Banach空间中常微分方程Cauchy问题的近似解与解的关系

THE RELATION BETWEEN APPROXIMATE FOLUTION ANDSOLUTION FOR THE CAUCHY PROBLEM IN THE BANACH SPACE

讨论Ｂａｎａｃｈ空间中常微分方程Ｃａｕｃｈｙ问题的近似解与解的关系，得到一个Ｃａｕｃｈｙ问题的近似解与解的关系的定理：定理设ｆ＿ｎ∈Ｃ［Ｒ＿０，Ｅ］（ｎ≥１），ｆ∈Ｃ［Ｒ＿０，Ｅ］，序列｛ｆ＿ｎ｝在Ｒ＿０上一致收敛于ｆ；又设０＜α≤ａ，ｘ＿ｎ∈Ｃ￣１［［ｔ＿０，ｔ＿０＋α］，Ｂ（ｘ＿０，ｂ）］，且满足Ｃａｕｃｈｙ问题ｘ＇＿ｎ（ｔ）＝ｆ＿ｎ（ｔ，ｘ＿ｎ（ｔ））ｘ＿ｎ（ｔ＿０）＝ｚ＿ｎ其中ｔ∈［ｔ＿０，ｔ＿０，ｔ＿０＋α］，ｎ＝１，２，…，ｚ＿ｎ∈Ｅ，ｚ＿ｎ→ｘ＿０（ｎ→∞），如果ｘ＿ｎ（ｔ）在［ｔ＿０，ｔ＿０＋α］上一致收敛于ｘ（ｔ），则ｘ∈Ｃ￣１［［ｔ＿０，ｔ＿０＋α］，Ｂ（ｘ＿０，ｂ）］，且对ｔ∈［ｔ＿０，ｔ＿０＋α］，有ｘ＇（ｔ）＝ｆ（ｔ。

We discuss the relation between approximate solu tion for the canchy problem of ordinary differen-tial equation on Banach space and obtain a theorem about it.Theorem Let f_n∈C[R_0,E](n≥1),f∈C[R_0,E],the seqence{f_n}uniformly converges to fon R_0;and let 0<α<a,x_n∈C￣1[[t_0,t_0+a],B(x_0,b)] and satisfy the Cauchy problemwhere t∈[t_0,t_0+a],n=1,2,…, if x_n(t) uniformly converges to x(t)on[t_0,t_0+a],x∈C￣1[[t_0,t_0+a],B(x_0,b)]and x′(t)=f(t,x(t))x(t_0)=x_0 for every t∈[t_0,t_0+a]。