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z_N%_A@*&-o@7t_gQu*=Km7klfTvquN=*sVjR<5Z0`R#7hlAWwvRrSwxC2)(CYpOvp zu0+1Fa$U7*vMX^Po$(E=-%w4M>`LPQ|JqH}r2pr+rJCmFR_i~va$7Y$%$25nof#Xg z-%)M$(B>zqEza6}SAR7#+Rfgb39&F$^nUI9A38jKVmqjtQ8EX{w)pi2MWYET;|%Wo~41beEgs0U8N5H3}sqMNdWw Drq4-B diff --git a/zavislost-na-podmince.tex b/zavislost-na-podmince.tex index 8e02217..8be8f4b 100644 --- a/zavislost-na-podmince.tex +++ b/zavislost-na-podmince.tex @@ -100,7 +100,7 @@ Označme pro účely následující věty $\pdv{}{w}$ derivaci ve směru $w \in Dle Věty \ref{thm-unique-sol-lineq} existuje právě jedno řešení a je definované na celém intervalu, kde je definovaná $A(t)$. Chceme dokázat, že $u(t) = \pdv*{\varphi}{w}(t, t_0, x_0)$. Z definice máme, že $$ \pdv*{\varphi}{w}(t, t_0, x_0) = \lim_{h \rightarrow 0} \frac{1}{h} (\varphi(t, t_0, x_0 + hw) - \varphi(t, t_0, x_0). $$ - Vezmeme $t$ pevné tak, aby $(t, t_0, x_0) \in G$, tedy $x(t)$ je dobře definované. Vezmeme dost malé tak, aby $\varphi(t, t_0, x_0 + hw)$ bylo definované. Položme $y_h(t) := \varphi(t, t_0, x_0 + hw)$. + Vezmeme $t$ pevné tak, aby $(t, t_0, x_0) \in G$, tedy $x(t)$ je dobře definované. Vezmeme dost malé $h$ tak, aby $\varphi(t, t_0, x_0 + hw)$ bylo definované. Položme $y_h(t) := \varphi(t, t_0, x_0 + hw)$. Definujeme funkci $\eta_h(t) = \frac{1}{h}(y_h(t) - x(t)) - u(t)$. Ukážeme, že $\lim_{h \rightarrow 0} \eta_h(t) = 0$. Pišme $$ \eta'_h(t) = \frac{1}{h}(y_h'(t) - x'(t)) - u'(t) = \frac{1}{h} (f(y_h(t), t) - f(x(t), t)) - \nabla_x f(x(t), t) u(t). $$