Existence and uniqueness of solution of a certain boundary-value problem for a convolution integral equation with monotone non-linearity

We study the existence and uniqueness as well as the asymptotic behaviour of solutions of a certain boundary-value problem for a convolution integral equation on the whole line with monotone non-linearity. In some special cases, there are concrete applications to $p$-adic string theory, the mathematical theory of the geographical spread of an epidemic, the kinetic theory of gases and the theory of radiation transfer. We prove \linebreak the existence and uniqueness of an odd bounded continuous solution. The monotonicity and the integral asymptotics of this solution is also discussed. We finally give particular application-oriented examples of the equations considered, which illustrate the special nature of our results.