On positive solutions of a boundary value problem for a nonlinear integro-differential equation on a semi-infinite interval

The article is devoted to the study of a boundary value problem for a first order nonlinear integro-differential equation on the positive semi axis with a Hammerstein type noncompact integral operator. Such a problem arises in kinetic theory of plasma. In particular, this nonlinear integro-differential equation describes the problem of stationary distribution of electrons in semi infinite plasma in the presence of an external potential electric field. This boundary value problem can be derived from nonlinear Boltzmann model equation, where the role of unknown function plays the first coordinate of an electric field. Depending on a physical parameter, involved in the equation, some constructive existence theorems of one-parametric family of positive solutions in Sobolev's $W_1^1(\mathbb{R}^+)$ space are proved. The asymptotic behavior of the constructed solutions at infinity is also investigated. The proofs of the above statements are based on the construction of a one-parametric family of conic segments, which are invariant with respect to a convolution type nonlinear monotone operator. Further, using some a priori estimates, which are of independent interest, as well as some results from linear theory of conservative homogenous Wiener–Hopf integral equations, the asymptotic properties of obtained results are studied. At the end of the article, some important applications and examples are presented.