Approximation Properties of Solutions of a Mean Value-Type Functional Inequality, II

Mathematics

Sung-Mo Yang¹  Ki-Suk Lee¹  MichaelTh. Rassias²  Soon-Mo Jung³ 

[1] Department of Mathematics Education, Korea National University of Education, Cheongju 28173, Korea;Institute of Mathematics, University of Zurich, CH-8057 Zurich, Switzerland;Mathematics Section, College of Science and Technology, Hongik University, Sejong 30016, Korea;

关键词: Hyers-Ulam stability;    Hyers-Ulam-Rassias stability;    generalized Hyers-Ulam stability;    mean value-type functional equation;   

DOI  :  10.3390/math8081299

来源: DOAJ

【 摘 要 】

Let X be a commutative normed algebra with a unit element e (or a normed field of characteristic different from 2), where the associated norm is sub-multiplicative. We prove the generalized Hyers-Ulam stability of a mean value-type functional equation, $f\left(x\right)-g\left(y\right)=(x-y)h(sx+ty)$, where $f,g,h:X\to X$ are functions. The above mean value-type equation plays an important role in the mean value theorem and has an interesting property that characterizes the polynomials of degree at most one. We also prove the Hyers-Ulam stability of that functional equation under some additional conditions.

【 授权许可】

Unknown