Approximate Euler Lagrange quadratic mappings

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R E S E A R C H

Open Access

Approximate Euler-Lagrange quadratic mappings

Hark-Mahn Kim, Juri Lee

*

and Eunyoung Son

* Correspondence: [email protected] Department of Mathematics, Chungnam National University, 79 Daehangno, Yuseong-gu, Daejeon 305-764, Korea

Abstract

For any fixed integerkwithk ≠0, 1, we prove the Hyers-Ulam stability of an Euler-Lagrange-type quadratic functional equation

f(kx+y) +f(kx−y) =kf(x+y) +kf(x−y) + 2k(k−1)f(x)−2(k−1)f(y)

in normed spaces and in non-Archimedean normed spaces.

1 Introduction

The problem of stability of functional equations was originally stated by Ulam [1]. In 1941, Hyers [2] gave an affirmative answer to Ulam’s problem for the case of approxi-mate additive mappings on Banach spaces. In 1950, Aoki discussed the Hyers-Ulam stability theorem in [3]. His result was further generalized and rediscovered by Rassias [4] in 1978. The stability problem for functional equation has extensively been investi-gated by a number of mathematicians [5-9].

The quadratic functionf(x) =cx2satisfies the functional equation

f(x+y) +f(x−y) = 2f(x) + 2f(y) (1)

and therefore Equation (1) is called the quadratic functional equation. Every solution of Equation (1) is said to be a quadratic mapping. The Hyers-Ulam stability theorem for the quadratic functional equation (1) was by Skof [9] for the functionsf: E1 ®E2 whereE1 is a normed space andE2 is a Banach space. The result of Skof is still true if the relevant domain E1 is replaced by an Abelian group and this was dealt with by Cholewa [10]. Czerwik [11] proved the Hyers-Ulam stability of the quadratic functional equation (1). This result was further generalized by Rassias [12], Borelli and Forti [13]. During the last three decades, a number of papers and research monographs have been published on various generalizations and applications of the Hyers-Ulam stability of several functional equations, and there are many interesting results concerning this problem [14-20]. In particular, Rassias investigated the Hyers-Ulam stability for the relative Euler-Lagrange functional equation

f(ax+by) +f(bx−ay) = (a2+b2)[f(x) +f(y)] (2)

in [21-23].

In 2008, Ravi et al. [24] investigated the Hyers-Ulam stability of a quadratic func-tional equation

f(2x+y) +f(2x−y) = 2f(x+y) + 2f(x−y) + 4f(x)−2f(y). (3)

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In this article, we generalize the functional equation (3) to a more general form

f(kx+y) +f(kx−y) =kf(x+y) +kf(x−y) + 2k(k−1)f(x)−2(k−1)f(y) (4)

and investigate the Hyers-Ulam stability of the equation for any fixed integer kwith k ≠ 0, 1. As results, we improve the generalized stability results given in [24] in normed spaces and in non-Archimedean normed spaces.

2 General solution of (4)

First of all, if k= -1 in Equation (4), then it is easy to see that

f(−x+y) +f(−x−y) =−f(x+y)−f(x−y) + 4f(x) + 4f(y)

is equivalent to Equation (1), and so the solution of Equation (4) with k= -1 is a quadratic mapping. Thus, we consider general solutions of Equation (4) for any fixed integerkwith |k| > 1 in the following theorem. The following lemma can be found in [25-27].

Lemma 2.1. A mapping f: X ® Y between linear spaces satisfies the functional equation

f(2x+y) +f(2x−y) = 4[f(x+y) +f(x−y)] + 2[f(2x)−4f(x)]−6f(y)

if and only if fis quadratic and quartic.

Theorem 2.2. A mappingf: X ® Ybetween linear spaces satisfies the functional equation (4) with |k| > 1 if and only iffis quadratic.

Proof. Letfbe a solution of Equation (4). Lettingx=y= 0 in (4), we havef(0) = 0. Putting y= 0 in (4), we getf(kx)= k2f(x). Putting x= 0 in (4), we getf(-y)= f(y). Thus, the mappingfis even. Therefore, it suffices to prove that if a mappingfsatisfies Equa-tion (4) for any fixed integer kwith |k| > 1, thenfis quadratic. Now, replacingy byx +yin (4), we have

f((k+ 1)x+y) +f((k−1)x−y) =kf(2x+y) +kf(y) + 2k(k−1)f(x)−2(k−1)f(x+y) (5)

for all x,yÎX. Replacingyby -yin (5), we obtain

f((k+ 1)x−y) +f((k−1)x+y) =kf(2x−y) +kf(y) + 2k(k−1)f(x)−2(k−1)f(x−y) (6)

for all x, yÎX. Adding (5) to (6), we get

f((k+ 1)x+y) +f((k+ 1)x−y) +f((k−1)x+y) +f((k−1)x−y) =k[f(2x+y) +f(2x−y)]−2(k−1)[f(x+y) +f(x−y)] =−2(k−1)[f(x+y) +f(x−y)] + 4k(k−1)f(x) + 2kf(y)

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for all x, yÎX. From the substitutiony=kx+yin (4), we have

f(2kx+y) +f(y) =k[f((k+ 1)x+y) +f((k−1)x+y)]

+ 2k(k−1)f(x)−2(k−1)f(kx+y) (8)

for all x, yÎX. Replacingyby -yin (8), we get

f(2kx−y) +f(y) =k[f((k+ 1)x−y) +f((k−1)x−y)]

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for all x, yÎX. Adding (8) to (9), we get

f(2kx+y) + (2kx−y) =k[f((k+ 1)x+y) +f((k+ 1)x−y)] +k[f((k−1)x+y) +f((k−1)x−y)]

−2(k−1)[f(kx+y) +f(kx−y)] + 4k(k−1)f(x)−2f(y)

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for all x, yÎX. It follows from (10), by using (4) and (7), that

f(2kx+y) +f(2kx−y) =k2[f(2x+y) +f(2x−y)]

−4x(k−1)[f(x+y) +f(x−y)] + 8k(k−1)f(x) + 2(k−1)(3k−1)f(y)

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for all x, yÎX. If we replacexby 2xin (4), then we obtain that

f(2kx+y) +f(2kx−y) =k[f(2x+y) +f(2x−y)]

+ 2k(k−1)f(2x)−2(k−1)f(y) (12)

for all x,y Î X. Associating (11) with (12), we conclude that the mapping fsatisfies the equation

f(2x+y) +f(2x−y) = 4[f(x+y) +f(x−y)] + 2[f(2x)−4f(x)]−6f(y)

for all x, yÎ X. Therefore, it follows from Lemma 2.1 that fis quadratic because of the propertyf(kx) =k2f(x).

Conversely, if a mappingfis quadratic, then it is obvious thatfsatisfies (4).

3 Hyers-Ulam stability of (4) in banach spaces

In this section, let Xbe a normed space andYa Banach space. We will investigate the Hyers-Ulam stability problem for the functional equation (4). For notational conveni-ence, we define an operatorDkf(x, y) as

Dkf(x,y) :=f(kx+y) +f(kx−y)−kf(x+y)−kf(x−y)

−2k(k−1)f(x) + 2(k−1)f(y)

for all x, yÎX, wherekis a fixed integer with |k| > 1.

Theorem 3.1. Letψ:X2 ®[0,∞) be a function such that ∞

i=0

ψ(ki_x,_ki_y)

k2i <∞ (13)

for all x, yÎX. If a mappingf:X ®Ysatisfies the inequality

Dkf(x,y)≤ψ(x,y) (14)

for all x, yÎ X, then there exists a unique quadratic mapping Q1 : X ® Y which satisfies Equation (4) and the inequality

f(x)− f(0)

k+ 1 −Q1(x)

≤ 1

2k2

∞

i=0

ψ(ki_{x, 0)}

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for all xÎX, wheref(0)≤ ψ(0, 0)

2k(k−1). The mappingQ1is defined by

Q1(x) = lim

n→∞

f(kn_x)

k2n (16)

for all xÎX.

Proof. Lettingx=y:= 0 in (14), we get

f(0)≤ ψ(0, 0) 2k(k−1).

Putting y:= 0 in (14) and dividing by 2k2, we obtain

f(kx)_k2 −f(x) + k−1

k2 f(0)

≤ ψ(x, 0)

2k2 (17)

for all xÎX. Setting f¯(x) :=f(x)− f(0)

k+ 1, we lead to a functional inequality

¯

f(kx) k2 − ¯f(x)

≤

ψ(x, 0)

2k2 (18)

for allx Î X. Using the induction argument on a positive integern we may obtain that

f¯(x)−

¯

f(kn_x)

k2n

≤

1 2k2

n−1

i−0

ψ(ki_{x, 0)}

k2i (19)

for all xÎX. Now, it follows from (19) that form>n> 0,

¯

f(km_x)

k2m −

¯

f(kn_x)

k2n

= ¯

f(km−n+n_x)

k2(m−n+n) −

¯

f(kn_x)

k2n

= 1 k2n

¯f(knx)−

¯

f(km−n_kn_x)

k2(m−n)

≤ 1

2k2

m−n−1

i=0

ψ(ki+n_{x, 0)}

k2(i+n)

(20)

for allxÎ X. Since the right-hand side of the inequality (20) tends to 0 asn® ∞, a

sequence

_¯ f(kn_x)

k2n

is Cauchy. Therefore, we may define a mappingQ1 :X®Yas

Q1(x) = lim

n→∞ ¯

f(kn_x)

k2n = limn→∞

f(kn_x)

k2n

for all xÎX. Lettingn® ∞in (19), we lead to the approximation (15).

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1 k2nf(k

n_(kx₊_{y)) +}_f_(kn_(kx₋_y))₋_kf_(kn_(x₊_y))₋_kf_(kn_(x₋_y))

−2k(k−1)f(knx) + 2(k−1)f(xny)≤ 1 k2nψ(k

n_x,_kn_y)

for allx, yÎX. Taking the limit asn®∞, we see from (13) and (16) that the map-pingQ1satisfies Equation (4) and so it is quadratic by Theorem 2.2.

To prove the uniqueness of the quadratic mapping Q1 satisfying the inequality (15), let us assume that there exists a quadratic mappingQ₁:X→Y which satisfies the inequality (15). Then, we haveQ1(knx) =k2nQ1(x) andQ₁(knx) =k2nQ₁(x)for allxÎ X and allnÎN. Hence, it follows from (15) that

Q1(x)−Q1(x)= 1 k2nQ1(k

n_x)₋_Q

1(knx)

≤ 1

k2n

Q1(knx)− ¯f(knx)+f¯(knx)−Q1(knx)

≤ 1

k2

∞

i=n

ψ(kix, 0) k2i

which tends to zero as n®∞. This completes the proof of the theorem.

The following theorem is an alternative stability result concerning the stability of functional equation (4).

i=1 k2iψ

_x ki,

y ki

<∞ (21)

Dkf(x,y)≤ψ(x,y) (22)

f(x)−Q2(x)≤ 1 2k2

∞

i=1 k2iψ(x

ki, 0) (23)

for all xÎX. The mapping Q2 is defined by

Q2(x) = lim

n→∞k

2n_f₍x

kn) (24)

for all xÎX.

Corollary 3.3. Letε1≥0,ε2≥0 andp, qbe real numbers such that either 0 <p, q< 2 or p, q> 2. If a mappingf:X® Ysatisfies the inequality

Dkf(x,y)≤ ε1xp+ε2y

q

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⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

f(x)−Q1(x)≤ ε1 xp

2(k2₋_kp₎, if 0<p,q<2;

2(kp₋_k2₎, if p,q>2.

for all xÎ X. Furthermore, for each fixedxÎ X if f(tx) is continuous for alltÎ R, then f(tx)= t2f(x) for alltÎ R.

Proof. Taking ψ(x, y) = ε1║x║p +ε2║y║q and applying Theorems 3.1 and 3.2, we obtain the desired approximations.

The following is a simple example that the quadratic functional equationDkf(x, y)= 0,k≥2 is not stable forp= 2 =qin Corollary 3.3.

Example 3.4. Letj:R ®Rbe defined by

φ(x) =

μx2if |x|<1,

μ otherwise.

whereμ> 0 is a positive constant, and definef:R ®Rby

f(x) =

∞

i=0

φ(2i_x)

4i , for allx ∈ R.

Then, fsatisfies the functional inequality

f(2x+y) +f(2x−y)x−2f(x+y)−2f(x−y)−4f(x) + 2f(y)

≤44μ(|x|2+y2) (25)

for all x, yÎ R, but there do not exist a quadratic functionQ: R ®Rand a con-stantb> 0 such that

f(x)−Q(x)≤β|x|2for all x ∈ R. (26)

Proof. It is easy to see thatfis bounded by4μ

3onR. If|x|

2₊_y2_≥ 1

4or 0, then the

left side of (25) is less than 16μ, and thus (25) is true. Now suppose that

0<|x|2+y2< 1

4. Then there exists a positive integerksuch that 1

4k+2 ≤ |x|

2₊_y2_< 1

4k+1, (27)

so that 44|x|2< 1 4, 4

k_y2_< 1

4 and 2

k - 1_{(2x ± y), 2}k - 1_{(x ± y), 2}k - 1_{x, 2}k - 1_y all

belong to the interval (-1, 1). Hence, fori= 0, 1,...,k- 1,

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Therefore, it follows from the definition of fand the inequality (27) that

f(2x+y) +f(2x−y)−2f(x+y)−2f(x−y)−4f(x) + 2f(y)

≤∞

i=0 1 4iφ(2

i_(2x₊_{y)) +}_φ₍₂i_(2x₋_y))

−2φ(2i(x+y))−2φ(2i(2x−y))−4φ(2ix) + 2φ(2iy)

≤∞

i=k

1

4i12μ= 16μ4−

k_≤₄4_μ₍_|_x_|2₊_y2 )

(28)

for all x, yÎRwith0<|x|2+y2< 1

4. Thusfsatisfies (25) for allx, yÎ R.

We claim that the quadratic functional equationD2f(x, y)=0 is not stable forp= 2 =qin Corollary 3.3. Suppose on the contrary that there exist a quadratic functionQ:

R ® Rand a constant b> 0 satisfying (26). Sincefis bounded and continuous for all x ÎR,Qis bounded on any open interval containing the origin and continuous at the origin. Therefore, Qmust have the form Q(x)=hx2 for anyx inR. Thus, we obtain that

f(x)≤(β+|η|)|x|2 for allx ∈ R. (29)

However, we can choose a positive integerm with mμ>b+ |h|. If x ∈

0, 1 2m−1

,

then 2ixÎ(0, 1) for alli= 0, 1,...,m- 1, and for thisxwe get

f(x) =

∞

i=0

φ(2ix) 4n ≥

m−1

i=0

μ(2ix)2 4i =mμx

2_>₍_β₊_|η|_)x2_,

which contradicts (29). Therefore, the quadratic functional equationD2f(x, y) = 0 is not stable if p= 2= qin Corollary 3.3.

Corollary 3.5. Let εbe a nonnegative real number. If a mappingf:X ® Ysatisfies the inequality

Dkf(x,y)≤ε

f(x)− f(0)

k+ 1−Q1(x)

≤ ε

2(k2₋₁₎

for allx ÎX. Furthermore, for each fixed xÎ Xif f(tx) is continuous for all tÎR, then f(tx)= t2f(x) for alltÎ R.

Proof. Takingψ(x, y) =εand applying Theorem 3.1, we lead to the approximation. In the last part, we consider a singular case k= -1, which is not investigated in The-orems 3.1 and 3.2 concerning the stability of the functional equation (4).

i=0

ψ(2i_{x, 2}i_y)

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D₋1f(x,y)≤ψ(x,y) (31)

for all x, yÎ X, then there exists a unique quadratic mapping Q1 : X ® Y which satisfies the equationD-1Q1(x, y)=0 and the inequality

f(x)−2f(0)

3 −Q1(x) ≤ 1

8

∞

i=0

ψ(2i_{x, 2}i_x)

4i +

1 4

∞

i=0

ψ(2i+1_{x, 0)}

4i+1 (32)

4 . The mappingQ1 is defined by

Q1(x) = lim

n→∞

f(2nx)

4n (33)

for all xÎX.

Proof. Lettingx=y:= 0 in (31), we getf(0)≤ ψ(0, 0)

4 . Puttingy:= 0 in (31) and

dividing by 2, we obtain

f(−x)−f(x)−2f(0)≤ψ(x, 0) 2

for all xÎX. Setting y:=xin (31), one has

f(−2x) +f(2x)−8f(x) + 2f(0)≤ψ(x,x)

for all x Î X. Combining two inequalities above and then letting

¯

f(x) :=f(x)− 2f(0)

3 , we lead to a functional inequality

¯

f(2x) 4 − ¯f(x)

≤

1

8ψ(x,x) + 1

42ψ(2x, 0) (34)

for all xÎ X. Using the induction argument on a positive integern, we may obtain that

f¯(x)−

¯

f(2nx) 4n

≤

1 8

n−1

i=0

ψ(2ix, 2ix)

4i +

1 4

n−1

i=0

ψ(2i+1x, 0)

4i+1 (35)

for all xÎX. The remaining proof of this theorem follows similarly from the corre-sponding part of Theorem 3.1.

The following theorem is an alternative stability result concerning the stability of the functional equation (4) withk= -1.

i=1 4iψ(x

2i,

y

2i)<∞ (36)

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for all x, yÎ X, then there exists a unique quadratic mapping Q2 : X ® Y which satisfies the equationD-1Q2(x, y)=0 and the inequality

f(x)−Q2(x)≤ 1 8

∞

i=1 4iψ

_x 2i,

x 2i

+1

4

∞

i=1 4i−1ψ

_x

2i−1, 0

(38)

for all xÎX. The mapping Q2 is defined by

Q2(x) = lim

n→∞4

n_f₍x

2n) (39)

for all xÎX.

Corollary 3.8. Letε1 ≥0,ε2 ≥0 andp, qbe real numbers such that either 0 <p, q < 2 or p, q> 2. If a mappingf:X® Ysatisfies the inequality

D₋1f(x,y)≤ε1xp+ε2yq

for all x, yÎ X, then there exists a unique quadratic mappingQi: X® Y(i = 1,2) which satisfies the equationD-1Qi(x, y) = 0 and inequality

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

f(x)−Q1(x)≤ ε1

(2 + 2p₎_xp

4(22₋₂p₎ +

ε2xq

2(22₋₂q₎, if 0<p,q<2;

f(x)−Q2(x)≤ ε1

(2 + 2p)xp 4(2p₋₂2₎ +

ε2xq

2(2q₋₂2₎, if p,q>2.

for allx ÎX. Furthermore, for each fixed xÎ Xif f(tx) is continuous for all tÎR, then f(tx)=t2f(x) for alltÎ R.

4 Hyers-Ulam stability of (4) in non-archimedean spaces

In this section, letX be a vector space andYa complete non-Archimedean space. We recall that a fieldK, equipped with a function (non-Archimedean absolute value, valua-tion) | · | from Kinto [0,∞), is called a non-Archimedean field if the function | · |:K ® [0,∞) satisfies the following conditions:

(1) |r| = 0 if and only ifr = 0; (2) |rs| = |r||s|;

(3) the strong triangle inequality, namely, |r+s|≤max{|r|, |s|} for allr, sÎK.

Clearly, |1| = 1 = | - 1| and |n|≤1 for all nonzero integer n.

Let Ybe a vector space over the non-Archimedean field Kwith a trivial non-Archimedean valuation | · |. A function ║·║:Y®[0, ∞) is called a non-Archimedean norm (valuation) if it satisfies the following conditions:

(1)║x║= 0 if and only ifx= 0;

(2) |rx| = |r|║x║for allxÎYand allrÎK; (3) the strong triangle inequality, namely,

x+y≤maxx,y

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In this case, the pair (Y,║ ·║) is called a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent. It follows from the strong triangle inequality that

xn−xm ≤maxxj+1−xj:m≤j<n−1

for allxn,xmÎ Yand allm,nÎ Nwithn>m. Therefore, a sequence {xn} is a Cauchy sequence in non-Archimedean space (Y,║ ·║) if and only if the sequence {xn+1-xn} converges to zero in the space (Y,║·║). Now, we will investigate the generalized the Hyers-Ulam stability problem for the functional equation (4) in a complete non-Archi-medean spaceY.

Theorem 4.1. Letψ:X2 ®[0,∞) be a function such that

1(x) := lim

n→∞max

ψ(kix, 0)

|k|2i : 0≤i<n

<∞,

lim

n→∞

ψ(kn_x,_kn_y)

|k|2n = 0

(40)

Dkf(x,y)≤ψ(x,y) (41)

for all x, y Î X, then there exists a quadratic mapping Q1 : X ® Ywhich satisfies Equation (4) and the inequality

f(x)−Q1(x)≤ 1

|2| |k|21(x) (42)

2k(k−1). The mapping Q1 is defined by

Q1(x) = lim

n→∞

f(knx)

k2n (43)

for all xÎX. Moreover, if

lim

l→∞

1(klx)

|k|2l = liml→∞nlim→∞max

_ψ (kjx, 0)

|k|2j :l≤j<l+n

= 0

for all xÎX, then Q1 is a unique quadratic mapping satisfying (42).

Proof. Lettingx=y:= 0 in (41), we getf(0) = 0 becauseψ(0,0) = 0 by the condition (40). Puttingy:= 0 in (41) and dividing by 2|k|2, we obtain

f(kx)_k2 −f(x)

≤ ψ(x, 0)

2|k|2 (44)

for allx ÎX, where |k| ≤1 is a non-Archimedean valuation. Replacing xby knxin (44) and dividing by |k|2n_,

f(k_k2nn+1+2x)− f(kn_x)

k2n

≤ ψ(knx, 0)

2|k|2n+2 (45)

for allxÎ X. Since the right-hand side of the inequality (45) tends to 0 asn® ∞, a

sequence_{f(k

n_x)

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may define a mappingQ1:X ®Yas

Q1(x) = lim

n→∞

f(kn_x)

k2n

for all xÎ X. Using the induction argument and the strong triangle inequality, we may obtain that

f(x)−f(k

n_x)

k2n

≤ 1

2|k|2max

ψ(ki_{x, 0)}

|k|2i : 0≤i<n

(46)

for all xÎX. Lettingn®∞in (46), we lead to the approximation (42).

Next, we have to show thatQ1 satisfies Equation (4). Replacing x, y by knx, kny in (41) and dividing by |k|2n

, it then follows that

1

|k|2nf(k

n_(kx₊_{y)) +}_f_(kn_(kx₋_y))₋_kf_(kn_(x₊_y))₋_kf(kn_(x₋_y))

−2k(k−1)f(knx) + 2(k−1)f(kny)≤ 1

|k|2nψ(k

n_x,_kn_y)

for all x, yÎX. Taking the limit asn®∞, we see from (40) and (43) that

Q1(kx+y) +Q1(kx−y)

=kQ1(x+y) +kQ1(x−y) + 2k(k−1)Q1(x)−2(k−1)Q1(y)

for allx, yÎX. Therefore, the mapping Q1 satisfies Equation (4) and so it is quadra-tic by Theorem 2.2.

Moreover, to prove the uniqueness of the quadratic mapping Q1 satisfying the inequality (42), let us assume that there exists a quadratic mapping Q₁:=X→Y

which satisfies the inequality (42). Then, we have Q1(klx) = k2lQ1(x) and

Q₁(kl_{x) =}_k2l_Q

1(x)for all xÎXand all lÎN. Hence, it follows from (42) that for allx

Î X

Q1(x)−Q1(x)= 1

|k|2l

Q1(klx)−Q1(klx)

≤ 1 |k|2lmax

_Q

1(klx)−f(klx),f(klx)−Q1(klx)

≤ 1

|k|2nlim→∞max

ψ(kl+i_{x, 0)}

|k|2(l+1) : 0≤i<n

= 1

|k|2nlim→∞max

_ψ (kj_{x, 0)}

|k|2j :l≤j<l+n

= 1

|k|2

1(klx)

|k|2l , ∀l ∈ N,

which tends to zero as l®∞. This completes the proof of the theorem.

Theorem 4.2. Letψ:X2 ®[0,∞) be a function such that

2(x) := lim

n→∞max

|k|2iψ(x

ki, 0) : 1≤i≤n

<∞

lim

n→∞|k|

2n_ψ

(x kn,

y kn) = 0

(12)

Dkf(x,y)≤ψ(x,y) (48)

for all x, y Î X, then there exists a quadratic mapping Q2 : X ® Ywhich satisfies Equation (4) and the inequality

f(x)− f(0)

k+ 1 −Q2(x)

≤ 1

|2| |k|22(x) (49)

2k(k−1). The mapping Q2 is defined by

Q2(x) = lim

n→∞k

2n

f(x

kn) (50)

for all xÎ X. Moreover, ifliml→∞|k|2l2( x

kl) = 0for all xÎ X, thenQ2 is a unique

quadratic mapping satisfying (49).

Proof. Lettingx=y:= 0 in (48), we get

f(0)≤ ψ(0, 0)

|2k(k−1)|,

where |2k(k- 1)|≤1 is a non-Archimedean valuation. Puttingy = 0 in (48), one has

|2|f(kx)−k2f(x) + (k−1)f(0)≤ψ(x, 0),

i.e., ¯f(x)−k2¯f _x

k

_≤ 1

|2| |k|2|k| 2_ψx

k, 0

₍₅₁₎

for allxÎ X, where f¯(x) =f(x)− f(0)

k+ 1. Replacingxby x

knin (51) and multiplying |k|

2n

, we have

k2nfx

kn

−k2n+2f x kn+1

_≤ 1

|2||k|

2n+2_ψ x kn+1, 0

(52)

for allx Î X. Since the right-hand side of the inequality (52) tends to 0 asn ®∞,

the sequence{k2nf(x

kn)}is Cauchy. Therefore, one may define a mappingQ2 : X® Y

by

Q2(x) = lim

n→∞k

2n_f₍x

kn)

for all xÎX. Using induction on positive integersn, we obtain that

f(x)−k2nf(x kn)

≤ 1

|2| |k|2max

|k|2iψ(x

ki, 0) : 1≤i≤n

(53)

for all xÎX. Lettingn®∞in (53), we arrive at the estimation (49). The remaining assertion is similar to that of Theorem 4.1.

Corollary 4.3. Letε1 ≥0,ε2 ≥0 andp, qbe real numbers such that either 0 <p, q < 2 or p, q> 2. If a mappingf:X® Ysatisfies the inequality

Dkf(x,y)≤ε1xp+ε2y

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for all x, yÎ X, then there exists a unique quadratic mappingQi: X® Y(i = 1,2) which satisfies (4) and inequality

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

|2| |k|2 ≤

ε1xp

|2| |k|p, if p,q>2;

f(x)− f(0)

k+ 1−Q2(x)

≤ ε1xp

|2| |k|P ≤

ε1xp

|2| |k|2, if 0<p,q<2.

for all xÎX.

Proof. Takingψ(x, y) = ε1║x║p+ ε2║y║q and applying Theorems 4.1 and 4.2, we obtain the desired approximations.

Corollary 4.4. Let εbe a nonnegative real number. If a mappingf:X ® Ysatisfies the inequality

Dkf(x,y)≤ε

f(x)− f(0)

k+ 1 −Q1(x) ≤ _|ε

2|

for all xÎX.

We remark that stability results of the Euler-Lagrange-type quadratic equation (4) in normed spaces are very different from those of Equation (4) in non-Archimedean normed spaces, of which stability results in non-Archimedean normed spaces maybe come from the opposite direction to stability results in normed spaces in view of Cor-ollaries 4.3 and 4.4.

Acknowledgements

The authors would like to thank the referees and the editors for carefully reading this article and for their valuable comments. This study was supported by the Basic Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (No. 2011-0002614).

Authors_’contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 18 April 2011 Accepted: 7 March 2012 Published: 7 March 2012

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doi:10.1186/1029-242X-2012-58

Cite this article as:Kimet al.:Approximate Euler-Lagrange quadratic mappings.Journal of Inequalities and

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