Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations

In this article, we study the asymptotic behavior of strongly decreasing solutions of the first-order nonlinear functional differential equation $$ x'(t)+p(t)| x(g(t))| ^{\alpha -1}x(g(t))=0, $$ where $\alpha $ is a positive constant such that $0<\alpha <1$, p(t) is a positive continuous function on $[a,\infty )$, a>0 and g(t) is a positive continuous function on $[a,+\infty )$ such that $\lim_{t\to \infty }g(t)=\infty $. Conditions which guarantee the existence of strongly decreasing solutions are established, and theorems are stated on the asymptotic behavior of such solutions, at infinity. The problem it is studied in the framework of regular variation, assuming that the coefficient p(t) is a regularly varying function, and focusing on strongly decreasing solutions that are regularly varying. In addition, g(t) is required to satisfy the condition $$ \lim_{t\to \infty }\frac{g(t)}{t}=1. $$ Examples illustrating the results are also given