## Asymptotic behavior of solutions of retarded differential equations

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- by G. Ladas, Y. G. Sficas and I. P. Stavroulakis PDF
- Proc. Amer. Math. Soc.
**88**(1983), 247-253 Request permission

## Abstract:

In this paper we obtain sufficient conditions under which every solution of the retarded differential equation \[ (1)\quad x’(t) + p(t)x(t - \tau ) = 0,\quad t \geqslant {t_0},\], where $\tau$ is a nonnegative constant, and $p(t) > 0$, is a continuous function, tends to zero as $t \to \infty$. Also, under milder conditions, we prove that every oscillatory solution of (1) tends to zero as $t \to \infty$. More precisely the following theorems have been established. Theorem 1.*Assume that*$\int _{{t_0}}^\infty {p(t)dt = + \infty }$

*and*${\lim _{t \to \infty }}\int _{t - \tau }^t {p(s)ds < \pi /2}$

*or*$\lim {\sup _{t \to \infty }}\int _{t - \tau }^t {p(s)ds < 1}$.

*Then every solution of (1) tends to zero as*$t \to \infty$. Theorem 2.

*Assume that*$\lim {\sup _{t \to \infty }}\int _{t \to \tau }^t {p(s)ds < 1}$.

*Then every oscillatory solution of (1) tends to zero as*$t \to \infty$.

## References

- R. D. Driver,
*Ordinary and delay differential equations*, Applied Mathematical Sciences, Vol. 20, Springer-Verlag, New York-Heidelberg, 1977. MR**0477368** - Gerasimos Ladas,
*Sharp conditions for oscillations caused by delays*, Applicable Anal.**9**(1979), no. 2, 93–98. MR**539534**, DOI 10.1080/00036817908839256

## Additional Information

- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**88**(1983), 247-253 - MSC: Primary 34K25
- DOI: https://doi.org/10.1090/S0002-9939-1983-0695252-7
- MathSciNet review: 695252