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General Solutions of Two Quadratic Functional Equations of Pexider Type on Orthogonal Vectors
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.
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 , Cholewa , Czerwik , Rassias , 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 ). 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 , 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. ; its stability has been studied, among others, by Jung and Sahoo  and Yang  and its orthogonal stability has been studied by Mirzavaziri and Moslehian , 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, , 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.
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.
Now consider (3.9): the characterization of its solutions is given by the following theorem.
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.
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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.