Solvability of some integro-differential equations with the logarithmic Laplacian

TL;DR

This paper investigates the existence of solutions for a class of integro-differential equations involving the logarithmic Laplacian, using fixed point methods and kernel convergence analysis.

Contribution

It introduces a new approach to prove solution existence for equations with the logarithmic Laplacian without the Fredholm property, based on kernel convergence.

Findings

Solutions exist under technical conditions

Kernel convergence implies solution convergence in L^2

Method applicable to operators lacking Fredholm property

Abstract

We address the existence in the sense of sequences of solutions for a certain integro-differential type problem involving the logarithmic Laplacian. The argument is based on the fixed point technique when such equation contains the operator without the Fredholm property. It is established that, under the reasonable technical conditions, the convergence in L^1(R^d) of the integral kernels yields the existence and convergence in L^2(R^d) of the solutions.