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Existence and Uniqueness Theorems for a Three-step Newton-type Method under $L$-Average Conditions
KeyWord:Banach space; Nonlinear equation; Lipschitz condition; $L$-average; Convergence ball
In this paper, we study the local convergence of a three-step Newton-type method for solving nonlinear equations in Banach spaces under weaker hypothesis. More precisely, we derive the existence and uniqueness theorems, when the first-order derivative of nonlinear operator satisfies the $L$-average conditions instead of the usual Lipschitz condition, which have been discussed in the earlier study.