Semilocal convergence analysis for a three-step scheme in Banach spaces with a new type majorant: using average Lipschitz conditions

We analyze the semilocal convergence (S.C.) of the Newton-Traub scheme (NTS) used for finding solutions of nonlinear problems in Banach spaces. The analysis is based on the assumption that a generalized Lipschitz condition is satisfied by the first derivative of the relevant operator. The analysis establishes the fifth-order convergence of the NTS under an additional condition. Furthermore, we consider two special cases. The findings contribute to the theoretical understanding of NTS in Banach spaces and have practical applications, for example to integral equations.