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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(ax+by)-αF(x)-βF(y), where all number parameters satisfy the conditions a>1,b>1,α+β>1, FC(I,B) is a compact supported Banach space-valued continuous function of a single variable, and the points (x,y) fill out the triangle D={(x,y)(ax+by)1,0x,y1}. 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 𝒫 ^ is considered, and the stability problems as well as some solvability problems for 𝒫 ^ are studied.
39B82Stability, separation, extension, and related topics
39B22Functional equations for real functions
39B52Functional equations for functions with more general domains and/or ranges