An improved convergence theorem for a class of Secant-like methods

An improved semilocal convergence theorem for a class of Secant-like methods in Banach spaces is provided for solving a nonlinear operator equation. It is assumed that the first-order divided difference of the nonlinear operator is Hölder continuous. Our convergence conditions, strategically proposed, are weaker than some existing ones and are easily satisfied. Finally, two simple examples are provided to show that our results apply, while earlier ones fail.