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 R¨atz, and the second equation is considered on the orthogonal vectors in the inner product spaces with the usual orthogonality.


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, and its relative conditional form is Although the Hyers-Ulam stability of the conditional quadratic functional equation 1.3 has been studied by Moslehian 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.

Orthogonality Spaces in the Sense of Ratz
In the class of real functionals f, g, h defined on an orthogonality space in the sense of Rätz, f, g, h : X, ⊥ → R, let us consider the conditional equation 1.3 .
We describe its solutions first assuming that f is an odd functional, then an even functional, finally, using the decomposition of the functionals f, g, h into their even and odd parts, we describe the general solutions.
If f is an odd functional, then the solutions of 1.3 are given by If f is an even functional, then the solutions of 1.3 are given by Proof. Let us first consider f an odd functional. Letting x 0 and y 0 in 1.3 , by f 0 0 for the oddness of f, we obtain Now, putting x, 0 in place of x, y in 1.3 , we have f x g x h 0 , then putting again 0, x in place of x, y we get g 0 h x 0 for all x ∈ X, since f is odd.
Now writing 1.3 with x, y replaced, respectively, first by x, 0 , then by 0, y , we get for all x, y ∈ X, since f is even. From 1.3 , using 2.7 , 2.8 , and 2.6 , we obtain

2.11
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.

2.12
Abstract and Applied Analysis 5 where A : X, ⊥ → R is an additive function and Q : X, ⊥ → R is an orthogonally quadratic function.
In the case of an inner product space H, ·, · dim H > 2 which is a particular orthogonality space in the sense of Rätz, with the ordinary orthogonality given by ⊥ y ⇔ x, y 0, we have the characterization of the orthogonally quadratic mappings from 11, Theorem 2 . Hence we have the following corollary.

2.13
where A : H, ·, · → R is an additive function and Q : H, ·, · → R is a quadratic function.

The Conditional Equation x ⊥ y ⇒ f x y g x − y h x k y in Inner Product Spaces
Consider now H an inner product space with dim H > 2 and the usual orthogonality given by ⊥ y ⇔ x, y 0. In the class of real functionals f, g, h, k defined on H, we consider the conditional equation 1.5 .
First prove the following lemma.
where A : H → R is an additive function and Q : H → R is a quadratic function.
Proof. Replacing in 1.5 x, y by 0, 0 , then by x, 0 and finally by 0, y , we obtain Hence 1.5 may be rewritten as So that, setting F t f t − f 0 and G t g t − g 0 , we infer Abstract and Applied Analysis Now, substituting −y in 3.3 in place of y, we have Adding 3.3 and 3.4 , we get So, defining the functional S : H → R by S t F t G t , 3.6 the above equation becomes Using ii and i , the left-hand side of the above equation may be written in the following way: 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 f, g, h, k into their even and odd parts. Using the same approach of Lemma 2.2, we easily prove the following lemma.
x ⊥ y ⇒ f e x y g e x − y h e x k e y .

3.10
Now consider 3.9 : the characterization of its solutions is given by the following theorem. Consider now x, y ∈ H with x ⊥ y. Writing 3.14 with x y instead of x and 3.15 with x − y instead of x, we get 2f o x y A x y k o x y , Adding the above equations, from 3.9 , the additivity of A and h o x A x , we obtain k 0 x y − k 0 x − y 2k 0 y 3.17 for x ⊥ y. By the symmetry of the orthogonality relation, we get, changing x and y and from the oddness of the function, k 0 x y k 0 x − y 2k 0 x , 3.18