Open AccessLinear and nonlinear non-Fredholm operators and their applications
Author(s) -
Messoud Efendiev,
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Vitali Vougalter,
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Publication year - 2022
Publication title -
electronic research archive
Language(s) - English
Resource type - Journals
ISSN - 2688-1594
DOI - 10.3934/era.2022027
Subject(s) - mathematics , verifiable secret sharing , nonlinear system , fredholm integral equation , scalar (mathematics) , pure mathematics , fredholm theory , type (biology) , dimension (graph theory) , laplace operator , mathematical analysis , integral equation , computer science , quantum mechanics , biology , programming language , ecology , physics , geometry , set (abstract data type)
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.