On the constructive solvability of a class of nonlinear multidimensional integral equations in the theory of $p$-adic strings

A. Kh. Khachatryan^ab, Kh. A. Khachatryan^c, H. S. Petrosyan<sup>a

a</sup> Armenian National Agrarian University, Yerevan, Armenia
^b Institute of Mathematics of the National Academy of Sciences of the Republic of Armenia, Yerevan, Armenia
^c Yerevan State University, Yerevan, Armenia

Abstract: We study multidimensional integral equations with monotonic and odd nonlinearity. These equations have applications in the dynamic theory of $p$-adic strings. In particular, when the nonlinearity is power-law and the kernels are represented as Gaussian distributions, these equations describe the dynamics (rolling) of $p$-adic open or open–closed strings for a scalar tachyon field. Under certain restrictions on nonlinearity and kernels, we prove the constructive solvability of the equation in the space of continuous and bounded functions. We establish the uniform convergence of the corresponding successive approximations (with a rate of infinitely decreasing geometric progression) to the solution. We prove that the equation under study can simultaneously have alternating and sign-preserving bounded solutions. The results obtained are applied to specific problems in the dynamic theory of $p$-adic strings and in solving a nonlinear boundary value problem for the heat conduction equation.

Keywords: monotonicity, bounded solution, uniform convergence, iterations, $p$-adic string.

Received: 21.10.2025
Revised: 21.10.2025

DOI: 10.4213/tmf11102

 English version: Theoretical and Mathematical Physics, 2026, 226:2, 298–312