where

m± ¤ x± (t) ¤ M± , t ∈ [0, T ], ± > 0,

and hence

T

1

±m± + G(m± ) ¤ {±x± (t) + g(t, x± (t))}dt ¤ ±M± + G(M± ), ± > 0.

T 0

Finally we get

T

1

{±x± (t)+g(t, x± (t))}dt = ±u± +G(u± ), u± ∈]m± , M± [, ± > 0.

G(x0 ) =

T 0

(30)

Multiplying by u± ’ x0 we obtain

±u± (u± ’ x0 ) = ’(G(x0 ) ’ G(u± ))(x0 ’ u± ), ± > 0.

Since G is increasing we deduce that |u± |2 ¤ u± x0 ¤ |u± | · |x0 |, ± > 0 and hence

(u± )±>0 is bounded:

|u± | ¤ |x0 |, ± > 0.

Now using (30) it follows

T T

1 1

{±x± (t) + g(t, x± (t))}dt = {±u± + g(t, u± )}dt,

T T

0 0

and thus there is t± ∈]0, T [ such that

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

Since ±(x± (t± ) ’ u± )2 = ’[g(t± , x± (t± )) ’ g(t± , u± )][x± (t± ) ’ u± ] ¤ 0 we ¬nd

that x± (t± ) = u± , ± > 0 and thus (x± (t± ))±>0 is also bounded

|x± (t± )| ¤ |x0 |, ± > 0.

Now by standard calculations we can write

1d

|x± (t) ’ x± (t± )|2 + [g(t, x± (t)) ’ g(t, x± (t± ))][x± (t) ’ x± (t± )]

2 dt

¤ [f (t) ’ ±x± (t± ) ’ g(t, x± (t± ))][x± (t) ’ x± (t± )], t ∈ R,

16 Periodic solutions for evolution equations

and thus

t

|x± (t) ’ x± (t± )| ¤ |f (s) ’ ±x± (t± ) ’ g(s, x± (t± ))|ds, t > t± , ± > 0,

t±

which implies

T

|x± (t)| ¤ |x0 | + |f (t) ’ ±x± (t± ) ’ g(t, x± (t± ))|dt, t ∈ [0, T ], ± > 0. (31)

0

Since (x± (t± ))±>0 is bounded we have

u± = x± (t± ) ’ x1 ,

such that

G(x0 ) = lim {±u± + G(u± )} = G(x1 ).

±’0

Moreover, if x0 ¤ x1 we have

T

1

0¤ [g(t, x1 ) ’ g(t, x0 )]dt = G(x1 ) ’ G(x0 ) = 0,

T 0

and hence g(t, x1 ) = g(t, x0 ) for all t ∈ [0, T ]. Obviously the same equalities

hold if x0 > x1 . Now by passing to the limit in (31) we ¬nd

T

|x(t)| ¤ |x0 | + |f (t) ’ g(t, x1 )|dt (32)

0

T

t ∈ [0, T ], ∀ x0 ∈ G’1 f ,

= |x0 | + |f (t) ’ g(t, x0 )|dt,

0

and therefore (x± )±>0 is uniformly bounded in L∞ (R).

3 Periodic solutions for evolution equations on

Hilbert spaces

In this section we analyze the existence and uniqueness of periodic solutions for

general evolution equations on Hilbert spaces

x (t) + Ax(t) = f (t), t > 0, (33)

where A : D(A) ‚ H ’ H is a maximal monotone operator on a Hilbert

space H and f ∈ C 1 (R; H) is a T -periodic function. As known by the theory

of Hille-Yosida, for every initial data x0 ∈ D(A) there is an unique solution

x ∈ C 1 ([0, +∞[; H) © C([0, +∞[ ; D(A)) for (33), see [6, p. 101]. Obviously,

the periodic problem reduces to ¬nd x0 ∈ D(A) such that x(T ) = x0 . As

in the one dimensional case we demonstrate uniqueness for strictly monotone

operators. We state also necessary and su¬cient condition for the existence

in the linear symmetric case. Finally the case of non-linear sub-di¬erential

operators is considered. Let us start with the de¬nition of periodic solutions for

(33).

Mihai Bostan 17

De¬nition 3.1 Let A : D(A) ‚ H ’ H be a maximal monotone operator

on a Hilbert space H and f ∈ C 1 (R; H) a T -periodic function. We say that

x ∈ C 1 ([0, T ]; H) © C([0, T ]; D(A)) is a periodic solution for (33) if and only if

t ∈ [0, T ],

x (t) + Ax(t) = f (t),

and x(0) = x(T ).

3.1 Uniqueness

Generally the uniqueness does not hold (see the example in the following para-

graph). However it occurs under the hypothesis of strictly monotony.

Proposition 3.2 Assume that A is strictly monotone ((Ax1 ’Ax2 , x1 ’x2 ) = 0

implies x1 = x2 ). Then (33) has at most one periodic solution.

Proof Let x1 , x2 be two di¬erent periodic solutions. By taking the di¬erence

of (33) and multiplying both sides by x1 (t) ’ x2 (t) we ¬nd

1d 2

x1 (t) ’ x2 (t) + (Ax1 (t) ’ Ax2 (t), x1 (t) ’ x2 (t)) = 0, t ∈ [0, T ].

2 dt

2

By the monotony of A we deduce that x1 ’ x2 is decreasing and therefore

we have

x1 (0) ’ x2 (0) ≥ x1 (t) ’ x2 (t) ≥ x1 (T ) ’ x2 (T ) , t ∈ [0, T ].

Since x1 and x2 are T -periodic we have

x1 (0) ’ x2 (0) = x1 (T ) ’ x2 (T ) ,

which implies that x1 (t) ’ x2 (t) is constant for t ∈ [0, T ] and thus

(Ax1 (t) ’ Ax2 (t), x1 (t) ’ x2 (t)) = 0, t ∈ [0, T ].

Now uniqueness follows by the strictly monotony of A.

3.2 Existence

In this section we establish existence results. In the linear case we state the

following necessary condition.

Proposition 3.3 Let A : D(A) ‚ H ’ H be a linear maximal monotone

operator and f ∈ L1 (]0, T [; H) a T -periodic function. If (33) has T -periodic

solutions, then the following necessary condition holds.

T

1

f (t)dt ∈ Range(A),

f :=

T 0

(there is x0 ∈ D(A) such that f = Ax0 ).

18 Periodic solutions for evolution equations

Proof Suppose that x ∈ C 1 ([0, T ]; H)©C([0, T ]; D(A)) is a T -periodic solution

for (33). Let us consider the divisions ∆n : 0 = tn < tn < · · · < tn = T such

n

0 1

that

lim max |tn ’ tn | = 0. (34)

i i’1

n’∞ 1¤i¤n

We can write

(tn ’ tn )x (tn ) + (tn ’ tn )Ax(tn ) = (tn ’ tn )f (tn ), 1 ¤ i ¤ n.

i i’1 i’1 i i’1 i’1 i i’1 i’1

Since A is linear we deduce

n n n

1 1 1

(tn ’tn )x (tn )+A (tn ’tn )x(tn ) (tn ’tn )f (tn ),

=

i i’1 i’1 i i’1 i’1 i i’1 i’1

T T T

i=1 i=1 i=1

and hence

n n

1 1

(tn tn )x(tn )), (tn ’ tn )[f (tn ) ’ x (tn )] ∈ A.

’

i i’1 i’1 i i’1 i’1 i’1

T T

i=1 i=1

By (34) we deduce that

n T

1 1

(tn ’ tn )x(tn )) ’ x(t)dt,

i i’1 i’1

T T 0

i=1

and

n T

1 1

(tn tn )[f (tn ) (tn )]

’ ’x ’ [f (t) ’ x (t)]dt

i i’1 i’1 i’1

T T 0

i=1

T

1 1

x(t)|T

f (t)dt ’

= 0

T T

0

T

1

= f (t)dt.

T 0

Since A is maximal monotone Graph(A) is closed and therefore

T T

1 1

f (t)dt ∈ A.

x(t)dt,

T T

0 0

T T

1 1

Thus T 0 x(t)dt ∈ D(A) and f = A( T 0 x(t)dt). Generally the previous

condition is not su¬cient for the existence of periodic solutions. For example

let us analyse the periodic solutions x = (x1 , x2 ) ∈ C 1 ([0, T ]; R2 ) for

x (t) + Ax(t) = f (t), t ∈ [0, T ], (35)

where A : R2 ’ R2 is the orthogonal rotation:

A(x1 , x2 ) = (’x2 , x1 ), (x1 , x2 ) ∈ R2 ,

Mihai Bostan 19

and f = (f1 , f2 ) ∈ L1 (]0, T [; R2 ) is T -periodic. For a given initial data x(0) =

x0 ∈ R2 the solution writes

t

’tA

e’(t’s)A f (s)ds,

x(t) = e x0 + t > 0, (36)

0

where the semigroup e’tA is given by

cos t sin t

e’tA = . (37)

’ sin t cos t

Since e’2πA = 1 we deduce that the equation (35) has 2π-periodic solutions if

and only if

2π

etA f (t)dt = 0. (38)

0

2π 2π

Thus if 0 {f1 (t) cos t ’ f2 (t) sin t}dt = 0 or 0 {f1 (t) sin t + f2 (t) cos t}dt = 0

equation (35) does not have any 2π-periodic solution and the necessary condition

still holds because Range(A) = R2 . Moreover if (38) is satis¬ed then every

solution of (35) is periodic and therefore uniqueness does not occur (the operator

A is not strictly monotone). Let us analyse now the existence. As in the one

dimensional case we have

Proposition 3.4 Suppose that A : D(A) ‚ H ’ H is maximal monotone and

f ∈ C 1 (R; H) is T -periodic. Then for every ± > 0 the equation

t ∈ R,

±x(t) + x (t) + Ax(t) = f (t), (39)

has an unique T -periodic solution in C 1 (R; H) © C(R; D(A)).

Proof Since ±+A is strictly monotone the uniqueness follows from Proposition

3.2. Indeed,

2

± x’y + (Ax ’ Ay, x ’ y) = 0, x, y ∈ D(A),

implies ± x ’ y 2 = 0 and hence x = y.

Consider now an arbitrary initial data x0 ∈ D(A). By the Hille-Yosida™s theo-

rem, there is x ∈ C 1 ([0, +∞[; H) © C([0, +∞[; D(A)) solution for (39). Denote