Eigenvalue of higher-order semipositone multi-point boundary value problems with derivatives

摘要

Let ξi∈(0,1) with 0<ξ1<ξ2<⋯<ξm-2<1, ai,bi∈[0,+∞) with 0<∑i=1m-2ai<1 and 0<∑i=1m-2bi<1. We consider the eigenvalue of higher-order multi-point boundary value problem with derivativesx(n)(t)+λf(t,x(t),x′(t),…,x(n-2)(t))=0,t∈(0,1),n⩾3,x(i)(0)=0,0⩽i⩽n-3,x(n-1)(0)=∑i=1m-2aix(n-1)(ξi),x(n-2)(1)=∑i=1m-2bix(n-2)(ξi),where f∈C[(0,1)×Rn-1,R] satisfies f(t,u1,u2,…,un-1)⩾-p(t) and p∈L1[(0,1),(0,+∞)]. In the case where f can be singular at t=0 and/or t=1 and be allowed to change sign, we show the existence of positive solutions by using the fixed point theorem in cone, the main technique is approximating the singular higher-order boundary value problem to the singular second-order boundary value problem by constructing some available integral operators. In doing so, the usual restriction that f⩾0 will be removed.