Existence of positive solutions for multi-point boundary value problems

The existence of positive solutions are established for the multi-point boundary value problems {(-1)(n) u((2n)) (x) = lambda p(x) f(u(x)), 0 < x < 1 u((2i)) (0) = m Sigma(j=1) a(j) u((2i)) (eta(j)), u((2i+1)) (1) = (m)Sigma(j=1) b(j) u((2i+1)) (eta(j)), i = 0, 1, ..., n-1 where a(j), b(j) is an element of [0, infinity), j = 1, 2, ..., m, with 0 < (m)Sigma(j)=1 a(j) < 1, 0 < (m)Sigma(j=1) b(j) < 1, and eta(j) is an element of (0, 1) with 0 < eta(1) < eta(2) < ... < eta(m) < 1, under certain conditions on f f and p using the Krans'sskii fixed point theorem for certain values of lambda. We use the positivity of the Green's function and cone thoery to prove our results.