Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

Abstract

We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term -??????(??)=??(??)??(??,??(??),????1??(??),????2??(??),…,??????-1??(??)),0<??<1,????????(0)=0,1=??=??-1,??????-1+1??(0)=0, ??????-1???(1)=??-2??=1??????????-1??(????), where ??-1<??=??, ???N and ??=3 with 0<??1<??2<?<????-2<????-1 and ??-3<????-1<??-2, ?????R,0<??1<??2<?<????-2<1 satisfying ?0<??-2??=1????????-????-1??-1<1, ???? is the standard Riemann-Liouville derivative, ??:[0,1]×R???R is a sign-changing continuous function and may be unbounded from below with respect to ????, and ??:(0,1)?[0,8) is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.