We extend the applicability of a cubically convergent nonlinear system solver using
Lipschitz continuous first-order Fréchet derivative in Banach spaces. This analysis avoids
the usual application of Taylor expansion in convergence analysis and extends the
applicability of the scheme by applying the technique based on the first-order
derivative only. Also, our study provides the radius of convergence ball and
computable error bounds along with the uniqueness of the solution. Furthermore, the
generalization of this analysis using Hölder condition is provided. Various numerical
tests confirm that our analysis produces better results and it is useful in solving such
problems where previous methods can not be implemented.