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# can be satisfied by requiring 2 and 4

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Unformatted text preview: ! ! ! !". This must be true for all !. Replace ! by ! and solve for !! ! (!) to obtain !! ! ! = − ! (! ! ! )! ! !! ! ! ! !". [9] When these last two expressions are substituted back into [8] an identity is obtained, showing that it is a solution for [8]. This expression for !! ! ! is also bounded since, using [3]: ! ! ! !! ! ! !! ! ! ! !" ≤ ! ! ! !! ! !! ! ≤ !" ! !" !! ! ! !" ! !!" ! !" ! 10 Ch5Lecs ≤ !" /!. Adding [7] and [9] gives [5], the promised bounded solution. Note that [9] is required in order that !! ! (!) be bounded, so our solution is unique. □ To construct the integral operator !: !0 ℝ+ , ℝ! → !0 ℝ+ , ℝ! whose fixed point will give us the local stable manifold, replace !(!) by !(! ! ) in [5]: ! ! ! = ! !" ! + ! ! !! ! ! !! ! ! ! ! !" − ! ! !! ! ! ! !! ! ! ! !", [10] where ! = !! ! (0). Exercise 5 shows that a fixed point of [10] is also a solution of [1]. Theorem 5.3 (Local Stable Manifold) Let ! be hyperbolic, ! ∈ ! ! ! , ! ≥ 1, for some neighborhood ! of 0, and ! ! = !(! ) as ! → 0. Denote the linear stable and unstable subspaces of ! by ! ! and ! ! . Then there is a ! ⊂ ! such that the local stable manifold of [1] ! !!"# 0 = {! ∈ ! ! 0 :!! ! ⊂ !, ! ≥ 0} , is a Lipschitz graph over ! ! that is tangent to ! ! at 0. Moreover, ! ! (0) is a ! ! manifold. Note that the phrase “Lipschitz graph over ! ! ” only means that the unstable component of ! !!"# is a Lipschitz function of the stable component. ! Remark The existence and pro...
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