Palmers Cocoa For Acne Scars

Global Solution (on IR) differential equations of Lipschitz type.

Exercise.

1.Question preliminary.

Let (E, II) a Banach space, C = CB (IR, E) denotes

the space of continuous functions and bounded on IR and

with values in E. It is known that CB, equipped with the Standard

IIxII = sup {Ix (t) I t IR}

Banach.Soit is c> 0 and consider the space:

CB '= C' (IR, E) = {x

: IR ---> E, continuous and sup (exp (-Citi) Ix (t) I, t in IR) is finite}

or ITI denotes the absolute value of t

demonstrate that CB ', with the norm:

IIIxIII = sup {exp (-Citi) Ix (t) I , t IR} is

Banach space and CB injected continuously and strictly in BC '.

2.Soit f be a Lipschitz from E to E, ie d:

There exists k> 0 such that If (x)-f (y) I

Consider the problem Cauchy:

x '= f (x) (1) x (0) = x_ {0} belongs to E (2). For x in

CB 'and t in IR, or the application:

Tx (t) = x_ {0} + Integral (0 to t) f (x (s)) ds.

i) Show that BC is stable T.

ii) Take c> k.Montrer that T is strictly contracting CB "in CB '.

deduce that the Cauchy problem (1) - (2) admits a unique solution in

CB '.