Let f :E→E be a mapping between Banach spaces such that f(x+y)−f(x)−f(y)δ for all x,y∈E, and for some δ >0

Mathematical Inequalities

Volume 14, Number 2 (2020), 421–436 doi:10.7153/jmi-2020-14-27

ULAM STABILITY OF AN ADDITIVE–QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES

INHOHWANG ANDCHOONKILPARK∗

(Communicated by M. Krni´c)

Abstract. Using thefixed point method and the direct method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation

f(x+y,z+w) +f(x−y,z−w)−2f(x,z)−2f(x,w) =0. (0.1)

1. Introduction and preliminaries

The stability problem of functional equations originated from a question of Ulam [25] in 1940, concerning the stability of group homomorphisms. Let(G1,.)be a group and let(G2,∗)be a metric group with the metric d(.,.). Given ε>0, does there ex- ist aδ0, such that if a mapping h:G₁→G₂ satisfies the inequality d(h(x.y),h(x)∗ h(y))<δ ^{for all x},y∈G₁, then there exists a homomorphism H:G₁→G₂ with d(h(x),H(x))<ε ^{for all x}∈G₁? In the other words, Under what condition does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. In 1941, Hyers [15] gave thefirst affirma- tive answer to the question of Ulam for Banach spaces. Let f :E→E be a mapping between Banach spaces such that

f(x+y)−f(x)−f(y)δ

for all x,y∈E, and for some δ >0. Then there exists a unique additive mapping T:E→E such that

f(x)−T(x)δ

for allx∈E. In 1978, Rassias [22] proved the following theorem.

Mathematics subject classification(2010): 39B52, 47H10, 39B62.

Keywords and phrases: Hyers-Ulam stability,fixed point method, additive-quadratic functional equa- tion, direct method.

∗Corresponding author.

c , Zagreb

Paper JMI-14-27 421

THEOREM1.1. [22]Let f :E→Ebe a mapping from a normed vector space E into a Banach space E subject to the inequality

f(x+y)−f(x)−f(y)ε(x^p+y^p) (1.1) for all x,y∈E, whereε and p are constants withε>0 and p<1. Then there exists a unique additive mapping T:E→E such that

f(x)−T(x) 2ε

2−2^px^p (1.2)

for all x∈E.If p<0then inequality(1.1)holds for all x,y=0, and(1.2)for x=0. Also, if the function t→ f(tx) from R into E is continuous in t∈Rfor eachfixed x∈E,then T is R-linear.

A generalization of the Rassias’ theorem was obtained by Gavruta [12] by replac-˘ ing the unbounded Cauchy difference by a general control function.

THEOREM1.2. [12]Suppose (G,+)is an abelian group, E is a Banach space, and that the so-called admissible control functionϕ^:G×G→Rsatisfies

ϕ˜(x,y):=2⁻¹

∑

^∞

n=0

2⁻ⁿϕ(2ⁿx,2ⁿy)<∞ for all x,y∈G. If f:G→E is a mapping with

f(x+y)−f(x)−f(y)ϕ(x,y)

for all x,y∈G, then there exists a unique mapping T :G→E such that T(x+y) = T(x) +T(y) andf(x)−T(x)ϕ˜(x,x)for all x,y∈G.

Gil´anyi [13] showed that if f satisfies the functional inequality

2f(x) +2f(y)−f(x−y)f(x+y) (1.3) then f satisfies the Jordan-von Neumann functional equation

2f(x) +2f(y) =f(x+y) +f(x−y).

See also [23]. Fechner [11] and Gil´anyi [14] proved the Hyers-Ulam stability of the functional inequality (1.3). Park [18,19] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Ba- nach spaces and non-Archimedean Banach spaces. The stability problems of various functional equations and functional inequalities have been extensively investigated by a number of authors (see [6,7,9,10,18,24]).

We recall a fundamental result infixed point theory.

THEOREM1.3. [2,5]Let(X,d)be a complete generalized metric space and let J:X →X be a strictly contractive mapping with Lipschitz constant α<1. Then for each given element x∈X , either

d(Jⁿx,Jⁿ⁺¹x) =∞

for all nonnegative integers n or there exists a positive integer n0such that (1) d(Jⁿx,Jⁿ⁺¹x)<∞, ∀nn₀;

(2) the sequence{Jⁿx}converges to afixed point y^∗of J ;

(3) y^∗is the uniquefixed point of J in the set Y={y∈X|d(Jⁿ⁰x,y)<∞}; (4) d(y,y^∗)_1−α¹ d(y,Jy)for all y∈Y .

In 1996, Isac and Rassias [16] were thefirst to provide applications of stability the- ory of functional equations for the proof of newfixed point theorems with applications.

By usingfixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [3,4,8,20,21]).

This paper is organized as follows: In Section 2, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (0.1) in Banach spaces by using the direct method. In Section 3, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (0.1) in Banach spaces by using thefixed point method.

Throughout this paper, let X be a complex normed space and Y be a complex Banach space.

2. Hyers-Ulam stability of the additive-quadratic functional equation (0.1):

direct method

We investigate the additive-quadratic functional equation (0.1) in complex normed spaces.

LEMMA2.1. If a mapping f :X²→Y satisfies f(0,z) =f(x,0) =0and f(x+y,z+w) +f(x−y,z−w)−2f(x,z)−2f(x,w) =0 (2.1) for all x,y,z,w∈X , then f :X²→Y is additive in thefirst variable and quadratic in the second variable.

Proof. Ifw=0, then f(x+y,z) +f(x−y,z)−2f(x,z) =0 for allx,y,z∈X. So f is additive in thefirst variable.

Ify=0, then f(x,z+w)+f(x,z−w)−2f(x,z)−2f(x,w) =0 for allx,z,w∈X. So f is quadratic in the second variable.

Note that if f :X→Y satisfies (2.1), then the mapping f :X→Y is called an additive-quadraticmapping.

Now we prove the Hyers-Ulam stability of the additive-quadratic functional in- equality (0.1) in complex Banach spaces.

THEOREM2.2. Letϕ^:X2→[0,∞)be a function satisfying Φ(x,y):=

∑

^∞

j=1

4^jϕ₂^x_j, y 2^j

<∞ (2.2)

for all x,y∈X and f:X²→Y be a mapping satisfying f(x,0) = f(0,z) =0 and f(x+y,z+w) +f(x−y,z−w)−2f(x,z)−2f(x,w)ϕ(x,y)ϕ(z,w) (2.3) for all x,y,z,w∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z)min 1

2Ψ(x,x)ϕ(z,0),1

4ϕ(x,0)Φ(z,z)

for all x,z∈X , where

Ψ(x,y):=

∑

^∞

j=1

2^jϕ₂^x_j, y 2^j

for all x,y∈X .

Proof. Lettingy=xandw=0 in (2.3), we get

f(2x,z)−2f(x,z)ϕ(x,x)ϕ(z,0) (2.4) and so

f(x,z)−2fx

2,zϕ₂^x,x 2

ϕ(z,0)

for allx,z∈X. Hence 2^lfx

2^l,z

−2^mf x

2^m,z^m−1

∑

j=l

2^jfx 2^j,z

−2^j+1f x

2^j+1,z (2.5)

1

2

∑

m j=l+1

2^jϕ₂^x_j, x 2^j

ϕ(z,0)

for all nonnegative integersmandl withm>l and all x,z∈X. It follows from (2.5) that the sequence{2^kf(₂^xk,z)} is Cauchy for allx,z∈X. SinceY is a Banach space, the sequence{2^kf(₂^xk,z)} converges. So one can define the mappingP:X²→Y by

P(x,z):=lim

k→∞2^kfx 2^k,z

for allx,z∈X. Moreover, lettingl=0 and passing the limitm→∞in (2.5), we get f(x,z)−P(x,z)1

2Ψ(x,x)ϕ(z,0) (2.6)

for allx,z∈X.

It follows from (2.2) and (2.3) that

P(x+y,z+w) +P(x−y,z−w)−2P(x,z)−2P(x,w)

= lim

n→∞

2ⁿ

f x+y

2ⁿ ,z+w

+f x−y

2ⁿ ,z−w

−2fx 2ⁿ,z

−2f x 2ⁿ,w

lim

n→∞2ⁿϕ₂^x_n, y 2ⁿ

ϕ(z,w) =0

for allx,y,z,w∈X. So

P(x+y,z+w) +P(x−y,z−w)−2P(x,z)−2P(x,w) =0

for all x,y,z,w∈X. By Lemma2.1, the mappingP:X²→Y is additive in thefirst variable and quadratic in second variable.

Now, letT:X²→Y be another additive-quadratic mapping satisfying (2.6). Then we have

P(x,z)−T(x,z)=2^qPx 2^q,z

−2^qTx 2^q,z 2^qPx

2^q,z

−2^qf x

2^q,z+2^qTx 2^q,z

−2^qfx 2^q,z 2^qΨ x

2^q, x 2^q

ϕ(z,0),

which tends to zero asq→∞for allx,z∈X. So we can conclude thatP(x,z) =T(x,z) for allx,z∈X. This proves the uniqueness ofP.

On the other hand, letting y=0 andw=z in (2.3), we get

f(x,2z)−4f(x,z)ϕ(x,0)ϕ(z,z) (2.7) and so

f(x,z)−4f x,z

2ϕ(x,0)ϕ₂^z,z 2

for allx,z∈X. Hence 4^lf

x, z 2^l

−4^mf x, z

2^mm−1

∑

j=l

4^jf x, z

2^j

−4^j+1f x, z

2^j+1 (2.8)

1

4

∑

m j=l+1

4^jϕ(x,0)ϕ₂^z_j, z 2^j

for all nonnegative integersmandl withm>l and all x,z∈X. It follows from (2.8) that the sequence{4^kf(x,₂^zk)} is Cauchy for all x,z∈X. SinceY is a Banach space, the sequence{4^kf(x,₂^zk} converges. So one can define the mapping Q:X²→Y by

Q(x,z):=lim

k→∞4^kf x, z

2^k

for allx,z∈X. Moreover, lettingl=0 and passing the limitm→∞in (2.8), we get f(x,z)−Q(x,z)1

4ϕ(x,0)Φ(z,z) (2.9)

for allx,z∈X.

It follows from (2.2) and (2.3) that

Q(x+y,z+w) +Q(x−y,z−w)−2Q(x,z)−2Q(x,w)

= lim

n→∞

4ⁿ

f

x+y,z+w 2ⁿ

+f

x−y,z−w 2ⁿ

−2f x, z

2ⁿ

−2f x, w

2ⁿ

lim

n→∞4ⁿϕ(x,y)ϕ₂^z_n,w 2ⁿ

=0

for allx,y,z,w∈X. So

Q(x+y,z+w) +Q(x−y,z−w)−2Q(x,z)−2Q(x,w) =0

for all x,y,z,w∈X. By Lemma2.1, the mapping Q:X²→Y is additive in thefirst variable and quadratic in second variable.

Now, letT:X²→Y be another additive-quadratic mapping satisfying (2.9). Then we have

Q(x,z)−T(x,z)=4^qQ x, z

2^q

−4^qT x, z

2^q 4^qQ

x, z 2^q

−4^qf x, z

2^q+4^qT x, z

2^q

−4^qf x, z

2^q 4^q

2 ϕ(x,0)Φ z 2^q, z

2^q ,

which tends to zero asq→∞for allx,z∈X. So we can conclude thatQ(x,z) =T(x,z) for allx,z∈X. This proves the uniqueness ofQ.

It follows from (2.9) that 2ⁿf x

2ⁿ,z

−Q x

2ⁿ,z2ⁿ

4ϕ₂^x_n,0 Φ(z,z),

which tends to zero as n→∞ for all x,z∈X. Since Q:X²→Y is additive in the first variable, we get P(x,z)−Q(x,z)=0, i.e., F(x,z):=P(x,z) =Q(x,z) for all x,z∈X. Thus there is an additive-quadratic mappingF:X²→Y such that

f(x,z)−F(x,z)min 1

2Ψ(x,x)ϕ(z,0),1

4ϕ(x,0)Φ(z,z)

for allx,z∈X.

COROLLARY2.3. Let r>2andθ be nonnegative real numbers and f:X²→Y be a mapping satisfying f(x,0) = f(0,z) =0 and

f(x+y,z+w) +f(x−y,z−w)−2f(x,z)−2f(x,w)

θ(x^r+y^r)(z^r+w^r) (2.10) for all x,y,z,w∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z) 2θ

2^r−2x^rz^r for all x,z∈X .

Proof. The proof follows from Theorem2.2by takingϕ(x,y) =√

θ(x^r+y^r) for all x,y∈X, since min{₂^2θr₋₂x^rz^r,₂^2θr₋₄x^rz^r}=₂^2θr₋₂x^rz^r for allx,z∈ X.

THEOREM2.4. Letϕ^:X2→[0,∞)be a function satisfying Ψ(x,y):=

∑

^∞

j=0

1

2^jϕ ^2jx,2^jz

<∞ (2.11)

for all x,y∈X and let f:X²→Y be a mapping satisfying f(x,0) = f(0,z) =0 and (2.3)for all x,z∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z)min 1

2Ψ(x,x)ϕ(z,0),1

4ϕ(x,0)Φ(z,z)

(2.12) for all x,z∈X , where

Φ(x,y):=

∑

^∞

j=0

1

4^jϕ ^2jx,2^jy for all x,y∈X .

Proof. It follows from (2.4) that f(x,z)−1

2f(2x,z) 1

2ϕ(x,x)ϕ(z,0) for allx,z∈X. Hence

1 2^l f

2^lx,z

− 1

2^mf(2^mx,z) ^m−1

∑

j=l

1

2^j f 2^jx,z

− 1

2^j+1f 2^j+1x,z (2.13)

1

2

m−1

∑

j=l

1

2^jϕ ^2jx,2^jx ϕ(z,0)

for all nonnegative integersmandl withm>l and allx,z∈X. It follows from (2.13) that the sequence{₂¹kf(2^kx,z)} is Cauchy for allx,z∈X. SinceY is a Banach space, the sequence{₂¹kf(2^kx,z)} converges. So one can define the mappingP:X²→Y by

P(x,z):=lim

k→∞

1 2^kf

2^kx,z

for allx,z∈X. Moreover, lettingl=0 and passing the limitm→∞in (2.13), we get f(x,z)−P(x,z)1

2Ψ(x,x)ϕ(z,0) (2.14)

for allx,z∈X.

It follows from (2.3) and (2.11) that

P(x+y,z+w) +P(x−y,z−w)−2P(x,z)−2P(x,w)

= lim

n→∞

1

2ⁿ(f(2ⁿ(x+y),z+w)+f(2ⁿ(x−y),z−w)−2f(2ⁿx,z)−2f(2ⁿx,w))

lim

n→∞

1

2ⁿϕ(2ⁿx,2ⁿy)ϕ(z,w) =0 for allx,y,z,w∈X. So

P(x+y,z+w) +P(x−y,z−w)−2P(x,z)−2P(x,w) =0

for all x,y,z,w∈X. By Lemma2.1, the mappingP:X²→Y is additive in thefirst variable and quadratic in second variable.

Now, let T :X²→Y be another additive-quadratic mapping satisfying (2.14).

Then we have

P(x,z)−T(x,z)= 1

2^qP(2^qx,z)− 1

2^qT(2^qx,z)

1

2^qP(2^qx,z)− 1

2^qf(2^qx,z) +

1

2^qT(2^qx,z)− 1

2^qf(2^qx,z)

1

2^qΨ(2^qx,2^qx)ϕ(z,0),

which tends to zero asq→∞for allx,z∈X. So we can conclude thatP(x,z) =T(x,z) for allx,z∈X. This proves the uniqueness ofP.

It follows from (2.15) that

f(x,2z)−4f(x,z)ϕ(x,0)ϕ(z,z) (2.15) and so

f(x,z)−1

4f(x,2z) 1

4ϕ(x,0)ϕ(2z,2z) for allx,z∈X. Hence

1

4^lf(x,2^lz)− 1

4^mf(x,2^mz) ^m−1

∑

j=l

1

4^jf x,2^jz

− 1

4^j+1f x,2^j+1z (2.16)

1

4

m−1

∑

j=l

1

4^jϕ(x,0)ϕ ^2jz,2^jz

for all nonnegative integersmandl withm>l and allx,z∈X. It follows from (2.16) that the sequence{₄¹kf(x,2^kz)} is Cauchy for allx,z∈X. SinceY is a Banach space, the sequence{₄¹kf(x,2^kz} converges. So one can define the mapping Q:X²→Y by

Q(x,z):=lim

k→∞

1 4^kf

x,2^kz

for allx,z∈X. Moreover, lettingl=0 and passing the limitm→∞in (2.16), we get f(x,z)−Q(x,z)1

4ϕ(x,0)Φ(z,z) (2.17)

for allx,z∈X.

It follows from (2.3) and (2.11) that

Q(x+y,z+w) +Q(x−y,z−w)−2Q(x,z)−2Q(x,w)

= lim

n→∞

1

4ⁿ(f(x+y,2ⁿ(z+w)) +f(x−y,2ⁿ(z−w))−2f(x,2ⁿz)−2f(x,2ⁿw))

lim

n→∞

1

4ⁿϕ(x,y)ϕ(2ⁿz,2ⁿw) =0 for allx,y,z,w∈X. So

Q(x+y,z+w) +Q(x−y,z−w)−2Q(x,z)−2Q(x,w) =0

for all x,y,z,w∈X. By Lemma2.1, the mapping Q:X²→Y is additive in thefirst variable and quadratic in second variable.

Now, let T :X²→Y be another additive-quadratic mapping satisfying (2.17).

Then we have

Q(x,z)−T(x,z)= 1

4^qQ(x,2^qz)− 1

4^qT(x,2^qz)

1

4^qQ(x,2^qz)− 1

4^qf(x,2^qz) +

1

4^qT(x,2^qz)− 1

4^qf(x,2^qz)

1

2·4^qϕ(x,0)Φ(2^qz,2^qz),

which tends to zero asq→∞for allx,z∈X. So we can conclude thatQ(x,z) =T(x,z) for allx,z∈X. This proves the uniqueness ofQ.

It follows from (2.17) that 1

2ⁿf(2ⁿx,z)−Q(2ⁿz,z) 1

4·2ⁿϕ(2ⁿx,0)Φ(z,z),

which tends to zero as n→∞ for all x,z∈X. Since Q:X²→Y is additive in the first variable, we get P(x,z)−Q(x,z)=0, i.e., F(x,z):=P(x,z) =Q(x,z) for all x,z∈X. Thus there is an additive-quadratic mappingF:X²→Y such that

f(x,z)−F(x,z)min 1

2Ψ(x,x)ϕ(z,0),1

4ϕ(x,0)Φ(z,z) for allx,z∈X.

COROLLARY2.5. Let r<1andθ be nonnegative real numbers and f:X²→Y be a mapping satisfying(2.10)and f(x,0) = f(0,z) =0 for all x,z∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z) 2θ

4−2^rx^rz^r for all x,z∈X .

Proof. The proof follows from Theorem2.4by takingϕ(x,y) =√

θ(x^r+y^r) for all x,y∈X, since min{₂₋₂^2θrx^rz^r,₄₋₂^2θrx^rz^r}=₄₋₂^2θrx^rz^r for allx,z∈ X.

3. Hyers-Ulam stability of the additive-quadratic functional equation (0.1):fixed point method

Using thefixed point method, we prove the Hyers-Ulam stability of the additive- quadratic functional equation (0.1) in complex Banach spaces.

THEOREM3.1. Letϕ^:X2→[0,∞)be a function such that there exists an L<1 with

ϕ^x₂,y 2

L

4ϕ(x,y)L

2ϕ(x,y) (3.1)

for all x,y∈X . Let f:X²→Y be a mapping satisfying satisfying(2.3)and f(x,0) = f(0,z) =0 for all x,z∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z)min L

2(1−L)ϕ(x,x)ϕ(z,0), L

4(1−L)ϕ(x,0)ϕ(z,z)

(3.2) for all x,z∈X .

Proof. Lettingw=0 and y=xin (2.3), we get

f(2x,z)−2f(x,z)ϕ(x,x)ϕ(z,0) (3.3) for allx,z∈X.

Consider the set

S:={h:X²→Y, h(x,0) =h(0,z) =0 ∀x,z∈X} and introduce the generalized metric onS:

d(g,h) =inf{μ∈R+:g(x,z)−h(x,z)μϕ(x,x)ϕ(z,0), ∀x,z∈X}, where, as usual, infφ= +∞. It is easy to show that(S,d)is complete (see [17]).

Now we consider the linear mapping J:S→S such that Jg(x,z):=2gx

2,z for allx,z∈X.

Let g,h∈S be given such thatd(g,h) =ε^{. Then} g(x,z)−h(x,z)εϕ(x,x)ϕ(z,0) for allx,z∈X. Hence

Jg(x,z)−Jh(x,z)=2gx 2,z

−2hx

2,z2εϕ₂^x,x 2

ϕ(z,0)

2ε^L₂ϕ(x,x)ϕ(z,0) =Lεϕ(x,x)ϕ(z,0) for allx,z∈X. Sod(g,h) =ε implies thatd(Jg,Jh)Lε. This means that

d(Jg,Jh)Ld(g,h) for allg,h∈S.

It follows from (3.3) that f(x,z)−2fx

2,zϕ^x₂,x 2

ϕ(z,0)L

2ϕ(x,x)ϕ(z,0) for allx,z∈X. Sod(f,J f)^L₂.

By Theorem1.3, there exists a mappingP:X²→Y satisfying the following:

(1) P is afixed point ofJ, i.e.,

P(x,z) =2Px 2,z

(3.4) for allx,z∈X. The mappingPis a uniquefixed point ofJ. This implies thatP is a unique mapping satisfying (3.4) such that there exists a μ∈(0,∞)satisfying

f(x,z)−P(x,z)μϕ(x,x)ϕ(z,0) for allx,z∈X;

(2) d(J^lf,P)→0 asl→∞. This implies the equality

l→∞lim2^lfx 2^l,z

=P(x,z) for allx,z∈X;

(3) d(f,P)_1−L¹ d(f,J f), which implies f(x,z)−P(x,z) L

2(1−L)ϕ(x,x)ϕ(z,0) for allx,z∈X.

By the same reasoning as in the proof of Theorem2.2, one can show that the mapping P:X²→Y is additive in thefirst variable and quadratic in the second variable.

Letting z=wandy=0 in (2.3), we get

f(x,2z)−4f(x,z)ϕ(x,0)ϕ(z,z) (3.5) for allx,z∈X.

Consider the set

S:={h:X²→Y, h(x,0) =h(0,z) =0 ∀x,z∈X} and introduce the generalized metric onS:

d(g,h) =inf{μ∈R+:g(x,z)−h(x,z)μϕ(x,0)ϕ(z,z), ∀x,z∈X}, where, as usual, infφ= +∞. It is easy to show that(S,d)is complete (see [17]).

Now we consider the linear mapping J:S→S such that Jg(x,z):=4gx

2,z for allx,z∈X.

Let g,h∈S be given such thatd(g,h) =ε^{. Then} g(x,z)−h(x,z)εϕ(x,0)ϕ(z,z) for allx,z∈X. Hence

Jg(x,z)−Jh(x,z)=4g x,z

2

−4h x,z

24εϕ(x,0)ϕ^z₂,z 2

4ε^L₄ϕ(x,0)ϕ(z,z) =Lεϕ(x,0)ϕ(z,z)

for allx,z∈X. Sod(g,h) =ε implies thatd(Jg,Jh)Lε. This means that d(Jg,Jh)Ld(g,h)

for allg,h∈S.

It follows from (3.5) that f(x,z)−4f

x,z

2ϕ(x,0)ϕ₂^z,z 2

L

4ϕ(x,0)ϕ(z,z) for allx,z∈X. Sod(f,Jf)^L₄.

By Theorem1.3, there exists a mappingQ:X²→Y satisfying the following:

(1) Qis afixed point ofJ, i.e.,

Q(x,z) =4Q x,z

2

(3.6)

for allx,z∈X. The mappingQis a uniquefixed point ofJ. This implies thatQ is a unique mapping satisfying (3.6) such that there exists a μ∈(0,∞)satisfying

f(x,z)−Q(x,z)μϕ(x,0)ϕ(z,z) for allx,z∈X;

(2) d(J^lf,Q)→0 as l→∞. This implies the equality

l→∞lim4^lf x, z

2^l

=Q(x,z)

for allx,z∈X;

(3) d(f,Q)_1−L¹ d(f,Jf), which implies f(x,z)−Q(x,z) L

4(1−L)ϕ(x,0)ϕ(z,z) for allx,z∈X.

By the same reasoning as in the proof of Theorem2.2, one can show that the mapping Q:X²→Y is additive in thefirst variable and quadratic in the second variable.

By the same reasoning as in the proof of Theorem2.2, we getP(x,z)−Q(x,z)= 0, i.e.,F(x,z):=P(x,z) =Q(x,z) for all x,z∈X. Thus there is an additive-quadratic mappingF:X²→Y such that

f(x,z)−F(x,z)min L

2(1−L)ϕ(x,x)ϕ(z,0), L

4(1−L)ϕ(x,0)ϕ(z,z)

for allx,z∈X.

COROLLARY3.2. Let r>2andθ be nonnegative real numbers and f:X²→Y be a mapping satisfying(2.10)and f(x,0) = f(0,z) =0 for all x,z∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z) 2θ

2^r−4x^rz^r for all x,z∈X .

Proof. The proof follows from Theorem3.1by taking L=2^2−r and ϕ(x,y) =

√θ(x^r+y^r)for allx,y∈X, since min{₂²r−^θ4x^rz^r,₂⁴r−^θ4x^rz^r}=₂²r−^θ4x^rz^r for allx,z∈X.

THEOREM3.3. Letϕ^:X2→[0,∞)be a function such that there exists an L<1 with

ϕ(x,y)2Lϕ₂^x,y 2

4Lϕ₂^x,y 2

(3.7)

for all x,y∈X . Let f:X²→Y be a mapping satisfying satisfying(2.3)and f(x,0) = f(0,z) =0 for all x,z∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z)min 1

2(1−L)ϕ(x,x)ϕ(z,0), 1

4(1−L)ϕ(x,0)ϕ(z,z)

for all x,z∈X .

Proof. Consider the complete metric spaces(S,d) and(S,d) given in the proof of Theorem3.1.

Now we consider the linear mapping J:S→S such that Jg(x,z):=1

2g(2x,z) for allx,z∈X.

It follows from (3.3) that f(x,z)−1

2f(2x,z) 1

2ϕ(x,x)ϕ(z,0) for allx,z∈X. Sod(f,J f)¹₂.

By the same reasoning as in the proof of Theorem2.2, one can show that there exists a unique additive-quadratic mappingP:X²→Y such that

f(x,z)−P(x,z) 1

2(1−L)ϕ(x,x)ϕ(z,0) for allx,z∈X.

By the same reasoning as in the proof of Theorem 2.2, one can show that the mappingP:X²→Y is additive in thefirst variable and quadratic in the second variable.

Now we consider the linear mapping J:S→S such that Jg(x,z):=4gx

2,z for allx,z∈X.

It follows from (3.5) that f(x,z)−1

4f(x,2z) 1

4ϕ(x,0)ϕ(z,z) for allx,z∈X. Sod(f,Jf)¹₄.

By the same reasoning as in the proof of Theorem2.2, one can show that there exists a unique additive-quadratic mappingQ:X²→Y such that

f(x,z)−Q(x,z) 1

4(1−L)ϕ(x,0)ϕ(z,z) for allx,z∈X.

By the same reasoning as in the proof of Theorem 2.2, one can show that the mappingQ:X²→Y is additive in thefirst variable and quadratic in the second variable.

By the same reasoning as in the proof of Theorem2.2, we getP(x,z)−Q(x,z)= 0, i.e.,F(x,z):=P(x,z) =Q(x,z) for all x,z∈X. Thus there is an additive-quadratic mappingF:X²→Y such that

f(x,z)−F(x,z)min 1

2(1−L)ϕ(x,x)ϕ(z,0), 1

4(1−L)ϕ(x,0)ϕ(z,z)

for allx,z∈X.

COROLLARY3.4. Let r<1andθ be nonnegative real numbers and f:X²→Y be a mapping satisfying(2.10)and f(x,0) = f(0,z) =0 for all x,z∈X . Then there exists a unique additive-quadratic mapping F:X²→Y such that

f(x,z)−F(x,z) θ

2−2^rx^rz^r for all x,z∈X .

Proof. The proof follows from Theorem3.3by taking L=2^r−1 and ϕ(x,y) =

√θ(x^r+y^r)for allx,y∈X, since min{₂₋^2θ₂rx^rz^r,₂₋^θ₂rx^rz^r}=₂₋₂^θrx^rz^r for allx,z∈X.

Competing interests

The authors declare that they have no competing interests.

R E F E R E N C E S

[1] T. AOKI,On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan2(1950), 64–66.

[2] L. C˘ADARIU, V. RADU,Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math.4, no. 1, Art. ID 4 (2003).

[3] L. C˘ADARIU, V. RADU,On the stability of the Cauchy functional equation: afixed point approach, Grazer Math. Ber.346(2004), 43–52.

[4] L. CADARIU˘ , V. RADU,Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl.2008, Art. ID 749392 (2008).

[5] J. DIAZ, B. MARGOLIS,Afixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Am. Math. Soc.74(1968), 305–309.

[6] Y. DING, T.-Z. XU,Approximate solution of generalized inhomogeneous radical quadratic functional equations in 2-Banach spaces, J. Inequal. Appl.2019, 2019:31, 13 pp.

[7] A. EBADIAN, S. ZOLFAGFHARI, S. OSTADBASHI, C. PARK,Approximation on the reciprocal func- tional equation in several variables in matrix non-Archimedean random normed spaces, Adv. Differ- ence Equ.2015, 2015:314, 13 pp.

[8] I. EL-FASSI,Generalized hyperstability of a Drygas functional equation on a restricted domain using Brzdek’sfixed point theorem, J. Fixed Point Theory Appl.19(2017), 2529–2540.

[9] I. EL-FASSI,Solution and approximation of radical quintic functional equation related to quintic mapping in quasi-β-Banach spaces, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat.113(2019), no. 2, 675–687.

[10] M. ESHAGHIGORDJI, N. GHOBADIPOUR,Stability of(α,β,γ)-derivations on Lie C^∗-algebras, Int.

J. Geom. Meth. Mod. Phys.7(2010), 1097–1102.

[11] W. FECHNER,Stability of a functional inequalities associated with the Jordan-von Neumann func- tional equation, Aequationes Math.71(2006), 149–161.

[12] P. GAVRUTAˇ ,A generalization of the Hyers-Ulam-Rassias stability of approximately additive map- pings, J. Math. Anal. Appl.184(1994), 431–436.

[13] A. GILANYI´ ,Eine zur Parallelogrammgleichung ¨aquivalente Ungleichung, Aequationes Math.62 (2001), 303–309.

[14] A. GILANYI´ ,On a problem by K. Nikodem, Math. Inequal. Appl.5(2002), 707–710.

[15] D.H. HYERS,On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A.27 (1941), 222–224.

[16] G. ISAC, TH. M. RASSIAS,Stability of ψ-additive mappings: Applications to nonlinear analysis, Int. J. Math. Math. Sci.19(1996), 219–228.

[17] D. MIHET¸, V. RADU,On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl.343(2008), 567–572.

[18] C. PARK,Additiveρ-functional inequalities and equations, J. Math. Inequal.9(2015), 17–26.

[19] C. PARK,Additiveρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal.9 (2015), 397–407.

[20] C. PARK,Fixed point method for set-valued functional equations, J. Fixed Point Theory Appl.19 (2017), 2297–2308.

[21] V. RADU,Thefixed point alternative and the stability of functional equations, Fixed Point Theory4 (2003), 91–96.

[22] TH. M. RASSIAS,On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc.72 (1978), 297–300.

[23] J. R ¨ATZ,On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math.66(2003), 191–200.

[24] L. SZEKELYHIDI´ ,Superstability of functional equations related to spherical functions, Open Math.

15(2017), no. 1, 427–432.

[25] S. M. ULAM,A Collection of the Mathematical Problems, Interscience Publ. New York, 1960.

(Received August 7, 2019) Inho Hwang

Department of Mathematics Incheon National University Incheon 22012, Korea e-mail:ho818@inu.ac.kr Choonkil Park Research Institute for Natural Sciences Hanyang University Seoul 04763, Korea e-mail:baak@hanyang.ac.kr

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