  In this paper, the existence and uniqueness of solution of a specific
differential equation is studied. This equation originates from the description
of a coupled process by totally asymmetric simple exclusion process (TASEP) and
Langmuir kinetics (LK). In the fields of physics and biology, the properties of
the TASEP-LK coupled process have been extensively studied by Monte Carlo
simulations and numerical calculations, as well as detailed experiments.
However, so far, no rigorous mathematical analysis has been given to the
corresponding differential equations, especially their existence and uniqueness
of solution. In this paper, using the upper and lower solution method, the
existence of solution of the steady state equation is obtained. Then using a
generalized maximum principle, we show that the solution constructed from the
upper and lower solution method is actually the unique solution in Câ space.
Moreover, the existence and uniqueness of solution of the time dependent
differential equation are also obtained in one specific space X\b{eta}. Our
results imply that the previous results obtained by numerical calculations and
Monte Carlo simulations are theoretically correct, especially the most
important phase diagram of particle density along the travel track under
different model parameters. The study in this paper provides theoretical
foundations for the analysis of TASEP-LK coupled process. At the same time, the
methods used in this paper may be instructive for studies about the more
general cases of the TASEP-LK process, such as the one with multiple travel
tracks or the one with multiple particle species.
