Consider the the forward recurrent time proc

Consider the the forward recurrent time process; i.e. jumps from $0$ to $x+1$ with probability $f(x)$ and from $x+1$ jumps to $x$ and then to $x-1$ and so on until a return to zero.

Let $f(x)=\exp(\alpha x) $.
a) Find $V=[0,\infty)$, $b<\infty$ such that
$$\Delta V(x)\leq -f(x)+b\chi_C(x).$$
where $\Delta V(x)=PV(x)-V(x)\;\;\text{and}\;\; PV(x)=\int_\Omega V(y)P(x,dy)$ and $C=\{0\}$

b) Can we conclude on how $K^n$ (where $K$ is the kernel function) converges to $\pi$ (what is $pi$) in norm $||\cdot||_f=\sup_{f:|f|<=1}||\cdot(f)||$?

My approach:
If I choose $V$ and $b$ such that
$$V(x)=\frac{e^{\alpha}}{e^{\alpha}-1}(e^{\alpha x}-1)\;\;\; \text{and}\;\;\;b=\phi(\alpha)=\sum_x e^{\alpha x}f(x)$$
question a) can be written as
$f(x)+ PV(x)\leq V(x) +b\chi_C(x)$
where $PV(x)$ is a conditional expected value ie $PV(x)=E(V(X_1)/X_0=x)$

Now from here i'm blocked. Your help is very appreciated.

For question b) just need a starting point( I know that $K^n(x)=E(V(X_n)/X_0=x)$.

Thanks in advance.

Merci d'écrire en français sur ce forum. --JLT