TY - JOUR

T1 - On nonnegative solvability of linear integral equations

AU - Jang, Ruey Jen

AU - Victory, Harold Dean

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1992/3/1

Y1 - 1992/3/1

N2 - Let (Ω,Σ,μ) denote a σ-finite measure space, and Lp(Ω,Σ,μ) (1≤p<∞) the usual Banach lattices of pth summable real-valued functions. Suppose, moreover, K is an integral operator whose nonnegative kernel k(·,·) is (Σ×Σ)-measurable on Ω×Ω and which maps Lp(Ω,Σ,μ) into itself while possessing a compact iterate. We present necessary and sufficient conditions for the integral operator equation λf{hook} = Kf{hook}+g to possess a nonnegative solution f{hook}ε{lunate}Lp(Ω,Σ,μ) whenever g is a given nontrival and nonnegative element of Lp(Ω,Σ,μ) and λ is any given positive parameter. This analysis extends that by Victory [SIAM J. Algebraic Discrete Methods 6:406-412 (1985)] for the matrix case.

AB - Let (Ω,Σ,μ) denote a σ-finite measure space, and Lp(Ω,Σ,μ) (1≤p<∞) the usual Banach lattices of pth summable real-valued functions. Suppose, moreover, K is an integral operator whose nonnegative kernel k(·,·) is (Σ×Σ)-measurable on Ω×Ω and which maps Lp(Ω,Σ,μ) into itself while possessing a compact iterate. We present necessary and sufficient conditions for the integral operator equation λf{hook} = Kf{hook}+g to possess a nonnegative solution f{hook}ε{lunate}Lp(Ω,Σ,μ) whenever g is a given nontrival and nonnegative element of Lp(Ω,Σ,μ) and λ is any given positive parameter. This analysis extends that by Victory [SIAM J. Algebraic Discrete Methods 6:406-412 (1985)] for the matrix case.

UR - http://www.scopus.com/inward/record.url?scp=0013478514&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(92)90238-6

DO - 10.1016/0024-3795(92)90238-6

M3 - Article

AN - SCOPUS:0013478514

VL - 165

SP - 197

EP - 228

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -