Nonlinear singular integral equations in Lebesgue spaces
type: Item
instance of: scholarly article
title: Nonlinear singular integral equations in Lebesgue spaces
zbMATH Open document ID: 1228.45005
author: S. N. Askhabov
published in: Journal of Mathematical Sciences (New York)
publication date: 2011-07-18T00:00:00Z
review text: Using the theory of monotone operators, existence and uniqueness for singular integral equations of the type \[ \lambda_1a(x)u(x)+{\lambda_2\over\pi}\int_a^b {\bigl(w(x)+w(s)\bigr){\mathcal K}(x,s)u(s)\over s-x}ds+ \lambda_3F\bigl(x,u(x)\bigr)=f(x), \] of singular Hammerstein integral equations of the type \[ u(x)+{\lambda\over\pi}\int_a^b {\bigl(w(x)+w(s)\bigr){\mathcal K}(x,y)F\bigl(s,u(s)\bigr)\over s-x}ds=f(x), \] and of \[ u(x)+\lambda F\Bigl(x,{1\over\pi}\int_a^b {\bigl(w(x)+w(s)\bigr){\mathcal K}(x,s)u(s)\over s-x}ds\Bigr)=f(x) \] are established in a weighted \(L_p\) space \(L_p(\rho)\). The main hypotheses are that \(\lambda_1\lambda_3\geq0\), that \({\mathcal K}\) is Hölder continuous (and in some cases symmetric), that the weight function \(\rho\) and the functions \(w,a\geq0\) satisfy certain integral estimates, and that the nonlinearity \(F(x,t)\) is monotone with respect to \(t\) and satisfies some growth estimates such that, roughly speaking, it grows like \(\rho(x)| t|^{p}/t\) or, for the last equation, like \((| t|/\rho(x))^{1/(p-1)}\,\text{sgn}\,t\). For the first equation, also an analogue for complex-valued function is obtained. Besides standard theory of monotone operators, the proof for the first equation makes use of a remarkable integral identity involving Riesz type integral operators by which the term involving \(\lambda_2\) can be eliminated for monotonicity considerations.
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45E05
zbMATH DE Number: 5924759
zbMATH Keywords: singular integral operator
zbMATH Keywords: Riesz type integral operators
zbMATH Keywords: nonlinear integral equation
zbMATH Keywords: Browder-Minty theorem
zbMATH Keywords: monotone operator
zbMATH Keywords: Hammerstein operator
zbMATH Keywords: singular integral equations
zbMATH Keywords: singular Hammerstein integral equations
MaRDI profile type: Publication
full work available at URL: s10958-011-0237-6
OpenAlex ID: W1977061478
cites work: Q5715735
cites work: Q4748577
cites work: Nonlinear integral equations and systems of Hammerstein type
cites work: Q4345444
cites work: Q3253229
cites work: Q5689742
cites work: Q4121022
cites work: Q3966601
cites work: Q3329996
cites work: Some results of analysis of Hammerstein- and Uryson-type equations
cites work: Some new approaches to the investigation of a system of nonlinear equations of the second kind with singular integrals
cites work: Q4001740
cites work: Q3995224
cites work: Q4726489
cites work: Fredholm determinants, differential equations and matrix models
cites work: ON EIGENFUNCTIONS OF AN OPERATOR CORRESPONDING TO THE POLES OF THE ANALYTIC CONTINUATION OF THE RESOLVENT THROUGH THE CONTINUOUS SPECTRUM
cites work: Monotonicity Methods for Nonlinear Singular Integral and Integro‐Differential Equations
cites work: Some recent developments in the theory of nonlinear singular integral equations
DOI: 10.1007/S10958-011-0237-6
Recommended article: Nonlinear singular integral equations on a finite interval
Recommended article: Q3984836
Recommended article: Singular Lebesgue-Stieljes integral equations
Recommended article: On a Class of Nonlinear Singular Integral Equations
Recommended article: Singular integral equations with monotone nonlinearity in complex Lebesgue spaces
Recommended article: Q4221670
Recommended article: On the solvability of nonlinear singular integral equations
Recommended article: On the solvability of nonlinear integral equations in Lebesgue space
Recommended article: Q3542606
Recommended article: Nonlinear Singular Integral and Integro‐Differential Equations on the Positive Real Axis
description: scientific article; zbMATH DE number 5924759