Unique fixed point

Lemma 58 If C is a contraction defined on a complete metric space X then C has a unique fixed point which can be determined by the method of successive approximations as follows. 

Show that the map = 1 -i- sinx has a unique fixed point. Is it stable ... 

Based on the given function f, we will develop an auxiliary function g. We will show it to be a contraction, which is associated to a unique fixed point. This property will then lead to the existence of the inverse function f " ". We start with the description of a contraction and its fixed point. 

In the next three steps, we will show that there is a unique fixed point x in X such that x = 0(x). 

Having shown that a contraction has a unique fixed point, we consider the givens of the Inverse Function Theorem. Based on f(x), we propose an auxiliary function and prove it to be a contraction. 

Hence, g(x) is a contraction as defined by Inequality (4.22). Being a contraction, it has a unique fixed point. 

It is obvious that dx/dt = 0 when x = 0 or x = The unique fixed point x = 0 existing for < 0 is stable while for > 0 is unstable. Furthermore, for ju, > 0, the fixed points x = are stable. This result can be expressed diagrammatically as follows. 

Since the derivative of the 1-D map in eq. (8.13) is just A.(l — lx), the fixed point at 0 is stable for A. < 1, that is, when it is the unique fixed point, and it is unstable for all larger values of A.. Thus, for small values of A., the sequence should approach zero. A bit more algebra shows that the second fixed point, at X = 1 — 1/A, is stable for 1 < A, < 3. What happens if we pick a value of A. greater than 3 At this point, it seems to be time to plug in some numbers, so let us choose a few representative values of A. and a starting point xq (the choice of xq will not affect the asymptotic solution—try it yourself ), say, 0.25, and see what happens. Some results are shown in Table 8.2. 

For all X and p sufficiently small, the map (11.4.6) has a unique fixed point. This point (at the origin) is stable when // < 0 and unstable when 0] when /i 0, it does not undergo bifurcations. Besides this fixed point, the map (11.4.6) may have points of period two. Therefore, to examine the bifurcations one should consider the second iteration of the map. There may not be periodic points of other periods more than two because the second iteration of (11.4.6) is a monotonically increasing one-dimensional map — periodic orbits of such maps are fixed points. 

To prove the above lemma, let us continue the map T on V U so that TM M " for all M G V U. This extended map remains contracting and takes V into V. Thus, it has a unique fixed point M which attracts forward orbits of all points of V. This fixed point is a fixed point of the original map T provided it belongs to U otherwise, it is a virtual fixed point if it lies in V p. In the latter case, M = TM = by construction. Vice versa, if M" " G V U, then TM — M. This means that — M (by uniqueness... 

C.3. 19. Determine the condition imder which the above map has (1) a unique fixed point and, (2) no fixed points. 

Consider now the case where cj is not an integer. According to C.3. 31, the map (C.3.12) has a unique fixed point close to zero in this case. 

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