SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS

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)-m1∑i=1 αi x(ξi) = 0, x′(0)-m2∑i=1 βi x′(ηi) = 0, x′(1)=0,where 0 ≤ ai≤m1∑i=1 αi < 1, i = 1, 2, ···, m1, 0 < ξ1< ξ2< ··· < ξm1< 1, 0 ≤βj≤m2∑i=1βi<1,J=1,2, ···, m2, 0 < η1< η2< ··· < ηm2< 1. And we obtain some necessa βi <=11, j = 1,ry 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.