The existence of odd solution for one boundary-value problem with power nonlinearity

In the present work we investigate a boundary value problem for a nonlinear convolution type singular integral equation on the whole axis with power nonlinearity. The above-mentioned problem has direct application in $p$-adic open-closed strings theory. We prove the existence of a rolling odd solution for the considered problem. We also establish an integral asymptotic for the constructed solution. At the end we list particular examples of the given equation, having separate interest.