On the Continous Dependence of Continuous Solutions of a Functional Equation of n-th Order on Given Functions (original) (raw)

On some extensions of the Gołąb-Schinzel functional equation

1994

Composite functional equations (arising in applications) are pre sented that may be interpreted as extensions of the Goląb-Schinzel equation and as modifications of d'Alembert's equation. Depending on the type of the considered equation, continuous, and finite rate of growth solutions are discussed. Geometric interpretations are given. Introduction. The Golab-Schinzel equation has been the topic of many papers. The equation was introduced by J. Aczel [1] and was treated in considerable detail by S. Gołąb and A. Schinzel [14] and by J. Aczel and S. Gołąb [5]. The general solution and general continuous or measurable solutions have been dealt with by several authors (cf. [2], [4], [6], [10], [12], [14], [15], [17], [19]). Certain generalizations have been obtained recently (cf. [7], [8], [9], [10]). On the other hand, d'Alembert's equation has been a classical theme in theory and applications (cf. [2], [3], [4]). In this paper we present some functional equations that are related both to Golab-Schinzel and d'Alembert's equation, but may be viewed properly as extensions of the Golab-Schinzel equation. In particular, in Section 3, we deal with the functional equation and we prove that if / : R -• R is a continuous solution then, either / = 0 or f(x) = ex 2 + 1 for some c > 0. For each c < 0 the function f(x) = ex 2 + 1 satisfies this functional equation for all x and y in a neighbourhood of zero. It seems to be of interest that, in contrast to the well known property of

On continuous solutions of a problem of R.Schilling

Results in Mathematics, 1995

then the zero function is the only solution f : IR → IR of (1) satisfying (2) and right-hand-side or left-hand-side continuous at each point of the interval (−q/(1 − q), −q/(1 − q) + δ) or of the interval (q/(1 − q) − δ, q/(1 − q)) with some δ > 0.