Positive solutions to singular semipositone m-point n-order boundary value problems

Abstract

In this paper, we consider the existence of positive solutions to the following Singular Semipositone m-Point n-order Boundary Value Problems (SBVP):

$$\left\{\begin{array}{l@{\quad}l}(-1)^{(n-k)}x^{(n)}(t)=\lambda f(t,x(t)),&0<t<1,\\[4pt]x(1)=\sum_{i=1}^{m-2}a_ix(\eta_i),\qquad x^{(i)}(0)=0,&0\leq i\leq k-1,\\[4pt]x^{(j)}(1)=0,&1\leq j\leq n-k-1,\end{array}\right.$$

where m≥3, λ>0, a _i ∈[0,∞),(i=1,2,…,m−2),0<η ₁<η ₂<⋅⋅⋅<η _m−2<1 are constants, f:(0,1)×[0,+∞)→R is continuous and may have singularity at t=0 and/or 1. Without making any monotone-type assumption, we obtain the positive solution of the problem for λ lying in some interval, based on fixed-point index theorem in a cone.