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From: Arslan Ibrai on 10 Feb 2010 22:58
Let's consider a system of ODE

dx/dt = -ax + f(y,z),
dy/dt = -by + g(x,z),
dz/dt = -cz + h(x,y),

where a>0,b>0,c>0, f,g,h are bounded functions.

Is it true that solutions of the system are bounded functions? If it's true then how to prove this fact?

Thank you very much for attention!
From: achille on 11 Feb 2010 09:41
On Feb 11, 9:58 pm, Arslan Ibrai <arslan.ib...(a)gmail.com> wrote:
> Let's consider a system of ODE
>
> dx/dt = -ax + f(y,z),
> dy/dt = -by + g(x,z),
> dz/dt = -cz + h(x,y),
>
> where a>0,b>0,c>0,  f,g,h are bounded functions.
>
> Is it true that solutions of the system are bounded functions? If it's true then how to prove this fact?
>
> Thank you very much for attention!

Hint:
| x(t) exp(at) - x(0) | <= \int^{t}_0 | d( exp(at) x(t) ) |
From: Alfred Flaßhaar on 11 Feb 2010 11:38
achille wrote:
> On Feb 11, 9:58 pm, Arslan Ibrai <arslan.ib...(a)gmail.com> wrote:
>> Let's consider a system of ODE
>>
>> dx/dt = -ax + f(y,z),
>> dy/dt = -by + g(x,z),
>> dz/dt = -cz + h(x,y),
>>
>> where a>0,b>0,c>0, f,g,h are bounded functions.
>>
>> Is it true that solutions of the system are bounded functions? If
>> it's true then how to prove this fact?
>>
>> Thank you very much for attention!
>
> Hint:
>> x(t) exp(at) - x(0) | <= \int^{t}_0 | d( exp(at) x(t) ) |

Second hint:

A proof should be done by using Differential-/Integralinequalities (Lit.:
Gronwall, Szarski, Walter, ...).

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