On nonnegative solvability of linear integral equations

Abstract

Let (Ω,Σ,μ) denote a σ-finite measure space, and L p (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 L p (Ω,Σ,μ) into itself while possessing a compact iterate. We present necessary and sufficient conditions for the integral operator equation λƒ = Kƒ+g to possess a nonnegative solution ƒϵ L p (Ω,Σ,μ) whenever g is a given nontrival and nonnegative element of L p (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.