Contraction mapping principle

Let us now consider the case A < 0 in more detail. First, let us examine the associated Lamerey diagram (Fig. 13.2.6) with A < 0 when i/ > 1 (A -h7 < 0). Just as in the orientable case, a stable fixed point exists for // > 0 indeed, the map T is decreasing and contracting, so the interval [0, a p) = T(0)] is mapped into the inside itself by T, which implies the existence of a unique stable fixed point on this interval by virtue of the Banach contraction mapping principle. 

By applying the principle of contracting mappings, one can easily prove that the equation (2.15) has a solution continuous in t and x provided that... 

By virtue of the principle of contracting mappings the equation (3.12) has a unique periodic solution z-zjf) satisfying the inequality... 

Generally speaking, one must prove that the successive approximations converge it can be easily checked if the time interval t is finite and if iyo is sufficiently small, we can apply the Banach principle of contraction mappings. 

Since the contraction in the local map can be made arbitrarily strong and the derivative of the global map is bounded, the superposition T = To oTi inherits the contraction of the local map for all small p as well. It then follows from the Banach principle of contracting mappings (Sec. 3,15) that the map T has a unique stable fixed point on So- As this is a map defined along the trajectories of the system, it follows that the system has a stable periodic orbit in V which attracts all trajectories in V. The period of this orbit is the sum of two times the dwelling time t of local transition from Sq to S and the flight time from Si to Sq. The latter is always finite for all small p. It now follows from (12.1.4) that the period of the stable periodic orbit increases asymptotically of order tt/x/a This completes the proof. 

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