We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability. In particular we provide some information on hyperstability and the fixed point methods.

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In the present paper, we consider the linear functional equationwhere E and G are normed linear spaces over K, K is either R or C, α and β are given scalars in K, F : E → G is a given function and Φ : E → G is the unknown function.In an earlier paper, we studied the case E = R or [0, +∞) or (0, +∞), G = R, and we gave there the general solution of (1) and also its continuous and differentiable solutions by using elementary direct methods. The results presented here extend the previous ones to the general case. Mathematics Subject Classification (2000). Primary 39B12, 39B52.