Does the following non-linear recurrence equation has an explicit solution with given boundary conditions $x_0$ and $x_\infty$? $$ x_n = a + b x_{n-1}x_{n+1} $$ $a$ and $b$ are constants.
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Remarks: Is $L=x_{\infty}$ finite ? In this case, its 2 possible values are known because $L$ is necessarily solution to the quadratic equation $L=a+bL^2$. Another thing: are you sure you dont need also to give $x_1$ ? – Jean Marie May 01 '16 at 06:44
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Yes, $L$ is finite and as you mentioned it has at most two possible values. – Erfan Salavati May 01 '16 at 06:46
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You have not answered concerning the fact that $a_1$ should as well be given. If $a_1$ is chosen at our own will, how can you await a unique general explicit solution ? – Jean Marie May 01 '16 at 06:52
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No, $x_1$ is not given. But if an explicit solution is found in terms of $x_0$ and $x_1$ then it may be useful too. – Erfan Salavati May 01 '16 at 06:57