Solution of the Ulam stability problem for Euler-Lagrange \((\alpha,\beta;k)\)-quadratic mappings (Q2794926)

The stability problem of functional equations originated from a question of \textit{S. M. Ulam} [A collection of mathematical problems. (Interscience Tracts in Pure and Applied Mathematics. No. 8.) New York and London: Interscience Publishers (1960; Zbl 0086.24101)] concerning the stability of group homomorphisms. \textit{D. H. Hyers} [Proc. Natl. Acad. Sci. USA 27, 222--224 (1941; Zbl 0061.26403)] gave a first affirmative partial answer to the question of Ulam for Banach spaces.NEWLINENEWLINEUsing the direct method, the authors prove the Hyers-Ulam stability of the Euler-Lagrange \((\alpha, \beta; k)\)-quadratic functional equation NEWLINE\[NEWLINEk f(\alpha x + \beta y) + f(k \beta x - \alpha y) = ( \alpha^2 + k \beta^2 ) (k f(x) + f(y)) NEWLINE\]NEWLINE in Banach spaces.NEWLINENEWLINEThe techniques and the ideas are simple and traditional. The paper contains a lot of grammatical mistakes, for example, `one find,' `one get' and `mixed Quadratic-Quartic.' The most serious problem is that the main theorems and the proofs are not clearly divided.