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J. Nonlinear Sci. Appl. 8 (2015), 64–85
Research Article
Stability of an ACQ-functional equation in various
matrix normed spaces
Zhihua Wanga,∗, Prasanna K. Sahoob
a
b
School of Science, Hubei University of Technology, Wuhan, Hubei 430068, P.R. China.
Department of Mathematics, University of Louisville,Louisville, KY 40292, USA.
Communicated by Choonkil Park
Abstract
Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the following
additive-cubic-quartic (ACQ ) functional equation
11[f (x + 2y) + f (x − 2y)]
= 44[f (x + y) + f (x − y)] + 12f (3y) − 48f (2y) + 60f (y) − 66f (x)
in matrix Banach spaces. Furthermore, using the fixed point method, we also prove the Hyers-Ulam stability
c
of the above functional equation in matrix fuzzy normed spaces. 2015
All rights reserved.
Keywords: Fixed point method, Hyers-Ulam stability, matrix Banach space, matrix fuzzy normed space,
additive-cubic-quartic functional equation.
2010 MSC: 47L25, 39B82, 39B72, 46L07.
1. Introduction
In 1940, Ulam [28] posed the first stability problem concerning group homomorphisms. In the next year,
Hyers [8] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ result was
generalized by Aoki [1] for additive mappings and by Rassias [24] for linear mappings. Găvruta [5] obtained
generalized Rassias’ result which allows the Cauchy difference to be controlled by a general unbounded
function in the spirit of Rassias’ approach. Since then, the stability of several functional equations has been
extensively investigated by several mathematicians (see [9, 10, 11, 25, 26] and references therein); as well
∗
Corresponding author. The first author is supported by BSQD12077, NSFC 11401190 and NSFC 11201132
Email addresses: matwzh2000@126.com (Zhihua Wang), sahoo@louisville.edu (Prasanna K. Sahoo)
Received 2014-11-1
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
65
as various fuzzy stability results concerning Cauchy, Jensen, quadratic and cubic functional equations (cf.
[16, 17, 18, 19]). Furthermore some stability results of functional equations and inequalities were investigated
[13, 14, 20, 21, 22] in matrix normed spaces, matrix paranormed spaces and matrix fuzzy normed spaces.
In this paper, we consider the following functional equation derived from additive, cubic and quaritc
mappings:
11[f (x + 2y) + f (x − 2y)]
= 44[f (x + y) + f (x − y)] + 12f (3y) − 48f (2y) + 60f (y) − 66f (x).
(1.1)
It is easy to see that the function f (x) = ax + bx3 + cx4 satisfies the functional equation equation (1.1),
where a, b, c are arbitrary constants. In [6], the authors established the general solution and proved the
generalized Hyers-Ulam stability of the functional equation (1.1) in Banach spaces. And using the fixed
point method, the Hyers-Ulam stability results for the functional equation (1.1) in fuzzy Banach spaces and
multi-Banach spaces were established in [12, 27], respectively.
The main purpose of this paper is to apply the direct method and fixed point method to investigate the
Hyers-Ulam stability of functional equation (1.1) in matrix Banach spaces. We also prove the Hyers-Ulam
stability of the functional equation (1.1) in matrix fuzzy normed spaces by using the fixed point method.
2. Preliminaries
In this section, some definitions and preliminary results are given which will be used in this paper.
Following [2, 16, 17], we give the following notion of a fuzzy norm.
Definition 2.1. Let X be a real vector space. A function N : X × R → [0, 1] is said to be a fuzzy norm on
X if for all x, y ∈ X and all s, t ∈ R:
(N1) N (x, c) = 0 for c ≤ 0;
(N2) x = 0 if and only if N (x, c) = 1 for all c > 0;
t
(N3) N (cx, t) = N (x, |c|
) if c 6= 0;
(N4) N (x + y, s + t) ≥ min{N (x, s), N (y, t)};
(N5) N (x, ·) is a non-decreasing function on R and lim N (x, t) = 1;
t→∞
(N6) for x 6= 0, N (x, ·) is continuous on R.
In this case (X, N ) is called a fuzzy normed vector space.
Definition 2.2 ([2, 16, 17]). Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is said to
be convergent if there exists x ∈ X such that lim N (xn − x, t) = 1 (t > 0). In that case, x is called the
n→∞
limit of the sequence {xn } and we denote by N − lim xn = x.
n→∞
Definition 2.3 ([2, 16, 17]). Let (X, N ) be a fuzzy normed vector space. A sequence {xn } in X is called
Cauchy if for each ε > 0 and t > 0, there exists n0 ∈ N such that N (xm − xn , t) > 1 − ε (m, n ≥ n0 ). If each
Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is
called a fuzzy Banach space.
We will also use the following notations. The set of all m × n-matrices in X will be denoted by Mm,n (X).
When m = n, the matrix Mm,n (X) will be written as Mn (X). The symbols ej ∈ M1,n (C) will denote the
row vector whose jth component is 1 and the other components are 0. Similarly, Eij ∈ Mn (C) will denote
the n × n matrix whose (i, j)-component is 1 and the other components are 0. The n × n matrix whose
(i, j)-component is x and the other components are 0 will be denoted by Eij ⊗ x ∈ Mn (X).
Let (X, k · k) be a normed space. Note that (X, {k · kn }) is a matrix normed space if and only if
(Mn (X), k·kn ) is a normed space for each positive integer n and kAxBkk ≤ kAkkBkkxkn holds for A ∈ Mk,n ,
x = [xij ] ∈ Mn (X) and B ∈ Mn,k , and that (X, {k · kn }) is a matrix Banach space if and only if X is a
Banach space and (X, {k · kn }) is a matrix normed space.
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
66
Let E, F be vector spaces. For a given mapping h : E → F and a given positive integer n, define
hn : Mn (E) → Mn (F ) by
hn ([xij ]) = [h(xij )]
for all [xij ] ∈ Mn (E).
We introduce the concept of a matrix fuzzy normed space.
Definition 2.4 ([22]). Let (X, N ) be a fuzzy normed space.
(1) (X, {Nn }) is called a matrix fuzzy normed space if for each positive integer n, (Mn (X), Nn ) is a fuzzy
t
normed space and Nk (AxB, t) ≥ Nn (x, kAk·kBk
) for all t > 0, A ∈ Mk,n (R), x = [xij ] ∈ Mn (X) and
B ∈ Mn,k (R) with kAk · kBk =
6 0.
(2) (X, {Nn }) is called a matrix fuzzy Banach space if (X, N ) is a fuzzy Banach space and (X, {Nn }) is a
matrix fuzzy normed space.
Example 2.5. Let (X, {k · kn }) be a matrix normed space and α, β > 0. Define
(
αt
t > 0, x = [xij ] ∈ Mn (X),
αt+βkxkn ,
Nn (x, t) =
0,
t ≤ 0, x = [xij ] ∈ Mn (X).
Then (X, {Nn }) is a matrix fuzzy normed space.
3. Stability of the functional equation (1.1) in matrix Banach spaces: Direct method
Throughout this section, let (X, {k · kn }) be a matrix normed space, (Y, {k · kn }) be a matrix Banach
space and let n be a fixed positive integer. In this section, we prove the Hyers-Ulam stability of the ACQ
-functional equation (1.1) in matrix Banach spaces by using the direct method. We need the following
Lemmas:
Lemma 3.1 ([6]). Let V and W be real vector spaces. If an odd mapping f : V → W satisfies (1.1), then
f is cubic-additive.
Lemma 3.2 ([6]). Let V and W be real vector spaces. If an even mapping f : V → W satisfies (1.1), then
f is quartic.
Lemma 3.3 ([13, 14, 20, 21]). Let (X, {k · kn }) be a matrix normed space. Then
(1) kEkl ⊗ xkn = kxk for x ∈ X;
n
P
(2) kxkl k ≤ k[xij ]kn ≤
kxij k for [xij ] ∈ Mn (X);
i,j=1
(3) lim xn = x if and only if lim xijn = xij for xn = [xijn ], x = [xij ] ∈ Mk (X).
n→∞
n→∞
For a mapping f : X → Y , define Df : X 2 → Y and
Dfn : Mn (X 2 ) → Mn (Y )
by
Df (a, b) := 11[f (a + 2b) + f (a − 2b)] − 44[f (a + b) + f (a − b)]
− 12f (3b) + 48f (2b) − 60f (b) + 66f (a),
Dfn ([xij ], [yij ]) := 11[fn ([xij ] + 2[yij ]) + fn ([xij ] − 2[yij ])]
− 44[fn ([xij ] + [yij ]) + fn ([xij ] − [yij ])]
− 12fn (3[yij ]) + 48fn (2[yij ]) − 60fn ([yij ]) + 66fn ([xij ])
for all a, b ∈ X and all x = [xij ], y = [yij ] ∈ Mn (X).
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
67
Theorem 3.4. Let ϕ : X 2 → [0, ∞) be a function such that
∞
∞
X
X
1
1
l+1
l
ϕ(2 a, 2 a) +
ϕ(0, 2l a) < +∞,
l+1
l+1
8
8
(3.1)
1
ϕ(2k a, 2k b) = 0
k→∞ 8k
(3.2)
l=0
l=0
lim
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying
kDfn ([xij ], [yij ])kn ≤
n
X
ϕ(xij , yij )
(3.3)
i,j=1
for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y such that
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn
∞
∞
n X
1 X ϕ(2l+1 xij , 2l xij ) 14 X ϕ(0, 2l xij )
+
≤
11
33
8l+1
8l+1
i,j=1
l=0
(3.4)
l=0
for all x = [xij ] ∈ Mn (X).
Proof. When n = 1, (3.3) is equivalent to
kDf (a, b)k ≤ ϕ(a, b)
(3.5)
for all a, b ∈ X. Letting a = 0 in (3.5), we get
k12f (3b) − 48f (2b) + 60f (b)k ≤ ϕ(0, b)
(3.6)
for all b ∈ X. Replacing a by 2b in (3.5), we get
k11f (4b) − 56f (3b) + 114f (2b) − 104f (b)k ≤ ϕ(2b, b)
(3.7)
for all b ∈ X. It follows from (3.6) and (3.7) that
kf (4b) − 10f (2b) + 16f (b)k ≤
14
1
ϕ(2b, b) + ϕ(0, b)
11
33
(3.8)
for all b ∈ X. Replacing b by a and g(a) := f (2a) − 2f (a) in (3.8), we get
kg(2a) − 8g(a)k ≤
1
14
ϕ(2a, a) + ϕ(0, a)
11
33
(3.9)
for all a ∈ X. Replacing a by 2l a and dividing both sides by 8l+1 in (3.9), we have
g(2l+1 a) g(2l a)
1 ϕ(2l+1 a, 2l a) 14 ϕ(0, 2l a)
−
k≤
+
l+1
l
11
33 8l+1
8
8
8l+1
for all a ∈ X. Hence
k
g(2q a) g(2p a)
−
k
k
8q
8p
(3.10)
q−1
X
g(2l a) g(2l+1 a)
≤
k
−
k
8l
8l+1
l=p
q−1
m−1
1 X ϕ(2l+1 a, 2l a) 14 X ϕ(0, 2l a)
≤
+
11
33
8l+1
8l+1
l=p
(3.11)
l=p
for all nonnegative integers p, q with p < q and all a ∈ X. It follows from (3.1) and (3.11) that the sequence
k
k
{ g(28k a) } is a Cauchy sequence in Y for all a ∈ X. Since Y is complete, the sequence { g(28k a) } converges. So
one can define the mapping C : X → Y by
1
C(a) = lim k g(2k a)
(3.12)
k→∞ 8
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
68
for all a ∈ X. Moreover, letting p = 0 and passing the limit q → ∞ in (3.11), we get
∞
kg(a) − C(a)k ≤
∞
1 X ϕ(2l+1 a, 2l a) 14 X ϕ(0, 2l a)
+
11
33
8l+1
8l+1
l=0
(3.13)
l=0
for all a ∈ X.
Now, we show that the mapping C is cubic. By (3.1), (3.9) and (3.12),
1
1
g(2n+1 a) − n−1 g(2n a)k
n
n→∞ 8
8
1
1
n+1
a) − n g(2n a)k = 0
= lim 8k n+1 g(2
n→∞
8
8
for all a ∈ X. Therefore, we obtain
kC(2a) − 8C(a)k
= lim k
(3.14)
C(2a) = 8C(a)
(3.15)
for all a ∈ X. On the other hand it follows from (3.2), (3.5) and (3.12) that
1
Dg(2k a, 2k b)k
k→∞ 8k
1
= lim k kDf (2k+1 a, 2k+1 b) − 2Df (2k a, 2k b)k
k→∞ 8
1
≤ lim k (ϕ(2k+1 a, 2k+1 b) + 2ϕ(2k a, 2k b)) = 0
k→∞ 8
for all a, b ∈ X. Hence the mapping C satisfies (1.1). So by Lemma 3.1, the mapping
kDC(a, b)k
= lim k
(3.16)
a 7→ C(2a) − 2C(a)
is cubic. Hence (3.15) implies that the mapping C is cubic.
To prove the uniqueness of C, let C 0 : X → Y be another cubic mapping satisfying (3.13). Let n = 1.
Then we get
1
1
kC(a) − C 0 (a)k
= k q C(2q a) − q C 0 (2q a)k
8
8
1
1
1
1
q
≤ k q C(2 a) − q g(2q a)k + k q C 0 (2q a) − q g(2q a)k
8
8
8
8
∞
∞
1 X ϕ(2l+q+1 a, 2l+q a) 14 X ϕ(0, 2l+q a)
≤ 2(
+
)
11
33
8l+q+1
8l+q+1
= 2(
1
11
l=0
∞
X
l=0
ϕ(2l+1 a, 2l a)
8l+1
l=q
+
14
33
∞
X
l=q
ϕ(0, 2l a)
)
8l+1
for all a ∈ X. Letting q → ∞ in the above inequality, we get C(a) = C 0 (a) for all a ∈ X, which gives the
conclusion. Thus the mapping C : X → Y is a unique cubic mapping.
By Lemma 3.3 and (3.13), we get
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn ≤
n
X
kf (2xij ) − 2f (xij ) − C(xij )k
i,j=1
≤
n ∞
∞
X
1 X ϕ(2l+1 xij , 2l xij ) 14 X ϕ(0, 2l xij )
+
11
33
8l+1
8l+1
i,j=1
l=0
l=0
for all x = [xij ] ∈ Mn (X). Thus C : X → Y is a unique cubic mapping satisfying (3.4), as desired. This
completes the proof of the theorem.
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
69
Corollary 3.5. Let r, θ be positive real numbers with r < 3. Suppose that f : X → Y is an odd mapping
satisfying
kDfn ([xij ], [yij ])kn ≤
n
X
θ(kxij kr + kyij kr )
(3.17)
i,j=1
for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y such that
n
1 X 17 + 3 · 2r
θkxij kr
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn ≤
33
8 − 2r
(3.18)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. The proof follows immediately by taking ϕ(a, b) = θ(kakr + kbkb ) for all a, b ∈ X in Theorem 3.4.
Theorem 3.6. Let ϕ : X 2 → [0, ∞) be a function such that
∞
X
8l−1 ϕ(
l=1
lim 8k ϕ(
k→∞
a
2
,
l−1
∞
X
a
a
)+
8l−1 ϕ(0, l ) < +∞,
l
2
2
(3.19)
l=1
a b
, )=0
2k 2k
(3.20)
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (3.3) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique cubic mapping C : X → Y such that
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn
n ∞
∞
X
xij
14 X l−1
1 X l−1 xij xij
≤
8 ϕ( l−1 , l ) +
8 ϕ(0, l )
11
33
2
2
2
i,j=1
l=1
(3.21)
l=1
for all x = [xij ] ∈ Mn (X).
Proof. The proof of the theorem is similar to the proof of Theorem 3.4 and thus it is omitted.
Corollary 3.7. Let r, θ be positive real numbers with r > 3. Suppose that f : X → Y is an odd mapping
satisfying (3.17) for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y
such that
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn ≤
n
1 X 3 · 2r + 17
θkxij kr
33
2r − 8
(3.22)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. The asserted result in Corollary 3.7 can be easily derived by considering ϕ(a, b) = θ(kakr + kbkr ) for
all a, b ∈ X in Theorem 3.6.
Theorem 3.8. Let ϕ : X 2 → [0, ∞) be a function such that
∞
∞
X
X
1
1
l+1
l
ϕ(2
a,
2
a)
+
ϕ(0, 2l a) < +∞,
l+1
l+1
2
2
(3.23)
1
ϕ(2k a, 2k b) = 0
k→∞ 2k
(3.24)
l=0
lim
l=0
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
70
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (3.3) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique additive mapping A : X → Y such that
kfn (2[xij ]) − 8fn ([xij ]) − An ([xij ])kn
n ∞
∞
X
1 X ϕ(2l+1 xij , 2l xij ) 14 X ϕ(0, 2l xij )
≤
+
11
33
2l+1
2l+1
i,j=1
l=0
(3.25)
l=0
for all x = [xij ] ∈ Mn (X).
Proof. As in the proof of Theorem 3.4, we have
kf (4b) − 10f (2b) + 16f (b)k ≤
1
14
ϕ(2b, b) + ϕ(0, b)
11
33
(3.26)
for all b ∈ X. Replacing b by a and h(a) := f (2a) − 8f (a) in (3.26), we get
1
14
ϕ(2a, a) + ϕ(0, a)
11
33
for all a ∈ X. The rest of the proof is similar to the proof of Theorem 3.4.
kh(2a) − 2h(a)k ≤
(3.27)
Corollary 3.9. Let r, θ be positive real numbers with r < 1. Suppose that f : X → Y is an odd mapping
satisfying (3.17) for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique additive mapping A : X →
Y such that
n
1 X 17 + 3 · 2r
kfn (2[xij ]) − 8fn ([xij ]) − An ([xij ])kn ≤
θkxij kr
(3.28)
33
2 − 2r
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. The asserted result in Corollary 3.9 can be easily derived by considering ϕ(a, b) = θ(kakr + kbkr ) for
all a, b ∈ X in Theorem 3.8.
Theorem 3.10. Let ϕ : X 2 → [0, ∞) be a function such that
∞
X
2l−1 ϕ(
l=1
a
2
,
l−1
∞
X
a
a
)+
2l−1 ϕ(0, l ) < +∞,
l
2
2
(3.29)
l=1
a b
lim 2 ϕ( k , k ) = 0
(3.30)
k→∞
2 2
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (3.3) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique additive mapping A : X → Y such that
k
kfn (2[xij ]) − 8fn ([xij ]) − An ([xij ])kn
n ∞
∞
X
xij
14 X l−1
1 X l−1 xij xij
2 ϕ( l−1 , l ) +
2 ϕ(0, l )
≤
11
33
2
2
2
i,j=1
l=1
(3.31)
l=1
for all x = [xij ] ∈ Mn (X).
Proof. The proof of the theorem is similar to the proof of Theorem 3.6 and 3.8 thus it is omitted.
Corollary 3.11. Let r, θ be positive real numbers with r > 1. Suppose that f : X → Y is an odd mapping
satisfying (3.17) for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique additive mapping A : X →
Y such that
n
1 X 3 · 2r + 17
kfn (2[xij ]) − 8fn ([xij ]) − An ([xij ])kn ≤
θkxij kr
(3.32)
33
2r − 2
i,j=1
for all x = [xij ] ∈ Mn (X).
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
71
Proof. Letting ϕ(a, b) = θ(kakr + kbkb ) in Theorem 3.10, we obtain the result.
Theorem 3.12. Let ϕ : X 2 → [0, ∞) be a function such that
∞
X
l=0
∞
X 1
1
l
l
ϕ(2
a,
2
a)
+
ϕ(0, 2l a) < +∞,
16l+1
16l+1
(3.33)
l=0
1
ϕ(2k a, 2k b) = 0
k→∞ 16k
lim
(3.34)
for all a, b ∈ X. Suppose that f : X → Y is an even mapping satisfying (3.3) and f (0) = 0 for all
x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic mapping Q : X → Y such that
kfn ([xij ]) − Qn ([xij ])kn ≤
n ∞
∞
X
6 X ϕ(2l xij , 2l xij )
1 X ϕ(0, 2l xij )
+
11
16l+1
22
16l+1
i,j=1
l=0
(3.35)
l=0
for all x = [xij ] ∈ Mn (X).
Proof. Putting a = 0 in (3.5), we get
k − 12f (3b) + 70f (2b) − 148f (b)k ≤ ϕ(0, b)
(3.36)
for all b ∈ X. On the other hand, substituting a = b in (3.5), we obtain the following
k − f (3b) + 4f (2b) + 17f (b)k ≤ ϕ(b, b)
(3.37)
for all b ∈ X. By (3.36) and (3.37), we have
kf (2b) − 16f (b)k ≤
1
6
ϕ(b, b) + ϕ(0, b)
11
22
(3.38)
for all b ∈ X. Replacing a by 2l a and dividing both sides by 16l+1 in (3.38), we have
f (2l+1 a) f (2l a)
6 ϕ(2l a, 2l a)
1 ϕ(0, 2l a)
−
k
≤
+
11 16l+1
22 16l+1
16l+1
16l
for all a ∈ X. Hence
k
k
f (2q a) f (2p a)
−
k
16q
16p
≤
(3.39)
q−1
X
f (2l a) f (2l+1 a)
k
−
k
16l
16l+1
l=p
≤
q−1
m−1
1 X ϕ(0, 2l a)
6 X ϕ(2l a, 2l a)
+
11
22
16l+1
16l+1
l=p
(3.40)
l=p
for all nonnegative integers p, q with p < q and all a ∈ X. It follows from (3.33) and (3.40) that the sequence
k a)
k a)
} is a Cauchy sequence in Y for all a ∈ X. Since Y is complete, the sequence { f (2
} converges. So
{ f (2
16k
16k
one can define the mapping Q : X → Y by
1
Q(a) = lim
f (2k a)
(3.41)
k→∞ 16k
for all a ∈ X. Moreover, letting p = 0 and passing the limit q → ∞ in (3.40), we get
∞
kf (a) − Q(a)k ≤
∞
6 X ϕ(2l a, 2l a)
1 X ϕ(0, 2l a)
+
l+1
11
22
16
16l+1
l=0
(3.42)
l=0
for all a ∈ X. By (3.5), (3.34) and (3.41), we get
kDQ(a, b)k = lim k
k→∞
1
1
Df (2k a, 2k b)k ≤ lim
ϕ(2k a, 2k b) = 0
k
k→∞ 16k
16
(3.43)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
72
for all a, b ∈ X. Hence by Lemma 3.2, Q is quartic.
Now, Let Q0 : X → Y be another quartic mapping satisfying (3.42). Let n = 1. Then we get
kQ(a) − Q0 (a)k
1
1
Q(2q a) − q Q0 (2q a)k
q
16
16
1
1
1
1
q
≤ k q C(2 a) − q f (2q a)k + k q Q0 (2q a) − q f (2q a)kk
16
16
8
8
∞
∞
l+q
l+q
l+q
X
X
6
ϕ(2 a, 2 a)
1
ϕ(0, 2 a)
≤ 2(
+
)
l+q+1
11
22
16
16l+q+1
=k
l=0
l=0
∞
∞
6 X ϕ(2l a, 2l a)
1 X ϕ(0, 2l a)
= 2(
+
)
11
22
16l+1
16l+1
l=q
l=q
which tends to zero as q → ∞ for all a ∈ X. So we can conclude that Q(a) = Q0 (a) for all a ∈ X. This
proves the uniqueness of Q. Thus the mapping Q : X → Y is a unique quartic mapping.
By Lemma 3.3 and (3.42), we get
kfn ([xij ]) − Qn ([xij ])kn ≤
≤
n
X
kf (2xij ) − Q(xij )k
i,j=1
n X
i,j=1
∞
∞
l=0
l=0
1 X ϕ(0, 2l xij )
6 X ϕ(2l xij , 2l xij )
+
11
22
16l+1
16l+1
for all x = [xij ] ∈ Mn (X). Thus Q : X → Y is a unique quartic mapping satisfying (3.35), as desired. This
completes the proof of the theorem.
Corollary 3.13. Let r, θ be positive real numbers with r < 4. Suppose that f : X → Y is an even mapping
satisfying (3.17) and f (0) = 0 for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic
mapping Q : X → Y such that
kfn ([xij ]) − Qn ([xij ])kn ≤
n
25 X
θ
kxij kr
22
16 − 2r
(3.44)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. The proof follows immediately by taking ϕ(a, b) = θ(kakr +kbkb ) for all a, b ∈ X in Theorem 3.12.
Theorem 3.14. Let ϕ : X 2 → [0, ∞) be a function such that
∞
X
∞
X
a a
a
,
)
+
16l−1 ϕ(0, l ) < +∞,
l
l
2 2
2
(3.45)
a b
, )=0
2k 2k
(3.46)
16l−1 ϕ(
l=1
l=1
lim 16k ϕ(
k→∞
for all a, b ∈ X. Suppose that f : X → Y is an even mapping satisfying (3.3) and f (0) = 0 for all
x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic mapping Q : X → Y such that
n ∞
∞
X
6 X l−1 xij xij
xij
1 X l−1
kfn ([xij ]) − Qn ([xij ])kn ≤
16 ϕ( l , l ) +
16 ϕ(0, l )
11
2 2
22
2
i,j=1
l=1
l=1
for all x = [xij ] ∈ Mn (X).
Proof. The proof of the theorem is similar to the proof of Theorem 3.12 and thus it is omitted.
(3.47)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
73
Corollary 3.15. Let r, θ be positive real numbers with r > 4. Suppose that f : X → Y is an even mapping
satisfying (3.17) and f (0) = 0 for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic
mapping Q : X → Y such that
n
25 X
θ
kfn ([xij ]) − Qn ([xij ])kn ≤
kxij kr
22
2r − 16
(3.48)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. Letting ϕ(a, b) = θ(kakr + kbkb ) in Theorem 3.14, we obtain the result.
4. Stability of the functional equation (1.1) in matrix Banach spaces: Fixed point method
Throughout this section, let (X, {k · kn }) be a matrix normed space, (Y, {k · kn }) be a matrix Banach
space and let n be a fixed positive integer. In this section, we prove the Hyers-Ulam stability of the ACQ functional equation (1.1) in matrix Banach spaces by using Fixed point method. We begin with the definition
of a generalized metric on a set.
Let E be a set. A function d : E × E → [0, ∞] is called a generalized metric on E if d satisfies
(1) d(x, y) = 0 if and only if x = y;
(2) d(x, y) = d(y, x), ∀x, y ∈ E;
(3) d(x, z) ≤ d(x, y) + d(y, z), ∀x, y, z ∈ E.
Before proceeding to the proof of the main results, we begin with a result due to Diaz and Margolis [4].
Lemma 4.1 ([4] or [23]). Let (E, d) be a complete generalized metric space and J : E → E be a strictly
contractive mapping with Lipschitz constant L < 1. Then for each fixed element x ∈ E, either
d(J n x, J n+1 x) = ∞
∀n ≥ 0,
or
d(J n x, J n+1 x) < ∞
∀n ≥ n0 ,
for some natural number n0 . Moreover, if the second alternative holds then:
(i) The sequence {J n x} is convergent to a fixed point y ∗ of J;
(ii) y ∗ is the unique fixed point of J in the set E ∗ := {y ∈ E | d(J n0 x, y) < +∞} and d(y, y ∗ ) ≤
1
∗
1−L d(y, Jy), ∀x, y ∈ E .
Theorem 4.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
a b
ϕ(a, b) ≤ 8αϕ( , )
2 2
(4.1)
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (3.3) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique cubic mapping C : X → Y such that
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn
n
X
1
1
14
≤
ϕ(2xij , xij ) + ϕ(0, xij )
8(1 − α) 11
33
(4.2)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. When n = 1, similar to the proof of Theorem 3.4, and by (3.9),
1
1 1
14
ϕ(2a, a) + ϕ(0, a)
kg(a) − g(2a)k ≤
8
8 11
33
for all a ∈ X.
(4.3)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
74
Let S1 := {q1 : X → Y }, and introduce a generalized metric d1 on S1 as follows:
1
14
d1 (q1 , k1 ) := inf λ ∈ R+ kq1 (a) − k1 (a)k ≤ ϕ(2a, a) + ϕ(0, a), ∀a ∈ X .
11
33
It is easy to prove that (S1 , d1 ) is a complete generalized metric space [3, 7, 15].
Now we consider the mapping J1 : S1 → S1 defined by
1
J1 q1 (a) := q1 (2a), for all q1 ∈ S1 and a ∈ X.
(4.4)
8
Let q1 , k1 ∈ S1 and let λ ∈ R+ be an arbitrary constant with d1 (q1 , k1 ) ≤ λ. From the definition of d1 ,
we get
1
14
kq1 (a) − k1 (a)k ≤ λ
ϕ(2a, a) + ϕ(0, a)
11
33
for all a ∈ X. Therefore, using (4.1), we get
1
1
kJ1 q1 (a) − J1 k1 (a)k = k q1 (2a) − k1 (2a)k
8
8
14
λ 1
2
ϕ(2 a, 2a) + ϕ(0, 2a)
≤
8 11
33
14
1
ϕ(2a, a) + ϕ(0, a)
≤ αλ
11
33
(4.5)
for some α < 1 and for all a ∈ X. Hence, it holds that d1 (J1 q1 , J1 k1 ) ≤ αλ, that is, d1 (J1 q1 , J1 k1 ) ≤
αd1 (q1 , k1 ) for all q1 , k1 ∈ S1 .
It follows from (4.3) that d1 (g, J1 g) ≤ 81 . Therefore according to Lemma 4.1, the sequence J1n g converges
to a fixed point C of J1 , that is,
1
C : X → Y, lim n g(2n a) = C(a)
n→∞ 8
for all a ∈ X, and
C(2a) = 8C(a)
(4.6)
for all a ∈ X. Also C is the unique fixed point of J1 in the set S1∗ = {q1 ∈ S1 : d1 (g, q1 ) < ∞}. This implies
that C is a unique mapping satisfying (4.6) such that there exists a λ ∈ R+ such that
1
14
kg(a) − C(a)k ≤ λ
ϕ(2a, a) + ϕ(0, a)
11
33
for all a ∈ X. Also,
1
1
d1 (g, C) ≤
d1 (g, J1 g) ≤
.
1−α
8(1 − α)
So
kg(a) − C(a)k ≤
1
14
1
ϕ(2a, a) + ϕ(0, a)
8(1 − α) 11
33
for all a ∈ X.
It follows from (3.5) and (4.1) that
kDC(a, b)k
1
kDg(2l a, 2l b)k
l→∞ 8l
1
≤ lim l (ϕ(2l · 2a, 2l · 2b) + 2ϕ(2l a, 2l b))
l→∞ 8
8l α l
≤ lim l (ϕ(2a, 2b) + 2ϕ(a, b)) = 0
l→∞ 8
= lim
(4.7)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
75
for all a, b ∈ X. Hence DC(a, b) = 0. So by Lemma 3.1, the mapping x 7→ C(2a) − 2C(a) is cubic. Hence
(4.6) implies that the mapping C : X → Y is cubic.
By Lemma 3.3 and (4.7),
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn ≤
≤
n
X
i,j=1
n
X
i,j=1
kf (2xij ) − 2f (xij ) − C(xij )k
1
1
14
ϕ(2xij , xij ) + ϕ(0, xij )
8(1 − α) 11
33
for all x = [xij ] ∈ Mn (X). Thus C : X → Y is a unique cubic mapping satisfying (4.2), as desired. This
completes the proof of the theorem.
Corollary 4.3. Let r, θ be positive real numbers with r < 3. Suppose that f : X → Y is an odd mapping
satisfying (3.17) for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y
satisfying (3.18) for all x = [xij ] ∈ Mn (X).
Proof. The proof follows immediately by taking
ϕ(a, b) = θ(kakr + kbkr )
for all a, b ∈ X and choosing α = 2r−3 in Theorem 4.2.
Theorem 4.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
ϕ(a, b) ≤
α
ϕ(2a, 2b)
8
(4.8)
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (3.3) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique cubic mapping C : X → Y such that
kfn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ])kn
n
X
1
14
α
≤
ϕ(2xij , xij ) + ϕ(0, xij )
8(1 − α) 11
33
(4.9)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. Let (S1 , d1 ) be the generalized metric space defined in the proof of Theorem 4.2.
Now, we consider the mapping J1 : S1 → S1 defined by
a
J1 q1 (a) := 8q1 ( ), for all q1 ∈ S1 and a ∈ X.
2
It follows from (3.9) that
α 1
14
a
kg(a) − 8g( )k ≤
ϕ(2a, a) + ϕ(0, a)
2
8 11
33
(4.10)
(4.11)
for all a ∈ X. Thus d1 (g, J1 g) ≤ α8 . So
d1 (g, C) ≤
1
α
d1 (g, J1 g) ≤
.
1−α
8(1 − α)
The rest of the proof is similar to the proof of Theorem 4.2.
Corollary 4.5. Let r, θ be positive real numbers with r > 3. Suppose that f : X → Y is an odd mapping
satisfying (3.17) for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y
satisfying (3.22) for all x = [xij ] ∈ Mn (X).
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
76
Proof. The asserted result in Corollary 4.5 can be easily derived by considering
ϕ(a, b) = θ(kakr + kbkr )
for all a, b ∈ X and α = 23−r in Theorem 4.4.
Theorem 4.6. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
a b
ϕ(a, b) ≤ 2αϕ( , )
2 2
(4.12)
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (3.3) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique additive mapping A : X → Y such that
kfn (2[xij ]) − 8fn ([xij ]) − An ([xij ])kn
n
X
1
14
1
≤
ϕ(2xij , xij ) + ϕ(0, xij )
2(1 − α) 11
33
(4.13)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. When n = 1, similar to the proof of Theorem 3.8, and by (3.27),
1 1
14
1
ϕ(2a, a) + ϕ(0, a)
kh(a) − h(2a)k ≤
2
2 11
33
for all a ∈ X. Let (S1 , d1 ) be the generalized metric space defined in the proof of Theorems 4.2.
Now we consider the mapping J1 : S1 → S1 defined by
1
J1 q1 (a) := q1 (2a), for all q1 ∈ S1 and a ∈ X.
2
(4.14)
(4.15)
Thus d1 (h, J1 h) ≤ 21 . So
d1 (h, A) ≤
1
1
d1 (h, J1 h) ≤
.
1−α
2(1 − α)
The rest of the proof is similar to the proof of Theorem 4.2.
Corollary 4.7. Let r, θ be positive real numbers with r < 1. Suppose that f : X → Y is an odd mapping
satisfying (3.17) for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique additive mapping A : X →
Y satisfying (3.28) for all x = [xij ] ∈ Mn (X).
Proof. The proof follows from Theorem 4.6 by taking asserted ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X.
Then we can choose α = 2r−1 and we get the desired result.
Theorem 4.8. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
ϕ(a, b) ≤
α
ϕ(2a, 2b)
2
(4.16)
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (3.3) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique additive mapping A : X → Y such that
kfn (2[xij ]) − 8fn ([xij ]) − An ([xij ])kn
n
X
α
1
14
≤
ϕ(2xij , xij ) + ϕ(0, xij )
2(1 − α) 11
33
i,j=1
for all x = [xij ] ∈ Mn (X).
(4.17)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
Proof. Let (S1 , d1 ) be the generalized metric space defined in the proof of Theorem 4.2.
Now we consider the mapping J1 : S1 → S1 defined by
a
J1 q1 (a) := 2q1 ( ), for all q1 ∈ S1 and a ∈ X.
2
It follows from (3.27) that
α 1
14
a
ϕ(2a, a) + ϕ(0, a)
kh(a) − 2h( )k ≤
2
2 11
33
77
(4.18)
(4.19)
for all a ∈ X. Thus d1 (h, J1 h) ≤ α2 . So
d1 (h, A) ≤
1
α
d1 (h, J1 h) ≤
.
1−α
2(1 − α)
The rest of the proof is similar to the proof of Theorem 4.2 and 4.6.
Corollary 4.9. Let r, θ be positive real numbers with r > 1. Suppose that f : X → Y is an odd mapping
satisfying (3.17) for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique additive mapping A : X →
Y satisfying (3.32) for all x = [xij ] ∈ Mn (X).
Proof. By choosing ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and α = 21−r in Theorem 4.8, we obtain the
inequality (3.32).
Theorem 4.10. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
a b
ϕ(a, b) ≤ 16αϕ( , )
2 2
(4.20)
for all a, b ∈ X. Suppose that f : X → Y is an even mapping satisfying (3.3) and f (0) = 0 for all
x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic mapping Q : X → Y such that
kfn ([xij ]) − Qn ([xij ])kn ≤
n
X
i,j=1
1
6
1
ϕ(xij , xij ) + ϕ(0, xij )
16(1 − α) 11
22
(4.21)
for all x = [xij ] ∈ Mn (X).
Proof. When n = 1, similar to the proof of Theorem 3.12, and by (3.38),
1 6
1
1
kf (a) − f (2a)k ≤
ϕ(a, a) + ϕ(0, a)
16
16 11
22
(4.22)
for all a ∈ X.
Let S2 := {q2 : X → Y }, and introduce a generalized metric d2 on S2 as follows:
6
1
d2 (q2 , k2 ) := inf µ ∈ R+ kq2 (a) − k2 (a)k ≤ ϕ(a, a) + ϕ(0, a), ∀a ∈ X .
11
22
It is easy to prove that (S2 , d2 ) is a complete generalized metric space [3, 7, 15].
Now we consider the mapping J2 : S2 → S2 defined by
1
J2 q2 (a) := q2 (2a), for all q2 ∈ S2 and a ∈ X.
(4.23)
16
Let q2 , k2 ∈ S2 and let µ ∈ R+ be an arbitrary constant with d2 (q2 , k2 ) ≤ µ. From the definition of d2 ,
we get
1
6
ϕ(a, a) + ϕ(0, a)
kq2 (a) − k2 (a)k ≤ µ
11
22
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
78
for all a ∈ X. Therefore, using (4.20), we get
1
1
q1 (2a) − k1 (2a)k
16
16
µ 6
1
≤
ϕ(2a, 2a) + ϕ(0, 2a)
16 11
22
1
6
ϕ(a, a) + ϕ(0, a)
≤ αµ
11
22
kJ2 q2 (a) − J2 k2 (a)k = k
(4.24)
for some α < 1 and for all a ∈ X. Hence, it holds that d2 (J2 q2 , J2 k2 ) ≤ αµ, that is, d2 (J2 q2 , J2 k2 ) ≤
αd2 (q2 , k2 ) for all q2 , k2 ∈ S2 .
1
It follows from (4.22) that d2 (f, J2 f ) ≤ 16
. Therefore according to Lemma 4.1, the sequence J2n g
converges to a fixed point Q of J2 , that is,
1
f (2n a) = Q(a)
Q : X → Y, lim
n→∞ 16n
for all a ∈ X, and
Q(2a) = 16Q(a)
(4.25)
for all a ∈ X. Also Q is the unique fixed point of J2 in the set S2∗ = {q2 ∈ S2 : d2 (f, q2 ) < ∞}. This implies
that Q is a unique mapping satisfying (4.25) such that there exists a µ ∈ R+ such that
6
1
kf (a) − Q(a)k ≤ µ
ϕ(a, a) + ϕ(0, a)
11
22
for all a ∈ X. Also,
1
1
d2 (f, J2 f ) ≤
.
d2 (f, Q) ≤
1−α
16(1 − α)
So
1
6
1
kf (a) − Q(a)k ≤
ϕ(a, a) + ϕ(0, a)
16(1 − α) 11
22
(4.26)
for all a ∈ X.
It follows from (3.5) and (4.20) that
1
1
kDf (2l a, 2l b)k ≤ lim l ϕ(2l a, 2l b)
l→∞ 16
16l
16l αl
≤ lim
ϕ(a, b) = 0
l→∞ 16l
for all a, b ∈ X. Hence DQ(a, b) = 0. So by Lemma 3.2, the mapping Q : X → Y is quartic.
By Lemma 3.3 and (4.26),
kDQ(a, b)k
= lim
l→∞
kfn ([xij ]) − Qn ([xij ])kn ≤
≤
n
X
i,j=1
n
X
i,j=1
kf (2xij ) − Q(xij )k
6
1
1
ϕ(xij , xij ) + ϕ(0, xij )
16(1 − α) 11
22
for all x = [xij ] ∈ Mn (X). Thus Q : X → Y is a unique quartic mapping satisfying (4.21), as desired. This
completes the proof of the theorem.
Corollary 4.11. Let r, θ be positive real numbers with r < 4. Suppose that f : X → Y is an even mapping
satisfying (3.17) and f (0) = 0 for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic
mapping Q : X → Y satisfying (3.44) for all x = [xij ] ∈ Mn (X).
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
79
Proof. The proof follows immediately by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and choosing
α = 2r−4 in Theorem 4.10.
Theorem 4.12. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
ϕ(a, b) ≤
α
ϕ(2a, 2b)
16
(4.27)
for all a, b ∈ X. Suppose that f : X → Y is an even mapping satisfying (3.3) and f (0) = 0 for all
x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic mapping Q : X → Y such that
kfn ([xij ]) − Qn ([xij ])kn ≤
n
X
i,j=1
6
1
α
ϕ(xij , xij ) + ϕ(0, xij )
16(1 − α) 11
22
(4.28)
for all x = [xij ] ∈ Mn (X).
Proof. Let (S2 , d2 ) be the generalized metric space defined in the proof of Theorem 4.10.
Now we consider the mapping J2 : S2 → S2 defined by
a
J2 q2 (a) := 16q2 ( ), for all q2 ∈ S2 and a ∈ X.
2
It follows from (3.38) that
a
α 6
1
kf (a) − 16f ( )k ≤
ϕ(a, a) + ϕ(0, a)
2
16 11
22
for all a ∈ X. Thus d2 (f, J2 f ) ≤
d2 (f, Q) ≤
α
16 .
(4.29)
(4.30)
So
α
1
d2 (f, J2 f ) ≤
.
1−α
16(1 − α)
The rest of the proof is similar to the proof of Theorem 4.10.
Corollary 4.13. Let r, θ be positive real numbers with r > 4. Suppose that f : X → Y is an even mapping
satisfying (3.17) and f (0) = 0 for all x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic
mapping Q : X → Y satisfying (3.48) for all x = [xij ] ∈ Mn (X).
Proof. The proof follows immediately by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and choosing
α = 24−r in Theorem 4.12.
5. Stability of the functional equation (1.1) in matrix fuzzy normed spaces
Throughout this section, let (X, {Nn }) be a matrix fuzzy normed space, (Y, {Yn }) be a matrix fuzzy
Banach space and let n be a fixed positive integer. Using the fixed point method, we prove the HyersUlam stability of the ACQ-functional equation (1.1) in matrix fuzzy normed spaces. We need the following
Lemma:
Lemma 5.1 ([22]). Let (X, {Nn }) be a matrix fuzzy normed space. Then
(1) Nn (Ekl ⊗ x, t) = N (x, t) for all t > 0 and x ∈ X;
n
P
(2) For all [xij ] ∈ Mn (X) and t =
tij ,
i,j=1
N (xkl , t) ≥ Nn ([xij ], t) ≥ min{N (xij , tij ) : i, j = 1, 2, . . . , n},
t
N (xkl , t) ≥ Nn ([xij ], t) ≥ min{N (xij , 2 ) : i, j = 1, 2, . . . , n};
n
(3) lim xn = x if and only if lim xijn = xij for xn = [xijn ], x = [xij ] ∈ Mk (X).
n→∞
n→∞
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
80
Theorem 5.2. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
a b
ϕ(a, b) ≤ 8αϕ( , )
2 2
(5.1)
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying
t
Nn (Dfn ([xij ], [yij ]), t) ≥
n
P
t+
(5.2)
ϕ(xij , yij )
i,j=1
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y such
that
Nn (fn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ]), t)
(264 − 264α)t
≥
n
P
(264 − 264α)t + 17n2
(ϕ(2xij , xij ) + ϕ(0, xij ))
(5.3)
i,j=1
for all t > 0 and x = [xij ] ∈ Mn (X).
Proof. Let n = 1 in (5.2). Then (5.2) is equivalent to
N (Df (a, b), t) ≥
t
t + ϕ(a, b)
(5.4)
for all t > 0 and a, b ∈ X. By the same reasoning as in the proof of [12, Theorem 3], one can show that
there exists a unique cubic mapping C : X → Y such that
N (f (2a) − 2f (a) − C(a), t) ≥
(264 − 264α)t
(264 − 264α)t + 17(ϕ(2a, a) + ϕ(0, a))
(5.5)
for all t > 0 and a ∈ X. The mapping C : X → Y is given by
f (2l+1 a) − 2f (2l a)
l→∞
8l
C(a) = N − lim
for all a ∈ X.
By Lemma 5.1 and (5.5),
Nn (fn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ]), t)
t
≥ min{N (f (2xij ) − 2f (xij ) − C(xij ), 2 ) : i, j = 1, 2, . . . , n}
n
(264 − 264α)t
≥ min{
: i, j = 1, 2, . . . , n}
(264 − 264α)t + 17n2 (ϕ(2xij , xij ) + ϕ(0, xij ))
(264 − 264α)t
≥
n
P
(264 − 264α)t + 17n2
(ϕ(2xij , xij ) + ϕ(0, xij ))
i,j=1
for all t > 0 and x = [xij ] ∈ Mn (X). Thus C : X → Y a unique cubic mapping satisfying (5.3), as desired.
This completes the proof of the theorem.
Corollary 5.3. Let r, θ be positive real numbers with r < 3. Suppose that f : X → Y is an odd mapping
satisfying
t
Nn (Dfn ([xij ], [yij ]), t) ≥
t+
n
P
i,j=1
θ(kxij kr + kyij kr )
(5.6)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
81
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y such
that
Nn (fn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ]), t)
(264 − 33 · 2r )t
≥
n
P
(264 − 33 · 2r )t + 17n2 (2r + 2)
θkxij kr
(5.7)
i,j=1
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X).
Proof. The proof follows immediately by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and choosing
α = 2r−3 in Theorem 5.2.
Theorem 5.4. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
α
ϕ(a, b) ≤ ϕ(2a, 2b)
(5.8)
8
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (5.2) for all t > 0 and x = [xij ], y =
[yij ] ∈ Mn (X). Then there exists a unique cubic mapping C : X → Y such that
Nn (fn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ]), t)
(264 − 264α)t
≥
n
P
(264 − 264α)t + 17n2 α
(ϕ(2xij , xij ) + ϕ(0, xij ))
(5.9)
i,j=1
for all t > 0 and x = [xij ] ∈ Mn (X).
Proof. The proof is similar to the proof of Theorem 5.2.
Corollary 5.5. Let r, θ be positive real numbers with r > 3. Suppose that f : X → Y is an odd mapping
satisfying (5.6) for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique cubic mapping
C : X → Y such that
Nn (fn (2[xij ]) − 2fn ([xij ]) − Cn ([xij ]), t)
(33 · 2r − 264)t
≥
n
P
(33 · 2r − 264)t + 17n2 (2r + 2)
θkxij kr
(5.10)
i,j=1
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X).
Proof. By choosing ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and α = 23−r in Theorem 5.4, we obtain the
inequality (5.10).
Theorem 5.6. Let ϕ : X 2 → [0, ∞) is a function such that there exists an α < 1 with
a b
ϕ(a, b) ≤ 2αϕ( , )
(5.11)
2 2
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (5.2) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique additive mapping A : X → Y such that
Nn (fn (2[xij ]) − 8fn ([xij ]) − An ([xij ]), t)
(66 − 66α)t
≥
n
P
(66 − 66α)t + 17n2
(ϕ(2xij , xij ) + ϕ(0, xij ))
i,j=1
for all x = [xij ] ∈ Mn (X).
(5.12)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
82
Proof. Let n = 1 in (5.2). Then (5.2) is equivalent to (5.4) for all t > 0 and a, b ∈ X. By the same reasoning
as in the proof of [12, Theorem 5], one can show that there exists a unique additive mapping A : X → Y
such that
(66 − 66α)t
N (f (2a) − 8f (a) − A(a), t) ≥
(5.13)
(66 − 66α)t + 17(ϕ(2a, a) + ϕ(0, a))
for all t > 0 and a ∈ X. The mapping C : X → Y is given by
f (2l+1 a) − 8f (2l a)
l→∞
2l
A(a) = N − lim
for all a ∈ X.
By Lemma 5.1 and (5.13),
Nn (fn (2[xij ]) − 8fn ([xij ]) − An ([xij ]), t)
t
≥ min{N (f (2xij ) − 8f (xij ) − A(xij ), 2 ) : i, j = 1, 2, . . . , n}
n
(66 − 66α)t
≥ min{
: i, j = 1, 2, . . . , n}
(66 − 66α)t + 17n2 (ϕ(2xij , xij ) + ϕ(0, xij ))
(66 − 66α)t
≥
n
P
(66 − 66α)t + 17n2
(ϕ(2xij , xij ) + ϕ(0, xij ))
i,j=1
for all t > 0 and x = [xij ] ∈ Mn (X). Thus A : X → Y a unique additive mapping satisfying (5.12), as
desired. This completes the proof of the theorem.
Corollary 5.7. Let r, θ be positive real numbers with r < 1. Suppose that f : X → Y is an odd mapping
satisfying (5.6) for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique additive mapping
A : X → Y such that
Nn (fn (2[xij ]) − 8fn ([xij ]) − An ([xij ]), t)
(66 − 33 · 2r )t
≥
n
P
(66 − 33 · 2r )t + 17n2 (2r + 2)
θkxij kr
(5.14)
i,j=1
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X).
Proof. The proof follows immediately by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and choosing
α = 2r−1 in Theorem 5.6.
Theorem 5.8. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
ϕ(a, b) ≤
α
ϕ(2a, 2b)
2
(5.15)
for all a, b ∈ X. Suppose that f : X → Y is an odd mapping satisfying (5.2) for all x = [xij ], y = [yij ] ∈
Mn (X). Then there exists a unique additive mapping A : X → Y such that
Nn (fn (2[xij ]) − 8fn ([xij ]) − An ([xij ]), t)
(66 − 66α)t
≥
n
P
(66 − 66α)t + 17n2 α
(ϕ(2xij , xij ) + ϕ(0, xij ))
i,j=1
for all x = [xij ] ∈ Mn (X).
(5.16)
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
83
Proof. The proof is similar to the proof of Theorem 5.6.
Corollary 5.9. Let r, θ be positive real numbers with r > 1. Suppose that f : X → Y is an odd mapping
satisfying (5.6) for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique additive mapping
A : X → Y such that
Nn (fn (2[xij ]) − 8fn ([xij ]) − An ([xij ]), t)
(33 · 2r − 66)t
≥
n
P
θkxij kr
(33 · 2r − 66)t + 17n2 (2r + 2)
(5.17)
i,j=1
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X).
Proof. The proof follows immediately by taking ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and choosing
α = 21−r in Theorem 5.8.
Theorem 5.10. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
a b
ϕ(a, b) ≤ 16αϕ( , )
2 2
(5.18)
for all a, b ∈ X. Suppose that f : X → Y is an even mapping satisfying (5.2) and f (0) = 0 for all t > 0 and
x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic mapping Q : X → Y such that
Nn (fn ([xij ]) − Qn ([xij ]), t) ≥
(352 − 352α)t
n
P
(352 − 352α)t + 13n2
(ϕ(xij , xij ) + ϕ(0, xij ))
(5.19)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. Let n = 1 in (5.2). Then (5.2) is equivalent to (5.4) for all t > 0 and a, b ∈ X. By the same reasoning
as in the proof of [12, Theorem 7], one can show that there exists a unique quartic mapping Q : X → Y
such that
(352 − 352α)t
(5.20)
N (f (a) − Q(a), t) ≥
(352 − 352α)t + 13(ϕ(a, a) + ϕ(0, a))
for all t > 0 and a ∈ X. The mapping C : X → Y is given by
f (2l a)
l→∞ 16l
Q(a) = N − lim
for all a ∈ X.
By Lemma 5.1 and (5.13),
Nn (fn ([xij ]) − Qn ([xij ]), t)
t
) : i, j = 1, 2, . . . , n}
n2
(352 − 352α)t
≥ min{
: i, j = 1, 2, . . . , n}
(352 − 352α)t + 13n2 (ϕ(xij , xij ) + ϕ(0, xij ))
(352 − 352α)t
≥
n
P
(352 − 352α)t + 13n2
(ϕ(xij , xij ) + ϕ(0, xij ))
≥ min{N (f (xij ) − Q(xij ),
i,j=1
for all t > 0 and x = [xij ] ∈ Mn (X). Thus Q : X → Y a unique quartic mapping satisfying (5.19), as
desired. This completes the proof of the theorem.
Z. Wang, P. K. Sahoo, J. Nonlinear Sci. Appl. 8 (2015), 64–85
84
Corollary 5.11. Let r, θ be positive real numbers with r < 4. Suppose that f : X → Y is an even mapping
satisfying (5.6) and f (0) = 0 for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique
quartic mapping Q : X → Y such that
Nn (fn ([xij ]) − Qn ([xij ]), t) ≥
(352 − 22 · 2r )t
n
P
(352 − 22 · 2r )t + 39n2
θkxij kr
(5.21)
i,j=1
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X).
Proof. By choosing ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X and α = 2r−4 in Theorem 5.10, we obtain the
inequality (5.21).
Theorem 5.12. Let ϕ : X 2 → [0, ∞) be a function such that there exists an α < 1 with
ϕ(a, b) ≤
α
ϕ(2a, 2b)
16
(5.22)
for all a, b ∈ X. Suppose that f : X → Y is an even mapping satisfying (5.2) and f (0) = 0 for all t > 0 and
x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique quartic mapping Q : X → Y such that
Nn (fn ([xij ]) − Qn ([xij ]), t) ≥
(352 − 352α)t
n
P
(352 − 352α)t + 13n2 α
(ϕ(xij , xij ) + ϕ(0, xij ))
(5.23)
i,j=1
for all x = [xij ] ∈ Mn (X).
Proof. The proof is similar to the proof of Theorem 5.10.
Corollary 5.13. Let r, θ be positive real numbers with r > 4. Suppose that f : X → Y is an even mapping
satisfying (5.6) and f (0) = 0 for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X). Then there exists a unique
quartic mapping Q : X → Y such that
Nn (fn ([xij ]) − Qn ([xij ]), t) ≥
(22 · 2r − 352)t
n
P
(22 · 2r − 352)t + 39n2
θkxij kr
(5.24)
i,j=1
for all t > 0 and x = [xij ], y = [yij ] ∈ Mn (X).
Proof. The proof follows from Theorem 5.12 by taking asserted ϕ(a, b) = θ(kakr + kbkr ) for all a, b ∈ X.
Then we can choose α = 24−r and we get the desired result.
Acknowledgements
The research work was done during the first author’s study to University of Louisville as a Postdoctoral
Fellow in 2013-14.
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