Lipschitz Operators and the Solvability of Non-linear Operator Equations

Let U and V be Banach spaces,L and N be non-linear operators from U into V.L issaid to be Lipschitz if L(L):=sup{‖Lx-Ly‖·‖x-y‖:x≠y}is finite.In this paper,we givesome basic properties of Lipschitz operators and then discuss the unique solvability,exact solvability,approximate solvability of the operator equations Lx=y and Lx+Nx=y.Under some conditionswe prove the equivalence of these solvabilities.We also give an estimation for the relative-errors of thesolutions of these two systems and an application of our method to a non-linear control system.