A priori estimates for the existence of a solution for a multi-point boundary value problem

Let a(i) is an element of R, xi(i)E(0, 1), i = 1, 2,...,m - 2, 0 < xi(1) < xi(2) < ... < xi(m-2) < 1, with alpha = Sigma(i=1)(m-2) a(i) not equal 1 be given. Let x(t) is an element of W-2,W-1(0, 1) be such that x'(0) = 0, x(1 ) = Sigma(i=1)(m-2) a(i)x(xi(i)) (*) be given. This paper is concerned with the problem of obtaining Poincare type a priori estimates of the form \\x\ \(infinity) less than or equal to C\\x "\\(1). The study of such estimates is motivated by the problem of existence of a solution for the Caratheodory equation x "(t) = f(t,x(t),x'(t)) + e(t), 0 < t < 1, satisfying boundary c onditions (*). This problem was studied earlier by Gupta et al. (Jour. Math . Anal. Appl. 189 (1995), 575-584) when the a(i)'s, all had the same sign.