Solution of the Ulam stability problem for cubic mappings

In 1968 S.M. Ulam proposed the general problem: When is it true taht by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or spproximately true. In 1978 P.M. Gruber stated that this kind of stability problems are of particular interest in probability theory and in the case of functional equations of different types. In 1982-1998 we solved above Ulam problem for linear mappings and also established analogous stability problems for quadratic mappings. In this paper we introduce the new cubic mappings C : X → Y, satisfying the cubic functional equation C(x1 + 2x2) + 3C(x1) = 3C(x1 + x2) + C(x1 - x2) + 6C(x2) for all 2-dimensional vectors (x1,x2) ∈ X2, with X a linear space (Y a real complete linear space), and then solve the Ulam stability problem for the above-said mappings C

Solution of a quadratic stability Ulam type problem

summary:In 1940 S. M. Ulam (Intersci. Publ., Inc., New York 1960) imposed at the University of Wisconsin the problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist”. According to P. M. Gruber (Trans. Amer. Math. Soc. 245 (1978), 263–277) the afore-mentioned problem of S. M. Ulam belongs to the following general problem or Ulam type problem: “Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this objects by objects, satisfying the property exactly?” In 1941 D. H. Hyers (Proc. Nat. Acad. Sci. 27 (1941), 411–416) established the stability Ulam problem with Cauchy inequality involving a non-negative constant. Then in 1989 we (J. Approx. Theory, 57 (1989), 268–273) solved Ulam problem with Cauchy functional inequality, involving a product of powers of norms. Finally we (Discuss. Math. 12 (1992), 95–103) established the general version of this stability problem. In this paper we solve a stability Ulam type problem for a general quadratic functional inequality. Moreover, we introduce an approximate eveness on approximately quadratic mappings of this problem. These problems, according to P. M. Gruber (1978), are of particular interest in probability theory and in the case of functional equations of different types. Today there are applications in actuarial and financial mathematics, sociology and psychology, as well as in algebra and geometry

Solution of the Ulam stability problem for quartic mappings

In 1940, S. M. Ulam proposed at the University of Wisconsin the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist." In 1968, S. M. Ulam proposed the general problem: "When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?" In 1978, P. M. Gruber proposed the Ulam type problem: "Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?" According to P. M. Gruber, this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982-1998, we solved the above Ulam problem, or equivalently the Ulam type problem for linear mappings and also established analogous stability problems for quadratic and cubic mappings. In this paper we introduce the new quartic mappings F : X → Y, satisfying the new quartic functional equation F(x1 + 2x2) + F(x1 - 2x2) + 6F(x1) = 4[F(x1 + x2) + F(x1 - x2) + 6F(x2)] for all 2-dimensional vectors (x1,x2) ∈ X2, with X a linear space (Y := a real complete linear space), and then solve the Ulam stability problem for the above mappings F

Solution of the Ulam stability problem for cubic mappings

In 1968 S.M. Ulam proposed the general problem: When is it true taht by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or spproximately true. In 1978 P.M. Gruber stated that this kind of stability problems are of particular interest in probability theory and in the case of functional equations of different types. In 1982-1998 we solved above Ulam problem for linear mappings and also established analogous stability problems for quadratic mappings. In this paper we introduce the new cubic mappings C : X → Y, satisfying the cubic functional equation C(x1 + 2x2) + 3C(x1) = 3C(x1 + x2) + C(x1 - x2) + 6C(x2) for all 2-dimensional vectors (x1,x2) ∈ X2, with X a linear space (Y a real complete linear space), and then solve the Ulam stability problem for the above-said mappings C

Homomorphisms and derivations on C*-ternary algebras assocoated with a generalized Cauchy-Jensen type additive functional equation

In this paper, we prove the generalized Hyers-Ulam stability of C∗-ternary homomorphisms and C∗-ternary derivations on C∗-ternary algebras associated with the generalized Cauchy-Jensen type additive functional equation for all xi; xj ∈ X where n ∈ Z+ is a fixed integer with n ≥ 3. &nbsp