Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

Abstract EN

We study the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation -?tμx(t) = λf(t,x(t),?tμ1x(t),?tμ2x(t),… ,?tμn-1x(t))…?tμix(0) = 0, 1≤i≤n-1,?tμn-1+1x(0)=0, ?tμn-1x(1)=∑j=1m-2aj?tμn-1x(ξj), where n-1<μ≤n, n≥3, 0<μ1<μ2<⋯<μn-2<μn-1, n-3<μn-1<μ-2, aj∈ℝ, 0<ξ1<ξ2<⋯<ξm-2<1 satisfying 0<∑j=1m-2ajξjμ-μn-1-1<1, ?tμ is the standard Riemann-Liouville derivative, f∈C((0,1)×ℝn,(-∞,+∞)), and f is allowed to be changing-sign.

By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.