The asymptotic properties of solutions of the equation $u^{\prime \prime \prime }(t)=p_1(t)u(\tau _1(t))+p_2(t)u^{\prime }(\tau _2(t))$, are investigated where $p_i:[a,+\infty [\rightarrow R \;\;\;\;(i=1,2)$ are locally summable functions, $\tau _i:[a,+\infty [\rightarrow R\;\;\;(i=1,2)$ measurable ones and $\tau _i(t)\ge t\;\;\;(i=1,2)$. In particular, it is proved that if $p_1(t)\le 0$, $p^2_2(t)\le \alpha (t)|p_1(t)|$, \[\int _a^{+\infty }[\tau _1(t)-t]^2p_1(t)dt<+\infty \;\;\;\text{and}\;\;\; \int _a^{+\infty }\alpha (t)dt<+\infty ,\] then each solution with the first derivative vanishing at infinity is of the Kneser type and a set of all such solutions forms a one-dimensional linear space.
@article{107495,
author = {Ivan Kiguradze},
title = {On asymptotic properties of solutions of third order linear differential equations with deviating arguments},
journal = {Archivum Mathematicum},
volume = {030},
year = {1994},
pages = {59-72},
zbl = {0806.34063},
mrnumber = {1282113},
language = {en},
url = {http://dml.mathdoc.fr/item/107495}
}
Kiguradze, Ivan. On asymptotic properties of solutions of third order linear differential equations with deviating arguments. Archivum Mathematicum, Tome 030 (1994) pp. 59-72. http://gdmltest.u-ga.fr/item/107495/
Contributi allo studio asintotico dell’ equazione $x^{\prime \prime \prime }(t)+p(t)x(t)=0 $, Ann. Math. Pura ed Appl., 51(1960), 301-328. | MR 0121528 | Zbl 0095.06903
On the Kneser problem for functional differential equations, (Russian) Differentsial’nie Uravneniya 27 (1991), No 11, 1879-1892. | MR 1199212
On some properties of solutions of second order linear functional differential equations, Proc. of the Georgian Acad. of Sciences, Mathematics 1 (1993), No 5, 545-553. | MR 1288650 | Zbl 0810.34067