Stability on Quinquevigintic Functional Equation in Different Spaces

The paper defines H via the 25th forward difference and proves a generalized Hyers–Ulam stability theorem in matrix normed spaces using a fixed-point operator Jf(x)=2^{-25q}f(2^q x) and a control σ that scales by 2^{25q} (Theorem 3.1). It also shows that any exact solution satisfies ζ(2u)=2^{25}ζ(u) by combining shifted instances of (1) (eqs. (2)–(14)). The candidate solution mirrors this logic: it recognizes H as a 25th finite difference, derives the same weighted identity that expresses f(2x)−2^{25}f(x) as a linear combination of H_f at specific arguments, defines the same contraction J, and applies the Diaz–Margolis fixed point alternative to obtain a unique limit V with H_V≡0 and the expected error bound. The main difference is stylistic: the model justifies the weights via an operator identity involving T (shift), while the paper derives them by explicit elimination. Both yield the same contraction scheme and estimates. Minor issues in the paper include typographical slips (e.g., “223q” clearly meaning 2^{25q}) and that the section titled “general solution” only proves the 2^{25}-homogeneity property rather than a full structural characterization, but these do not affect the core stability result the model addresses. Citations: definition of H and σ∗ in Theorem 3.1 and its bound (17) ; fixed-point consequences in the proof of Theorem 3.1 (21) and H_V≡0 ; derivation of ζ(2u)=2^{25}ζ(u) from (2)–(14) ; and the constants appearing in σ∗/corollaries .