By Leonard Lovering Barrett
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Additional info for An Introduction to Tensor Analysis
1, J is nonempty, compact, connected and invariant. 26 3 Functional Diﬀerential Equations on Manifolds To prove that J is the maximal compact invariant set, suppose H is any compact invariant set. Since K attracts H and H is invariant, it follows that H ⊂ Φnr K and, therefore, H ⊂ J . It remains to prove that J is independent of the choice of the compact set K which attracts all compact sets of C 0 . For this, denote J = J (K) and J (K1 ) = n≥0 Φnr K1 where K1 is a compact set which attracts all compact sets of C 0 .
21) written in the form 0 x0 (·, ϕ) = ϕ; x(t) = t f (xs (·, ϕ))ds, t ∈ [0, αϕ ). [dµ(θ)]x(t+θ)+Dϕ+ −r 0 In this case, it is not possible to use the contraction principle to get a ﬁxed point of a map deﬁned by the right hand side of the equalities in this equation. However, it is possible to show that, on an appropriate class of functions on an interval [−r, a] with a small, one has an α-contraction. This gives existence. The maximal interval of existence is obtained by invoking Zorn’s lemma in the usual way.
Thus, for a certain t∗ , we ∗ have u ¯(t ) = 0 and v¯(t∗ ) = c. Both solutions deﬁne then the same periodic orbit and there exists α ∈ (0, 4) such that u(t), v(t) = u ¯(t + α), v¯(t + α) = −v(t + α), u(t + α) , ∀t ∈ R. But u(t) = −v(t + α) = −u(t + 2α) = v(t + 3α) = u(t + 4α) and, since u(t) has period 4, we need to have α = 1. Then u(t), v(t) satisfy the conditions required above since u(t) = v(t + 3) = v(t − 1) v(t + 2) = u(t + 3) = u(t − 1) = −v(t). 14) with period T = 4 and such that θ(t) = π − θ(t − 2).
An Introduction to Tensor Analysis by Leonard Lovering Barrett