Constructive Study of Solvability for a Class of NonlinearIntegral Equations with a Symmetric Kernel

We consider a class of nonlinear integral equations of Hammerstein type with a symmetric and substochastic kernel. Equations of this class arise in various branches of physics and mathematical biology; namely, the dynamic theory of $$p $$ -adic strings, the kinetic theory of gases, and in model problems of the mathematical theory of the spread of epidemic diseases. We prove a constructive theorem on existence of a nontrivial, bounded, nonnegative, continuous, and monotone nondecreasing solution. We also prove a uniqueness theorem for such a solution in the class of nontrivial nonnegative bounded functions. We use the obtained results in studying nonlinear integral equations on the whole line with almost difference kernels. In conclusion, we present specific examples of kernels and nonlinearities that illustrate applications in the above-mentioned theories.