On the qualitative properties of the solution of a nonlinear boundary value problem in the dynamic theory of $p$-adic strings

The article considers a boundary value problem for a class of singular integral equations with an almost total-difference kernel and convex nonlinearity on the positive half-line. This problem arises in the dynamic theory of $ p $-adic open-closed strings. It is proved that any non-negative and bounded solution of a given boundary value problem is a continuous function and the difference between the limit and the solution is itself an integrable function on the positive half-line. For a particular case, it is proved that the solution is a monotonically non-decreasing function. A uniqueness theorem is established in the class of nonnegative and bounded functions. At the conclusion of the article, a specific applied example of this boundary problem is given.