Electron. J. Diff. Equ., Vol. 2014 (2014), No. 206, pp. 1-14.

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

George E. Chatzarakis, Kusano Takasi, Ioannis P. Stavroulakis

Abstract:
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.

Submitted June 26, 2014. Published October 2, 2014.
Math Subject Classifications: 34C11, 26A12
Key Words: Functional differential equation; strongly decreasing solution; regularly varying function; slowly varying solution.

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George E. Chatzarakis
Department of Electrical and Electronic Engineering Educators
School of Pedagogical and Technological Education (ASPETE)
14121, N. Heraklio, Athens, Greece
email: geaxatz@otenet.gr, gea.xatz@aspete.gr
  Kusano Takasi
Professor Emeritus at: Department of Mathematics
Faculty of Science, Hiroshima University
Higashi-Hiroshima 739-8526, Japan
email: kusanot@zj8.so-net.ne.jp
Ioannis P. Stavroulakis
Department of Mathematics, University of Ioannina
451 10 Ioannina, Greece
email: ipstav@cc.uoi.gr

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