Integral equations with substochastic kernels

The non-homogeneous or homogeneous integral equation of the second kind with a substochastic kernel $W(x,t)=K(x-t)+T(x,t)$ is considered on the semi axis, where $K$ is the density of distribution of some variate, and $T\ge0$ satisfies the condition $\lambda(t)=\int^\infty_{-t}K(y)\,dy+\int^\infty_0T(x,t)\,dx<1$, $\sup\lambda(t)=1$.
The existence of a minimal positive solution of the non-homogeneous equation is proved. The existence of a positive solution of the homogeneous equation is also proved under some simple additional conditions. The results may be applied to the study of Random Walk on the semi axis with the reflection at the boundary.