Solvability of Quasilinear and Nonlinear Integral Equations with a Sum-Difference Kernel on the Half-Line

This paper investigates two classes of quasilinear and essentially nonlinear integral equations with a sum-difference kernel on the half-line. Such equations arise in various areas of physics, including the theory of radiative transfer in spectral lines, the dynamic theory of p-adic strings, and the kinetic theory of gases within the framework of the modified Bhatnagar–Gross–Krook (BGK) model. Under specific conditions on the kernel and the nonlinear terms, we establish constructive existence theorems for non-negative, nontrivial, and continuous solutions. For the quasilinear case, we construct a one-parameter family of non-negative, nontrivial, linearly growing, and continuous solutions. For the class of essentially nonlinear equations, we prove the uniform convergence-at a geometric rate-of a specially constructed sequence of successive approximations to a non-negative, nontrivial, continuous, and bounded solution. We also study the asymptotic behavior of the constructed solutions at infinity. Additionally, for the second class of equations, a uniqueness theorem is established within a certain subclass of non-negative, bounded, and nontrivial functions. The paper concludes with concrete examples of kernels and nonlinearities that satisfy all conditions of the proven theorems.