boundary value problem with complementary slackness

I have an ODE system with complementary slackness condition.

• y'(z) = f(x(z),y(z),z)
• y(zmin)=y(zmax)=0
• x'(z)>=0
• y(z)>=0
• x'(z)*y(z)=0

It seems the solution takes the form of

• x'(z)=0 for [zmin,some z] and x'(z)>0 for [some z, zmax].

but I am not sure if matlab can solve such problem using bvp4c or something. Any suggestion would be appreciated.