Analyzing the Semilocal Convergence of a Fourth-Order Newton-Type Scheme with Novel Majorant and Average Lipschitz Conditions

The main focus of this paper is the analysis of the semilocal convergence (S.C.) of a three-step Newton-type scheme (TSNTS) used for finding the solution of nonlinear operators in Banach spaces (B.S.). A novel S.C. analysis of the TSNTS is introduced, which is based on the assumption that a generalized Lipschitz condition (G.L.C.) is satisfied by the first derivative of the operator. The findings contribute to the theoretical understanding of TSNTS in B.S. and have practical implications in various applications, such as integral equation further validating our results.