Recursive function's value between iterations

Question

I have the following function:

$$ x_{t=1} = x_{t=0} - \frac{c_1* y_{t=0}^{c_2} }{x_{t=0} + y_{t=0}}$$

$c_1$ and $c_2$ are constants, and we know the values $x_{t=0}$ and $y_{t=0}$. How can I calculate $x_{t=0.1}$ for example?

Answer 1

Your recurrence defines $x_1$ in terms of $x_0, y_0, c_1$ and $c_2$. However, it does not specify any kind of behavior for $x_t$ with $t \in (0,1)$. Hence, these values are undefined. Perhaps if you need to assume some form for them, one could consider linear interpolation. Then you set $$ x_t = tx_1 + (1-t)x_0, \quad \forall t \in (0,1), $$ and in particular, $$ x_{0.1} = 0.1x_1 + 0.9x_0 = \frac{x_1 + 9x_0}{10}. $$