On nonlinear convolution-type integral equations in the theory of $p$-adic strings

We study a class of integral equations of convolution type on the whole line with a monotone and odd nonlinearity. We prove constructive existence and absence theorems for nonnegative (nontrivial) and bounded solutions. We study the asymptotic behavior of the constructed solution at $\pm\infty$. We also prove the uniqueness of the solution in the class of nonnegative (nonzero) and bounded functions and present specific examples of this class of equations that can be applied in various fields of mathematical physics.