On the solvability of a class of nonlinear two-dimensional integral equations Hammerstein–Nemytskii type on the plane

We consider a class of nonlinear integral equations with a stochastic and symmetric kernel on the whole line. With certain particular representations of the kernel and nonlinearity, equations of the above character arise in many branches of mathematical natural science. In particular, such equations occur in the theory $p$-adic strings, in the kinetic theory of gases, in mathematical biology and in the theory of radiative transfer. Constructive existence theorems are proved for non-negative non-trivial and bounded solutions under various restrictions on the function describing the nonlinearity in the equation. Under additional restrictions on the kernel and on the nonlinearity, a uniqueness theorem is also proved in a certain class of bounded and non-negative functions that have a finite limit in $\pm\infty$. Specific applied examples of the kernel and non-linearity are given that satisfy all the restrictions of the proven statements.