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Existence and Uniqueness Theorems for a Three-step Newton-type Method under $L$-Average Conditions
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KeyWord:Banach space  Nonlinear equation  Lipschitz condition  $L$-average  Convergence ball
Author NameAffiliation
Jai Prakash Jaiswal Department of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur, C.G., India-495009
Faculty of Science, Barkatullah University, Bhopal, M.P.-462026, India
Regional Institute of Education, Bhopal, M.P.-462013, India 
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      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.