This article investigates the existence of a solution of first-order periodic boundary value problem (Equation Presented) where T > 0 and f : I x ℝ → ℝ is a continuous function, in a space where self-distance of a point may be non-zero. To accomplish this goal, we launch a new α-weakly contractive mapping which involve rational terms and then build up fixed point results for weakly α-admissible mapping. To illustrate our results, we give throughout the paper some examples. Finally, we suggest some future work to find positive solutions of three classes of nonlinear fractional differential equations. Copyright ©2016 Watam Press.