The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) , is established, where , is the Caputo fractional derivative, and is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.
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