Y → X satisfying (1.1) such that
15
13 + 2 · 3r
r
P 2f (x, y) − F (x, y) ≤ 2θ r
kxkr +
kyk
2 −6
3r − 6
for all x, y ∈ Y .
(2.2)
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
712
Proof. Letting y = x in (2.1), we gain
z + w
P 2f 2x,
− 2f (x, z) − 2f (x, w) ≤ θ(2kxkr + kzkr + kwkr )
2
(2.3)
for all x, z, w ∈ Y . Letting w = −z in (2.3)), we get
P (2f (x, z) + 2f (x, −z)) ≤ 2θ(kxkr + kzkr )
(2.4)
for all x, z ∈ Y . Replacing z by −z and w by −z in (2.3)), we have
P (2f (2x, −z) − 4f (x, −z)) ≤ 2θ(kxkr + kzkr )
(2.5)
for all x, z ∈ Y . By (2.4) and (2.5), we obtain
P (4f (x, z) + 2f (2x, −z)) ≤ 2P (2f (x, z) + 2f (x, −z)) + P (2f (2x, −z) − 4f (x, −z))
≤ 6 θ(kxkr + kzkr )
for all x, z ∈ Y . Putting w = −3z in (2.3)), we gain
P (2f (2x, −z) − 2f (x, z) − 2f (x, −3z)) ≤ θ [ 2kxkr + (1 + 3r )kzkr ]
for all x, z ∈ Y . By the above two inequalities, we see that
P (6f (x, z) + 2f (x, −3z)) ≤ θ [ 8kxkr + (7 + 3r )kzkr ]
(2.6)
for all x, z ∈ Y . Replacing z by 3z in (2.5), we gain
P (2f (2x, −3z) − 4f (x, −3z)) ≤ 2θ(kxkr + 3r kzkr )
for all x, z ∈ Y . By (2.6) and the above inequality, we get
P (12f (x, z) + 2f (2x, −3z)) ≤ 2P (6f (x, z) + 2f (x, −3z)) + P (2f (2x, −3z) − 4f (x, −3z))
≤ 2θ [ 9kxkr + (7 + 2 · 3r )kzkr ]
for all x, z ∈ Y . Replacing z by −z in the above inequality, we have
P (12f (x, −z) + 2f (2x, 3z)) ≤2P (6f (x, −z) + 2f (x, 3z)) + P (2f (2x, 3z) − 4f (x, 3z))
≤2θ [ 9kxkr + (7 + 2 · 3r )kzkr ]
for all x, z ∈ Y . By (2.4) and the above inequality, we obtain
P (12f (x, z) − 2f (2x, 3z)) ≤6P (2f (x, z) + 2f (x, −z)) + P (−12f (x, −z) − 2f (2x, 3z))
≤2θ [ 15kxkr + (13 + 2 · 3r )kzkr ]
for all x, z ∈ Y . Replacing x by
x
2j+1
and z by
z
3j+1
in the above inequality, we see that
x
x z z 15
13 + 2 · 3r
r
r
P 12f j+1 , j+1 − 2f j , j
≤ 2θ (j+1)r kxk +
kzk
2
3
2 3
2
3(j+1)r
for all nonnegative integers j and all x, z ∈ Y . For given integers l, m(0 ≤ l < m), we obtain that
x z x z m−1
x
x z X z l
j+1
j
≤
P 2 · 6 f j+1 , j+1 − 2 · 6 f j , j
P 2 · 6 f m, m − 2 · 6 f l, l
2 3
2
3
2 3
2 3
m
j=l
≤ 2θ
m−1
X
j=l
j
6
13 + 2 · 3r
kxk
+
kzkr
2(j+1)r
3(j+1)r
15
r
(2.7)
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
713
for all x, z ∈ Y . By (2.7), the sequence {2 · 6j f ( 2xj , 3zj )} is a Cauchy sequence in X for all x, z ∈ Y .
x z
j
Since X is complete, the sequence
{2 · 6 f ( 2j , 3j )} converges for all x, z ∈ Y . Define F : Y × Y → X by
x z
j
F (x, z) := limj→∞ 2 · 6 f 2j , 3j for all x, z ∈ Y . By (2.1), we see that
z + w
P 2F x + y,
− F (x, z) − F (x, w) − F (y, z) − F (y, w)
2
h x z
x w
y z
y w i
x + y z + w
j
= lim P 6 4f
,
− 2f j , j − 2f j , j − 2f j , j − 2f j , j
j→∞
2j
3j
2 3
2 3
2 3
2 3
x+y z+w
x z
x w
y z
y w
j
≤ lim 2 · 6 P 2f
,
−f j, j −f j, j −f j, j −f j, j
j→∞
2j
3j
2 3
2 3
2 3
2 3
kxkr + kykr
r + kwkr kzk
≤ 2θ lim 6j
+
=0
j→∞
2jr
3jr
for all x, y, z, w ∈ Y . Since X is total, F satisfies (1.1). Setting l = 0 and taking m → ∞ in (2.7), one can
obtain the inequality (2.2).
Let F 0 : Y × Y → X be another mapping satisfying (1.1) and (2.2). By [7], there exist bi-additive
mappings B, B 0 : Y × Y → X and additive mappings A, A0 : Y → X such that F (x, y) = B(x, y) + A(x)
and F 0 (x, y) = B 0 (x, y) + A0 (x) for all x, y ∈ Y . Since r > log2 6, we obtain that
h i
x
x y
y
0
n
0 x
0 x
P (F (x, y) − F (x, y)) = P 6 B n , n + A n − B
,
−A n
2 3
2
2n 3n
2
x y
x y
x y
y
n
0 x
≤ 6 P F n , n − 2f n , n
+ P 2f n , n − F
,
2 3
2 3
2 3
2n 3n
15
13 + 2 · 3r
r
r
≤ 4 · 6n θ
→ 0 as n → ∞
kxk
+
kyk
(2r − 6)2nr
(3r − 6)3nr
for all x, y ∈ Y . Hence F is a unique mapping satisfying (1.1) and (2.2), as desired.
Theorem 2.2. Let r be a positive real number with r < log3 6, and let f : X × X → Y be a mapping
satisfying f (x, 0) = 0 for all x ∈ X such that
z + w
− f (x, z) − f (x, w) − f (y, z) − f (y, w) ≤ P (x)r + P (y)r + P (z)r + P (w)r
(2.8)
2f x + y,
2
for all x, y, z, w ∈ X. Then there exists a unique mapping F : X × X → Y satisfying (1.1) such that
f (x, y) − F (x, y) ≤
18
15 + 3r+1
r
P
(x)
+
P (y)r
6 − 2r
6 − 3r
(2.9)
for all x, y ∈ X.
Proof. Letting y = x in (2.8), we gain
z + w
− 2f (x, z) − 2f (x, w) ≤ 2P (x)r + P (z)r + P (w)r
2f 2x,
2
(2.10)
for all x, z, w ∈ X. Putting w = −z in (2.10), we get
k2f (x, z) + 2f (x, −z)k ≤ 2 P (x)r + P (z)r
for all x, z ∈ X. Replacing z by −z and w by −z in (2.10), we have
kf (2x, −z) − 2f (x, −z)k ≤ 2 P (x)r + P (z)r
(2.11)
(2.12)
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
714
for all x, z ∈ X. By (2.11) and (2.12), we obtain
kf (2x, −z) + 2f (x, z)k ≤ 4 P (x)r + P (z)r
(2.13)
for all x, z ∈ X. Setting w = −3z in (2.10), we gain
k2f (2x, −z) − 2f (x, z) − 2f (x, −3z)k ≤ 2P (x)r + (1 + 3r )P (z)r
for all x, z ∈ X. By (2.13) and the above inequality, we get
k6f (x, z) + 2f (x, −3z)k ≤ 10P (x)r + (9 + 3r )P (z)r
(2.14)
for all x, z ∈ X. Replacing z by 3z in (2.12), we have
kf (2x, −3z) − 2f (x, −3z)k ≤ 2 P (x)r + 3r P (z)r
for all x, z ∈ X. By (2.14) and the above inequality, we gain
k6f (x, z) + f (2x, −3z)k ≤ 12P (x)r + (9 + 3r+1 )P (z)r
for all x, z ∈ X. Replacing z by −z in the above inequality, we get
k6f (x, −z) + f (2x, 3z)k ≤ 12P (x)r + (9 + 3r+1 )P (z)r
for all x, z ∈ X. By (2.11) and the above inequality, we have
k6f (x, z) − f (2x, 3z)k ≤ 18P (x)r + (15 + 3r+1 )P (z)r
for all x, z ∈ X. Replacing x by 2j x and z by 3j z in the above inequality and dividing 6j+1 , we see that
1
1
1
j
j
j+1
j+1
f (2 x, 3 z) −
f (2 x, 3 z)
≤ j+1 [18 · 2jr P (x)r + (15 + 3r+1 )3jr P (z)r ]
6j
j+1
6
6
for all nonnegative integers j and all x, z ∈ X. For given integers l, m(0 ≤ l < m), we obtain that
m−1
1
X 1
f (2l x, 3l z) − 1 f (2m x, 3m z) ≤
[18 · 2jr P (x)r + (15 + 3r+1 )3jr P (z)r ]
6l
6m
6j+1
(2.15)
j=l
for all x, z ∈ X. By (2.15), the sequence { 61j f (2j x, 3j y)} is a Cauchy sequence for all x, y ∈ X. Since
Y is complete, the sequence { 61j f (2j x, 3j y)} converges for all x, y ∈ X. Define F : X × X → Y by
F (x, y) := limj→∞ 61j f (2j x, 3j y) for all x, y ∈ X.
By (2.8), we see that
j (z + w)
1
3
j
j
j
j
j
j
j
j
j
2f 2 (x + y),
− f (2 x, 3 z) − f (2 x, 3 w) − f (2 y, 3 z) − f (2 y, 3 w)
j
6
2
1
≤ j [P (2j x)r + P (2j y)r + P (3j z)r + P (3j w)r ]
6
1
≤ j 2rj [P (x)r + P (y)r ] + 3rj [P (z)r + P (w)r ]
6
for all x, y, z, w ∈ X. Letting j → ∞, F satisfies (1.1). By Theorem 4 in [7], F is a Cauchy-Jensen mapping.
Setting l = 0 and taking m → ∞ in (2.15), one can obtain the inequality (2.9). Let G : X × X → Y be
another Cauchy-Jensen mapping satisfying (2.9). Since 0 < r < log3 6, we obtain that
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
kF (x, y) − G(x, y)k =
=
≤
≤
≤
715
1
kF (2n x, y) − F (2n x, 0) + G(2n x, 0) − G(2n x, y)k
2n
1
kF (2n x, 3n y) − F (2n x, 0) + G(2n x, 0) − G(2n x, 3n y)k
6n
1
kF (2n x, 3n y) − F (2n x, 0) − f (2n x, 3n y) + f (2n x, 0)k
6n
1
+ n k − f (2n x, 0) + f (2n x, 3n y) + G(2n x, 0) − G(2n x, 3n y)k
6
1
(kF (2n x, 3n y) − f (2n x, 3n y)k + k − F (2n x, 0) + f (2n x, 0)k)
6n
1
+ n (k − f (2n x, 0) + G(2n x, 0)k + kf (2n x, 3n y) − G(2n x, 3n y)k)
6
2 36 · 2nr
3nr (15 + 3r+1 )
r
r
P (x) +
P (y) → 0 as n → ∞
6n 6 − 2r
6 − 3r
for all x, y ∈ X. Hence F is a unique Cauchy-Jensen mapping, as desired.
3. Ulam stability of the additive-quadratic functional equation (1.2)
Theorem 3.1. Let r, θ be positive real numbers with r > log2 8 = 3, and let f : Y × Y → X be a mapping
satisfying f (x, 0) = 0 for all x ∈ Y such that
P (f (x + y,z + w) + f (x + y, z − w) − 2f (x, z) − 2f (x, w) − 2f (y, z) − 2f (y, w))
≤ θ(kxkr + kykr + kzkr + kwkr )
(3.1)
for all x, y, z, w ∈ Y . Then there exists a unique mapping F : Y × Y → X satisfying (1.2) such that
P f (x, y) − F (x, y) ≤
2θ
(kxkr + kykr )
−8
2r
for all x, y ∈ Y .
Proof. Letting y = x and w = z in (3.1), we gain
P (f (2x, 2z) − 8f (x, z)) ≤ 2θ(kxkr + kzkr )
z
x
for all x, z ∈ Y . Replacing x by 2j+1
and z by 2j+1
in the above inequality, we see that
x z
x
z
2θ
P f j , j − 8f j+1 , j+1
≤ (j+1)r (kxkr + kzkr )
2 2
2
2
2
for all nonnegative integers j and all x, z ∈ Y . Thus we obtain that
x z
x
z
j
j+1
P 8 f j , j − 8 f j+1 , j+1
2 2
2
2
x z
x
z
2 8 j
j
≤ r
θ(kxkr + kzkr )
≤ 8 P f j , j − 8f j+1 , j+1
2 2
2
2
2 2r
for all nonnegative integers j and all x, z ∈ Y . For given integers l, m(0 ≤ l < m), we have
(3.2)
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
m−1
X 2 8 j
x z
x z
m
l
P 8 f l, l − 8 f m, m
≤
θ(kxkr + kzkr )
2 2
2r 2r
2 2
716
(3.3)
j=l
for all x, z ∈ Y . By (3.3), the sequence {8j f ( 2xj , 2zj )} is a Cauchy sequence in X for all x, z ∈ Y .
Since X is complete, the sequence
{8j f ( 2xj , 2zj )} converges for all x, z ∈ Y . Define F : Y × Y → X by
x z
j
F (x, z) := limj→∞ 8 f 2j , 2j for all x, z ∈ Y . By (3.1), we see that
P F (x + y, z + w) + F (x + y, z − w) − 2F (x, z) − 2F (x, w) − 2F (y, z) − 2F (y, w)
h x + y z − w
x + y z + w
= lim P 8j f
,
+
f
,
j→∞
2j
2j
2j
2j
x z
x w
y z
y w i
− 2f j , j − 2f j , j − 2f j , j − 2f j , j
2 2
2 2
2 2
2 2
x+y z+w
x+y z−w
≤ lim 8j P f
,
+f
,
j→∞
2j
2j
2j
2j
x z
x w
y z
y w − 2f j , j − 2f j , j − 2f j , j − 2f j , j
2 2
2 2
2 2
2 2
8 j
≤ θ(kxkr + kykr + kzkr + kwkr ) lim
=0
j→∞ 2r
for all x, y, z, w ∈ Y . Since X is total, F satisfies (1.2). Setting l = 0 and taking m → ∞ in (3.3), one can
obtain the inequality (3.2).
Let F 0 : Y × Y → X be another mapping satisfying (1.2) and (3.2). By [8], there exist multi-additive
mappings M, M 0 : Y × Y × Y → X such that F (x, y) = M (x, y, y), F 0 (x, y) = M 0 (x, y, y), M (x, y, z) =
M (x, z, y) and M 0 (x, y, z) = M 0 (x, z, y) for all x, y, z ∈ Y . Since r > 3, we obtain that
h x y y i
x y y
P (F (x, y) − F 0 (x, y)) = P 8n M n , n , n − M 0 n , n , n
2 2 2
2 2 2
x y y
y y
n
0 x
≤ 8 P M n, n, n − M
, ,
2 2 2
2n 2n 2n
x y x y
x y
y
n
0 x
+ P f n, n − F
,
≤ 8 P F n, n − f n, n
2 2
2 2
2 2
2n 2n
8 n 4θ
≤ r
(kxkr + kykr ) → 0 as n → ∞
2
2r − 8
for all x, y ∈ Y . Hence F is a unique mapping satisfying (1.2) and (3.2), as desired.
Theorem 3.2. Let r be a positive real number with r < log2 8 = 3, and let f : X × X → Y be a mapping
satisfying f (x, 0) = 0 for all x ∈ X such that
kf (x + y, z + w) + f (x + y, z − w) − 2f (x, z) − 2f (x, w) − 2f (y, z) − 2f (y, w)k
≤ P (x)r + P (y)r + P (z)r + P (w)r
(3.4)
for all x, y, z, w ∈ X. Then there exists a unique mapping F : X × X → Y satisfying (1.2) such that
f (x, y) − F (x, y) ≤
for all x, y ∈ X.
2
[P (x)r + P (y)r ]
8 − 2r
(3.5)
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
717
Proof. Letting y = x and w = z in (3.4), we gain
kf (2x, 2z) − 8f (x, z)k ≤ 2[P (x)r + P (z)r ]
for all x, z ∈ X. Replacing x by 2j x and z by 2j z in the above inequality, we see that
jr
1
j+1
j+1
j
j
f (2 x, 2 z) − f (2 x, 2 z) ≤ 2 [P (x)r + P (z)r ]
8
4
for all nonnegative integers j and all x, z ∈ X. Thus we obtain that
1
1 2r j
1
j+1
j+1
j
j
r
r
≤
f
(2
x,
2
z)
−
f
(2
x,
2
z)
8j+1
4 8 [P (x) + P (z) ]
8j
for all nonnegative integers j and all x, z ∈ X. For given integers l, m(0 ≤ l < m), we have
m−1
1
X1
1
1
l
l
m
m
j
j
j+1
j+1
f (2 x, 2 z) −
f (2 x, 2 z) −
f (2 x, 2 z)
≤
f (2 x, 2 z)
8l
m
j
j+1
8
8
8
j=l
≤
m−1
X
j=l
1 2r j
[P (x)r + P (z)r ]
4 8
(3.6)
for all x, z ∈ X. By (3.6), the sequence { 81j f (2j x, 2j z)} is a Cauchy sequence in Y for all x, z ∈ X.
Since Y is complete, the sequence { 81j f (2j x, 2j z)} converges for all x, z ∈ X. Define F : X × X → Y by
F (x, z) := limj→∞ 81j f (2j x, 2j z) for all x, z ∈ X. By (3.4), we see that
F (x + y, z + w) + F (x + y, z − w) − 2F (x, z) − 2F (x, w) − 2F (y, z) − 2F (y, w)
1 j
f (2 (x + y), 2j (z + w)) + f (2j (x + y), 2j (z − w))
= lim j→∞ 8j
− 2f (2j x, 2j z) − 2f (2j x, 2j w) − 2f (2j y, 2j z) − 2f (2j y, 2j w) 1
= lim j f (2j (x + y), 2j (z + w)) + f (2j (x + y), 2j (z − w))
j→∞ 8
− 2f (2j x, 2j z) − 2f (2j x, 2j w) − 2f (2j y, 2j z) − 2f (2j y, 2j w)
2r j
≤ [P (x)r + P (y)r + P (z)r + P (w)r ] lim
=0
j→∞ 8
for all x, y, z, w ∈ X. Thus F is a mapping satisfying (1.2). Setting l = 0 and taking m → ∞ in (3.6), one
can obtain the inequality (3.5).
Let G : X × X → Y be another additive-quadratic mapping satisfying (3.5). Since 0 < r < 3, we have
1
kF (2n x, 2n y) − G(2n x, 2n y)k
8n
1
1
≤ n kF (2n x, 2n y) − f (2n x, 2n y)k + n kf (2n x, 2n y) − G(2n x, 2n y)k
8 8
2r n 4
≤
[P (x)r + P (y)r ] → 0 as n → ∞
8
8 − 2r
kF (x, y) − G(x, y)k =
for all x, y ∈ X. Hence F is a unique additive-quadratic mapping, as desired.
Acknowledgements:
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(grant number 2014014135).
J.-H. Bae, W.-G. Park, J. Nonlinear Sci. Appl. 8 (2015), 710–718
718
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