On Ulam's type stability of the Cauchy additive equation (original) (raw)
Stability of Functional Equations and Properties of Groups
Annales Mathematicae Silesianae
Investigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups involved.
Hyers-Ulam-Rassias stability of the -quadratic functional equation
Journal of Inequalities in Pure & Applied Mathematics, 2007
In this paper we obtain the Hyers-Ulam-Rassias stability for the functional equation 1 |K| ∑ k∈K f(x + k · y) = f(x) + f(y), x, y ∈ G, whereK is a finite cyclic transformation group of the abelian group (G, +), acting by automorphisms ofG. As a consequence we can derive the Hyers-Ulam-Rassias stability of the quadratic and the additive functional equations.
Ulam's type stability of a functional equation deriving from quadratic and additive functions
Journal of Mathematical Inequalities, 2007
In this paper, we continue the investigation of functional equation which is begun by the authors in the first part. We also prove the Hyers-Ulam stability for the following mixed quadratic-additive functional equation in quasi-Banach spaces. f (x + my) + f (x − my) = 2 f (x) − 2m 2 f (y) + m 2 f (2y) m is even f (x + y) + f (x − y) − 2(m 2 − 1) f (y) + (m 2 − 1) f (2y), m is odd.
Ulam-Hyers Stability of Additive and Reciprocal Functional Equations: Direct and Fixed Point Methods
In this paper, the authors established the generalized Ulam - Hyers stability of additive functional equation which is originating from arithmetic mean of n consecutive terms of an arithmetic progression in Intuitionistic fuzzy normed spaces and reciprocal functional equation originating from n-consecutive terms of a harmonic progression in Non - Archimedean Fuzzy normed spaces using direct and fixed point methods. Applications of the above functional equations are also given.
Stability of generalized additive Cauchy equations
International Journal of Mathematics and Mathematical Sciences, 2000
A familiar functional equation f (ax + b) = cf (x) will be solved in the class of functions f : R → R. Applying this result we will investigate the Hyers-Ulam-Rassias stability problem of the generalized additive Cauchy equation
The stability of a Cosine-Sine Functional Equation on abelian groups
arXiv (Cornell University), 2018
In this paper we establish the stability of the functional equation f (x − y) = f (x)g(y) + g(x)f (y) + h(x)h(y)), x, y ∈ G, where G is an abelian group. f (xy) = f (x)g(y) + g(x)f (y), x, y ∈ G and cosine functional equation g(xy) = g(x)g(y) − f (x)f (y), x, y ∈ G on amenable group G. Chung, choi and Kim [8] poved the Hyers-Ulam stability of f (x + σ(y)) = f (x)g(y) − g(x)f (y), x, y ∈ G where σ : G → G is an involution of the abelian group G. Recently, in [3, 4] the authors obtained the stability of the functional equations f (xy) = f (x)g(y) + g(x)f (y) + h(x)h(y), x, y ∈ G,
Generalized Hyers–Ulam stability for general additive functional equations in quasi-β-normed spaces
In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. The first author of this paper investigated the Hyers-Ulam stability of Cauchy and Jensen type additive mappings. In this paper we generalize results obtained for Jensen type mappings and establish new theorems about the Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces.
Mathematics and Statistics, 2024
Functional equations are important and exciting concepts in mathematics. They make it possible to investigate fundamental algebraic operations and create fascinating solutions. The concept of functional equations develops further creative methods and techniques for resolving issues in information theory, finance, geometry, wireless sensor networks, and other domains. These include geometry, algebra, analysis, and so on. In recent decades, several writers in many domains have covered the study of various types of stability. Many authors have studied the stability of various functional equations in great detail, with the traditional case (Archimedean) revealing more fascinating results. Recently, researchers have used NANS to study the equivalent conclusions of stability problems from various functional equations. In this research, we examine the Hyers-Ulam stability of the hexic-quadraticadditive mixed-type functional equation g(mx + ny) + g(mx − ny) + g(nx + my) + g(nx − my) = m 2 n 2 (m 2 + n 2)[g(x + y) + g(x − y) − 2g(x)− g(y) − g(−y) + 2[g(mx) + g(nx) + g(my) + g(ny)]− (m + n)[g(y) − g(−y)] where m, n ∈ Z, m is fixed such that m, n ̸ ∈ {−1, 0, 1} and m + n ̸ = 0 in NANS and also provided some suitable counterexamples.
The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping. Recently, Park 9 investigated the approximate additive mappings, approximate Jensen mappings, and approximate quadratic mappings in 2-Banach spaces and proved the 2 Abstract and Applied Analysis generalized Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation, and the quadratic functional equation in 2-Banach spaces. This is the first result for the stability problem of functional equations in 2-Banach spaces.