Stability of the Exponential Functional Equation in Riesz Algebras

We deal with the stability of the exponential Cauchy functional equation in the class of functions mapping a group ( , +) into a Riesz algebra . The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem. 1. Introduction In 1979 Baker et al. (cf. [1]) proved that the exponential functional equation in the class of functions mapping a vector space to the real numbers is superstable; that is, any function satisfying, with given , the inequality is either bounded or exponential (satisfies (1)). Then Baker generalized this famous result in [2]. We quote this theorem here since it will be used in the sequel. Theorem 1 (cf. [2, Theorem？？1]). Let be a semigroup and let be given. If a function satisfies the inequality for all , then either for all or for all . After that the stability of the exponential functional equation has been widely investigated (cf., e.g., [3–6]). This paper will primarily be concerned with the question if similar result holds true in the class of functions taking values in Riesz algebra with the common notion of the absolute value of an element stemming from the order structure of . The main aim of the present paper is to show that the superstability phenomenon does not hold in such an order setting. However, we prove that the exponential functional equation (1) is stable in the Ulam-Hyers sense; that is, for any given satisfying inequality (3) there exists an exponential function which approximates uniformly on in the sense that the set is bounded in . As a method of investigation we apply spectral representation theory for Riesz spaces; to be more precise, we use the Yosida Spectral Representation Theorem for Riesz spaces with a strong order unit. For some recent results concerning stability of functional equations in vector lattices we refer the interested reader to [7–12]. 2. Preliminaries Throughout the paper , , , and are used to denote the sets of all positive integers, integers, real numbers and nonnegative real numbers, respectively. For the readers convenience we quote basic definitions and properties concerning Riesz spaces (cf. [13]). Definition 2 (cf. [13, Definitions？？11.1 and？？22.1]). We say that a real linear space , endowed with a partial order , is a Riesz space if exists for all and We define the absolute value of by the formula . A Riesz space is called Archimedean if, for each , the inequality holds whenever the set is bounded above.