The Dominance Property of Solutions

Abstract

Examples 1.13–1.16 demonstrate that the solutions of equation (1.1) differ not only in their zero distributions but also in their relative size near infinity. It is obvious how does one compare the order of magnitude of two nonvanishing functions near infinity, but it is not self explanatory how to compare the ‘sizes’ of oscillatory solutions whose values change between positive and negative numbers and they attain the value zero infinitely many times. Nevertheless, it is clear that some terms of a linear combination of solutions may dictate the essential behaviour of the sum more than others, i.e., they dominate the sum due to their relatively large extremal values.