  The class of nonlinear integral equations on the positive half-line with a
monotone operator of Hammerstein type is studied. With various partial
representations of the corresponding kernel and nonlinearity, this class of
equations has applications in the dynamic theory of $p$-adic strings, in the
kinetic theory of gases, in the theory of radiation transfer and in the
mathematical theory of the geographical spread of epidemic diseases. A
constructive theorem for the existence of a nontrivial bounded solution is
proved. The asymptotic behavior of the constructed solution at infinity is
studied. We also prove a theorem for the uniqueness of a solution in the class
of nonnegative nontrivial and bounded functions. At the end of the work,
specific particular examples of the kernel and nonlinearity of this class of
equations are given, which are of independent interest.
