How can we show that&nbsp;x(t)−y(t)=∫s−εtf(x(r),a)−f(y(r),a(r)):dr&nbsp;implies&nbsp;x(s)−y(s)=(f(x(s),a)−f(x(s),α(s)))ε+o(ε)

Question

How can we show that&amp;nbsp;x(t)−y(t)=∫s−εtf(x(r),a)−f(y(r),a(r)):dr&amp;nbsp;implies&amp;nbsp;x(s)−y(s)=(f(x(s),a)−f(x(s),α(s)))ε+o(ε)

Daisy Patrick · Accepted Answer

Step 1Given:&amp;nbsp;‖x(s)-y(s)-ε(f(x(s),a)-f(x(s),α(s)))‖≤∫s-εs‖f(x(t),a)-f(x(s),a)‖dt+∫s-εs‖f(y(t),α(t))-f(x(s),α(s))‖dt.Using continuity the first term is bounded byεsupt∈[s-ε;s]‖f(x(t),a)-f(x(s),a)‖=o(ε).Step 2For the second term we use‖f(y(t),α(t))-f(x(s),α(s))‖≤‖f(y(t),α(t))-f(y(s),α(s))‖+‖f(y(s),α(s))-f(x(s),α(s))‖.We only need to show that the expression above is o(1). For the first term this follows from continuity and for the second term we use that f is differentiable to get‖f(y(s),α(s))-f(x(s),α(s))‖≤C‖x(s)-y(s)‖=o(1).

How can we show that x(t)-y(t)=\int_{s-ε}^tf(x(r),a)-f(y(r),a(r))\:{\rm d}r implies x(s)-y(s)=(f(x(s),a)-f(x(s),\alpha(s)))\varepsilon+o(ε)

Answered question

Answer & Explanation

New Questions in Calculus 2