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This paper is concerned with a class of anti-periodic boundary value problems for fractional differential equations with the Riesz-Caputo derivative, which can reflected both the past and the future nonlocal memory effects. By means of new fractional Gronwall inequalities and some fixed point theorems, we obtain some existence results of solutions under the Lipschitz condition, the sublinear growth condition, the nonlinear growth condition and the comparison condition. Three examples are given to illustrate the results.doi:10.1186/s13662-019-2001-z fatcat:nlhnhmxu3vh5bcst523fpfkjxi