we establish the existence for the equation

t ∈ R,

±x± (t) + x± (t) + g(x± (t)) = f (t), ± > 0. (11)

Proposition 2.7 Suppose that g is increasing Lipschitz continuous and f is

T -periodic and continuous. Then for every ± > 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 g± = ±1R + g. Since g is increasing, is clear that g±

is strictly increasing and by the Proposition 2.1 we deduce that the uniqueness

holds. Moreover since Range(g± ) = R, the necessary condition (10) is trivially

veri¬ed and therefore, in this case we can expect to prove existence.

Proof First of all remark that the existence of periodic solutions reduces to

¬nding x0 ∈ R such that the solution of the evolution problem

t ∈ [0, T ],

±x± (t) + x± (t) + g(x± (t)) = f (t),

(12)

x(0) = x0 ,

veri¬es x(T ; 0, x0 ) = x0 . Here we denote by x(· ; 0, x0 ) the solution of (12)

(existence and uniqueness assured by Picard™s theorem). We de¬ne the map

S : R ’ R given by

S(x0 ) = x(T ; 0, x0 ), x0 ∈ R. (13)

We demonstrate the existence and uniqueness of the periodic solution of (12)

by showing that the Banach™s ¬xed 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

±|x(t ; 0, x1 ) ’ x(t ; 0, x2 )|2 + |x(t ; 0, x1 ) ’ x(t ; 0, x2 )|2 ¤ 0,

0 0 0 0

2 dt

which implies

1 d 2±t

{e |x(t ; 0, x1 ) ’ x(t ; 0, x2 )|2 } ¤ 0,

0 0

2 dt

and therefore,

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

0 0 0 0 0 0

For ± > 0 S is a contraction and the Banach™s ¬xed point theorem applies.

Therefore S(x0 ) = x0 for an unique x0 ∈ R and hence x(· ; 0, x0 ) is a periodic

™¦

solution of (3).

Naturally, in the following proposition we inquire about the convergence of

(x± )±>0 to a periodic solution of (3) as ± ’ 0. In view of the Proposition 2.6

this convergence does not hold if (10) is not veri¬ed. 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 (x± )±>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

¬nd

1d

±|x± (t) ’ x(t)|2 + |x± (t) ’ x(t)|2 ¤ ’±x(t)(x± (t) ’ x(t)), t ∈ R, (17)

2 dt

Mihai Bostan 7

which can be also written as

1 d 2±t

{e |x± (t) ’ x(t)|2 } ¤ ±e±t |x(t)| · e±t |x± (t) ’ x(t)|, t ∈ R. (18)

2 dt

Therefore, by integration on [0, t] we deduce

t

1 ±t 1

{e |x± (t) ’ x(t)|}2 ¤ |x± (0) ’ x(0)|2 + ±e±s |x(s)| · e±s |x± (s) ’ x(s)|ds.

2 2 0

(19)

Using Bellman™s lemma, formula (19) gives

t

±t

±e±s |x(s)|ds,

e |x± (t) ’ x(t)| ¤ |x± (0) ’ x(0)| + t ∈ R. (20)

0

Let us consider ± > 0 ¬xed for the moment. Since x is periodic and continuous,

it is also bounded and therefore from (20) we get

|x± (t) ’ x(t)| ¤ e’±t |x± (0) ’ x(0)| + (1 ’ e’±t ) x t ∈ R.

L∞ (R) , (21)

By periodicity we have

|x± (t) ’ x(t)| = |x± (nT + t) ’ x(nT + t)|

¤ e’±(nT +t) |x± (0) ’ x(0)| + (1 ’ e’±(nT +t) ) x L∞ (R)

¤ e’±(nT +t) |x± (0) ’ x(0)| + x t ∈ R, n ≥ 0.

L∞ (R) ,

By passing to the limit as n ’ ∞, we deduce that (x± )±>0 is uniformly bounded

in L∞ (R):

|x± (t)| ¤ |x± (t) ’ x(t)| + |x(t)| ¤ 2 x t ∈ R, ± > 0.

L∞ (R) ,

The derivatives x± are also uniformly bounded in L∞ (R) for ± ’ 0:

|x± (t)|

= |f (t) ’ ±x± (t) ’ g(x± (t))|

¤ f L∞ (R) + 2± x L∞ (R) + max{g(2 x L∞ (R) ), ’g(’2 x L∞ (R) )}.

The uniform convergence of (x± )±>0 follows now from the Arzela-Ascoli™s the-

orem. Denote by u the limit of (x± )±>0 as ± ’ 0. Obviously u is also periodic

u(0) = lim x± (0) = lim x± (T ) = u(T ).

±’0 ±’0

To prove that u veri¬es (3), we write

t

{f (s) ’ g(x± (s)) ’ ±x± (s)}ds, t ∈ R.

x± (t) = x± (0) +

0

Since the convergence is uniform, by passing to the limit for ± ’ 0 we obtain

t

{f (s) ’ g(u(s))}ds,

u(t) = u(0) +

0

8 Periodic solutions for evolution equations

and hence u ∈ C 1 (R; R) and

t ∈ 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∞ (R) for (x± )±>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 (x± )±>0 is uniformly bounded in L∞ (R). Moreover, in this

case (x± )±>0 is convergent in C 0 (R; R) and the limit is a periodic solution for

(3).

Note that generally we can not estimate (x± )±>0 uniformly in L∞ (R). In-

deed, by standard computations we obtain

1d

±(x± (t) ’ u)2 + (x± (t) ’ u)2 ¤ |f (t) ’ ±u ’ g(u)| · |x± (t) ’ u|, t, u ∈ R

2 dt

and therefore

1 d 2±t

{e (x± (t) ’ u)2 } ¤ e±t |f (t) ’ ±u ’ g(u)| · e±t |x± (t) ’ u|, t, u ∈ R.

2 dt

Integration on [t, t + h], we get

t+h

1 2±(t+h)

(x± (t + h) ’ u)2 e2±s |f (s) ’ ±u ’ g(u)| · |x± (s) ’ u|ds

¤

e

2 t

1

+ e2±t (x± (t) ’ u)2 , t < t + h, u ∈ R.

2

Now by using Bellman™s lemma we deduce

t+h

’±h

e’±(t+h’s) |f (s)’±u’g(u)|ds,

|x± (t+h)’u| ¤ e |x± (t)’u|+ t < t+h.

t

Since x± is T -periodic, by taking h = T we can write

T

1

e’±(T ’s) |f (s) ’ ±u ’ g(u)|ds,

|x± (t) ’ u| ¤ t ∈ R,

1 ’ e’±T 0

and thus for u = 0 we obtain

T

1 1

¤ |f (s) ’ g(0)|ds ∼ O

x± , ± > 0.

L∞ (R)

1 ’ e’±T ±

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 ∈ Range(g) (there is x0 ∈ R such that f = g(x0 )).

Moreover in this case we have the estimate

T

∀ x0 ∈ g ’1 f ,

¤ |x0 | + |f (t) ’ f |dt,

x L∞ (R)

0

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

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

Proof The condition is necessary (see Proposition 2.6). We will prove now

that it is also su¬cient. Let us consider the sequence of periodic solutions

(x± )±>0 of (11). Accordingly to the Proposition 2.10 we need to prove uniform

estimates in L∞ (R) for (x± )±>0 . Since x± is T -periodic by integration on [0, T ]

we get

T

{±x± (t) + g(x± (t))}dt = T f , ± > 0.

0

Using the average formula for continuous functions we have

T

{±x± (t) + g(x± (t))}dt = T {±x± (t± ) + g(x± (t± ))}, t± ∈]0, T [, ± > 0.

0

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

±x± (t± ) + g(x± (t± )) = g(x0 ), ± > 0. (22)

Since g is increasing, we deduce

±x± (t± )[x0 ’ x± (t± )] = [g(x0 ) ’ g(x± (t± ))][x0 ’ x± (t± )] ≥ 0, ± > 0,

and thus

|x± (t± )|2 ¤ x± (t± )x0 ¤ |x± (t± )||x0 |.

Finally we deduce that x± (t± ) is uniformly bounded in R:

|x± (t± )| ¤ |x0 |, ∀ ± > 0.

Now we can easily ¬nd uniform estimates in L∞ (R) for (x± )±>0 . Let us take in

the previous calculus u = x± (t± )and integrate on [t± , t]:

t

1 2±t

e (x± (t)’x± (t± ))2 ¤ e2±s |f (s)’±x± (t± )’g(x± (t± ))|·|x± (s)’x± (t± )|ds.

2 t±

By using Bellman™s lemma we get

t

e’±(t’s) |f (s) ’ ±x± (t± ) ’ g(x± (t± ))|ds,

|x± (t) ’ x± (t± )| ¤ t > t± ,

t±

10 Periodic solutions for evolution equations

and hence by (22) we deduce

T

|x± (t)| ¤ |x0 | + |f (t) ’ ±x± (t± ) ’ g(x± (t± ))|dt

0

T

= |x0 | + |f (t) ’ f |dt, t ∈ R, ± > 0. (23)

0

Now by passing to the limit in (23) we get

T

t ∈ R, ∀ x0 ∈ g ’1 f .

|x(t)| ¤ |x0 | + |f (t) ’ 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