Linear and nonlinear non-Fredholm operators and their applications

Messoud Efendiev; Vitali Vougalter; Messoud Efendiev; Vitali Vougalter

doi:10.3934/era.2022027

Electronic Research Archive

2022, Volume 30, Issue 2: 515-534. doi: 10.3934/era.2022027

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Research article Special Issues

Linear and nonlinear non-Fredholm operators and their applications

Messoud Efendiev ^{1,2
,
,},
Vitali Vougalter ³

1.
Helmholtz Zentrum München, Institut für Computational Biology, Germany
2.
Department of Mathematics, Marmara University, Istanbul, Turkey
3.
Department of Mathematics, University of Toronto, Canada

Received: 13 August 2021 Revised: 13 December 2021 Accepted: 14 December 2021 Published: 10 February 2022

In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.
- solvability conditions,
- non-Fredholm operators,
- anomalous diffusion,
- nonnegativity of solutions
Citation: Messoud Efendiev, Vitali Vougalter. Linear and nonlinear non-Fredholm operators and their applications[J]. Electronic Research Archive, 2022, 30(2): 515-534. doi: 10.3934/era.2022027

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Abstract

In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.

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