Aequationes mathematicae

We present an approach to solving a number of functional equations for functions with values in abstract convex cones. Such cones seem to be good generalizations of, e.g., families of nonempty compact and convex subsets or nonempty closed, bounded and convex subsets of a normed space. Moreover, we study some related stability problems.

Two semidiscrete collocation approximations using smooth cubic splines are developed as approximations to the solution of two-point linear parabolic boundary value problems.L ∞-convergence results are presented for these two approximations as well as the piecewise linear Galerkin approximation. Several computational examples are given to illustrate the convergence results and demonstrate the applicability of the method.

It is proved that if f and g are complex-valued multiplicative functions such that $ g(2n + 1) - Af(n) \to 0 (n \to \infty) $ , for some $ A \neq 0 $ , then either $ f(n) \to 0 (n \to \infty) $ , or $ f(n) = n^s $ , $ 0 \le {\rm Re} s < 1 $ and $ A = f(2), g(n) = f(n) $ for every odd n.

SoitG un groupe abélien. Thèorème 1:Une fonction f: G → ℝ est solution de l'équation: $$\max \{ f(x + y),f(x - y)\} = f(x) + f(y)(x,y \in G)$$ si et seulement si: f(x) = |a(x)|, où a: G → ℝ est additive. Théorème 2:On suppose que tout élément de G est divisible par 6. Alors f: G → ℝ est solution de: $$\max \{ f(x + y),f(x - y)\} = f(x)f(y)(x,y \in G)$$ si et seulement si: f(x) ≡ 0 ou f(x) = exp|a(x)|, où a: G → ℝ est additive.