Integral equations with substochastic kernels

Abstract

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 \( T \geq 0 \) satisfies the condition \( \lambda(t) = \int_{−t}^{\infty} K(y) \,dy + \int_0^{\infty} T(x, t) \,dx < 1, \sup \lambda(t) = 1 \). 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.