Nonlinearly weighted \(L^1\)-solvability of the first kind Fredholm integral equation
type: Item
instance of: scholarly article
title: Nonlinearly weighted \(L^1\)-solvability of the first kind Fredholm integral equation
zbMATH Open document ID: 0971.45001
author: Nassar H. S. Haidar
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-05-08T00:00:00Z
review text: This paper is devoted to the study of the first kind Fredholm integral equation \[ g(\nu)= \int^b_aK(\nu,x) f(x)dx, \tag{1} \] in which \(x\in I= [a,b]\), \(\nu\in J=[c,d]\), and \(J\times J=\Omega\). The author derives an explicit solution for (1) with kernels in a dual space which is a certain restriction of a nonlinearly weighted \(L^1\)-space \(L_C^1(\Omega, \gamma(K))\), via a perturbational auxiliary construction involving a generalized moment problem. It is illustrated how the auxiliary problem, which has a maximum a posteriori solution, can be utilized in a certain inverse-problem formulation to establish that some variational perturbation of the kernel of (1) in its dual space may yield for it that a closed-form primal solution \(\psi(x)\). The main result is that this analytical solution must satisfy an equivalent, well-posed, but nonlinear homogeneous functional equation of the second-kind represented by a Urysohn-type regularizing integral equation with a movable bifurcation point. The main result is utilized to prove the inclusion \(L^1 (I)\cap L^\infty (I;x^{-1}) \subset L^2(I;x^{-1})\) and to illustrate that this Urysohn integral equation is equivalent to a unique Fredholm integral equation of the second kind with a degenerate kernel and having the solution space \(L^1(I; \rho(\psi)) =L^2(I;x^{-1}) \cap L^1_C(\Omega; \gamma(K))\).
Mathematics Subject Classification ID: 45B05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1449164
zbMATH Keywords: nonlinearly weighted \(L^1\)-solvability
zbMATH Keywords: first kind Fredholm integral equation
zbMATH Keywords: explicit solution
zbMATH Keywords: generalized moment problem
zbMATH Keywords: inverse-problem formulation
zbMATH Keywords: analytical solution
zbMATH Keywords: Urysohn-type regularizing integral equation
zbMATH Keywords: bifurcation point
zbMATH Keywords: degenerate kernel
reviewed by: Pavel Talpalaru
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6736
OpenAlex ID: W2045854934
cites work: On Iteration Procedures for Equations of the First Kind, Ax = y, and Picard's Criterion for the Existence of a Solution
cites work: On Existence Criteria and Approximation Procedures for Integral Equations of the First Kind
cites work: Q3685144
cites work: A nonlinear method for a severely ill-posed generalized moment problem
cites work: Q4704749
cites work: The collocational double series inverse in quasilinear regularizer form
cites work: Q5586854
cites work: Iterative methods for best approximate solutions of linear integral equations of the first and second kinds
cites work: Q4875411
DOI: 10.1006/JMAA.2000.6736
description: scientific article; zbMATH DE number 1449164