New PDF release: Advanced Calculus, Revised Edition

By Lynn H. Loomis, Shlomo Sternberg

ISBN-10: 0867201223

ISBN-13: 9780867201222

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21) < ∞. Then K has a unique fixed point x such that   ∞ K n (x) − x ≤  θj  K(x) − x , x ∈ C. 22) j=n Our first objective is to give some concrete values for the existence time T0 . 3. 5 (Picard-Lindel¨of). Suppose f ∈ C(U, Rn ), where U is an open subset of Rn+1 , and f is locally Lipschitz continuous in the second argument. Choose (t0 , x0 ) ∈ U and δ > 0, T > t0 such that [t0 , T ] × Bδ (x0 ) ⊂ U . 23) t0 x∈Bδ (x0 ) L(t) = |f (t, x) − f (t, y)| . 24) Note that M (t) is nondecreasing and define T0 via T0 = sup{T > t0 | M (T ) = δ}.

84) ∂h where we have added h as a subscript to emphasize the dependence on the parameter h. 85) 1 − (1 − x)e and there are two fixed points x1 = 0 and x1 = 1. As h increases these points will approach each other and collide at some critical value hc . Above this value there are no periodic orbits and all orbits converge to −∞ since P (x) < x for all x ∈ R (show this). To complete our analysis suppose h < hc and denote by x1 < x2 the two fixed points of P (x). Define the iterates of P (x) by P 0 (x) = x and P n (x) = P (P n−1 (x)).

Now we come to the proof of our anticipated result. 16. Suppose U = R × Rn and for every T > 0 there are constants M (T ), L(T ) such that |f (t, x)| ≤ M (T ) + L(T )|x|, (t, x) ∈ [−T, T ] × Rn . 11) are defined for all t ∈ R. Proof. Using the above estimate for f we have (t0 = 0 without loss of generality) t |φ(t)| ≤ |x0 | + (M + L|φ(s)|)ds, t ∈ [0, T ] ∩ I. 7) M LT (e − 1). 68) L Thus φ lies in a compact ball and the result follows by the previous lemma. 69) where M (t), L(t) are locally integrable.

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Advanced Calculus, Revised Edition by Lynn H. Loomis, Shlomo Sternberg


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