## 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_

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