On integral equations of stationary distributions for biological systems

In this paper, properties of solutions of the convolution-type integral equation $(1+w(x))P(x)=(m*P)(x)+Cm(x)$ on the real axis are studied. The main concern is to find conditions for the function $w(x)$ and the kernel $m(x)$ sufficient for the existence of an admissible solution $P(x)$, i.e., a solution which has a nonzero limit at infinity. The main results of the paper are the uniqueness theorem for the admissible solution for rapidly decreasing kernels $m$ and the existence theorem for one-sided compactly supported kernels m.