Growth properties of solutions to a linear differential equation

Abstract

SynopsisDominance properties of solutions to Lny + p(x)y = 0, where Ln is a disconjugate operator, are compared to dominance properties of solutions to its adjoint.

Zeros of solutions and of Wronskians for the differential equation $L\sb ny+p(x)y=0$

“…, Un_I} which is defined in this way, exists, it is unique, and it is a basis of solutions of (1.1). This basis had been used extensively in [1][2][3][4][5].…”

“…Accordingly, we defined Skin [1] as the set of solutions of (1.1) on (a, 00) such that S(y, x+) == k on some ray (x o ' 00). The partition of the solution space into the sets Sk for admissible values of k is discussed in [1,5]. One property of Sk that we are going to use here is that the elements of Sk are either all oscillatory or all nonoscillatory solutions of (1.1).…”

“…Let the disconjugate differential operator Ln be Our results will be formulated in terms of a special basis {uo(x) , ... , Un_I (x)} of the solution space of (1.1), which has been used in [1][2][3][4][5]. Since uo(x) , ... , Un_I (x) will be defined in the next section by boundary value problems (2.2)0-(2.2)n_1 ' we first mention a known property [7] of boundary value problems of equation (1.1): If k quasiderivatives of a solution y vanish at the endpoint a and n -k quasiderivatives of y vanish at b, a < b , then n -k must be odd if p(x) > 0 and even if p(x) < O.…”

Minors of the Wronskian of the differential equation Lny + p(x)y = 0. II. Dominance of solutions

“…are integers of admissible parities. Some of their useful properties are now summarised: For various applications of S(y, x + ) see, for example, [5][6][7]12]. Now we define a basis for the solution space of equation (1.1) by means of boundary value problems.…”

“…Their results were formulated in terms of dominance of sets as denned by Dolan and Klaasen [1] as follows. The set A of solutions dominates another set B if u e A and v e B imply that u + cv e A for every c. Jones [7] studied questions of dominance for equation (1.1) and its adjoint. In [3] we made several attempts to compare solutions one with another; however, none of them now seems satisfactory.…”