<< œÂ‰˚‰Û˘ýˇ ÒÚ. 2(ËÁ 10 ÒÚ.)Œ√À¿¬À≈Õ»≈ —ÎÂ‰Û˛˘ýˇ >>
existence of periodic solutions. We will prove this result in several steps. First
we establish the existence for the equation

t ‚àà R,
Œ±xŒ± (t) + xŒ± (t) + g(xŒ± (t)) = f (t), Œ± > 0. (11)

Proposition 2.7 Suppose that g is increasing Lipschitz continuous and f is
T -periodic and continuous. Then for every Œ± > 0 the equation (11) has exactly
one periodic solution.
6 Periodic solutions for evolution equations

Remark 2.8 Before starting the proof let us observe that (11) reduces to an
equation of type (3) with gŒ± = Œ±1R + g. Since g is increasing, is clear that gŒ±
is strictly increasing and by the Proposition 2.1 we deduce that the uniqueness
holds. Moreover since Range(gŒ± ) = R, the necessary condition (10) is trivially
veriÔ¬Åed and therefore, in this case we can expect to prove existence.

Proof First of all remark that the existence of periodic solutions reduces to
Ô¬Ånding x0 ‚àà R such that the solution of the evolution problem

t ‚àà [0, T ],
Œ±xŒ± (t) + xŒ± (t) + g(xŒ± (t)) = f (t),
(12)
x(0) = x0 ,

veriÔ¬Åes x(T ; 0, x0 ) = x0 . Here we denote by x(¬· ; 0, x0 ) the solution of (12)
(existence and uniqueness assured by Picard‚Ä™s theorem). We deÔ¬Åne the map
S : R ‚Ü’ R given by
S(x0 ) = x(T ; 0, x0 ), x0 ‚àà R. (13)
We demonstrate the existence and uniqueness of the periodic solution of (12)
by showing that the Banach‚Ä™s Ô¬Åxed point theorem applies. Let us consider
two solutions of (12) corresponding to the initial datas x1 and x2 . Using the
0 0
monotony of g we can write
1d
Œ±|x(t ; 0, x1 ) ‚à’ x(t ; 0, x2 )|2 + |x(t ; 0, x1 ) ‚à’ x(t ; 0, x2 )|2 ‚â¤ 0,
0 0 0 0
2 dt
which implies
1 d 2Œ±t
{e |x(t ; 0, x1 ) ‚à’ x(t ; 0, x2 )|2 } ‚â¤ 0,
0 0
2 dt
and therefore,

|S(x1 ) ‚à’ S(x2 )| = |x(T ; 0, x1 ) ‚à’ x(T ; 0, x2 )| ‚â¤ e‚à’Œ±T |x1 ‚à’ x2 |.
0 0 0 0 0 0

For Œ± > 0 S is a contraction and the Banach‚Ä™s Ô¬Åxed point theorem applies.
Therefore S(x0 ) = x0 for an unique x0 ‚àà R and hence x(¬· ; 0, x0 ) is a periodic
‚™¦
solution of (3).
Naturally, in the following proposition we inquire about the convergence of
(xŒ± )Œ±>0 to a periodic solution of (3) as Œ± ‚Ü’ 0. In view of the Proposition 2.6
this convergence does not hold if (10) is not veriÔ¬Åed. Assume for the moment
that (3) has at least one periodic solution. In this case convergence holds.

Proposition 2.9 If equation (3) has at least one periodic solution, then (xŒ± )Œ±>0
is convergent in C 0 (R; R) and the limit is also a periodic solution of (3).

Proof Denote by x a periodic solution of (3). By elementary calculations we
Ô¬Ånd
1d
Œ±|xŒ± (t) ‚à’ x(t)|2 + |xŒ± (t) ‚à’ x(t)|2 ‚â¤ ‚à’Œ±x(t)(xŒ± (t) ‚à’ x(t)), t ‚àà R, (17)
2 dt
Mihai Bostan 7

which can be also written as
1 d 2Œ±t
{e |xŒ± (t) ‚à’ x(t)|2 } ‚â¤ Œ±eŒ±t |x(t)| ¬· eŒ±t |xŒ± (t) ‚à’ x(t)|, t ‚àà R. (18)
2 dt
Therefore, by integration on [0, t] we deduce
t
1 Œ±t 1
{e |xŒ± (t) ‚à’ x(t)|}2 ‚â¤ |xŒ± (0) ‚à’ x(0)|2 + Œ±eŒ±s |x(s)| ¬· eŒ±s |xŒ± (s) ‚à’ x(s)|ds.
2 2 0
(19)
Using Bellman‚Ä™s lemma, formula (19) gives
t
Œ±t
Œ±eŒ±s |x(s)|ds,
e |xŒ± (t) ‚à’ x(t)| ‚â¤ |xŒ± (0) ‚à’ x(0)| + t ‚àà R. (20)
0

Let us consider Œ± > 0 Ô¬Åxed for the moment. Since x is periodic and continuous,
it is also bounded and therefore from (20) we get

|xŒ± (t) ‚à’ x(t)| ‚â¤ e‚à’Œ±t |xŒ± (0) ‚à’ x(0)| + (1 ‚à’ e‚à’Œ±t ) x t ‚àà R.
L‚àû (R) , (21)

By periodicity we have

|xŒ± (t) ‚à’ x(t)| = |xŒ± (nT + t) ‚à’ x(nT + t)|
‚â¤ e‚à’Œ±(nT +t) |xŒ± (0) ‚à’ x(0)| + (1 ‚à’ e‚à’Œ±(nT +t) ) x L‚àû (R)

‚â¤ e‚à’Œ±(nT +t) |xŒ± (0) ‚à’ x(0)| + x t ‚àà R, n ‚â• 0.
L‚àû (R) ,

By passing to the limit as n ‚Ü’ ‚àû, we deduce that (xŒ± )Œ±>0 is uniformly bounded
in L‚àû (R):

|xŒ± (t)| ‚â¤ |xŒ± (t) ‚à’ x(t)| + |x(t)| ‚â¤ 2 x t ‚àà R, Œ± > 0.
L‚àû (R) ,

The derivatives xŒ± are also uniformly bounded in L‚àû (R) for Œ± ‚Ü’ 0:

|xŒ± (t)|
= |f (t) ‚à’ Œ±xŒ± (t) ‚à’ g(xŒ± (t))|
‚â¤ f L‚àû (R) + 2Œ± x L‚àû (R) + max{g(2 x L‚àû (R) ), ‚à’g(‚à’2 x L‚àû (R) )}.

The uniform convergence of (xŒ± )Œ±>0 follows now from the Arzela-Ascoli‚Ä™s the-
orem. Denote by u the limit of (xŒ± )Œ±>0 as Œ± ‚Ü’ 0. Obviously u is also periodic

u(0) = lim xŒ± (0) = lim xŒ± (T ) = u(T ).
Œ±‚Ü’0 Œ±‚Ü’0

To prove that u veriÔ¬Åes (3), we write
t
{f (s) ‚à’ g(xŒ± (s)) ‚à’ Œ±xŒ± (s)}ds, t ‚àà R.
xŒ± (t) = xŒ± (0) +
0

Since the convergence is uniform, by passing to the limit for Œ± ‚Ü’ 0 we obtain
t
{f (s) ‚à’ g(u(s))}ds,
u(t) = u(0) +
0
8 Periodic solutions for evolution equations

and hence u ‚àà C 1 (R; R) and

t ‚àà R.
u (t) + g(u(t)) = f (t),

From the previous proposition we conclude that the existence of periodic solu-
tions for (3) reduces to uniform estimates in L‚àû (R) for (xŒ± )Œ±>0 .

Proposition 2.10 Assume that g is increasing Lipschitz continuous and f is
T -periodic and continuous. Then the following statements are equivalent:
(i) equation (3) has periodic solutions;
(ii) the sequence (xŒ± )Œ±>0 is uniformly bounded in L‚àû (R). Moreover, in this
case (xŒ± )Œ±>0 is convergent in C 0 (R; R) and the limit is a periodic solution for
(3).

Note that generally we can not estimate (xŒ± )Œ±>0 uniformly in L‚àû (R). In-
deed, by standard computations we obtain

1d
Œ±(xŒ± (t) ‚à’ u)2 + (xŒ± (t) ‚à’ u)2 ‚â¤ |f (t) ‚à’ Œ±u ‚à’ g(u)| ¬· |xŒ± (t) ‚à’ u|, t, u ‚àà R
2 dt
and therefore
1 d 2Œ±t
{e (xŒ± (t) ‚à’ u)2 } ‚â¤ eŒ±t |f (t) ‚à’ Œ±u ‚à’ g(u)| ¬· eŒ±t |xŒ± (t) ‚à’ u|, t, u ‚àà R.
2 dt
Integration on [t, t + h], we get
t+h
1 2Œ±(t+h)
(xŒ± (t + h) ‚à’ u)2 e2Œ±s |f (s) ‚à’ Œ±u ‚à’ g(u)| ¬· |xŒ± (s) ‚à’ u|ds
‚â¤
e
2 t
1
+ e2Œ±t (xŒ± (t) ‚à’ u)2 , t < t + h, u ‚àà R.
2
Now by using Bellman‚Ä™s lemma we deduce
t+h
‚à’Œ±h
e‚à’Œ±(t+h‚à’s) |f (s)‚à’Œ±u‚à’g(u)|ds,
|xŒ± (t+h)‚à’u| ‚â¤ e |xŒ± (t)‚à’u|+ t < t+h.
t

Since xŒ± is T -periodic, by taking h = T we can write
T
1
e‚à’Œ±(T ‚à’s) |f (s) ‚à’ Œ±u ‚à’ g(u)|ds,
|xŒ± (t) ‚à’ u| ‚â¤ t ‚àà R,
1 ‚à’ e‚à’Œ±T 0

and thus for u = 0 we obtain
T
1 1
‚â¤ |f (s) ‚à’ g(0)|ds ‚àº O
xŒ± , Œ± > 0.
L‚àû (R)
1 ‚à’ e‚à’Œ±T Œ±
0

Now we can state our main existence result.
Mihai Bostan 9

Theorem 2.11 Assume that g is increasing Lipschitz continuous, and f is T -
periodic and continuous. Then equation (3) has periodic solutions if and only
1T
if f := T 0 f (t)dt ‚àà Range(g) (there is x0 ‚àà R such that f = g(x0 )).
Moreover in this case we have the estimate
T
‚àÄ x0 ‚àà g ‚à’1 f ,
‚â¤ |x0 | + |f (t) ‚à’ f |dt,
x L‚àû (R)
0

and the solution is unique if and only if Int(O f ) = ‚à… or

diam(g ‚à’1 f ) ‚â¤ diam(Range {f (t) ‚à’ f }dt).

Proof The condition is necessary (see Proposition 2.6). We will prove now
that it is also suÔ¬Écient. Let us consider the sequence of periodic solutions
(xŒ± )Œ±>0 of (11). Accordingly to the Proposition 2.10 we need to prove uniform
estimates in L‚àû (R) for (xŒ± )Œ±>0 . Since xŒ± is T -periodic by integration on [0, T ]
we get
T
{Œ±xŒ± (t) + g(xŒ± (t))}dt = T f , Œ± > 0.
0
Using the average formula for continuous functions we have
T
{Œ±xŒ± (t) + g(xŒ± (t))}dt = T {Œ±xŒ± (tŒ± ) + g(xŒ± (tŒ± ))}, tŒ± ‚àà]0, T [, Œ± > 0.
0

By the hypothesis there is x0 ‚àà R such that f = g(x0 ) and thus

Œ±xŒ± (tŒ± ) + g(xŒ± (tŒ± )) = g(x0 ), Œ± > 0. (22)

Since g is increasing, we deduce

Œ±xŒ± (tŒ± )[x0 ‚à’ xŒ± (tŒ± )] = [g(x0 ) ‚à’ g(xŒ± (tŒ± ))][x0 ‚à’ xŒ± (tŒ± )] ‚â• 0, Œ± > 0,

and thus
|xŒ± (tŒ± )|2 ‚â¤ xŒ± (tŒ± )x0 ‚â¤ |xŒ± (tŒ± )||x0 |.
Finally we deduce that xŒ± (tŒ± ) is uniformly bounded in R:

|xŒ± (tŒ± )| ‚â¤ |x0 |, ‚àÄ Œ± > 0.

Now we can easily Ô¬Ånd uniform estimates in L‚àû (R) for (xŒ± )Œ±>0 . Let us take in
the previous calculus u = xŒ± (tŒ± )and integrate on [tŒ± , t]:
t
1 2Œ±t
e (xŒ± (t)‚à’xŒ± (tŒ± ))2 ‚â¤ e2Œ±s |f (s)‚à’Œ±xŒ± (tŒ± )‚à’g(xŒ± (tŒ± ))|¬·|xŒ± (s)‚à’xŒ± (tŒ± )|ds.
2 tŒ±

By using Bellman‚Ä™s lemma we get
t
e‚à’Œ±(t‚à’s) |f (s) ‚à’ Œ±xŒ± (tŒ± ) ‚à’ g(xŒ± (tŒ± ))|ds,
|xŒ± (t) ‚à’ xŒ± (tŒ± )| ‚â¤ t > tŒ± ,
tŒ±
10 Periodic solutions for evolution equations

and hence by (22) we deduce
T
|xŒ± (t)| ‚â¤ |x0 | + |f (t) ‚à’ Œ±xŒ± (tŒ± ) ‚à’ g(xŒ± (tŒ± ))|dt
0
T
= |x0 | + |f (t) ‚à’ f |dt, t ‚àà R, Œ± > 0. (23)
0

Now by passing to the limit in (23) we get
T
t ‚àà R, ‚àÄ x0 ‚àà g ‚à’1 f .
|x(t)| ‚â¤ |x0 | + |f (t) ‚à’ f |dt,
0

2.3 Sub(super)-periodic solutions
In this part we generalize the previous existence results for sub(super)-periodic
solutions. We will see that similar results hold. Let us introduce the concept of
 << œÂ‰˚‰Û˘ýˇ ÒÚ. 2(ËÁ 10 ÒÚ.)Œ√À¿¬À≈Õ»≈ —ÎÂ‰Û˛˘ýˇ >>