Extending the applicability of improved Chebyshev–Secant-type methods

In this work, we present a new semilocal convergence for the family of improved Chebyshev–Secant-type methods (ICSTM) using auxiliary points under generalized convergence conditions on divided differences for non-differentiable operators. The existence and uniqueness theorems are established for the solution using recurrence relations. The parameter domain is also analyzed for both differentiable and non-differentiable operators. Finally, the theoretical results are validated by considering a nonlinear integral equation of the Hammerstein type and a nonlinear elliptic PDE that arise in electromagnetic fluid dynamics and in the theory of gas dynamics, respectively. The improved convergence domains are obtained under weaker conditions compared to the existing approach (Kumar et al. in Numer Algorithms 86(3):1051–1070, 2021). Moreover, the convergence domains are also established where the existing results are not applicable.