A generalized mixed type of quartic, cubic, quadratic and additive functional equation

We determine the general solution of the functional equation f(x+ky)+f(x−ky) = g(x+y)+g(x−y)+ +h(x)+h˜(y) for fixed integers k with k 6= 0, ±1 without assuming any regularity condition on the unknown functions f, g, h, h˜. The method used for solving these functional equations is elementary but exploits an important result due to Hosszu. The solution of this functional equation can also be determined in certain type ´ of groups using two important results due to SzekelyhidiВизначено загальний розв’язок функцiонального рiвняння f(x + ky) + f(x − ky) = g(x + y) + + g(x − y) + h(x) + h˜(y) для фiксованих цiлих k при k 6= 0, ±1 без припущення наявностi будь-якої умови регулярностi для невiдомих функцiй f, g, h, h˜. Метод, що використано для розв’язку цих функцiональних рiвнянь, елементарний, але базується на важливому результатi Хозу. Розв’язок цього функцiонального рiвняння може бути визначений у певному типi груп з використанням двох важливих результатiв Чекелiхiдi

Asymptotic behavior of alternative Jensen and Jensen type functional equations

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-2005 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1998 S.-M. Jung and in 2002-2005 the authors of this paper investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve our bounds and thus our results obtained, in 2003 for Jensen type mappings and establish new theorems about the Ulam stability of additive mappings of the second form on restricted domains. Besides we introduce alternative Jensen type functional equations and investigate pertinent stability results for these alternative equations. Finally, we apply our recent research results to the asymptotic behavior of functional equations of these alternative types. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology. © 2005 Elsevier SAS. All rights reserved

On the Hyers-Ularn stability problem for quadratic multi-dimensional mappings

In 1940 S. M. Ulam proposed the well-known Ulam stability problem. In 1941 D. H. Hyers solved this problem for linear mappings. According to P. M. Gruber (1978) this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982-1999 we solved the above Ulam problem for different mappings. In this paper we solve the Hyers-Ulam stability problem for quadratic multi-dimensional mappings. © Birkhäuser Verlag, 2002

Six-dimensional Landau inequalities

Let X be a complex Banach space, and let t → T(t)(||T(t)|| ≤ 1, t ≥ 0) be a strongly continuous contraction semigroup (on X) with infinitesimal generator A.In this paper we prove that the following five inequalities (Equation presented) hold for every x ∈ D(A6) and for a fixed number i = 1,2,3,4,5, where Ri1(6) = (Equation presented) such that our symbol (Equation presented) holds as well as ϵ1 = 213655, ϵ2 = 5676, ϵ3 = 215, ϵ4 = 243676, ϵ5 = 2555. Analogous inequalities hold for strongly continuous contraction cosine functions. In this case of cosine functions constants Ri1 (6) are replaced by Ri3 (6) = (Equation presented)' Inequalities are established also for uniformly bounded strongly continuous semigroups and cosine functions. © 1999 Demonstratio Mathematica. All rights reserved

On the Ulam stability of mixed type mappings on restricted domains

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-1998 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1983 F. Skof was the first author to solve the Ulam problem for additive mappings on a restricted domain. In 1998 S.M. Jung investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve the bounds and thus the results obtained by S.M. Jung, in 1998. Besides we establish the Ulam stability of mixed type mappings on restricted domains. Finally, we apply our recent results to the asymptotic behavior of functional equations of different types. © 2002 Elsevier Science (USA). All rights reserved

On the Ulam stability of mixed type mappings on restricted domains

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-1998 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1983 F. Skof was the first author to solve the Ulam problem for additive mappings on a restricted domain. In 1998 S.M. Jung investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve the bounds and thus the results obtained by S.M. Jung, in 1998. Besides we establish the Ulam stability of mixed type mappings on restricted domains. Finally, we apply our recent results to the asymptotic behavior of functional equations of different types. © 2002 Elsevier Science (USA). All rights reserved

ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

Let G be a commutative group, Y a real Banach space and f : G → Y. We prove the Ulam-Hyers stability theorem for the cyclic functional equation 1 |H|∑h∈H f(x + h � y) = f(x) + f(y) for all x, y ∈ Ω, where H is a finite cyclic subgroup of Aut(G) and Ω ⊂ G � G satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation 1/N ∑Nk=1 f(z + ωkζ) = f(z) + f(ζ) for all (z, ζ) ∈ Ω, where f : ℂ → Y, ω = e2πi/N and Ω ⊂ ℂ2 has four-dimensional Lebesgue measure 0. � 2015 Australian Mathematical Publishing Association Inc

Quadratic functional equations in a set of Lebesgue measure zero

Let R be the set of real numbers, Y a Banach space and f: R → Y. We prove the Hyers-Ulam stability theorem for the quadratic functional inequality. {norm of matrix}f(x+y)+f(x-y)-2f(x)-2f(y){norm of matrix}≤ε for all (x, y). ∈ Ω, where Ω⊂R2 is of Lebesgue measure 0. Using the same method we dealt with the stability of two more functional equations in a set of Lebesgue measure 0. © 2014 Elsevier Inc