De Haan type increasing solutions of half-linear differential equations

We study asymptotic behavior of eventually positive increasing solutions to the half-linear equation ( r ( t ) | y ′ | α − 1 sgn y ′ ) ′ = p ( t ) | y | α − 1 sgn y , where r , p are positive continuous functions and α ∈ ( 1 , ∞ ) . We give conditions which guarantee that any such a solution is in the class Γ (in the de Haan sense). We also discuss regularly varying solutions and connections with a generalized regular variation and other related concepts. The results can be viewed as a half-linear extension of existing statements for linear equations, but in some aspects they are new also in the linear case.