  This paper is devoted to solving a time-inconsistent risk-sensitive control
problem with parameter $\e$ and its limit case ($\e\rightarrow0^+$) for
countable-stated Markov decision processes (MDPs for short). Since the cost
functional is time-inconsistent, it is impossible to find a global optimal
strategy for both cases. Instead, for each case, we will prove the existence of
time-inconstant equilibrium strategies which verify the so-called
step-optimality. Moreover, we prove the convergence of $\e$-equilibriums and
the corresponding value functions as $\e\rightarrow0^+$.
