A one-parameter family of unbounded positive solutions for a class of nonlinear three-dimensional integral equations with an operator of the Hammerstein–Nemytsky type

This paper studies a class of three-dimensional integral equations with a nonlinear monotone integral operator of the Hammerstein–Nemytsky type. This class of equations has applications in many areas of mathematical physics and mathematical biology.
In particular, such equations arise in gamma-quantum theory, in problems of neutron moderation in nuclear reactors, in queuing theory, and in the mathematical theory of spatiotemporal spread of epidemics. Under certain restrictions on the kernel of the equation and on the functions describing the nonlinearity (that are natural from the point of view of applications in the indicated areas), the existence of a one-parameter family of unbounded positive solutions is proved, and the set of corresponding parameters is explicitly described. It is also possible to study the asymptotic behavior of the constructed solutions at infinity. At the end, specific applied examples of the indicated equations are given to illustrate the importance of the results obtained.