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The existence andmultiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) Dα0+yðxÞ + f ðx, yðxÞÞ = 0, 0 < x < 1, yð0Þ = y′ð1Þ = y′′ð0Þ = 0 is established, where 2 < α ≤ 3, D0+ is the Caputo fractional derivative, and f : 1⁄20, 1 × 1⁄20,∞Þ⟶ 1⁄20,∞Þ 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.

Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem