On solvability of one class of Urysohn type nonlinear integral equation on the whole line

In present work one class of Urysohn type nonlinear integral equation on whole line is studied. Equations observed have applications in various fields of mathematical physics. It is assumed that Hammerstein type nonlinear integral operator with a difference kernel serves local minorant in terms of M. A. Krasnoselskii for the Urysohn initial operator. Combination of construction methods of invariant cone segments for initial Urysohn nonlinear operator with the methods of monotone operator theory and convolution type conservative integral equations in the case of some restrictions on nonlinearity allows us to prove constructive existence theorems about one parametric positive solutions. A set of parameters is described and the behavior of constructed solutions at infinity is examined. At the еnd of the work specific examples are given for which conditions of formulated theorems are satisfied.