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# Article

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Keywords:
differential equation with deviating arguments; Kneser type solutions; vanishing at infiniting solution
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.
References:
[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. MR 0121528 | Zbl 0095.06903
[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
[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. MR 1288650 | Zbl 0810.34067

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