Remarks on the Conditional Quadratic and the Conditional D’Alembert Functional Equations on Particular Normed Spaces

Abstract

Let $X$ be a real normed space with dimension greater than 2 and let $f$ be a real functional defined on $X$. Applying some ideas from the studies made on the conditional Cauchy functional equation on the restricted domain of the vectors of equal norm and the isosceles orthogonal vectors, the conditional quadratic equation and the D’Alembert one, namely, $\parallel x\parallel =\parallel y\parallel \Rightarrow f(x+y)+f(x-y)=2f(x)+2f(y)$ and $\parallel x\parallel =\parallel y\parallel \Rightarrow f(x+y)+f(x-y)=2f(x)f(y)$, have been studied in this paper, in order to describe their solutions. Particular normed spaces are introduced for this aim.