On two conditional Pexider functional equations and their stabilities

We solve conditional Pexider functional equations on prescribed sets being generalizations of spheres, or defined for vectors satisfying a generalized James orthogonality condition, namely, we solve the conditional equations of the form γ(x)=γ(y)impliesf(x+y)=g(x)+h(y), or γ(x+y)=γ(x−y)impliesf(x+y)=g(x)+h(y) with a given function γγ and unknown functions f,g,hf,g,h.Moreover, we study the stability problems for these equations, both in the sense of Hyers and Ulam, and in the sense of Bourgin.