Existence and Uniqueness of Positive Solutions of Boundary-Value Problems for Fractional Differential Equations with p-Laplacian Operator and Identities on the Some Special Polynomials

Abstract

We consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator D0+β(ϕp(D0+αu(t)))+a(t)f(u)=0, 01, ϕp-1=ϕq, 1/p+1/q=1,0⩽γ0 are parameters, a:(0,1)→[0,+∞), and f:[0,+∞)→[0,+∞) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters λ and μ are obtained. The uniqueness of positive solution on the parameters λ and μ is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative.