Summary. Let H be a real (complex) Hilbert space with dim H >/3, A : H --* H a continuous selfadjoint operator with dim A(H) > 2; we introduce on H a suitable A-orthogonality relation and study, in the class of the real (complex) functionals defined on H, two conditional functional equations --the Cauchy and the quadratic one --on the restricted domain of the A-orthogonal vectors.In this paper we determine the general solutions of these equations by theorems in which we establish the equivalence between each equation postulated on the whole space and the respective conditional equation.Our investigations have been motivated by incomplete studies on these conditional functional equations made in 1986 and 1966 by H. Drljevi6 and F. Vajzovi6, respectively.

Let X be a real inner product space with dimension greater than 2 and f be a real functional defined on X. In this paper we study an alternative functional equation defined on the whole space and the same equation on the restricted domain of all orthogonal vectors, in order to establish the equivalence between the alternative equation and its conditional form. This research follows previous studies concerning alternative functional equations and the Cauchy functional equation on the restricted domain of the orthogonal vectors.

Summary. In the class of functionals f: X --* ~, where X is an inner product space with dim X > 3, we study the D'Alembert functional equation( 1) on the restricted domainsand x= = {(x, y) ~ xZ/llxll = Ilyll}.In this paper we prove that the equation (1) restricted to X~ is not equivalent to (1) on the whole space 3(. We also succeed in characterizing all common solutions if we add the condition f(2x) = 2f2(x) -1.Using this result, we prove the equivalence between (1) restricted to X 2 and (1) on the whole space X.This research follows similar previous studies concerning the additive, exponential and quadratic functional equations.

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.

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