Stability of nonlinear Urysohn integral equations via global diffeomorphisms and implicit function theorems

Abstract

In the paper, we prove the existence, uniqueness and differentiable dependence of solutions for some nonlinear Urysohn integral equations on parameters. Some sufficient conditions for the nonlinear integral operator of the Urysohn type to be a diffeomorphism are stated. Global invertibility of the Urysohn operator in a certain Sobolev space is ascertained. Consequently, global solvability of Urysohn equations is claimed. Similar results are obtained for some nonlinear Urysohn integral equations with controls by the use of the global implicit function theorem published in the recent paper by Idczak. The proofs of global diffeomorphisms and global implicit functions theorems, the main tools used in the paper, rely in an essential way on the mountain pass theorem. Applications of results to some specific nonlinear Urysohn integral equations are also presented.