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# Now suppose that yo â‚¬ R and v' (c) = F(v(z),2) , v(0) = yo has unique solution U: [0, 1] = R and that Un (0) = yo: Prove that un 57 uniformly. (Hint: Sin...

## Question

###### Now suppose that yo â‚¬ R and v' (c) = F(v(z),2) , v(0) = yo has unique solution U: [0, 1] = R and that Un (0) = yo: Prove that un 57 uniformly. (Hint: Since uniform convergence is given metric, previous homework problem shows that it is enough to show that given any subsequence Unk there is a further subsequence Unki with Unki 4 uniformly: To prove this, argue that the Arzela-Ascoli theorem applies to {un n â‚¬ N} and use uniqueness of the solution of the ODE): (d) Now suppose that for a

Now suppose that yo â‚¬ R and v' (c) = F(v(z),2) , v(0) = yo has unique solution U: [0, 1] = R and that Un (0) = yo: Prove that un 57 uniformly. (Hint: Since uniform convergence is given metric, previous homework problem shows that it is enough to show that given any subsequence Unk there is a further subsequence Unki with Unki 4 uniformly: To prove this, argue that the Arzela-Ascoli theorem applies to {un n â‚¬ N} and use uniqueness of the solution of the ODE): (d) Now suppose that for all y â‚¬ R, the differential equation v (c) = F(v(x),x) , v(0) = y has a unique solution %y: [0,1] = R Explain why part (c) implies that the map R + C([0, 1],R) given by y + %y is continuous when we give C([0,1], R) the metric induced by the supremum norm_  #### Similar Solved Questions

##### Test: Final Exam This Queation:Tlma Romainlng 022977 tincompietuThis Te9t- 20 blecossitCunnnecongoc Cd nvanoqvalgolmmana connun%i0l JG Inueonee sndud Curiahon 0l S15C0 Tra uere Jtoyo dacho 4 tho nverju Ncowassijn-conuk ciketunl Irotti Fir rogcawnl & Usa # Ieyur Rentnc4iJuineme0l58310EornpdashanIdert thuotatAlustole TRrd Demnl nincosJin-dud;,Thu P-4ue (Reund to thue doomzl Elacus as needud ) Tvele" IIna cn-Iubon Leal7 Mhs arnge Icss Obsotved < crafizanty â‚¬ Norent Irt P_IUuc e sn ial
Test: Final Exam This Queation: Tlma Romainlng 022977 tin compietu This Te9t- 20 blecossit Cunnn econgoc Cd nvanoqvalgolmmana connun%i0l JG Inueonee sndud Curiahon 0l S15C0 Tra uere Jtoyo dacho 4 tho nverju Ncowassijn-conuk ciketunl Irotti Fir rogcawnl & Usa # Ieyur Rentnc4i Juineme0l58310 Eornp...
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