# Stability of additive mappings in generalized normed spaces.

1. Introduction and Preliminaries

It is well known that the Ulam's (12) question in 1940: "Under what conditions does there exist an additive mapping near an approximately additive mapping?", is the origin of the stability problem of functional equations. Many authors have extended, generalized and improved the answer to Ulam's question such as, Hyers (4) in the context of Banach space, K. Ravi, R. Murali and M. Arunkumar (10) for quadratic functional equation, T. Aoki (1) for additive mappings and Th.M. Rassias (8) for linear mappings in 1978 by considering the unbounded Cauchy difference. It states as follows:

Theorem 1.1. ((8)) Let E, E' be two Banach spaces and let [theta][member of][0, [infinity]) and p [member of] [0, 1). If a function f: E [right arrow] E' satisfies the inequality

||f(x + y) - f(x) - f(y)|| [less than or equal to] [theta][[||x||.sup.p] + [||y||.sup.p]]

for all x, y [member of] E. Then there exists a unique additive mapping T: E [right arrow] E' such that

||f(x) - T(x)|| [less than or equal to] 2[theta]/(2 - [2.sup.p])[||x||.sup.p]

for all x [member of] E. Moreover, if f(tx) is continuous in t for each fixed x [member of] E then T is linear.

In the following theorem J.M. Rassias replaced the sum by the product of powers of norms.

Theorem 1.2. ((7)) Let f: E [right arrow] E' be a mapping from a normed vector space E into a Banach space E' subject to the inequality

||f(x + y) - f(x) - f(y)|| [less than or equal to] [member of][||x||.sup.p][||y||.sup.p] (1.1)

for all x, y [member of] E, where [member of] and p are constants with [member of] > 0 and 0 [less than or equal to] p [less than or equal to] 1/2. Then the limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

exists for all x [member of] E and L: E [right arrow] E' is the unique additive mapping which satisfies

||f(x) - L(x)|| [less than or equal to] [member of]/(2 - [2.sup.2p])[||x||.sup.2p] (1.2)

for all x [member of] E. If p < 0, then the inequality (1.1) holds for x, y [not equal to] 0 and (1.2) for x [not equal to] 0. If p > 1/2 then the inequality (1.1) holds for x, y [member of] E and the limit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

exists for all x [member of] E and A: E [right arrow] E' is the unique additive mapping which satisfies

||f(x) - A(x)|| [less than or equal to] [member of]/([2.sup.2p] - 2)[||x||.sup.2p]

for all x [member of] E. If in addition f: E [right arrow] E' is a mapping such that the transformation t [right arrow] f(tx) is continuous in t [member of] R for each fixed x [member of] E, then L is R--linear mapping.

Also, the topic of stability of functional equations has been studied by a number of mathematicians (see (6), (2), (3), (9), (5) for more detailed information). Before giving the main results, we recall the definition of normed-binary operation and some examples and lemmas were used in (11).

Definition 1.3. ((11))A normed-binary operation is a mapping [??]: [0, [infinity]) x [0, [infinity]) [right arrow] [0, [infinity]) which satisfies the following conditions:

(i) [??] is associative and commutative,

(ii) [??] is continuous,

(iii) a [??] 0 = a for all a [member of] [0, [infinity]),

(iv) a [??] b [less than or equal to] c [??] d whenever a [less than or equal to] c and b [less than or equal to] d, for each a, b, c, d [member of] [0, [infinity]).

Example 1.4. ((11))Let a, b [member of] [0, [infinity]). Five typical examples of [??] are:

(a) a [[??].sub.1] b = max{a, b}

(b) a [[??].sub.2] b = [square root of ([a.sup.2] + [b.sup.2])]

(c) a [[??].sub.3] b = a + b

(d) a [[??].sub.4] b = ab + a + b

(e) a [[??].sub.5] b = [[([square root of a] + [square root of b])].sup.2].

For a, b [member of] [0, [infinity]), straight forward calculations lead to the following relations among normed binary operations giving above

a [[??].sub.1] b [less than or equal to] a [[??].sub.2] b [less than or equal to] a [[??].sub.3] b [less than or equal to] a [[??].sub.4] b,

and

a [[??].sub.3 b [less than or equal to] a [[??].sub.5] b.

The follwing lemma defines a normed binary operation exploting some properties of a self map on [0, [infinity]).

Lemma 1.5. ((11))Let f: [0, [infinity]) [right arrow] [0, [infinity]) be a continuous, onto, and increasing map. Let [??]: [0, [infinity]) x [0, [infinity]) [right arrow] [0, [infinity]) be defined by

a [??] b = [f.sup.-1](f(a) + f(b)) for a, b [member of] [0, [infinity]),

then [??] is a normed binary operation.

Example 1.6. ((11))Let f: [0, [infinity]) [right arrow] [0, [infinity]) defined by f(x) = [e.sup.x]-1. Then a [??] b = Ln([e.sup.a] + [e.sup.b] - 1) for a, b [member of] [0, [infinity]) defines a normed binary operation.

We have the following simple observations about normed binary operation.

Lemma 1.7. ((11)) (i) If r, r' [greater than or equal to] 0, then r [less than or equal to] r [??] r'.

(ii) If [delta] [member of] (0, r), there exist [delta]' [member of] (0, r) such that [delta]' [??] [delta] < r.

(iii) For every [epsilon] > 0, there exists [delta] > 0 such that [delta] [??] [delta] < [epsilon].

In this paper all vector spaces are real.

Now we are set to generalize the concept of a normed space.

Definition 1.8. Let X be vector space and [??] be a binary operation. A generalized norm on X is a function: N: X [right arrow] R that satisfies the following properties:

(1) N(x) [greater than or equal to] 0 for each x in X,

(2) N(x) = 0 if and only if x = 0,

(3) N([alpha]x) = [|[alpha]|.sup.t]N(x) for some t [member of] (0, [infinity]), for each x in X and every [alpha] [member of] R.

(4) N(x + y) [less than or equal to] N(x) [??] N(y), for each x, y [member of] X.

The 3--tuple (X,N, [??]) is called a generalized normed space or a G--normed space.

Example 1.9. Let (X, ||.||) be a normed space, a, b [member of] [0, [infinity]), and x [member of] X. If we define [??]: [0, [infinity]) x [0, [infinity]) [right arrow] [0, [infinity]),

(i) a [??] b = a + b, and N is defind by N(x) = ||x||, then (X, N, [??]) is a G--normed space for t = 1.

(ii) a [??] b = [square root of [a.sup.2] + [b.sup.2]], and N is defined by N(x) = [square root of ||X||], then (X, N, [??]) is a G--normed space for t = 1/2.

(iii) a [??] b = [([square root of a] + [square root of b]).sup.2], and N is defined by N(x) = [||x||.sup.2], then (X, N, [??]) is a G--normed space for t = 2.

Remark 1.10. From Example 1.9 (i), we see that:

every normed space is a G--normed space.

Remark 1.11. In (3) of Definition 1.8 t is unique.

Example 1.12. Let X = [R.sup.2], if we define [??]: [0, [infinity]) x [0, [infinity]) [right arrow] [0, [infinity]) by a [??] b = [([4th root of a] + [4th root of b]).sup.4] for a, b [member of] [0, [infinity]), and define N: X [right arrow] R by N(x, y) = [x.sup.4] + [y.sup.4] for x, y [member of] R, then (X, N, [??]) is a G--normed space for t = 4.

Definition 1.13. Let (X, N, [??]) be a G--normed space. For r > 0, the ball [B.sub.N](x, r) with center x [member of] X and radius r is defined by

[B.sub.N](x, r) = {y [member of] X: N(x - y) < r}.

Definition 1.14. Let (X, N, [??]) be a G--normed space. A subset A [??] X is open if for every x [member of] A, there exists r > 0 such that [B.sub.N](x, r) [??] A.

Let [tau] be the set of all open subsets A [??] X. It can be verified that [tau] is a topology on X, called a topology induced by generalized norme N.

Lemma 1.15. Let (X, N, [??]) be a G--normed space. Then

(i) N(ax) [less than or equal to] N(x) for all real scalars a with |a| [less than or equal to] 1.

(ii) if X is convex, then we get

N(ax + (1 - a)y) [less than or equal to] N(x) [??] N(y)

for all x, y [member of] X and every a [member of] (0, 1).

Proof. Proof immediately follows from Definition 1.8.

Definition 1.16. Let (X, N, [??]) be a G--normed space. A sequence {[x.sub.n]} in X is said to be convergent to x [member of] X if for each [member of] > 0 there exists [n.sub.0] [member of] N such that

n [greater than or equal to] [n.sub.0] [??] N([x.sub.n] - x) < [member of].

We denote this by N([x.sub.n] - x) [right arrow] 0 as n [right arrow] [infinity] or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1.17. Let (X, N, [??]) be a G--normed space. A sequence {[x.sub.n]} in X is called a Cauchy sequence if for each [member of] > 0, there exists [n.sub.0] [member of] N such that N([x.sub.n] - [x.sub.m]) < [member of] for each n,m [greater than or equl to] [n.sub.0].

The generalized normed space (X, N, [??]) is said to be generalized Banach space or G--Banach space if every Cauchy sequence is convergent in X.

Now we prove the following basic lemmas needed in the sequel.

Lemma 1.18. Let (X, N, [??]) be a G--normed space. If r > 0, then the ball [B.sub.N](x, r) is open.

Proof. Let y [member of] [B.sub.N](x, r), so that we have N(x - y) < r. Put, N(x - y) = [theta] then by Lemma 1.7 there exists [theta]' > 0 such that [theta]' [??] [theta] < r. Now, we prove that [B.sub.N](y, [theta]') [??] [B.sub.N](x, r). For this, let z [member of] [B.sub.N](y, [theta]'). By triangle inequality we have

N(x - z) [less than or equal to] N(x - y) [??] N(y - z) < [theta] [??] [theta]' < r.

This implies that

[B.sub.N](y, [theta]') [??] [B.sub.N](x, r).

Hence [B.sub.N](x, r) is an open set.

Lemma 1.19. Every G--normed space (X, N, [??]) is a Hausdorff space.

Proof. Let x, y [member of] X and x [??] y. If we set N(x - y) = r then for 0 < [theta] < r by Lemma 1.7 there exists 0 < [theta]' < r such that [theta]' [??] [theta] < r. We prove that [B.sub.N](x, [theta]) [intersection] [B.sub.N](y, [theta]') = [empty set]. Let z [member of] [B.sub.N](x, [theta]) [intersection] [B.sub.N](y, [theta]'). Now, by triangle inequality, we get that

r = N(x - y) [less than or equal to] N(x - z) [??] N(z - y) < [theta] [??] [theta]' < r,

which is a contradiction. Hence (X, N, [??]) is a Hausdorff space.

Lemma 1.20. Let (X, N, [??]) be a G--normed space, then every convergent sequence in X is Cauchy in X.

Proof. Let {[x.sub.n]} be a sequence in X which converges to x [member of] X. For [member of] > 0, by Lemma 1.7 we can choose a [theta] > 0 such that [theta] [??] [theta] < [member of]. Since [x.sub.n] [right arrow] x, there exists [n.sub.0] [member of] N such that for every n [greater than or equal to] [n.sub.0], we obtain that N([x.sub.n] - x) < [theta].

Thus for every n,m [greater than or equal to] [n.sub.0], we have

N([x.sub.n] - [x.sub.m]) [less than or equal to] N([x.sub.n] - x) [??] N(x - [x.sub.m]) < [theta] [??] [theta] < [member of].

Hence {[x.sub.n]} is a Cauchy sequence.

Lemma 1.21. Let (X, N, [??]) be a G--normed space, then addition +: X x X [right arrow] X defined by + (x, y) = x + y and scalar multiplication *: R x X [right arrow] X defined by * ([alpha], x) = [alpha] * x are continuous.

Proof. First we prove continuity of addition. Let [x.sub.n] [right arrow] x, [y.sub.n] [right arrow] y. By Lemma 1.7 for each [member of] > 0 there exists [theta] > 0 such that [theta] [??] [theta] < [member of]. Also, there exists [n.sub.0] [member of] N such that

n [greater than or equal to] [n.sub.0] [??] N([x.sub.n] - x) < [theta],

and

n [greater than or equal to] [n.sub.0] [??] N([y.sub.n] - y) < [theta].

By triangle inequality we have

N(([x.sub.n + [y.sub.n]) - (x + y)) [less than or equal to] N([x.sub.n] - x) [??] N([y.sub.n] - y) < [theta] [??] [theta] < [member of].

Now we prove that scalar multiplication is continuous. Let [[alpha].sub.n] [right arrow] [alpha], and [x.sub.n] [right arrow] x( which means that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

Triangle inequality gives that

N([[alpha].sub.n] * [x.sub.n] - [[alpha] * x) = N([[alpha.sub.n] * ([x.sub.n] - x) + ([[alpha].sub.n] - [alpha]) * x) [less than or equal to] [|[[alpha.sub.n]|].sup.t] N([x.sub.n] - x) [??] [|[[alpha.sub.n] - [alpha]|].sup.t] N(x).

and so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 1.22. Let a [??] b = max{a, b}, then there is not any t [member of] (0, [infinity]) such that

N([alpha] * x) = [|[alpha]|.sup.t] * N(x).

Because If we assume that (on contrary), there exists t [member of] (0, [infinity]) and N([alpha] * x) = [|[alpha]|.sub.t] * N(x). Then by taking [alpha] = 2 we obtain:

[|2|.sup.t] * N(x) = N(2x) = N(x + y) [less than or equal to] N(x) [??] N(x) = N(x),

Henceforth, we assume that the normed binary operation [??] on [0, [infinity]) x [0, [infinity]) satisfy the following properties:

(PI): [alpha] * (a [??] b) = [alpha] * a [??] [alpha] * b for every [alpha] [member of] [R.sup.+] and

(PII): there exists h [greater than or equal to] 0 such that 1 [??] 1 [??] ... [??] 1 [less than or equal to] [n.sub.h], for every n [member of] N.

In the following example, we give some normed binary operations [??] on [0, [infinity]) x [0, [infinity]) with properties (PI) and P(II).

Example 1.23. Let a [??] b = max{a, b} or a [??] b = [square root of ([a.sup.2] + [b.sup.2])] or a [??] b = a + b or a [??] b = ([square root of a] + [square root of b].sup.2]

then in each case, [??] satisfies properties (PI) and (PII).

The next example includes a normed binary operation [??] on [0, [infinity]) x [0, [infinity]) which does not satisfy (PI) and P(II) properties.

Example 1.24. Define [??]: [0, [infinity]) x [0, [infinity]) [right arrow] [0, [infinity]) by a [??] b = a + b + ab, for a, b [member of] [0, [infinity]). Obviously [??] is not have (PI) and (PII) properties.

2. Main Results

In the rest of this paper, we will assume that (X, N', [??]) is G--normed space and (Y, N, [??]) is G--Banach space.

Let [empty set] be a function from X x X to X.. A mapping f: X [right arrow] Y is called a [empty set]--approximately Cauchy function, if

N(f(x + y) - f(x) - f(y)) [less than or equal to] N'([empty set](x, y)) (2.1)

for all x, y [member of] X.

Example 2.1. Let X = Y = R and N, N' be usual norm. Let [empty set] be a function from X x X [right arrow] X defined by [empty set](x, y) = xy((x + y)/2).

Let mapping f: R [right arrow] R is defined by f(x) = sinx. Then one can easily see that f is a [empty set]--approximately Cauchy function, because:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] R.

Hence f: R [right arrow] R is a [empty set]--approximately Cauchy function.

In the sequel, all binary operation [??] satisfy (PI) and (PII) properties.

Theorem 2.2. Let [empty set]: X x X [right arrow] X be a function and f: X [right arrow] Y be a [empty set]--approximately Cauchy function and for some 0 < [alpha] < 1/2 assume that,

N'([empty set](x/2, y/2)) [less than or equal to] N'([alpha][empty set](x, y))

then there exists an additive mapping T: X [right arrow] Y.

Morover, if a [??] b [less than or equal to] a + b for every a, b [member of] [0, [infinity]), then

N(f(x) - T(x))[less than or equal to][[alpha].sup.t]/[(1 - (2[alpha])).sup.t]N'([empty set](x, y)) (2.2)

for all x [membe of] X and T is unique.

Proof. Since f is [empty set]--approximately Cauchy function, put x = y in (2.1) to obtain

N(f(2x) - 2f(x)) [less than or equal to] N'([empty set](x, y)) (x[member of]X) (2.3)

Replacing x by [2.sup.-n-1]x in inequality (2.3) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If set [a.sub.n](x) = [2.sup.n]f([2.sup.-n]x), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also for n [less than or equal to] m (n, m[member of]M)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to see that for every m [greater than or equal to] n, there exists s > 0 such that m [less than or equal to] [n.sup.s]. Thus

N([a.sub.m](x) - [a.sub.n](x)) [less than or equal to] [|2[alpha]|.sup.nt] * [n.sup.s] * N'([empty set](x, y)) [right arrow] 0.

Which implies that [a.sub.n](x) = [2.sup.n]f([2.sup.-n]x) is a Cauchy sequence. Since Y is G--Banach space, hence, for every x [member of] X there exists [y.sub.v] [member of] Y such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Indeed we can define a mapping T: X [right arrow] Y by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, we show that T is an additive mapping. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As n [right arrow] [infinity] we get

N(T(x + y) - [T.sub.x] - [T.sub.y]) [right arrow] 0.

Hence T(x+y) = T(x)+T(y). Now we show that the mapping T satisfies in the inequality (2.2).

We have

N(f(x) - T(x)) [less than or equal to] N(f(x) - [2.sup.n]f([2.sup.-n]x)) [??] N([2.sup.n]f([2.sup.-n]x) - T(x)).

Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore

N(f(x) - T(x)) [less than or equal to] [[alpha].sup.t]/(1 - [(2[alpha]).sup.t]N'([empty set](x, x)) [??] N([2.sup.n]f([2.sup.-n]x) - T(x)).

As n tends to infinity we have

N(f(x) - T(x)) [less than or equal to] [alpha].sup.t]/(1 - [(2[alpha]).sup.t]N'([empty set](x, x))

for every x [member of] X

Also, since T and T' are additive we have

T(x) = [2.sup.n]T([2.sup.-n]x) and T'(x) = [2.sup.n]T'([2.sup.-n]x)

Hence we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follwes that T = T'.

Corollary 2.3. Let (X, [||*||.sub.2]) be normed space and (Y, [||*||.sub.1]) be Banach Space. Let [empty set] be a function from X x X to X, and mapping f: X [right arrow] Y be a [empty set]--approximately Cauchy function. If for some 0 < [alpha] < 1/2 assume that,

[||[empty set](x/2, y/2)||.sub.2] [less than or equal to] [||[alpha][empty set](x, y)||.sub.2]

then there exists a unique additive mapping T: X [right arrow] Y such that

[||f(x) - T(x)||.sub.1][less than or equal to] [alpha]/(1 - 2[alpha])[||[empty set](x, y)||.sub.2]

for all x [member of] X.

Proof. If we take (X, [||*||.sub.2]) = (X, N', [??]) and (Y, [||*||.sub.1]) = (Y, N, [??]) we get the proof by Theorem 2.2 and Remark 1.10.

Example 2.4. Let X = Y = R and N, N' be usual norm. Let [empty set] be a function from R x R [right arrow] R defined by [empty set](x, y) = xy((x + y)/2)). Let f: R [right arrow] R is defind by f(x) = sinx. By Example 2.1 f is a [empty set]--approximately Cauchy function and

|[empty set](x/2, y/2)| [less than or equal to] |[alpha][empty set](x, y)| for 1/8 [less than or equal to] [alpha] < 1/2.

Hence all conditions of Corollary 2.3 are hold. So there exists a unique additive mapping T: R [right arrow] R such that

|f(x) - T(x)| [less than or equal to] [alpha]/(1 - 2[alpha])|[x.sup.3]|.

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

And

|f(x) - T(x)| = |sin x - x| [less than or equal to] [[alpha]/[1 - 2[alpha]]] |[x.sup.3]|.

We show that corollary (2.3) extends the theorems 1.1 ((8)) and 1.2 ((7)).

Corollary 2.5. Let X, X' be two Banach spaces and let [theta] [member of] [0, [infinity]). If a function f : X [right arrow] or [vector] X' satisfies the inequality

||f(x + y) - f(x) - f(y)|| [less than or equal to] [theta][[||x||.sup.p] + [||y||.sup.p]] (2.4)

for all x, y [member of] X. Then there exists a unique additive mapping T : X [right arrow] or [vector] X' such that

||f(x) - T(x)|| [less than or equal to] [2[theta]/[[2.sup.p]-2]][||x||.sup.p], for p > 1. (2.5)

And

||f(x) - T(x)|| [less than or equal to] [2[theta]/[2 - [2.sup.p]]][||x||.sup.p], for p > 1. (2.6)

Proof. We show that, if set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, all conditions of corollary (2.3) are established. Case of (x, y) = (0, 0) is obviously. In case of (x, y) [not equal to] (0, 0) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Also we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i) Let p > 1. If set [alpha] = 1/[2.sup.p], then [alpha] < 1/2 and by corollary (2.3) there exists an additive mapping T : X [right arrow] or [vector] Y such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ii) Let 0 < p < 1. If set [alpha] = [2.sup.p]/4, then [alpha] < 1/2 and by corollary (2.3) there exists an additive mapping T : X [right arrow] or [vector] Y such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Corollary 2.6. Let f: X [right arrow] or [vector] X' be a mapping from a normed vector space X into a Banach space X' subject to the inequality

||f(x + y) - f(x) - f(y)|| [less than or equal to] [member of] [||x||.sup.p] [||y||.sup.p]

for all x, y [member of] X, where [member of] > 0.

If p > 1/2, then there exists T : X [right arrow] or [vector] X' such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] X and T : X [right arrow] or [vector] X' is the unique additive mapping which satisfies

||f(x) - T(x)|| [less than or equal to] [[member of]/[[2.sup.2p] - 2]][||x||.sup.2p].

If 0 < p < 1/2, then there exists T : X [right arrow] or [vector] X' such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x [member of] X and T : X [right arrow] or [vector] X' is the unique additive mapping which satisfies

||f(x) - T(x)|| [less than or equal to] [[member of]/[2 - [2.sup.2p]]][||x||.sup.2p].

Proof. We show that if set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then all conditions of corollary (2.3) are established.

Case of (x, y) = (0, 0) is obviously. In case of (x, y) [not equal to] (0, 0) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

i) Let p > 1/2. If set [alpha] = 1/[2.sup.2p], then [alpha] < 1/2 and by corollary (2.3) there exists an additive mapping T : X [right arrow] or [vector] Y such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ii) Let 0 < p < 1/2. If set [alpha] = [2.sup.2p]/4. then [alpha] < 1/2 and by corollary (2.3) there exists an additive mapping T : X [right arrow] or [vector] Y such that

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Received January 18, 2012, Accepted April 8, 2013.

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Alireza Mohammadpour [dagger] and Nabi Shobe [double dagger]

Department of Mathematics, Islamic azad university-Babol Branch, Babol, Iran

* 2000 Mathematics Subject Classification. Primary 39B82, 39B70, 39B52.

[dagger] Corresponding author. E-mail: mohammadpour ar@yahoo.com

[double dagger] E-mail: nabi shobe@yahoo.com