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

摘要

We consider the following boundary-value problem of nonlinear fractional differential equation withp-Laplacian operatorD0+β(ϕp(D0+αu(t)))+a(t)f(u)=0,0<t<1,u(0)=γu(h)+λ,u′(0)=μ,ϕp(D0+αu(0))=(ϕp(D0+αu(1)))′=(ϕp(D0+αu(0)))′′=(ϕp(D0+αu(0)))′′′=0, where1<α⩽2,3<β⩽4are real numbers,D0+α,D0+βare the standard Caputo fractional derivatives,ϕp(s)=|s|p-2s,p>1,ϕp-1=ϕq,1/p+1/q=1,0⩽γ<1,0⩽h⩽1,λ,μ>0are parameters,a:(0,1)→[0,+∞),andf:[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 parametersλandμare obtained. The uniqueness of positive solution on the parametersλandμis 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.