On nonnegative solvability of linear integral equations

AbstractLet (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 λƒ = Kƒ+g to possess a nonnegative solution ƒϵLp(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