On solvability of class of nonlinear integral equations in $p$-adic string theory

In this paper, we study a class of integral equations with power nonlinearity on the entire real line. This class of equations arises in the $p$-adic theory of open-closed strings. Using the method of successive approximations and justifying their convergence, we prove the existence of a nontrivial continuous odd bounded solution on the entire line. The asymptotic behavior of the solution is studied as the argument increases unboundedly. We obtain integral estimates and some properties of approximations of the solution to the considered equation. Under some additional restrictions, we also establish the uniqueness of the constructed solution in a certain class of continuous functions. We provide examples of integral kernels of the equation satisfying all assumptions of the formulated theorems. As the nuclear function is a Gaussian distribution, from the proven results we obtain Vladimirov-Volovich theorem as a special case.