Abstract :
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.
