A method of proving the existence of solution curves based on the numerical computation with guaranteed accuracy

A new method is presented for proving the existence of solution curves of a system of nonlinear equations. In this method, the existence of a solution curve is verified by numerically checking the sufficient condition of the implicit function theorem. For the purpose, the numerical computation with guaranteed accuracy is used. The algorithm proposed in this paper outputs regions in which the existence of a solution curve is guaranteed locally by the implicit function theorem. Under suitable conditions, it is proved that the algorithm always succeeds to prove the existence of a solution curve globally in a given region within finite steps provided that the total length of the solution curve is finite. A few numerical examples are presented to demonstrate the validity of this algorithm.