# On the generalized Hyers-Ulam stability of the generalized polynomial function of degree 3.

1. Introduction

Throughout this paper, let V be a vector space and Y a Banach space. Let n be a positive integer. For a given mapping f : V [right arrow] Y, define mappings [C.sub.n]f, [D.sub.n]f : V x V [right arrow] Y by

[E.sub.n]f(x,y) := [n.summation over (i = 0)][.sub.n][C.sub.i][(-1)].sup.n-i]f(ix + y),

[D.sub.n]f(x,y) := [n.summation over (i = 0)][.sub.n][C.sub.i][(-1)].sup.n-i]f(ix + y) - n!f(x)

for all x, y [member of] X, where [.sub.n][C.sub.i] = [n!/i!(n-i)!]. A mapping f : X [right arrow] Y is called a generalized polynomial(monomial, respectively) function of degree n [member of] N if f satisfies the functional equation [E.sub.n+1]f(x, y) = 0([D.sub.n]f(x, y) = 0, respectively). The functional equation [E.sub.n+1]f(x, y) = 0([D.sub.n]f(x, y) = 0, respectively) is called a generalized polynomial(monomial, respectively) functional equation of degree n [member of] N. In particular, a mapping f : V [right arrow] Y is called an additive (quadratic, cubic, quartic, respectively) mapping if f satisfies the functional equation [D.sub.1]f = 0 ([D.sub.2]f = 0, [D.sub.3]f = 0, [D.sub.4]f = 0, respectively). The functional equation [D.sub.1]f = 0 ([D.sub.2]f = 0, [D.sub.3]f = 0, [D.sub.4]f = 0, respectively) is called a Cauchy equation(quadratic functional equation, cubic functional equation, quartic functional equation, respectively). The functions f : R [right arrow] R defined by f(x) = [ax.sup.n] and f(x) = [[SIGMA].sub.i=0.sup.n-1][a.sub.i][x.sup.i] satisfy the functional equation [D.sub.n]f = 0 and [E.sub.n]f = 0 respectively, where a, [a.sub.i] are real constants and R is the set of real numbers.

If we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality must be close to the solutions of the given equation? If the answer is affirmative, we would say that a given functional equation is stable.

In 1941, D.H.Hyers (8) proved the stability of Cauchy equation [D.sub.1]f = 0 and in 1978, Th.M.Rassias (19) gave a significant generalization of the Hyers' result. Th.M.Rassias (20) during the 27th International Symposium on Functional Equations, that took place in Bielsko-Biala, Poland, in 1990, asked the question whether such a theorem can also be proved for a more general setting. Z.Gadja (6) following Th.M.Rassias's approach (19) gave an affirmative solution to the question. Recently, P.Gavruta (7) obtained a further generalization of Rassias' theorem, the so-called generalized Hyers-Ulam-Rassias stability(See also (4),(5),(9-11),(16-18)).

A stability problem for the quadratic functional equation [D.sub.2]f = 0 was proved by F.Skof (21) for a function f : X [right arrow] Y, where X is a normed space. P.W. Cholewa (2) noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. S.Czerwik (3) proved the Hyers-Ulam-Rassias stability of the quadratic functional equation.

J. C. Parnami, H. L. Vasudeva (14) and J.M. Rassias (15) investigated the stability of the functional equation [D.sub.3]f = 0. Also, Jun and Kim (12) proved the stability of the functional equation [E.sub.4]f = 0 under the approximately cubic condition and Baker (1) proved the stability of the functional equation [E.sub.n]f = 0.

In this paper, I solve the general solution of [E.sub.4]f = 0 and prove the generalized Hyers-Ulam stability of the functional equation [E.sub.4]f = 0 on the punctured domain V \ {0}without the approximately cubic condition.

2. General Solution of [E.sub.4]f = 0

In this section I establish the general solution of [E.sub.4]f = 0. First I obtain the general solution for the odd cases. Throughout this section, let V and W be vector spaces.

Theorem 2.1 Suppose that the odd function f : V [right arrow] W satisfies

[E.sub.4]f(x,y) = f(4x + y)-4f(3x + y) + 6f(2x + y)-4f(x + y) + f(y) = 0

(2.1) for all x,y [member of] V \ {0} and

f(2x) = 2f(x)

for all x [member of] V. Then f is an additive function.

Proof. Note that f(0) = 0 and f(x) + f(-x) = [-[E.sub.4]f(x,-2x)/2] = 0 for all x [member of] V. From (2.1) and f(2x) = 2f(x), we have

[D.sub.1]f(x,y) = [1/132](-28[E.sub.4]f(x,y - 2x)-7[E.sub.4]f(y,2x - 2y)-16[E.sub.4]f(x,y - 3x) + 4[E.sub.4]f(y - x,4x - 2y)-28[E.sub.4]f(y,x - 2y) -7[E.sub.4]f(x,2y - 2x)-16[E.sub.4]f(y,x - 3y) + 4[E.sub.4]f(x - y,4y - 2x)) = 0

for all x, y [member of] V with x [not equal to] 0, y, 2y, 3y and y [not equal to] 0, x, 2x, 3x. Since f(2x) = 2f(x) and f(3x) = [E.sub.4]f(x,-x) + 3f(x) = 3f(x), f is an additive function.

Theorem 2.2 Suppose that the odd function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0} and

f(2x) = 8f(x)

for all x [member of] V. Then f is a cubic function.

Proof. Note that f(0) = 0 and f(x) + f(-x) = [-[E.sub.4]f(x,-2x)/2] = 0 for all x [member of] V. From (2.1) and f(2x) = 8f(x), we easily get the equality

[D.sub.3]f(x,y) = [E.sub.4]f(x,y - x)-[[E.sub.4]f(y + x,-2y)/4] = 0

for all x, y [member of] V \ {0} with x [not equal to] y,-y. Since f(2x) = 8f(x) and f(3x) = [E.sub.4]f(x,-x) + 27f(x) = 27f(x), f is a cubic function.

Theorem 2.3 Suppose that the odd function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0}. Then there exist a cubic function C : V [right arrow] W and an additive function A : V [right arrow] W such that

f(x) = C(x) + A(x)

for all x [member of] V, where

C(x) = [-1/3] [f(x) - [1/2] f(2x)] = [4/3] [f(x) -2f ([1/2]x)]

A(x) = [4/3] [f(x) - [1/8] f(2x)] = [-1/3] [f(x) -8f ([1/2]x)].

Proof. Since

f(4x)-10f(2x) + 16f(x) = [1/4] (11[E.sub.4]f(x,-x) + [E.sub.4]f(2x,-3x)-[E.sub.4](x,x)) = 0

for all x [member of] V \ {0}, we have

f(x) = C(x) + A(x), C(2x) = 8C(x), A(2x) = 2A(x)

for all x [member of] V, where

C(x): = [-1/3] [f(x) - [1/2] f(2x)] and A(x): = [4/3] [f(x)- [1/8] f(2x)].

By Theorem 2.2 and the equalities

[E.sub.4]C(x,y) = [-1/6](2[E.sub.4]f(x,y) - [E.sub.4]f(2x,2y)) = 0, [E.sub.4]A(x,y) = [1/6](8[E.sub.4]f(x,y)-[E.sub.4]f(2x,2y)) = 0

for all x, y [member of] V \ {0}, C is a cubic function and A is an additive function.

In the following theorem we obtain the general solution for the even case.

Theorem 2.4

Suppose that the even function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0} and

f(2x) = 4f(x)

for all x [member of] V. Then f is a quadratic function.

Proof. Note that f(0) = 0. From (2.1) and f(2x) = 4f(x), we get the equality

[D.sub.2]f(x,y) = -[1/12](4[E.sub.4]f(x,y-2x) + [E.sub.4]f(y,2x - 2y)) = f(x + y) + f(x-y)-2f(x)-2f(y) = 0

for all x, y [member of] V \ {0} with y [not equal to] x, 2x. Using 4f(x) = f(2x) and f(3x) + f(x) = [E.sub.4]f(x, -x) + 10f(x) = 10f(x) for all x [member of] V, we get

[D.sub.2]f(x,y) = 0

for all x, y [member of] V.

Theorem 2.5 Suppose that the even function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0}. Then f - f(0) is a quadratic function.

Proof. Let g = f - f(0). Then g satisfies (2.1) and g(0) = 0. Since g(2x) = [[E.sub.4]g(x,-2x)/2] + 4g(x) = 4g(x) for all x [member of] V {0}, by Theorem 2.4, g is a quadratic function.

Now I establish the general solution of [E.sub.4]f = 0.

Theorem 2.6 Suppose that the function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0}. Then there exist a cubic function C : V [right arrow] W, a quadratic function Q : V [right arrow] W, and an additive function A : V [right arrow] W such that

f(x) = C(x) + Q(x) + A(x) + f(0)

for all x [member of] V. The functions C, Q, A : V [right arrow] W are given by

C(x): = [-1/12](2f(x) - 2f(-x) - f(2x) + f(-2x))

Q(x): = [[f(x) + f(-x)]/2] - f(0)

A(x): = [1/12](8f(x) - 8f(-x) - f(2x) + f(-2x))

for all x [member of] V.

Proof. Since f(x) = [[f(x) - f(-x)]/2[ + [[f(x) + f(-x)]/2], we can apply Theorem 2.3 and 2.4

3. Stability of the Equation [E.sub.4]f = 0

The following lemma is seen in (13).

Lemma 3.1. Let a be a positive real number and [PHI] : X \ {0} [right arrow] [0, [infinity]) a map. Suppose that the function f : X [right arrow] Y satisfies the inequality

||f(x) - [[f(2x)]/a]|| [less than or equal to] [[PHI](x)/a] and f(0) = 0.

1. If [[SIGMA].sub.l=0.sup.[infinity]][1/[a.sup.l+1]][PHI]([2.sup.l]x) < [infinity] for all x [member of] X \ {0}, then there exists a unique function F : X [right arrow] Y satisfying

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (l=0)][1/[a.sup.[l+1]][PHI]([2.sup.l]x)

for all x [member of] X \ {0} and F is given by F(x) = [lim.sub.n[right arrow][infinity]][f([2.sup.n]x)]/[a.sup.n]] for all x [member of] X.

2. If [[SIGMA].sub.l=0.sup.[infinity]][a.sup.l][PHI](x/[2.sup.l+1]) < [infinity] for all x [member of] X \ {0}, then there exists a unique function F : X [right arrow] Y satisfying

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (l=0)][a.sup.l][PHI](x/[2.sup.l+1]) < [infinity]

for all x [member of] X \ {0} and F is given by F(x) = [lim.sub.n[right arrow][infinity]][a.sup.n]f(x/[2.sup.n]) for all x [member of] X.

Theorem 3.2 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the condition

[[infinity].summation over (i=0)][[phi]([2.sup.i]x, [2.sup.i]y)/[2.sup.i]] < [infinity] (3.1)

If a function f : V [right arrow] Y satisfies

||[E.sub.4]f(x, y)|| [less than or equal to][phi](x, y) (3.2)

for all x, y [member of] V \ {0}, then there exists a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j=0)]([[psi]([2.sup.j]x)/3 * [2.sup.j-2]] + [[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[phi]([2.sup.j]x, -[2.sup.j+1]x)/[2.sup.2j+3]]) (3.3)

for all x, y [member of] V \ {0}, where

[psi](x) = [1/128](11[phi](x, -x) + [phi](2x, -3x) + [phi](x, x) + 11[phi](-x, x) + [phi](-2x, 3x) + [phi](-x, -x))

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Proof.Note that if [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) satisfies the condition (3.1) then [phi] satisfies the condition [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x,[2.sup.i]y)/[4.sup.i]] < [infinity] and [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x,[2.sup.i]y)/[8.sup.i]] < [infinity]. From (3.2), we get the inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0}. By Lemma 3.1, there exist functions [C.sub.0], [A.sub.0], Q : V [right arrow] Y defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V and the functions [C.sub.0], [A.sub.0], Q satisfy the inequalities

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/4] - [C.sub.0](x)|| [less than or equal to] [[infinity].summation over (j=0)][[psi]([2.sup.j]x)/[8.sup.j]], (3.4)

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/16] - [A.sub.0](x)|| [less than or equal to] [[infinity].summation over (j=0)][[psi]([2.sup.j]x])/[2.sup.j]], (3.5)

||[[f(x) + f(-x)]/2] - f(0) - Q(x)|| [less than or equal to] [[infinity].summation over (j=0)][[phi]([2.sup.j]x, -[2.sup.j+1]x)/[2.sup.2j+3]] (3.6)

for all x [member of] V \ {0}. From (3.1) and (3.2), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] V \ {0}. Since [C.sub.0](2x) = [8C.sub.0](x)([A.sub.0](2x) = [2A.sub.0](x) and Q(2x) = 4Q(x), respectively), [C.sub.0] is a cubic function ([A.sub.0] is an additive function and Q is a quadratic function, respectively) by Theorem 2.2(Theorem 2.1 and Theorem 2.4, respectively). From (3.4), (3.5), (3.6) and the inequality

||f(x) - F(x)|| [less than or equal to] [1/3]||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/4] - [C.sub.0](x)|| + [4/3]||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/16] - [A.sub.0](x)|| + ||[[f(x) + f(-x)]/2] - f(0) - Q(x)|| (3.7)

for all x [member of] V \ {0}, we get the inequality (3.3), where F = -[[C.sub.0](x)/3] + Q(x) + [[4A.sub.0]/3] + f(0). Now, let F' be another generalized polynomial function of degree 3 satisfying (3.3) with F'(0) = f(0). Then there are cubic functions C, C' : V [right arrow] Y, quadratic functions Q, Q' : V [right arrow] Y and additive functions A, A' : V [right arrow] Y such that F(x) = C(x) + Q(x) + A(x) + f(0) and F'(x) = C'(x) + Q'(x) + A'(x) + f(0). Since C, C' : V [right arrow] Y are cubic functions, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0} and n [member of] N. As n [right arrow] [infinity], we may conclude that C(x) = C'(x) for all x, y [member of] V. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0} and n [member of] N. As n [right arrow] [infinity], we may conclude that Q(x) = Q'(x) for all x, y [member of] V. Similarly, we get A(x) = A'(x) for all x, y [member of] V as we desired.

By the similar method in the proof of Theorem 3.2, I can prove the following theorem.

Theorem 3.3 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the condition

[[infinity].summation over (i=0)][8.sup.i][phi]([x/[2.sup.i]], [y/[2.sup.i]]) < [infinity] (3.8)

for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies (3.2) for all x, y [member of] V \ {0}, then there exists a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j=1)]([[[8.sup.j] + [2.sup.j+2]]/3] [psi](x/[2.sup.j]) + [2.sup.2j-3][phi]([x/[2.sup.j]], -[x/[2.sup.j-1])) (3.9)

for all x [member of] V \ {0}, where

[psi](x) = [1/128](11[phi](x, -x) + [phi](2x, -3x) + [phi](x, x) + 11[phi](-x, x) + [phi](-2x, 3x) + [phi](-x, -x))

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Proof. Note that if [phi] : V \ {0} x V \ {0} [right arrow][0, [infinity]) satisfies the condition (3.8) then [phi] satisfies the condition [[SIGMA].sub.i=0.sup.[infinity]] [4.sup.i][phi]([x/[2.sup.i]], [y/[2.sup.i]])<[infinity] and [[SIGMA].sub.i=0.sup.[infinity]][2.sup.i][phi]([x/[2.sup.i]], [y/[2.sup.i]]) < [infinity]. From (3.2), we get the inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0}. By Lemma 3.1, there exist functions [C.sub.0], [A.sub.0], Q : V [right arrow] Y defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V and the functions [C.sub.0], [A.sub.0], Q satisfy the inequalities

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x))]/4] - [C.sub.0](x)|| [less than or equal to] [[infinity].summation over (j = 1)][8.sup.j][psi](x/[2.sup.j]), (3.10)

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x))]/16] - [A.sub.0](x)|| [less than or equal to] [[infinity].summation over (j = 1)][2.sup.j][psi](x/[2.sup.j]), (3.11)

||[[f(x) - f(-x)]/2] - f(0) - Q(x)|| [less than or equal to] [[infinity].summation over (j = 1)][2.sup.2j-3][phi]([x/[2.sup.j][, - [x/[2.sup.j-1]]) (3.12)

for all x [member of] V\{0}. From (3.2) and (3.8), we obtain [E.sub.4][C.sub.0](x, y) = 0, [E.sub.4][A.sub.0](x, y) = 0, [E.sub.4]Q(x, y) = 0 for all x, y [member of] V \ {0}. Since [C.sub.0](2x) = 8[C.sub.0](x)([A.sub.0](2x) = [2A.sub.0](x) and Q(2x) = 4Q(x), respectively), [C.sub.0] is a cubic function ([A.sub.0] is an additive function and Q is a quadratic function, respectively) by Theorem 2.2(Theorem 2.1 and Theorem 2.4, respectively). From (3.8), (3.10), (3.11) and (3.12), we get the inequality (3.9), where F = [[-[C.sub.0](x)]/3] + Q(x) + [[4A.sub.0](x)]/3] + f(0) Now, let F' be another generalized polynomial function of degree 3 satisfying (3.9) with F'(0) = f(0). Then there are cubic functions C, C' : V [right arrow] Y, quadratic functions Q, Q' : V [right arrow] Y and additive functions A, A' : V [right arrow] Y such that F(x) = C(x) + Q(x) + A(x) + f(0) and F'(x) = C'(x) + Q'(x) + A'(x) + f(0). Since A, A' : V [right arrow] Y are additive functions, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V\{0} and n [member of] N. As n [right arrow] [infinity], we may conclude that A(x) = A'(x) for all x, y [member of] V. Similarly, we get Q(x) = Q'(x) and C(x) = C'(x) for all x, y [member of] V as we desired.

Theorem 3.4 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the conditions [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x, [2.sup.i]y)/[8.sup.i+1]) and [[SIGMA].sub.i=0.sup.[infinity]][4.sup.i][phi]([x/[2.sup.i+1], [y/[2.sup.i+1]) < [infinity] for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies (3.2) for all x, y [member of] V \ {0}, then there exists a generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j = 0)][[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[infinity].summation over (j = 1)]([[2.sup.j + 2]/3][psi](x/[2.sup.j]) + [2.sup.2j-3][phi]([x/[2.sup.j]], - [x/[2.sup.j-1]])) (3.13)

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Proof.Let the function [C.sub.0] be as in the proof of Theorem 3.2 and let Q, [A.sub.0] as in the proof of Theorem 3.3. We easily get [C.sub.0], Q, [A.sub.0] and the inequalities (3.4), (3.11) and (3.12) for all x [member of] V \ {0}. From (3.4), (3.11) and (3.12), we obtain (3.13), where F = -[1/3][C.sub.0]] + [4/3][A.sub.0]] + Q + f(0). Theorem 3.5 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the conditions [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x, [2.sup.i]y/[4.sup.i + 1]]) and [[SIGMA].sub.i=0.sup.[infinity]][2.sup.i][phi]([x/[2.sup.i+1]], [y/[2.sup.i+1]]) < [infinity] for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies (3.2) for all x, y [member of] V \ {0}, then there exists a generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j = 0)]([[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[phi]([2.sup.j]x, - [2.sup.j+1]x)/[2.sup.2j+3]]) + [[infinity].summation over (j = 1)][[2.sup.j+2]/3][psi](x/[2.sup.j])

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Corollary 3.6 Let p [not equal to] 1, 2, 3 and [epsilon] > 0. Suppose that the function f : V [right arrow] Y satisfies

||[E.sub.4]f(x, y)|| [less than or equal to] [epsilon]([||x||.sup.p] + [||y||.sup.p])

for all x, y [member of] V \ {0}. Then there exists a generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] ([[24 + [2.sup.p] + [3.sup.p]]/24]([1/[absolute value of 2 - [2.sup.p]]] + [1/[absolute value 8 - [2.sup.p]]]) + [[1 + [2.sup.p]]/[2[absolute value 4 - [2.sup.p]]]])[epsilon][||x||.sup.p])

for all x [member of] V \ {0}.

Proof. Applying Theorem 3.2, 3.3, 3.4 and 3.5, the following corollary can be proved easily.

Corollary 3.7 Let [epsilon] > 0. Suppose that the function f : V [right arrow] Y satisfies

||[E.sub.4]f(x,y)|| [less than or equal to] [epsilon]

for all x, y [member of] V \ {0}. Then there exists a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [11/14][epsilon]

for all x [member of] V \ {0}.

4. Superstability of the Equation [E.sub.4]f = 0

Theorem 4.1 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies

||[E.sub.4]f(x,y)|| [less than or equal to] [phi](x,y)

for all x, y [member of] V \ {0}, then f is a generalized polynomial function of degree 3.

Proof. Note that if [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) satisfies the condition (4.1) then [phi] satisfies the condition (4.1). By Theorem 3.1, there exist a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that the inequality (3.3) holds for all x [member of] V \ {0}. Hence the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0} and k [member of] N, where [PHI] is defined by

[PHI](x) := [[infinity].summation over (j = 0)]([[psi]([2.sup.j]x)/3 * [2.sup.j-2]] + [[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[phi]([2.sup.j]x, - [2.sup.j+1]x)/[2.sup.2j+3]]).

Taking as k [right arrow] [infinity], we conclude f(x) = F(x) for all x [member of] V \ {0}.

References

(1) J. A. Baker, A general functional equation and its stability, Proc. Amer. Math. Soc.133(2005), 1657-1664.

(2) P. W. Cholewa, Remarks on the stability of functional equations, Aeq. Math. 27(1984), 76-86.

(3) S. Czerwik, On the stability of the quadratic mapping in the normed space, Abh. Math. Sem. Hamburg, 62(1992), 59-64.

(4) V. Faiziev and Th. M. Rassias, The space ([psi], [gamma])-pseudocharacters on semigroups, Nonlinear Functional Analysis and Applications, 5(1)(2000), 107-126.

(5) V. A. Faiziev, Th. M. Rassias and P. K. Sahoo, The space of ([psi], [gamma])-additive mappings on semigroups,Transactions of the American Mathematical Society, 354(11)(2002), 4455-4472.

(6) Z. Gajda, On the stability of additive mappings, Internat. J. Math. and Math. Sci. 14(1991), 431-434.

(7) P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. and Appl. 184(1994), 431-436.

(8) D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A. 27(1941), 222-224.

(9) D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional equations in Several Variables, Birkhauser, Boston, Basel, Berlin, 1998.

(10) D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Mathematicae, 44(1992), 125-153.

(11) D. H. Hyers, G. Isac and Th. M. Rassias, On the asymptoticity aspect of Hyers-Ulam stability of mappings, Proceedings of the American Mathematical Society, 126(2)(1998), 425-430.

(12) K.-W. Jun and H.-M. Kim, On the Hyers-Ulam-Rassias Stability of a general cubic functional equation, Math. Ineq. Appl. 6(2)(2003), 289-302.

(13) K.-W. Jun Y.-H. Lee, and J.-R. Lee, On the Stability of a new Pexider type functional equation, J. Ineq. and App. 2008, ID 816963, 22pages.

(14) J. C. Parnami and H. L. Vasudeva, On Jensen's functional equation, Aeq. Math. 43(1992), 211-218.

(15) J. M. Rassias, Solution of the Ulam stability problem for Cubic mappings, Glasnik Matematicki, 36(56)(2001), 63-72.

(16) Th. M. Rassias, Functional Equations and Inequalities, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.

(17) Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003.

(18) Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158(1991), 106-113.

(19) Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72(1978), 297-300.

(20) Th. M. Rassias, Report of the 27th Internat. Symposium on Functional Equations, Aeq. Math. 39(1990), 292-292. Problem 16, 2[degrees], Same report, p. 309.

(21) F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano, 53(1983), 113-129.

Yang-Hi Lee [dagger]

Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea

Received June 16, 2008, Accepted June 16, 2008.

* 2000 Mathematics Subject Classification. Primary 39B52.

[dagger] E-mail: yanghi2@hanmail.net

Throughout this paper, let V be a vector space and Y a Banach space. Let n be a positive integer. For a given mapping f : V [right arrow] Y, define mappings [C.sub.n]f, [D.sub.n]f : V x V [right arrow] Y by

[E.sub.n]f(x,y) := [n.summation over (i = 0)][.sub.n][C.sub.i][(-1)].sup.n-i]f(ix + y),

[D.sub.n]f(x,y) := [n.summation over (i = 0)][.sub.n][C.sub.i][(-1)].sup.n-i]f(ix + y) - n!f(x)

for all x, y [member of] X, where [.sub.n][C.sub.i] = [n!/i!(n-i)!]. A mapping f : X [right arrow] Y is called a generalized polynomial(monomial, respectively) function of degree n [member of] N if f satisfies the functional equation [E.sub.n+1]f(x, y) = 0([D.sub.n]f(x, y) = 0, respectively). The functional equation [E.sub.n+1]f(x, y) = 0([D.sub.n]f(x, y) = 0, respectively) is called a generalized polynomial(monomial, respectively) functional equation of degree n [member of] N. In particular, a mapping f : V [right arrow] Y is called an additive (quadratic, cubic, quartic, respectively) mapping if f satisfies the functional equation [D.sub.1]f = 0 ([D.sub.2]f = 0, [D.sub.3]f = 0, [D.sub.4]f = 0, respectively). The functional equation [D.sub.1]f = 0 ([D.sub.2]f = 0, [D.sub.3]f = 0, [D.sub.4]f = 0, respectively) is called a Cauchy equation(quadratic functional equation, cubic functional equation, quartic functional equation, respectively). The functions f : R [right arrow] R defined by f(x) = [ax.sup.n] and f(x) = [[SIGMA].sub.i=0.sup.n-1][a.sub.i][x.sup.i] satisfy the functional equation [D.sub.n]f = 0 and [E.sub.n]f = 0 respectively, where a, [a.sub.i] are real constants and R is the set of real numbers.

If we replace a given functional equation by a functional inequality, when can one assert that the solutions of the inequality must be close to the solutions of the given equation? If the answer is affirmative, we would say that a given functional equation is stable.

In 1941, D.H.Hyers (8) proved the stability of Cauchy equation [D.sub.1]f = 0 and in 1978, Th.M.Rassias (19) gave a significant generalization of the Hyers' result. Th.M.Rassias (20) during the 27th International Symposium on Functional Equations, that took place in Bielsko-Biala, Poland, in 1990, asked the question whether such a theorem can also be proved for a more general setting. Z.Gadja (6) following Th.M.Rassias's approach (19) gave an affirmative solution to the question. Recently, P.Gavruta (7) obtained a further generalization of Rassias' theorem, the so-called generalized Hyers-Ulam-Rassias stability(See also (4),(5),(9-11),(16-18)).

A stability problem for the quadratic functional equation [D.sub.2]f = 0 was proved by F.Skof (21) for a function f : X [right arrow] Y, where X is a normed space. P.W. Cholewa (2) noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. S.Czerwik (3) proved the Hyers-Ulam-Rassias stability of the quadratic functional equation.

J. C. Parnami, H. L. Vasudeva (14) and J.M. Rassias (15) investigated the stability of the functional equation [D.sub.3]f = 0. Also, Jun and Kim (12) proved the stability of the functional equation [E.sub.4]f = 0 under the approximately cubic condition and Baker (1) proved the stability of the functional equation [E.sub.n]f = 0.

In this paper, I solve the general solution of [E.sub.4]f = 0 and prove the generalized Hyers-Ulam stability of the functional equation [E.sub.4]f = 0 on the punctured domain V \ {0}without the approximately cubic condition.

2. General Solution of [E.sub.4]f = 0

In this section I establish the general solution of [E.sub.4]f = 0. First I obtain the general solution for the odd cases. Throughout this section, let V and W be vector spaces.

Theorem 2.1 Suppose that the odd function f : V [right arrow] W satisfies

[E.sub.4]f(x,y) = f(4x + y)-4f(3x + y) + 6f(2x + y)-4f(x + y) + f(y) = 0

(2.1) for all x,y [member of] V \ {0} and

f(2x) = 2f(x)

for all x [member of] V. Then f is an additive function.

Proof. Note that f(0) = 0 and f(x) + f(-x) = [-[E.sub.4]f(x,-2x)/2] = 0 for all x [member of] V. From (2.1) and f(2x) = 2f(x), we have

[D.sub.1]f(x,y) = [1/132](-28[E.sub.4]f(x,y - 2x)-7[E.sub.4]f(y,2x - 2y)-16[E.sub.4]f(x,y - 3x) + 4[E.sub.4]f(y - x,4x - 2y)-28[E.sub.4]f(y,x - 2y) -7[E.sub.4]f(x,2y - 2x)-16[E.sub.4]f(y,x - 3y) + 4[E.sub.4]f(x - y,4y - 2x)) = 0

for all x, y [member of] V with x [not equal to] 0, y, 2y, 3y and y [not equal to] 0, x, 2x, 3x. Since f(2x) = 2f(x) and f(3x) = [E.sub.4]f(x,-x) + 3f(x) = 3f(x), f is an additive function.

Theorem 2.2 Suppose that the odd function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0} and

f(2x) = 8f(x)

for all x [member of] V. Then f is a cubic function.

Proof. Note that f(0) = 0 and f(x) + f(-x) = [-[E.sub.4]f(x,-2x)/2] = 0 for all x [member of] V. From (2.1) and f(2x) = 8f(x), we easily get the equality

[D.sub.3]f(x,y) = [E.sub.4]f(x,y - x)-[[E.sub.4]f(y + x,-2y)/4] = 0

for all x, y [member of] V \ {0} with x [not equal to] y,-y. Since f(2x) = 8f(x) and f(3x) = [E.sub.4]f(x,-x) + 27f(x) = 27f(x), f is a cubic function.

Theorem 2.3 Suppose that the odd function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0}. Then there exist a cubic function C : V [right arrow] W and an additive function A : V [right arrow] W such that

f(x) = C(x) + A(x)

for all x [member of] V, where

C(x) = [-1/3] [f(x) - [1/2] f(2x)] = [4/3] [f(x) -2f ([1/2]x)]

A(x) = [4/3] [f(x) - [1/8] f(2x)] = [-1/3] [f(x) -8f ([1/2]x)].

Proof. Since

f(4x)-10f(2x) + 16f(x) = [1/4] (11[E.sub.4]f(x,-x) + [E.sub.4]f(2x,-3x)-[E.sub.4](x,x)) = 0

for all x [member of] V \ {0}, we have

f(x) = C(x) + A(x), C(2x) = 8C(x), A(2x) = 2A(x)

for all x [member of] V, where

C(x): = [-1/3] [f(x) - [1/2] f(2x)] and A(x): = [4/3] [f(x)- [1/8] f(2x)].

By Theorem 2.2 and the equalities

[E.sub.4]C(x,y) = [-1/6](2[E.sub.4]f(x,y) - [E.sub.4]f(2x,2y)) = 0, [E.sub.4]A(x,y) = [1/6](8[E.sub.4]f(x,y)-[E.sub.4]f(2x,2y)) = 0

for all x, y [member of] V \ {0}, C is a cubic function and A is an additive function.

In the following theorem we obtain the general solution for the even case.

Theorem 2.4

Suppose that the even function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0} and

f(2x) = 4f(x)

for all x [member of] V. Then f is a quadratic function.

Proof. Note that f(0) = 0. From (2.1) and f(2x) = 4f(x), we get the equality

[D.sub.2]f(x,y) = -[1/12](4[E.sub.4]f(x,y-2x) + [E.sub.4]f(y,2x - 2y)) = f(x + y) + f(x-y)-2f(x)-2f(y) = 0

for all x, y [member of] V \ {0} with y [not equal to] x, 2x. Using 4f(x) = f(2x) and f(3x) + f(x) = [E.sub.4]f(x, -x) + 10f(x) = 10f(x) for all x [member of] V, we get

[D.sub.2]f(x,y) = 0

for all x, y [member of] V.

Theorem 2.5 Suppose that the even function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0}. Then f - f(0) is a quadratic function.

Proof. Let g = f - f(0). Then g satisfies (2.1) and g(0) = 0. Since g(2x) = [[E.sub.4]g(x,-2x)/2] + 4g(x) = 4g(x) for all x [member of] V {0}, by Theorem 2.4, g is a quadratic function.

Now I establish the general solution of [E.sub.4]f = 0.

Theorem 2.6 Suppose that the function f : V [right arrow] W satisfies (2.1) for all x, y [member of] V \ {0}. Then there exist a cubic function C : V [right arrow] W, a quadratic function Q : V [right arrow] W, and an additive function A : V [right arrow] W such that

f(x) = C(x) + Q(x) + A(x) + f(0)

for all x [member of] V. The functions C, Q, A : V [right arrow] W are given by

C(x): = [-1/12](2f(x) - 2f(-x) - f(2x) + f(-2x))

Q(x): = [[f(x) + f(-x)]/2] - f(0)

A(x): = [1/12](8f(x) - 8f(-x) - f(2x) + f(-2x))

for all x [member of] V.

Proof. Since f(x) = [[f(x) - f(-x)]/2[ + [[f(x) + f(-x)]/2], we can apply Theorem 2.3 and 2.4

3. Stability of the Equation [E.sub.4]f = 0

The following lemma is seen in (13).

Lemma 3.1. Let a be a positive real number and [PHI] : X \ {0} [right arrow] [0, [infinity]) a map. Suppose that the function f : X [right arrow] Y satisfies the inequality

||f(x) - [[f(2x)]/a]|| [less than or equal to] [[PHI](x)/a] and f(0) = 0.

1. If [[SIGMA].sub.l=0.sup.[infinity]][1/[a.sup.l+1]][PHI]([2.sup.l]x) < [infinity] for all x [member of] X \ {0}, then there exists a unique function F : X [right arrow] Y satisfying

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (l=0)][1/[a.sup.[l+1]][PHI]([2.sup.l]x)

for all x [member of] X \ {0} and F is given by F(x) = [lim.sub.n[right arrow][infinity]][f([2.sup.n]x)]/[a.sup.n]] for all x [member of] X.

2. If [[SIGMA].sub.l=0.sup.[infinity]][a.sup.l][PHI](x/[2.sup.l+1]) < [infinity] for all x [member of] X \ {0}, then there exists a unique function F : X [right arrow] Y satisfying

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (l=0)][a.sup.l][PHI](x/[2.sup.l+1]) < [infinity]

for all x [member of] X \ {0} and F is given by F(x) = [lim.sub.n[right arrow][infinity]][a.sup.n]f(x/[2.sup.n]) for all x [member of] X.

Theorem 3.2 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the condition

[[infinity].summation over (i=0)][[phi]([2.sup.i]x, [2.sup.i]y)/[2.sup.i]] < [infinity] (3.1)

If a function f : V [right arrow] Y satisfies

||[E.sub.4]f(x, y)|| [less than or equal to][phi](x, y) (3.2)

for all x, y [member of] V \ {0}, then there exists a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j=0)]([[psi]([2.sup.j]x)/3 * [2.sup.j-2]] + [[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[phi]([2.sup.j]x, -[2.sup.j+1]x)/[2.sup.2j+3]]) (3.3)

for all x, y [member of] V \ {0}, where

[psi](x) = [1/128](11[phi](x, -x) + [phi](2x, -3x) + [phi](x, x) + 11[phi](-x, x) + [phi](-2x, 3x) + [phi](-x, -x))

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Proof.Note that if [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) satisfies the condition (3.1) then [phi] satisfies the condition [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x,[2.sup.i]y)/[4.sup.i]] < [infinity] and [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x,[2.sup.i]y)/[8.sup.i]] < [infinity]. From (3.2), we get the inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0}. By Lemma 3.1, there exist functions [C.sub.0], [A.sub.0], Q : V [right arrow] Y defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V and the functions [C.sub.0], [A.sub.0], Q satisfy the inequalities

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/4] - [C.sub.0](x)|| [less than or equal to] [[infinity].summation over (j=0)][[psi]([2.sup.j]x)/[8.sup.j]], (3.4)

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/16] - [A.sub.0](x)|| [less than or equal to] [[infinity].summation over (j=0)][[psi]([2.sup.j]x])/[2.sup.j]], (3.5)

||[[f(x) + f(-x)]/2] - f(0) - Q(x)|| [less than or equal to] [[infinity].summation over (j=0)][[phi]([2.sup.j]x, -[2.sup.j+1]x)/[2.sup.2j+3]] (3.6)

for all x [member of] V \ {0}. From (3.1) and (3.2), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] V \ {0}. Since [C.sub.0](2x) = [8C.sub.0](x)([A.sub.0](2x) = [2A.sub.0](x) and Q(2x) = 4Q(x), respectively), [C.sub.0] is a cubic function ([A.sub.0] is an additive function and Q is a quadratic function, respectively) by Theorem 2.2(Theorem 2.1 and Theorem 2.4, respectively). From (3.4), (3.5), (3.6) and the inequality

||f(x) - F(x)|| [less than or equal to] [1/3]||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/4] - [C.sub.0](x)|| + [4/3]||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x)]/16] - [A.sub.0](x)|| + ||[[f(x) + f(-x)]/2] - f(0) - Q(x)|| (3.7)

for all x [member of] V \ {0}, we get the inequality (3.3), where F = -[[C.sub.0](x)/3] + Q(x) + [[4A.sub.0]/3] + f(0). Now, let F' be another generalized polynomial function of degree 3 satisfying (3.3) with F'(0) = f(0). Then there are cubic functions C, C' : V [right arrow] Y, quadratic functions Q, Q' : V [right arrow] Y and additive functions A, A' : V [right arrow] Y such that F(x) = C(x) + Q(x) + A(x) + f(0) and F'(x) = C'(x) + Q'(x) + A'(x) + f(0). Since C, C' : V [right arrow] Y are cubic functions, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0} and n [member of] N. As n [right arrow] [infinity], we may conclude that C(x) = C'(x) for all x, y [member of] V. Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0} and n [member of] N. As n [right arrow] [infinity], we may conclude that Q(x) = Q'(x) for all x, y [member of] V. Similarly, we get A(x) = A'(x) for all x, y [member of] V as we desired.

By the similar method in the proof of Theorem 3.2, I can prove the following theorem.

Theorem 3.3 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the condition

[[infinity].summation over (i=0)][8.sup.i][phi]([x/[2.sup.i]], [y/[2.sup.i]]) < [infinity] (3.8)

for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies (3.2) for all x, y [member of] V \ {0}, then there exists a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j=1)]([[[8.sup.j] + [2.sup.j+2]]/3] [psi](x/[2.sup.j]) + [2.sup.2j-3][phi]([x/[2.sup.j]], -[x/[2.sup.j-1])) (3.9)

for all x [member of] V \ {0}, where

[psi](x) = [1/128](11[phi](x, -x) + [phi](2x, -3x) + [phi](x, x) + 11[phi](-x, x) + [phi](-2x, 3x) + [phi](-x, -x))

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Proof. Note that if [phi] : V \ {0} x V \ {0} [right arrow][0, [infinity]) satisfies the condition (3.8) then [phi] satisfies the condition [[SIGMA].sub.i=0.sup.[infinity]] [4.sup.i][phi]([x/[2.sup.i]], [y/[2.sup.i]])<[infinity] and [[SIGMA].sub.i=0.sup.[infinity]][2.sup.i][phi]([x/[2.sup.i]], [y/[2.sup.i]]) < [infinity]. From (3.2), we get the inequalities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0}. By Lemma 3.1, there exist functions [C.sub.0], [A.sub.0], Q : V [right arrow] Y defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V and the functions [C.sub.0], [A.sub.0], Q satisfy the inequalities

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x))]/4] - [C.sub.0](x)|| [less than or equal to] [[infinity].summation over (j = 1)][8.sup.j][psi](x/[2.sup.j]), (3.10)

||[[f(x) - f(-x)]/2] - [[f(2x) - f(-2x))]/16] - [A.sub.0](x)|| [less than or equal to] [[infinity].summation over (j = 1)][2.sup.j][psi](x/[2.sup.j]), (3.11)

||[[f(x) - f(-x)]/2] - f(0) - Q(x)|| [less than or equal to] [[infinity].summation over (j = 1)][2.sup.2j-3][phi]([x/[2.sup.j][, - [x/[2.sup.j-1]]) (3.12)

for all x [member of] V\{0}. From (3.2) and (3.8), we obtain [E.sub.4][C.sub.0](x, y) = 0, [E.sub.4][A.sub.0](x, y) = 0, [E.sub.4]Q(x, y) = 0 for all x, y [member of] V \ {0}. Since [C.sub.0](2x) = 8[C.sub.0](x)([A.sub.0](2x) = [2A.sub.0](x) and Q(2x) = 4Q(x), respectively), [C.sub.0] is a cubic function ([A.sub.0] is an additive function and Q is a quadratic function, respectively) by Theorem 2.2(Theorem 2.1 and Theorem 2.4, respectively). From (3.8), (3.10), (3.11) and (3.12), we get the inequality (3.9), where F = [[-[C.sub.0](x)]/3] + Q(x) + [[4A.sub.0](x)]/3] + f(0) Now, let F' be another generalized polynomial function of degree 3 satisfying (3.9) with F'(0) = f(0). Then there are cubic functions C, C' : V [right arrow] Y, quadratic functions Q, Q' : V [right arrow] Y and additive functions A, A' : V [right arrow] Y such that F(x) = C(x) + Q(x) + A(x) + f(0) and F'(x) = C'(x) + Q'(x) + A'(x) + f(0). Since A, A' : V [right arrow] Y are additive functions, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V\{0} and n [member of] N. As n [right arrow] [infinity], we may conclude that A(x) = A'(x) for all x, y [member of] V. Similarly, we get Q(x) = Q'(x) and C(x) = C'(x) for all x, y [member of] V as we desired.

Theorem 3.4 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the conditions [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x, [2.sup.i]y)/[8.sup.i+1]) and [[SIGMA].sub.i=0.sup.[infinity]][4.sup.i][phi]([x/[2.sup.i+1], [y/[2.sup.i+1]) < [infinity] for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies (3.2) for all x, y [member of] V \ {0}, then there exists a generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j = 0)][[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[infinity].summation over (j = 1)]([[2.sup.j + 2]/3][psi](x/[2.sup.j]) + [2.sup.2j-3][phi]([x/[2.sup.j]], - [x/[2.sup.j-1]])) (3.13)

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Proof.Let the function [C.sub.0] be as in the proof of Theorem 3.2 and let Q, [A.sub.0] as in the proof of Theorem 3.3. We easily get [C.sub.0], Q, [A.sub.0] and the inequalities (3.4), (3.11) and (3.12) for all x [member of] V \ {0}. From (3.4), (3.11) and (3.12), we obtain (3.13), where F = -[1/3][C.sub.0]] + [4/3][A.sub.0]] + Q + f(0). Theorem 3.5 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the conditions [[SIGMA].sub.i=0.sup.[infinity]][[phi]([2.sup.i]x, [2.sup.i]y/[4.sup.i + 1]]) and [[SIGMA].sub.i=0.sup.[infinity]][2.sup.i][phi]([x/[2.sup.i+1]], [y/[2.sup.i+1]]) < [infinity] for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies (3.2) for all x, y [member of] V \ {0}, then there exists a generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [[infinity].summation over (j = 0)]([[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[phi]([2.sup.j]x, - [2.sup.j+1]x)/[2.sup.2j+3]]) + [[infinity].summation over (j = 1)][[2.sup.j+2]/3][psi](x/[2.sup.j])

for all x [member of] V \ {0}. In particular, F is represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V.

Corollary 3.6 Let p [not equal to] 1, 2, 3 and [epsilon] > 0. Suppose that the function f : V [right arrow] Y satisfies

||[E.sub.4]f(x, y)|| [less than or equal to] [epsilon]([||x||.sup.p] + [||y||.sup.p])

for all x, y [member of] V \ {0}. Then there exists a generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] ([[24 + [2.sup.p] + [3.sup.p]]/24]([1/[absolute value of 2 - [2.sup.p]]] + [1/[absolute value 8 - [2.sup.p]]]) + [[1 + [2.sup.p]]/[2[absolute value 4 - [2.sup.p]]]])[epsilon][||x||.sup.p])

for all x [member of] V \ {0}.

Proof. Applying Theorem 3.2, 3.3, 3.4 and 3.5, the following corollary can be proved easily.

Corollary 3.7 Let [epsilon] > 0. Suppose that the function f : V [right arrow] Y satisfies

||[E.sub.4]f(x,y)|| [less than or equal to] [epsilon]

for all x, y [member of] V \ {0}. Then there exists a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that

||f(x) - F(x)|| [less than or equal to] [11/14][epsilon]

for all x [member of] V \ {0}.

4. Superstability of the Equation [E.sub.4]f = 0

Theorem 4.1 Let [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) be a mapping satisfying the condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

for all x, y [member of] V \ {0}. If a function f : V [right arrow] Y satisfies

||[E.sub.4]f(x,y)|| [less than or equal to] [phi](x,y)

for all x, y [member of] V \ {0}, then f is a generalized polynomial function of degree 3.

Proof. Note that if [phi] : V \ {0} x V \ {0} [right arrow] [0, [infinity]) satisfies the condition (4.1) then [phi] satisfies the condition (4.1). By Theorem 3.1, there exist a unique generalized polynomial function F : V [right arrow] Y of degree 3 with f(0) = F(0) such that the inequality (3.3) holds for all x [member of] V \ {0}. Hence the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] V \ {0} and k [member of] N, where [PHI] is defined by

[PHI](x) := [[infinity].summation over (j = 0)]([[psi]([2.sup.j]x)/3 * [2.sup.j-2]] + [[psi]([2.sup.j]x)/3 * [8.sup.j]] + [[phi]([2.sup.j]x, - [2.sup.j+1]x)/[2.sup.2j+3]]).

Taking as k [right arrow] [infinity], we conclude f(x) = F(x) for all x [member of] V \ {0}.

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Yang-Hi Lee [dagger]

Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea

Received June 16, 2008, Accepted June 16, 2008.

* 2000 Mathematics Subject Classification. Primary 39B52.

[dagger] E-mail: yanghi2@hanmail.net

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Author: | Lee, Yang-Hi |
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Publication: | Tamsui Oxford Journal of Mathematical Sciences |

Date: | Dec 15, 2008 |

Words: | 4838 |

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