Secant method with regularly continuous divided differences

We offer a convergence analysis of the secant method for solving nonlinear operator equations in Banach spaces using Kantorovich's technique of majorization. In contrast with other known convergence analyses of this method, ours is based on a different continuity characteristic of the divided difference operator (called regular continuity) which is more general (but not too general) and more flexible than those used by other researchers. As we show, it allows to obtain broader convergence domains and tighter error bounds. Another distinctive feature of our analysis is the use of a functional equation for precise description of convergence domain of the majorant generator (a system of difference equations).