常微分方程的比较定理及其应用

<正> 本文§1讨论由驻定二阶系统(1.1)确定非驻定二阶系统(1.2)解的各种性态的比较定理.由于对(1.2)的定性研究,不能使用象Poincare-Bendixson理论那样的有力工具,显得办法很少,这种比较定理的意义之一在于,通过它可以将更多的工具间接用于(1.2).文1]首先给出了许多这种比较定理,伹尚未得到使用,而且存在错误,本节将予纠正然

COMPARISION THEOREM OF ORDINARY DIFFERENTIAL EQUATIONS AND ITS APPLICATION

1. Consider the comparision theorem of the qualities of solutions of two-dimensional non-autonomous systems. These qualities, boundedness, stability, stability in the large, etc., are determined by two-dimensional autonomous systems.With the theorem. more tools, such as Poincare-Bendixson theory, can be indirectly put to use in twodimensional non-autonomous systems.2. We give a general theorem on non-autonomous system, the trival solution of which is stable in the large.3. Consider the equations: x+ q(t, x, x)f(x)(x)x + g(x)h(x) = 0 x+ q(t, x, x)f(x)x + p(t)g(x) = e(t, x, x) x+ A(t, x, x)x + g(x)h(x) = 0 x+ f(x) + h(x)x]x + p(t)g(x) = 0 etc. We give sufficient and necessary conditions of the trivial solution that is stable in the large.The method given in this paper can also be adopted to carry out reseaches into other qualities in the large of equations with variable coefficients such as boundedhess, momotonicity, and the tendency of all solutions towards zero. Its main use lies in helping loosen the requirements and obtain sufficient and necessary conditions.