POSITIVE SOLUTIONS OF MULTI-POINT BOUNDARY VALUE PROBLEMS

This article concerns the boundary value problem consisting of the nonlinear differential equation u '' +g(t)f(t,u(t)) = 0, t is an element of (0, 1) and the multi-point boundary conditions u(0) = alpha u'/(0), u(1) = Sigma(m)(i=1) beta(i)u(eta(i)) broken vertical bar Sigma(m)(i=1) gamma(i)u'(eta(i)), where 0 <= alpha <= infinity, 0 < eta(1) <= eta(1) < eta(2) < ... < eta(m) < 1, beta(i), > 0, gamma(i) < 0 (i - 1, 2,..., m). By using the fixed point index theory, we establish the existences of at least one positive solution and at least two positive solutions.