The role of a priori estimates in the method of non-local continuation of solution by parameter

An iterative method for continuation of solutions with respect to a parameter is proposed. The nonlocal case is studied when the parameter belongs to the segment of the real axis. An iterative scheme for continuing the solution is constructed for a linear equation in Banach spaces with a linear operator continuously depending on the parameter, satisfying the Lipschitz condition with respect to the parameter. The generalization of this result on a nonlinear equation in Banach spaces is proposed. The iterative scheme of the method of continuation of the solution by parameter using the Newton-Kantorovich method is constructed. An priori estimates of solutions enable solution construction for arbitrary parameters.