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Some remarks on stability and solvability of linear functional equations. (English) Zbl 1130.39026
The author studies the functional operator $𝒫F:=F\left(ax+by\right)-\alpha F\left(x\right)-\beta F\left(y\right)$, where all number parameters satisfy the conditions $a>1,\phantom{\rule{1.em}{0ex}}b>1,\phantom{\rule{1.em}{0ex}}\alpha +\beta >1$, $F\in C\left(I,B\right)$ is a compact supported Banach space-valued continuous function of a single variable, and the points $\left(x,y\right)$ fill out the triangle $D=\left\{\left(x,y\right)\mid \left(ax+by\right)\le 1,\phantom{\rule{1.em}{0ex}}0\le x,y\le 1\right\}$. He also investigates the strong and weak stability of the $𝒫$ [see B. Paneah, Another approach to the stability of linear functional operators. Preprint 2006/13, ISSN 14437-739X, Institut für Mathematik. Uni Postdam. (2006)]. By analogy with the Cauchy and Jensen operators once more model operator $\stackrel{^}{𝒫}$ is considered, and the stability problems as well as some solvability problems for $\stackrel{^}{𝒫}$ are studied.
##### MSC:
 39B82 Stability, separation, extension, and related topics 39B22 Functional equations for real functions 39B52 Functional equations for functions with more general domains and/or ranges