The Cake Eating Problem with Depreciation (Modelling difficulties)
It would seem that the way you've formulated your production function/law of motion has introduced double counting into the problem. Note that substituting 1 and 2 into 3 gives:
$$k_{t+1}=(1-\delta)(c_t+x_t)+x_t$$
Where investment in period t is counted twice.

The correct law of motion is simply:

$$ 3. \: k_{t+1}=(1-\delta)x_t $$

And the general form of the Bellman equation would be:
$$v(k_t)=\max_{k_{t+1}}\left\{\ln\left(k_t-\frac{k_{t+1}}{1-\delta}\right)+\beta v\left(\frac{k_{t+1}}{1-\delta}\right)\right\}$$

Hope this helps!