Solvability in the sense of sequences for some linear and nonlinear Fredholm operators with the logarithmic Laplacian

Abstract

We study the solvability of certain linear and nonlinear nonhomogeneous equations in one dimension involving the logarithmic Laplacian and the transport term. In the linear case we show that the convergence in L 2 ( R ) of their right sides yields the existence and the convergence in L 2 ( R ) of the solutions. We generalize the results obtained in the earlier article of Efendiev and Vougalter [Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian. Monatsh Math. 2023] in the non-Fredholm case without the drift. In the nonlinear part of the work we demonstrate that, under the reasonable technical assumptions, the convergence in L 1 ( R ) of the integral kernels implies the existence and the convergence in L 2 ( R ) of the solutions.