Almost solubility of classes of non-linear integral equations of the first kind on cones

Using convexity properties of the images of completely continuous non-linear integral operators, we describe the closed convex cones lying either in the recessive cone, or in the tangent cone of the closed image of the operator being studied (depending on the nature of the integrand). These cones are determined by the principal part of the asymptotics of the integrand at infinity, independently of the variation of the subordinate part. We discuss applications to the generalized solubility of non-linear integral equations of the first kind.