Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives

In this work, we establish the existence of multiple positive solutions for a general higher order fractional differential equation with derivatives and a sign-changing Carathèodory perturbed term-Dαx(t)=p(t)ft,x(t),Dμ1x(t),Dμ2x(t),…,Dμn-1x(t)-gt,x(t),Dμ1x(t),Dμ2x(t),…,Dμn-1x(t),Dμix(0)=0,1⩽i⩽n-1,Dμn-1+1x(0)=0,Dμn-1x(1)=∑j=1m-2ajDμn-1x(ξj),where n-1<α⩽n,n∈Nn-1<α⩽n,n∈N and n⩾3n⩾3 with 0<μ1<μ2<⋯<μn-2<μn-10<μ1<μ2<⋯<μn-2<μn-1 and n-3<μn-1<α-2,aj∈R,0<ξ1<ξ2<…<ξm-2<1 satisfying 0<∑j=1m-2ajξjα-μn-1-1<1, DαDα is the standard Riemann–Liouville derivative, f∈C([0,1]×Rn,[0,+∞))f∈C([0,1]×Rn,[0,+∞)). This equation is viewed as a perturbation of a general higher order fractional differential equation, where the perturbed term g:[0,1]×Rn→(-∞,+∞)g:[0,1]×Rn→(-∞,+∞) only satisfies the global Carathéodory conditions, which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points.