| ZHANG BINGGEN.[J].数学年刊A辑,1981,2(1):25~32 |
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| OSCILLATION AND NONOSCILLATION FOR SECOND-ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS |
| Received:November 16, 1979 |
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| In this paper，we consider oscillation and nonoscillation of the solution for the more general second-order functional differential equations. Some sufficient conditions which keep all the solutions oscillating have been obtained. They are theorems 1—3. Here is an example for the equation
\[(r(t){y^'}{(t)^'} + f(t,y(t),y(g(t)),{y^'}(t),{y^'}(h(t))) = 0,t \geqslant A \geqslant 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (5)\]
where \[g(t) \to \infty \] and h(t)->\[\infty \] if \[t \to \infty \].
Theorem I. Besides the conditions on the continuity and the existence and uniqueness of solution, suppose that the following conditions are also satisfied:
I. f(t, u, v, w, z) has the same sign as u, v, if u, v have the same sign;
II. r(t) is a continuous non-decreasing and positive function, where \[t \geqslant A\]. Suppose
\[R(t) = \int_A^t {\frac{{ds}}{{r(s)}}} \] and \[\mathop {\lim }\limits_{t \to \infty } R(t) = \infty \]
III. For every positive monotonic non-decreasing function or negative non- increasing function y(t), the following equation
\[\frac{1}{{R(t)}}\int_T^t {(\frac{1}{{r(\tau )}}} \int_T^\tau {f(s,y(s),y(g(s)),{y^'}(s),{y^'}(h(s)))ds)d\tau \to \infty \cdot sgn{\kern 1pt} {\kern 1pt} y,(t \to \infty )} \](6)
is correct, then all the solutions of equation (5) oscillate.
These theorems contain partial results of the articles [2—6].
The necessary, and sufficient condition in which eq. (5) has a non-oscillatory bounded solution is established in this paper. This is theorem 4.
In the present paper, we have proved six theorems which contain some results of foreign articles. |
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