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Unformatted text preview: Notes on cobbwebbing Patrick De Leenheer * January 15, 2009 Here we will consider the scalar nonlinear discrete-time system: x n +1 = f ( x n ) , x n ∈ [0 , + ∞ ) , n = 0 , 1 ,... (1) under certain assumptions about the function f . We say that x is a fixed point of (1) if f ( x ) = x . Denote the k th composition of f with itself as f k (e.g. f 2 = f ◦ f ), and thus x n = f n ( x ). We call a point y a period 2 point for (1) if f 2 ( y ) = y 6 = f ( y ). f is non-decreasing Let f : [0 , + ∞ ) → [0 , + ∞ ) be continuous and non-decreasing, i.e. x < y implies that f ( x ) ≤ f ( y ). We have the following result. Theorem 1. Every bounded solution sequence x n converges to a fixed point as n → + ∞ . Proof. Let x n be a bounded solution sequence. If f ( x ) = x , then x is a fixed point. If x < f ( x ), then x n = f n ( x ) is an non-decreasing sequence which is bounded, and therefore it converges to, say x * . Let’s show that x * must be a fixed point. Since f is continuous, the sequence f n +1 ( x ) converges to f ( x * ). But since f n +1 ( x ) is a subsequence of f n ( x ), it follows that...
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- Summer '06
- Continuous function, Fixed points, Patrick De Leenheer, solution sequence xn