A generalized multi-point boundary value problem for second order ordinary differential equations

Let f: [0,1] x R-2 --> R be a function satisfying Caratheodory's conditions and e(t)is an element of L-1[0,1]. Let xi(i), tau(j) is an element of (0,1), a(i), b(j) is an element of R, i=1,2,..., m-2, j=1,2,..., n- 2, 0<xi(1)<xi(2)<...<xi(m-2)<1,0<tau(1)<tau(2) <...<tau(n-2)<1 be given. This paper is concerned with the problem of existence of a solution for the generalized multi-point boundary value problems x(")(t)=f(t,x(t),x(')(t))+e(t), 0<t<1, x(0)=(m-2)Sigma(i=1)a(i)x(xi(i)), x(1)=(n-2)Sigma(j=1)b(j)x(tau(j)), and x(")(t)=f(t,x(t), x(')(t))+e(t), 0<t<1, x(0)=(m-2)Sigma(i-1)a(i)x(xi(i)), x(')(1)=(n-2)Sigma(j=1)b(j)x(')(tau(j)). (C) Elsevier Science Inc., 1998.