On the solvability of a boundary value problem in $p$-adic string theory

This paper is devoted to the study and solution of a boundary value problem for a convolution-type integral equation with cubic nonlinearity. The above problem has a direct application to the $p$-adic theory of open-closed strings for the scalar tachyon field. It is shown that a one-parameter family of monotone continuous bounded solutions exists. Under additional conditions on the kernel of the equation, an asymptotic formula for the solutions thus constructed is established. Using these results, as particular cases we obtain Zhukovskaya's theorem on rolling solutions of the nonlinear equation in the $p$-adic theory of open-closed strings and the Vladimirov–Volovich theorem on the existence of a nontrivial solution between certain vacua.
The results are extended to the case of a more general nonlinear boundary value problem.