Research Article
Choonkil Park, Kandhasamy Tamilvanan, Ganapathy Balasubramanian, Batool Noori, and Abbas Najati*
On a functional equation that has the quadratic - multiplicative property
https://doi.org/10.1515/math-2020-0032 received May 13, 2020; accepted June 22, 2020
Abstract:In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation:
( − ) + ( + ) = [ ( ) + ( )][ ( ) + ( )]
ϕ st uv ϕ sv tu ϕ s ϕ u ϕ t ϕ v by using the direct method and thefixed point method.
Keywords:fixed point method, Hyers-Ulam stability, quadratic functional equation, direct method MSC 2020:39B52, 47H10, 39B72, 39B82
1 Introduction
The problem of the stability of functional equations has been posed by Ulam[1]. Hyers[2]gave a positive answer for the stability of the Cauchy equation:
( + ) = ( ) + ( ) f x y f x f y .
From that time many other functional equations and inequalities have been studied(see[3–9]).
Let E1 andE2 be real vector spaces. A mapping f:E1→E2 is called a quadratic mapping if f is a solution of the quadratic functional equation:
( + ) + ( − ) = ( ) + ( )
f x y f x y 2f x 2f y . (1.1)
It is well known that for each quadratic mapping f:E1→E2, there exists a unique symmetric biadditive mappingB:E1×E1→E2 satisfyingf x( ) = (B x x, )for allx∈E1.
The Hyers-Ulam stability of the quadratic functional equation was first proved by Skof [10] for mappings f:E1→E2, whereE1is a normed space andE2 is a Banach space. Cholewa[11]demonstrated that Skof’s theorem is also valid ifE1is replaced by an abelian groupG.
Choonkil Park:Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Korea, e-mail: [email protected] Kandhasamy Tamilvanan, Ganapathy Balasubramanian:Department of Mathematics, Government Arts College for Men, Krishnagiri, Tamil Nadu, 635001, India, e-mail: [email protected], [email protected]
Batool Noori:Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran, e-mail: [email protected]
* Corresponding author: Abbas Najati,Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran, e-mail: [email protected]
Open Access. © 2020 Choonkil Parket al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License.
Theorem 1.1.[10]Let G be an abelian group and E be a real Banach space. If a mappingf: G→Esatisfies the inequality
∥ ( + ) + ( − ) −f x y f x y 2f x( ) −2f y( )∥ ⩽δ
for someδ> 0and allx y, ∈G, then there exists a unique quadratic mappingq:G→E such that
∥ ( ) − ( )∥ ⩽f x q x 1δ for all x∈G
2 .
Thereafter, Czerwik[12]proved the Hyers-Ulam stability of the quadratic functional equation with nonconstant bound.
Theorem 1.2.[12]LetE1andE2be a real normed space and a real Banach space, respectively, and letp≠2 be a positive constant. If a mapping f:E1→E2 satisfies the inequality
∥ ( + ) + ( − ) −f x y f x y 2f x( ) −2f y( )∥ ⩽ (∥ ∥ + ∥ ∥ )ε x p y p
for someε>0and all x y, ∈E1, then there exists a unique quadratic mappingq: E1→E2 such that
∥ ( ) − ( )∥ ⩽
| − |∥ ∥ ∈
f x q x ε
x for all x E 2
4 2p p 1.
In 1991, Hammer and Volkmann[13]investigated mappings f : → which satisfy(1.1)and
( ) = ( ) ( ) ( ∈ )
f xy f x f y x y, . (1.2)
It will turn out thatf x( ) =0andf x( ) =x2 are the only ones, which are continuous solutions of system of two functional equations(1.1)and(1.2).
In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation:
( − ) + ( + ) = [ ( ) + ( )][ ( ) + ( )]
ϕ st uv ϕ sv tu ϕ s ϕ u ϕ t ϕ v (1.3)
by using the direct method and thefixed point method.
Throughout this article,stands for the set of all real numbers andstands for the set of all complex numbers.
2 General solution of the functional equation ( 1.3 )
In this section, we obtain the general solution of the functional equation(1.3). We need the following results.
Theorem 2.1.[13]A function f :→ satisfies the system of the following functional equations:
( + ) + ( − ) = ( ) + ( ) ( ) = ( ) ( ) ∈ f x y f x y 2f x 2f y, f xy f x f y , x y, ,
if and only if there exists an additive-multiplicative functionw : →such thatf x( ) = | ( )|w x 2for allx∈. Theorem 2.2.[14]A functionf :→is additive-multiplicative, i.e., f satisfies the system of the following functional equations:
( + ) = ( ) + ( ) ( ) = ( ) ( ) ∈ f x y f x f y, f xy f x f y , x y, , if and only if f≡0or f x( ) =xfor allx∈.
Theorem 2.3.If a functionϕ :→ satisfies(1.3)for alls t u v, , , ∈, thenϕ≡0, 2 ϕ≡1orϕ t( ) =t2 for allt∈.
Proof.Letϕ be a constant function. Then,ϕ≡0 or2ϕ≡ 1. We assume that ϕ is not constant. Letting
= = =
t u v 0in(1.3), we obtain
( ) = [ ( ) + ( )] ( ) ∈
ϕ ϕ s ϕ ϕ s
2 0 2 0 0 , .
This implies thatϕ 0( ) =0, sinceϕis not constant. Lettingu=v=0in (1.3), we get
( ) = ( ) ( ) ∈
ϕ st ϕ s ϕ t s t, . (2.1)
Puttingt=1in(2.1), we getϕ 1( ) =1, sinceϕis not constant. Lettingt=v=1in(1.3)and usingϕ 1( ) =1, we obtain
( − ) + ( + ) = ( ) + ( ) ∈ ϕ s u ϕ s u 2ϕ s 2ϕ u s u, .
By Theorem 2.1, there exists an additive-multiplicative functionw : →such thatϕ t( ) = | ( )|w t 2for all
t∈ . Sinceϕ 1( ) =1, lettingu=v=1in(1.3), we get
[ ( ) ( )] =w s w t [ ( ) ( )]w s w t , s t, ∈ .
R R
Then,w s( ) ∈ orw t( ) ∈. This implies w( ) ⊆ . Therefore, w :→ is an additive-multiplicative function, and by Theorem 2.2 we concludew t( ) =tfor allt∈. Hence,ϕ t( ) =t2for allt∈. □ Corollary 2.4. If a nonzero mapping ϕ : → with ϕ 0( ) =0 satisfies (1.3) for all s t u v, , , ∈, then
( ) =
ϕ t t2for allt∈.
Proposition 2.5.Letϕ :→ be a derivation and satisfy(1.3)for alls t u v, , , ∈. Then,ϕ≡0.
Proof.Sinceϕis a derivation, we getϕ( ) =0 ϕ( ) =1 0. Lettingu=v=0in(1.3), we getϕ st( ) =ϕ s ϕ t( ) ( )for alls t, ∈. Hence,ϕ s( ) =ϕ s ϕ 1( ) ( ) =0for alls∈. □
3 Hyers - Ulam stability of the functional equation ( 1.3 ) : direct method
In this section, we prove the Hyers-Ulam stability of the functional equation(1.3)by using the direct method.
For notational convenience, we use the following abbreviation, for a given mappingϕ :→, ( ) = ( − ) + ( + ) − [ ( ) + ( )][ ( ) + ( )]
D s t u vϕ , , , ϕ st uv ϕ sv tu ϕ s ϕ u ϕ t ϕ v . Theorem 3.1.Letε∈ {−1, 1}andϑ :4→ [0,∞)be a function such that
∑
( ) < ∞=
∞ ϑ 2 s, 2 t, 2 u, 2 v
n 4
nε nε nε nε
nε 0
for alls t u v, , , ∈. Suppose thatϕ :→ is a function satisfyingϕ 0( ) =0and
|D s t u vϕ( , , , )| ⩽ (ϑ , , ,s t u v), s t u v, , , ∈ . (3.1) Then, there exists a unique quadratic functionQ :→ which satisfies(1.3)and
∑ ∑
∑ ∑
| ( ) − ( )| ⩽
( )
+ ( )
=
+ = −
=
∞
=
∞
=
∞
=
ϕ s Q s ∞
s s s
ε
s s s
ε 1
4
ϑ 2 , 1, 2 , 1 4
ϑ 2 , 1, 0, 0
4 , 1,
1
4 4 ϑ
2 , 1,
2 , 1 4 ϑ
2 , 1, 0, 0 , 1
n
n n
n
n n
n
n n
n n
n n
n
0 0
1 1
(3.2)
for alls∈, whereQ s( ) =s2 ifQ 1( ) =1andQ≡0ifQ 1( ) ≠1.
Proof.We assume thatε=1. Lettingu=v= 0in(3.1), we get
| ( ) −ϕ st ϕ s ϕ t( ) ( )| ⩽ (ϑ , , 0, 0 ,s t ) s t, ∈ . (3.3) Puttingt=1in(3.3), we get
| ( ) −ϕ s ϕ s ϕ( ) ( )| ⩽ (1 ϑ , 1, 0, 0 ,s ) s∈ . (3.4) Lettingt=v=1in(3.1), we get
| ( + ) +ϕ s u ϕ s( − ) −u 2ϕ( )[ ( ) +1 ϕ s ϕ u( )]| ⩽ (ϑ , 1, , 1 ,s u ) s u, ∈ . (3.5) Then, it follows from(3.4)and(3.5)that
| ( + ) +ϕ s u ϕ s( − ) − [ ( ) +u 2ϕ s ϕ u( )]| ⩽ψ s u( , ), s u, ∈ ,
whereψ s u( , ) = (: ϑ , 1, , 1s u ) +2ϑ , 1, 0, 0(s ) +2ϑ , 1, 0, 0(u ). By[15], there exists a unique quadratic function
→
Q : such that
∑
( ) = ( )
∥ ( ) − ( )∥ ⩽ ( )
∈
→∞ =
∞
Q s ϕ s
ϕ s Q s ψ s s
s lim 2
4 , 1
4
2 , 2
4 , .
n
n n
n
n n
n 0
Then from(3.1)and(3.3), we obtain
( − ) + ( + ) = [ ( ) + ( )][ ( ) + ( )] ∈ Q st uv Q sv tu Q s Q u Q t Q v , s t u v, , , .
SinceQ 0( ) =0, we inferQ 1( ) ∈ {0, 1}. Hence by Theorem 2.3, we getQ s( ) =s2 for alls∈ ifQ 1( ) =1and
≡
Q 0ifQ 1( ) =0. Then, we obtain(3.2).
Similarly, we can obtain the result forε= −1. Hence, the proof of the theorem is now complete. □ Corollary 3.2.Suppose thatϕ :→ is a function satisfying
|D s t u vϕ( , , , )| ⩽ε, s t u v, , , ∈ (3.6) for someε⩾0. Then, we have the following assertions:
(i) Ifϕ 0( ) ≠0, thenϕis bounded.
(ii) Ifϕ 0( ) =0, then there exists a unique quadratic functionQ : →, which satisfies(1.3)and
| ( ) −ϕ s Q s( )| ⩽ 5ε s∈
3 , ,
whereQ s( ) =s2 ifQ 1( ) =1andQ≡0ifQ 1( ) ≠1.
Proof.Letϕ 0( ) ≠0. Lettingt=u= v=0in(3.6), we obtain
|2ϕ( ) − [ ( ) +0 2 ϕ s ϕ( )] ( )| ⩽0 ϕ0 ε, s∈ . Then, we get
| ( )| ⩽ | ( ) − | +
| ( )| ∈
ϕ s ϕ ε
ϕ s
0 1
2 0 , .
For the caseϕ 0( ) =0, the result follows from Theorem 3.1. □
Corollary 3.3.Letε⩾0andϑ :4→ [0,∞)be a function. Suppose thatϕ : →is a function satisfying
|D s t u vϕ( , , , )| ⩽ε s t u vϑ , , ,( ), s t u v, , , ∈ .
Let{e e e e1, 2, 3, 4}be the standard basis in4 andϑ :i → be a function defined byϑi( ) = (s ϑsei)for a giveni∈ {1, 2, 3, 4}. Ifϕ 0( ) ≠0andϑiis bounded for some i, thenϕis bounded.
Corollary 3.4.Fixε⩾0and p∈ (0, 2). Suppose thatϕ :→ is a function satisfying
|D s t u vϕ( , , , )| ⩽ (| | + | | + | | + | | )ε sp tp up vp , s t u v, , , ∈ .
Then, we have the following assertions:
(i) Ifϕ 0( ) ≠0, then
( ) − ⩽ | | ∈
ϕ s 1 ε s s
2 p, .
(ii) Ifϕ 0( ) =0, then there exists a unique quadratic functionQ :→ which satisfies(1.3)and
| ( ) − ( )| ⩽
− | | + ∈
ϕ s Q s ε
s ε s
6
4 2p p 2 , ,
whereQ s( ) =s2ifQ 1( ) =1andQ≡ 0ifQ 1( ) ≠1.
Theorem 3.5.Fixε⩾0and p>4. Suppose thatϕ : → is a function satisfying
|D s t u vϕ( , , , )| ⩽ (| | + | | + | | + | | )ε sp tp up vp , s t u v, , , ∈ . (3.7) Then, we have the following assertions:
(i) Ifϕ 0( ) ≠0, then
( ) − ⩽ | | ∈
ϕ s 1 ε s s
2 p, .
(ii) Ifϕ 0( ) =0, then there exists a unique quadratic functionQ :→, which satisfies(1.3)and
| ( ) − ( )| ⩽
− ( + | | ) ∈
ϕ s Q s ε
s s
6
2p 4 1 p , , (3.8)
whereQ s( ) =s2ifQ 1( ) =1andQ≡ 0ifQ 1( ) ≠1.
Proof.It suffices to prove(ii). Lettingu=v=0in(3.7), we get
| ( ) −ϕ st ϕ s ϕ t( ) ( )| ⩽ (| | + | | )ε sp tp , s t, ∈ . (3.9) Puttingv=tandu=sin(3.7), we obtain
| (ϕ st2 ) −4ϕ s ϕ t( ) ( )| ⩽2ε s(| | + | | )p tp , s u v, , ∈ . (3.10) Using(3.9)and(3.10), we obtain
| (ϕ st2 ) −4ϕ st( )| ⩽6ε s(| | + | | )p tp , s u v, , ∈ . Replacings andtby ( + )/2
2n 1 2 and ( + )/1
2n 1 2, respectively, in the last inequality and multiplying the resulting inequality by4n, we obtain
− + + ⩽ / ( + | | ) ∈
+
ϕ s
ϕ s
ε s s
4 2 4
2
3 2
4
2 1 , .
n n
n
n p
n
p 1
1 2
1
(3.11)
Then,
∑ ∑
− = − ⩽ ( + | | ) ∈
> ⩾
=
−
+
+ = + /
ϕ s
ϕ s
ϕ s
ϕ s
ε s s m
n
4 2 4
2 4
2 4
2
3
2 1 4
2 , ,
0.
n n
m
m i n
m i
i i
i
p i n
m p 1 i
1 1
1 2
(3.12)
This implies that the sequence{4nϕ( )}s
2n is a Cauchy sequence and so the sequence is convergent for all
∈
s . We define a functionQ :→ by ( ) =
→∞
Q s ϕ s
lim 4 2 .
n n
n
Sinceϕ 0( ) =0, we getQ 0( ) =0. By Assumption(3.7)and Inequality(3.9), we obtain thatQsatisfies(1.3) andQ st( ) =Q s Q t( ) ( )for alls t, ∈. Hence by Theorem 2.3, we getQ s( ) =s2 ( ∈s )ifQ 1( ) =1andQ≡ 0if
( ) =
Q 1 0. Lettingn=0 andm→ ∞in(3.12), we obtain(3.8). □
4 Hyers - Ulam stability of the functional equation ( 1.3 ) : fi xed point method
In this section, we will adopt the idea of[16]to prove the Hyers-Ulam stability of the functional equation (1.3)by using thefixed point method.
Letbe a set. A functiond : × → [0,+∞]is called a generalized metric onifdsatisfies 1. d x y( , ) =0if and only if x=y;
2. d x y( , ) = (d y x, )for allx y, ∈;
3. d x z( , ) ⩽ (d x y, ) + (d y z, )for allx y z, , ∈.
It should be noted that the only difference between the generalized metric and the metric is that the generalized metric accepts the infinity.
We will use the following fundamental result in thefixed point theory.
Theorem 4.1.[17]Let(,d)be a generalized complete metric space andΛ :→be a strictly contractive function with the Lipschitz constantL<1. Suppose that for a given elementa∈there exists a nonnegative integer k such thatd Λ( k+1a Λ a, k ) < ∞. Then,
(i) the sequence{Λ an }∞n 1= converges to afixed pointb∈ofΛ;
(ii) b is the uniquefixed point ofΛin the set= { ∈y : (d Λ a yk , ) < ∞};
(iii) ( ) ⩽ ( )
d y b, 1−1Ld y Λy, for ally∈.
Theorem 4.2.Let0<L<1andϑ :4→ [0,∞)be a function such that
( )
= ( ) ⩽ ( ) ∈
→∞
s t u v
ψ s t Lψ s t s t u v lim ϑ 2 , 2 , 2 , 2
4 0, 2 , 2 4 , , , , , ,
n
n n n n
n (4.1)
whereψ s t( , ) ≔ (ϑ , 1, , 1s t ) +2ϑ , 1, 0, 0(s ) +2ϑ , 1, 0, 0(t ). Suppose thatϕ : → is a function satisfying ( ) =
ϕ 0 0and
|D s t u vϕ( , , , )| ⩽ (ϑ , , ,s t u v), s t u v, , , ∈ . (4.2) Then, there exists a unique quadratic functionQ :→, which satisfies(1.3)and
| ( ) − ( )| ⩽
( − ) ( ) ∈
ϕ s Q s
L ψ s s s 1
4 1 , , , (4.3)
whereQ s( ) =s2 ifQ 1( ) =1andQ≡0ifQ 1( ) ≠1.
Proof.It is clear that =
→∞
( )
lim 0
n
ψ2 , 2s t 4 n n
n for alls t, ∈. Using a similar argument of the proof of Theorem 3.1, we obtain
| ( + ) +ϕ s u ϕ s( − ) − [ ( ) +u 2ϕ s ϕ u( )]| ⩽ψ s u( , ), s u, ∈ ,
whereψ s u( , ) ≔ (ϑ , 1, , 1s u ) +2ϑ , 1, 0, 0(s ) +2ϑ , 1, 0, 0(u ). Hence by[18, Theorem 4], there exists a unique quadratic functionQ :→ such that
( ) = ( )
| ( ) − ( )| ⩽
( − ) ( ) ∈
→∞
Q s ϕ s
ϕ s Q s
L ψ s s s
lim 2
4 , 1
4 1 , , .
n
n
n (4.4)
From the definition ofQand using(4.1)and(4.2), we obtain thatQsatisfies(1.3). SinceQ 0( ) =0, it follows from Theorem 2.3 thatQ s( ) =s2 ( ∈s )ifQ 1( ) =1andQ≡0ifQ 1( ) =0. Hence, we get(4.3), which ends
the proof. □
Obviously, Corollary 3.2 can be obtained by using Theorem 4.2. By a similar argument as in the proof of Theorem 4.2, we can obtain the following result.
Theorem 4.3.Let0<L<1andϑ :4→ [0,∞)be a function such that
= ( ) ⩽ ( ) ∈
→∞
s t u v
ψ s t L
ψ s t s t u v lim 4 ϑ
2 , 2 ,
2 ,
2 0, ,
4 2 , 2 , , , , ,
n n
n n n n
whereψ s t( , ) ≔ (ϑ , 1, , 1s t ) +2ϑ , 1, 0, 0(s ) +2ϑ , 1, 0, 0(t ). Suppose thatϕ :→ is a function satisfying ( ) =
ϕ 0 0 and
|D s t u vϕ( , , , )| ⩽ (ϑ , , ,s t u v), s t u v, , , ∈ . Then, there exists a unique quadratic functionQ :→, which satisfies(1.3)and
| ( ) − ( )| ⩽
( − ) ( ) ∈
ϕ s Q s L
L ψ s s s
4 1 , , ,
whereQ s( ) =s2 ifQ 1( ) =1andQ≡0ifQ 1( ) ≠1.
Theorem 4.4.Letd>0andε ⩾0. Assume that a functionϕ : →withϕ 0( ) =0satisfies the inequality
|D s t u vϕ( , , , )| ⩽ε (4.5) for alls t u v, , , ∈withmax{| | | | | | | |} ⩾s, t, u, v d. Then, there exists a unique quadratic functionQ :→, which satisfies(1.3)and
| ( ) −ϕ s Q s( )| ⩽ε, s∈ , whereQ s( ) =s2 ifQ 1( ) =1andQ≡0ifQ 1( ) ≠1.
Proof.Lettingu=v=0in(4.5), we get
| ( ) −ϕ st ϕ s ϕ t( ) ( )| ⩽ε, s t, ∈ , | | ⩾t d. (4.6) Settingv=tin(4.5), we obtain
| (ϕ st− tu) +ϕ st( +tu) −2ϕ t ϕ s( )[ ( ) +ϕ u( )]| ⩽ε, s t u, , ∈ , | | ⩾t d. (4.7) It follows from(4.6)and(4.7)that
| (ϕ st−tu) +ϕ st( +tu) −2ϕ st( ) −2ϕ tu( )| ⩽3 ,ε s t u, , ∈ , | | ⩾t d. (4.8) Lettingt=dand replacingsanduby s
d and u
d in (4.8), respectively, we get
| ( − ) +ϕ s u ϕ s( + ) −u 2ϕ s( ) −2ϕ u( )| ⩽3 ,ε s u, ∈ . Hence by[12, Theorem 1], there is a unique quadratic functionQ : → such that
( ) = ( )
| ( ) − ( )| ⩽ ∈
→∞
Q s ϕ s
ϕ s Q s ε s
lim 2
4 , , .
n
n
n (4.9)
Applying(4.5)and(4.9), we obtain
| ( )| = | ( )| ⩽ = ∈
→∞ →∞
D s t u v, , , lim 1 D s t u v ε s t u v
16 2 , 2 , 2 , 2 lim 1
16 0, , , , .
Q n n ϕ n n n n
n n
SinceQ 0( ) =0, by Theorem 2.3, we getQ s( ) =s2 ifQ 1( ) =1andQ≡0ifQ 1( ) =0. □
Jung[19]proved an asymptotic property of the quadratic mapping. It is a natural thing to expect such a property also for the functional equation(1.3).
Corollary 4.5.Assume that a nonzero functionϕ :→ withϕ 0( ) =0satisfies
| ( )| =
{| | | | | | | |}→+∞
D s t u v
lim sup , , , 0.
s t u v
ϕ
max , , , (4.10)
Then,ϕ s( ) =s2 for alls∈.
Proof.Let{ }εn be a sequence of positive real numbers which decreases monotonically to zero. Then, by (4.10)there exists an increasing sequence{ }dn of positive real numbers such that = +∞
→+∞
d lim
n n and
|D s t u vϕ( , , , )| ⩽εn, max{| | | | | | | |} ⩾s, t, u, v dn.
According to Theorem 4.4, there exists a unique quadratic function Q :n → satisfying (1.3) and
| ( ) −ϕ s Q sn( )| ⩽εnfor alls∈. Then, the quadratic functionQ :n →satisfies| ( ) −ϕ s Q sn( )| ⩽ε1for all
∈
s and alln⩾1. Hence, the uniqueness impliesQn=Q1 for alln. Therefore,| ( ) −ϕ s Q s1( )| ⩽εnfor all
∈
s and alln⩾1. Lettingn→ +∞, we getϕ s( ) =Q s1( )for all s∈. Soϕsatisfies(1.3)andϕ 0( ) =0.
According to Corollary 2.4, we getϕ s( ) =s2 for alls∈. □
Applying the same argument given in the proof of Theorem 4.4, we obtain the following result.
Theorem 4.6.Letd>0and ε⩾0. Assume that a functionϕ :→ withϕ 0( ) =0satisfies one of the following inequalities:
1. |D s t u vϕ( , , , )| ⩽ε,s t u v, , , ∈ with| | ⩾s d;
2. |D s t u vϕ( , , , )| ⩽ε,s t u v, , , ∈ with| | ⩾t d;
3. |D s t u vϕ( , , , )| ⩽ε,s t u v, , , ∈ with| | ⩾u d;
4. |D s t u vϕ( , , , )| ⩽ε,s t u v, , , ∈ with| | ⩾v d;
5. |D s t u vϕ( , , , )| ⩽ε,s t u v, , , ∈ with| | + | | + | | + | | ⩾s t u v d.
Then, there exists a unique quadratic functionQ :→, which satisfies(1.3)and
| ( ) −ϕ s Q s( )| ⩽ε, s∈ , whereQ s( ) =s2 ifQ 1( ) =1andQ≡0ifQ 1( ) ≠1.
Corollary 4.7.Assume that a nonzero functionϕ : →withϕ 0( ) =0satisfies one of the following conditions:
1. | ( )| = ∈
| |→+∞
D s t u v s t u v
lim sup , , , 0, , , ,
s
ϕ ;
2. | ( )| = ∈
| |→+∞
D s t u v s t u v
lim sup , , , 0, , , ,
t
ϕ ;
3. | ( )| = ∈
| |→+∞
D s t u v s t u v
lim sup , , , 0, , , ,
u
ϕ ;
4. | ( )| = ∈
| |→+∞
D s t u v s t u v
lim sup , , , 0, , , ,
v
ϕ ;
5. | ( )| = ∈
| |+| |+| |+| |→+∞
D s t u v s t u v
lim sup , , , 0, , , ,
s t u v
ϕ .
Then,ϕ s( ) =s2 for alls∈.
5 Conclusions
Using the direct method and thefixed point method, we have obtained the general solution and have proved the Hyers-Ulam stability of the quadratic-multiplicative functional equation:
( − ) + ( + ) = [ ( ) + ( )][ ( ) + ( )]
ϕ st uv ϕ sv tu ϕ s ϕ u ϕ t ϕ v .
Acknowledgments:This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (NRF- 2017R1D1A1B04032937).
Availability of data and materials:Not applicable.
Conflict of interest:The authors declare that they have no competing interests.
Author contributions:The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved thefinal manuscript.
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