A Newton-like method and its application

In this paper we prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton's method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler–Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided.