Annales Polonici Mathematici

On the uniqueness of continuous solutions of functional equations by Boles law Gawe l (Katowice) Abstract. We consider the problem of the vanishing of non-negative continuous so-lutions ψ of the functional inequalities (1) ψ(f(x)) ≤ β(x, ψ(x)) and (2) α(x, ψ(x)) ≤ ψ(f(x)) ≤ β(x, ψ(x)), where x varies in a fixed real interval I. As a consequence we obtain some results on the uniqueness of continuous solutions ϕ: I → Y of the equation (3) ϕ(f(x)) = g(x, ϕ(x)), where Y denotes an arbitrary metric space. It is well known that the iterative properties of the given function f occurring in (3) play a fundamental role in the theory of continuous solutions of this equation. For the most part, the assumptions imposed on f in the literature imply very simple dynamics of f; it is usually assumed that f has exactly one fixed point which is, moreover, attractive (cf. [5] or [6]). Papers in which the dynamical behaviour of f plays a role and this assumption is not imposed appear quite seldom. (The author can only quote [1]–[4].) In [2] one can find results on the vanishing of non-negative continuous solutions of α(x, ψ(x)) ≤ ψ(f(x)) as well as on the uniqueness of continuous solutions of (3). Now we want to investigate (1), (2) and (3) in the spirit of [2] but under complementary assumptions on the given functions α and g.