Existence of positive solutions to higher order singular semi-positively definite conjugate boundary-value problems

By using the theory of index of fixed-point on a cone and Lebesgue's dominated convergence theorem,the existence of positive solutions to the higher-order conjugate boundary-velue problem(-1)n-kx(n)(t)=f(t,x(t)),0t1;x(i)(0)=0,i=0,1,…,k-1;x(j)(1)=0,j=0,1,…,n-k-1was investigated.In above equation system n≥2,1≤k≤n-1,f:(0,1)×(0,+∞)→R is continuous and may be taken as negative a have no lower bound.It,in addition,may be allowed to be singular at t=0,t=1,and x=0.