A version of the inverse function theorem for solving nonlinear equations

A version of the inverse function theorem for solving nonlinear equations

The operator equation F(u) = h, where F is a nonlinear operator in a Hilbert space H is studied. Suppose that y is a solution of F(u) = f. It is proved that the equation F(u) = h is uniquely solvable for any h in a sufficiently small neighborhood of f, if F is Fr´echet differentiable on a neighborhood of y, F0 is continuous at y and F0(y) is invertible. The method of the proof is similar to the proof of the inverse function theorem. Moreover, the convergence to the solution y by the Newton method un+1 = un − [F0(u0)]−1(F(un) − f) with an initial approximation u0, sufficiently close to y, is proved.

The operator equation F(u) = h, where F is a nonlinear operator in a Hilbert space H is studied. Suppose that y is a solution of F(u) = f. It is proved that the equation F(u) = h is uniquely solvable for any h in a sufficiently small neighborhood of f, if F is Fr´echet differentiable on a neighborhood of y, F0 is continuous at y and F0(y) is invertible. The method of the proof is similar to the proof of the inverse function theorem. Moreover, the convergence to the solution y by the Newton method un+1 = un − [F0(u0)]−1(F(un) − f) with an initial approximation u0, sufficiently close to y, is proved.