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

Abstract: 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.