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Zbl 1130.39026
Paneah, Boris
Some remarks on stability and solvability of linear functional equations.
(English)
[J] Banach J. Math. Anal. 1, No. 1, 56-65, electronic only (2007). ISSN 1735-8787/e

The author studies the functional operator $\Cal{P}F{:}=F(ax+by)-\alpha F(x)-\beta F(y)$, where all number parameters satisfy the conditions $a>1,\quad b>1,\quad \alpha+\beta>1$, $F \in C(I,B)$ is a compact supported Banach space-valued continuous function of a single variable, and the points $(x,y)$ fill out the triangle $D=\{(x,y)\mid (ax+by)\leq 1,\quad 0\leq x,y\leq 1\}$. He also investigates the strong and weak stability of the $\Cal{P}$ [see {\it 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 $\widehat{\Cal{P}}$ is considered, and the stability problems as well as some solvability problems for $\widehat{\Cal{P}}$ are studied.
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B22 Functional equations for real functions
39B52 Functional equations for functions with more general domains

Keywords: strong stability; weak stability; Ulam stability; Cauchy operator; Jensen operator; à priori estimate; Banach space

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