An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness

We propose a numerical method to verify the existence and local uniqueness of solutions to nonlinear elliptic equations. We numerically construct a set containing solutions which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space. By using the finite element approximation and constructive error estimates, we calculate the eigenvalue bound with smallest absolute value to evaluate the norm of the inverse of the linearized operator. Utilizing this bound we derive a verification condition of the Newton-Kantorovich type. Numerical examples are presented.