Caveat: I have a research problem in the numerical analysis of fluid dynamics similar to this one. I'm following ideas in some research papers (e.g. this one that I asked about on MathOverflow), but I have an Engineering background so there are definitely some glaring knowledge gaps, so I might be missing something basic and I might not use the correct terminology. I cross-posted this problem on MathOverflow here.
While trying to examine if a mathematical fluid dynamics system is stable, I have $$(1-k)f_{n+1} \le kf_n + (1+k)f_{n-1} + c, \quad n \ge 1$$ where each $f_i$ is a matrix norm of the system at time-step $i$, $k \in \mathbb{Q}$ is a fraction between $0$ and $1$ which I can control, $c \in \mathbb{R}$ is a positive force. I do not know how to resolve this recurrence inequality.
My goal: I hope to find a solution of the form below for all $n \ge 1$, where $p(k),q(k),r(k)$ are some constant depending on $n$. $$f_n \le p(k)f_1 + q(k)f_0 + r(k)c$$
My attempts:
(1) Mathematica has no method for solving recurrence inequalities, but if I change the $\le$ to an $==$ and pick $k=\frac12$ for example, I get, for constants $p$ and $q$ which I can relate to $f_0$ and $f_1$, $$f_n = p \left(\frac{1}{2} \left(1-\sqrt{13}\right)\right)^n +q \left(\frac{1}{2} \left(\sqrt{13}+1\right)\right)^n-\frac{2 c}{3}.$$ My understanding of recurrence inequalities is that a "tighter" solution is needed, than in the equality case. I also asked on Mathematica SE.
(2) I attempted using a method of generating functions similar to this, but I have the last term on the right, the constant term which I am not sure how to handle. If I ignore the c term, I can re-write it in the form of the other question, as below, and I obtain the same result accepted in the other question. $$f_{n+1} \le \frac{k}{1-k}f_n + \frac{1+k}{1-k}f_{n-1}, \quad n \ge 1$$ which can be written in the form $$f_{n+1} - \alpha{f_n} - \beta{f_{n-1}} \le 0$$ and by the method of generating functions where $F(x)=\displaystyle\sum^\infty_{n=0}f_nx^n$, \begin{align} \sum_{n=2}^\infty f_n x^n &= F(x) - f_0-f_1x \tag{1} \\-\alpha\sum_{n=2}^\infty f_{n-1} x^n &= -\alpha x(F(x)-f_0) \tag{2} \\-\beta\sum_{n=2}^\infty f_{n-2} x^n &= -\beta x^2 F(x) \tag{3} \end{align} which leads to $$F(x) \le \frac{f_0-f_0\alpha{x}+f_1x}{1-\alpha x-\beta x^2}$$ and it appears to be correct because it is the same result in the linked question. I don't know how to handle this in the presence of the $c$ term. i.e. I can't express it in terms of $F(x)$.
I overhauled this question, thanks to all the comments on MO about asking a more useful question.