Semilocal convergence of a family of third-order methods in Banach spaces under Hölder continuous second derivative

In this paper, the semilocal convergence of a family of multipoint third-order methods used for solving F(x)=0F(x)=0 in Banach spaces is established. It is done by using recurrence relations under the assumption that the second Fréchet derivative of FF satisfies Hölder continuity condition. Based on two parameters depending upon FF, a new family of recurrence relations is defined. Using these recurrence relations, an existence–uniqueness theorem is established to prove that the RR-order convergence of the method is (2+p)(2+p). A priori error bounds for the method are also derived. Two numerical examples are worked out to demonstrate the efficacy of our approach.