Solvability of some classes of nonlinear singular boundary value problems in the theory of $p$-adic open–closed strings

We investigate boundary value problems for a singular integral equation of the convolution type with a power-law nonlinearity. Such problems arise in the theory of $p$-adic open–closed strings. We prove constructive theorems on the existence of nontrivial solutions and also prove a uniqueness theorem in a certain weight class of functions. We study asymptotic properties of the constructed solutions and obtain the Vladimirov theorem on tachyon fields for open–closed strings as a particular case of the proved results.