Optimizing the applicability of a theorem by F. Potra for Newton-like methods

We present a new sufficient semilocal convergence conditions for Newton-like methods in order to approximate a locally unique solution of an equation in a Banach space setting. Thi s way, we expand the applicability of these methods in cases not covered in other studies such as Dennis (1971) [12], Ezquerro et al. (2000, 2010) [13,14], Kornstaedt (1975) [18], Potra and Pták (1984) [24], Potra (1985, 1979, 1982, 1981, 1984) [23,25,26,27,28], Proinov (2010) [29], Schmidt (1978) [31] or Yamamoto (1987) [32]. The advantages of our approach also include a tighter convergence analysis under the same computational cost. Applications, where the older convergence criteria are not satisfied but the new convergence criteria are satisfied are also given in this study.