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 is said to be Lipschitz if L₁(L) := sup{∥Lx - Ly∥ · ∥x - y∥^-1 : x ≠ y} is finite. In this paper, we give some 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 conditions we prove the equivalence of these solvabilities. We also give an estimation for the relative-errors of the solutions of these two systems and an application of our method to a non-linear control system.