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t âˆˆ [0, T ], n â‰¥ 0.

xn (t) = x(nT + t),

We have

t âˆˆ [0, T ],

Î±xn+1 (t) + xn+1 (t) + Axn+1 (t) = f ((n + 1)T + t),

and

t âˆˆ [0, T ].

Î±xn (t) + xn (t) + Axn (t) = f (nT + t),

20 Periodic solutions for evolution equations

Since f is T -periodic, after usual computations we get

1d2

xn+1 (t) âˆ’ xn (t) 2

Î± xn+1 (t) âˆ’ xn (t) +

2 dt

+(Axn+1 (t) âˆ’ Axn (t), xn+1 (t) âˆ’ xn (t)) t âˆˆ [0, T ].

= 0,

Taking into account that A is monotone we deduce

xn+1 (t) âˆ’ xn (t) â‰¤ eâˆ’Î±t xn+1 (0) âˆ’ xn (0) , t âˆˆ [0, T ],

and hence

xn+1 (0) âˆ’ xn (0) âˆ’ xnâˆ’1 (T )

= xn (T )

âˆ’Î±T

â‰¤ xn (0) âˆ’ xnâˆ’1 (0)

e

eâˆ’2Î±T xnâˆ’1 (0) âˆ’ xnâˆ’2 (0)

â‰¤

â‰¤ ...

eâˆ’nÎ±T x1 (0) âˆ’ x0 (0) , n â‰¥ 0.

â‰¤ (40)

Finally we get the estimate

xn+1 (t) âˆ’ xn (t) â‰¤ eâˆ’Î±(nT +t) SÎ± (T ; 0, x0 ) âˆ’ x0 , t âˆˆ [0, T ], n â‰¥ 0.

Here SÎ± (t; 0, x0 ) represents the solution of (39) for the initial data x0 . From the

previous estimate it is clear that (xn )nâ‰¥0 is convergent in C 0 ([0, T ]; H):

nâˆ’1

(xk+1 (t) âˆ’ xk (t)), t âˆˆ [0, T ],

xn (t) = x0 (t) +

k=0

where

nâˆ’1 nâˆ’1

(xk+1 (t) âˆ’ xk (t)) â‰¤ xk+1 (t) âˆ’ xk (t)

k=0 k=0

nâˆ’1

eâˆ’Î±(kT +t) SÎ± (T ; 0, x0 ) âˆ’ x0

â‰¤

k=0

eâˆ’Î±t

â‰¤ SÎ± (T ; 0, x0 ) âˆ’ x0 .

1 âˆ’ eâˆ’Î±T

Moreover xn (t) â‰¤ SÎ± (t; 0, x0 ) + 1âˆ’e1 SÎ± (T ; 0, x0 ) âˆ’ x0 . Denote by

âˆ’Î±T

xÎ± the limit of (xn )nâ‰¥0 as n â†’ âˆž. We should note that without any other

hypothesis (xÎ± )Î±>0 is not uniformly bounded in Lâˆž (]0, T [; H). We have only

1

estimate in O(1 + Î± ),

1 1

â‰¤C 1+ âˆ¼O 1+

xÎ± .

Lâˆž ([0,T ];H)

1 âˆ’ eâˆ’Î±T Î±

The above estimate leads immediately to the following statement.

Mihai Bostan 21

Remark 3.5 The sequence (Î±xÎ± )Î±>0 is uniformly bounded in Lâˆž ([0, T ]; H).

Let us demonstrate that xÎ± is T -periodic and solution for (39). Indeed,

xÎ± (0) = lim xn (0) = lim xnâˆ’1 (T ) = xÎ± (T ).

nâ†’âˆž nâ†’âˆž

Now let us show that (xn )nâ‰¥0 is also uniformly bounded in Lâˆž (]0, T [; H). By

taking the diï¬€erence between the equations (39) at the moments t and t + h we

have

Î±(x(t+h)âˆ’x(t))+x (t+h)âˆ’x (t)+Ax(t+h)âˆ’Ax(t) = f (t+h)âˆ’f (t), t < t+h.

Multiplying by x(t + h) âˆ’ x(t) we obtain

1d

2 2

Î± x(t + h) âˆ’ x(t) x(t + h) âˆ’ x(t) â‰¤ f (t + h) âˆ’ f (t) Â· x(t + h) âˆ’ x(t) ,

+

2 dt

which can be also rewritten as

t

1 2Î±t 2

eÎ±s f (s + h) âˆ’ f (s) Â· eÎ±s x(s + h) âˆ’ x(s) ds

x(t + h) âˆ’ x(t) â‰¤

e

2 0

1

x(h) âˆ’ x(0) 2 ,

+ t < t + h.

2

By using Bellmanâ€™s lemma we conclude that

t

1 1

eâˆ’Î±(tâˆ’s)

x(t + h) âˆ’ x(t) â‰¤ f (s + h) âˆ’ f (s) ds

h h

0

1

+eâˆ’Î±t x(h) âˆ’ x(0) , 0 â‰¤ t < t + h. (41)

h

By passing to the limit for h â†’ 0 the previous formula yields

t

âˆ’Î±t

eâˆ’Î±(tâˆ’s) f (s) ds

â‰¤e

x (t) x (0) +

0

1

â‰¤ eâˆ’Î±t f (0) âˆ’ Î±x0 âˆ’ Ax0 + (1 âˆ’ eâˆ’Î±t ) f Lâˆž (]0,T [;H)

Î±

1

â‰¤ f (0) âˆ’ Î±x0 âˆ’ Ax0 + f < +âˆž.

Lâˆž (]0,T [;H)

Î±

Therefore (xn )nâ‰¥0 is uniformly bounded in Lâˆž (]0, T [; H) since

â‰¤x

xn = x (nT + (Â·)) Lâˆž ([0,+âˆž[;H) ,

Lâˆž (]0,T [;H) Lâˆž (]0,T [;H)

yÎ± (t), t âˆˆ [0, T ]. We can write

and thus we have xn (t)

t

z âˆˆ H, t âˆˆ [0, T ], n â‰¥ 0,

(xn (t), z) = (xn (0), z) + (xn (s), z)ds,

0

22 Periodic solutions for evolution equations

and by passing to the limit for n â†’ âˆž we deduce

t

z âˆˆ H, t âˆˆ [0, T ],

(xÎ± (t), z) = (xÎ± (0), z) + (yÎ± (s), z)ds,

0

which is equivalent to

t

t âˆˆ [0, T ].

xÎ± (t) = xÎ± (0) + yÎ± (s)ds,

0

Therefore xÎ± is diï¬€erentiable and xÎ± = yÎ± . Finally we can write xn (t) xÎ± (t),

t âˆˆ [0, T ]. Let us show that xÎ± is also solution for (39). We have

[xn (t), f (t) âˆ’ Î±xn (t) âˆ’ xn (t)] âˆˆ A, n â‰¥ 0, t âˆˆ [0, T ].

Since xn (t) â†’ xÎ± (t), xn (t) xÎ± (t) and A is maximal monotone we conclude

that

[xÎ± (t), f (t) âˆ’ xÎ± (t)] âˆˆ A, t âˆˆ [0, T ], Î± > 0,

which means that xÎ± (t) âˆˆ D(A) and AxÎ± (t) = f (t) âˆ’ xÎ± (t), t âˆˆ [0, T ].

Now we establish for the linear case the similar result stated in Proposition

2.10. Before let us recall a standard result concerning maximal monotone oper-

ators on Hilbert spaces

Proposition 3.6 Assume that A is a maximal monotone operator (linear or

not) and Î±uÎ± + AuÎ± = f , uÎ± âˆˆ D(A), f âˆˆ H, Î± > 0. Then the following

statements are equivalent:

(i) f âˆˆ Range(A);

(ii) (uÎ± )Î±>0 is bounded in H. Moreover, in this case (uÎ± )Î±>0 is convergent in

H to the element of minimal norm in Aâˆ’1 f .

Proof it (i) â†’ (ii) By the hypothesis there is u âˆˆ D(A) such that f = Au.

After multiplication by uÎ± âˆ’ u we get

Î±(uÎ± , uÎ± âˆ’ u) + (AuÎ± âˆ’ Au, uÎ± âˆ’ u) = 0, Î± > 0.

Taking into account that A is monotone we deduce

2

â‰¤ (uÎ± , u) â‰¤ uÎ± Â· u ,

uÎ± Î± > 0,

and hence uÎ± â‰¤ u , Î± > 0, u âˆˆ Aâˆ’1 f which implies that uÎ± u0 . We have

[uÎ± , f âˆ’ Î±uÎ± ] âˆˆ A, Î± > 0 and since A is maximal monotone, by passing to the

limit for Î± â†’ 0 we deduce that [u0 , f ] âˆˆ A, or u0 âˆˆ Aâˆ’1 f . Moreover

âˆ€u âˆˆ Aâˆ’1 f.

u0 = w âˆ’ lim uÎ± â‰¤ lim inf uÎ± â‰¤ lim sup uÎ± â‰¤ u ,

Î±â†’0 Î±â†’0 Î±â†’0

In particulat taking u = u0 âˆˆ Aâˆ’1 f we get

w âˆ’ lim uÎ± = lim uÎ± ,

Î±â†’0 Î±â†’0

Mihai Bostan 23

and hence, since any Hilbert space is strictly convex, by Mazurâ€™s theorem we

deduce that the convergence is strong

uÎ± â†’ u0 âˆˆ Aâˆ’1 f, Î± â†’ 0,

where u0 = inf uâˆˆAâˆ’1 f u = minuâˆˆAâˆ’1 f u .

(ii) â†’ (i) Conversely, suppose that (uÎ± )Î±>0 is bounded in H. Therefore uÎ± u

in H. We have [uÎ± , f âˆ’ Î±uÎ± ] âˆˆ A, Î± > 0 and since A is maximal monotone

by passing to the limit for Î± â†’ 0 we deduce that [u, f ] âˆˆ A or u âˆˆ D(A) and

f = Au.

Theorem 3.7 Assume that A : D(A) âŠ‚ H â†’ H is a linear maximal monotone

operator on a compact Hilbert space H and f âˆˆ C 1 (R; H) is a T -periodic func-

tion. Then the following statements are equivalent:

(i) equation (33) has periodic solutions;

(ii) the sequence of periodic solutions for (39) is bounded in C 1 (R; H). Moreover

in this case (xÎ± )Î±>0 is convergent in C 0 (R; H) and the limit is also a T -periodic

solution for (33).

Proof (i) â†’ (ii) Denote by x, xÎ± the periodic solutions for (33) and (39). By

taking the diï¬€erence and after multiplication by xÎ± (t) âˆ’ x(t) we get:

1d

2 2

Î± xÎ± (t) âˆ’ x(t) xÎ± (t) âˆ’ x(t) â‰¤ Î± x(t) Â· xÎ± (t) âˆ’ x(t) , t âˆˆ R. (42)

+

2 dt

Finally, after integration and by using Bellmanâ€™s lemma, formula (42) yields

t

âˆ’Î±t

Î±eâˆ’Î±(tâˆ’s) x(s) ds

xÎ± (t) âˆ’ x(t) â‰¤e xÎ± (0) âˆ’ x(0) +

0

âˆ’Î±t

xÎ± (0) âˆ’ x(0) + (1 âˆ’ eâˆ’Î±t ) x

â‰¤e t âˆˆ R.

Lâˆž ,

Since xÎ± and x are T -periodic we can also write

xÎ± (t) âˆ’ x(t) = xÎ± (nT + t) âˆ’ x(nT + t)

â‰¤ eâˆ’Î±(nT +t) xÎ± (0) âˆ’ x(0) + (1 âˆ’ eâˆ’Î±(nT +t) ) x Lâˆž .

By passing to the limit for n â†’ âˆž we obtain

xÎ± âˆ’ x â‰¤x Lâˆž , Î± > 0,

Lâˆž

and hence

â‰¤2 x

xÎ± Lâˆž , Î± > 0.

Lâˆž

Since A is linear we can write

Î± 1 1

(xÎ± (t + h) âˆ’ xÎ± (t)) + (xÎ± (t + h) âˆ’ xÎ± (t)) + A(xÎ± (t + h) âˆ’ xÎ± (t))

h h h

1

(f (t + h) âˆ’ f (t)), t < t + h, Î± > 0,

=

h

24 Periodic solutions for evolution equations

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