【摘要】 建立了Banach空间常微分方程初值问题在弱拓扑下解的一个逼近定理:设fn(t,x)与f(t,x)在R0=[t0,t0+a]×B(x0,b)上是弱弱连续的(n=1,2,…),且{fn(t,x)}在R0上弱一致收敛于f(t,x),又设0<α≤a,xn(t):[t0,t0+α]B(x0,b)弱可微且满足方程:x′n(t)=fn(t,xn(t))xn(t0)=zn n=1,2,…这里x′n(t)表示xn(t)的弱导数,{zn}弱收敛到x0.如果{xn(t)}在[t0,t0+α]上弱一致收敛于x(t),则x(t)是初值问题x′(t)=f(t,x(t))x(t0)=x0的弱解.更多还原

【Abstract】 In this paper,we establish an approximation theorem of weak solution of an ordinary differential equation in Banach space as followsTheorem Let E be weakly sequentially complete Banach space,fn(t,x)(n=1,2,…) and f(t,x) be weakly weakly continuous on R0=[t0,t0+a]×B(x0,b),{fn(t,x)} be converge weakly uniformly to f(t,x) on R0,0<α≤a,xn(t): [t0,t0+α]B(x0,b) be weakly differentiable andx′n(t)=fn(t,xn(t))xn(t0)=zn n=1,2,…where x′n(t) is weakly derivative of xn(t) and {zn} is converge weakly to x0.If {xn(t)} is converge weakly uniformly to x(t) on [t0,t0+α],then x(t) is the weakly solution of Cauchy problemx′(t)=f(t,x(t))x(t0)=x0更多还原