Positive Solutions of m-point Conjugate Boundary Value Problems

In this paper, the conjugate boundary value problem -u´´(t) = f(t,u(t)) for t isin [0,1]{eta₁middot,eta₂,--- ,eta_m-2} subject to u´(0) = u(1) = 0 and u´₊(eta_i) - u´_(eta_i) = a_iu´(1) (i = 1, 2, ... , m - 2), of the m-point boundary value problem -v´´(t) = f(t,v(t)) subject to v(0) = 0 and v(1) = Sigma_i-1 ^m-2 alpha_iv(eta_i) is put forward and considered, where eta_i isin (0,1) with eta₁ < eta₂ < ... < eta_m-2, alpha_iisin[0,1) with 0 < Sigma_i-1 ^m-2 alpha_i < 1, f isin C([0,1] times [0, infin), [0, infin)). The problem is translated into Hammertein ´s integral equation with the use of Green´s function. Then the existence of single and multiple positive solutions of the conjugate problem is shown under some conditions concerning the first eigenvalue of the relevant linear operator by means of fixed point index theory. Similar conclusions hold for some other m-point boundary value conditions.