In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [6]-[8]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations in a Banach space setting. Finer error estimates, a larger convergence domain, and a more precise information on the

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... tion of the solution are obtained under the same or weaker hypotheses than before [6]-[8]. Numerical examples are also provided. MSC 2000. 65H10, 65G99, 47H10, 49M15.