Eigenvalue of Fractional Differential Equations with p-Laplacian Operator

Abstract

We investigate the existence of positive solutions for the fractional order eigenvalue problem with p-Laplacian operator -𝒟tβ(φp(𝒟tαx))(t)=λf(t,x(t)), t∈(0,1), x(0)=0, 𝒟tαx(0)=0, 𝒟tγx(1)=∑j=1m-2‍aj𝒟tγx(ξj), where 𝒟tβ, 𝒟tα, 𝒟tγ are the standard Riemann-Liouville derivatives and p-Laplacian operator is defined as φp(s)=|s|p-2s, p>1.f:(0,1)×(0,+∞)→[0,+∞) is continuous and f can be singular at t=0,1 and x=0. By constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of fractional differential equation is established.