On the convexity of images of nonlinear integral operators

We study continuous nonlinear Urysohn-type integral operators acting from the spaces of vector functions with integrable components to the space of continuous functions. We obtain conditions under which the images of sets defined by pointwise constraints have a convex closure under the action of these operators. The result is used to justify a method of constructive approximation of these images and to derive a necessary solvability condition for Urysohn-type integral equations. A numerical method for finding the residual of equations of this type on the sets under consideration is justified.