Half-linear equations and characteristic properties of the principal solution

Abstract

The characteristic properties of the principal solution for half-linear differential equation ðaðtÞFðx 0 ÞÞ 0 þ bðtÞFðxÞ ¼ 0;where the functions a; b are positive and continuous for tX0 and FðuÞ ¼ juj pÀ2 u; p41; are investigated. In the linear case it is well-known that the principal solution is the ''smallest one'' in a neighbourhood of infinity; we show that this property continues to hold in the half-linear case. In addition, it is proved that the principal solutions can be fully characterized by means of two different integral criteria, which reduce to that one well known in the linear case. r

Perturbations of the half-linear Euler–Weber type differential equation

“…Recall also that it was shown in [4,5] that under the assumption that c(t) = 0 for large t the principal solution is the least solution of (1) near ∞ in the sense (similarly as in the linear case, see [18, Chapter XI])…”

“…i.e., w g is the solutions of the Riccati equation associated with (5) w +c(t) + (p − 1)r 1−q (t)|w| q = 0 (23) generated by g. Note that nonoscillation of (5) implies that g(t) > 0 on [t 2 , t 3 ), if t 2 is sufficiently large. Using integration by parts we have…”

A remark on an oscillation constant in the half-linear oscillation theory

“…Condition (6) is closely related to the integral characterization of the principal solution of half-linear differential equations, we refer to [16,17] for discussion concerning this assumption. Condition (5) is technical, we needed it to prove an asymptotic formula for nonprincipal solution of (2); see [8].…”

On intermediate solutions and the Wronskian for half-linear differential equations

“…We refer, in particular, to [3,6,9,[12][13][14] and to the monographs [1,10,16]. Here we recall some basic results, which will be useful in the sequel.…”