Ulam-stability of a generalized linear functional equation, a fixed point approach
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Aequationes Mathematicae
Ulam-stability of a generalized linear functional equation, a fixed point approach Chaimaa Benzarouala and Lahbib Oubbi In memory of Professor Hha Mohamed Oudadess.
Abstract. We introduce the general functional equation m
Ai (f (ϕi (¯ x))) + b = 0,
x ¯ := (x1 , x2 , . . . , xn ) ∈ X n ,
i=1
and study its Ulam–Hyers-stability and hyperstability, using a fixed point approach, where m and n are positive integers, f is a mapping from a vector space X into a Banach space (Y, ), and, for every i ∈ {1, 2, . . . , m}, ϕi is a linear mapping from X n into X, Ai is a continuous endomorphism of Y and b ∈ Y . Our result covers most of the former ones in the literature concerning the stability and hyperstability of linear functional equations, as well as new situations. Mathematics Subject Classification. 39B05, 39B82, 47H10. Keywords. Hyers–Ulam stability, Hyers–Ulam hyperstability, Functional equation, Fixed point theorem.
1. Introduction The stability problem of functional equations started in 1940, when Ulam [14] asked the following question: Let X be a group, (Y, d) be a metric group, and be a positive number. Does there exist a number δ > 0 such that, whenever a function f : X → Y satisfies d(f (xy), f (x)f (y)) < δ, for all (x, y) ∈ X 2 , there exists a group homomorphism G : X → Y such that d(f (x), G(x)) < , for all x ∈ X? Whenever the answer to this problem is in the affirmative, one says that the equation of group-homomorphisms G(xy) = G(x)G(y) is Ulamstable. More generally, one says that a functional equation (E) is stable (resp.
C. Benzarouala and L. Oubbi
AEM
hyperstable), if any function satisfying “approximately” this equation must be “near” an exact solution (resp. a solution) of (E). The first affirmative answer to Ulam’s problem was given in the context of real Banach spaces in 1941 by Hyers [8], using what has subsequently come to be known as the direct method. Since then, the Ulam-stability problem has seen a huge expansion during the last decades. Actually, Hyers’ result has been extended to numerous equations, using different types of approximation functions and of methods (see for example [1,2,4,5,7,10–13] and the references therein). It is worthwhile to mention here the fixed point method introduced by Radu in [12]. In 2015, Bahyrycz and Olko considered in [1] the general linear functional equation ⎞ ⎛ m n (1.1) Bi f ⎝ bij xj ⎠ + B = 0, i=1
j=1
where f is a mapping from a linear space X into a Banach space (Y, ), Bi = 0 and bij are scalars, and B is a vector from Y . This equation generalizes most of the linear functional equations ever considered in the literature. The authors used a fixed point theorem due to Brzd¸ek et al. [3] to show the stability of (1.1). In the same year, Dong [6], using the same theorem of [3], proved the hyperstability of Eq. (1.1), whenever B = 0, with respect to two different approximation functions. In the same sense as in [6], Bahyrycz and Olko [2] showed the hyperstability of Eq. (1.1), with respect to a general approximation func
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