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# Article

 Title: On asymptotic properties of solutions of third order linear differential equations with deviating arguments (English) Author: Kiguradze, Ivan Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 30 Issue: 1 Year: 1994 Pages: 59-72 Summary lang: English . Category: math . Summary: 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. (English) Keyword: differential equation with deviating arguments Keyword: Kneser type solutions Keyword: vanishing at infiniting solution MSC: 34K15 MSC: 34K99 idZBL: Zbl 0806.34063 idMR: MR1282113 . Date available: 2008-06-06T21:25:39Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107495 . Reference: [1] C. Villari: 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. Zbl 0095.06903, MR 0121528 Reference: [2] I. T. Kiguradze and D. I. Chichua: On the Kneser problem for functional differential equations.(Russian) Differentsial’nie Uravneniya 27 (1991), No 11, 1879-1892. MR 1199212 Reference: [3] I .T. Kiguradze: 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. Zbl 0810.34067, MR 1288650 .

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