Uniqueness of the Solution of a Class of Integral Equations with Sum-Difference. Kernel and with Convex Nonlinearity on the Positive Half-Line

The paper is devoted to the study of the uniqueness and certain qualitative properties of the solution of a class of integral equations with sum-difference kernel on the positive half-line and with a convex nonlinearity. This class of equations arises in a particular case in the dynamical theory of $p$-adic closed-open strings for the scalar field of tachyons. Such equations also play a very important role in the study of the existence and uniqueness of solutions of nonlinear integral equations in the mathematical theory of the geographical distribution of an epidemic within the framework of the Diekmann–Kaper model.
We prove the uniqueness theorem for the solution of the equation under consideration for a class of nonnegative (nonzero) and bounded functions on $\mathbb{R}^+$, thereby obtaining a definitive solution of Vladimirov's open problem on the uniqueness of rolling solutions of nonlinear $p$-adic equations. Under an additional constraint on the kernel of the equation, we also prove that the solution is a concave function on $[0,+\infty)$ whose derivative belongs to the space $L_1(0,+\infty)$. At the end of the paper, we give specific model equations from the above-mentioned applications, to which our results are applied.