# General system of cubic functional equations in non-Archimedean spaces.

1. Introduction and Preliminaries

Hensel (13) has introduced a normed space which does not have the Archimedean property. During the last three decades theory of non-Archimedean spaces has gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings and superstrings (17). Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are different and require a rather new kind of intuition (3), (4), (8), (18), (25), (30). One may note that [absolute value of n] [less than or equal to] 1 in each valuation field, every triangle is isosceles and there may be no unit vector in a non-Archimedean normed space; cf. (25). These facts show that the non-Archimedean framework is of special interest.

Definition 1.1. Let K be a field. A valuation mapping on K is a function [absolute value of *]: K [right arrow] R such that for any a, b [member of] K we have

(i) [absolute value of a] [greater than or equal to] 0 and equality holds if and only if a = 0,

(ii) [absolute value of ab] = [absolute value of a][absolute value of b],

(iii) [absolute value of (a+b)] [less than or equal to] [absolute value of a] + [absolute value of b].

A field endowed with a valuation mapping will be called a valued field. If the condition (iii) in the definition of a valuation mapping is replaced with

(iii)' [absolute value of (a+b)] [less than or equal to] max{[absolute value of a], [absolute value of b]}

then the valuation [absolute value of *] is said to be non-Archimedean. The condition (iii)' is called the strict triangle inequality. By (ii), we have [absolute value of 1] = [absolute value of -1] = 1. Thus, by induction, it follows from (iii)' that [absolute value of n] [less than or equal to] 1 for each integer n. We always assume in addition that [absolute value of *] is non trivial, i.e., that there is an [a.sub.0] [member of] K such that [absolute value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].The most important examples of non-Archimedean spaces are p-adic numbers.

Example 1.2. Let p be a prime number. For any non-zero rational number a = [p.sup.r][m/n] such that m and n are coprime to the prime number p, define the p-adic absolute value [[absolute value of a].sub.p] = [p.sup.-r]. Then [absolute value of *] is a non-Archimedean norm on Q. The completion of Q with respect to [absolute value of *] is denoted by [Q.sub.p] and is called the p-adic number field.

Definition 1.3. Let X be a linear space over a scalar field K with a non-Archimedean non-trivial valuation [absolute value of *]. A function ||*||: X [right arrow] R is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(NA1) ||X|| = 0 if and only if x = 0;

(NA2) ||rx|| = ||r||||x|| for all r [member of] K and x [member of] X;

(NA3) the strong triangle inequality (ultrametric); namely, ||x + y|| [less than or equal to] max{||x||; ||y||} (x; y [member of] X).

Then (X, ||*||) is called a non-Archimedean space.

It follows from (NA3) that

||[x.sub.m] - [x.sub.l]|| [less than or equal to] max{||[x.sub.[l + 1]]- [x.sub.l||: l[less than or equal to] l [less than or equal to] m-1 (m > l);

therefore a sequence {[x.sub.m]} is Cauchy in X if and only if {[x.sub.m+1] - [x.sub.m]} converges to zero in a non-Archimedean space.

Probabilistic normed spaces were first defined by Serstnev in 1962 (see (28)). Their definition was generalized in (1). We recall and apply the definition of Menger probabilistic normed spaces briefly as given in (27).

Definition 1.4. A distance distribution function (briefly, a d.d.f.) is a non-decreasing function F from [0, +[infinity]] into [0, 1] that satisfies F(0) = 0 and F(+[infinity]) = 1, and is left-continuous on (0, +[infinity]). The space of d.d.f.'s will be denoted by [[DELTA].sup.+]; and the set of all F in [[DELTA].sup.+] for which [lim.sub.t[right arrow]+[infinity]] - F(t) = 1 by [D.sup.+]. The space [[DELTA].sup.+] is partially ordered by the usual pointwise ordering of functions, i.e., F [less than or equal to] G if and only if F(x) [less than or equal to] G(x) for all x in [0, +[infinity]]. For any a [greater than or equal to] 0, [[epsilon].sub.a] is the d.d.f. given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Definition 1.5. A triangular norm (briefly t-norm) is a binary operation T: [0, 1] x [0, 1] [right arrow] [0, 1] which is commutative, associative, non{decreasing in each variable and has 1 as the unit element. Basic examples are the Lukasiewicz t-norm [T.sub.L], [T.sub.L](a, b) = max(a + b -1, 0), the product t-norm [T.sub.P], [T.sub.P] (a, b) = ab and the strongest triangular norm [T.sub.M], [T.sub.M](a, b) = min(a, b).

Definition 1.6. A Menger Probabilistic Normed space is a triple (X, v, T), where X is a real vector space, T is continuous t-norm and X is a mapping (the probabilistic norm) from X into [[DELTA].sup.+], such that for every choice of p and q in X and a, s, t in (0, +[infinity]), the following hold:

(PN1) v(p) = [[epsilon].sub.0], if and only if, p = [theta] ([theta] is the null vector in X);

(PN2) v(ap)(t) = v(p)(t/[absolute value of a]);

(PN3) v(p + q)(s + t) [greater than or equal to] T (v (p)(s); v(q)(t)).

Now we introduce definition of a Menger probabilistic non-Archimedean normed space.

Definition 1.7. Let X be a vector space over a non-Archimedean field K and T be a continuous t-norm. A triple (X, v, T) is said to be a Menger probabilistic non-Archimedean normed space if (PN1) and (PN2) (in De_nition1.6) and

(PNA3) v (x + y)(max{s, t}) [greater than or equal to] T (v(x)(s), v(y)(t)),

for all x, y [member of] X and all s, t [member of] K, are satisfied.

It follows from v(x) [member of] [[DELTA].sup.+] that v(x) is non-decreasing for every x [member of] X. So one can show that the condition (PNA3) is equivalent to the following condition:

v(x + y)(t) [greater than or equal to] T (v(x)(t), v(y)(t)).

Definition 1.8. Let (X, v, T) be a Menger probabilistic non-Archimedean normed space. Let {[x.sub.n]} be a sequence in X. Then {[x.sub.n]} is said to be convergent if there exists x [member of] X such that [lim.sub.n[right arrow][infinity]] v ([x.sub.n] - x)(t) = 1, for all t > 0. In that case, x is called the limit of the sequence {[x.sub.n]}. A sequence {[x.sub.n]} in X is called Cauchy if for each [epsilon] > 0 and each t > 0 there exists [n.sub.0] such that for all n [less than or equal to] [n.sub.0] and all p > 0 we have v([x.sub.n+p] - [x.sub.n])(t) > 1 - [epsilon].

Let T be a given t-norm. Then (by associativity) a family of mappings [T.sup.n]: [0, 1] [right arrow] [0, 1], n [member of] N, is de_ned as follows:

[T.sup.1](x) = T(x, x), [T.sup.n](x) = T([T.sup.n-1](x), x), x [member of] [0, 1].

For three important t-norms [T.sub.M], [T.sub.P] and [T.sub.L] we have

[T.sub.M.sup.n] (x) = x, [T.sub.n.sup.P] (x) = [x.sup.n], [T.sub.L.sup.n] (x) = max{(n + 1)x - n, 0}, n [member of] N.

Definition 1.9. (Hadzic (11)) A t-norm T is said to be of H-type if a family of functions {[T.sup.n](t)}; n [member of] N, is equicontinuous at t = 1, that is,

[[for all].sub.[epsilon]] [member of] (0, 1) [there exists] [delta] [member of] (0, 1): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The t-norm [T.sub.M] is a trivial example of t-norm of H-type, but there are t-norms of H-type with T [not equal to] [T.sub.M] (see, e.g., Hadzic(10)).

Lemma 1.10. We consider the notations of the definition (1.8). Also assume that T is a t-norm of H-type. Then the sequence {[x.sub.n]} is Cauchy if for each [epsilon] > 0 and each t > 0 there exists [n.sub.0] such that for all n [greater than or equal to] [n.sub.0] we have v([x.sub.n+1]- [x.sub.n])(t) > 1 - [epsilon].

Proof. Due to

v([x.sub.n+p] - [x.sub.n])(t) [greater than or equal to] T (v([x.sub.n+p] - [x.sub.n+p-1])(t), v([x.sub.n+p-1] - [x.sub.n])(t)) [greater than or equal to] T (v([x.sub.n+p] - [x.sub.n+p-1])(t), T(v([x.sub.n+p-1] - [x.sub.n+p-2])(t), v([x.sub.n+p-2] - [x.sub.n])(t)) [greater than or equal to] ... [greater than or equal to] T(v([x.sub.n+p] - [x.sub.n+p-1])(t), T(v([x.sub.n+p-1] - [x.sub.n+p-2])(t), ..., T(v([x.sub.n+2] - [x.sub.n+1])(t), v([x.sub.n+1] - [x.sub.n])(t))) ...),

and by the assumption of T, which is an H-type t-norm, the sequence {[x.sub.n]} is Cauchy if for each [epsilon] > 0 and each t > 0 there exists [n.sub.0] such that for all n [greater than or equal to] [n.sub.0] we have v([x.sub.n+1] - [x.sub.n])(t) > 1 - [epsilon]. We will use this criterion in this paper.

It is easy to see that every convergent sequence in a (Menger probabilistic) non-Archimedean normed space is Cauchy. If each Cauchy sequence is convergent, then the (Menger probabilistic) non-Archimedean normed space is said to be complete and is called (Menger probabilistic) non-Archimedean Banach space.

The first stability problem concerning group homomorphisms was raised by Ulam (29) in 1940 and solved in the next year by Hyers (12). Hyers' theorem was generalized by Aoki (2) for additive mappings and by Rassias (26) for linear mappings by considering an unbounded Cauchy difierence. In 1994, a generalization of the Rassias?theorem was obtained by Gavruta (9) by replacing the unbounded Cauchy difference by a general control function.

Jun and Kim (14) introduced the following cubic functional equation

f(2x + y) + f(2x - y) = 2f(x + y) + 2f(x - y) + 12f(x) (1.1)

and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.1). They proved that a function f: X [right arrow] Y where X and Y are real vector spaces, is a solution of (1.1) if and only if there exists a unique function C: X x X x X [right arrow] Y such that f(x) = C(x, x, x) for all x [member of] X. Moreover, C is symmetric for each fixed one variable and is additive for fixed two variables. The function C is given by

C(x, y, z) = [1/24](f(x + y + z) + f(x - y - z) - f(x + y - z) - f(x - y + z)),

for all x, y, z [member of] X. Obviously, the function f(x) = c[x.sup.3] satisfies the functional equation (1.1) which is called the cubic functional equation. Jun et al.(15) investigated the solution and the Hyers? Ulam stability for the cubic functional equation

f(ax + by) + f(ax - by) = [ab.sup.2](f(x + y) + f(x - y)) + 2a([a.sup.2] - [b.sup.2])f(x)

where a, b [member of] Z \ {0} with a [not equal to] [+ or -]1, [+ or -]b.

In recent years, many authors have proved the stability of various functional equations in various spaces (see for instance (4-7), (16), (19-25)). Using the method of our paper, one can investigate the stability of many general systems of various functional equations with n functional equations and n variables (n [member of] N) and our paper notably generalizes previous papers in this area.

We assume that f: [X.sup.n] [right arrow] Y is a mapping and consider the following generalized system of cubic functional equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

for all [x.sub.i], [y.sub.i] [member of] X and [a.sub.i], [b.sub.i] [member of] K \ {0}, with [a.sub.i] [not equal to] [+ or -] 1, [+ or -] [b.sub.i], i = 1,..., n.

In the section (4), we establish the generalized Hyers-Ulam-Rassias stability of system (1.2) in non-Archimedean Banach spaces. In the section (3), we establish the generalized Hyers-Ulam-Rassias stability of system (1.2) in Menger probabilistic non-Archimedean Banach spaces.

2. Stability of System (1.2) in non-Archimedean Banach Spaces

In this section, we prove the generalized Hyers-Ulam-Rassias stability of system (1.2) in non-Archimedean Banach spaces. Throughout this section, we assume that i; k; m; n; p [member of] N [union] {0}, K is a non{Archimedean field, Y is a non-Archimedean Banach space over K and X is a vector space over K. Also assume that f: [X.sup.n] [right arrow] Y is a mapping.

Theorem 2.1. Let [phi]k: [X.sup.n+1] [right arrow] [0, [infinity]) for k [member of] k,..., {1,..., n} be a function such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

for all [x.sub.i], [y.sub.i] 2 X, i = 1,..., n. Let f: [X.sup.n] [right arrow] Y be a mapping satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [x.sub.i], [y.sub.i] 2 X, i = 1,..., n. Then there exists a unique mapping T: [X.sup.n] [right arrow] Y satisfying (1.2) and

||f([x.sub.1], ..., [x.sub.n]) - T([x.sub.1],..., [x.sub.n])|| [less than or equal to] [PHI] ([x.sub.1],..., [x.sub.n]) (2.4)

for all [x.sub.i] [member of] X, i = 1, ...,n.

Proof. Fix k [member of] {1, 2,..., n} and consider the following inequality.

||f([x.sub.1], [x.sub.2],..., [a.sub.k][x.sub.k] + [b.sub.k][y.sub.k],..., [x.sub.n]) + f([x.sub.1], [x.sub.2], ..., [a.sub.k] [x.sub.k] - [b.sub.k] [y.sub.k],..., [x.sub.n]) - [a.sub.k][b.sup.k.sub.2](f([x.sub.1], [x.sub.2],..., [x.sub.k] + [y.sub.k]..., [x.sub.n]) + [f(x.sub.1], [x.sub.2],..., [x.sub.k] - [y.sub.k],..., [x.sub.n])) - [2a.sub.k]([a.sub.k.sup.2] - [b.sub.k.sup.2])f([x.sub.1], [x.sub.2],..., [x.sub.n])|| [phi]k([x.sub.1],..., [x.sub.k-1], [x.sub.k], [y.sub.k], [x.sub.k+1],..., [x.sub.n]). (2.5)

Let [y.sub.k] = 0 in (2.5). Then we get

||f([x.sub.1],..., [x.sub.k],..., [x.sub.n]) - 1 / [a.sub.k.sup.3] f([x.sub.1], [x.sub.2], ..., [a.sub.k] [x.sub.k], ..., [x.sub.n])|| [less than or equal to] 1 / [absolute value of [2a.sub.k.sup.3]] [phi] k ([x.sub.1], ..., [x.sub.k-1], [x.sub.k], 0, [x.sub.k+1],..., [x.sub.n]):

Therefore one can obtain

||[1/[[a.sub.1.sup.3] ... [a.sub.k-1.sup.3]]]f([a.sub.1] [x.sub.1],..., [a.sub.k-1] [x.sub.k-1], [x.sub.k], [x.sub.k+1],..., [x.sub.n ]) - [1/[[a.sub.1.sup.3] ... [a.sub.k-1.sup.3] [a.sub.k.sup.3]]] f([a.sub.1] [x.sub.1],..., [a.sub.k-1] [x.sub.k-1], [a.sub.k] [x.sub.k], [x.sub.k+1],..., [x.sub.n])k|| [less than or equal to] [1/[absolute value of (2[a.sub.1.sup.3] ... [a.sub.k-1.sup.3] [a.sub.k.sup.3])]][phi]k ([a.sub.1] [x.sub.1],..., [a.sub.k-1] [x.sub.k-1], [x.sub.k], [x.sub.k+1],..., [x.sub.n])

So we have

||f([x.sub.1], [x.sub.2],..., [x.sub.n]) - [1/[[a.sub.1.sup.3] ... [a.sub.n.sup.3]]]f([a.sub.1] [x.sub.1],..., [a.sub.n] [x.sub.n])|| [less than or equal to] max [1/[absolute value of (2[a.sub.1.sup.3] ... [a.sub.k-1.sup.3] [a.sub.k.sup.3])]] [phi]k ([a.sub.1] [x.sub.1],..., [a.sub.k-1] [x.sub.k-1], [x.sub.k], [x.sub.k+1],..., [x.sub.n ]): k = 1,..., n

Therefore we get

||[1/[[a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]] f([a.sub.1.sup.m] [x.sub.1], ..., [a.sub.n.sup.3m][x.sub.n]) - [1/[[a.sub.1.sup.3(m+1)] ... [a.sub.n.sup.3(m+1)]]] f([a.sub.1.sup.(m+1)] [x.sub.1],..., [a.sub.n.sup.(m+1)][x.sub.n])|| [less than or equal to] max {1/[absolute value of (2[a.sub.1.sup.3(m+1)] ... [a.sub.k-1.sup.3(m+1)] [a.sub.k+1.sup.3m] ... [a.sub.n.sup.3m])] [phi]k([a.sub.1.sup.(m+1)][x.sub.1],..., [a.sub.(k-1).sup.(m+1)][x.sub.k-1], [a.sub.k.sup.m][x.sub.k], 0, [a.sub.(k+1).sup.m], [x.sub.k+1] ..., [a.sub.n.sup.m][x.sub.n]): k = 1,..., n} (2.6)

for all m [member of] N [union] {0}. It follows from (2.6) and (2.2) that the sequence

{[1/[[a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]] f([a.sub.1.sup.3m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]}

is Cauchy. Since the space Y is complete, it is convergent. Therefore we can dene T: [X.sup.n] [right arrow] Y by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

for all [x.sub.i] [member of] X, i = 1,..., n. Using induction with (2.6) one can show that

||f([x.sub.1], ..., [x.sub.n]) - [1/[[a.sub.1.sup.3(p+1)] ... [a.sub.n.sup.3(p+1)]]] f([a.sub.1.sup.(p+1)] [x.sub.1], ..., [a.sub.n.sup.3(p+1)[x.sub.n])|| [less than or equal to] max {max {[1/[[absolute value of ([2a.sub.1.sup.3(m+1)] ... [a.sub.k.sup.3(m+1)] [a.sub.(k+1).sup.3m] ... [a.sub.n.sup.3m])]] [phi]k([a.sub.1.sup.(m+1)][s.sub.1], ..., [a.sub.k-1.sup.m+1][x.sub.k-1], 0, [a.sub.k+1.sup.m][x.sub.k+1], ..., [a.sub.n.sup. m][x.sub.n]]|: k = 1, ..., n}, m = 0, 1,..., p} (2.8)

for all [x.sub.i] [member of] X; i = 1,..., n and p [member of] N [union] {0}. By taking p to approach infinity in (2.8) and using (2.3) one obtains (2.4).

For k [member of] {1, 2,..., n} and by (2.5) and (2.7), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

By (2.1) and (2.9), we conclude that T satisfies (1.2).

Suppose that there exists another mapping T': [X.sup.n] [right arrow] Y which satisfies (1.2) and (2.4). So we have

||T([x.sub.1], [x.sub.2], ..., [x.sub.n]) - T'([x.sub.1], [x.sub.2], ..., [x.sub.n])|| [less than or equal to] [1/[absolute value of ([a.sub.1.sup.3m] - [a.sub.n.sup.3m])]] max{||T ([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]) - f([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n])||, ||f([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]) - T'([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n])||} [less than or equal to] [1/[absolute value of ([a.sub.1.sup.3m] - [a.sub.n.sup.3m])]] max{[PHI]([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]), [PHI]([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n])}

which tends to zero as m [right arrow] [infinity] by (2.3). Therefore T = T'. This completes the proof.

3. Stability of System(1.2) in Menger Probabilistic non-Archimedean Banach Spaces

In this section, we prove the generalized Hyers-Ulam-Rassias stability of system (1.2) in Menger probabilistic non-Archimedean Banach spaces. Throughout this section, we assume that u [member of] R, i, k, m, n [member of] N [union] {0}, K is a non-Archimedean field, T is a continuous t-norm of H-type, (Y, n, T) is a Menger probabilistic non-Archimedean Banach space over K, (Z, [omega], T) is a Menger probabilistic non-Archimedean normed space over K and X is a vector space over K. Also assume that f: [X.sup.n] [right arrow] Y is a mapping.

Theorem 3.1. Let [[phi].sub.k]: [X.sup.n+1] [right arrow] Z for k [member of] {1, ..., n} be a mappings such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

for all u > 0 and [x.sub.i], [y.sub.i] [member of] X, i = 1, ..., n. Let f: [X.sup.n] [right arrow] Y be a mapping satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all u > 0 and [x.sub.i], [y.sub.i] [member of] X, i = 1, ..., n. Then there exists a unique mapping F: [X.sup.n] [right arrow] Y satisfying (1.2) and

v(f([x.sub.1], ..., [x.sub.n]) - F([x.sub.1], ..., [x.sub.n])) (u) [greater than or equal to] [PSI] (3.4)

for all u > 0 and [x.sub.i] [member of] X, i = 1,..., n.

Proof. Fix k [member of] {1, 2, ..., n} and consider the following inequality.

v(f([x.sub.1], [x.sub.2], ..., [a.sub.k][x.sub.k] + [b.sub.k][y.sub.k], ..., [x.sub.n]) + f([x.sub.1], [x.sub.2], ..., [a.sub.k][x.sub.k] - [b.sub.k][y.sub.k], ..., [x.sub.n]) - [a.sub.k][b.sub.k.sup.2] (f([x.sub.1], [x.sub.2], ..., [x.sub.k] + [y.sub.k], ..., [x.sub.n]) + f([x.sub.1], [x.sub.2], ..., [x.sub.k] - [y.sub.k], ..., [x.sub.n])) - [2a.sub.k]([a.sub.k.sup.2] - [b.sub.k.sup.2]])f([x.sub.1], [x.sub.2],..., [x.sub.n]) [greater than or equal to] [omega] ([phi] k([x.sub.1], ..., [x.sub.k], [y.sub.k], ..., [x.sub.n]) (u). (3.5)

Let [y.sub.k] = 0 in (3.5). Then we get

v(f([x.sub.1], ..., [x.sub.n]) - [1/[a.sub.k.sup.3]] f([x.sub.1], [x.sub.2], ..., [a.sub.k] [x.sub.k], ..., [x.sub.n]) (u) [greater than or equal to] [omega] ([1/[absolute value of [2a.sub.k.sup.3]]] [[phi].sub.k] ([x.sub.1], ..., [x.sub.k], 0, ..., [x.sub.n]) (u).

Therefore one can obtain

v([1/[[a.sub.1.sup.3] ... [a.sub.k-1.sup.3]]] - f([a.sub.1][x.sub.1], ..., [a.sub.k-1] [x.sub.k-1], [x.sub.k], [x.sub.k+1], ..., [x.sub.n]) - [1/[[a.sub.1.sup.3]... [a.sub.k-1.sup.3] [a.sub.k.sup.3]]] f([a.sub.1] [x.sub.1], ..., [a.sub.k-1][x.sub.k-1], [a.sub.k][x.sub.k], [x.sub.k+1], ..., [x.sub.n]) (u) [greater than or equal to] [omega] ([1/[absolute value of [2a.sub.1.sup.3] ... [a.sub.k-1.sup.3] [a.sub.k.sup.3]]] [[phi].sub.k] ([a.sub.1] [x.sub.1], ..., [a.sub.k-1] [x.sub.k-1], 0, [x.sub.k], [x.sub.k+1], ..., [x.sub.n])) (u) = [[~.[phi]].sub.k]

Therefore we get

v(f([x.sub.1], ..., [x.sub.n]) - [1/[[a.sub.1.sup.3] ... [a.sub.n.sup.3]]] f([a.sub.1] [x.sub.1], ..., [a.sub.n][x.sub.n])) (u) [greater than or equal to] [[PHI].sub.n]

So we have

v([1/[[a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]] f([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]) - [1/[[a.sub.1.sup.3(m+1)] ... [a.sub.n.sup.3(m+1)]]] f([a.sub.1.sup.(m+1)][x.sub.1], ..., [a.sub.n.sup.(m+1)][x.sub.n]))(u) [greater than or equal to] [[PHI].sub.n] ([a.sub.1.sup.m] [x.sub.1], ..., [a.sub.n.sup.m] [x.sub.n], [absolute value of [a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]u)

for all m [member of] N [union] {0}. Therefore by (3.1) the sequence

{[1/[[a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]] f([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n])}

is Cauchy. By completeness of Y, we conclude that it is convergent. Therefore we can define F: [X.sup.n] [right arrow] Y by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

for all u > 0 and [x.sub.i] [member of] X, i = 1,..., n. Using induction with (3.6) one can show that

v(f([x.sub.1], ..., [x.sub.n]) - [1/[[a.sub.1.sup.3(m+1)] ... [a.sub.n.sup.3(m+1)]]] f([a.sub.1.sup.(m+1)][x.sub.1], ..., [a.sub.n.sup.(m+1)][x.sub.n]) (u) [greater than or equal to] [[PSI].sub.m] (3.8)

By taking m to approach in_nity in (3.8) and using (3.3) one obtains (3.4).

For k [member of] {1; 2,..., n} and by (3.5) and (3.7), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

By (3.2) and (3.9), we conclude that F satisfies (1.2).

Suppose that there exists another mapping F': [X.sup.n] [member of] Y which satisfies (1.2) and (3.4). So we have

v(F([x.sub.1], [x.sub.2],..., [x.sub.n]) - F'([x.sub.1], [x.sub.2],..., [x.sub.n]))(u) v([1/[[a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]] F([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]) - [1/[[a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]] f([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]) + [1/[[a.sub.1.sup.3m]... [a.sub.n.sup.3m]]] f([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n]) - [1/[[a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]] F' ([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n])(u) [greater than or equal to] T {[PSI] ([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n], [absolute value of [a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]u, [PSI] ([a.sub.1.sup.m][x.sub.1], ..., [a.sub.n.sup.m][x.sub.n], [absolute value of [a.sub.1.sup.3m] ... [a.sub.n.sup.3m]]u)}

which tends to 1 as m [right arrow] [infinity] by (3.1) and (3.3). Therefore F = F'. This completes the proof.

4. Acknowledgement

* 2000 Mathematics Subject Classification. Primary 39B82, 54E70, 46S10.

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M. B. Ghaemi [dagger], H. Majani [double dagger]

Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran

and

M. E. Gordji [section]

Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan, Iran

Received May 6, 2011, Accepted December 1, 2011.

[dagger] E-mail: mghaemi@iust.ac.ir

[double dagger] Corresponding author. E-mail: majani.hamid@hotmail.com