On the global solvability of one class of nonlinear integral equations of Hammerstein–Volterra type on the nonnegative half-line

The class of nonlinear integral equations of the Hammerstein–Volterra type on the nonnegative half-line is studied. This class of equations, with various partial representations of the corresponding kernel and nonlinearity, has important applications in hydroaerodynamics, in the theory of radiative transfer, and in models of population genetics. The combination of the method of successive approximations with some geometric estimates for concave and monotone functions makes it possible to prove constructive theorems for the existence and uniqueness of a nonnegative bounded and continuous solution to the specified equation. Moreover, a uniform convergence of special successive approximations (at a rate of decreasing geometric progression) to the solution is established. The corresponding nonlinear “homogeneous” equation is also considered, and it is proven that in the class of nonnegative slowly growing functions, this equation has only a trivial (zero) solution. Lastly, particular applied examples of such equations that satisfy all the restrictions of the proven statements are given.