Abstract
Precise asymptotic behavior of regularly varying solutions of second order half-linear differential equations
Highlights
IntroductionRehák [10] considering only special case of the equation (A) with nonpositive differentiable coefficient q(t) established a condition which guarantees that all eventually positive increasing solutions are regularly varying
The second order half-linear differential equation (|x |αsgn x ) + q(t)|x|αsgn x = 0, (A)is considered under the assumption that (a) α > 0 is a constant, and (b) q : [a, ∞) → R, a > 0, is a continuous function.Note that (A) can be expressed as ((x )α∗) + q(t)xα∗ = 0, Corresponding author
The objective of this paper is to extend and improve results obtained in [7,11], by indicating assumptions that make it possible to determine the accurate asymptotic formulas for regularly varying solutions (1.3) of (A). This can be accomplished by elaborating the proof of Theorem A so as to gain insight into the interrelation between the asymptotic behavior of solutions of (A) and the rate of decay toward zero of the function
Summary
Rehák [10] considering only special case of the equation (A) with nonpositive differentiable coefficient q(t) established a condition which guarantees that all eventually positive increasing solutions are regularly varying. The objective of this paper is to extend and improve results obtained in [7,11], by indicating assumptions that make it possible to determine the accurate asymptotic formulas for regularly varying solutions (1.3) of (A). This can be accomplished by elaborating the proof of Theorem A so as to gain insight into the interrelation between the asymptotic behavior of solutions of (A) and the rate of decay toward zero of the function. For the convenience of the reader the definition and some basic properties of regularly varying functions are summarized in the Appendix at the end of the paper
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