On Newton-like method for solving generalized nonlinear operator equations in banach spaces

The purpose of this paper is to prove existence and uniqueness theorem for solving an operator equation of the form F(x)+G(x) = 0, where F is a Gˆateaux differentiable operator and G is a Lipschitzian operator defined on an open convex subset of a Banach space. Our result extends and improves the previously known results in recent literature.