Nicole Brillouët-Belluot

On Some Recent Developments in Ulam's Type Stability

We present a survey of some selected recent developments (results and methods) in the theory of Ulams type stability. In particular we provide some information on hyperstability and the fixed point methods.

Ulam's Type Stability

Quite often (e.g., in applications) we have to do with functions that satisfy some equations only approximately. There arises a natural question what errors we commit when we replace such functions by the exact solutions to those equations. Some tools to evaluate them are provided within the theory of the Ulam (also Hyers-Ulam) type stability. The issue of Ulams type stability of ( rst, functional, but next also di erence, di erential and integral) equations has been a very popular subject of investigations for the last nearly fty years (see, e.g., [3, 8, 9, 10]). The main motivation for it was given by S.M. Ulam in 1940. The following de nition somehow describes the main ideas of such stability notion for equations in n variables (R+ stands for the set of nonnegative reals). De nition 1. Let A be a nonempty set, (X, d) be a metric space, C ⊂ R+ n be nonempty, T map C into R+, and F1,F2 map a nonempty D ⊂ X into X n . We say that the equation F1φ(x1, . . . , xn) = F2φ(x1, . . . , xn) (1) is T stable provided for every ε ∈ C and φ0 ∈ D with

Multiplicative symmetry and related functional equations

SummaryLet (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G → G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)Equivalently, iff: G → G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y ∈ G), whereF: G × G → G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y ∈ G), (E) whereF: G × G → G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y ∈ K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K → K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K → K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G → G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1) Equivalently, iff: G → G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y ∈ G), whereF: G × G → G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *). In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y ∈ G), (E) whereF: G × G → G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y ∈ K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K → K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K → K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.

Ulam’s Type Stability 2013

1 Department of Mathematics, Pedagogical University, Podchorązych 2, 30-084 Krakow, Poland 2 Ecole Centrale de Nantes, Departement d’Informatique et de Mathematiques, 1 rue de la Noe, BP92101, 44321 Nantes Cedex 3, France 3Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea 4Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

On a class of iterative-difference equations

Solving the class of second order iterative-difference equations is a difficult problem, proposed as an open problem in Brillouët-Belluot [Aequationes Math. 61 (3) (2001), p. 304]. In this paper, we use three methods to approach such a class of equations, finding their affine solutions, proving the existence of their bounded continuous solutions and constructing their continuous and piecewise continuous solutions.