Eigenvalue of Fractional Differential Equations withp-Laplacian Operator

Abstract

We investigate the existence of positive solutions for the fractional order eigenvalue problem withp-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), where𝒟tβ,  𝒟tα,  𝒟tγare the standard Riemann-Liouville derivatives andp-Laplacian operator is defined asφp(s)=|s|p-2s,  p>1.f:(0,1)×(0,+∞)→[0,+∞)is continuous andfcan be singular att=0,1andx=0.By constructing upper and lower solutions, the existence of positive solutions for the eigenvalue problem of fractional differential equation is established.