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Keywords:
third order nonlinear differential equation; singular nonlinearity; positive solution; decaying solution; asymptotic behavior; regularly varying functions
third order nonlinear differential equation; singular nonlinearity; positive solution; decaying solution; asymptotic behavior; regularly varying functions
Summary:
This paper is concerned with asymptotic analysis of strongly decaying solutions of the third-order singular differential equation $x^{\prime \prime \prime }+q(t)x^{-\gamma }=0$, by means of regularly varying functions, where $\gamma $ is a positive constant and $q$ is a positive continuous function on $[a,\infty )$. It is shown that if $q$ is a regularly varying function, then it is possible to establish necessary and sufficient conditions for the existence of slowly varying solutions and regularly varying solutions of (A) which decrease to $0$ as $t\rightarrow \infty $ and to acquire precise information about the asymptotic behavior at infinity of these solutions. The main tool is the Schauder–Tychonoff fixed point theorem combined with the basic theory of regular variation.
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