September 12th, 2013, 08:58 PM  #1 
Newbie Joined: Sep 2013 Posts: 7 Thanks: 0  Recurrence relation
Given this recurrence relation: [attachment=0:2jibvwh8]recurrence.jpg[/attachment:2jibvwh8] with K>0, a parameter. If initial condition is K/4, show that the sequence converges to K/2 (unique positive equilibrium point). Can I show that using Banach fixedpoint theorem? Please help me. 
September 13th, 2013, 02:58 PM  #2  
Global Moderator Joined: May 2007 Posts: 6,728 Thanks: 689  Re: Recurrence relation Quote:
Need to verify that of x starts at K/4 it won't converge to 0.  
September 14th, 2013, 12:52 AM  #3 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Recurrence relation
As mathman has shown, computing the equilibrium point is not too hard. If you wish to confirm the uniqueness of this fixed point using Banach's contraction principle, you need to do a little further checking up. The function is differentiable and by the mean value theorem, it's a strict contraction whenever . Observe that . Thus . With little difficulty using your initial condition, you can verify that . Let . The function is a strict contraction since the absolute value of the derivative of on the interior of A is less than 1. Hence, by Banach's contraction principle, the fixed point in A is unique since A is a complete metric space. Any sequence defined by will converge to the unique fixed point of in , also by Banach's contraction principle. 

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