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18.034 at ESG Spring, 2003Problems Related to theExistence and Uniqueness Theorem(Starting with this Problem Set, I’ll be using the notation of Apostol, in thatelements of Rnwill be denoted by bold italics and quantities that are inherentlyscalars will be denoted by light italics. There is some overlap in that elements ofR1might be considered scalars, but in that case we will be using the notation todistinguish between a dependent variable and an independent variable.)(1) Consider Equation (4) on Page 78. The proof starts with an example,using|x |max=max{|x1|, ..., |xn|},and finds for this norm the constants in Equation (4) are A =1/√n, B =1. Taken = 2 and show this geometrically. A suggestion is to find the level sets in R2for anorm of one. For instance, the level set of the Euclidean norm is a circle of radiusone.Find the values of A and B corresponding to the “taxicab norm” (the text callsthis the “sum norm”),|x |sum= |x1| + ···+ |xn|.You may do this geometrically for n = 2 and extend the argument to higher dimen-sions. The method of considering level sets will be useful here as well.(2) As described in the text in §1 of Chapter 5 (Page 75), every inner prod-uct (not necessarily the simple “dot product”) can be used to generate a norm,specifically|x | = x , x 1/2.Using only the three properites of Symmetry, Bilinearity (find the typo!) andPositive definiteness, show that any norm defined thusly in terms of an inner prod-uct must satisfy the Paral lelogram Law :|x + y |2+ |x − y |2=2|x |2+2|y|2.Illustrate geometrically why this law has the name it does.1(3) The point of this sequence of problems in that while every inner productgenerates a norm, the converse is not true; not every normed space is an inner prod-uct space. Show this by demontrating that the two norms considered in Problem (1)cannot be generated from an inner product. You wish to show that something isnot true, so you can pick some very simple elements of the space.So, we see now why we have to be careful in dealing with norms and innerproducts, and why the text sometimes seems to be picky about the distinction.(4) (Borelli & Coleman, §A.2, Problem 2). Let m and n be positive integerswithout common factors (i.e., relatively prime). Consider the initial value problemy= |y|m/n, y(0) = 0 , ♣where y ∈ R1.a Show that ♣ has the unique solution y(t)=0 if m ≥ n.b Show that ♣ has infinitely many solutions if m<n.(5) A shape of things to come, illustrating things we’ve done. Consider thedifferential equationx= f (x ), x =x1x2, f (x )=−x2x1, x (0) =AB.(In the above, it’s implied that x , f ∈ R2,andA and B are certainly not the sameas in Problem (1).)Find enough of the Picard iterates so that you can recognize a pattern. Then,find a simple expression for f (f (x )) in terms of x , and so find an equation forx= f (f (x )) (what do you do with the initial condition?) that has the x1andx2parts “separated.” You should be able to solve these by methods already seen.Check the series expansions of these solutions to see that they agree with the resultsof the Picard


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MIT 18 034 - Existence and Uniqueness Theorem

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