On nonnegative solvability of linear operator equations

Let E be a Banach lattice having order continuous norm. Suppose, moreover, T is a nonnegative reducible operator having a compact iterate and which maps E into itself. The purpose of this work is to extend the previous results of the authors, concerning nonnegative solvability of (kernel) operator equations on general L^p-spaces. In particular, we provide necessary and sufficient conditions for the operator equation λx=Tx+y to possess a nonnegative solution xεE where y is a given nonnegative and nontrivial element of E and λ is any given positive parameter.