A Two-Point Boundary Value Problem with Reflection of the Argument

We consider the following two-point boundary value problems u '' x+u pi-x+gx,u pi-x=hx in 0,pi,u0=0=u pi, and u '' x+u pi-x-gx,u pi-x=-hx in 0,pi,u0=0=u pi, by setting h is an element of L10,pi and g:0,pi xR ⟶R being a Caratheodory function. When a,b is an element of L10,pi, ax <= 3 for x is an element of 0,pi a.e. with strict inequality on a positive measurable subset of 0,pi, and gx,u <= axu+bx for x is an element of 0,pi a.e. as well as sufficiently large u, several existence theorems will be obtained, with or without a sign condition.