Posted by J. Holmer on September 10, 2007, 12:41 am So there is one degree of freedom in the problem. Note if you select b_1, then b_2 will be determined, as will b_3, b_4, and so on. But the problem is that if you select b_1 incorrectly, then the resulting sequence b_n will not be bounded. One interpretation of the problem is that you have to prove that there exists a correct choice of b_1 leading to a bounded sequence b_n. But the idea of "solving for b_1" is not the right way to conceptualize the problem. You instead want to solve for the entire sequence. Think of a map that would take one "approximate" sequence b_n to a better approximate sequence \tilde b_n. Then iterating this map should, in the limit (limit taken in l^\infty) give the correct sequence.
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You are given ( a_1, a_2, a_3, . . . ), b_0 is set to 0, and you would like to find (b_1, b_2, . . . . ) bounded such that b_{n-1}+4b_n+b_{n+1}=a_n for all n=1, 2, 3, . . .
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