Abstract
In the present work we continue studying the solvability of the linear functional equations ${\sum_{j=1}^N }c_j F\circ a_j=H$ and also the strong and weak stability of the corresponding operator $\mathcal{P}$ (see the definition below). By analogy with the Cauchy and Jensen operators once more model operator $\widehat{\mathcal{P}}$ is considered, and the stability problems as well as some solvability problems for $\widehat{\mathcal{P}}$ are studied. Several unsolved problem of a general character are formulated.
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