John Michael Rassias

On approximation of approximately linear mappings by linear mappings

Abstract Assume A is a normed linear space, B is a Banach space, and f : A → B is a mapping “approximately linear.” We solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist.”

Solution of a problem of Ulam

Abstract In this paper we solve the following Ulam problem : “Give conditions in order for a linear mapping near an approximately linear mapping to exist” and establish results involving a product of powers of norms [ S. M. Ulam, “A Collection of Mathematical Problems,” Interscience, New York, 1961 ; “Problems in Modern Mathematics,” Wiley, New York, 1964 ; “Sets, Numbers, and Universes,” MIT Press, Cambridge, MA, 1974 ]. There has been much activity on a similar “ e-isometry” problem of Ulam [ J. Gervirtz, Proc. Amer. Math. Soc. 89 (1983) , 633–636; P. Gruber, Trans. Amer. Math. Soc. 245 (1978) , 263–277; J. Lindenstrauss and A. Szankowski, “Non-linear Perturbations of Isometries,” Colloquium in honor of Laurent Schwartz, Vol. I, Palaiseau, 1985 ]. This work represents an improvement and generalization of the work of D. H. Hyers [ Proc. Nat. Acad. Sci USA 27 (1941) , 222–224].

Approximate ternary Jordan derivations on Banach ternary algebras

Let A be a Banach ternary algebra over a scalar field R or C and X be a ternary Banach A-module. A linear mapping D:(A,[ ]A)→(X,[ ]X) is called a ternary Jordan derivation if D([xxx]A)=[D(x)xx]X+[xD(x)x]X+[xxD(x)]X for all x∊A. In this paper, we investigate ternary Jordan derivations on Banach ternary algebras, associated with the following functional equation f((x+y+z)/4)+f((3x−y−4z)/4)+f((4x+3z)/4)=2f(x). Moreover, we prove the generalized Ulam–Hyers stability of ternary Jordan derivations on Banach ternary algebras.

Solution and Stability of a Mixed Type Cubic and Quartic Functional Equation in Quasi-Banach Spaces

We obtain the general solution and the generalized Ulam-Hyers stability of the mixed type cubic and quartic functional equation in quasi-Banach spaces.

On the Ulam stability of Jensen and Jensen type mappings on restricted domains

In 1941 Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 Bourgin was the second author to treat this problem for additive mappings. In 1982–1998 Rassias established the Hyers–Ulam stability of linear and nonlinear mappings. In 1983 Skof was the first author to solve the same problem on a restricted domain. In 1998 Jung investigated the Hyers–Ulam stability of more general mappings on restricted domains. In this paper we introduce additive mappings of two forms: of “Jensen” and “Jensen type,” and achieve the Ulam stability of these mappings on restricted domains. Finally, we apply our results to the asymptotic behavior of the functional equations of these types.

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.

Lecture Notes on Mixed Type Partial Differential Equations

This book discusses various parts of the theory of mixed type partial differential equations with boundary conditions such as: Chaplygins classical dynamical equation of mixed type, the theory of regularity of solutions in the sense of Tricomi, Tricomis fundamental idea and one-dimensional singular integral equations on non-Carleman type, Gellerstedts characteristic problem and Frankls non-characteristic problem, Bitsadze and Lavrentjevs mixed type boundary value problems, quasi-regularity of solutions in the classical sense. Some of the latest results of the author are also presented in this book.

Solution and Stability of a Mixed Type Additive, Quadratic, and Cubic Functional Equation

We obtain the general solution and the generalized Hyers-Ulam-Rassias stability of the mixed type additive, quadratic, and cubic functional equation .

EXTENDED HYERS-ULAM STABILITY FOR A CAUCHY-JENSEN MAPPINGS

In 1940, Ulam proposed the famous Ulam stability problem. In 1941, Hyers solved the well-known Ulam stability problem for additive mappings subject to the Hyers condition on approximately additive mappings. In 2003–2006, the last author of this paper investigated the Hyers–Ulam stability of additive and Jensen type mappings. In this paper, we improve results obtained in 2003 and 2005 for Jensen type mappings and establish new theorems about the Ulam stability of additive and alternative additive mappings. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.

Stability of the Jensen-Type Functional Equation in ∗-Algebras: A Fixed Point Approach

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in 𝐶∗-algebras and Lie 𝐶∗-algebras and also of derivations on 𝐶∗-algebras and Lie 𝐶∗-algebras for the Jensen-type functional equation 𝑓((𝑥