Solvability conditions for some non-Fredholm operators with shifted arguments

TL;DR

The paper addresses solvability of non-Fredholm second-order operators with shifted arguments on the real line, focusing on the linear equation $L_h u = -u'' - a u(x-h)$ and its spectral properties. It shows that for $h\neq \frac{2\pi n}{\sqrt{a}}$ the linear problem has a unique solution $u\in H^{2}({\mathbb R})$, while at resonant shifts $h=\frac{2\pi n}{\sqrt{a}}$ solvability hinges on orthogonality to $e^{\pm i\sqrt{a}x}$ in $L^{2}$; it also proves stability under $L^{2}$-convergence of the data via Fourier-based estimates. The results extend to nonlocal integro-differential equations with a shifted argument and to nonlinear problems by employing a contraction-mapping framework under Carathéodory and Lipschitz conditions, with convergence results when kernels converge in $L^{1}$. An appendix collects the technical lemmas and definitions (e.g., $N_{a,h}$, $N_{a,h,m}$) that underpin the operator-norm bounds and convergence arguments. The work clarifies how Fredholm-property failures due to shifting interact with solvability, providing precise conditions for existence, uniqueness, and sequence convergence with potential applications to embedded solitons and nonlocal reaction-diffusion models.

Abstract

In the first part of the article we establish the existence in the sense of sequences of solutions in $H^{2}(R)$ for some nonhomogeneous linear differential equation in which one of the terms has the argument translated by a constant. It is shown that under the reasonable technical conditions the convergence in $L^{2}(R)$ of the source terms implies the existence and the convergence in $H^{2}(R)$ of the solutions. The second part of the work deals with the solvability in the sense of sequences in $H^{2}(R)$ of the integro-differential equation in which one of the terms has the argument shifted by a constant. It is demonstrated that under the appropriate auxiliary assumptions the convergence in $L^{1}(R)$ of the integral kernels yields the existence and the convergence in $H^{2}(R)$ of the solutions. Both equations considered involve the second order differential operator with or without the Fredholm property depending on the value of the constant by which the argument gets translated.