9 research outputs found

    Oscillation of differential systems of neutral type

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    summary:We study oscillatory properties of solutions of systems \[ \begin{aligned} {[y_1(t)-a(t)y_1(g(t))]}^{\prime }=&p_1(t)y_2(t), y_2^{\prime }(t)=&{-p_2}(t)f(y_1(h(t))), \quad t\ge t_0. \end{aligned} \

    Oscillatory properties of solutions of second order nonlinear neutral differential inequalities with oscillating coefficients

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    summary:This paper deals with the second order nonlinear neutral differential inequalities (Aν)(A_\nu ): (1)νx(t){z(t)+(1)νq(t)f(x(h(t)))}0, (-1)^\nu x(t)\,\lbrace \,z^{\prime \prime }(t)+(-1)^\nu q(t)\,f(x(h(t))) \rbrace \le 0,\ tt00,t\ge t_0\ge 0, where  ν=0 \ \nu =0\ or  ν=1, \ \nu =1,\  z(t)=x(t)+p(t)x(tτ), \ z(t)\,=\,x(t)\,+\,p(t)\,x(t-\tau ),\  0<τ= \ 0<\tau =\ const,  p,q,h:[t0,)R \ p,q,h:[t_0,\infty )\rightarrow R\  f:RR \ f:R\rightarrow R\ are continuous functions. There are proved sufficient conditions under which every bounded solution of (Aν)(A_\nu ) is either oscillatory or $\ \liminf \limits _{t\rightarrow \infty }|x(t)|=0.
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