New sufficient convergence conditions of the Secant method for nondifferentiable operators

We provide new sufficient conditions for the convergence of the Secant method to a locally unique solution of a nondifferentiable equation on a Banach space. Suppose x0, x−1 are two initial points, and x1 is the first approximation generated by the Secant method. Our new idea uses the conditions ∥x1 − xi∥ ⩽ η (i = 0, −1) instead of the classical conditions ∥x0 − x−1∥ ⩽ α and ∥x1 − x0∥ ⩽ η. Finally, a numerical example is provided to show that our result applies, where earlier one fails.