Asymptotic behavior of oscillatory solutions (Q1109214)

The aim in this paper is to study the asymptotic behavior of the oscillatory solutions of certain delay differential equations of the form \[ (1)\quad x'(t)+p(t)x(t-\tau)+q(t)x(t-\sigma)=0,\quad t\geq t_ 0 \] and of certain neutral equations of the form \[ (2)\quad (d/dt)[x(t)- px(t-\tau)]+q(t)x(t-\sigma)=0,\quad t\geq t_ 0. \] Our results, combined with known oscillaton results or with known results about the asymptotic behavior of the nonoscillatory solutions of Eqs. (1) and (2), lead to sufficient conditions for the trivial solution of Eqs. (1) and (2) to be asymptotically stable. In our opinion, the main contribution of this paper is that it shows how oscillation theory may be used, as another tool, in establishing new stability results for differential equations of diverse nature, like Eqs. (1) and (2) above.