Abstract and Applied Analysis

Volume 2012 (2012), Article ID 675810, 10 pages

doi:10.1155/2012/675810

Research Article

General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors

Margherita Fochi

Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Received 5 May 2012; Accepted 15 July 2012

Academic Editor: Janusz Brzdek

Copyright © 2012 Margherita Fochi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Based on the studies on the Hyers-Ulam stability and the orthogonal stability of some Pexider-quadratic functional equations, in this paper we find the general solutions of two quadratic functional equations of Pexider type. Both equations are studied in restricted domains: the first equation is studied on the restricted domain of the orthogonal vectors in the sense of Rätz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.

1. Introduction

Stability problems for some functional equations have been extensively investigated by several authors, and in particular one of the most important functional equation studied in this topic is the quadratic functional equation,

(Skof [1], Cholewa [2], Czerwik [3], Rassias [4], among others).

Recently, many articles have been devoted to the study of the stability or orthogonal stability of quadratic functional equations of Pexider type on the restricted domain of orthogonal vectors in the sense of Rätz.

We remind the definition of orthogonality space (see [5]). The pair is called orthogonality space in the sense of Rätz if is a real vector space with and is a binary relation on with the following properties:

(i), for all , (ii)if , , then the vectors are linearly independent,(iii)if , , then for all ,(iv)let be a 2-dimensional subspace of . If then there exists such that and .

The relation is called symmetric if implies that .

An example of orthogonality in the sense of Rätz is the ordinary orthogonality on an inner product space given by .

In the class of real functionals defined on an orthogonality space in the sense of Rätz, , a first version of the quadratic equation of Pexider type is

and its relative conditional form is

Although the Hyers-Ulam stability of the conditional quadratic functional equation (1.3) has been studied by Moslehian [6], we do not know the characterization of the solutions of the conditional equation (1.3).

In the same class of functions, , another version of the quadratic functional equation of Pexider type is

and its relative conditional form is

Equation (1.4) has been solved by Ebanks et al. [7]; its stability has been studied, among others, by Jung and Sahoo [8] and Yang [9] and its orthogonal stability has been studied by Mirzavaziri and Moslehian [10], but also in this case we do not know the general solutions of (1.5).

Based on those studies, we intend to consider the above-mentioned functional equations (1.3) and (1.5) on the restricted domain of orthogonal vectors in order to present the characterization of their general solutions.

Throughout the paper, the orthogonality in the sense of Rätz is assumed to be symmetric.

2. The Conditional Equation in Orthogonality Spaces in the Sense of Ratz

In the class of real functionals defined on an orthogonality space in the sense of Rätz, , let us consider the conditional equation (1.3).

We describe its solutions first assuming that is an odd functional, then an even functional, finally, using the decomposition of the functionals into their even and odd parts, we describe the general solutions.

Theorem 2.1. Let be real functionals satisfying (1.3).

If is an odd functional, then the solutions of (1.3) are given by

where is an additive function, that is, is solution of for all .

If is an even functional, then the solutions of (1.3) are given by

where is an orthogonally quadratic function, that is, solution of for .

Proof. Let us first consider an odd functional. Letting and in (1.3), by for the oddness of , we obtain

Now, putting in place of in (1.3), we have , then putting again in place of we get for all , since is odd. The first equation gives

from (2.3), and the last equation proves that

using (2.3) again.

From the above results, (1.3) may be rewritten in the following way: for all . Hence by Lemma 3.1, [6], we have where is an orthogonally additive functional. But since and from [5, Theorem 5], we deduce that is everywhere additive.

Consider now an even functional. Substituting in (1.3) in place of , we obtain

Now writing (1.3) with replaced, respectively, first by , then by , we get

for all , since is even. From (1.3), using (2.7), (2.8), and (2.6), we obtain

Hence, setting , we infer for , that is, is an orthogonally quadratic functional. So, , and from (2.7), using (2.6), , and from (2.8), =.

The theorem is so proved.

Lemma 2.2. Let be real functionals satisfying (1.3).

Then both the even parts and the odd parts of , namely, and , satisfy (1.3).

Proof. Denoting by and the even and odd parts, respectively, of , we have from (1.3)

From the homogeneity of the orthogonality relation (property (iii)), we have , so that, by (1.3), choosing , we get

Adding and then subtracting (2.10) and (2.11), we easily prove the lemma.

From Lemma 2.2 and Theorem 2.1, we may easily prove the following theorem.

Theorem 2.3. The general solution of the functional equation (1.3) is given by

where is an additive function and is an orthogonally quadratic function.

In the case of an inner product space () which is a particular orthogonality space in the sense of Rätz, with the ordinary orthogonality given by , we have the characterization of the orthogonally quadratic mappings from [11, Theorem 2]. Hence we have the following corollary.

Corollary 2.4. Let H be an inner product space with and. The general solution of the functional equation (1.3) is given by

where is an additive function and is a quadratic function.

3. The Conditional Equation in Inner Product Spaces

Consider now an inner product space with and the usual orthogonality given by . In the class of real functionals defined on , we consider the conditional equation (1.5).

First prove the following lemma.

Lemma 3.1. Let be solutions of (1.5); then

where is an additive function and is a quadratic function.

Proof. Replacing in (1.5) by , then by and finally by , we obtain

(i),(ii),(iii).

Hence (1.5) may be rewritten as

So that, setting and , we infer

Now, substituting in (3.3) in place of , we have

Adding (3.3) and (3.4), we get

So, defining the functional by

the above equation becomes

From [11, Theorem 3], we have

where is an additive function and is a quadratic function. From (3.6), we have, , that is, . Using (ii) and (i), the left-hand side of the above equation may be written in the following way: ; hence we get . The theorem is so proved.

Our aim is now to characterize the general solutions of (1.5): this is obtained using the decomposition of the functionals into their even and odd parts. Using the same approach of Lemma 2.2, we easily prove the following lemma.

Lemma 3.2. Let be real functionals satisfying (1.5).

Then both the even parts and the odd parts of , namely, and , satisfy (1.5), that is,

Now consider (3.9): the characterization of its solutions is given by the following theorem.

Theorem 3.3. Let be real odd functionals satisfying (3.9); then the solutions of (3.9) are given by

where and are additive functions.

Proof. Substituting in (3.9) first , then in place of , and by and by the oddness of the functions, we obtain

Adding and then subtracting the above equations, we get

By (3.1), , hence from the above equations,

Consider now with . Writing (3.14) with instead of and (3.15) with instead of , we get

Adding the above equations, from (3.9), the additivity of and , we obtain

for . By the symmetry of the orthogonality relation, we get, changing and and from the oddness of the function,

hence for . By [5, Theorem 5], is an additive function; consequently, there exists an additive function such that for all . Now (3.14) and (3.15) give and , so the theorem is proved.

Finally, consider equation (3.10): the characterization of its solutions is given by the following theorem.

Theorem 3.4. Let be real even functionals satisfying (3.10); then there exist a quadratic function and a function such that

Proof. From Lemma 3.1, we first notice that

Substituting now in (3.10) first then instead of , we obtain, respectively

Consequently, by subtraction and from (3.20), we have

Substitution of (3.20) and (3.22) in (3.10) gives

Then, we substitute in place of in (3.23) and have

for all . Hence, for in (3.24), we obtain

Subtracting now (3.23) and (3.24), we get for all . Consider with : it follows that , hence in the above equation we may replace with , , respectively. We obtain , that is, for all with . Thus the function is constant on each sphere with center , and is well defined by

Hence (3.25) and (3.26) lead to

which finishes the proof.

Finally, the general solution of (1.5) is characterized by the following theorem.

Theorem 3.5. Let be real functionals satisfying (1.5); then there exist additive functions, , a quadratic function , and a function such that

Conversely, the above functionals satisfy (1.5).

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