  In numerical existence proofs for solutions of the semi-linear elliptic
system, evaluating the norm of the inverse of a perturbed Laplace operator
plays an important role. We reveal an eigenvalue problem to design a method for
verifying the invertibility of the operator and evaluating the norm of its
inverse based on Liu's method and the Temple-Lehman-Goerisch method. We apply
the inverse-norm's estimation to the Dirichlet boundary value problem of the
Lotka-Volterra system with diffusion terms and confirm the efficacy of our
method.
