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This paper is devoted to investigating the fractional order of convergence of operator iteration schemes and, as a means of obtaining rapid convergence, rediscovering fixed-point methods by fractional calculus. In addition, a new definition for fractional Gateaux derivative is used which seems a key idea for accelerating the convergence of iterative methods. At the end, the relevance of the theory and applications of integral equations are given. Mathematics Subject Classification: 65M12, 26A33

A fractional order of convergence rate for successive methods: Examples on integral equations