Semilocal convergence on a family of root-finding multi-point methods in Banach spaces under relaxed continuity condition

In this paper, we consider the semilocal convergence on a family of root-finding multi-point methods. Compared with the results in reference (Hernandez, M.A., Salanova, M.A., J. Comput. Appl. Math. 126, 131---143 3), these multi-point methods do not require the second derivative, Holder continuity condition is relaxed, and the R-order is also enhanced. We prove an existence-uniqueness theorem of the solution. The R-order for these multi-point methods is at least 6 + q with relaxed continuous second derivative, where q?[0,1].