Geodesic
On asymptotic properties of solutions of third order linear differential equations with deviating arguments
Archivum mathematicum, Tome 30 (1994) no. 1, pp. 59-72

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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.
Classification : 34K15, 34K99
Keywords: differential equation with deviating arguments; Kneser type solutions; vanishing at infiniting solution 
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     title = {On asymptotic properties of solutions of third order linear differential equations with deviating arguments},
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     pages = {59--72},
     publisher = {mathdoc},
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     zbl = {0806.34063},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1994__30_1_a6/}
}
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Kiguradze, Ivan. On asymptotic properties of solutions of third order linear differential equations with deviating arguments. Archivum mathematicum, Tome 30 (1994) no. 1, pp. 59-72. http://geodesic.mathdoc.fr/item/ARM_1994__30_1_a6/