On the Solvability in the Sense of Sequences for Some Non-Fredholm Operators Related to the Anomalous Diffusion

Abstract

We study solvability of some linear nonhomogeneous elliptic problems and prove that under reasonable technical conditions the convergence in \(L^{2}({\mathbb R}^{d})\) of their right sides implies the existence and the convergence in \(H^{2s}({\mathbb R}^{d})\) of the solutions. The equations involve the second order non-Fredholm differential operators raised to certain fractional powers s and we use the methods of spectral and scattering theory for Schrodinger type operators developed in our preceding work (Volpert and Vougalter, Electron J Differ Equ 160:16 pp, 2013).