Solution to the Problem 2003 by Shangbing Ai, Georgia Institute of Technology.
The Problem. (Proposed by P. Korman)
(i) Consider the equation for x=x(t)
Assume that g(x) is continuous for all 
, and that the limits 
and 
exist (the limits are finite or infinite), with
Assume that f(t) is a continuous periodic function with period p. Show that problem (1) has a p-periodic solution if and only if
(ii) Generalize this result for the equation
where a(t) is a continuous p-periodic function satisfying
obtaining a version of the well known Landesman-Lazer result. (E.H. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, Jour. Math. Mech. 19, 609-623 (1970).) }
Proof: We first show that if (1) has a p-periodic solution x(t), then (3) holds. Integrating (1) over the interval [0,p] gives
and so
Which together with condition (2) yields (3).
Next we show that if (4) has a p-periodic solution x(t), then
In fact, from (4) and (5) we get
and then (6) follows easily from (2).
Assume that (3) and (6) hold. We want to show there exist p-periodic solutions for (1) and (4) respectively. We shall use the following lemma to show that.
Proof: Let 
and 
. Further let 
and 
. Then from (9) we have
Since 
is periodic with period p, it suffices to show that under the condition (8), the equation (9) has a p-periodic solution. To do that, we define two sets
and
where 
denotes the solution of (9) with 
. Since u'(0,C)>0 and u'(0,-C)<0, it follows that 
and 
. So A and B are both nonempty. The continuity of solutions with respect to initial data implies that A and B are both open sets. Observe from (9) that if u=C (or u=-C) for some 
then u'>0 (respectively, u'<0) for 
as long as u exists. Hence A and B are also disjoint. Then the connectedness of 
implies that there is at least one 
so that 
for all 
. Since (9) is a scaler equation, it follows from a well known result (which is easy to be shown) that the existence of the bounded solution 
over 
implies that there exists a p-periodic solution for (9). This shows the lemma.
We now use the above lemma to show the existence parts of (i) and (ii).
In (i), we let G(t,x):=g(x) and F(t):=f(t). Then it is obvious that the conditions (2) and (3) implies (8). Therefore (1) has a p-periodic solution.
In order to apply the above lemma for the existence part in (ii) under the assumption (6), we first make the following changes of variables. Let 
, 
, 
, and F(z):=f(t). Then (4) can be written as
Notice that G(z+Z,y)=G(z,y) and F(z+Z)=F(z) where 
. Since 
, it suffices to show the existence of a Z-periodic solution for (10). From the above lemma we need to verify that (8) holds for (10). It follows from (6) that
By the definition of G(z,y), we see that 
uniformly for 
, it follows from (11) that there is a constant C>0 such that 
for all 
. Similarly we can show 
for all if we take C sufficiently large. Therefore by the lemma we see that (10) has a Z-periodic solution y(z) so that 
gives a p-periodic solution of (4).
Thus we have proved the following version of Landesman-Lazer result:
Assume that (2) and (5) hold. Then the equation (4) has a p-periodic solution if and only if (6) holds.
Problem 2000-3 was also solved by Dian K. Palagachev, Polytech of Bari, ITALY
