Stability of Quadratic Functional Equations via the Fixed Point and Direct Method

Volume 2010, Article ID 635720,15pages doi:10.1155/2010/635720

Research Article

Stability of Quadratic Functional Equations via

the Fixed Point and Direct Method

Eunyoung Son, Juri Lee, and Hark-Mahn Kim

Department of Mathematics, Chungnam National University, Daejeon 305-764, South Korea

Correspondence should be addressed to Hark-Mahn Kim,[email protected]

Received 13 October 2009; Accepted 19 January 2010

Academic Editor: Yeol Je Cho

Copyrightq2010 Eunyoung Son et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

C˘adariu and Radu applied the fixed point theorem to prove the stability theorem of Cauchy and Jensen functional equations. In this paper, we prove the generalized Hyers-Ulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation.

1. Introduction

In 1940, Ulam 1gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

LetG be a group and letGbe a metric group with metricρ·,·. Given > 0, does there exist aδ > 0such that if f : G → Gsatisfiesρfxy, fxfy < δfor allx, y ∈ G, then a homomorphismh:G → Gexists withρfx, hx< for allx∈G?

The concept of stability for functional equations arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus we say that a functional equationE1f E2f is stable if any mappingg approximately satisfying

the equationdE1g, E2g≤ϕxis near to a true solutionf such thatE1f E2fand

dfx, gx≤Φxfor some functionΦdepending on the given functionϕ.In 1941, the first result concerning the stability of functional equations for the case whereG1andG2are Banach

spaces was presented by Hyers2. In fact, he proved that each solutionf of the inequality

fxy−fx−fy ≤ for allx, y ∈ G1 can be approximated by a unique additive

functionL : G1 → G2 defined byLx limn→ ∞f2nx/2nsuch that fx−Lx ≤

for every x ∈ G1. Moreover, if ftx is continuous in t ∈ R for each fixedx ∈ G1, then

stability theorems of functional equations which generalize the Hyers’ result. In 1978, Rassias

6attempted to weaken the condition for the bound of Cauchy difference controlled by a sum of unbounded function εxpyp,0 < p < 1, and provided a generalization of Hyers’ theorem. In 1991, Gajda7 gave an affirmative solution to this question for p > 1 by following the same approach as in6. Rassias8established a similar stability theorem for the unbounded Cauchy difference controlled by a product of unbounded functionεxp·

yq, pq /1. G˘avrut¸a9provided a further generalization of Rassias’ theorem by replacing the bound of Cauchy difference by a general control function. During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappingssee10–15.

Let E1 and E2 be real vector spaces. A functionf : E1 → E2 is called a quadratic

function if and only iffis a solution function of the quadratic functional equation

fxyfx−y2fx 2fy 1.1

for allx, y∈E1.It is well known that a functionfbetween real vector spaces is quadratic if

and only if there exists a unique symmetric biadditive functionBsuch thatfx Bx, xfor allx, where the mappingBis given byBx, y 1/4fxy−fx−y. See16,17for the details.

The Hyers-Ulam stability of the quadratic functional equation1.1was first proved by Skof 18for functions f : E1 → E2, where E1 is a normed space and E2 is a Banach

space. Cholewa noticed that Skof’s theorem is also valid if E1 is replaced by an Abelian

group. Czerwik 19 proved the generalized Hyers-Ulam stability of quadratic functional equation1.1in the spirit of Rassias approach. On the other hand, according to the theorem of Borelli and Forti 20, we know the following generalization of stability theorem for quadratic functional equation. LetG be a 2-divisible Abelian group andEa Banach space, and letf :G → Ebe a mapping withf0 0 satisfying the inequality

fxyfx−y−2fx−2fy≤ϕx, y 1.2

for allx, y∈G. Assume that one of the series

Φx, y:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

∞

k0

1 22k1ϕ

2k_x,2k_y<∞,

∞

k0

22k_ϕ

x

2k1,

y

2k1

<∞

1.3

holds for allx, y∈G, then there exists a unique quadratic functionQ:G → Esuch that

_{fx−Qx≤Φx, x 1.4}

In 1996, Isac and Rassias 28 applied the stability theory of functional equa-tions to prove fixed point theorems and study some new applicaequa-tions in nonlinear analysis.Radu29, C˜adariu and Radu30,31applied the fixed point theorem of alternative to the investigation of Cauchy and Jensen functional equations. Recently, Jung et al.32, Jung33,34, Jung and Lee35, Jung and Min36, Jung and Rassias37have obtained the generalized Hyers-Ulam stability of functional equations via the fixed point method.

Now, we see that the norm defined by a real inner product space satisfies the following equality:

2

n

i1

xi

2

i /j

xi−xj22n n

i1

xi2 1.5

for all vectors x1, . . . , xn. Thus employing the last equality, we introduce to consider the

following functional equation

2f

_n

i1

xi

i /j

fxi−xj

2n

n

i1

fxi 1.6

with several variables for any fixedn ∈ Nwith n ≥ 2. It is obvious that ifn 2 in1.6, then the solution function is even and thus it reduces to1.1. Conversely, we observe that the general solution of1.6in the class of all functions between vector spaces is exactly a quadratic function. In this paper, we are going to investigate the general solution of 1.6

and then we are to prove the generalized Hyers-Ulam stability of1.6for a large class of functions from vector spaces into completeβ-normed spaces by using fixed point method, and direct method.

2. Stability of

1.6

by Fixed Point Method

For notational convenience, given a mappingf :X → Y, we define the difference operator

Df :Xn _{→ Y of1.6by}

Dfx1, . . . , xn:2f

_n

i1

xi

i /j

fxi−xj

−2n

n

i1

fxi, n≥2 2.1

for allx1, . . . , xn ∈ X, which is called the approximate remainder of the functional equation

1.6and acts as a perturbation of the equation.

We now introduce a fundamental result of fixed point theory. We refer to38for the proof of it. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al.39.

Theorem 2.1. LetΩ, dbe a generalized complete metric space (i.e.,dmay assume infinite values). Assume thatΛ:Ω → Ωis a strictly contractive operator, that is, there exists a Lipschitz constantL

one of the following assertions is true: A1dΛk1x,Λkx ∞for allk≥0;

A2there exists a nonnegative integern0such that A2.1dΛn1x,Λnx<∞for alln≥n0;

A2.2the sequence{Λnx}converges to a fixed pointx∗ofΛ;

A2.3x∗is the unique fixed point ofΛinΔ {y∈Ω:dΛn0x, y<∞}; A2.4dy, x∗≤1/1−Ldy,Λyfor ally∈Δ.

Throughout this paper, we consider a β-Banach space. Let βbe a real number with 0< β≤1 and letKdenote either real fieldRor complex fieldC. SupposeEis a vector space overK. A function · _β:E → 0,∞is called aβ-norm if and only if it satisfies

N1xβ0, if and only ifx0;

N2λxβ|λ|βxβ, for allλ∈Kand allx∈E;

N3xy_β≤ xβ yβ, for allx, y∈E.

A β-Banach space is a β-normed space which is complete with respect to the β -norm. Now we are ready to investigate the generalized Hyers-Ulam stability problem for the functional equation1.6using the fixed point method. From now on, let X be a linear space and let Y be aβ-Banach space overKunless we give any specific reference whereβis a fixed real number with 0< β≤1.

Theorem 2.2. Letf : X → Y be a function withf0 0for which there exists a functionϕ :

Xn _{→ 0,∞such that there exists a constantL,0< L <1,satisfying the inequalities}

_{Dfx1, . . . , xn}

β≤ϕx1, . . . , xn, 2.2

ϕnx1, . . . , nxn≤n2βLϕx1, . . . , xn 2.3

for all x1, . . . , xn ∈ X. Then there exists a unique quadratic function Q : X → Y defined by

limk→ ∞fnkx/n2k Qxsuch that

fx−Qx_β≤ 1

2β_n2β_1−Lϕx, . . . , x 2.4

for allx∈X.

Proof. Let us defineΩto be the set of all functionsg : X → Y and introduce a generalized metricdonΩas follows:

dg, hinfC∈0,∞:gx−hx_β≤Cϕx, . . . , x, ∀x∈X. 2.5

Then it is easy to show thatΩ, dis completesee the proof of Theorem 3.1 of35. Now we define an operatorΛ:Ω → Ωby

Λgx gnx

for allx∈X.First, we assert thatΛis strictly contractive with constantLonΩ. Giveng, h∈Ω, letC∈0,∞be an arbitrary constant withdg, h≤C,that is,gx−hx_β≤Cϕx, . . . , x.

Then it follows from2.3that

_Λgx−Λhx

β

1

n2βgnx−hnxβ≤

1

n2βCϕnx, . . . , nx ≤LCϕx, . . . , x

2.7

for allx∈X,that is,dΛg,Λh≤LC.Thus we see thatdΛg,Λh≤Ldg, hfor anyg, h∈Ω

and soΛis strictly contractive with constantLonΩ.

Next, if we putx1, . . . , xn : x, . . . , xin2.2and we divide both sides by2n2β,

then we get

f_nnx2 −fx

β

1 2n2β

2fnx−2n2_fx

β

≤ 1

2n2βϕx, . . . , x

2.8

for allx∈X,which impliesdΛf, f≤1/2n2β_<∞.

Thus applying Theorem 2.1 to the complete generalized metric space Ω, d with contractive constantL, we see fromTheorem 2.1A2.2that there exists a functionQ:X → Y

which is a fixed point ofΛ, that is,Qx ΛQx Qnx/n2_{,such thatdΛ}k_{f, Q → 0 as}

k → ∞.By mathematical induction we know that

Λk_Qx Q

nkx

n2k Qx 2.9

for allk∈N.SincedΛk_{f, Q → 0 ask → ∞,there exists a sequence{Ck}such thatC} k → 0

ask → ∞,anddΛk_{f, Q≤C}

kfor everyk∈N.Hence, it follows from the definition ofdthat

Λk_fx−Qx

β≤Ckϕx, . . . , x 2.10

for allx∈X.This implies

lim

k→ ∞

Λk_fx−Qx

β0, that is, klim→ ∞

fnk_x

n2k Qx 2.11

for allx∈X.ByTheorem 2.1A2.4,we obtain

df, Q≤ 1

1−Ld

Λf, f≤ 1

2n2β_1−L, 2.12

In turn, it follows from2.2and2.3that

DQx1, . . . , xnβ lim k→ ∞

1

n2kβ

Dfnkx1, . . . , nkxn β

≤ lim

k→ ∞

1

n2kβϕ

nkx1, . . . , nkxn

≤ lim

k→ ∞L

k_ϕx

1, . . . , xn

0

2.13

for all x1, . . . , xn ∈ X, which implies thatQ is a solution of1.6and so the mapping Qis

quadratic.

To prove the uniqueness ofQ, assume now that Q1 : X → Y is another quadratic

mapping satisfying inequality2.4. Then Q1 is a fixed point ofΛandQ1 ∈ Δ {g ∈ Ω :

df, g < ∞}.Since the mapping Q is a unique fixed point ofΛ in the set Δ {g ∈ Ω :

df, g<∞},we see thatQQ1byTheorem 2.1A2.3.The proof is complete.

The following theorem is an alternative result ofTheorem 2.2.

Theorem 2.3. Letf : X → Y be a function withf0 0for which there exists a functionϕ :

Xn → 0,∞such that there exists a constantL,0< L <1,satisfying the inequalities

Dfx1, . . . , xn_β≤ϕx1, . . . , xn, 2.14

ϕ

x1

n, . . . , xn

n

≤ L

n2βϕx1, . . . , xn 2.15

for all x1, . . . , xn ∈ X. Then there exists a unique quadratic function Q : X → Y defined by

limk→ ∞n2kfx/nk Qxsuch that

_fx−Qx

β≤

L

2β_n2β_1−Lϕx, . . . , x 2.16

for allx∈X.

Proof. We use the same notations forΩanddas in the proof ofTheorem 2.2. ThusΩ, dis a complete generalized metric space. Let us define an operatorΛ:Ω → Ωby

Λgx n2g

x n

, g∈Ω 2.17

for allx∈X.

Then it follows from2.15that

_Λgx−Λhx

βn2β

g

x n

−h

x n

β

≤n2βCϕ

x n, . . . ,

x n

≤LCϕx, . . . , x

for allx∈X,that is,dΛg,Λh≤LC.Thus we see thatdΛg,Λh≤Ldg, hfor anyg, h∈Ω

and soΛis strictly contractive with constantLonΩ.

Next, if we putx1, . . . , xn: x/n, . . . , x/nin2.14and we multiply both sides by

1/2β_{, then we get by virtue of2.15}

fx−n2f

x n

β

1

2βϕ

x n, . . . ,

x n

≤ L

2β_n2βϕx, . . . , x

2.19

for allx∈X,which impliesdf,Λf≤L/2β_n2β_<∞.

Thus according toA2.2ofTheorem 2.1, there exists a functionQ:X → Y which is a

fixed point ofΛ, that is,Qx ΛQx n2_{Qx/n,such that}

lim

k→ ∞d

Λk_{f, Q0, that is, lim} k→ ∞n

2k_f

x nk

Qx, x∈X. 2.20

ByTheorem 2.1A2.4,we obtain

df, Q≤ 1

1−Ld

Λf, f≤ L

2β_n2β_1−L, 2.21

which yields the inequality2.16.

Replacingx/nk _{instead ofn}k_{xin the last part ofTheorem 2.2, we can prove thatQ :}

X → Y is a unique quadratic function satisfying2.16for allx∈X.

As applications, one has the following corollaries concerning the stability of1.6.

Corollary 2.4. Letεbe a real number withε≥0. Assume that a functionf:X → Y withf0 0

satisfies the inequality

Dfx1, . . . , xn_β≤ε 2.22

for allx1, . . . , xn ∈X. Then there exists a unique quadratic functionQ :X → Y given byQx

limk→ ∞fnkx/n2k,which satisfies the inequality _fx−Qx

β≤

ε

2β_n2β₋₁ 2.23

for allx∈X.

Proof. Letting ϕx1, . . . , xn : ε and then applyingTheorem 2.2 with contractive constant

1/n2β_{, we obtain easily the result.}

Corollary 2.5. LetXbe anα-normed space with0< α≤1andY aβ-Banach space, respectively. Let

eitherαmax{pi}< 2β,orαmin{pi} >2β. Assume that a functionf :X → Y withf0 0

satisfies the inequality

Dfx1, . . . , xn_β≤ n

i1

θixipi

α 2.24

for allx1, . . . , xn∈XandX\{0}ifpi<0. Then there exists a unique quadratic functionQ:X → Y

which satisfies the inequality

fx−Qx_β≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

n

i1θixpαi

2β_n2β_−nαmax{pi} ifα

maxpi

<2β,

_n

i1θixpαi

2β_nαmin{pi}−_n2β ifα

minpi

>2β,

2.25

for allx∈XandX\ {0}ifp <0. The functionQis given by

Qx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

lim

k→ ∞

fnk_x

n2k ifα

maxpi

<2β,

lim

k→ ∞n 2k_f

x nk

ifαminpi

>2β,

2.26

for allx∈X.

Proof. Letting ϕx1, . . . , xn : ni1θixipαi for all x1, . . . , xn ∈ X and then applying

Theorem 2.2 with contractive constant nαmax{pi}_/n2β _{and Theorem 2.3 with contractive}

constantn2β_/nαmin{pi}_{, we obtain easily the results.}

3. Stability of

1.6

by Direct Method

In the next two theorems, letϕ:Xn _{→ 0,∞be a mapping satisfying one of the conditions}

Φ1x1, . . . , xn:

∞

l0

1 2β_n2βl1ϕ

nlx1, . . . , nlxn

<∞, 3.1

Φ2x1, . . . , xn:

∞

l0

1 2βn

2βl_ϕ

x1

nl1, . . . ,

xn

nl1

<∞ 3.2

for allx1, . . . , xn∈X.

Theorem 3.1. Assume that a functionf:X → Ysatisfies

for allx1, . . . , xn∈Xandϕsatisfies the condition3.1. Then there exists a unique quadratic function

Q:X → Ysatisfying

fx− nf0

2n1−Qx

β

≤Φ1x, . . . , x 3.4

for allx∈X, wheref0_β≤ϕ0, . . . ,0/n−1βn2β. The functionQis given by

Qx lim

k→ ∞

fnk_x

n2k 3.5

for allx∈X.

Proof. Puttingx1, . . . ,xn0 in3.3, we getf0β≤ϕ0, . . . ,0/n−1βn2β. Putting

x1, . . . ,xnxin3.3, we obtain

2fnx nn−1f0−2n2fx

β≤ϕx, . . . , x 3.6

forx∈X. Dividing3.6by 2β_n2β_{, we get}

_n12fnx−fx

β ≤ 1

2β_n2βϕx, . . . , x, 3.7

wherefx fx−n/2n1f0for anyx∈X. Thus it follows from formula3.7and triangle inequality that

_n12kfn

k_x−fx β

≤k−1 l0

1 2β_n2βl1ϕ

nlx, . . . nlx 3.8

for allx∈Xand allk∈N,which is verified by induction. Therefore we prove from inequality

3.8that for any integersm, kwithm > k≥0

_n12mfn

m_x− 1

n2kf

nkx

β ≤ 1

n2βk

_n21m−kf

nm−knkx−fnkx

β

m−k−1

l0

1 2β_n2βlk1ϕ

nlkx, . . . , nlkx

m−1

lk

1 2β_n2βl1ϕ

nl_{x, . . . , n}l_x

for all x ∈ X. Since the right-hand side of 3.9 tends to zero as k → ∞, the sequence

{1/n2kfnkx}is a Cauchy sequence for allx∈Xand thus converges by the completeness ofY. DefineQ:X → Y by

Qx lim

k→ ∞

1

n2k

fnkx− n

2n1f0

lim

k→ ∞

fnk_x

n2k , x∈X.

3.10

Taking the limit in3.8ask → ∞, we obtain that

fx− n

2n1f0−Qx

β

≤Φ1x, . . . , x 3.11

for allx ∈ X. Lettingxi : nkxi for alli 1, . . . , nin3.3, respectively, and dividing both

sides byn2βkand after then taking the limit in the resulting inequality, we have

2Q

_n

i1

xi

i /j

Qxi−xj

−2n

n

i1

Qxi

β

lim

k→ ∞

1

n2βk

Dfnkx1, . . . , nkxn β

≤ lim

k→ ∞

1

n2βkϕ

nkx1, . . . , nkxn

0,

3.12

so the functionQis quadratic.

To prove the uniqueness of the quadratic function Qsubject to 3.4, let us assume that there exists a quadratic functionQ : X → Y which satisfies1.6and inequality3.4. Obviously, we obtain that

Qx n−2kQnkx, Qx n−2kQnkx 3.13

for allx∈X. Hence it follows from3.4that

_Qx−Qx

β

1

n2βk

Qnkx−Qnkx

β

≤ 2

n2βk

∞

l0

1

n2βl1ϕ

nlkx, . . . , nlkx

2

∞

lk

1

n2βl1ϕ

nlx, . . . , nlx, x∈X

3.14

Theorem 3.2. Assume that a functionf:X → Ysatisfies

Dfx1, . . . , xn_β≤ϕx1, . . . , xn 3.15

for allx1, . . . , xn ∈X andϕsatisfies condition3.2. Then there exists a unique quadratic function

Q:X → Ysatisfying

fx−Qx_β≤Φ2x, . . . , x 3.16

for allx∈X. The function Q is given by

Qx lim

k→ ∞n 2k_f

x nk

3.17

for allx∈X.

Proof. In this case, f0 0 since ∞_l01/2β_n2βl_{ϕ0, . . . ,0 < ∞ and so ϕ0, . . . ,0 0 by}

assumption. Replacingxbyx/nin3.6, we obtain

fx−n2f

x n

β ≤ 1

2βϕ

x n, . . . ,

x n

3.18

forx∈X.

Therefore we prove from inequality3.18that for any integersm, kwithm > k≥0

n2m_f

x nm

−n2k_f

x nk

β

n2βk_n2m−k_f

x nm

−f

x nk

β

≤n2βkm−k−1 l0

1 2βn

2βl_ϕ

x nlk1, . . . ,

x nlk1

m−1

lk

1 2βn

2βl_ϕ

x nl1, . . . ,

x nl1

3.19

for all x ∈ X. Since the right-hand side of 3.19 tends to zero as k → ∞, the sequence

{n2k_fx/nk_{}is a Cauchy sequence for allx∈X, and thus converges by the completeness of}

Y. DefineQ:X → Y by

Qx lim

k→ ∞n 2k_f

x nk

3.20

for allx∈X. Taking the limit in3.19withk0 asm → ∞, we obtain that

Replacingx1, . . . , xnin3.3byx1/nk, . . . , xn/nk, multiplying both sides byn2βk,

and then taking the limit ask → ∞in the resulting inequality, we have

2Q _n i1 xi

i /j

Qxi−xj

−2n

n

i1

Qxi 0 3.22

for allx1, . . . , xn∈X. Therefore the functionQis quadratic.

To prove the uniqueness, letQbe another quadratic function satisfying3.16. Then it is easy to see that the following identitiesQx n2k_Qx/nk_{andQx n}2k_Qx/nk_hold

for allx∈X. Thus we have

_Qx−Qx

βn2βk

Q x nk −Q x nk β ≤ 2 2β ∞ l0

n2βlkϕ

_x

nlk1, . . . ,

x nlk1

≤ 2

2β

∞

lk

n2βlϕ

x nl1, . . . ,

x nl1

3.23

for allx∈Xand allk∈N. Therefore lettingk → ∞, one hasQx−Qx 0 for allx∈X. This completes the proof.

In the following corollary, we have a stability result of1.6with difference operator

Df bounded by the sum of powers ofα-norms.

Corollary 3.3. LetX be anα-normed space with0 < α ≤ 1andY aβ-Banach space, respectively. Let{θi}n_i1 be real numbers withθi ≥ 0 for alli, and letε,{pi}ni1 be real numbers such that either

αmax{pi}<2β, ε≥0orαmin{pi}>2β, ε0. Assume that a functionf:X → Y satisfies the inequality

_{Dfx1, . . . , xn}

β≤ε n

i1

θixipi

α 3.24

for allx1, . . . , xn∈XandX\{0}ifpi<0. Then there exists a unique quadratic functionQ:X → Y

which satisfies the inequality

fx− nf0

2n1−Qx

β ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ε

2β_n2β₋₁ n

i1

θixpi

α

2β_n2β_−nαpi

ifαmaxpi

<2β, ε≥0,

n

i1

θixpi

α

2β_nαpi−_n2β

ifαminpi

>2β, ε0

for allx∈XandX\ {0}ifpi<0. The functionQis given by

Qx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

lim

k→ ∞

fnk_x

n2k , if α

maxpi

<2β, ε≥0

lim

k→ ∞n 2k_f

x nk

, if αminpi

>2β, ε0.

3.26

for allx∈X.

Proof. Letting ϕx1, . . . , xn : ε

_n

i1θixiαpi for all x1, . . . , xn ∈ X and then applying

Theorems3.1and3.2, we obtain easily the results.

We observe that iff0 0 andε0 inCorollary 3.3, then the stability result obtained by the fixed point method in Corollary 2.5 is somewhat different from the stability result obtained by direct method inCorollary 3.3. The stability result in Corollary 3.3 is sharper than that ofCorollary 2.5.

In the next corollary, we get a stability result of 1.6 with difference operator Df

bounded by the product of powers ofα-norms.

Corollary 3.4. LetX be anα-normed space with0 < α ≤ 1andY aβ-Banach space, respectively, and letθ,{pi}n_i1 be real numbers such thatθ ≥ 0andαp /2β, wherep : n_i1pi. Suppose that a

functionf:X → Y satisfies

_{Dfx1, . . . , xn}

β≤θ n

i1

xipi

α 3.27

for allx1, . . . , xn∈XandX\{0}ifpi<0. Then there exists a unique quadratic functionQ:X → Y

which satisfies the inequality

fx− n

2n1f0−Qx

β

≤ θxp

2β_n2β_−nαp 3.28

for allx∈X,and for allX\ {0}ifp <0, wheref0 0ifp >0.

Proof. We remark thatϕx1, . . . , xn:θni1xipαisatisfies condition3.1for the caseαp <2β

or condition3.2for the caseαp >2β. By Theorems3.1and3.2, we get the results.

We observe that if f0 0 in Corollary 3.4, then the stability result obtained by the fixed point method with contractive constants nαp_/n2β_{αp < 2β, n}2β_/nαp_{αp > 2β,}

respectively, coincides with the stability result3.28obtained by direct method.

Acknowledgment

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