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On the non-oscillation of solutions of some nonlinear differential equations of third order. (English) Zbl 1140.34415
From the introduction: We are concerned with non-oscillation of solutions of third-order nonlinear differential equations of the form
$(r(t)y''(t))' + q(t)y'(t) +p(t)y^\alpha(g(t)) = f(t),\quad t\geq t_0,\tag{1}$
and
$(r(t)y''(t))' + q(t) (y'(g_1(t)))^\beta+p(t)y^\alpha(g(t)) = f(t),\quad t\geq t_0,\tag{2}$
where $$t_0\geq 0$$ is a fixed real number, $$f$$, $$p$$, $$q$$, $$r$$, $$g$$ and $$g_1\in C([0,\infty),{\mathbb R})$$ such that $$r(t) > 0$$ and $$f(t)\geq 0$$ forall $$t\in [0,\infty)$$. Throughout the paper, it is assumed, for all $$g(t)$$, $$g_1(t)$$, $$\alpha$$ and $$\beta$$ appeared in (1) or (2), that $$g(t)\leq t$$ and $$g_1(t)\leq t$$ for all $$t\geq t_0$$; $$\lim_{t\to\infty}g(t) = \infty$$ and $$\lim_{t\to\infty}g_1(t) = \infty$$; both $$\alpha > 0$$ and $$\beta > 0$$ are quotients of odd integers.

##### MSC:
 34K11 Oscillation theory of functional-differential equations