Nonlinear convolution type integral equations in complex spaces

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 $L_p(\mathbf{R})$, $1<p<\infty$, into the dual space $L_q(\mathbf{R})$, $q=p/(p-1)$, and are monotone. In the case of the space $L_2(\mathbf{R})$, we combine the method of monotone operator and contracting mappings principle to show that the solutions can be found by the successive approximation method of Picard type and provide estimates for the convergence rate. Our proofs employ essentially the criterion of the Bochner positivity of a linear convolution integral operator in the complex space $L_p(\mathbf{R})$ as $1<p\leq 2$ and the coercitivity of the operator inverse to the Nemytskii operator. In the framework of the space $L_2(\mathbf{R})$, the obtained results cover, in particular, linear convolution integral operators.