Existence of Concave Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation with p‐Laplacian Operator

We consider the existence and multiplicity of concave positive solutions for boundary value problem of nonlinear fractional differential equation with p‐Laplacian operator , 0 < t < 1, u(0) = u′(1) = 0, u′′(0) = 0, , where 0 < γ < 1, 2 < α < 3, 0 < ρ ⩽ 1, denotes the Caputo derivative, and f : [0,1]×[0, +∞) × R → [0, +∞) is continuous function, ϕp(s) = |s|p−2s, p > 1, (ϕp) −1 = ϕq, 1/p + 1/q = 1. By using fixed point theorem, the results for existence and multiplicity of concave positive solutions to the above boundary value problem are obtained. Finally, an example is given to show the effectiveness of our works.