Some necessary and sufficient conditions for existence of positive solutions for third order singular sublinear multi-point boundary value problems

Abstract

We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem {x(3)(t)+f(t,x(t),x′(t))=0, 0<t<1,x(0)-∑i=1m1αix(ξi)=0, x′(0)-∑i=1m2βix′(ηi)=0, x′(1)=0,where 0≤αi≤∑i=1m1αi<1,i=1,2,…,m1,0<ξ1<ξ2<…<ξm1<1,0≤βj≤∑i=1m2βi<1,j=1,2,…,m2,0<η1<η2<…<ηm2<1. And we obtain some necessary and sufficient conditions for the existence of C1[0,1] and C2[0,1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f(t,x,y) may be singular at x,y,t=0 and/or t=1.