On generalized solvability of variable nonlinear integral equations on cones

Abstract We study the solvability of multidimensional nonlinear integral equations whose operator is a sum of the Nemytsky and Urysohn type operators, and the desired solution belongs to a cone in a space of summable functions. For a wide class of nonlocal variations of the functions defining the composing operators, we prove that the equation remains solvable in the sense that its residual is arbitrarily small with respect to a suitable norm or a seminorm. The classes of acceptable variations are described by convex cones in corresponding function spaces. We also prove that images of variable operators possess the stability property in the sense that their closures under any of these variations contain a fixed convex cone.