Nonlinear convolution type integral equations in complex spaces

Abstract

We study various classes of nonlinear convolution type integral equations appearing in the theory of feedback systems, models of population genetics and others. By the method of monotone in the Browder-Minty operators we prove global theorems on existence, uniqueness and estimates for the solutions to the considered equations in complex Lebesgue spaces $L_p(\mathbf{R})$ under rather simple restrictions for the nonlinearities. Subject to the considered class of equations, we assume that either $p\in (1,2]$ or $p\in [2,\infty)$. The conditions imposed on nonlinearities are necessary and sufficient to ensure that the generated superposition operators act from the space $L_p(\mathbf{R})$, $1