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**Extra resources for Approximation of Hilbert Space Operators: v. 2**

**Example text**

Another interesting difference, parallel to differential calculus, is the discrete analogue of the Fundamental Theorem of Calculus 2 , stated here. 1. The following statements hold. n-1 (i) L:~x(k) = x(n)- x(no). 3) x(n). 1, Problem 3. 2 The Fundamental Theorem of Calculus states that (i) t (ii) d df(x) (fa' = f(b)- f(a), f(t)dt) = f(x). 1 Difference Calculus 51 We would now like to introduce a third property that the operator !!.. has in common with the derivative operator D. Let be a polynomial of degree k.

In fact, x* is a repelling point. Proof (i) Suppose that 1/'(x*)l < M < 1. Then there is an interval J = (x*-y, x*+y) containing x* such that 1/'(x)l :::: M < 1 for all x E J. (Why? ) For x(O) E J, we have lx(l)- x*l = lf(x(O))- f(x*)l. By the Mean Value Theorem, there exists; between x(O) and x* such that lf(x(O))- f(x*)l = 1/'(;)llx(O)- x*l. Thus lf(x(O))- x*l :::: Mlx(O)- x*l. Hence lx(l)- x*l :::: Mlx(O)- x*l. 2) shows that x(1) is closer to x* than x(O). Consequently, x (1) E J. By induction we conclude that lx(n)- x*l:::: M"lx(O)- x*l.

C. Pielou [ 1] referred to the following equation as the discrete logistic equation: x(n + 1) = ax(n) 1 + f3x(n) , a> 1, f3 < 0. (a) Find the positive equilibrium point. (b) Demonstrate, using the stair step diagram, that the positive equilibrium point is asymptotically stable, taking a = 2 and f3 = 1. 4. Find the equilibrium points and determine their stability for the equation 6 x(n+l)=5--. x(n) 22 I. Dynamics of First Order Difference Equations 5. (a) Draw a stair step diagram for the Eq. 3.