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) @? L^1[0,1]. Let @x"i, @t"j @? (0,1), a"i, b"j @? R, i = 1, 2,..., m - 2, j = 1, 2,..., n-2, 0 < @x"1 < @x"2 < ... < @x"m"-"2 < 1, 0 < @t"1 < @t"2 < ... < @t"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