Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces

Abstract

The semilocal convergence of double step Secant method to approximate a locally unique solution of a nonlinear equation is described in Banach space setting. Majorizing sequences are used under the assumption that the first-order divided differences of the involved operator satisfy the weaker Lipschitz and the center-Lipschitz continuity conditions. A theorem is established for the existence-uniqueness region along with the estimation of error bounds for the solution. Our work improves the results derived in Ren and Argyros (2015) in more stringent Lipschitz and center Lipschitz conditions and gives finer majorizing sequences. Also, an example is worked out where the conditions of Ren and Argyros (2015) fail but our’s work. Numerical examples including nonlinear elliptic differential equations and integral equations are worked out. It is found that our conditions enlarge the convergence domain of the solution. Finally, taking a nonlinear system of $m$ equations, the Efficiency Index $\left(EI\right)$ and the Computational Efficiency Index $\left(CEI\right)$ of double step Secant method are computed and its comparison with respect to other similar existing iterative methods are summarized in the tabular forms.