Moderately growing solutions of third-order differential equations with a singular nonlinearity and regularly varying coefficients

This paper is concerned with asymptotic analysis of moderately growing solutions of the third-order differential equation with singular nonlinerity

((((x′)^α₁*)′)^α₂*)′ +q(t)x^-β= 0;

(A) where α₁, α₂ and β are positive constants and σ : [α;∞)→(0;∞) is a continuous regularly varying function of index σ, α > 0 and u^γ* = |u|^γ sgnu. An application of the theory of regular variation allows us to establish necessary and sufficient conditions for the existence of regularly varying solutions of (A) which are moderately growing and to acquire precise information about the asymptotic behavior at infinity of these solutions. The Schauder-Tychonoff fixed point technique is used.