Electron. J. Diff. Equ., Vol. 2016 (2016), No. 129, pp. 1-32.

Asymptotic behavior of intermediate solutions of fourth-order nonlinear differential equations with regularly varying coefficients

Aleksandra B. Trajkovic, Jelena V. Manojlovic

Abstract:
We study the fourth-order nonlinear differential equation
$$ \big(p(t)|x''(t)|^{\alpha-1}x''(t)\big)''+q(t)|x(t)|^{\beta-1}x(t)=0,\quad \alpha>\beta, $$
with regularly varying coefficient $p,q$ satisfying
$$ \int_a^\infty t\Big(\frac{t}{p(t)}\Big)^{1/\alpha}\,dt<\infty. $$
in the framework of regular variation. It is shown that complete information can be acquired about the existence of all possible intermediate regularly varying solutions and their accurate asymptotic behavior at infinity.

Submitted March 26, 2016. Published June 5, 2016.
Math Subject Classifications: 34C11, 34E05, 26A12.
Key Words: Fourth order differential equation; asymptotic behavior of solutions; positive solution, regularly varying solution, slowly varying solution.

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Aleksandra Trajkovic
University of Nis
Faculty of Science and Mathematics
Department of Mathematics
Visegradska 33, 18000 Nis, Serbia
email: aleksandra.trajkovic@pmf.edu.rs
Jelena V. Manojlovic
University of Nis
Faculty of Science and Mathematics
Department of Mathematics
Visegradska 33, 18000 Nis, Serbia
email: jelenam@pmf.ni.ac.rs

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