Asymptotic approximation of a recurrence relation (Akra-Bazzi doesn't seem to apply)

Suppose an algorithm has a runtime recurrence relation:

$ T(n) = \left\{
\begin{array}{lr}
g(n)+T(n-1) + T(\lfloor\delta n\rfloor ) & : n \ge n_0\\
f(n) & : n < n_0
\end{array}
\right.
$

for some constant $0 < \delta < 1$. Assume that $g$ is polynomial in $n$, perhaps quadratic. Most likely, $f$ will be exponential in $n$.

How would one go about analyzing the runtime ($\Theta$ would be excellent)? The master theorem and the more general Akra-Bazzi method do not seem to apply.

One possible approach might be by analogy to differential equations. Let $T'(n) = T(n)-T(n-1)$. Here $T'(n)$ is a discrete analog of the first derivative of $T(n)$. We get the following relationship:
$$T'(n) = T(\lfloor \delta n \rfloor) + g(n).$$
The continuous analog of this is the differential equation
$$t'(x) = t(\delta x) + g(x),$$
or, if you prefer to see it written differently:
$${d \over dx} t(x) = t(\delta x) + g(x).$$
That's a differential equation.

Now you could try to solve the differential equation for the continuous function $t(x)$, then hypothesize that a similar function will be the solution to your original recurrence relation, and try to prove your hypothesis. At least, this is one general approach you could follow.

I've forgotten everything I once knew about differential equations, so I don't know the solution that differential equation, but maybe you will be able to solve it by reviewing all the techniques for solving differential equations.

StackExchange Computer Science Q#14343, answer score: 5