Existence Of Solutions Of Boudary Value Problem For Nonlinear Fractional Functional Differential Equations

In this paper, we first give sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problemwhere Dαis the Riemann-Liouville differential operator of orderα, f:[0.1]×[0,∞)→[0,∞)连续.Next, we give sufficient conditions for the existence of at least one and at least three positive solutions to the nonlinear fractional boundary value problemwhere Da is the Riemann-Liouville differential operator of orderα,0≤β≤1,0≤α≤1,ξ∈(0,1),0≤α-β-1,f:[0,1]×[0,∞)→[0,∞) satisfies the following conditions of Caratheodory type.In the end, we study the existence of solutions for boundary value problems of fractional functional differential equations where 1＜α≤2 is a real number, cD0+αis the Caputo differential operator of orderα, f:[0,T]×CΥ→R,φ∈CΥ(:= C[-Υ,0]) and A∈R. The proof of our main result is based upon the fixed-point theorem.