The archetypal equation and its solutions attaining the global extremum
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Aequationes Mathematicae
The archetypal equation and its solutions attaining the global extremum Mariusz Sudzik
Abstract. Let (Ω, F , P) be a probability space and let α, β : F → R be random variables. We provide sufficient conditions under which every bounded continuous solution ϕ : R → R of the equation ϕ(x) = Ω ϕ (α(ω)(x − β(ω))) P(dω) is constant. We also show that any non-constant bounded continuous solution of the above equation has to be oscillating at infinity. Mathematics Subject Classification. Primary 39B22, Secondary 39B05. Keywords. Archetypal equation, Linear functional equation, Equation with rescaling, Iterative equation, Bounded continuous function, Global extremum.
1. Introduction The paper concerns the linear functional equation of infinite order ϕ(x) = ϕ (a(x − b)) μ(da, db),
(1.1)
R2
where μ is a given Borel probability measure on R2 . We can think about equation (1.1) also in the language of random variables. Given a probability space (Ω, F, P) let (α, β) : Ω → R2 be a fixed random vector with a distribution μ. Then the equation ϕ(x) = ϕ (α(ω) (x − β(ω))) P(dω) (1.2) Ω
is equivalent to (1.1). The study of equation (1.1) was initiated by Derfel [5] in 1989. He considered equation (1.2) under the additional assumption α > 0 a.s. and he noticed
M. Sudzik
AEM
that the behaviour of solutions of (1.1) crucially depends on the value of the integral ln |a|μ(da, db). (1.3) K := R2
He proved that, under some additional technical assumptions, if K ∈ (−∞, 0) then equation (1.1) has only trivial, i.e. constant, solutions in the class of bounded continuous functions. If K ∈ (0, ∞) and α > 0 a.s., then he constructed a non-trivial bounded continuous solution. The details of this theorem and its proof are included in [4], Theorem 1.1. Many deep results connected with equation (1.1) were obtained and collected by Bogachev et al. [3] and [4]. In those papers equation (1.1) was named by the authors as the archetypal equation since it is a rich source of many famous functional and differential-functional equations. For instance, one can obtain a balanced version of the pantograph equation; details can be found in [4]. It is worth noting that functional equations of a more general form than (1.2) were also considered in the literature; an interested reader, can find them e.g. among papers written by Polish mathematicians from the Silesian University: Baron, Kapica and Morawiec (see, for instance, [2] and [8]). Moreover, a lot of interesting remarks connected with the archetypal equation, its special cases and equations of similar forms can be found in [1] and [6]. Here we can confine ourselves to (1.1) since equations with linearly transformed arguments are of prominent interest in applications. We have to add that the main results obtained in this paper hold under the condition P(α < 0) > 0.
(1.4)
In this case we still do not have extensive knowledge about the existence of non-constant solutions of the archetypal equation in the class of bounded continuous functions. It is worth noting that every
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