关于方程y＇=f(x,y)解存在的一个定理

AN EXISTENCE THEOREM OF THE SOLUTION OF EQUATION y＇=f_(x,y)

本文证明了在以下条件: 若f(x,y)是区域D:|x-x_0|≤a,|y-y_0|≤b上的函数,并且|f(x,y)|≤M,当固定x,y∈[y_0-b,y_0+b]时,f(x,y)是y的左连续递增涵数;当固定y,x∈[x_0-a,x_0+a]时,f(x,y)是x的递增涵数时,那么(E)在(?){a,b/M}上有递增函数解。

In this paper an existence theorem of the solution of(E). (E) y′=f_(x,y) y(X_0)=Y_0 is shown, where f_(x,y) is a left continuous monotone increasing function for fixed x and y∈[y_0- b, y_0+b], monoton increasing function for fixed y and x∈[x_0-a, x_0+a], and |f_(x,y)|≤M in D: |x -x_0|≤a, [y-y_0|≤b. Under these conditions the solution of (E) is a monotone increasing function in |x-x_0|≤(?) 0≤(?)≤h=min(a,b/M).