Memory Access C0000005

Demarcation and periodicity of solutions of differential equations (based Massera)

Exercise, below, presents the first result

in the famous article Massera (1950), which highlights

relief, the link between the existence of bounded solutions and the existence

periodic solutions.

Exercise.

Consider the following scalar differential equation:

x '= f (t, x) (*)

where f is a continuous function of IR XIR ^ {+} in IR, and

p-periodic report at (p> 0).

(ie d, for x in IR, f (t + p, x) = f (t, x) <1.Sans utiliser le principe de

for all t in IR ^{+}).

Assume that equation (*), satisfies the conditions

the uniqueness of solutions with respect to conditions

cause will.

Let y be a solution of equation (*), bounded on IR ^{+}.

i) Show that if y (0) = y (p), then y is p-periodic.

ii) Assume that y (0) is different from y (p), for example

y (0)

{y_ {n} (t) = y (t + pn)} is strictly increasing on IR .

iii) In the space of continuous functions, C ([0, t], IR),

with the norm sup, demonstrate that the sequence {y_ {n}}

is uniformly convergent to a solution

p-periodic Equation (*).

(Using the Ascoli theorem and Dini's theorem)