I ran into the following problem but don't know how to solve it, anyone?

F: R -> R is a smooth function with F'(x) >= 0 for all x in R and u,v,w are all continuous functions (not differentiable) on R and I = [0,t] where t >= 0. The following relations hold:

u(t) <= w(t) + integral(F(u(s))) ds over I
v(t) >= w(t) + integral(F(v(s))) ds over I

the problem is that this implies that u(t) <= v(t) for all t >= 0. I've been struggling with it for quite a while now and would appreciate any help!