On nonnegative solvability of linear integral equations

Ruey Jen Jang, Harold Dean Victory

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6 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)197-228
Number of pages32
JournalLinear Algebra and Its Applications
Volume165
Issue numberC
DOIs
StatePublished - Mar 1 1992

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