How can we show that \(\displaystyle{x}{\left({t}\right)}-{y}{\left({t}\right)}={\int_{{{s}-ε}}^{{t}}}{f{{\left({x}{\left({r}\right)},{a}\right)}}}-{f{{\left({y}{\left({r}\right)},{a}{\left({r}\right)}\right)}}}:{\left\lbrace{r}{m}{d}\right\rbrace}{r}\) implies

Tony Mccarthy

Tony Mccarthy

Answered question

2022-03-25

How can we show that x(t)-y(t)=s-εtf(x(r),a)-f(y(r),a(r))dr implies x(s)-y(s)=(f(x(s),a)-f(x(s),α(s)))ε+o(ε)

Answer & Explanation

Janessa Foster

Janessa Foster

Beginner2022-03-26Added 12 answers

Step 1
Given: x(s)-y(s)-ε(f(x(s),a)-f(x(s),α(s)))s-εsf(x(t),a)-

-f(x(s),a)dt+s-εsf(y(t),α(t))-f(x(s),α(s))dt.
Using continuity the first term is bounded by
εsupt[s-ε;s]f(x(t),a)-f(x(s),a)=o(ε).
Step 2
For 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))Cx(s)-y(s)=o(1).

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?