Semilocal convergence of a continuation method with Hölder continuous second derivative in Banach spaces

In this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton’s method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frëchet derivative satisfies the Hölder continuity condition. This condition is mild and works for problems in which the second Frëchet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter α∈[0,1]$\alpha \in [0,1]$ for the solution x^∗$x^{\ast }$ is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Hölder continuity condition on F^″$F^{″}$, it is found that our analysis gives improved results. Further, we have observed that for particular values of the α$\alpha$, our analysis reduces to those for the Chebyshev method (α=0$\alpha =0$) and the convex acceleration of Newton’s method (α=1)$(\alpha =1)$ respectively with improved results.