Solvability conditions for some non-Fredholm operators with shifted arguments

TL;DR

This paper investigates conditions under which solutions exist and converge for certain non-Fredholm differential and integro-differential equations with shifted arguments, using sequence convergence in specific function spaces.

Contribution

It establishes solvability criteria and convergence results for non-Fredholm operators with shifted arguments in differential and integro-differential equations.

Findings

Convergence in $L^{2}(R)$ implies solution convergence in $H^{2}(R)$ for differential equations.

Convergence in $L^{1}(R)$ of kernels leads to solution convergence in $H^{2}(R)$ for integro-differential equations.

Existence of solutions depends on the translation parameter affecting Fredholm properties.

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.