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In this paper, we prove the existence of fixed points for every 2-rotative continuous mapping in Banach spaces to answer an open question raised by Goebel and Koter. Further, we modify <i>F</i>-contraction by developing <i>F</i>-rotative mapping and establish some fixed-point theorems. Finally, we apply our results to prove the existence of a solution of a non-linear fractional differential equation.