Post on 09-Aug-2020
International Journal of Mathematical Analysis
Vol. 14, 2020, no. 5, 219 - 239
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ijma.2020.91169
Hyers-Ulam Stability Additive β-Functional
Inequalities with Three Variables in
Non-Archimedean Banach Space and
Complex Banach Spaces
Ly Van An
Faculty of Mathematics Teacher Education
Tay Ninh University
Ninh Trung, Ninh Son, Tay Ninh Province, Vietnam
This article is distributed under the Creative Commons by-nc-nd Attribution License.
Copyright c© 2020 Hikari Ltd.
Abstract
In this paper we solve additive β-functional inequalities with three variables and their
Hyers-Ulam stability in non- Archimedean Banach spaces as well as in complex Banach
spaces. It is shown that the solutions of first and second inequalities are additive map-
pings. Then Hyers-Ulam stability of these inequalities is studied and proven.
Keywords: additive β-functional equation; additive β-functional inequality; non-
Archimedean normed space; complex Banach space; Hyers-Ulam stability
1. Introduction
Let X and Y be a normed spaces on the same field K, and f : X → Y . We usethe notation
∥∥ · ∥∥ for all the norm on both X and Y . In this paper, we investigatesome additive β-functional inequalities when X is non-Archimedean normed spaceand Y is non-Archimedean Banach space, or X is complex normed space and Y iscomplex Banach space.
220 Ly Van An
In fact, when X is non-Archimedean normed space and Y is non-Archimedean Ba-nach space we solve and prove the Hyers-Ulam stability of following additive β-functional inequality. Let
∣∣2∣∣ 6= 1 and let β be a non-Archimedean number∣∣β∣∣ < 1.∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f(z)
∥∥∥∥ ≤ ∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f(z)
)∥∥∥∥(1.1)
∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f(z)
∥∥∥∥≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f(z)
)∥∥∥∥(1.2)
when X is complex normed space and Y is complex Banach spaces we solve andprove the Hyers-Ulam stability of following additive β-functional inequality. Let βbe a complex number with
∣∣β∣∣ < 1.∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f(z)
∥∥∥∥ ≤ ∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f(z)
)∥∥∥∥(1.3)
∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f(z)
∥∥∥∥≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f(z)
)∥∥∥∥(1.4)
The notions of non-Archimedean normed space and complex normed spaces will
be reminded in the next section. The Hyers-Ulam stability was first investigated for
functional equation of Ulam in [6] concerning the stability of group homomorphisms.
The functional equation
f(x+ y) = f(x) + f(y)
is called the Cauchy equation. In particular, every solution of the Cauchy equation
is said to be an additive mapping.
The Hyers [7] gave first affirmative partial answer to the equation of Ulam in
Banach spaces. After that, Hyers’ Theorem was generalized by Aoki [1] for additive
mappings and by Rassias [8] for linear mappings considering an unbouned Cauchy
diffrence. A generalization of the Rassias theorem was obtained by Gavruta [4] by
replacing the unbounded Cauchy difference by a general control function in the spirit
Hyers-Ulam stability additive β-functional inequalities 221
of Rassias’ approach.
f
(x+ y
2
)=
1
2f(x)
+1
2f(y)
is called the Jensen equation. See [2, 3, 9, 10] for more information on functional
equations.
The Hyers-Ulam stability for functional inequalities have been investigated such
as in [14, 15]. Gilany showed that is if satisfies the functional inequality∥∥∥∥2f(x)
+ 2f(y)− f
(xy−1
)∥∥∥∥ ≤ ∥∥f(xy)∥∥Then f satisfies the Jordan-von Newman functional equation
2f(x)
+ 2f(y)− f
(xy−1
)(1.5)
See also [15,16]. Gilanyi [13] and Fechner [9] proved the Hyers-Ulam stability of the
functional inequality.
Choonkil Park [17] proved the Hyers-Ulam stability of additive β-functional inequal-
ities. Recently, in [18, 19, 20] the authors studied the Hyers-Ulam stability for the
following functional inequalities∥∥∥∥f(x+ y)− f
(x)− f(y)
∥∥∥∥ ≤ ∥∥∥∥ρ(2f
(x+ y
2
)− f
(x)− f
(y))∥∥∥∥ (1.6)
∥∥∥∥2f
(x+ y
2
)− f
(x)− f
(y)∥∥∥∥ ≤ ∥∥∥∥ρ(f(x+ y
)− f
(x)− f
(y))∥∥∥∥ (1.7)
in Non-Archimedean Banach spaces and complex Banach spaces
In this paper, we solve and prove the Hyers-Ulam stability for two β-functional
inequalities (1.1)-(1.2), i.e. the β -functional inequalities with three variables [4,
13,14]. Under suitable assumptions on spaces X and Y , we will prove that the
mappings satisfying the β-functional inequatilies (1.1) or (1.2). Thus, the results in
this paper are generalization of those in [17] for β-functional inequatilies with three
variables.
The paper is organized as follows: In section preliminaries we remind some basic
notations in [18, 19, 20] such as Non-Archimedean field, Non-Archimedean normed
space and Non-Archimedean Banach space.
Section 3 is devoted to prove the Hyers-Ulam stability of the addive β- functional
inequalities (1.1) and (1.2) when X Non-Archimedean normed space and Y Non-
Archimedean Banach space.
Section 4 is devoted to prove the Hyers-Ulam stability of the additive β- functional
inequalities (1.1) and (1.2) when X is complex normed space and Y is complex
Banach space.
222 Ly Van An
2. preliminaries
2.1. Non-Archimedean normed and Banach spaces. In this subsection we re-
call some basic notations [19, 20] such as Non-Archimedean fields, Non-Archimedean
normed spaces and Non-Archimedean normed spaces.
A valuation is a function∣∣ · ∣∣ from a field K into
[0,∞
)such that 0 is the unique
element having the 0 valuation,∣∣r.s∣∣ =∣∣r∣∣.∣∣s∣∣,∀r, s ∈ K
and the triangle inequality holds, ie;∣∣r + s∣∣ ≤ ∣∣r∣∣+
∣∣s∣∣,∀r, s ∈ K
A field K is called a valued filed if K carries a valuation. The usual absolute values
of R and C are examples of valuation. Let us consider a vavluation which satisfies
a stronger condition than the triangle inaquality. If the tri triangle inequality is
replaced by
∣∣r + s∣∣ ≤ max
{∣∣r∣∣, ∣∣s∣∣},∀r, s ∈ K
then the function∣∣ · ∣∣ is called a norm -Archimedean valuational, and filed. Clearly∣∣1∣∣ =
∣∣ − 1∣∣ = 1 and
∣∣n∣∣ ≤ 1, ∀n ∈ N . A trivial expamle of a non- Archimedean
valuation is the function∣∣ · ∣∣ talking everything except for 0 into 1 and
∣∣0∣∣ = 0 this
paper, we assume that the base field is a non-Archimedean filed, hence call it simply
a filed.
Definition 2.1.
Let be a vector space over a filed K with a non-Archimedean∣∣ · ∣∣. A function∥∥ · ∥∥ : X → [0,∞) is said a non-Archimedean norm if it satisfies the following
conditions:
(1)∥∥x∥∥ = 0 if and only if x = 0;
(2)∥∥rx∥∥ =
∣∣r∣∣∥∥x∥∥(r ∈ K, x ∈ X);
(3) the strong triangle inequlity
∥∥x+ y∥∥ ≤ max
{∥∥x∥∥, ∥∥y∥∥}, x, y ∈ Xhold. Then (X,
∥∥ · ∥∥) is called a norm -Archimedean norm space.
Hyers-Ulam stability additive β-functional inequalities 223
Definition 2.2.
(1) Let{xn}
be a sequence in a norm-Archimedean normed space X. Then
sequence{xn}
is called Cauchy if for a given ε > 0 there a positive integer
N such that ∥∥xn − x∥∥ ≤ ε
for all n,m ≥ N
(2) Let{xn}
be a sequence in a norm-Archimedean normed space X. Then
sequence{xn}
is called cauchy if for a given ε > 0 there a positive integer N
such that
∥∥xn − x∥∥ ≤ ε
for all n,m ≥ N. The we call x∈ X a limit of sequence xn and denote
limn→∞ xn = x.
(3) If every sequence Cauchy inX converger, then the norm-Archimedean normed
space X is called a norm-Archimedean Bnanch space.
2.2. Solutions of the inequalities. The functional equation
f(x+ y) = f(x) + f(y)
is called the cauchuy equation. In particular, every solution of the cauchuy equation
is said to be an additive mapping.
3. Additive β-functional inequality in Non-Archimedean Banach
space
Now, we first study the solutions of (1.1) and (1.2). Note that for these in-
equalities, X is non-Archimedean normed space and Y is non-Archimedean Banach
spaces. Under this setting, we can show that the mapping satisfying (1.1) and (1.2)
is additive. These results are given in the following.
Lemma 3.1. A mapping f:X→Y satilies∥∥∥∥f(x+ y
2+z
)−f(x+ y
2
)−f(z)∥∥∥∥ ≤ ∥∥∥∥β(2f
(x+ y
22+z
2
)−f(x+ y
2
)−f(z))∥∥∥∥
(3.1)
for all x,y,z∈ X if and only if f:X→Y is additive
224 Ly Van An
Proof. Assume that f : X →Y satisfies (3.1)
Letting x = y = z = 0 in (3.1), we get∥∥f(0)
∥∥ ≤ 0. So f(0) = 0
Letting x = y = z in (3.1), we get∥∥f(2x) − 2f
(x)∥∥ ≤ 0 and so f
(2x)
= 2f(x)
for
all x ∈X.
Thus
f(x
2) =
1
2f(x) (3.2)
for all x ∈ X It follows from (3.1) and (3.2) that:∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥ ≤ ∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥
=
∥∥∥∥β(2f
( x+y2
+ z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥
≤∣∣β∣∣∥∥∥∥(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥
and so
f
(x+ y
2+ z
)= f
(x+ y
2
)+ f(z)
. for all x, y, z ∈ X The coverse is obviously true. �
Lemma 3.2. A mapping f : X → Y satisfy f(0) = 0 and∥∥∥∥2f
(x+ y
22+z
2
)−f(x+ y
2
)−f(z)∥∥∥∥ ≤ ∥∥∥∥β(f(x+ y
2+z
)−f(x+ y
2
)−f(z))∥∥∥∥(3.3)
for all x, y ,z∈ X if and if f : X → Y is additive.
Proof. Assume that f : X → Y (3.3).
Letting x = y = 0 in (3.3), we get∥∥∥∥2f
(z
2
)− f
(z)∥∥∥∥ ≤ 0 (3.4)
and so
f
(z
2
)− 1
2f(z)
It follows from (3.3) and (3.4) that∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥ =
∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z)∥∥∥∥
=
∥∥∥∥2f
( x+y2
+ z
2
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∣∣β∣∣∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥
Hyers-Ulam stability additive β-functional inequalities 225
and so.
f
(x+ y
2+ z
)= f
(x+ y
2
)+ f(z)
for all x, y, z ∈ X The converse is obvoiusly true.
�
Theorem 3.3. Let ϕ : X3 →[0,∞
)be a function and let f : X → Y be a mapping
such that
ψ(x, y, z) :=∞∑j=1
∣∣2∣∣jψ(x2,y
2,z
2
)<∞ (3.5)
∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ ϕ
(x, y, z
)(3.6)
for all x, y, z ∈X.Then there exists a unique mapping h: X → Y such that∥∥f(x)− h(x)∥∥ ≤ 1∣∣2∣∣ψ(x, x, x) (3.7)
for all x ∈ X
Proof. Letting x = y = z in (3.6), we get∥∥f(2x)− 2f(x)∥∥ ≤ ϕ
(x, x, x
)(3.8)
for all x ∈ X.
So ∥∥∥∥f(x)− 2f
(x
2
)∥∥∥∥ ≤ ϕ
(x
2,x
2,x
2
)for all x ∈ X. Hence∥∥∥∥2lf
( x2k
)− 2mf
( x2m
)∥∥≤ max
{∥∥2lf
(x
2l
)− 2l+1f
(x
2l+1
)∥∥∥∥, ...,∥∥∥∥2m−1f
(x
2m−1
)− 2mf
(x
2m
)∥∥∥∥}= max
{∣∣2∣∣l∥∥∥∥f( x2l
)− 2f
(x
2l+1
)∥∥∥∥, ..., |2m−1|∥∥∥∥f( x
2m−1
)− 2f
(x
2m
)∥∥∥∥}≤
∞∑j=l
∣∣2∣∣jϕ( x
2j+1,x
2j+1,x
2j+1
)(3.9)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from
(3.9) that the sequence{
2nf(
x2n
)}is a cauchy sequence for all x ∈ X. Since Y is a
226 Ly Van An
Non-Archimedean Banach space, the sequence{
2nf(
x2n
)}coverges.
So one can define the mapping h : X → Y by
h(x)
:= limn→∞
2nf
(x
2n
)for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (3.9), we get
(3.7).
Now, let T : X → Y be another additive mapping satistify (3.7) then we have
∥∥h(x)− T(x)∥∥ =
∥∥∥∥2qh
(x
2q
)− 2qT
(x
2q
)∥∥∥∥≤ max
{∥∥∥∥2qh
(x
2q
)− 2qf
(x
2q
)∥∥∥∥,∥∥∥∥2qT
(x
2q
)− 2qf
(x
2q
)∥∥∥∥}≤∣∣2∣∣q−1ψ( x
2q,x
2q,x
2q
)(3.10)
which tends to zero as q→∞ for all x ∈ X. So we can conclude that h(x)
= T(x)
for all x ∈ X. The proves the uniqueness of h. It follows from (3.5) and (3.6) that∥∥∥∥h(x+ y
2+ z
)− h(x+ y
2
)− h(z)∥∥∥∥ = lim
n→∞
∥∥2n(f
(x+ y
2n+1+
z
2n
)− f
(x+ y
2n+1
)− f
(z
2n
))∥∥∥∥≤ lim
n→∞
∥∥∥∥2nβ
(f
(x+ y
2n+2+
z
2n+1
)− f
(x+ y
2n+1
)− f
(z
2n
)∥∥∥∥+ lim
n→∞|2|nψ(
x
2n,y
2n,z
2n
))=
∥∥∥∥β(2h
(x+ y
2+ z
)− h(x+ y
2
)− h(z))∥∥∥∥ (3.11)
for all x,y,z ∈X
∥∥∥∥h(x+ y
2+ z
)− h(x+ y
2
)− h(z)∥∥∥∥ ≤ ∥∥∥∥β(2h
(x+ y
2+ z
)− h(x+ y
2
)− h(z))∥∥∥∥
for all x,y∈ X. By lemma 3.1, the mapping h : X → Y is additive. �
Theorem 3.4.
Let ϕ : X →[0,∞
)be a function and let f : X → Y be a mapping satisfying
ψ(x, y, z
):=
∞∑j=0
1∣∣2∣∣jϕ(2jx, 2jy, 2jz)<∞ (3.12)
Hyers-Ulam stability additive β-functional inequalities 227
and∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ ϕ
(x, y, z
)(3.13)
for all x,y,z∈ X. Then there exists a unique additive mapping h : X → Y such
that ∥∥f(x)− h(x)∥∥ ≤ 1∣∣2∣∣ψ(x, x, x) (3.14)
for all x ∈ X
Proof. . Letting x = y = z in (3.13), we get∥∥∥∥f(x)− 1
2f(2x)∥∥∥∥ ≤ 1∣∣2∣∣ψ(x, x, x)
for all x ∈ X.
Hence∥∥∥∥ 1
2lf(2lx)− 1
2mf(2mx
)∥∥∥∥≤ max
{∥∥∥∥ 1
2lf(2lx
)− 1
2l+1f(2l+1x
)∥∥∥∥, ...,∥∥∥∥ 1
2m−1f(2m−1x
)− 1
2mf(2mx
)∥∥∥∥}= max
{1∣∣2∣∣l∥∥∥∥f(2lx
)− 1
2f(2l+1x
)∥∥∥∥, ..., 1∣∣2∣∣m−1∥∥∥∥f(2m−1x
)− 1
2f(2mx
)∥∥∥∥}
≤∞∑j=1
1∣∣2∣∣j+1ϕ(2jx, 2jy, 2jz
)(3.15)
for all nonnegative integers m and l with m > l and all x ∈ X . It folows (3.15)
that the sequence
{12nf(2nx)}
is sequence for all x ∈ X. Since Y is complete, the
sequence
{12nf(2nx)}
converger so one can define the mapping h : X → Y by
h(x)
:= limn→∞
1
2nf(2nx)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.15), we
get (3.14). the rest of the proof is similar to the proof of the Theorem 3.3.
�
228 Ly Van An
Theorem 3.5.
Let ϕ : X3 →[0,∞
)be a function and let f : X → Y be a mapping satisfying
f(0) = 0 and
ψ(x, y, z
):=
∞∑j=1
∣∣2∣∣jψ(x2,y
2,z
2
)<∞ (3.16)
∥∥∥∥2f
(x+ y
22+z
2
)−f(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥ +ϕ
(x, y, z
)(3.17)
for all x, y, z ∈ X. Then there exists a unique mapping h : X → Y such that∥∥f(z)− h(z)∥∥ ≤ ψ(0, 0, z
)(3.18)
for all z ∈ X
Proof. Letting x = y = 0 in (3.17)∥∥∥∥2f
(z
2
)− f
(z)∥∥∥∥ ≤ ϕ
(0, 0, z
)(3.19)
for all x ∈ X .
So∥∥∥∥2lf
(z
2l
)− 2mf
(z
2m
)∥∥∥∥≤ max
{∥∥∥∥2lf
(z
2l
)− 2l+1f
(z
2l+1
)∥∥∥∥, ...,∥∥∥∥2m−1f
(z
2m−1
)− 2mf
(z
2m
)∥∥∥∥}= max
{∣∣2∣∣l∥∥∥∥f( z2l
)− 2f
(z
2l+1
)∥∥∥∥, ..., ∣∣2∣∣m−1∥∥∥∥f( z
2m−1
)− 2f
(z
2m
)∥∥∥∥}≤
∞∑j=l+1
∣∣2∣∣j+1ϕ(0, 0,
z
2j
)<∞ (3.20)
for all nonnegativers integers m and l with m > l and all x ∈ X. It follows from
(3.20) that the sequence{
2kf(
z2k
)}is Cauchy for all x ∈ X. Since Y is a non -
Archimedean Banach space, the sequence,{
2kf(
z2k
)}converges. So one can define
the mapping h : X → Y by
h(z)
:= limk→∞
2kf
(z
2k
)for all z ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.20), we
get (3.18).
Hyers-Ulam stability additive β-functional inequalities 229
The rest of the proof is similar to the proof of the theorem (3.3).
�
Theorem 3.6.
Let ϕ : X3 →[0,∞
)be a function and let f : X → Y be a mapping satisfying
f(0) = 0 and
ψ(x, y, z) :=∞∑j=1
1
|2|jψ(2jx, 2jy2jz) <∞; (3.21)
∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ ϕ
(x, y, z
)(3.22)
for all x, y, z ∈X. Then there exists a unique mapping h: X → Y such that
∥∥f(x)− h(x)∥∥ ≤ ψ(0, 0, z
)(3.23)
for all z ∈ X
Proof. . Letting x = y = 0 in (3.22)∥∥∥∥2f
(z
2
)− f
(z)∥∥∥∥ ≤ ϕ
(0, 0, z
)(3.24)
for all z ∈ X .
So ∥∥∥∥f(z)− 1
2f(2z)∥∥∥∥ ≤ 1∣∣2∣∣ϕ(0, 0, 2z)
for all z ∈ X. Hence
∥∥∥∥ 1
2lf(2lz)− 1
2mf(2mz
)∥∥∥∥≤ max
{∥∥∥∥ 1
2lf(2lz)− 1
2l+1f(2l+1z
)∥∥∥∥, ...,∥∥∥∥ 1
2m−1f(2m−1z
)− 1
2mf(2mz
)∥∥∥∥}= max
{1∣∣2∣∣l∥∥∥∥f(2lz
)− 1
2f(2l+1z
)∥∥∥∥, ..., 1∣∣2∣∣m−1∥∥∥∥f(2m−1z
)− 1
2f(2mz
)∥∥∥∥}
≤∞∑
j=l+1
1∣∣2∣∣jϕ(0, 0, 2jz) <∞ (3.25)
230 Ly Van An
for all nonnegative integers m and l with m > l and all z ∈ X. It follows from (3.25)
that the sequence{
2nf(
z2n
)}is a cauchy sequence for all z ∈ X. Since is complete,
the sequence{
2nf(
z2n
)}coverges. So one can define the mapping h : X → Y by
h(z)
:= limn→∞
1
2nf(2nz)
for all z ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (3.25), we
get (3.24).
The rest of the proof is similar to the proof of theorem 3.3.
�
Corollary 3.7.
let r < 1 and θ be nonnegative real numbers, and let f:X → Y be a mapping such
that∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ θ(‖x‖r + ‖y‖r + ‖z‖r)
for all x, y,z∈ X.
Then there exits a unique additive maaping h : X → Y such that∥∥∥∥f(x)− h(x)∥∥∥∥ ≤ 2θ∣∣2∣∣r − ∣∣2∣∣∥∥x∥∥rfor all x ∈ X
Corollary 3.8.
let r > 1 and θ be nonnegative real numbers, and let f:X → Y be a mapping such
that∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ θ(‖x‖r + ‖y‖r + ‖z‖r)
for all x, y,z∈ X.
Then there exits a unique additive maaping h : X → Y such that∥∥∥∥f(x)− h(x)∥∥∥∥ ≤ 2θ∣∣2∣∣− ∣∣2∣∣r∥∥x∥∥rfor all x ∈ X.
Hyers-Ulam stability additive β-functional inequalities 231
Corollary 3.9.
Let r < 1 and θ be a nonnegativers real number and let f : X → Y be a mapping
satisfying f(0)
= 0 and∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ θ
(∥∥x∥∥r +∥∥y∥∥r +
∥∥z∥∥r)for all x, y ,z∈ X. The there exists a unique additive mapping h : X → Y such that∥∥f(x)− h(x)‖ ≤ ∣∣2∣∣rθ∣∣2∣∣r − ∣∣2∣∣∥∥x∥∥rfor all x ∈ X.
Corollary 3.10.
Let r > 1 and θ be a nonnegativers real number and let f : X → Y be a mapping
satisfying f(0)
= 0 and∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ θ
(∥∥x∥∥r +∥∥y∥∥r +
∥∥z∥∥r)for all x, y ,z∈ X. The there exists a unique additive mapping h : X → Y such that∥∥f(x)− h(x)‖ ≤ ∣∣2∣∣rθ∣∣2∣∣− ∣∣2∣∣r∥∥x∥∥rfor all x ∈ X.
4. Additive β-functional inequality in complex Banach space
Now, we study the solutions of (1.1) and (1.2). Note that for these inequalities,
X is complex normed space and Y is complex Banach spaces. Under this setting,
we can show that the mapping satisfying (1.1) and (1.2) is additive. These results
are give in the following.
Lemma 4.1.
Amapping f : X → Y satisfies∥∥∥∥f(x+ y
2+z
)−f(x+ y
2
)−f(z)∥∥∥∥ ≤ ∥∥∥∥β(2f
(x+ y
22+z
2
)−f(x+ y
2
)−f(z))∥∥∥∥(4.1)
for all x, y, z ∈ X if and only if f : X → Y is additive
Proof. The proof is simlar to the proof of lemma 3.1
232 Ly Van An
Lemma 4.2.
Amapping f : X → Y satisfies f(0) = 0 and∥∥∥∥2f
(x+ y
22+z
2
)−f(x+ y
2
)−f(z)∥∥∥∥ ≤ ∥∥∥∥β(f(x+ y
2+z
)−f(x+ y
2
)−f(z))∥∥∥∥(4.2)
for all x,y,z∈ X if and only if f:X→Y is additive
Proof. The proof is simlar to the proof of lemma 3.2
Theorem 4.3.
Let ϕ : X3 →[0,∞
)be a function and let f : X → Y be a mapping such that
ψ(x, y, z
):=
∞∑j=1
2jϕ
(x
2j,y
2j,z
2j
)<∞ (4.3)
∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ ϕ
(x, y, z
)(4.4)
for all x, y, z ∈X. Then there exists a unique mapping h : X → Y such that∥∥f(x)− h(x)∥∥ ≤ 1
2ψ(x, x, x
)(4.5)
for all x ∈ X
Proof. Let x = y = z in (4.4), we get∥∥f(2x)− 2f(x)∥∥ ≤ ϕ
(x, x, x
)(4.6)
for all x ∈ X. So ∥∥∥∥f(x)− 2f
(x
2
)∥∥∥∥ ≤ ϕ
(x
2,y
2,z
2
)for all x ∈ X. Hence∥∥∥∥2lf
(x
2l
)− 2mf
(x
2m
)∥∥∥∥≤
m−1∑j=1
∥∥∥∥2jf
(x
2j
)− 2j+1f
(x
2j+1
)∥∥∥∥ ≤ m−1∑j=1
‖2jϕ
(x
2j+1,x
2j+1,x
2j+1
)∥∥∥∥(4.7)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from
(4.7) that the sequence
{2nf
(x2n
)}is a cauchy sequence for all x ∈ X. Since
Hyers-Ulam stability additive β-functional inequalities 233
is complete, the sequence
{2nf
(x2n
)}coverges. So one can define the mapping
h : X → Y by
h(x)
:= limn→∞
1
2nf(2nx)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m→∞ in (4.7), we get
(4.5).
Now, let T : X → Y be another additive mapping satisfying (4.5). Then we have∥∥∥∥h(x)− T(x)∥∥∥∥ =
∥∥∥∥2qh
(x
2q
)− 2qT
(x
2q
)∥∥∥∥≤∥∥∥∥2qh
(x
2q
)− 2qf
(x
2q
)∥∥∥∥+
∥∥∥∥2qT
(x
2q
)− 2qf
(x
2q
)∥∥∥∥≤ 2qψ
(x
2q,x
2q,x
2q
)which tends to zero as q →∞ for all x ∈ X.So we can conclude that h
(x)
= T(x)
for all x ∈ X. This proves the uniqueness of h. It follows from (4.3) and (4.4) that∥∥∥∥h(x+ y
2+ z
)− h(x+ y
2
)− h(z)∥∥∥∥
= limn→∞
∥∥∥∥2n
(f
(x+ y
2n+2+
z
2n+1
)− f
(x+ y
2n+1
)− f
(z
2n
))∥∥∥∥≤ lim
n→∞
∥∥∥∥2nβ
(2f
(x+ y
2n+3+
z
2n+2
)− f
(x+ y
2n+1
)− f
(z
2n
))∥∥∥∥+ lim
n→∞2nϕ
(x
2n,y
2n,z
2n
)=
∥∥∥∥β(2h
(x+ y
22+z
2
)− h(x+ y
2
)− h(z))∥∥∥∥ (4.8)
for all x, y, z ∈ X.
So∥∥∥∥h(x+ y
2+ z
)−h(x+ y
2
)−h(z)∥∥∥∥ ≤ ∥∥∥∥β(2h
(x+ y
22+z
2
)−h(x+ y
2
)−h(z))∥∥∥∥
for all x, y, z ∈ X. By Lemma 4.1, the mapping h : X → Y is additive. �
Theorem 4.4.
Let ϕ : X3 →[0,∞
)be a function and let f : X → Y be a mapping such that
ψ(x, y, z
):=
∞∑j=1
1
2jϕ(2jx, 2jy, 2jz) <∞ (4.9)
234 Ly Van An
and∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ ϕ
(x, y, z
)(4.10)
for all x, y, z ∈X. Then there exists a unique mapping h : X → Y such that∥∥f(x)− h(x)∥∥ ≤ 1
2ψ(x, x, x
)(4.11)
for all x ∈ X
Proof. . It follows from (4.10) that∥∥∥∥f(x)− 1
2f(2x)∥∥∥∥ ≤ 1
2ϕ(x, x, x
)for all x ∈ X. Hence
∥∥∥∥ 1
2lf(2lx)− 1
2mf(2mx
)∥∥∥∥ ≤ m−1∑j=1
∥∥∥∥ 1
2jf(2jx
)− 1
2j+1f(2j+1x
)∥∥∥∥≤
m−1∑j=1
1
2jϕ(2jx, 2jx, 2jx
)∥∥ (4.12)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (4.12)
that the sequence { 12nf(2nx)} is a cauchy sequence for all x ∈ X. Since is complete,
the sequence { 12nf(2nx)} coverges. So one can define the mapping h : X → Y by
h(x)
:= limn→∞
1
2nf(2nx)
for all x ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (4.12), we
get (4.11).
The rest of the proof is similar to the proof of theorem 4.3.
�
Theorem 4.5.
Let ϕ : X3 →[0,∞
)be a function and let f : X → Y be a mapping such that
ψ(x, y, z
):=
∞∑j=1
2jϕ( x
2j,y
2j,z
2j
)<∞ (4.13)
Hyers-Ulam stability additive β-functional inequalities 235∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ ϕ
(x, y, z
)(4.14)
for all x, y, z ∈X. Then there exists a unique mapping h: X → Y such that∥∥f(x)− h(x)∥∥ ≤ 1
2ψ(x, x, x
)(4.15)
for all x ∈ X
Proof. . Letting y = x = 0 in (4.14), we get∥∥∥∥f(z)− 2f(z
2
)∥∥∥∥ =
∥∥∥∥2f(z
2
)− f
(z)∥∥∥∥ ≤ ϕ
(0, 0, z
)(4.16)
for all x ∈ X. So
∥∥∥∥2lf( z
2l
)− 2mf
( z2m
)∥∥≤
m−1∑j=1
∥∥∥∥2jf
(z
2j
)− 2j+1f
(z
2j+1
)∥∥∥∥ ≤ m−1∑j=1
2jϕ(0, 0,
z
2j
)(4.17)
for all nonnegative integers m and l with m > l and all x ∈ X. It follows from (4.17)
that the sequence{
2nf(
z2n
)}is a cauchy sequence for all z ∈ X. Since is complete,
the sequence{
2nf(
z2n
)}coverges. So one can define the mapping h : X → Y by
h(z)
:= limn→∞
2nf( z
2n
)(4.18)
for all z ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (4.17), we
get (4.15). The rest of the proof is similar to the proof of theorem 4.3. �
Theorem 4.6.
Let ϕ : X3 →[0,∞
)be a function and let f : X → Y be a mapping satisfying
f(0) = 0, and
ψ(x, y, z) :=∞∑j=1
1
2jϕ(2jx, 2jy, 2jz
)<∞ (4.19)
∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z)∥∥∥∥
≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ ϕ
(x, y, z
)(4.20)
236 Ly Van An
for all x, y, z ∈X. Then there exists a unique mappingh : X → Y such that∥∥f(z)− h(z)∥∥ ≤ 1
2ψ(0, 0, z
)(4.21)
for all z ∈ X
Proof. Letting y = x = 0 in (4.20), we get∥∥∥∥f(z)− 2f(z
2
)∥∥∥∥ =
∥∥∥∥2f(z
2
)− f
(z)∥∥∥∥ ≤ ϕ
(0, 0, z
)(4.22)
. It follows from (4.20) that∥∥∥∥f(z)− 1
2f(2z)∥∥∥∥ ≤ 1
2ϕ(0, 0, 2z
)for all z ∈ X. Hence∥∥∥∥ 1
2lf(2lz)− 1
2mf(2mz
)∥∥∥∥≤
m−1∑j=1
∥∥∥∥ 1
2jf(2jz)− 1
2j+1f(2j+1z
)∥∥∥∥ ≤ m−1∑j=1
1
2jϕ(0, 0, 2jx
)(4.23)
for all nonnegative integers m and l with m > l and all z ∈ X. It follows from
(4.23) that the sequence{
12nf(2nz)}
is a cauchy sequence for all z ∈ X. Since
is complete, the sequence{
12nf(2nz)}
coverges. So one can define the mapping
h : X → Y by
h(z)
:= limn→∞
1
2nf(2nz)
(4.24)
for all z ∈ X. Moreover, letting l = 0 and passing the limit m → ∞ in (4.23), we
get (4.22). The rest of the proof is similar to the proof of theorem 4.3.
�
Corollary 4.7.
Let r > 1 and θ be positive real numbers, and let f : X → Y be a mapping such
that∥∥∥∥f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ θ(‖x‖r + ‖y‖r + ‖z‖r)
for all x, y, z ∈ X. Then there exists a unique additive mapping h : X → Y such that
‖f(x)− h(x)‖ ≤ 2θ
2r − 2‖x‖r
Hyers-Ulam stability additive β-functional inequalities 237
for all x ∈ X
Corollary 4.8.
Let r < 1 and θ be positive real numbers, and let f : X → Y be a mapping such
that∥∥∥∥(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥
≤∥∥∥∥β(2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ θ
(∥∥x∥∥r +∥∥y∥∥r +
∥∥z∥∥r)for all x, y, z ∈ X. Then there exists a unique additive mapping h : X → Y such
that ∥∥f(x)− h(x)∥∥ ≤ 2θ
2r − 2
∥∥x∥∥rfor all x ∈ X
Corollary 4.9.
Let ϕ : X3 →[0,∞) be a function and let f : X → Y be a mapping satisfying
f(0) = 0,
ψ(x, y, z
):=
∞∑j=1
2jϕ( x
2j,y
2j,z
2j
)<∞
∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z)− β
(f
(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥
≤ ϕ(x, y, z
)for all x, y, z ∈X. Then there exists a unique mapping h: X → Y such that
∥∥f(z)− h(z)∥∥ ≤ 1
2ψ(0, 0, z
)for all z ∈ X
Corollary 4.10.
Let r > 1 and θ be positive real numbers, and let f : X → Y be a mapping such
that
238 Ly Van An
∥∥∥∥2f
(x+ y
22+z
2
)− f
(x+ y
2
)− f
(z))∥∥∥∥
≤∥∥∥∥β(f(x+ y
2+ z
)− f
(x+ y
2
)− f
(z))∥∥∥∥+ θ
(∥∥x∥∥r +∥∥y∥∥r +
∥∥z∥∥r)for all x, y, z ∈ X. Then there exists a unique additive mapping h : X → Y such that
∥∥f(x)− h(x)∥∥ ≤ 2θ
2r − 2
∥∥x∥∥rfor all x ∈ XRemark: If β is a real number such that −1 < β < 1 and is Y is a real Banach
space, then all the assertions in this sections remain valid.
5. Conclusion
In this paper, it is shown that the solutions of the first and second β-functional
inequalities are additive mappings and the Hyers-Ulam stability is proved. These
are the main results of the paper, which are a generalization of the results in [17,
20].
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Received: December 11, 2019; Published: June 12, 2020