NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE AND UNIQUENESS OF SOLUTIONS OF SECOND-ORDER AUTONOMOUS DIFFERENTIAL EQUATIONS

The main result ensures that the scalar problem x =f(x), x(0)=x0, x (0)=x1, has a nonconstant locallyW2,1 solution if and only if there exists a nontrivial interval J such that x0 ∈ J, f ∈ L1 loc(J), x21 + 2 y x0 f(s) ds > 0 for almost all y ∈ J and max{1, |f|} x21 + 2 · x0 f(s) ds ∈ L1 loc(J). Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.